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Finite Elements in Analysis and Design 40 (2004) 1729–1751 www.elsevier.com/locate/ÿnel A spectral ÿnite element for wave propagation and structural diagnostic analysis of composite beam with transverse crack D. Sreekanth Kumar, D. Roy Mahapatra, S. Gopalakrishnan ∗ Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India Received 12 March 2003; accepted 16 January 2004 Abstract A spectral ÿnite elemen
  Finite Elements in Analysis and Design 40 (2004) 1729–1751 www.elsevier.com/locate/ÿnel A spectral ÿnite element for wave propagation and structuraldiagnostic analysis of composite beam with transverse crack D. Sreekanth Kumar, D. Roy Mahapatra, S. Gopalakrishnan ∗ Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India Received 12 March 2003; accepted 16 January 2004 Abstract A spectral ÿnite element with embedded transverse crack is developed and implemented to simulate the di-agnostic wave scattering in composite beams with various forms of transverse crack, such as surface-breakingcracks, matrix cracking and ÿber fracture. The cracked region is discretized into few internal elements,which are modeled as one-dimensional (1D) waveguides. First-order shear deformable kinematics in eachof these waveguides is assumed. Appropriate displacement continuity at the element-internal waveguides areenforced. The equilibrium equations are represented using compact matrix notations. After assembly of theelement-internal system of waveguides, the internal nodes are condensed out and ÿnally a two-node ÿniteelement in frequency domain is obtained. Using this element, namely a single transverse crack is modeledthrough only three input parameters, the span-wise crack location and the thickness-wise locations of thecrack-tips. Although, the proposed element is not suited for dynamic fracture analysis at the local level, itis best suited for narrow-band as well as broad-band wave-based diagnostic simulations for structural healthmonitoring applications. Numerical simulations and comparison with detail 2D ÿnite element prediction showhighly ecient performance of the proposed element to predict the crack location and overall trends due tovarious crack conÿgurations. Important conclusions are drawn on the advantages of the proposed approach,limitations of the element and further scope of improved diagnostic analysis of cracked beam. ? 2004 Elsevier B.V. All rights reserved. Keywords: Spectral element; FFT; Transverse crack; Laminated composite; Waveguide; Wavenumber; Diagnostics;Sensitivity; Damage force; Identiÿcation 1. Introduction Surface-breaking cracks in metallic structures, matrix cracking and ÿber fracture in laminatedcomposite structures are considered to be the most severe mode of damage. The objective of this ∗ Corresponding author. Tel.: +91-80-309-2757; fax: +91-80-360-0134. E-mail address: krishnan@aero.iisc.ernet.in(S. Gopalakrishnan).0168-874X/$-see front matter ? 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.ÿnel.2004.01.001  1730 D. Sreekanth Kumar et al./Finite Elements in Analysis and Design 40 (2004) 1729–1751  paper is to develop a Spectral Finite Element Model (SFEM) to eciently capture the behavior of such transverse cracks during wave-based diagnostics, which can be used to improve the capabilityof the available structural health monitoring (SHM) software.The structural diagnostic analysis of cracked structure can have two main objectives, one at a timeor simultaneously.The ÿrst objective is to analyze the diagnostic signal (single or multi-frequency waveforms) whenit gets scattered due to the existing cracks in the structure, and then to identify the crack conÿgurationusing this scattered signal. The overall complexity may involve damage diagnostic Levels 1–3 [1,2]. As mentioned in these papers, Level 1 involves the detection of the existence of the damaged site;Level 2 involves Level 1 plus determination of damage location and size; Level 3 involves Level 2 plus quantiÿcation of severity of damage.The second objective is to analyze the stress waves generated from the damage source duringits formation. Damage initiation and progression in the form of matrix cracking, ÿber fracture anddelamination have been under immense research. Matrix cracking, which is regarded as the primaryand initial mode of damage in composite, causes stiness degradation [3]. The associated problem of  determining the eective constitutive model has been investigated by many researchers (e.g. Avestonand Kelly [4]and Hahn and Tsai [5]). As discussed in [2,6], the analytical and numerical treatment for determination of the eective constitutive model of cracked laminate can be categorized as (1)self-consistent method, (2) variational method, (3) continuum damage mechanics, (4) shear lag, (5)stress transfer mechanism and (6) method based on equivalent constraint. A majority of the methodsdeveloped for predicting the reduced properties of cracked laminates are applicable to only cross-plylaminates as discussed in [7]. Later Gudmundson and Zang [8] and Adolfsson and Gudmundson [9] developed models for equivalent degradation of the thermoelastic properties of laminate of moregeneral conÿgurations. Here, the strain energy release rate as a function of the transverse crackdensity was used to obtain the change in elastic coecients. This approach was later adopted in [10]to study the transverse crack-induced damage evolution. In another approach proposed by Abergand Gudmundson [11], the matrix crack and ÿber fracture initiation was modeled as equivalent volume force causing transient wave. Wave-based diagnostics is ideal for identiÿcation of suchdamage initiation. However, ecient modeling technique is required to analyze the interaction of thetransient wave with the transverse crack.In the present study, we consider the ÿrst of the two objectives stated above, where transient waveis introduced (rather than intrinsically generated due to energy release) to study the scattered wavefrom the transverse crack. Single transverse crack within a laminate is considered for modeling.Accuracy in modeling of such single transverse crack has more importance compared to modelingthe eect of multiple transverse crack through crack density or similar parameters, especially toidentify the location and size of the damage in wave-based diagnostics. As pointed out in [6], acracked lamina with known number and location of cracks behaves in a dierent manner comparedto a cracked lamina deÿned by the crack density. 1.1. Techniques for modeling of transverse crack  Transverse cracks in composite structures are usually modeled using beams, plates or shells withappropriate kinematics. Such kinematics can be implemented in continuous model or ÿnite elementmodel for global–local analysis.  D. Sreekanth Kumar et al./Finite Elements in Analysis and Design 40 (2004) 1729–1751 1731 Semi-analytic models for cracked Euler–Bernoulli beam were developed by Christides and Barr[12]and Shen and Pierre [13,14]. Similar approach along with crack function approximating the crack-tip singular ÿeld was used by Chondros et al. [15]. Carneiro and Inman [2] used the prescribed surface traction and the displacement at the crack surfaces and the Hu–Washizu–Barr variational principle to develop continuous model for dynamic analysis of Timoshenko beam with transversecrack. However, for a particular cracked beam conÿguration, a convergence study on the number of terms in the test functions is necessary while adopting such technique.Chinchalkar [16]developed a numerical technique for ÿnding the location of crack in a cracked metallic beam with varying cross-section considering the ÿrst three natural frequencies. The crackregion was modeled using a rotational spring and the intersection in the plot of spring stinessvs. crack location provided the correlation to the location of the crack. However, while adoptingthis technique, the chosen natural frequencies should be such that they are aected by the localdamage conÿguration. Marur [17] has derived mode shape and frequency equation of edge-crackedfree–free Timoshenko beam. The coupled dierential equations were solved by treating the crack asa discontinuity in terms of the moment of inertia at the location of the crack. Finite element forstatic and dynamic analysis of cracked and notched prismatic beams was developed by Viola et al.[18], where the crack closure eect is neglected and the cracked section is modeled as an elastic hinge.While investigating the dynamic behavior and crack detection in a cracked beam, Quian et al. [19]derived the stiness matrix of a beam element with a crack considering the stress intensity factor(SIF) in the energy expression. Sinha et al. [20]proposed a Euler–Bernoulli beam ÿnite element with small modiÿcation to the local exibility in the vicinity of a surface-breaking crack. Such changein the exibility, following the earlier study by Christides and Barr [12], yields a triangular-shaped variation in the exural rigidity around the span-wise location of the surface-breaking crack in a beam. It can be seen in the formulations based on simpliÿed one-dimensional (1D) models that afairly acceptable response of the cracked beam can be obtained under static and steady-state loading.However, these models may not be adequate for ecient analysis in wave-based diagnostics, wherethe excitation frequency is very high, depending on the desired nature of propagating wave tointerrogate the crack.Models based on 2D elastodynamics is the ideal candidate for wave propagation in cracked beam problems under plane stress or plane strain condition. The strip-element method for wave scatteringin cracked composite was ÿrst proposed by Liu and Achenbach [21]. The strip-element method usesthe characteristic solution (wavenumber dispersion) of the 2D elastodynamic equation and eigenvectors to compute the displacement in terms of integral expression. This method was later extended by Xi et al. [22]for analyzing wave scattering by an axisymmetric crack in a laminated composite cylindrical shell ÿlled with uid under axisymmetric conÿguration. A hybrid method by combiningÿnite element discretization in propagation direction and strip element discretization in thicknessdirection in cracked laminates was developed by Liu [23]. A spectral super-element model was used in [24] to model transverse crack in isotropic beam and the dynamic stress intensity factor wasobtained accurately under impact type loading. Krawczuk et al. [25]developed a spectral element for detecting cracks in beam structure, where the crack region was idealized using equivalent exibilityderived from a crack function.The spectral element technique used in [26]to model horizontal crack or delamination in laminated composite beam was based on interface kinematics between the sub-laminates and the base-laminate.  1732 D. Sreekanth Kumar et al./Finite Elements in Analysis and Design 40 (2004) 1729–1751 In the present paper, a more complex constrained kinematics is used to form the ÿnite element havinga single surface-breaking or embedded transverse crack. However, the eect of crack-tip singularityis not included in the present model, and the transverse crack conÿguration is modeled as structuraldiscontinuity in a particular laminate of the beam. The main objective is to automate the elementconstruction considering a depth-wise single transverse crack with generic conÿguration embeddedin it, so that the interaction of the incident wave with the crack can be simulated with sucientaccuracy and less computational cost. 2. Spectral ÿnite element model for uncracked beam We consider ÿrst-order shear deformable composite beam in (  x;z ) coordinate system (  x the beamaxis). In the following derivations, bold face capital letters indicate matrices and the bold face smallletters indicate for vectors. Using Discrete Fourier Transform (DFT), the spectral solution for primarydisplacement ÿeld variables can be expressed as u (  x;t  ) =  N   n =1 ˆ u (  x;! n )e i ! n t  =  N   n =1  6   j =1 ˜ u  j e − i k   j  x  e i ! n t  ; (1)where i = √− 1, ! n is the frequency at n th sampling point, N  is the Nyquist frequency in Fast FourierTransform (FFT) used for numerical implementation. k   j is the wavenumber associated with the j thmode of wave (forward or backward propagating or decaying mode). ˆ u = { ˆ u 0 ˆ w ˆ  } T representsthe spectral amplitude vector corresponding to the generic displacement vector as a function of (  x;! n ). u 0 is the in-plane displacement, w is the transverse displacement and  is the cross-sectionalrotation. ˜ u  j = { ˜ u  j ˜ w  j ˜   j } T represents the wave coecient vector associated with j th mode of wave.The applied force history is expressed as f  (  x;t  ) =  N   n =1 ˆ f  (  x;! n )e i ! n t  ; (2)where the spatial variation of the spectral amplitude of force vectorˆ f  (  x;! n ) can be lumped consis-tently using the spectral displacement interpolation function [27]. Substituting the ÿeld variables in governing dierential equations, we get the characteristic equation, which is then solved for k   j ateach ! n , n =1 ;:::;N  . A matrix called amplitude ratio matrix consisting of the base vectors from thenull space of the characteristic system is then used to form the element interpolation function. Ana-lytical expressions of the elements of the amplitude ratio matrix for the ÿrst-order shear deformable beam is given in [27]. Using the force boundary conditions we eliminate the wave coecients and form a complex dynamic stiness matrix. The assembled system of spectral elements is solved ateach sampling frequency. Time history of the ÿeld variables can then be post-processed using in-verse FFT. The detail derivation and the numerical implementation aspect can be found in [27,28]. Therefore we shall skip this formulation and show only the steps required to assemble the waveguidesand condense out the internal degrees of freedom near the crack.
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