A WAVELET SPECTRAL ELEMENT FOR LAMINATED COMPOSITE PLATE WITH DELAMINATION AND TRANSVERSE DAMAGE

Transcription

1 A WAVELET SPECTRAL ELEMENT FOR LAMINATED COMPOSITE PLATE WITH DELAMINATION AND TRANSVERSE DAMAGE S. GOAPALKRISHNAN a and RATNESHWAR JHA b a Department of Aerospace Engineering, Indian Institute of Science, India, Member AIAA, krishnan@aero.iisc.ernet.in b Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY, USA, Associate Fellow AIAA., rjha@clarkson.edu A new spectral plate element (SPE) with an embedded delamination or transverse crack using Daubechies compactly supported wavelet basis functions is developed to analyze wave propagation in an anisotropic laminated composite media. The element is based on the Classical Laminated Plate Theory (CLPT). The element is formulated using the recently developed methodology of spectral finite element formulation based on the solution of a Polynomial Eigenvalue Problem (PEP). By virtue of its frequency-wavenumber domain formulation, single element is sufficient to model large structures, where conventional finite element method will incur heavy cost of computation. The modeling of cracks is based on the use of flexibility functions. The developed element, which is well suited for performing Structural Health Monitoring studies, is validated with solutions obtained by conventional 2-D finite element method. I. Introduction Aerospace structures are subjected to high design loads under complex operational environments as well as discrete damaging loads such as impacts. The use of composites for aerospace structures is increasing rapidly due to several advantages such as lighter weight, fewer joints, improved fatigue life, and higher resistance to corrosion. However, composite structures have several forms of damage such as delamination between plies, fiber-matrix debonding, fiber breakage, and matrix cracking. These damages often occur below the surface due to fatigue, foreign object impact, etc., and may not be visible. The existing paradigm for structural safety leads to expensive inspections, unnecessary downtime and retirement, and sometimes catastrophic failures without any warning. Therefore, a structural health management (SHM) system capable of performing both diagnostics and prognostics for composite structures is urgently needed. A reliable SHM system will provide tremendous benefits in terms of life-cycle costs by detecting damages early and allowing a much more efficient maintenance schedule (condition-based maintenance). This technology will also allow designers to adopt more efficient designs, for example, through reduced factor of safety. An onboard SHM system will provide on-demand health state of a vehicle and will enable reconfiguration of onboard flight control laws in real time for best possible use of the residual control effectiveness following damage. Although SHM has been an active research field for over a decade -4, fundamental research is needed in several aspects of SHM. In this paper, we present a new spectral finite element formulated using Daubechies compactly supported wavelet basis functions for laminated composite plates with embedded damages like transverse cracks or delaminations. SPE, unlike conventional finite elements, are formulated using interpolating functions that are exact solutions to the governing wave equations and as a result, the inertial properties are exactly preserved, which leads to system size that are many orders smaller than conventional FEM. In this paper, first the formulation of the healthy composite plate element is given, which is based on CLPT. The transverse crack is modeled using some flexibility functions, which simulate a transverse through depth crack in a composite plate. These functions are then used to simulate the wave propagation in a transversely cracked plate. The delamination and arbitrarily oriented cracks are modeled using Bloch theorem. This theorem, which is extensively used to study the dynamics of lattice periodic structures, is used to construct the dynamic stiffness of the cracked region, which is then coupled to regular SPE to study the wave propagation in a damaged cracked plate.

2 II. Daubechies Compactly Supported Wavelets In this section, a concise review of orthogonal basis of Daubechies wavelets 5,6 is provided. Wavelets ϕ ( ) form a compactly supported orthonormal basis for L 2 (R). The wavelets and associated scaling functions ϕ ( ) are obtained by translation and dilation of single functions Ψ(t) and ϕ (t), respectively, The scaling functions ϕ (t), are derived from the dilation or scaling equation, j, k t j, k t and the wavelet function ψ ( t) is obtained as where, a k are the filter coefficients, which are fixed for a specific wavelet or scaling function basis. For compactly supported wavelets, only a finite number of a k are nonzero. The filter coefficients a k are derived by imposing certain constraints on the scaling functions. Let P j ( f )( t) be the approximation of a function f(t) in L 2 (R) using ( ) ϕ as the basis, at a certain level (resolution) j, then j, k t where c j,k are the approximation coefficients. III. Reduction of Wave Equations to ODEs A. Governing Differential Equations. Using classical laminated plate theory (CLPT) 7 the three governing equations with respect to the three degrees of freedom u, and w are given below. Here, u 0 (x, y, t), v o (x, y, t), and w(x, y, t) are the axial and transverse ov o displacements in x, y and z directions, respectively, along the mid plane, which is at z = 0. 2

3 The stiffness coefficients A ij, B ij, D ij and the inertial coefficients I 0, I 2 are defined as where Q ij is the stiffness constant of the lamina and ρ is the mass density. The associated boundary conditions for edges parallel to Y-axis are where N x and N y are the normal forces in x and y direction, respectively. M y and M x (no eqn given) are the moments about x- and y-axis. The shear resultant or the Kirchoff shear 6 V is obtained as where Q is the transverse shear force in the z direction. Next, governing PDEs and the associated boundary conditions derived here are reduced to a set of ODEs using Daubechies scaling function approximation in time and one spatial (Y) dimension. B. Temporal Approximation. The first step in the formulation of the 2D WSFE is the reduction of each of the three governing differential equations given by Eqs. (6 8) to a set of PDEs by Daubechies scaling function-based transformation in time. The procedure is exactly similar to that in the formulation of the D WSFE 8. However, the key steps are stated here very briefly for completeness. Let u 0 (x, y, t) be discretized at n points in the chosen time window. Let τ =0,,...,n be the sampling points, then t = tτ (4) where t is the time interval between two sampling points. The function u 0 (x, y, t) can be approximated by ϕ τ at an arbitrary scale as scaling function ( ) 3

4 where u k ( x, y) ( ) ( τ ) ( ) φ ( τ ) u x, y, t = u x, y, = u x, y k, k Ζ 0 0 0k k (5) 0 (referred as u 0k hereafter) are the approximation coefficients at a certain spatial dimension x and y. The other displacements v 0 (x, y, t),w(x, y, t) can be transformed similarly. Substituting these approximations in Eq. (6), using the orthogonality property of the translates of the scaling functions and the definition of connection coefficients 9, the transformed coupled PDEs obtained are of the form where Γ is the first-order connection coefficient matrix obtained after using the wavelet extrapolation technique 6 These coupled PDEs are decoupled using eigenvalue analysis of form of the reduced PDEs given in Eqn (6) is Γ as given in Reference 8. The final decoupled where uˆoj and similarly other transformed displacements are ˆ u oj Φ u oj where Φ is the eigenvector matrix of = (8) Γ and γ j are the corresponding eigenvalues. Following exactly similar steps, the two other governing differential equations (Eqs. (7) and (8)) and the force boundary conditions (Eqs.(9) (2)) are transformed to PDEs in x and y. It should be mentioned here that the sampling rate t should be less than a certain value to avoid spurious dispersion in the simulation using WSFE. In Reference 0, a numerical study has been presented from which the required t can be determined depending on the order N of the Daubechies scaling function and frequency content of the load. C. Spatial (Y) Approximation. As said in Sec., the next step involved is to further reduce each of the transformed and decoupled PDEs given by Eq. (7) (similarly for the other transformed governing differential equations corresponding to (7) and (8)) for j=0,,...,n to a set of coupled ODEs using Daubechies scaling function approximation in one of the spatial (Y) direction. Similar to time approximation, the transformed variable uˆoj can be discretized at m points in the spatial window (0, L Y ), where L Y is the length in Y direction. Let ξ = 0,,..., m be the sampling points, then y = Yξ (9) where Y is the spatial interval between two sampling points. The function u oj ( x, y) scaling function ϕ ( ξ ) at an arbitrary scale as ˆ can be approximated by 4

5 ( ) ( ξ ) ( ) φ ( ξ ) uˆ x, y = uˆ x, = uˆ x l, l Ζ oj oj olj k where û olj ( y) other displacements ˆ (, ), ˆ (, ) (20) x, (referred as ûolj hereafter) are the approximation coefficients at a certain spatial dimension x. The v x y w x y can be similarly transformed. Following similar steps as the time oj j approximation, substituting the above approximations in Eq. (7) and taking inner product on both sides with the translates of scaling functions ϕ ( ξ i), where i =0,,..,m- and using their orthogonal properties, we get m simultaneous ODEs as follows: please check this eqn several terms need correction 2 where N is the order of Daubechies wavelet and Ω and are the connection coefficients for first- and i l i l second-order derivative defined in Reference 9. It can be seen from the ODEs given by Eq. (2), that, similar to time approximation, here also certain coefficients û oij Ω near the vicinity of the boundaries (i=0 and i=m ) lie outside the spatial window [L] defined by i=0,,...,m. These coefficients must be treated properly for finite domain analysis. However, here, unlike time approximation, these coefficients are obtained through periodic extension, but only for free lateral edges, while other boundary conditions may be imposed quite differently using a restraint matrix. The unrestrained, i.e., free-free boundary conditions may also be imposed in a similar way using a restraint matrix, but interestingly, it has been seen from the numerical experiments that the use of periodic extension gives accurate results. In addition, it allows decoupling of the ODEs using eigenvalue analysis and thus reduces the computational cost. 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. (2) can be written as a matrix equation of the form where Λ is the first-order connection coefficient matrix obtained after periodic extension. The coupled ODEs given by Eq.(22) are decoupled using eigenvalue analysis similar to that done in time approximation. It should be mentioned here that matrix Λ obtained after periodic extension has a circulant form and its eigenparameters are known analytically (9). Let the eigenvalues be βi, then the decoupled ODEs corresponding to Eqs. (22) are 5

6 where û and similarly other transformed displacements are oj u~ oj Ψ uˆ oj where?? is the eigenvector matrix of = (24) Λ. Following exactly similar steps, the final transformed and decoupled form of the Eqs. (7) and (8) (following reduction using temporal approximation) are Similarly, the transformed form of the force boundary conditions given by Eqs. (9) (2) (following reduction using temporal approximation) are 6

7 The final transformed ODEs given by Eqs. (23),(25), and (26) and the boundary conditions Eqs. (28) (30) are used for 2D. WSFE formulation is similar to the 2D FSFE technique 2. Similar to the temporal approximation, the spatial sampling rate Y is also determined from the order N of the scaling function used for spatial approximation and the spatial distribution of the load. The four degrees of freedom per node associated with the element formulation are u ~ oij, oij v ~, w ~ 3 and ~ / x ij w ij as shown in Fig. Figure : Spectral Plate Element with all the degrees of freedom and Stress Resultants The corresponding nodal forces are N ~ xij, yij N ~, M ~ yij and V ~ ij. From the previous sections, for unrestrained lateral edges we get a set of decoupled ODEs (Eqs. (23), (25), and (26)) for an isotopic plate using CLPT, in a transformed wavelet domain. These equations are required to be solved for u ~ oij, oij v ~, 7 w ~ oij, and the actual solutions u 0 (x, y, t), v 0( x, y, t), w(x, y, t) are obtained using inverse wavelet transform twice for spatial Y dimension and time. It can be seen that the transformed decoupled ODEs have a form that is similar to that in FSFE (??), and thus, the formulation of WSFE from here is similar to FSFE formulation or -D WSFE formulation given in Reference 8. Thus, the formulation is not repeated here. Finally, the transformed nodal forces { F ~ } e displacements { e } where e K u ~ are related as { F ɶ e } = K ɶ e { u ɶ e } (3) and transformed nodal ɶ is the exact elemental dynamic stiffness matrix. The solution of the Eq. (3) and the assembly of the elemental stiffness matrices to obtain the global stiffness matrix are exactly similar to conventional FE technique. IV Modeling of Transverse Cracks The approach presented here is similar to what is reported in Reference 3. Fig 2 shows a plate with transverse non propagating crack. The length of the plate element in the x direction is L, while the crack is located at a length of L from the left edge. The crack is of length 2c and it is symetric about the x axis. We assume different displacement

8 sets on the both sides of the crack, namely w (x,y) and φ ( x, y) along the left edge and w 2 (x,y) and φ ( x, y) 2 along the right edge. Following relations are enforced at the crack interface: Figure 2: The plate element with transverse and non propagating crack where M xx, M xy and V x are the moments and shear force obtained from w (x,y) and ( x, y) M xx2, M xy2 and V x2. The f 2 is the wavenumber transform of the slope discontinuity function f 2 ( y) flexibility function along the crack edge. The details of obtaining f ( y) 2 conditions can be written as φ and similarly for also called the is given in Reference 3..These boundary 8

9 Inverting the [M], a relationship can be established between the constants and the crack degrees of freedom, which can be integrated with regular spectral plate element. Figure 3 shows the variation of the slope for different crack lengths. Figure 3: Variation of crack flexibility as a function of crack location V Numerical Examples Here, the formulated 2D WSFE is used to study axial and transverse wave propagation in composite graphite-epoxy AS4/350 plates of different configurations and ply orientations. The analysis results are presented in both time and frequency domains. The responses simulated using the formulated element is first validated with 2D FE analysis. In addition, comparisons to corresponding responses obtained using FSFE are also provided. This highlights the advantages of WSFE over FSFE in modeling 2D structures with finite dimensions. The example used (shown in Fig. ) consists of uniform cantilever plate. The material properties are as follows: E =44.48 GPa, E 2 =E 3 =9.63 GPa, G 23 =G 3 =G 2 =4.28 GPa, v 23 =0.3, v 3 = v 2 =0.02, and ρ =389 kg/m 3. In all the examples provided, the load applied is a unit impulse of time duration 50 µ s and occurs between 00 and 50 µ s, with frequency content 44 khz. The load is applied at the edge along the Y-axis and has a spatial distribution of ( Y / α ) F( Y ) = e 2 (32) where α is a constant and can be varied to change the Y-axis variation of the load. 9

10 Figure axes barely readable Figure 4: The (a)real and (b) imaginary parts of the wave number of a plate with asymmetric ply layup of [0 4 /9 04 ] The 2D WSFE model is formulated with the Daubechies scaling function of order N=22 for temporal approximation and N =4 for spatial approximation. The time sampling rate is, unless otherwise mentioned, while the t = 2µ s spatial sampling rate Y is varied depending on L Y and load distribution F(Y). The uniform cantilever plate shown in Fig. is fixed at one edge (CD) and free at the other edge AB along Y-axis (Show A. B, C, D on figure). Numerical experiments are performed by considering the other two edges (AC and BD) along X-axis to be free-free. The dimensions are L X and L Y along X and Y axis, respectively, while the depth (=2h) is kept fixed at 0.0 m with eight laminates. A. Spectrum Relations. The spectrum relation for the plate with L Y =0.25 m and asymmetric ply lay up [0 4 /90 4 ] are plotted in Fig. 4. Figures 4(a) and 4(b) respectively show the real and imaginary parts of the wave numbers for a Y wave number of 50. It can be seen that the wave number has significant real and imaginary parts. This implies that the waves are inhomogeneous in nature, i.e., it attenuates as it propagates. The wave numbers have been obtained with t=4 µ s, i.e, for a Nyquist frequency of f nyq =25 khz. As said earlier 6, WSFE predicts accurate wave numbers only up to a certain fraction PN of Nyquist frequency f nyq. This fraction PN for N=22 is~ 0.6. Thus in Figs. 4(a) and 4(b), the wave numbers are plotted up to a frequency f N = PN f nyq =75 khz. There are three cutoff frequencies, which vary with the wave number. 0

11 Fig. 5 Axial (a) and transverse (b) velocities at midpoint of edge AB in a [0 4 /90 4 ] cantilever plate (see Fig. ) with L X =0.5 m and L Y =0.25 m due to tip impulse load applied in axial and transverse directions along AB, respectively (Fig b legend and line color mismatch) B. Response Analysis of a Healthy Plate. Next, the time domain responses of a plate with L X =0.5 m, L Y =0.25 m and asymmetric ply orientation of [0 4 /90 4 ], simulated using the WSFE method are validated with 2D FE results. In Figs. 5(a) and 5(b), respectively, the axial and transverse velocities measured at the midpoint of edge AB (see Fig. ) of the cantilever plate are plotted and compared to 2D FE results. The impulse loads are applied along AB correspondingly in axial and transverse directions. The Y variation of the load is obtained using??_=0.03 in Eq. (32). As mentioned earlier, only one WSFE is used to model the structure and the time window is kept to T w =52 µ s. The number of discretization points along Y-axis is m=64, and thus, the spatial sampling rate is Y =L y/(m )=0.004 m. A very refined mesh with 6432 four-noded plane stress quadrilateral elements were used for the 2D FE analysis, while Newmark s scheme with time step µ s was used for time integration. It can be seen that WSFE and FE results match very well. A comparison is also provided to FSFE results. As stated earlier, it can be seen from these results that unlike WSFE, FSFE is unable to accurately capture the reflections from the lateral edges AC and BD in this example. The velocities obtained from FSFE modeling show only the reflection from the fixed edge CD. Thus, for structures with finite or short dimensions, FSFE results will deviate substantially from the actual responses. In addition, simulation with FSFE requires a throw-off element to impart artificial damping to the structure and a large time window T w =6,384??_s to remove the distortions due to the wraparound problem. It should be restated here that the accuracy of the response simulated using WSFE is independent of the time window T w, which is chosen, as required, for observation.

12 Fig. 6 Snapshots of (a) axial velocities at time instance T =250 µ s and (b)transverse velocities at time instance T =000 µ s in a [0] 8 cantilever plate (see Fig. ) with L X =2.0 m and L Y =0.5 m due to tip impulse load applied in axial direction Figures 6(a) and 6(b) show the snapshots of the axial and transverse velocities of the cantilever plate shown in Fig. with a symmetric ply orientation of [0 8 ] at time instances T=250 µ s and T=000 µ s, respectively. The plate dimensions are L X =2.0 m and L Y =0.5 m, and is modeled using a single WSFE with m=64 sampling points in the Y direction. The impulse load as explained earlier is applied along edge AB in the axial and transverse directions, and the Y variation is obtained with α =0.05. The snapshot at T=250 µ s (Fig. 6(a)) shows the forward-moving axial wave. Similarly, Fig. 6(b) shows the forward-moving transverse waves, which are dissipative in nature. It should be mentioned here that the velocities at all the sampling points along Y direction and at any point along X direction used to obtain the snapshots are obtained from a single simulation. C. Response Analysis of a Transversely Cracked Plate. The effect of a nonpropagating transverse crack on the velocity field is studied in this section. A GFRP plate is taken for this purpose, which is.0 m long (L in Fig. 2). This large propagating length is taken to distinguish clearly the position of the crack (L in Fig 2.), where the crack length is taken as 0. m. In this example symmetric plysequence ($[0_{0}]$)?? is considered, where each ply is.0 mm thick. The plate is fixed at one end and is impacted by the same pulse loading used in the previous example at the other end. The response of the plate (transverse velocity) for various locations of the crack is shown in Fig 7 along with the response of a healthy plate at the bottom. 2

13 Figure 7: Wave scattering due to a transverse crack: Broadband pulse As the figure suggests the presence of crack does not alter the peak amplitude and the group velocity of the bending mode (as the reflection from the fixed end arrives at 500 µsecs in all the cases. However, due to the presence of the crack, there is an extra reflection from the crack front which arrives before the boundary reflection (at around 250, 200 and 50 µsecs, for L = 0.25, 0.5 and 0.75, respectively). Figure 8: Wave scattering due to a transverse crack: Narrow banded tone burst signal Next, a modulated pulse is applied at the free end and the transverse velocity is measured at the same point. Such a pulse has very low noise level and can be used to asses the location of the flaw by analyzing the reflections of this pulse. Fig 8 shows the time histories for both cracked (top) and healthy (bottom) plate. The extra reflections can be clearly seen in the subfigures, which change their positions with variation in L 3

14 VI Conclusions The paper presents the formulation of spectral plate element based on wavelet transforms, wherein the response analysis of both healthy and the damaged composite plate containing transverse crack is analyzed. The transverse damage is incorporated using a flexibility function that modifies the slope at the crack front. The spectral element approach presented in the paper shows the elegant way of capturing the transient response in a damaged plate, which can be used to predict the damage location. Acknowledgments The authors wish to thank the Air force Office of Scientific Research (AFOSR), Washington DC, USA and the Asian Office of Advanced Research and Development (AOARD), Toyo, Japan for the financial support to carry out this research. The authors also wish to thank the respective Program Managers Dr. David Stargel and Dr. Kumar Jata for their constant support and encouragement in carrying out this work. References Sohn, H., Farrar, C. R., Hemez, F. M., Shunk, D. D., Stinemates, D. W., and Nadler, B. R. (2003), A Review of Structural Health Monitoring Literature: , Los Alamos, National Laboratory Report, LA-3976-MS. 2 Chang, Fu-Kuo (ed), Structural Health Monitoring 2003: From Diagnostics & Prognostics to Structural Health Management, 4th International Workshop on Structural Health Monitoring, Stanford University, CA, September 5-7, Chang, Fu-Kuo (ed), Structural Health Monitoring 2005: Advancements and Challenges for Implementation, 5th International Workshop on Structural Health Monitoring, Stanford University, CA, September 2-4, Chang, Fu-Kuo (ed), Structural Health Monitoring 2007: Quantification, Validation, and Implementation, 6th International Workshop on Structural Health Monitoring, Stanford University, CA, September -3, Daubechies, I., 992, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia 6 Mira Mitra, Wavelet Based Spectral Finite Elements for Wave Propagation Analysis in Isotropic, Composite and Nano-composite Structures, July 2007, Ph.D Thesis, Indian Institute of Science, Bangalore, India. 7 Reddy, J. N., 997, Mechanics of Laminated Plates, CRC Press, Boca Raton. 8 Mitra, M., and Gopalakrishnan, S., Spectrally Formulated Wavelet Finite Element for Wave Propagation and Impact Force Identification in Connected D Waveguides, 2005, Int. J. Solids Struct., 42, Beylkin, G., On the Representation of Operators in Bases of Compactly Supported Wavelets,, 992, SIAM (Soc. Int. J. Appl. Math.) J. Numer. Anal., 6 (6), pp Mitra, M., and Gopalakrishnan, S., Extraction of Wave Characteristics from Wavelet Based Spectral Finite Element Formulation,, 2005, Mech. Syst. Signal Process., 20, pp Chen, M. Q., Hwang, C., and Shih, P., The Computation of Wavelet- Galerkin Approximation on a Bounded Interval,, 2006, Int. J. Numer. Methods Eng., 39, pp Gopalakrishnan, S., Chakraborty, A., and Roy Mahapatra, D., 2007, Spectral Finite Element, Springer, UK 3 Chakraborty, A. and Gopalakrishnan, S, A Spectral Finite Element Model for Wave Propagation Analysis in Laminated Composite Plate, ASME Journal of Vibration and Acoustics, 28(4), ,