GUDED WAVE ENERGY DSTRBUTON ANALYSS N NHOMOGENEOUS PLATES Krishnan Balasubramaniam and Yuyin Ji Department of Aerospae Engineering and Mehanis Mississippi State University, Mississippi State, MS 39762 NTRODUCTON An analysis of guided wave energy propagation in a inhomogeneous multi-layered omposite struture is presented. t has earlier been reported that ultrasoni guided wave energy within an inhomogeneous omposite materials tend to follow the orientation of the fibers, even when the plane of the inident wave is not along the fiber diretion [1-3]. n this paper, an exat analysis of a plane wave inident onto a generally anisotropi, viso-elasti, multi-layered omposite struture is use here to study the energy flow behavior of guided ultrasoni waves in inhomogeneous omposite plates [4] is used to predit this behavior. The refleted and refrated oeffiients are obtained by using the well know Thomson-Haskell transfer matrix method [5] whih transfer boundary onditions from one side of a solid medium to the other. Then, the power flow vetor is used to study the energy distributions within the plates as well as the generation mehanism of guided waves. This work also show that the assumption of the superposition of partial waves, used in the derivation of dispersion relationships for anisotropi plates, [6] may not be valid for the omputation of the power flow vetor (the magnitude represents energy), espeially in ases when the plane of inidene is not along the planes of material symmetry. This is aused due to the splitting of the fundamental wave mode propagation energy along preferred diretions, [7] thus not satisfying the superposition priniples. Theoretial results will be supported through experimental results on guided plate wave flow patterns. Review of Progress in Quantitative Nondestrutive Evaluation. Vol. 14 Edited by D.O. Thompson and D.E. Chimenti. Plenum Press. New York. 1995 227
THEORETCAL FORMULATON Let us first onsider the plane wave propagation in a generally anisotropi multi-layered struture as shown in Fig.l. The wave is assumed with the displaements in the form (1) where i=,2,3; A is the amplitude; Ui is the displaement vetor; ki are the wave number's osine omponents; ffi is the irular frequeny. Substituting eq.(l) into the equation of motion a 2 Ui _l a (auk au1 ) p at2 -:;,CijklaXj ax 1 + ax k (2) gives a system of equations where, rij=kjkmcijjm, and Cij1m are material onstants whih in a general form are omplex, ie. (3) 1 Cijlm=Cijlm+ ] ()11 ijlm (4) Here, llijjm are visosity oeffiients. Sine the inident diretion is known, the oordinate system an be hosen by onsidering X3 normal to the interfae between the layers and Xj within the inident plane. Thus k2= and k\ an be evaluated. Sine the unit displaement vetor Ui ;f., the determinant of the oeffiient matrix equals zero as represented below: (5) Expanding the above equation a 6th order of polynomial expression is obtained (6) Where ai (i=,1,2,3,4,5,6) are funtions of the material properties. Solving eq. (6), the six roots of k3 an be found. n general, these roots are omplex (7) where k' 3 defines the wave propagation diretion and k" 3 defines the attenuation of the wave amplitude in the X3 diretion. Substituting k3 to Eq.(3) one an get six unit displaement vetors Ui' with respet to six k3"(a,=1,2,...,6), then substitute k3 and Ui" to eq.(l) and using supposition, the displaement field is obtained.then by using the Thomson transfer matrix, the displaements an be obtained (8) where S'i=[U'j,U'2,U'3,"33,r'23,"3]T is the displaements and stresses at the top interfae, and Si=[U j,u2,u3''33''23''3]t is the displaements and stresses inside plate at depth 'z'. Tij is the 228
Thomson transfer matrix. t is known that the Sj is generally a ombination of two quasi-longitudinal partial wave modes and 4 quasi-shear partial wave modes. Hene, we get the following relationship whih must be solved to obtain the refletion and transmission fators for the amplitude of the six partial wave modes A r:j. (where <X=1,2,... 6) whih represent the refletion and transmission fator amplitudes. The refletion oeffiient and transmission oeffiient of power flow (energy) are alulated after obtaining the amplitude refletion and transmission oeffiients (Ar:J.). Thus, the displaements orresponding to eah wave an be obtained as below: After getting the partile displaement field, the partile veloity an easily be obtained through applying derivative with respet to 't' and the stress field an be obtained from Hook's law as shown below (9) (lo) and (11) The power flow (energy refletion and transmission oeffiients) using the superposition of partial waves is then defined as [1] (12) and the power flow with respet to eah partial wave mode is provided by (13) RESULTS AND DSCUSSON The theoretial model was used to analyze the magnitude (energy) and propagation diretion of power flow vetor. The 3 layer graphite epoxy omposite ase studies will be examined at a frequeny around 1 MHz. The experiments were onduted using the setup illustrated in Fig. 2 and the plate wave flow patterns (PWFP) thus obtained are well disussed in earlier publiations [1-3]. Both, the power flow vetors using superposition as well as without superpositions, were analyzed. The results obtained using superposition were unable to explain the PWFP experimental results and hene will not be presented in this paper. nstead, the results for the power flow (energy) of the individual partial wave modes for three graphite-epoxy omposite speimen (C, D, and E) are presented in Figs. 3-5. n Figure 3a, the magnitude of the power flow vetor (energy) for speimen C with {45sf135s}s ply layup is presented as a funtion of distane into the speimen (x 3 ). The three ply groups (upper - {45 degree}, lower - {135 degree}, and the bottom - {45 degree}) are 229
-- / Figure. Representation of the theoretial multilayered model. Figure 2. Plate Wave Flow Pattern method. [1] learly distinguishable. Here Ll represents the two partial modes of quasi-longitudinal bulk wave modes (dark lines are partial modes with wave vetor pointing down and light lines represent the upward pointing partial modes). Similiarly, Tf is the fast quasi-transverse partial wave modes while Ts is the slow quasi-transverse partial wave modes. Thus, the magnitude of all six partial modes are represented in the same plot. t is observed that the magnitude of energy is maximum for the Tf mode in the middle layer while the other two modes are relatively weak. Also in the outer layer, there is an even energy distribution of the three mode types with Tf and Ll higher than the Ts partial mode. Using Fig. 3b, the orientation of these partial modes in the azimuthal (1-2 or x-y) plane is predited. From this graph, the Tf in the middle layer is predited to be at an angle of approximately 132 (or-42) degree whih inidentally is along the fiber diretion in the middle ply group and is also observed in the experimentally obtained plate wave flow pattern results (Fig. 3). n the outer layers, the diretion of Tf and Ll are both approximately equal to 4 degrees whih again is almost along the fiber diretion. This is again visible in the PWFP. The weak Ts is also visible, although not as learly (due to low amplitude) along the 17 degree in the outer layers and -18 degree in the middle ply group as predited by theory in Figure 3b. For another speimen D {6/15 j }s shown in Fig. 4(a,b,), again the theory predits the plate wave flow pattern results. Here, the Tf partial mode at -25 degree azimuthal angle is the dominating partial mode in the middle ply group. Also, Ll partial mode is the predominant partial mode in the outer layers and propagates along +53 degree and has less energy when ompared with the Tf partial mode in the middle layer. This is learly manifested in the PWFP experiments and even the weak Ts partial mode along the -12 degree is visible. For speimen E {2/11 j }s, the energy is primarily onfined to Ll mode in the outer layers and is direted along the + 19 degree azimuthal angle. The energy along the middle layers is extremely weak. This is again exhibited in the PWFP results. 23
(a) :;:: u:: :;::.. - OJ ro (b) OJ :;:::; () OJ ro.....2.....1.. 6-1--_---- - --.., -- --- J..---- - r--------- 4 '--.4 2 - -------- -2-4 r- -6..4 - L -- Tf --- Ts 1--------- - - ---_%_--- -,.8 1.2 1.6 2. 2.4 ------ ------- - L --------------- -- Tf -.8 1.2 1.6 2. 2.4 Depth (mm ) --- Ts () Figure 3. Results for speimen C {45/-45 5 ls, (a) Magnitude of Power Flow for the six bulk wave modes, (b) Orientation of the diretion of energy flow in the 1-2 (x-y) plane, and () Plate Wave Flow Pattern at 9 to Transmitter. 231
(a) s: u:: s:... - (b) Ol t! :::2: Ol U (5 :;::::; t! Ol t!.....1.. 8 6-4 - 2-2 - 1= :::..:--= =,;:..--,:".4 ------ - - -- -- - --- -4..4.. - L -- Tf --- Ts.----- 1 --- -----1 1 k _ "'-':--T"-.8 1.2 1.6 2. 2.4. -----!-------- -- L -- Tf - - - -- -- ---- - -- --- Ts..8 1.2 1.6 2. 2.4 () Figure 4. Results for speimen D{6/l5 5 }s, (a) Magnitude of Power Flow for the six bulk wave modes, (b) Orientation of the diretion of energy flow in the 1-2 (x-y) plane, and () Plate Wave Flow Pattern at 9 to Transmitter. 232
(a).4 :;: u:: :;:.. - ) Cl.3.2.1 (b) _ ) :. U (5 ) Cl.......4 4. 2 i:------ :-.------- i" -2-4 -6-8...4.8 1.2 1.6 2. 2.4 r---------------...l -----,. 1 - L 1 -- Tf 1 1 --- Ts 1 ------ 1...8 1.2 1.6 2. 2.4 () Figure 5. Results for speimen E {2/11Os}s, (a) Magnitude of Power Flow for the six bulk wave modes, (b) Orientation of the diretion of energy flow in the 1-2 (x-y) plane, and () Plate Wave Flow Pattern at 9 to Transmitter. 233
From these results, it is shown that the power flow vetor for individual partial waves (without using superposition priniple) an be used to explain guided wave behavior in inhomogeneous anisotropi multi-layered plates suh as graphite-epoxy omposites. This ould be explained by the fat that the ultrasoni wave modes in the "zone of plate wave ineption", when it enounters off axis (plane of inidene not along material symmetrial axes), tends to separate into partial modes along different diretions (espeially azimuthal to the material axis) and hene annot interfere (or superpose) with eah other. t an also be hypothetially inferred using this study that these partial wave modes are then responsible for the launhing of guided plate waves in the omposite speimen. t must be noted that the Plate Wave Flow Patterns were produed using a frequeny*thikness produt around 2.75 mm.mhz. and hene represent guided plate wave regime. [9-11] ACKNOWLEDGEMENT The researh upon whih this material is based was supported in part by the National Siene Foundation through Grant No. ST-89264, the State of Mississippi, and the Mississippi State University. REFERENCES 1. Sullivan, R, et. ai., " Experimental maging of Fiber Orientation in Multi-layered Graphite Epoxy omposite strutures, " Review of Progress in Quantitative Nondestrutive Evaluation, Vol. 13, (Ed. D. o. Thompson and D. E. Chimenti), Plenum Press, NY, 1994, pp. 1313-132. 2. Sullivan, Rand Balasubramaniam, K., " Plate Wave Flow Patterns for Ply Orientation maging in Fiber Reinfored Composites," Submitted to Materials Evaluation, 1994. 3. Sullivan, R, "Ultrasoni maging of Ply Orientation in Graphite Epoxy Laminates using Oblique nidene Tehniques, " M.S. Thesis, 1993, Mississippi State University, MS 39762. 4. Balasubramaniam, K. and Ji, Y., "Analysis of the Refletionffransmission Fator Response from a Generally Anisotropi Viso-elasti Layered Media," submitted to 1. Aoust. So. of Am.(l994). 5. Thomson W.T., 195, " Transmission of elasti waves through a stratified solid medium", J. Appl. Phys. n. pg 89. 6. Auld, B. A, "Aousti Fields and Waves in Solids," ( edition, Vol. 1 and 2), Robert E. Krieger Publishing Company, Malabar, Florida, 199. 7. M. J. P. Musgrave, "Crystal aoustis" Holden Day, San Franiso, 197. 8. Pilarski, A and Rose, J.L. "Surfae and Plate Waves in Layered Strutures." Materials Evaluation, Vol.46, (1988). 9. Lamb, H. "On Waves in Elasti Plates." Pro. Royal So. London, 93A, (1914). 1. Miklowitz,1. "Elasti Waves and Waveguides." North-Holland Pub. Co., (1978). 11. Nayfeh, AH. and Chimenti, D.E. "Propagation of Guided Waves in Fluid-Coupled Plates of Fiber-Reinfored Composite." 1. Aoust. So. of Am., Vol. 86(2), (1989). 234