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Journal of Nondestructive Evaluation, VoL 13, No. 3, 1994
Leaky Guided Wave Propagation Along Imperfectly Bonded Fibers in Composite Materials Peter B. Nagy ~
Received December 20, 1993; revised July 24, 1994
Leaky guided modes propagating along embedded fibers in a composite material can be used for characterizing the fiber-matrix interface. This principle can be applied to real composites containing small-diameter fibers by using laser interferometric detection of very fine lateral resolution on the order of a few microns. The main purpose of this paper is to develop the analytical tools needed to assess the sensitivity of guided wave inspection to interface properties in composite materials. Typically, the sound velocity is much lower in the matrix than in the fiber and the guided modes are strongly attenuated by leaking their energy into the matrix as they propagate. As a result, the velocity of the lowest-order axisymmetric longitudinal mode decreases while its attenuation increases with increasing interfacial stiffness between the fiber and the matrix. It is shown that loose fibers can be readily identified from early signals produced by fast guided modes. In the case of a well-bonded fiber-matrix interface, these guided modes are slowed down and strongly attenuated by the loading of the matrix depending on the fiber diameter and the interracial stiffness of the interface. -Interestingly, the relative difference between the well-bonded and free fibers is greater at low frequencies. Therefore, good sensitivity to the sought interracial stiffness can be achieved at a few MHz, i.e., when the fiber diameter is still much smaller than the acoustic wavelength. Our analytical results show that the leaky guided mode technique is mainly sensitive to the transverse interfacial stiffness of the fiber-matrix interface. At typical ultrasonic frequencies between 1 and 20 MHz, the technique works best in the 101~-1013N/m 3 interfacial stiffness range which is one or two orders of magnitude lower than the optimal sensitivity range of the more conventional bulk velocity and reflection methods.
KEY WORDS: Leaky guided modes; interface characterization; fiber-matrix interface; guided waves; composites; imperfect bonding.
1. I N T R O D U C T I O N
it can be accurately assessed from macroscopic velocity measurements. (1~ However, in most other types of composites, the mechanical properties of the crucial fibermatrix interface have only a negligible effect on the macroscopic elastic properties of the material, therefore they can be much better evaluated at a microscopic scale. Addison and Sinclair studied the scattering spectra of silicon carbide fibers in titanium metal matrix composites at normal incidence32,3) Although they found that the frequency response of the embedded fiber is very sensitive to imperfections in the interfacial bond, this technique cannot be easily adapted to multiple-ply com-
One of the most challenging problems in nondestructive evaluation of composite materials is the inspection of the fiber-matrix interface. In certain ceramic matrix composites, the effective interfacial stiffness between the fiber and the matrix is deliberately made very low to prevent fiber fracture due to matrix cracking. In such cases, the interfacial stiffness significantly affects the overall elastic properties of the composite, therefore Department of Welding Engineering, The Ohio State University, Columbus, Ohio 43210.
137 0195-9298/94/0900-0137507.00/0 9 1994 PlenumPublishingCorporation
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.--.--q.
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Fig. 5. Transition of the lowest-orderaxisymmetficguided mode from "rigidly held" behavior at low-frequencies to "free" behavior a~ high-frequencies (silicon carbide fiber in aluminum matrix, imperfect interface: S, = S,, = 10~ N/m3).
3. A N A L Y T I C A L RESULTS Figure 4 shows the axisymmetric guided modes along a silicon carbide fiber embedded in an aluminum matrix for both rigidly held and free fibers (the material parameters used in these calculations were taken from Ref. 8). As one would expect, the mode structure is markedly different in these extreme cases of perfectlybonded an entirely misbonded interfaces. The propagation properties (phase and group velocities, attenuation, leakage angle, vibration profile, etc.) of these modes were discussed in great detail in Ref. 8. Our current goal is to study the feasibility of characterizing the fiber-matrix interface in real composite materials by measuring the propagation parameters of leaky guided waves along the embedded fibers. Since typical fiber diameters are 150 um or less and the inspection frequency is limited to approximately 20 MHz, the crucial frequency-radius product is usually less than 1.5 MHz mm. Therefore, from the infinite variety of theoretically-predicted guided modes, the only one of practical importance in our study variety of theoretically predicted guided modes is the lowest-order axisymmetric mode. All of the following results presented in this paper are limited to this particular mode. Figure 5 shows the frequency-dependent phase velocity of this mode for a silicon carbide fiber embedded in an aluminum matrix. The dashed line represents the well-known dispersion of a completely free fiber. At low frequencies, the mode asymptotically approaches the socalled rod velocity which is just the square-root of the ratio between Young's modulus and the density of the fiber (the fiber is assumed to be isotropic). At higher
frequencies, due to the Poisson effect, ti~e phase velocity starts to decrease and, at very high frequencies, asymptotically approaches the Rayleigh velocity. The solid line represents the dispersion of the rigidly held fiber. The most interesting feature of this mode is that at low frequencies the phase velocity drops below the shear velocity of the matrix and approaches zero. This unusual behavior is caused by the increasing loading of the matrix via the transverse component of the surface vibration. When the frequency decreases, the fiber diameter becomes negligible with respect to the acoustic wavelength in the fiber and the surrounding matrix. As a result, the elastic singularity presented by the finitediameter fiber embedded in the infinite matrix vanishes and the guided mode disappears with it. In comparison, the loading effect on a thin fiber immersed in a viscosib free fluid diminishes at low frequencies since the normal surface vibration (radial motion) vanishes. In a thin fiber embedded in an elastic solid, the remaining transverse surface vibration (axial motion) is also coupled to the surrounding medium therefore the loading effect of the matrix does not disappear but rather increases at low frequencies. In this region, the attenuation increases without limit and the guided mode cannot be regarded as a propagating mode any more. There is only one axisymmetric propagating mode at very t~ow frequencies, namely the longitudinal mode of the matrix running parallel to the fiber. Naturally, this is not a guided mode since its amplitude does not decay with increasing distance from the fiber and therefore it is not predicted by our dispersion equation.
Leaky Guided Wave Propagation Beside the free and rigidly bonded fibers, Fig. 5 also shows the lowest-order axisymmetric guided mode along an imperfectly bonded fiber (S, = S, = 1013 N/ m3). At very low frequencies, an imperfectly bonded interface of finite interfacial stiffness behaves as a perfect bond and the guided mode asymptotically approaches the previously discussed case of a rigidly bonded fiber. At very high frequencies, the same imperfect interface acts like an ideal misbond and the guided mode asymptotically approaches the case of a free fiber. In between, there is a smooth transition from the low-frequency "rigidly held" behavior to the high-frequency " f r e e " behavior. Naturally, the frequency at which this transition occurs is determined by the value of the interfacial stiffness. At a given inspection frequency, below a minimum interfacial stiffness, all boundaries appear to be perfectly bonded. Similarly, above a maximum interracial stiffness, the fibers appear to be perfectly free. The interfacial stiffness range between these limiting values is the actual measuring range at that particular frequency. In the following, we shall determine this measuring range where the propagation parameters of the lowest-order axisymmetric guided mode are sensitive to changes in the interface properties. Figure 6 shows the frequency-dependent propagation parameters of the same leaky guided mode along a silicon carbide fiber in an aluminum matrix for different interfacial stiffnesses. At any particular frequency, the phase velocity increases while the attenuation decreases with increasing interfacial stiffness. As we have demonstrated above, at sufficiently low and high frequencies, the imperfect interface asymptotically approaches a perfectly rigid bond and a free surface (delaminated interface), respectively. As far as the phase velocity is concerned, the difference between these two limiting cases decreases with frequency therefore the velocity is more sensitive to small changes in the interfacial stiffness at low frequencies. The leaky loss is measured in terms of the normalized attenuation, i.e., the attenuation coefficient divided by frequency. The frequency-dependence of the attenuation coefficient is markedly different from the behavior of the leaky loss in an immersed rod where the attenuation greatly increases with frequency (the normalized attenuation in an immersed rod is proportional to p). From a practical point of view, guided wave propagation is feasible only above a minimum frequency where the finite interfacial stiffness produces sufficiently weak coupling to the matrix so that the leaky loss is acceptably low. It is quite clear that more or less attenuation-free guided wave propagation is feasible even when the normal interfacial stiffness is very high if the transverse
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tenuated by leaking their energy into the matrix as they propagate. We found that the lowest-order axisymmetric mode lends itself the most easily for ultrasonic assessment of the fiber-matrix interface properties. The velocity of this mode decreases while its attenuation increases with increasing interfacial stiffness between the fiber and the matrix. Interestingly, the relative difference between the well-bonded and free fibers is greater at low frequencies therefore good sensitivity to the sought interfacial stiffness can be achieved at a few MHz, i.e., where the fiber diameter is still much smaller than the acoustic wavelength. Our analytical results show that the previously discussed leaky guided mode technique is mainly sensitive to the transverse interfacial stiffness of the fiber-matrix interface and it works best in the Sr ~- 10"-10 ~2 N/m 3 range. In this region, the displacement discontinuity between the fiber tip and the surrounding matrix is more than 20 dB and the velocity of the leaky guided mode is directly related to the sought interracial stiffness. The technique can be extended to the Sr ~" 1012-1013 N / m 3 stiffness range by increasing the inspection frequency. For example, at 13 MHz, a typical 150-gm-diameter (fa 1 MHz mm) silicon carbide fiber embedded in an aluminum matrix can propagate a leaky guided mode over 7.5 mm (100 radii) with as little as 12 dB loss if the transverse interfacial stiffness is Sr = 10 ~3 N/m 3. In this higher stiffness range, the attenuation is more sensitive to the interface properties than the velocity. We also showed that at sufficiently high frequencies significant displacement discontinuity occurs at the fiber-matrix interface, which might be also used to characterize the interface. Assuming that as little as 5% "interface
opening displacement" is stili detectable, the maximum interfacial stiffness which can be measured in this way is roughly Sz = 10 ta N/m 3 at fa = I MHz ram. tn comparison, at the same frequency, shear wave reflectivity measurement at normal incidence is most sensitive to the transverse interfacial stiffness around Sr = 10 js N/m -~, i.e., one order of magnitude higher. Low-frequency precision velocity measurements can be also used to characterize the interfacial stiffness. This technique is most sensitive around Sr = G/a. where G denotes the shear modulus of the fiber. For a typical SCS6 fiber, this transition point is around S r ~ 2 10 L5 N/m 3, i.e., again roughly one-two orders of magnitude higher than in the case of leaky guided wave inspection. The significantly lower sensitivity range of the guided wave technique is due to the fact that even relatively weak coupting between the fiber and the surrounding matrix through the interface can cause very substantial leakage over a long propagation path. Therefore, we have to conclude tha~ the leaky guided wave technique cannot be used directly to quantitatively verify the results of these more conventional measurements, but rather it is a complementary approach which extends the total inspection range toward low interracial stiffnesses.
ACKNOWLEDGMENTS This work was supported by the Center ;for Advanced Nondestructive Evaluation, operated by the Ames Laboratory, USDOE for the Air Force Wright Laboratory/Materials Directorate under Contract No. W7405-ENG-82 with Iowa State University. The author gratefully acknowledges many fruitful discussions with Stanislav Rokhlin and Eva Drescher-Krasicka.
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