NDT.net - February 2001, Vol. 6 No. 2 |
Fig 1: A Schlieren image of the LLW phenomenon showing a tone-burst before (right) and after impinging on a graphite/epoxy laminate. |
Bar-Cohen and Chimenti [1984] investigated the characteristics of the LLW phenomenon and its application to NDE, focusing on the experimental documentation of observed modes and the effects of various types of defects on the measurements. Their study was followed by numerous theoretical and experimental investigations of the phenomenon [e.g., Nagy, et al, 1987, Nayfeh & Chimenti, 1988, Mal & Bar-Cohen, 1988, Rohklin, et al, 1989, Dayal & Kinra, 1989, subramanian & Rose, 1991, Jansen & Hutchins, 1992]. A method was also developed to invert the elastic properties of composite laminates from the LLW dispersion data [Mal, 1988; Mal & Bar-Cohen, 1988; and Shih, et al, 1998] and the study was expanded to the NDE of bonded joints [Bar-Cohen, et al, 1989].
The experimental acquisition of dispersion curves for composite materials requires accurate control of the angle of incidence/reception and the polar angle with the fibers. To perform these measurements rapidly and accurately, a LLW scanner was designed and constructed by Bar-Cohen, et al, [1993] in collaboration with QMI (Costa Mesa, CA). With the aid of a personal computer, the scanner controls the height, angle of incidence and polar angle of the pitch-catch setup. It also manipulates the angle of incidence/reception simultaneously while maintaining a pivot point on the specimen surface.
A photograph of the scanner installed on a C-scan unit is shown in Figure 2. A computer code was written to control the incidence and polar angles, the height of the transducers from the sample surface, and the transmitted frequency. The data acquisition involves the use of sequentially transmitted tone-bursts, each of a specific frequency, spanning a selected frequency range (within 20dB level of the transducer pair). The reflected signals are acquired as a function of the polar and incidence angles and saved in a file for analysis and comparison with theoretical predictions. The minima in the acquired reflection spectra represent the LLW modes and are used to determine the dispersion curves (phase velocity as a function of frequency). The incident angle is changed incrementally within the selected range and the reflection spectra are acquired. For graphite/epoxy laminates the modes are identified for each angle of incidence in the range of 12^{o} to 50^{o} allowing the use of free-plate theoretical calculations. At each given incidence angle, the minima are identified, added to the accumulating dispersion curves, and plotted simultaneously on the computer display. While the data acquisition is in progress, the acquired minima are identified on both the reflected spectra and the dispersion curves.
Fig 2: A view of the LLW scanner (bridge right side) installed on the JPL's C-scan system. |
A follow-on study by Bar-Cohen, et al, [1991] showed that the capability to invert the elastic properties using LLW data is limited to the matrix-dominated stiffness constants. This limitation can be partially overcome if the incidence angles of 10 o or smaller can be used in the experiments, but this is difficult if not impossible to achieve in practice with a single scanner. An alternative methodology based on pulsed ultrasonics was developed by Bar-Cohen, Mal and Lih [1993]. Using pulses in pitch-catch and pulse-echo xperimental, it was shown that all five elastic constants of a unidirectional laminate could be determined with a fairly high degree of accuracy. This chapter provides a summary of the analytical model, the experimental procedure and test results used in he LLW method. It also discusses the challenges that need to be overcome in order to transfer the technology to practical applications.
Plate wave theory
The behavior of ultrasonic waves propagating through fiber-reinforced composites is determined by the material stiffness and dissipative characteristics. Theoretical modeling of this behavior requires several simplifying assumptions regarding the properties of the material. First, the unidirectional bulk material is treated as homogeneous, since the fiber diameter (e.g., graphite 5-10 mm and glass 10-15 mm) is significantly smaller than the wavelength (for frequencies up to 20 MHz the wavelength is larger than 100mm). Each layer of a composite laminate is assumed to be transversely isotropic, bonded to its neighboring layers with a thin layer of an isotropic resin at their interfaces. The mechanical behavior of the material of each lamina is described by an ensemble average of the displacements, the stresses and the strains over a representative volume element [Christensen, 1981]. The average strains are related to the average stresses through the effective elastic moduli. As a transversely isotropic material, unidirectional fiber-reinforced composites are characterized by five independent effective stiffness constants c_{ij}. These constants dependent on the elastic properties of the fiber and matrix materials as well as their volume fraction. The stress components, s_{ ij} are related to the strain components e_{ ij} through a linear constitutive equation. For a transversely isotropic elastic solid with its symmetry axis along the x_{1}-axis this equation can be expressed in the form [Mal and Singh, 1991],
(1) |
The geometry of the general theoretical problem is shown in Figure 3, where a plane acoustic wave is incident on a composite laminate immersed in a fluid. The thickness of the fluid layers above and below the specimen is large compared to the wavelength and the thickness of the laminate, so that they can be assumed to be semi-infinite. The interaction of the ultrasonic waves with the plate excites a variety of elastic waves including reflected and transmitted body waves and multimode guided waves. All of these waves are affected by the properties of the material as well as the nature of the interfaces. Some of their characteristics can be xtracted from the reflected and transmitted wave data that is acquired as a function of frequency and angle of incidence through comparison with theoretical results.
Fig 3: The general theoretical problem associated with the LLW method. |
Application of the LLW technique to the NDE of composite structures requires theoretical calculation of the Lamb wave dispersion curves for a given model of laminates that are often used to construct these structures. A number of authors have dealt with this problem [see, e.g., Nayfeh and Chimenti, 1988; Mal and Bar-Cohen, 1988] and have developed efficient and accurate computer codes for generating the dispersion in presence or absence of the surrounding fluid. The effect of fluid loading has been found to be negligible at incident angles and frequencies of interest in most applications. For a homogeneous but anisotropic plate the dispersion curves are obtained through the solution of an equation of the form
G(v,f, n_{i}, c_{ij}, r , h) = 0
where v is the phase velocity of the guided waves at frequency, f , along the direction vector, n, on the plate surface, c_{ij} is the elastic tensor and r is the density of the material, and h is the plate thickness. The dispersion function, G, is obtained from the solution of the boundary value problem depicted in Figure 3. It should be noted that for a given value of v, there are multiple values of f associated with different modes of propagation of the Lamb waves in the plate, and these frequencies correspond to the minima in the reflected spectra for a given incident angle.
For a multilayered laminate with n layers, each of the material constants, c_{ij}, r, and h is a vector of dimension n. At low frequencies, where the wavelengths are long compared to the total plate thickness, it is possible to develop approximate theories based on kinematic assumptions regarding the variation of the strains across the thickness of the plate. The approximate dispersion equation is of the form
(2) |
The Simplex Algorithm
The locations of the minima in the spectra of the reflected waves have been found highly sensitive to the thickness and the stiffness constants of the plate and are relatively insensitive to the damping parameters as well as the presence of water in a broad frequency range. Thus the dispersion data can, in principle, be used to determine accurately these properties and any changes in their values during service. The phase velocity of guided waves in a composite laminate in absence of water loading is obtained from the theoretical model as a transcendental equation and its function requires minimization. The minimization can be carried out through a variety of available optimization schemes. We have used the Simplex algorithm, which is based on a curve-fitting technique for a given set of data points to any function, no matter how complex. We illustrate the basic concept behind the method through its application to a simple function first.
Let us consider a function f of two variables x and y, and two unknown parameters a and b:
f ( a,b; x,y) = 0
Let us assume that there are a total of n data sets, x_{i}, y_{i}, i = 1, 2, ... n. Since the data sets are not exact, the error, e_{i}associated with the i ^{th}set is given by
e _{ i} = f( a,b;x_{i},y_{i} )
An objective function is defined in the form
(3) |
A simplex algorithm is a geometric figure that has one more vertex than the dimension of the space in which it is defined. For example, a simplex on a plane (a two-dimensional space) is a triangle; a simplex in a three-dimensional space is a tetrahedron, and so on. Returning to the simple example, we build a plane with a and b as the two axes; then create a simplex (a triangle) as shown in the Figure 4. Each vertex of the triangle is characterized by three values: a, b and S.
Fig 4: A 2-D simplex BWO illustrating the four mechanisms of movement: reflection, expansion, contraction, and shrinkage. B = best vertex, W = worst vertex, R = reflected vertex, E = expanded vertex, C = contracted vertex, and S = shrinkage vertexes. |
These four mechanisms are illustrated in the Figure 4. The above steps are repeated until satisfied convergence is achieved. The locations of the minima in the reflection coefficients are highly sensitive to the thickness and the stiffness constants of the plate and are insensitive to the damping parameters as well as the presence of water in a broad frequency range. Thus the dispersion data can, in principle, be used to determine accurately these properties and any changes in their values during service. Other details of the inversion process can be found in [Karim et al, 1990], and will not be repeated here. It should be noted that the simplex method is extremely efficient for the data inversion problem associated with the LLW technique.
The phase velocity of guided waves in a composite laminate in absence of water loading is obtained from the theoretical model as a transcendental equation given in equations (2) and (3). For a given data set {f_{k}, v_{k}}, c_{ij} and h can be determined by minimizing the objective function
(4) |
Data Inversion
The experimental system used in our studies consists of an LLW scanner (shown schematically
in Figure 5, which is computer controlled for changing the transducers' height, rotation angle and the angle of incidence. The LLW scanner is an add-on to standard ultrasonic scanning systems as shown photographically in Figure 2. The control of the angle of incidence allows
simultaneous changes in the transmitter and receiver angles while maintaining a pivot point on
the part surface and assuring accurate measurement of the reflected ultrasonic signals. The
signals are transmitted by a function generator that frequency-modulates the required spectral
range. This generator also provides a reference frequency marker for the calibration of data
acquisition when converting the signal from time to frequency domain. A digital scope is used
to acquire the reflection spectra after being amplified and rectified by an electronic hardware. The signals that are induced by the transmitter are received, processed and analyzed by a personal computer after being digitized. As discussed earlier, the reflected spectra for each of the desired angles of incidence is displayed on the monitor and the location of the minima (LLW modes) are marked by the computer on the reflection spectrum. These minima are accumulated on the dispersion curve, which is shown in the lower part of the display in Figure 6. The use of the frequency modulation approach introduces a significant increase in the speed of acquiring the dispersion curves. In this approach 20 different angles of incidence can be acquired in about 45 seconds in contrast to over 15-minutes using the former approach. Once the dispersion data is ready, the inversion option of the software is activated and the elastic stiffness constants are determined as shown in Figure 6.
Fig 5: A schematic view of the rapid LLW test system. | Fig 6: Computer display after the data acquisition and inversion completion. The elastic stiffness constants are inverted from the dispersion curve and are presented on the left of the screen. |
Fig 7a: The refection at 39.5 degrees incidence angle and the dispersion curve for a Gr/Ep [0]_{24} laminate with no defects. | Fig 7b: The response at a defect area where simulated porosity was introduced in the middle layer using microballoons. |
The LLW method can also be used characterize materials degradation of composites. A sample made of AS4/3501-6 [0]_{24} laminate was tested after it was subjected to heat treatment. The sample was exposed to a heat ramp from room temperature to 480^{o} F for 15 minutes, and then was taken out of the oven to cool in open air at room temperature. It was tested at a specific location before and after heat treatment. The measured dispersion curves for the two cases are shown in Figure 8. It can be seen that there are distinct differences in the dispersion data for the specimen before and after heat treatment. Since the heat damage occurs mostly in the matrix, the effect is expected to be more pronounced in the matrix dominated stiffness constants. The constants c_{11}, c_{12}, c_{22}, c_{23} and c_{55} obtained from the inversion process are 127.9, 6.32, 11.85, 6.92 and 7.43 GPa, before heat treatment, and 128.3, 6.35, 10.55, 6.9 and 7.71 GPa, after heat treatment. The most noticeable and significant change is in the stiffness constant c_{22}, which is the property most sensitive to the matrix, and it resulted in a significant reduction in the transverse Young’s modulus.
It should be noted that the dispersion equation is strongly nonlinear in c_{ij} (stiffness matrix) and h(thickness), and its solution is non-unique. Thus, extreme care must be taken in interpreting the numerical results obtained from the inversion of the dispersion data. Extensive parametric studies of the inversion process have shown that only the thickness and the matrix dominated constants c_{22}, c_{23}and c_{55} can be determined accurately from the inversion of dispersion data. This is due to the fact that the dispersion function G is not very sensitive to the fiber dominated constants c_{11} and c_{12}. These two constants can be determined accurately from the travel times and amplitudes of the reflected short-pulse signals in the oblique insonification experiment [Bar-Cohen, Mal and Lih, 1993], to be discussed below. For a thin symmetric cross-ply laminate, the guided data must be supplemented with direct measurement of other wave speeds in order to determine all three in-plane laminate constants [Shih, Mal and Vemuri, 1998].
Fig 8: The measured dispersion curves of a [0]_{24} graphite-epoxy panel before and after heat treatment. |
One of our recent enhancements is the use of transmitted signals that have been modified to frequency-modulated pulses that are induced sequentially within the required spectral range. The specific signal consists of a time-dependent voltage function V(t) that is amplified and transmitted at an induced frequency f(t) of the form
(5) |
Optimal Adjustment of the Wave Propagation orientation
The direction of the wave propagation in relation to the fibers is important for quantitative
evaluation of the test results. Generally, the top layer is used in the LLW studies as the reference orientation or as the 0^{o} layer. Visual adjustment of the orientation can lead to errors as high as ±5^{o} . In order to minimize this error, the polar backscattering that was discovered by the principal author [Bar-Cohen and Crane, 1982] was used to determine the direction of the first layer. The approach takes advantage of the fact that maximum backscattering is obtained from the fibers when insonifying the laminate within the plan that is normal to the fiber orientation. Practically, the transducer needs to be rotated about 200^{o} around an axis that is normal to the laminate and recording the amplitude of the backscattered signals while searching for the maximum value. A program that the authors wrote to search for the maximum amplitude provided about ±0.1^{o} accuracy of the first layer orientation.
Optimal Adjustment of Transducers’ Height
During Leaky Lamb Wave (LLW) experiments, the height of the transducer pair is an important
issue. For best results, the crossing point of the beam from the two transducers must fall on the surface of the specimen. If this point is located outside the surface, the sharpness of the minima in relation to the adjacent data points becomes weaker as shown in Figure 9a. In such cases, the identification of the minima becomes difficult and increasingly inaccurate. To adjust the height of the transducers precisely and efficiently, an algorithm was developed and implemented into the computer control program for the LLW experiment. The algorithm optimizes the contrast of the minima by determining the maximum variations in the reflection spectrum using adaptive adjustment of the height of the transducer pair.
Fig 9a: "Out of focus" LLW reflection spectrum. | Fig 9b: Optimized reflection spectrum using adaptively adjusted transducers pair height. |
As indicated earlier, the use of the frequency modulation and automatic optimal adjustment of the transducer height significantly increase the speed and accuracy of acquiring LLW dispersion curves.
Fig 10a: A view of the LLW dispersion curve for a unidirectional Gr/Ep for wave propagation along the fibers generated by an imaging method. | Fig 10b: Sam as in Fig 10a for propagation at 90^{o} to the fibers. |
Fig 11: A view of the first symmetric and antisymmetric modes of Lamb wave dispersion curves in an aluminum plate using the imaging method. |
Pulsed Oblique Insonification
Time-domain experiments in certain directions offer an alternative method to determine the
elastic constants c_{ij} for unidirectional composites accurately and efficiently. The ray analysis of the various reflected waves in the leaky Lamb wave experiment for a unidirectional composite plate immersed in water has been given in [Mal, et al, 1992]. The ray diagram for some of the early reflections is shown in Figure 12a. Here R^{o} indicates the first reflected wave from the top surface of the plate, the rays labeled 1, 2, 3 are associated with the three transmitted waves inside the plate in a decreasing order of their speeds, the rays labeled 11, 12, ..., and 33 are associated with the waves reflected from the bottom of the plate, and T_{1}, T_{2}, T_{3} indicate the waves transmitted into the fluid through the bottom of the plate. From Snell's law, the velocities V_{k}, V_{l} and the angles q_{k}, q_{l} in the diagram are related through
(6) |
Fig 12: Ray diagram of the reflected waves in a unidirectional composite plate |
Two possible ray paths leading to the same point on the receiver are sketched in Figure 12b. The difference in the arrival times, t_{kl}, between rays along paths DO and O¢BO can be expressed as
(7) |
Since experiment (d) is difficult to carry out due to the small incident angles involved, an alternate method is proposed to determine the remaining constants c_{11} andc_{12}. Our calculations have shown that the reflected field changes significantly near the critical angle [Mal, et al, 1993], and the arrival times of these pulses are strongly affected by c_{11} near the critical angle. We use
this critical angle phenomenon to determine the constantsc_{11} and c_{12}. Recall that the equation for the bulk wave speed V associated with the constantsc_{11}andc_{12}can be written as
where n_{1} = cosf, n_{2} = sinf. The constants a_{1}, a_{4}, a_{5} can be derived from the known constantsc_{22}, c_{23}, c_{55}, and the remaining unknowns a_{2} and a_{3} can be solved and the constantsc_{11}andc_{12}can be calculated for two measured critical angles f_{c} at different incident angles
Results From Pulsed Oblique Insonification
Polymer and metal matrix composites were tested to examine the capability of the ultrasonic oblique insonification method as a means to detect defects in them and to determine their material properties. Using the experimental setup and the LLW inversion algorithm described earlier the elastic properties of graphite/epoxy AS4/3501-6 composite using laminate specimens of different thickness. The result for a 24-layer, 3.378 mm thick unidirectional laminate are give in Table 1.
z< (g/cm ^{2} ) | c_{11} (GPa) | c_{12} (GPa) | c_{22} (GPa) | c_{23} (GPa) | c_{55} (GPa) |
1.578 | 166.07 | 7.09 | 16.19 | 7.52 | 7.79 |
Table 1: Inverted stiffness constants for graphite/epoxy AS4/3501-6 [0]_{24} laminate. |
Fig13: Reflected signal from [0]_{8}laminate at 20^{o}angle of incidence for 0^{o} , 30^{o}and 90^{o} orientations |
It should be noted that the effects of the transducer and other factors related to the experiment are eliminated in this study by convolving the calculated impulse response with a reference reflected pulse from a tungsten film placed over the specimen. The tungsten film produces nearly total reflection of the incident waves and the convolution with the reflected waves from it eliminates the effects of the transducer response and the propagation effects in the fluid medium. The effects of attenuation is included in the theoretical model through the introduction of three dissipation constants [Mal, et al, 1992]. The damping constants were varied until a good visual fit of the measured and calculated results could be obtained.
Using pulse-echo data for a unidirectional 3.378 mm thick AS4/3501-6 graphite/epoxy composite the value of c_{22} is found to be 16.3 GPa. The result for incidence at q = 15^{o}and f = 90^{o} allowed the determination of c_{23}as 7.95 GPa. Testing at q= 15^{o} and f = 30^{o} allowed to determine c_{55} as 7.74 GPa. The identification of the reflections 22 and 23 requires the use of a relatively low incidence angle; this angle is difficult to obtain experimentally with a single LLW setup. Therefore, this pulse technique was used and two critical angles were identified: f_{c} = 58.1^{o} for ? = 15^{o} , and f_{c} = 68.7^{o} for ? = 20^{o} , and thus c_{11} and c_{12} were calculated as 166.05 and 7.06 GPa, respectively. Detailed description of the procedure and the error analysis can be found in [Mal, et al, 1993] and will be omitted.
To produce an image of inherent defects, C-scans were prepared using the LLW setup and pulses as well as tone-bursts. Pulses provided much more effective defects imaging capability than using a single LLW mode in tone-burst. C-scan images were made using pulses where the variables that were examined included the time-of-flight and amplitude. A 24-layer unidirectional graphite/epoxy T300/CG914 laminate was tested at 20^{o} incidence angle along the critical polar angle. The sample was made with series of Teflon foils 25.4, 12.7 and 6.35 mm in diameter and were placed at 1/4, 1/2 and 3/4 of the laminate thickness. At the same depth, an area of 25.4 mm diameter was covered with 40 mm diameter microballoons. To simulate ply-gap, a 6.35 ´ 12.7 mm cut was made in a single layer at the above three depths along the rows of the simulated-delaminations and -porosity. Examination of the time-domain response from the areas with the simulated defects have shown a response that characterize the particular defect type
Fig 14:Pulsed oblique insonification C-scan showing amplitude variations for a graphite/epoxy [0]_{24} laminate |
Fig 15:Pulsed oblique insonification C-scan showing time-of-flight variations for a graphite/epoxy [0]_{24} laminate |
Figure 14 presents a C-scan with the variations of the highest signal obtained beyond the specular reflection (first reflection). As shown, all three types of embedded defects - delamination, porosity and ply-gap were clearly identified. Further, the changes of the resin/fiber ratio significantly affected the amplitude. On the other hand, Figure 15 shows a C-scan image presenting the variations of the time-of-flight values for the [0]_{24} sample. Different colors were assigned to the various time-of-flight ranges in order to present the depth of the defects in the sample. The horizontal strips along the C-scan image are parallel to the fiber orientation. Polar backscattering test of these areas indicated a distribution of localized scatterers. Most probable source of such indications is microballons that migrated along the fibers during the laminate cure stage. The depth distribution of the simulated porosity can be identified from the color of the dots representing their reflected signal.
a)Data acquisition - The current LLW data acquisition setup is complex and not very user friendly. Significant improvement of the data acquisition process has recently been made byautomating the process of aligning the height of the setup as discussed above. Also, the use of the polar backscattering technique [Bar-Cohen and Crane, 1982] polar angle has made the polar angle alignment more precise. However, development of user-friendly software is needed to allow interactive control of the system, minimizing the need for manual setup and alignment.
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