SPIE's 10th Annual International Symposium on Smart Structures and Materials and 8th Annual International Symposium on NDE for Health Monitoring and Diagnostics, 2-6 March 2003, San Diego, CA. paper # 5057-51
N-SCAN®: New Vibro-Modulation System for Detection and Monitoring of Cracks and Other Contact-Type Defects Dmitri Donskoya, Alexander Ekimova, Emile Luzzatob, Jean-Louis Lottiauxb, Stanislav Stoupina and Andrei Zagraia a Civil Engineering Department, Davidson Laboratory, Stevens Institute of Technology Hoboken, NJ 07030, 201-216-5316,
[email protected] b Electricite de France, 1 Avenue du General-de-Gaulle, Clamart Cedex, 92141, France
ABSTRACT In recent years, innovative vibro-modulation technique has been introduced for detection of contact-type interfaces such as cracks, debondings, and delaminations. The technique utilizes the effect of nonlinear interaction of ultrasound and vibrations at the interface of the defect. Vibration varies on the contact area of the interface modulating passing through ultrasonic wave. The modulation manifests itself as additional side-band spectral components with the combination frequencies in the spectrum of the received signal. The presence of these components allows for detection and differentiation of the contact-type defects from other structural and material inhomogeneities. Vibro-modulation technique has been implemented in N-SCAN damage detection system. The system consists of a digital synthesizer, high and low frequency amplifiers, a magnetostrictive shaker, ultrasonic transducers and a PC-based data acquisition/processing station with N-SCAN® software. The ability of the system to detect contact-type defects was experimentally verified using specimens of simple and complex geometries made of steel, aluminum, composites and other structural materials. N-SCAN® proved to be very effective for nondestructive testing of full-scale structures ranging from 24 foot-long gun barrels to stainless steel pipes used in nuclear power plants. Among advantages of the system are applicability for the wide range of structural materials and for structures with complex geometries, real time data processing, convenient interface for system operation, simplicity of interpretation of results, no need for sensor scanning along structure, onsite inspection of large structures at a fraction of time as compared with conventional techniques. This paper describes the basic principles of nonlinear vibro-modulation NDE technique, some theoretical background for nonlinear interaction and justification of signal processing algorithm. It is also presents examples of practical implementation and application of the technique. Keywords: nonlinear interaction, vibration, ultrasound, defect, diagnostics.
1.
INTRODUCTION
Defects of various types are known to significantly influence nonlinear properties of solids. There is a growing interest in utilization of different nonlinear acoustic effects for nondestructive evaluation (NDE). These effects can be separated into three major groups. In the first group we include nonlinear effects associated with the appearance of new spectral components in the received signal. Historically, the first nonlinear technique proposed for NDE was based on the effect of multiple harmonics generation (Buck et al. 1978, Antonetz et al. 1986). Subsequently, generations of the new frequencies due to nonlinearity was employed for detection of various defects: dislocation substructures in fatigued metals (Cantrell and Yost 1993), cracks and other contact-type defects (Donskoy and Sutin 1998), and weak adhesive joints (Hirsekorn 2001). The second group is associated with the acousto-elastic effect which manifests itself as a nonlinear dependence of the speed of sound in inspected material on the applied stress. Nagy (1998) obtained second order nonlinear correction for sound speed and proposed utilization of this phenomenon for quality control. NDE methods which are based on evaluation of the amplitude dependent frequency shift of a resonance curve are included in the third group. Recent developments in this area were reported by Van Den Abeele et al. (2000). Among the discussed effects, nonlinear manifestation of defects through generation of multiple harmonics is the most promising for nondestructive inspection though direct correspondence of newly appeared harmonics to defect or damage
1
presence is not always obvious. It is quite common that electronic equipment used for signal generation and acquisition produces its own multiple harmonics, which contaminate the spectrum and prevent clear observation of the effects due to damage presence. In addition, any structural material possesses a certain degree of nonlinearity even in an absence of a localized defect. This material nonlinearity might produce a strong background level of multiple harmonics exceeding effects attributed to damage. In contrast to multiple harmonics generation, vibromodulation effect produces spectral components at combination frequencies that could be easily identified in the spectrum. Low frequency, f1, flexural vibrations applied to a specimen vary a contact area between opposite surfaces of a crack or other contacttype defect. At the same time, high frequency, f0, ultrasound wave passes through this interface. As a result, ultrasonic energy transmitted through interface with varying contact area is modulated and additional spectral components at the combination frequencies, f0 ± f1, appear in the spectrum (Fig. 1). It is apparent that the presence and the level of the modulation intensity could be used for damage assessment.
Figure 1 Principles of nonlinear vibro-modulation effect
Utilization of harmonics generation at combination frequencies distinguishes vibro-modulation technique (VMT) from other nonlinear NDE technologies. The physical mechanism behind VMT is also quite different because in addition to ultrasonic signal it uses low frequency flexural vibrations. Vibration exerts much higher stresses on a defect, opening and closing defect’s interface, very effectively modulating probing ultrasonic signal and making VMT highly sensitive and selective to contact-type defects, such as cracks, debondings and delaminations. Damage discrimination capabilities of this method offer a substantial advantage over conventional linear acoustic NDE techniques. The majority of these techniques are based on linear acoustic phenomena including reflection, scattering, absorption and transmission of acoustic wave. Therefore, any structural inhomogeneity, a notch for example, is not distinguishable from a crack. In contrast, vibro-modulation method allows for differentiating these objects because of the specific contact-type interaction. It should be mentioned that fatigue cracks are developed at stress concentration areas most often due to structural vibration. On the other hand, vibro-modulation technique uses the same structural vibration (at much lower level) to exert stress on the crack. Therefore, VMT intrinsically applies more stress to defect-prone area substantially increasing selectivity and sensitivity for fatigue crack detection. Very important feature of VMT is that it does not require spatial scanning of ultrasonic transmitters and receivers along a tested structure. Generally, vibro-modulation technique can use only two ultrasonic transducers to test the entire structure. This is possible because the modulation effect does not depend on ultrasonic frequency, so VMT can utilize relatively low ultrasonic frequencies that can propagate through the whole or substantial part of a structure without significant attenuation. Typical excitation frequencies utilized in the experiments described herein lie within 100-200 kHz frequency range. Low attenuation and utilization of continuous signals instead of short pulses, used in conventional ultrasonic testing, allows for insonification of the entire structures as illustrated in the Fig. 2 presenting an acoustic field for a particular excitation frequency and nonlinear response measured in the 155-185 kHz frequency range.
2
Linear response
Nonlinear response
Fundamental frequency Difference frequency Sum frequency
Acoustic field
Figure 2 Acoustic field distribution at fundamental frequency and measured linear and nonlinear frequency responses.
2.
NONLINEAR VIBRO-ACOUSTIC INTERACTION
The response of structural components at high frequencies features densely resonance nature. At such frequencies, complex distribution of resonances and antiresonances within the spectrum is difficult to predict accurately. Figure 2 exemplifies typical linear (magnitude of fundamental or probing frequency signal vs. frequency) and nonlinear (magnitude of combination frequency components vs. frequency) responses of the tested structure in the specified frequency range. It is noticeable that the responses are significantly varying leading to respective variation of the modulation index (ratio of magnitudes at combination and fundamental frequencies). Practical application of VMT, however, requires the use of a stable parameter related to the modulation index but independent on the spectral variability. In order to introduce such a parameter, the understanding of the spectral variability as a function of transmitter/receiver location as well as position of damage is quite important. This section of the paper investigates these effects on VMT through theoretical analysis of structure’s response functions using simplified one-dimensional model. Vibro-modulation effect reveals itself through additional spectral components at combination frequencies which appear due to contact type nonlinearity at crack interface. The first model to account for such phenomenon was proposed by Richardson in 1979. The model described separation of the crack surfaces in the tensile phase of the passing longitudinal wave, while in the compression phase the surfaces moved together. It is separation of the surfaces during tensile phase that causes modulation of the longitudinal wave. More realistic models that incorporated a contact between two rough elastic surfaces were proposed later (Rudenko and Chin 1994, Nazarov and Sutin 1997). According to the authors, applied stress varies the contact area due to deformation of contacting irregularities and thus, crack can be effectively
3
represented as a nonlinear spring with stiffness depending on the contact area. Variation of losses due to cracks under applied low z frequency flexural vibrations and pure elastic nonlinearity was considered by Zaitsev and Sas 1999. Donskoy et al. (2001) utilized vibro-modulation effect for nondestructive inspection and obtained direct correlation between modulation intensity and x severity of damage. An analytical solution for a bar with a crack was presented in terms of the nonlinear components allowing for xt x0 xr prediction of damage size and modulation intensity as a function of the amplitudes of applied low and high frequency signals. The Figure 3. Schematic representation of onedimensional structure with a crack. authors developed averaging and normalization procedures to account for unique dynamic response of inspected specimens. However, contribution of this response was not included in the theoretical analysis and effect of transmitter/receiver as well as defect positions was not considered. This work further advances this development by introducing structural frequency response into theoretical analysis and considering arbitrary location of damage and transmitter/receiver couple. In this regard, we attempt to generalize theoretical analysis and include example discussed by Donskoy et al. 2001 as a special case. The analysis presented in this paper is based upon the assumption that nonlinear effect leading to the generation of combination frequencies is essentially caused by the variation of contact area at crack interface. Let us consider a one-dimensional structure (i.e. a beam), presented in Fig. 3, with a crack located at x= x0. The cross section area of the crack (s) and crack thickness (t) are assumed small in comparison with the cross-section of the bar (S). A transmitter at location x = xt excites high frequency longitudinal vibrations picked up by a receiver at x = xr. Steady state excitation F at frequency ω yields a longitudinal displacement u. u = F ⋅ K(xt ,xr ,k)
(1)
where K(xt,xr,F,k)= u / F denotes receptance (Inman, 1994). In this work we prefer to utilize a general term “response” in order to differentiate “linear response” at fundamental frequency ω and “nonlinear responses” revealed at combination frequencies ω±. The energy losses are accounted in the Eq. (1) by introducing complex wave number. k = k 0 (1 + iδ ) ,
(2)
where k0=ω / c is a wave number of the bulk medium and c denotes the speed of sound for longitudinal waves. Using Eq. (1) one can find an axial strain along the structure: ξ(xt ,xr ,F,k) = F
∂K(xt ,x,k) . ∂x x = xr
(3)
Low frequency flexural vibrations of structure are realized at the frequency Ω. We suppose that flexural displacement ζ(x) is governed by Euler-Bernoulli beam equation and axial strain due to flexural vibrations can be written as: η(x,z) = z
d 2 ζ(x) dx 2
,
(4)
where z denotes the distance to neutral axis. Crack behavior under the axial strains (3), (4) can be described by the nonlinear stress-strain relationship (Donskoy and Sutin, 1998): σ = E( 1 + βε)ε ,
(5)
where σ is the total stress, E is the Young’s Modulus, β is a nonlinear parameter that characterizes the crack, and ε(x0)= ξ(xt,x0,F,k) + η(x0,z0) is the total strain. Here z0 denotes distance between neutral axis and a crack. Nonlinear interaction at the crack interface leads to generation of the combination frequencies at ω± = ω ± Ω. These components are excited by a force which amplitude can be obtained by substituting Eqs. (3) and (4) into the stress-strain relationship (5):
4
f(xt ,x0 ,F,k) = sz0 βEF
∂K(xt ,x,k) d 2ξ(x) 2 ∂x x = x 0 dx x= x
(6) 0
The displacements at the combination frequencies are small in comparison with displacements governed by Eq.(1) and the nonlinear components can be calculated using first approximation of the perturbation theory. Following this theory, we obtain a solution by substituting the source of the combination frequencies (6) into the expression for the linear response (1). Thus, we arrive into the following relationship for structural displacements at combination frequencies ω±. u± = f ( xt , x0 , F , k ) ⋅ K ( x0 , xr , k ± )
(7)
where k± are given by the expression (2) at ω± = ω ± Ω. It should be mentioned that displacements defined by (7) correspond to the nonlinear components of the acoustic signal received at the location x=xr. Equations (1) and (7) describe the frequency responses for the fundamental and the combination frequencies respectively. Equation (7) suggests that the nonlinear responses differ from each other and from the linear one defined by (1). To illustrate the behavior of both linear and nonlinear responses we consider a problem previously presented by Donskoy et al. 2001 and supplement the solution with the results obtained in this section. The response of a beam having length L and cross-sectional area S can be represented as follows: cos (kx) 2 kES cos (kx ) tan (k(L − x )) + sin (kx ) , x < xt t t t K(xt ,x,k) = cos (k(L − x)) 2 , x > xt . kES cos (k(L − xt )) tan (kxt ) + sin (k(L − xt ))
(8)
The flexural displacement distribution along the simply supported beam excited at the first natural frequency of flexural vibrations (first mode) is given by expression: ξ(x) = B sin k1x ,
(9)
where B is the amplitude of vibration and k1=π/L is the wave number for the first mode. Using expressions (1) and (6) to (9), we performed numerical calculation for several sets of transmitter/receiver and crack locations. The following structural parameters were used to obtain responses depicted in Fig. 4: L= 0.15m, S=0.000322 m2, δ = 0.002, s = 0.1S, β = 10000. Linear response xt=0, xr=L Linear response xt=0.1L, xr=0.7L Nonlinear response xt=0, xr=L, x0=0.5L Nonlinear response xt=0.1L, xr=0.7L,
0
0
-20
-20
-40 -60 -80 -100 130
(a)
Amplitude, dB
Amplitude, dB
Linear response xt=0, xr=L Nonlinear response xt=0, xr=L, x0=0.5L Nonlinear response xt=0, xr=L, x0=0.2L
140
150
160
170
Frequency, kHz
180
-40 -60 -80 -100 130
190
(b)
140
150
160
170
180
190
Frequency, kHz
Figure 4. Linear and nonlinear responses of a simply supported cracked beam vibrating at the first flexural mode; (a) positions of the transducers are the same, while crack location is changed; (b) Crack location is the same, while positions of the transducers are changed.
5
Linear response
Linear response 0 Amplitude, dB
Amplitude, dB
0 -20 -40
Nonlinear response
-60 -80 -100 155
160
165
170
175
180
-20
-60 -80 -100 155
185
Nonlinear response
-40
160
Frequency, kHz
(a)
165
170
175
180
185
Frequency, kHz
(b)
Figure 5 Linear and nonlinear responses obtained with N-SCAN® vibro-modulation system for (a) pristine plate, (b) plate with a fatigue crack in the middle.
Fig. 4 illustrates a change of the responses with variation of crack and transducers positions. Since the sum frequency component, u+, behaves similarly to the difference frequency one, u- , only u- is shown. As crack is moved to a different location, the linear response does not change while the nonlinear one is noticeably changed as shown in Fig. 4a. This fact reflects the assumption that the crack is small and does not influence linear characteristics of the structure. On the other hand, variation of transmitter and receiver positions influences not only the linear response but the nonlinear response as well, as depicted on Fig. 4b. These numerical examples demonstrate that the linear and nonlinear responses of the structure are not identical. According to the proposed model, positions of the transducers on the structure as well as damage location may significantly alter linear and nonlinear responses. This implies that measuring the modulation index at the fixed frequency may lead to substantial error because it will depend on positions of transducers and crack location. Real structures have even more complicated responses (Fig. 5) because of 3-D geometry and a presence of transversal ultrasonic waves. For such complicated responses, stable modulation parameter can be obtained using statistical analysis. In its simplest form, such analysis incorporates averaging procedure across the chosen frequency range and finally yields averaged modulation index (MI), as proposed by Donskoy et al. 2001: 1 q A m−n + Am+ n MI = 20 Log10 q m=1 2 × Am
∑
(10)
where Am is the spectral amplitude at the fundamental frequency f m , Am ± n represents spectral amplitudes at the combination frequencies, fm ± fn , fn is the frequency of n-vibration mode, and q is the total number of the frequency steps in the chosen ultrasonic frequency range. Using proper number and value of the frequency steps, this modulation index provides stable and reliable measure of damage and does not depend on geometry of a tested specimen, positions of transducers and location of the damage.
3.
N-SCAN - PRACTICAL IMPLEMENTATION OF VIBRO-MODULATION TECHNIQUE
Unique features of VMT (high sensitivity and selectivity to the contact type defects, global inspection without transducer scanning, real time and simple interpretation of the results) motivated the design of a portable system capable of conducting laboratory and field measurements in a fraction of the time to demonstrate practical utility of the developed technique. These efforts resulted in N-SCAN 1000 detection system, developed by Intelligent Sensing Technologies, LLC (www.isensing.com). The components of the system include a digital synthesizer, high and low frequency amplifiers, a PC-based data acquisition and processing station with N-SCAN® software, a magnetostrictive shaker and ultrasonic transmitting/receiving transducers. The system has two modes of operations: Spectrum Analyzing (SA) mode and Vibro-Modulation testing (VM) mode. In the SA mode structural vibration response is measured and vibration modulating frequency is determined. VM mode allows for MI measurements upon selection of vibration and ultrasonic frequencies. Typical test takes five - ten minutes for installation of the transducers and approximately a minute for SA and VM measurements regardless of the size and
6
complexity of the structure: be it 24 foot long gun barrel or 500 lbs train coupler depicted in Fig.6. Data acquired by the N-SCAN® system consists of linear response at a fundamental frequency, nonlinear responses at sum and difference frequencies, and calculated Modulation Index. These characteristics are monitored in real time during the measurements and can be saved on a hard drive for documentation. A typical example of the results presented by NSCAN® application display is shown in Fig. 2. N-SCAN® is an evolving technology employing advanced engineering concepts and state-of-the-art electronic components, hardware instrumentation, and software algorithms. Driven by “open system” design concept, we understand that flexibility is a key issue for modern NDE. The N-SCAN® system allows for integration of the whole system or its selected components into varieties of tests and experimental setups. It is capable of performing linear and nonlinear testing of critical structural components in the automotive and aerospace industry, power plants, pipelines, etc. The system proved to be an effective tool for Figure 6 Typical inspection of large structural testing a wide range of structures made of different structural component using N-SCAN® vibromodulation system. materials such as steels, aluminum, plastics, composites, and many others. In the following chapter we present an example of NSCAN® application for detection of fatigue cracks in pipes typical to nuclear power plants.
4.
FATIGUE CRACKS DETECTION WITH N-SCAN®
Applicability of the N-SCAN® system for crack detection in pipes was independently evaluated by Stevens Institute of Technology and Electricite de France (EDF). Continuous heath monitoring and early damage detection in nuclear power plant piping network is critical for proper network maintenance and compliance with high levels of safety standards. The N-SCAN® advantages made it a system of choice due to high sensitivity to fatigue cracks, ability to inspect complex and large piping structures in a very short time or monitor them continuously, and very simple interpretation of results. Here we describe tests conducted at EDF R&D facilities in Clamart Cedex, France. Six specimens were used for investigation of performance of N-SCAN®: one reference specimen without cracks and five cracked specimens provided by PG&E-TES. These specimens have been fatigue cracked in a weld between a small bore pipe (1) and the pipe socket weld (2) as shown in Fig. 7a. Location of fatigue cracks in such welds is quite typical due to high stress concentration in this area. The size and location of the cracks were measured with conventional UT techniques and the results in terms of percentage of cracked areas in a weld are shown in the first column of Table 1. As can be seen from the table, the experimental specimens could be divided into three major groups: reference, partially cracked, and cracked through the pipe wall. One can associate these three groups with the severity of structural damage. Crack detection tests were performed using a specialized testing facility resembling a pipeline network in a nuclear power plant. Each specimen was installed into the pipeline using bolted flange connections. Two sets of measurements were carried out for each specimen: at the first mounting and after it was dismounted and reinstalled back for the next set of measurements. Experimental set up is presented in Fig. 8a. Two ultrasonic transducers/sensors were mounted with epoxy glue: first on top of a small bore pipe (transmitter 1), and the second on the main pipe near the pipe socket weld (receiver 2). Each transducer can be used either as a transmitter or as a receiver. The applied ultrasonic signals were scanned in 160 – 190 kHz frequency range. Low frequency excitation at 3 kHz was applied using the magnetostrictive shaker as shown in Fig. 8a. The results of the tests, presented in Table 1, clearly demonstrate high damage detection sensitivity. The tests are also indicates that average modulation index increases monotonically with the percentage of cracked areas (Fig. 8b). For the purpose of illustration, we calculated Damage Index (DI) in Fig.8b as a difference between averaged modulation indexes of damaged specimens and a reference specimen. The resulted dependence follows almost linear pattern proving a direct correlation between value of DI and damage severity.
7
Small Bore Pipe
1
Cracks Weld
Crack
2
Pipe Socket Weld
3
Main Pipe
(a)
(b)
Figure 7 Experimental specimens for crack detection in pipes: (a) schematics of a pipeline junction and crack location; (b) picture of two specimens with designated crack locations 40
y = 2.3185x + 2.8901 2 R = 0.9215
Transmitter (1) Damage index, dB
Receiver (2)
Accelerometer
30
20
10
Shaker 0 0
2
4
6
8
10
12
14
Estimated percentage of cracked area, %
(b)
(a)
Figure 8 Experimental set up and results of measurements for structural component of a pipeline network: (a) section of a pipeline under test; (b) modulation index vs. estimated percentage of cracked area
Specimen ID #2 #1 #4 #5 #3 reference
Table 1 Modulation Indexes for tested specimens Damage Estimated area of Test #1 Test #2 type damage, % Modulation Index, dB Modulation Index, dB through wall crack 12,7 -48 -46 through wall crack 12,3 -51 -48 partial crack 6,6 -53 -57 partial crack 5,9 -57 -62 partial crack 5,7 -63 -65 No cracks 0 -80 -77
8
The results of the testing have shown that the N-SCAN® crack detection system allows for reliable and reproducible crack detection and estimation of damage severity. For instance, dismounting and re-mounting of experimental specimens did not alter results of measurements ensuing accuracy within app. 3 dB. The averaged modulation index for damaged specimens increased substantially with damage severity. The difference between reference specimen, specimen with partial cracks (6%) and specimen with through-wall cracks (12%) was around 15 dB and 30 dB respectively. These tests have shown that the N-SCAN® system allows for fast and reliable nondestructive evaluations of fatigue cracks. Comparison of reference and damaged specimens revealed a substantial difference in averaged modulation indexes which was app. 20 dB and 30 dB for partial and through-wall cracks. The damage index evolved monotonically with the percentage of the cracked area for the tested pipes. For the piping configurations where the reference baseline cannot be established, 10 dB increase in the modulation index is a trustworthy indication of a defect presence. Consistent results were obtained for various test set ups involving installation and reinstallation of experimental specimens. The use of N-SCAN® vibro-modulation system for crack detection in piping network was very convenient due to the simplicity of installation and straightforward interpretation of results.
5.
CONCLUSION
Vibro-modulation technique is based upon interaction of low frequency vibrations with ultrasonic waves on the contacttype interfaces such as crack, debonding, delaminations. This interaction manifests itself as additional (side-band) components in the spectrum of ultrasonic signal. The presence of damage and its severity is indicated by the ratio of the magnitudes of the side-band to fundamental (transmitted) spectral component. This concept was used in design of the first practical system: N-SCAN. This is a portable detection system for real time online inspection and/or monitoring of structures made of various materials with complicated geometries. The system allows for very fast global inspection of rather large structures without spatial scanning of the transducers along structural surfaces. The system is especially sensitive to fatigue defects because the applied vibration stress inherently concentrates in the area of fatigue damage. Among others advantages of the system is very simple output (modulation index) which does not require any interpretation. Simple one dimensional model of a beam with a crack, considered in the paper, demonstrates the importance of the frequency averaging in assessing of damaged severity and provides an insight to proper measuring procedure. Application of N-SCAN® system for fatigue crack detection in welded pipes demonstrated that the system is capable not only to detect damage but also evaluate its severity. The tests independently conducted by Electricite de France revealed direct correlation between obtained modulation indexes and crack sizes. Numerous laboratory and field tests reliably confirm that N-SCAN® system provides unique capabilities for diagnostics of structures made out of wide range of materials: metals, plastics, composites, concrete, etc. with complicated geometries. Due to its flexibility, the developed system can be conveniently integrated into other applications, testing and monitoring procedures. Fast algorithms implemented in N-SCAN® allow for acquiring, processing, and displaying information in real time. Overall, the N-SCAN® system can be characterized as a very versatile, simple to use and reliable tool for global inspection and damage assessment.
REFERENCES Antonets, V.A., Donskoy, D.M., and Sutin, A.M. (1986) “Nonlinear Vibro-Diagnostics of Flaws in Multilayered Structures”, Mechanics of Composite Materials, Vol. 15, pp. 934-937. Buck, O., Morris, W.L., and Richardson, J.M. (1978) “Acoustic Harmonic Generation at Unbonded Interfaces and Fatigue Cracks”, Applied Physics Letters Vol. 33, N. 5, pp. 371-372. Cantrell, J.H., and Yost, W.T. (1993) “Acoustic Harmonic Generation and Dislocation Dynamics of Fatigued Aluminum Alloys”, Review of Progress in Quantitative Nondestructive Evaluation, Plenum Press, NY., Vol. 12, pp. 2059-2066. Donskoy, D., Sutin A., and Ekimov A. (2001) “Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing”, NDT&E International, Vol. 34, pp. 231-238, 2001. Donskoy, D.M., and Sutin, A.M. (1998) “Vibro-Acoustic Modulation Nondestructive Evaluation Technique”, Journal of Intelligent Material Systems and Structures, Vol. 9, September 1998, pp. 765-771. Hirsekorn, S. (2001) “Nonlinear transfer of ultrasound by adhesive joints – a theoretical description”, Ultrasonics, Vol. 39, pp. 57-68,
9
2001. Inman, D. J. (1994) “Engineering Vibrations”, Prentice Hall, 1994 Korotkov, A.S., and Sutin A.M. (1994) “Modulation of ultrasound by vibrations in metal constructions with cracks”, Applied Physics Letters Vol. 18, N. 4, pp. 59-62. Nagy, P.B. (1998) “Fatugue damage assessment by nonlinear ultrasonic materials characterization”, Ultrasonics, Vol. 36, pp. 375-381, 1998. Nazarov, V.E., Sutin, A.M. (1997), “Nonlinear elastic constants of solids with cracks”, Journal of Acoustical Society of America, Vol. 102, N. 6, pp. 3349-3354, December 1997. Richardson, M. (1979) “Harmonic Generation at an Unbonded Interface. I. Planar Interface Between Semi-infinite Elastic Media”, International Journal of Engineering Science, Vol. 17, pp. 73-75. Rudenko, O., and Chin An.Vu. (1994) “Nonlinear Acoustic Properties of a Rough Surface Contact and Acousto-Diagnostics of a Roughness Height distribution”, Acoustical Physics, Vol. 40, N. 4, pp. 593-596. Van Den Abeele, K.E. -A., Carmeliet, J., Ten Cate, J.A. and Johnson, P. A. (2000) “Nonlinear Elastic Wave Spectroscopy (NEWS) techniques to discern material damage, Part II: Single Mode Nonlinear Resonance Acoustic Spectroscopy”, Research in Nondestructive Evaluation, Vol. 12, N. 1, pp. 31-42, 2000. Zaitsev, V., and Sas, P. (2000) “Nonlinear Response of a Weakly Damaged Metal Sample: A Dissipative Modulation Mechanism of Vibro-Acoustic Interaction”, Journal of Vibration and Control, Vol. 6, pp. 803-822.
10
