акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en

When the amplitude of tangential traction is greater than the contact, static friction for the micro-slip motion changes for sliding. An oscillating shear wave drive is accompanied by a cyclic transition between static and kinematic friction (stick-and-slide mode) so that the contact stress–strain relation follows a hysteresis loop [25]. Independent of the direction of motion, the contact tangential stiffness changes symmetrically between the static (for a stick phase) and dynamic (for a slide phase) values twice over the input strain period, which provides the generation of odd higher harmonics similar to the above, the CAN features sinc-spectrum modulation, and nonpower dynamics.

6.2.3Nonlinear Resonance Modes

In addition to the generation of higher harmonics, the experiment of Solodov et al. [29, 30] also revealed different scenarios of CAN dynamics, which considerably expands the nonlinear spectrum of cracked defects. These scenarios exhibit forms of dynamic instability – that is, an abrupt change of output for a slight variation in the input parameters. This is a chaos phenomenon. To illustrate the feasibility of the new nonlinear vibration modes, and to ascertain their basic spectral pattern, we assume that the crack exhibits both resonance and nonlinear properties and is thus identified as a nonlinear oscillator according to Solodov et al. [29]. Its characteristic frequency (ω0) is determined by a linear stiffness and an associated mass of material inside the damaged area. The contact nonlinearity is introduced as a displacement-dependent (X) nonlinear interaction force FNL (X ). The driven vibration (driving force f (t ) = f0 cos νt) of the nonlinear oscillator are found as a solution to the nonlinear equation:

Using a second-order perturbation approach, FNL cos(ν − ω0 ), which accounts for the interaction between the driving and natural frequency vibration. If ν − ω0 ≈ ω0, a resonance increase in the output at ω0 ≈ ν/2 is observed (subharmonic generation). The higher-order terms in the interaction correspond to the frequency relation mν − nω0 that provides a resonance output at ω0 ≈ mν/(n + 1). For n = 1, the crack generates an USB of the second order mν/2; the higher-order USB corresponds to the higher values of n. In reality, a damaged area has a more complicated structure that can be conceived as a set of coupled nonlinear oscillators. If the frequency of the driving acoustic wave is ν = ωα + ωβ , the difference frequency components, ν − ωα = ωβ and ν − ωβ ≈ ωα , provide a cross-excitation of the coupled oscillation. This results in a resonant generation of the frequency pair, ωα , ωβ , centred around the subharmonic position. The higher-order nonlinear terms in (6.35) expand the CAN spectrum, which comprises a multiple ultra-frequency pairs (UFP) centred around the higher harmonics and the USB according to Solodov et al. [29].

The USB and UFP belong to a class of instability mode and can be interpreted, respectively, as the half-frequency and combination frequency decay of a high-frequency (HF) phonon (driving frequency signal). The resonance instability manifests in the avalanche-like, amplitude growth beyond the input threshold. The reverse amplitude excursion results in bistability ([30]: the input amplitudes for the up and down transitions are different (amplitude hysteresis). Such

the position of the second (40 kHz) and third (60 kHz) harmonics as well as USBs (50 and 70 kHz).

6.2.5CAN Application for Nonlinear Acoustical Imaging and NDE

The nonlinear spectra shown in Figures 6.18 and 6.19 are produced locally in the damaged area while the intact part of material outside the defects vibrates linearly, that is, with no frequency variation in the output spectrum. Thus, nonlinear defects are active sources of new frequency components rather than passive scatterers in conventional ultrasonic testing. This makes nonlinearity a defect-selective indicator of the presence of damage or its development. The high localization of nonlinear spectral components around the origin is a basis for the nonlinear imaging of damage. NLV uses a sensitive scanning laser interferometer for detecting the nonlinear vibrating of defects. The excitation system includes piezo-stack transducers operating at 20 and 40 kHz. After a 2D scan and FFT of the signal received, the C-scan images of the sample area are obtained for any spectral line within the frequency bandwidth of 1 MHz.

Figure 6.20 shows the image of an oval examination on top of a piezo-actuator embedded into a GFRP composite. Such smart structures are likely to be used for active structural health monitoring for aerospace components.

The actuator itself was used as an internal excitation source fed with an input of a few volts. The higher harmonic images selectively reveal the boundary ring of the delamination where clapping and rubbing of the contact surfaces are expected. On the contrary, the driving frequency (50 kHz) image indicates only a standing wave pattern over the area of the actuator.

Figure 6.21 (left) shows fatigue cracking produced by cyclic loading in an Ni-based superalloy. Such a crack of 1.5 mm length, with an average distance between the edge of only ≈ 5 μm, is clearly visualized in the USB image (Figure 6.21, right) whereas linear NDE using a slanted ultrasonic reflection failed to detect such small cracks. Similar to all nonlinear modes discussed, the UFP components generally display a strong spatial localization around the defects and are applicable for the detection of damage. The benefit of the UFP-mode is illustrated in Figure 6.22 for a 14-ply epoxy-based GFRP composite with a 9.5 J-impact damage in the central part. The linear image at a driving frequency of 20 kHz reveals only a standing wave pattern over the whole sample (Figure 6.22, left). The image at the first UFP

Figure 6.20 Fundamental frequency (ω) and higher harmonic imaging of a delamination in a ‘smart’ structure (Solodov [29])

Figure 6.21 Right: USB image of 5 μm-wide fatigue crack in Ni-based super-alloy; left: crack photo (Solodov [29])

side-lobe of the 10th harmonic of the driving frequency (198.8 kHz) yields a clear indication of the damaged area (Figure 6.22, right).

The scanning laser vibrometry suffers from variation of optical reflectivity, for example the measurements fail in the damaged areas with particularly strong scattering of laser light. Experiments of Solodov and Bussey [32] demonstrated that planar defects as localized sources of nonlinear vibration effectively radiate a nonlinear airborne ultrasound. Solodov and Bussey propose such a nonlinear air-coupled emission (NACE) as an alternative (and in many cases superior) methodology to locate and visualize the defects in NDE [32].

A practical version of the NACE for nonlinear imaging of defects uses a HF focused aircoupled (AC) ultrasonic transducer as a receiver [33]. In Figure 6.23, the NACE imaging results are compared with the NLV of multiple impact damage on the reverse side of a carbon fibre reinforced (CFR) multiply (+45◦; −45◦) composite plate (175 × 100 × 1 mm). Both techniques reliably visualize the defects with similar sensitivity.

Figure 6.24 (left) shows the (9th–11th) harmonic NACE image of the 50 μm-wide fatigue crack in a steel plate (150 × 75 × 5 mm) with two horizontally located grip holes for cyclic loading at some distance from the crack. The image reveals that the NACE detects not only

Figure 6.22 Nonlinear imaging of impact damage in central part of GFRP plate; left: linear (20 kHz image); right: UFP image (Solodov [29])

On the other hand, the acoustic wave interaction with a cracked defect is accompanied by a strong stepwise variation in local stiffness: the stiffness of the crack can be substantially greater for compression than for tensile stress if the wave amplitude is high enough to cause an intermittent contact between the crack surfaces. In this case, the intact material outside the defect can be considered as a ‘linear carrier’ of the acoustic wave while the defect is a localized source of CAN.

At moderate driving amplitude, the CAN suggests a fully deterministic scenario with higher harmonic generation and/or wave modulation. These effects feature, anomalously, a high efficiency, specific dynamic characteristics, modulated spectra, and an unconventional ‘rectified’ waveform distortion. For higher excitation amplitudes, the mechanical instability phenomena that are well known in other branches of nonlinear physics can develop in the defect area. The lower stiffness of the cracked area makes it behave as a localized oscillator which is, apparently, strongly nonlinear due to CAN. Therefore, for an intense acoustic excitation it can manifest such effects as subharmonic generation, instability, and a transition to chaotic dynamics. As a result, the spectrum of local oscillations acquires a number of new frequencies (USBs and UFPs). All the nonlinear spectral components demonstrate a strong localization in the defect areas.

This feature of localized CAN enables 2D-imaging of the nonlinear excitations confined inside the defect areas. Thus, nonlinear NDT (NNDT) of imperfect materials via CAN is inherently defect-selective, that is, it distinctively responds to fractured flaws. Fortunately, this group of flaws includes the most malignant defects for material strength: microand macro-cracks, delaminations, debondings, impact and fatigue damages.

Numerous case studies prove their applicability for NNDE and defect-selective imaging in various materials by using scanning NLV and NACE. Particularly successful examples include hi-tech and constructional materials: impact damage and delamination in fibre-reinforced plastics, fatigue micro-cracking and cold work in metals, delaminations in fibre-reinforced metal laminates and concrete.

6.3Modulation Method of Nonlinear Acoustical Imaging

6.3.1Introduction

Nonlinear acoustical imaging is the extension of acoustical imaging from linear to the nonlinear regime using techniques from nonlinear acoustics. The nonlinear acoustical approach to acoustical imaging is concerned not only with the application of a finite-amplitude sound wave, but also with the nonlinear material response that is inherently related to frequency changes of the input signal.

Various methods in nonlinear acoustics are being applied to nonlinear acoustical imaging. Based on these techniques, various forms of nonlinear acoustical imaging are classified as follows:

1.Modulation method

2.Nonclassical application of nonlinear acoustics

3.Subharmonics imaging

4.Higher harmonics imaging; second harmonics; third harmonics; the significance of contrast agents, especially in medical imaging

5.Fractal imaging and applications: (a) NDE; (b) medical imaging.

6.3.2Principles of Modulation Acoustic Method

Since the early 1990s, increasing activities in nonlinear acoustical imaging have been applied to NDE and medical imaging. Here we will describe the modulation acoustic method [34] applicable to the NDE for the location of a crack in a testing sample. Cracks exhibit a high acoustic nonlinearity [35, 36]. A crack is acoustically a highly nonlinear object. It produces different nonlinear acoustic responses, such as the generation of higher harmonics or the nonlinear modulation of an ultrasound wave passing through or reflected from a crack by the low-frequency (LF) vibration of tested objects. So far, the modulation method has only been applied to NDT. Simple nonlinear techniques can be used for the detection of damaged objects; however, they cannot give information on crack location. The modulation acoustic method to be described here is capable of crack location in a testing sample. The nonlinear acoustic parameter of solids are much more sensitive to crack-like defects than the linear acoustic parameters such as sound velocity. A crack can produce different nonlinear acoustic responses such as higher harmonic generation or frequency mixing when ultrasonic waves of different frequencies pass through or are reflected from a crack and some others. The modulation method is based on the effect of the modulation of a HF acoustic wave passing through a crack by the LF oscillations of testing samples. A low-frequency wave changes the parameters of the HF probe wave propagating through the crack. The modulation coefficient (index) therefore contains information on the interactions of highand LF waves at the crack. The modulation index depends on the position of the crack relative to the nodes and antinodes of the excited LF oscillation modes in a sample.

6.3.3The Modulation Mode Method of Crack Location

The modulation mode method of crack location [34] is based on the fact that the interactions of HF acoustic waves with the LF oscillations of a testing object take place at the crack. Consequently, the nonlinear force, which generates the modulation frequency components, acts at the position of the crack. By exciting different LF modes in the object, one can obtain a set of modulation amplitude responses from the testing sample. The resonance properties of HF acoustic waves are usually not as strong as for LF oscillation, and by the use of additional local spatial or frequency averaging, the influence of those resonances on modulation indexes can be easily avoided. As a result, by measuring the modulation indexes for different LF modes, one can reconstruct the crack position.

Let us take an illustration for the one-dimensional case: for example, a rod. LF longitudinal modes in such a sample for the free-boundary conditions at both ends can be written as:

where un (z, t ) is the displacement in the sample, l is the sample length, An is the amplitude, and ndenote the resonance frequencies of the modes. Let uω be the displacement in the rod caused by the HF wave propagating in t.

Without a crack in the rod, the LF and HF waves do not interact with each other. If there is a crack in the sample they interact, producing a modulation effect which is proportional, in the approximation of quadratic nonlinearity, to the product of un (z, t ) and uω .