Ihr Ansprechpartner: Thomas Berer
Optical excitation and detection of zero group velocity Lamb waves
Our proposal deals with non-contact excitation and detection of a Lamb wave in a particular point of the dispersion relation. By using spatially and temporally modulated laser-ultrasound we want to utilize the zero group velocity (ZGV) point of the first order symmetrical mode Lamb wave to characterize isotropic plates non-destructively.
Figure 1. (a) Lamb wave dispersion curves for a 1mm thick aluminum plate. The zero group velocity (ZGV) point of the S1 mode is marked with a black dot. ZGV point of the S4 mode is also visible but the zero slope of this mode is less distinct. (b) Complex dispersion relation with a complex mode originating from the saddle point with ZGV of the real mode.
Lamb waves and ZGV points
It is well–known that for certain wave modes within the Lamb wave dispersion relation segments with negative group velocity exist (Fig.1). The turning points of these curves between negative and positive slope and positive group velocity are of significant interest as here the group velocity becomes zero and at this point of the dispersion relation a strong, well-detectable resonance occurs. In contrast to thickness resonances in plates which occur with k=0, where k denotes the wave number, this mode is associated with a finite wave number. The mode shape of the S1 mode at the ZGV point is shown in Fig.2. The Poynting vector reveals that the power flow within the cross section cancels out for this particular frequency leading to a resonance-like behavior. This can be explained by the fact that the propagation velocity of the energy of a wave packet is equal to the group velocity; hence, the introduced energy cannot propagate away from the location of the generation, making them well detectable.
Figure 2. Mode shape of the S1 mode at the ZGV point.
In a cooperation with the University of Colorado at Boulder we experimentally investigated the coupling of a laser pulse into ZGV modes of tungsten plates. In the experiment, the diameter of a laser spot exciting a ZGV mode was varied. The wavelength of a ZGV mode relates to the optimum spot diameter and can therefore be investigated in this way.
Figure 3. Experimental setup to measure dependence of coupling laser excitation into a ZGV mode on the diameter of the laser spot. The spot size was varied by translating the lens which focused the laser on the sample surface. The acoustic waves were detected on the opposite side of the sample with a Michelson interferometer.
Frequency spectra of the time domain signals from simulations (Fig.4; left); the ZGV points at 2.28MHzmm, 4.00MHzmm, 9.33MHzmm are apparent. The frequency domain response of the tungsten plate is shown as a function of the D/h ratio. The inset shows the amplitude of the Fourier transforms at the S1, A3, and A7 ZGV frequencies from the numerical simulations.
Results of the experimental investigations of the excitability of ZGV modes with constant pulse energy (Fig. 4; right): (b) Spectra for the different excitation spot diameters. (c) Variation of the spectra of the measured signals with increasing spot diameter. The vertical plane and the line within the plane mark the position of the S1 ZGV mode. (d) Variation of the amplitude of the S1 ZGV mode normalized to constant energy density for each pulse diameter. The maximum at D=h approx. 1.4 is apparent.
Location of the ZGV points
For the practical use of the ZGV points, one must determine their location in the k—w plane. ZGV points represent mathematically
well-defined points within the dispersion relation as the wave number is nonzero and the group velocity vanishes. The latter is given through the implicit derivative for the symmetric and antisymetric modes. The system of equations are given as:
The solution of the system of non-linear equations can be obtained numerically. Here, a Newton–Raphson iteration is used and the corresponding iteration process is given as:
The Poisson’s ratio of an isotropic plate can be determined with high accuracy by using two ZGV frequencies . In addition, if the plate thickness is known then both elastic properties can be found. Although the basic idea is simple, the calculation of the locations of the ZGV points is more challenging. In our approach four non-linear equations are solved directly, yielding the unknown wave velocities cL and cT (or Poisson’s ratio and one wave velocity, for example).
The inverse problem is demonstrated with a tungsten plate with 0.25 mm thickness and with an aluminum plate with a thickness of 0.125 mm.
Figure 5. Experimental signal and its Fourier transform for tungsten (a) and (b) and for aluminum (c) and (d) plates. The first two resonances indicated in the spectra are associated with symmetric and antisymmetric modes, respectively.
Spatially and temporally modulated Laser-Ultrasound (STeMoLUS)
We experimentally and theoretically pursue a unique combination of adjustable spatial modulation13 with freely tunable temporal modulation.8,9 Our technique—spatial and temporal modulated laser-ultrasound (STeMoLUS)—combines the previous methods, allowing an arbitrary frequency resolution in the measurement without the necessity of using scanning optomechanics. Moreover, the dispersion relation is obtained directly from the maxima of the amplitudefrequency scan, without the transformation techniques required for the evaluation of a spatio-temporal scans.
Figure 6. Working principle of STeMoLUS: (a) Spatially modulated laser beams are realized using a SLM leading to excitation of interfering SAWs. The interference is constructive if the SAW’s wavelength k is equal to the periodicity D. For reasons of clarity, the reverse propagating wave is omitted in the figure. (b) Excitation pattern and detection spot on the sample surface (c) Simulated interference amplitudes for excitation frequencies in the range of three interference maxima and different numbers of excitation lines.
Figure 7. (a)–(d) Comparison of experimentally (top) and theoretically (bottom) obtained interference plots: (a) and (b) non-dispersive SAWs in aluminum, (c) and (d) dispersive SAWs in molybdenum coated glass. The first interference maxima correspond to the dispersion relations. As the interference condition D=mk is fulfilled for arbitrary integers m, several interference maxima arise. (e) Experimentally obtained SAW velocities compared to a theoretical dispersion curve for the system in (c) and (d). (f) Experimental verification of constructive interference on the aluminum sample: the amplitude of the observed nterference for D=54,5mm remains constant using 2, 3, or 4 excitation lines whilst keeping the total energy on the sample constant. The magnitudes are normalized by the single line frequency response to remove the damping effects.
The working principle is shown in Fig. 6(a): by utilizing a SLM, an intensity modulated laser beam with frequency fis imaged onto the sample surface as a parallel line pattern. The surface displacement produced by the superposition of SAWs emitted from the line array is detected in a point outside of the array along a line perpendicular to the array as shown in Fig. 6(b).
We verified the Stemolus with a non-dispersive sample (uncoated aluminum) and with a dispersive system (glass substrate coated with a 3 mm thick molybdenum layer). In the measurement, the spatial periods in the excitation pattern were varied in steps of 6.05 mm. For each pattern, a frequency scan between 20 MHz and 150MHz with a frequency step of 500 kHz was carried out. Subsequently, each frequency scan was filtered with a five point moving average filter. Figure 2(a) shows the recovered amplitudes for the measurement on aluminum. Since the interference condition is fulfilled for integer multiples of the SAW wavelengths, several maxima are obtained in the investigated frequency range, besides the distinct first maximum for k = D.
Numerical modeling of ultrasonic propagation and the generation of ultrasound by laser irradiation
Experimental application of laser-generated ultrasound in solids has been extensively used in experimental acoustics in the past decades since it enables contactless excitation of different elastic waves, such as bulk, surface or guided waves. In search of better understanding and optimization of the generation mechanism, the modeling by a thermoelastic source has attracted great attention from the very beginning of the experimental application. Thus, the generation process is described by the coupled heat and wave equations, whereby various coupling terms are taken into account. Within laser ultrasonics the coupling of the heat and wave equations is due to thermal expansion leading to a semi-coupled problem.
In the generalized thermoelasticity, however, also the thermal feedback of the propagating stress pulses is included, hence, the problem becomes fully coupled and both thermal and elastic waves arise:
The spatial discretisation of the two sets of equations in is done with finite differences using staggered grids. Hence, the temperature, displacement and stress components are discretized on different grids which are shifted by a half cell in one or in both directions. The grids of the temperatures and normal stresses are identical and the grids of the displacements and shear stresses are shifted by a half grid dimension (Dx=2;Dy=2).
The spatially discretised equations could be written in the following matrix form:
Where M, C, K are the coefficient matrices. In the current case the wave equation is discretized by the conditionally stable, explicit Euler-method; therefore, the displacements in the (n+1)th time step are explicitly solved as a function of the displacements and stresses in previous time steps. Unconditionally stable, implicit integration techniques enable the use of the same temporal and spatial discretization and the heat equation is therefore integrated by the Wilson H method. During the irradiation of a metallic surface with a short laser pulse the electromagnetic waves penetrate a thin skin close to the surface (in the range of 10 nm) which is modeled as a surface flux.
Figure 8. Out-of plane surface displacements with Rayleigh and leaky-Rayleigh waves for aluminum (a) and for zinc (b). (c) Sections of the generated wave fields in the models for aluminum (left) and for zinc (right). Besides the generated bulk longitudinal (L), quasi longitudinal (QL) and shear waves (S) also Rayleigh (R) and leaky-Rayleigh (LR) waves are visible.
Numerical results represented by the vertical surface displacements are shown above. The generated Rayleigh waves are unipolar with no attenuation, as expected. The small, attenuating wave in front of the Rayleigh wave is assigned to the longitudinal bulk wave, or ’’surface skimming longitudinal wave’’. The magnitude of the displacements are shown in Fig. 7c at t = 4.48 ns (aluminum) and t = 5.8 ns (zinc) after excitation from a 1mm wide line source with 20 ps temporal duration. Besides the generated bulk longitudinal (L) and shear waves (S) also surface waves (R) are visible in aluminum. The longitudinal bulk wave shows an almost cylindrical wave front in the isotropic case with strongly varying pulse widths (and frequency content), as a consequence of the finite width of the excitation. The wave forms in zinc are different due to the anisotropy. In the transversely isotropic case in the investigated plane there exist three different wave types: quasi longitudinal (QL), quasi transverse (QT) and pure transverse (PT) modes.
Characterization of optical fibers for the detection of ultrasound
In this study a theoretical framework for calculating the acoustic response of optical fiberbased ultrasound sensors is presented. The acoustic response is evaluated for optical fibers with several layers of coating assuming a harmonic point source with arbitrary position and frequency. First, the fiber is acoustically modeled by a layered cylinder on which spherical waves are impinged. The scattering of the acoustic waves is calculated analytically and used to find the normal components of the strains on the fiber axis. Then, a strain-optic model is used to calculate the phase shift experienced by the guided mode in the fiber owing to the induced strains.
Berer and Veres et al, J. Biophotonics (5), 518-528, 2012
Figure 9. Acoustic response of a bare optical fiber to an incident plane wave up the 100MHz. The peaks correspond to cross-sectional resonances of the fiber. The framework is showcased for a silica fiber with two layers of coating for frequencies in the megahertz regime, commonly used in medical imaging applications. The theoretical results are compared to experimental data obtained with a sensing element based on a piphase-shifted fiber Bragg grating and with photoacoustically generated ultrasonic signals.
Figure 10. Acoustic response of a layered optical fiber to an incident spherical wave.
Integrated response of a layered optical fiber by the combined phase shifts along 15 and 30 Deg compared to experimental results. Response of the fiber by shifting the source parallel to the fiber axis as the combined phase shifts along 15 and 30 Deg compared to experimental results.
Veres et al, JASA 135(4), 1853-1862, 2014.
Figure 11. Problem statement: Spherical waves generated from an opto-acoustical source (R=50 mm) scatter from the optical fiber which is located at a distance of d from the source. The radii of the glass fiber and the coatings are 62.5, 110, and 130 mm, respectively. The detection is achieved by an FBG sensor in a distance of d.
Waves in complex media
SHM - Structural Health Monitoring
Gerade beim Einsatz von Composite-Materialien (bzw. allgemein im Leichtbau) ist eine integrierte Qualitätsüberwachung (SHM - Structural Health Monitoring) ein sehr wertvolles Hilfsmittel um die Leichtbaupotentiale voll ausschöpfen zu können. Eine Struktur, deren Integrität kontinuierlich überwacht ist kann z.B. durch die Integration von Lichtwellenleitern mit FBG-Sensoren realisiert werden. Bei derartigen Lösungen unterstützen Sie gerne unsere Experten aus dem Bereich der Physical & Computational Acoustics mit entsprechender Expertise.
Bleche & Rohre
Auch zur Materialcharakterisierung von z.B. Blechen kann die Laserultraschall-Technologie gut eingesetzt werden. Ein großer Vorteil ist dabei, dass das Material nur einseitig zugänglich sein muss und derartige Messungen zum Teil auch im Fertigungsprozess erfolgen können. Somit ist mittels LUS auch eine Wanddickenmessung von Rohren online im Prozess möglich.
Unterschiedliche Materialien (Stahl, Aluminium, Messing, Kupfer) - unterschiedlichste Fragestellungen (Blechdicke, elastische Eigenschaften, Anisotropien): Die LUS-Technologie gepaart mit innovativer Datenauswertung ist höchst flexibel einsetzbar.
|Berer Thomas, Dipl.-Ing. Dr.
||interimistischer Head of Physical & Computational Acoustics|
Altenberger Straße 69, 4040 Linz
|Scherleitner Edgar, Dipl.-Ing. Dr.
||Area Manager Acoustics|
Altenberger Straße 69, 4040 Linz