Composite testing with Lamb waves

Viktoria Götz, summer semester 2016

Composite testing with Lamb waves is a non-destructive method to detect defects. Composites offer several advantages like their light weight, resistance of corrosion and uniqueness in their mechanical and thermal properties. ^[1] However, their characteristics make the use of common ultrasonic testing difficult. As a result the benefits of Lamb waves are used to detect damages inside the sample without destroying it. ^{[2] [3]}

Lamb waves

Based on Rayleigh waves Horace Lamb described in 1917 a new form of waves which propagate "in a solid bounded by parallel planes". ^[4] The solid boundaries are parallel arranged „free isotropic plates […] “ ^[4] and create so called Lamb waves. In contrast to Raleigh waves which are surface waves, Lamb waves are plate waves with a longitudinal and transversal character. In other words, a combination of longitudinal (P waves) and transversal waves (S waves) forms Lamb waves. ^[5]

The propagation depends „on the excitation frequencies and the thickness of the plate“^[3] and describes accordingly the dispersive equation. Using numeric methods, the solution shows that Lamb waves exists in two different modes: symmetrical and anti-asymmetrical. ^[2][3]

The equations

$//<![CDATA[ \begin{array}{l}\dfrac{tan⁡(\beta d/2)}{tan⁡(\alpha d/2)} = -\dfrac{4 \alpha \beta k^2}{(k^2-\beta^2 )^2}\end{array} //]]>$ (1)

$//<![CDATA[ \begin{array}{l}\dfrac{tan⁡(\beta d/2)}{tan⁡(\alpha d/2)} = -\dfrac{(k^2 - \beta^2)^2}{4 \alpha \beta k^2}\end{array} //]]>$ (2)

with

$//<![CDATA[ \begin{array}{l}\alpha^2 = \frac{\gamma^2}{(c_l^2) - k^2}, \beta^2 = \frac{\gamma^2}{(c_t^2) - k^2}\end{array} //]]>$,

$//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ is the wave number,

$//<![CDATA[ \begin{array}{l}c_l\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}c_t\end{array} //]]>$ are the velocities of longitudinal and transversal waves,

$//<![CDATA[ \begin{array}{l}c_P\end{array} //]]>$ is the phase velocity of Lamb waves and

$//<![CDATA[ \begin{array}{l}\gamma = 2 \pi f\end{array} //]]>$ is the angular frequency, displays mathematically the symmetrical mode (1) and asymmetrical (2). ^[6]

In case of a symmetrical mode, the displacement of the upper plate is in reverse direction than the lower plate. The asymmetrical mode displays the displacement in the same direction (see Figure 1).^[3]

As already mentioned Lamb waves have dispersive character. That means the phase velocity depends on the frequency. At every frequency there are always symmetrical (S) and asymmetrical (A) Lamb modes. At lower frequencies or very thing plates only two modes are excited in the plate: $//<![CDATA[ \begin{array}{l}S_0\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}A_0\end{array} //]]>$. As a result the number of modes increases with the frequency and plate thickness (see Figure 2) .^[6][7]

Rayleigh waves are easily influenced by surface conditions. They start to deform und and transform into Lamb waves. Like Rayleigh waves Lamb waves are only measured in their vertical displacement. However it resumes every condition over the complete surface of the plate. Therefore defects in a specimen disturb the propagation of Lamb waves. Change of velocities, reflections and mode conversions are interactions of Lamb waves with imperfections. ^[8] To detect and analyse damages inside the specimen, Lamb waves are induced into the element by actuator.

Inspection

Defects lead to material collapse. Typical damages in fibre reinforced composites include matrix cracks, fibre fracture and fibre – matrix debonding. In order to detect these defects during inspection or monitoring, Lamb waves can be an advancement to various conventional testing methods. ^[9]

Ultrasonic testing

Ultrasonic testing is an effective method to inspect specimen non-destructively. Based on the difference of the acoustic impedance, two types of methods can be applied:

• Ultrasonic pulse – echo method (see Ultrasonic Pulse-Echo Method)
• Ultrasonic through – transmission method (see Ultraschall-Durchschallung)

As Lamb waves propagate extensively through the composite specimen there are a few differences to the conventional ultrasonic testing on for example concrete specimen. At every frequency a symmetrical and asymmetrical mode can be seen. The higher the frequency the more modes develop. Therefore lower frequencies below 100 kHz are preferred, so symmetrical and asymmetrical modes can be better distinguished. The explanation lies in their different velocities. Symmetrical Lamb waves propagate with a higher speed than asymmetrical waves. Contrary to standard ultrasonic testing which uses broadband signaimls at frequencies between 0,5 and 20 MHz, testing with Lamb waves require narrow-band signals to induce only a few modes. Common devices for ultrasonic testing mostly work at higher frequencies, e.g. around 500 kHz, so devices with different filters are required. Transducer and receiver have to filter upper waves plus the receiver eliminates with the help of lowpass filter noise signals and highpass filter other low frequent disturbances. The device HFUS 2400 is specialised on working with Lamb waves. ^[10]

As coupling between specimen and probe air or water are being used (see Ultrasonic Pulse-Echo Method).

Impact- Echo testing method (see Impact-Echo)

Impact-echo (IE) testing method works similar to ultrasonic testing, with ultrasonic and audible sound. After a mechanic impact on the surface elastic waves propagate through the specimen by reflexion on interfacial. In a previous approach only p-waves are reflected on the upper and lower surface of the material and travel by that through the specimen.

A sensor receives the signal and a computer analysis it. Thereby the thickness of the material has an important role whereas it correlates with the frequency. That means by evaluating the frequency of the multiple reflexion the thickness of the specimen can be determined if the velocity of the p-waves are known or vice versa if the thickness is known, p-waves can be determined. ^{[5] [11]}

The equation $//<![CDATA[ \begin{array}{l}f_R = \dfrac{(\beta v_p)}{(2d)}\end{array} //]]>$ displays this correlation with $//<![CDATA[ \begin{array}{l}f_R\end{array} //]]>$ as resonant frequency, $//<![CDATA[ \begin{array}{l}v_p\end{array} //]]>$ as velocity of the p-wave, $//<![CDATA[ \begin{array}{l}d\end{array} //]]>$ as thickness of the specimen and $//<![CDATA[ \begin{array}{l}\beta\end{array} //]]>$ as correction factor. ^[5]

An explanation for the equation, especially for the correction factor lies in the Lamb waves theory. Taking a closer look the signal shows a symmetrical Lamb wave mode character. The so called “S1” is the most excited signal and its group velocity is zero. That implies “no energy of this mode [propagates] out of the area between source and sensor”. ^[5]

After the testing procedure (see Impact-Echo) the frequency spectrum is analysed. However so called geometric effects have to be considered as they affect the signal. The influence can be recognized since the resonance frequency cannot be clearly seen. Geometric effects develop because of Rayleigh waves which are reflected on upper and lower plate of the specimen. These surface waves start to interfere with p- and s-waves and especially with the S1 Lamb wave. ^[5] This interference prevents the precise identification of the propagation of the S1 Lamb wave. ^[11] By summarizing the signals with the impact echo array technique S1 Lamb waves will be intensified. Hence geometric effects are minimized by destructive interference of every wave which is different to S1 Lamb waves. Those S1 Lamb waves interfere in turn constructively. ^[11]

The IE testing method can be used to detect delamination of composites specimen. ^[5]

Monitoring

Acoustic emission analysis

The acoustic emission (AE) analysis is a non-destructive testing method and more specifically a method for structural health monitoring. ^[5] That implies that this method is used during loading of a structure. While a composite is under working load defects like cracks or fractures can develop. These damages emit acoustic waves, in this particular case, Lamb waves. AE analysis is a passive technique to measure the signal. That means the defect generates a signal in the form of lamb waves which propagate through the material to the surface where sensors are placed. ^[12]

High sensitive sensors, based on the piezoelectric effect, receive the vibration and transform it into an analogue electrical signal. ^{[5] [12]}

Literature

1. Lindemann, U.: Grundlagen der Entwicklung und Produktion. Produktentwicklung. Skript, p. 330 - 33. Lehrstuhl für Produktentwicklung der TU München. München, 2015.
2. Berning, F.: Möglichkeiten der Plattenwellenanalyse zur Schadencharakterisierung in Faserverbundwerkstoffen. Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen der Universität Stuttgart, p. 2 - 4. Stuttgart, 2007.
3. Ahmad, Z.A.B: Numerical simulations of Lamb waves in plates using a semi-analytical finite element method. VDI Verlag GmbH Düsseldorf, p. 1, 4 - 5. 2011.
4. Lamb, H.: On waves in an elastic plate. Royal Society of London Proceedings Series A., Nr. 93, p. 114 - 128. 1917.
5. Große, C.: Grundlagen der zerstörungsfreien Prüfung. Skript, p. 3 - 5, 38-39, 57, 83 - 87, 153. Lehrstuhl für Zerstörungsfreie Prüfung der TU München. 2005.
6. Liu, Z.: Lamb Wave Analysis of Acousto-Ultrasonic Signals in Plate. Institute of Acoustics, Tongji University Shanghai P.R.China.
7. Prager, J.; Köppe, E.; Bartholmai, M.: Früherkennung von Strukturschäden mittels geführter Lamb-Wellen. Bundesanstalt für Materialforschung und -prüfung (BAM), Fachbereich 8.1 Sensorik, Mess- und prüftechnische Verfahren. Berlin, 2012.
8. Mook, G.; Pohl, J.; Willberg, C.; Simonin, J.: Ankopplung, Ausbreitung und Wechselwirkung von Lambwellen zur strukturintegrierten Bauteilüberwachung von Faserverbunden. Otto-von-Guericke-Universität Magdeburg, Fachbereich EMW, Hochschule Anhalt, p. 5 - 8. 2011.
9. Sause, M.G.R; Horn, S.: Einfluss der Signallaufzeit auf die Unterscheidbarkeit von Schallemissionsquellen in Faserverbundwerkstoffen. p. 1 - 2. Institut für Physik der Universität Augsburg. Augsburg, 2011.
10. Hillger, W.: Lamb-Wellen zur Schadensanzeige in faserverstärkten Kunststoffen. p. 1 - 3. DLR Institut für Faserverbundleichtbau und Adaptronik. Braunschweig, 2005.
11. Große, C.; Groschup, R.: Neue Ansätze zur Anwendung der Impakt-Echo Methode. p. 2 - 4, DGZfP-Jahrestagung 2014. TU München.
12. Große, C.; Ohtsu, M.: Acoustic Emission Testing. p. 3 - 5, 20. Springer Verlag. 2008.