Lamb waves for the structural monitoring in civil engineering

Alex Martiner, winter semester 2016/17

Non Destructive Testing (NDT) with Lamb waves can provide a practical and cost effective way for the verification of material properties in old and new constructions, like the determination of thickness and stiffness of each layer in a pavement construction ^[1], the detection of delamination between steel bars and concrete ^[2] or pipe inspection.

Theoretical background

Lamb Waves are elastic body waves propagating along thin solid plates. Those are a combination of longitudinal- and transversal-waves in plates, where the thickness has to be the same order as the wavelength $//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$. In 1917 Horace Lamb was the first who described those waves in infinite elastic plates. Therefore those are also called Lamb waves ^[3]. In reality infinite elastic plates doesn’t exist, the dimensions of the media approach the order of the wavelength and there the waves are called guided Lamb waves. Unlike Rayleigh waves, which propagate in a shallow zone below the surface of a material, Lamb waves have through-thickness displacement permitting the detection of defects within and close to the surface of the plate. This property, together with the ability to propagate over long distances, make the waves suitable for inspection and monitoring of plate structures ^[4]. There exist two types of independently propagating Lamb Waves modes:

• $//<![CDATA[ \begin{array}{l}S_i\end{array} //]]>$-mode causing a symmetric-vertical deflection respect to the midlife (Figure 1, left).

• $//<![CDATA[ \begin{array}{l}A_i\end{array} //]]>$-mode causing an antisymmetric-vertical deflection respect to the midlife (Figure 1, right).

The different modes appear in different, as well independently propagating, orders. Every mode has a phase velocity $//<![CDATA[ \begin{array}{l}c_p\end{array} //]]>$ at a given frequency $//<![CDATA[ \begin{array}{l}f\end{array} //]]>$ and plate thickness $//<![CDATA[ \begin{array}{l}d\end{array} //]]>$. This phase velocity describe the velocity assumed by any given phase of the wave (for example the crest). The different phase velocities of every mode can be shown as function of plate thickness and frequency in a dispersion diagram (Figure 2) and be calculated with the following equations (1) and (2) where $//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ is the wavelength and $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ the wavenumber ^[5]:

$//<![CDATA[ \begin{array}{l}c_p=λ∙f \quad \quad \quad \quad (1)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}c_p= \frac{2πf}{k} \quad \quad \quad \quad (2)\end{array} //]]>$

In infinite isotropic plates the Rayleigh-Lamb-Equation describes the Lamb waves:

$//<![CDATA[ \begin{array}{l}\frac{tan⁡(qh)}{tan⁡(ph)} + {\left(\frac{4k^2 pq}{(q^2-k^2 )^2} \right)}^{±1}=0 \quad \quad \quad \quad (3)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}p^2=\frac {ω^2}{c_l^2} -k^2 \quad \quad \quad \quad (4)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}q^2=\frac {ω^2}{c_t^2} -k^2 \quad \quad \quad \quad (5)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}h\end{array} //]]>$… half plate thickness

$//<![CDATA[ \begin{array}{l}ω=2πf\end{array} //]]>$… angular frequency

Here $//<![CDATA[ \begin{array}{l}c_l\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}c_t\end{array} //]]>$ are the respective longitudinal and transversal phase velocities:

$//<![CDATA[ \begin{array}{l}c_l^2= \frac{E(1-ν)}{ρ(1+ν)(1-2ν)} \quad \quad \quad \quad (6)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}c_t^2=\frac{E}{ρ2(1+ν)} \quad \quad \quad \quad (7)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}E\end{array} //]]>$… elastic modulus

$//<![CDATA[ \begin{array}{l}ν\end{array} //]]>$… poisson’s ratio

$//<![CDATA[ \begin{array}{l}ρ\end{array} //]]>$… density


The phase velocity can be calculated in equation (2) with the numerical solution of the equation (3) for $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$. The exponent +1 is used for the symmetric mode $//<![CDATA[ \begin{array}{l}S_i\end{array} //]]>$ and the exponent -1 is used for the antisymmetric mode $//<![CDATA[ \begin{array}{l}A_i\end{array} //]]>$. The ascending order of the solutions describe their order $//<![CDATA[ \begin{array}{l}i\end{array} //]]>$ ^[6].

Lamb waves consist of wave packs constructed out of waves with different frequencies. The wave velocity is dispersive, hence the group velocity $//<![CDATA[ \begin{array}{l}c_g\end{array} //]]>$ is different from the phase velocity and can be described trough the differential quotient of $//<![CDATA[ \begin{array}{l}ω\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ ^[6]:

$//<![CDATA[ \begin{array}{l}c_g=\frac{dω}{dk} \quad \quad \quad \quad (8)\end{array} //]]>$

As consequence the shape of a wave packet changes during its propagation ^[4]. The most important modes are the $//<![CDATA[ \begin{array}{l}S_0\end{array} //]]>$- and $//<![CDATA[ \begin{array}{l}A_0\end{array} //]]>$-modes. Only these two modes exist for every frequency and don’t have a critical frequency. Additionally, both zero-modes have the biggest amplitude and are also therefore the most suitable for NDT ^[5].

Application of Lamb waves in civil engineering

Investigating Lamb wave propagation for NDT in civil engineering is not common. Structures to be analysed often do not fulfill the boundary conditions for the excitation of lamb waves like the plate thickness or the free boundaries approximation. Nevertheless, Lamb waves can be used for very specific purposes like providing some extra information about a structure to be analysed. Regarding the free boundaries condition it has been showed that the deviation of a coupled plate to the free plate approximation is less than 10% if two conditions are fulfilled:

1. The wavelength has to be smaller than five times the thickness of the stiff top layer.
2. The transversal wave velocity of the top layer has to be higher than the longitudinal wave velocity in the underlying medium ^[7].

NDT Testing of Pavements

It is possible to determine thickness and stiffness of multilayer pavements using guided Lamb Waves. Traditional methods like the impact-echo-method can’t i.e. provide the empirical correction factor (β) used to obtain thickness mode frequency ^[8].

Measurement setup

The data acquisition occurs with the Multichannel Simulation with One Receiver (MSOR) method. As receiver an accelerometer, a piezo ceramic, a laser vibrometer or an acoustic microphone can be used. The receiver is mounted on a fix point on the surface of the testing object. The signals are generated through several hammer impacts at incremental offsets (Figure 3) ^[9]. Then all recorded signals are compiled and transformed using the MASW processing technique ^[10] to the frequency-phase velocity domain to obtain multimodal dispersion curves.

Implementation

A point source on the surface of a multilayer pavement generates a dominating Lamb Wave that drives the complete system. This dominating Lamb wave generates continuously higher modes in the next layers across the boundaries. These interactions create branches in the dispersion curve measurable at the surface. Those branches can be individually resolved using the multichannel processing technique in the lower frequency range. Each branch correspond to a layer of different stiffness in the pavement system and has the potential to generate information about stiffness and thickness of the layer ^[12]. At higher frequencies there are theoretically substantially more modes merging together form different dispersion curves. The created mode corresponds now to a stiff plate with a lower velocity ^[1]. Stiffness and thickness of the top layer can be determined by matching theoretical dispersion curves to the measured curve ^[11].

Detecting delamination between steel bars and concrete

Lamb waves showed to be useful in the detection of degradation and separation of steel bars in concrete beams while traditional ultrasonic methods are not efficient for this application. The waves can propagate a long distance, reinforcing steel bars while being sensitive to the interface bonding condition between the steel bar and the concrete ^[2].

Measurement setup

There are several transmitter–receiver arrangements that can be used for delamination detection:

• Contact with different transducer angles (Figure 3).
• Direct contact at the bar ends (Figure 4).
• Coupler between concrete and ultrasonic transducers resulting 25° incident and receiving angle(Figure 5).

Literature

1. N. Ryden and M. J. S. Lowe, “Guided wave propagation in three-layer pavement structures,” The Journal of the Acoustical Society of America, vol. 116, no. 5, pp. 2902–2913, 2004..
2. W.-B. Na, T. Kundu, and M. R. Ehsani, “Lamb Waves for Detecting Delamination between Steel Bars and Concrete,” Comp-aided Civil Eng, vol. 18, no. 1, pp. 58–63, 2003.
3. H. Lamb, “On Waves in an Elastic Plate,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 93, no. 648, pp. 114–128, 1917.
4. M. Engholm, Ultrasonic Arrays for Sensing and Beamforming of Lamb Waves. Uppsala: Acta Universitatis Upsaliensis, 2010.
5. F. Berning, Möglichkeiten der Plattenwellenanalyse zur Schadenscharakterisierung in Faserverbundwerkstoffen. Zugl.: Stuttgart, Univ., Diss., 2007. Stuttgart: Inst. für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, 2007.
6. J. L. Rose, Ultrasonic waves in solid media. Cambridge: Cambridge University Press, 1999.
7. R. Jones and E. N. Thrower, “Effect of interfacial contact on the propagation of flexural waves along a composite plate,” Journal of Sound and Vibration, vol. 2, no. 2, pp. 167–174, 1965.
8. A. Gibson and J. S. Popovics, “Lamb Wave Basis for Impact-Echo Method Analysis,” J. Eng. Mech., vol. 131, no. 4, pp. 438–443, 2005.
9. Nils Ryden, Peter Ulriksen, Choon B. Park, Richard D. Miller, Jianghai Xia and Julian Ivanov, “High frequency MASW for non-destructive testing of pavements - accelerometer approach, 2006.”
10. Choon Byong Park, Richard D. Miller, and Jianghai Xia, and Kansas Geological Survey, “Imaging dispersion curves of surface waves on multi-channel record, 1998.”
11. N. Ryden, C. B. Park, P. Ulriksen, and R. D. Miller, “Multimodal Approach to Seismic Pavement Testing,” J. Geotech. Geoenviron. Eng., vol. 130, no. 6, pp. 636–645, 2004.
12. Nils Ryden, Choon Park, Peter Ulriksen and Richard Miller, “Branching of dispersion curves in surface wave testing of pavements, 2002.”
13. Guo, D. and Kundt, T., Ed., Progress in Quantitative Nondestructive Evaluation: Sperical sensors for generaing Lamb waves in Pipes, 18th ed. New York: Plenum Press, 1999.