On the crossing points of the Lamb modes and the maxima and minima of displacements observed at the surface.

This article elaborates on the crossing points of the frequency-wavenumber branches for the symmetric and anti-symmetric Lamb modes in a homogeneous plate. It is shown both theoretically as well as experimentally that at these crossing points either the normal or the longitudinal components of modal displacement attain an extreme value, i.e. a maximum or it vanishes. This behavior is assessed herein using a method due to Mindlin, who showed that the dispersion curves for a plate with mixed boundary conditions - which are associated with uncoupled shear and dilatational modes - provide bounds to the spectral lines of the free plate. Therefore, a subset of the crossing points of the symmetric and antisymmetric Lamb modes for a free plate coincide with the crossing points for a plate with mixed boundary conditions.

Lamb waves in isotropic plates possess some fascinating properties. In previous works it has been shown [1], [2] that certain components of the surface displacements of the Lamb waves vanish for particular values of the phase velocity. The normal component of the displacements of the symmetric branches vanishes at the surface – leading to a predominantly longitudinal motion – when the phase velocity becomes equal to the longitudinal bulk wave velocity [1]. The opposite happens for every mode (symmetric and antisymmetric branches), hence the longitudinal component vanishes at the surface, if , where is the shear wave velocity [2]. The latter points correspond to Lame’s equivoluminal modes [3]. Such points are of great practical interest as the interaction of the plates with the surroundings – such as excitability or detectability of the guided waves [7] – is strongly coupled to the normal component of the wave. Transmission of waves into a fluid loaded plate or the interferometric detection of the wave, for instances, require surface normal displacement component. A vanishing normal component, on the other hand, can reduce the leakage into the fluid [1]. The better understanding of this behavior and the possibility to predict frequencies with such particular properties can be great help in the choice of Lamb modes for long range inspection [4]. Here, we will show that the previously identified locations – frequencies for particular wave modes – with vanishing normal or longitudinal components [1], [2] can be related to the crossings of the uncoupled dilatational and shear modes or so-called bounds, based on Mindlin’s approach [3]. This also leads to further points with similar behavior at the crossings of the symmetric and antisymmetric branches giving a general rule for the locations of such points. At these crossings for one mode the longitudinal component vanishes with a maximal normal component and for the other mode the opposite occurs. Although, several cases with practical relevance, such as fluid loaded plate, are only approximated by the dispersion relation for plate with free boundaries. The investigated theory, however, reveals the fundamental behavior of the surface displacements of plates and allow the prediction of frequencies with particular properties for a particular wave mode.

For our investigations we will utilize the approach developed by Mindlin [3]. In this work, besides an approximate theory for plates, also a simple method was presented to evaluate the exact frequency spectra for plates. This method is based on a simple set of uncoupled dilatational and shear modes [3], given as:where k denotes the wave number, the thickness of the plate, and are integer numbers. The spectrum of an isotropic plate with mixed boundary conditions (i.e. plane strain boundary conditions) is described by these modes and Mindlin has shown that for the spectrum of a plate with free boundaries these modes become the bounds. Hence, the crossings of the Lamb modes in a plate with free boundaries coincides with some of the crossings of the bounds [3], [6]. The coordinates of these crossings are straightforwardly evaluated (see later) and by calculating the slopes of the dispersion curves in these points Mindlin could simply draw the frequency spectra of the plate without calculating the curves directly.

The most important result what we will utilize from this simple technique is that the crossings of the Lamb modes can be predicted by the bounds as their crossings are identical if are both even or odd, for Poisson’s ratios larger than 0.263 [6]. For Poison’s ratios smaller than 0.263 the false Rayleigh roots change from complex to real [3], [5] and further crossings also arise. These crossings of the symmetric and antisymmetric modes between the crossings of the bounds, however, do not lead to additional maximum or vanishing of the surface displacements and are therefore irrelevant for this investigation. The dispersion curves and the bounds are shown in Fig. 1(a) for an aluminum plate.1 The dispersion relation splits into two relationships for symmetric and antisymmetric modes. The displacements at the surface for symmetric modes are given as [5]:and for antisymmetric modes are given as:where and D are constants, x denotes the longitudinal and y the normal component of the displacement. The propagation constants are defined as and is the angular frequency. By using the relationship between the constants [5] the displacements reduce to:with and . It can be seen that for both symmetric and antisymmetric modes the longitudinal and the normal components of the displacements are in the opposite phase. Moreover, the symmetric and antisymmetric modes themselves are also in the opposite phase, hence, a maximum in the longitudinal component for one type of modes coincides with the vanishing of the normal component for the other types. The coordinates of the crossings of the bounds – and therefore also for the Lamb waves – are well-known and are given as [3]:where and are either both even or odd integers and are wave number and angular frequency coordinates of the crossings, respectively. The corresponding propagation constant reduces to . Since the obtained propagation constant is independent of m, for the nth shear bound every crossings with the dilatational bounds will show the identical behavior at the surface. By calculating the displacements of the Lamb waves at the intersections for even bounds we receive, and , hence, for symmetric modes and for antisymmetric modes . For odd bounds and , hence, for symmetric modes and for antisymmetric modes . This shows that at the crossings of the symmetric and antisymmetric modes one component of the displacements vanishes and the other component is maximal for one of the crossing modes, having the opposite for the displacements of the other mode.

Fig. 1

(a) Symmetric and antisymmetric Lamb wave dispersion curves with bounds. (b) Displacement components of the symmetric and antisymmetric Lamb wave dispersion curves at the crossings of the modes and the bounds.

The vanishing surface displacement components observed in previous works [1], [2] can be reinterpreted as two special cases of the regular crossings with . One such case is the intersections of the uncoupled dilatational mode with the even bounds which are crossed by the symmetrical modes as well (Fig. 1), except the zero order mode [3], [6]. This is a special case since the dilatational mode is bound only for the symmetrical modes. The coordinates of the intersections for the even bounds and therefore for the symmetrical modes are given as [3]:

The corresponding propagation constant at the coordinates given in Eq. (10) reduces to and the displacements in Eq. (6) become , hence, the motion becomes longitudinally dominated as expected [1]. Mindlin has also pointed out that the intersections of the odd bounds and the dilatational mode approximate well the intersections of the antisymmetrical modes with the dilatational mode but they are not identical [3].

A second special case arises in the points where the shear bounds become tangential to both symmetrical and antisymmetrical modes for [3]. These points are given at the intersection of the bounds with the line; where also Lame’s equivoluminal modes arise [2], [3]. The coordinates of these intersections are given as . The single index shows that these points are not real crossings of the bounds or the Lamb modes but only crossings between the shear bounds and the Lamb modes. The corresponding propagation constant reduces to . The symmetric modes converge to the odd bounds and the antisymmetrical ones to the even bounds . Since and the resulting displacements in Eq. (6) become and and which shows that the longitudinal component of the displacements vanishes for both symmetric and antisymmetric modes and the normal component of the displacement is maximal.

This is illustrated on Fig. 1(b) for the and modes by numerical calculations. At the crossing with the line (10.971 and 8.777 MHz) the longitudinal components vanish at the surface with maximal normal components. The mode crosses the dilatational mode at 7.201 MHz where the longitudinal component is maximal with a vanishing normal component at the surface. The crossing of the and modes at 5.092 MHz leads to a vanishing normal component for and to a vanishing longitudinal component of the modes at the surface.

Practical implications of this behavior are demonstrated experimentally in Fig. 2. The dispersion relation of an aluminum plate was evaluated experimentally by a laser-ultrasound technique.2 The dispersion relation in Fig. 2(a) reveals the Lamb spectrum in the plate and the magnitude of the Fourier transform is proportional to the normal velocities at the surface.

Fig. 2

(a) Experimentally evaluated dispersion relation of an aluminum plate with 1.80 mm thickness. (b) Enlarged view of the dispersion relation (marked area in part (a)) incorporating the bounds of the Lamb waves as well.

An enlarged view with the uncoupled dilatational and shear modes are shown in Fig. 2(b). The dilatational mode with and the shear mode with are shown with dashed lines. The maximal normal displacements (or velocities) are clearly visible for the Lame’s modes for every wave mode (symmetrical and antisymmetrical) due to the strong Fourier amplitudes. The area around the uncoupled dilatational mode with shows the opposite behavior; the wave modes in this region of the dispersion relation are hardly visible. For the symmetric modes this is the range where the modes become longitudinally dominated [1]. The antisymmetrical modes are also very weak; an explanation can be given by considering their crossings. The uncoupled dilatational mode is very close to the curve after the first few crossings ( mm−1). The antisymmetric modes converge to even shear bounds, hence at the crossing with the dilatational bound they cross also a symmetrical mode and an odd shear bound. This leads to a vanishing normal displacement of the antisymmetric mode whereby the normal displacement of the symmetric mode remains small. For low values of k the even symmetric modes dominate; they have a predominantly normal displacement at the surface. For higher k values also anticrossings become visible. During these avoided crossings the even symmetric modes interchange with the subsequent symmetric modes leading to well-detectable large normal displacements within these branches.

In conclusion we have investigated the behavior of the surface displacements of Lamb waves. We found that the crossings of the symmetric and antisymmetric Lamb modes in the dispersion relation coincides with the locations of the points with vanishing surface displacements. We used Mindlin’s technique to predict these points and to calculate the surface displacement components and showed that the vanishing of one component of the displacements coincides with the maximum of the other component. These maxima and minima have also strong experimental influence as was shown by evaluating the dispersion relation of an aluminum plate. Further application areas such as the characterization of sensors are provided by the fact that uncoupled dilatational and shear bounds partly also describe the wave spectra in cylindrical rods. Optical fiber-based sensors, for instance, are increasingly used to detect ultrasonic waves. Since their sensitivity strongly depends on the transmission and scattering of the incident waves, the corresponding transfer function is governed by the surface displacement components.