Generation of Lamb Waves for Nondestructive Inspection of Plate Specimens

Lamb waves are named after Horace Lamb who studied the motion of two-dimensional waves in solids bounded by two parallel planes in his classic work Ref. 1. He obtained two sets of analytical solutions for symmetric and antisymmetric modes propagating in infinite plates with a finite thickness. Lamb waves belong to guided waves that propagate in long solid objects (waveguides), such as rods; plates; or pipes, but also in more complex structures, like rails, where it is impossible to find an analytical solution.

Numerical modeling of Lamb waves is essential for the analysis and design of the structural health monitoring (SHM) systems for long range ultrasonic testing. The design of an SHM system based on guided waves consists of two main parts. In the first place, it is required to know the dispersion curves for the modes that can propagate in the waveguide of a certain cross section. In the second place, the propagation of a chosen mode through the waveguide and its reflection from a possible irregularity in the cross-section area (a fracture or a corrosion defect) is analyzed in the time domain.

This tutorial studies the propagation of guided waves in a steel plate with a finite width and thickness. In the first part, the Mode Analysis study for the Solid Mechanics physics interface applied on the 2D cross section of the plate computes the propagating modes. This approach is also sometimes referred to as the semi-analytical finite element (SAFE) method. In the second part, an angle beam wedge excites the desired mode in the plate with a flaw, which is modeled in 3D using the Elastic Wave, Time Explicit physics interface.

Model Definition

The steel plate used here is 15 mm wide and 5 mm thick. An angle beam wedge transducer placed on top of the plate generates the antisymmetric or flexural zero-order mode (A0) in the plate. The wedge is made of plastic and has the angle of incidence of 70°. A normal velocity applied on the excitation area of the wedge generates a longitudinal wave that travels through the wedge. The setup is shown in Figure 1. This tutorial thus omits the details on the transducer modeling which are, for example, given in Angle Beam Nondestructive Testing tutorial model.

The source signal is a five-cycle tone burst with the center frequency f0 = 200 kHz as shown in Figure 2. The longitudinal wave velocity in plastic is cp = 2080 m/s and the estimated velocity (see the next section) of the refracted A0-mode in the plate at the chosen frequency is vLamb = 2300 m/s. The wave refraction is computed according to Snell’s law as

(1)

The refraction angle β = 90°, which yields the critical angle of about α = 65°. Since the angle of incident of the chosen wedge is larger than the critical angle, the wave sent through the wedge will be refracted into the desired mode in the plate.

Figure 1: Model setup including wedge and test plate.

The plate has a flaw, which is a fracture with the dimensions of the plate thickness and half of its width. The orientation of the fracture makes the guided wave hit it at a right angle. Part of the wave will pass further along the plate, and part of it will be reflected back to the wedge. The transmitted and reflected signals are recorded at four observation points placed on the top surface of the plate at equal distance before and after the fracture.

Figure 2: Five-cycle tone burst source signal.

Results and Discussion

First, look at the dispersion diagrams, Figure 3 and Figure 4, obtained from the mode analysis for the plate cross section. The plots show the out-of-plane wave numbers, kn, and the phase velocities, vp, of the propagating modes, respectively. The phase velocities are defined as

and the group velocities as

Note that the phase velocities in Figure 4 are normalized to the shear wave speed in steel. The red lines depict the wave numbers and normalized velocities of the pure longitudinal and shear waves in steel.

Figure 3 and Figure 4 indicate that each propagating mode exists above a certain nascent frequency (cut on/off frequency of the mode). Four of the computed modes have nascent frequencies of zero and thus exist throughout the whole frequency range. The number of such modes in this example is higher than that from the classic result of Lamb (two zero-order antisymmetric, A0, and symmetric, S0, modes), which is due to the finite (and comparable) dimensions of the plate cross section. The more complex the waveguide cross section, the higher number of modes will be able to propagate in it.

Figure 3: Dispersion curves of Lamb waves in the plate: out-of-plane wave number.

The lowermost curve in Figure 4 corresponds to the vertically polarized A0-mode: its vertical displacements are symmetric while the out-of-plane ones are antisymmetric (see the orientation of the coordinate axes in Figure 1). It is the slowest propagating mode in the plate and is depicted in the upper-left corner in Figure 5. Figure 5 shows the profiles of the modes propagating at f0 = 200 kHz that are symmetric with respect to the sagittal plane of the system, as those are the only ones that can be excited by the current design of the wedge.

Figure 4 exhibits the known property of Lamb waves: they are dispersive, that is, their speed depends on the frequency. In this regards, the choice of the excitation frequency is no accident For example, the estimated velocity of the A0-mode at 100 kHz is about 1720 m/s. This value is less than the longitudinal wave speed in the wedge, thus making the desired wave refraction unreachable (see Equation 1) for the current dimension of the plate and the wedge material, and the angle of incidence.

Figure 4: Dispersion curves of Lamb waves in the plate: normalized phase velocity.

The second part of this tutorial is dedicated to the analysis of the selected mode propagation in the 3D plate and its reflection from the flaw. The evolution of the traveling wave is illustrated in Figure 6. Note that Figure 6 shows the velocity magnitude profiles only in the plate, as those in the wedge are of no interest here.

The wave packet travels toward the defect (t = 80 μs) until it hits it (t = 110 μs). Then a portion of the packet passes further down the plate while another portion is reflected back to the wedge (t = 140 μs). The reflected part of the signal travels back toward the wedge (t = 170 μs). Note that the signal is no longer symmetric with respect to the sagittal plane after it has hit the flaw.

The vertical displacements of the signal computed at the listening points placed before and after the fracture are depicted in Figure 7 on the left and on the right, respectively. The blue lines correspond to the side of the plate with the fracture; the green lines, to the opposite side. As the wave is dispersive, the wave packet travels with the group velocity. Measuring the time delay between the incident and reflected waves yields the group velocity of 3153 m/s. The group velocity of the A0-mode estimated from the dispersion diagram is about 3130 m/s, which is close to one computed from the 3D time-domain analysis.

Figure 5: Profiles of the modes propagating at f0 = 200 kHz and symmetric with respect to the sagittal plane.

The A0-mode is clearly visible when it propagates in the plate, especially at the times before it has reached the flaw. Although this mode has the highest magnitude, it is not the only refracted wave that is exited by the transducer. The other mode is faster than the A0-mode and is vaguely seen in Figure 6 at t = 80 μs. Its distinct shape is available from Figure 8 that shows the vertical (y-component) and longitudinal (z-component) velocity components on the top and the bottom of the plate at t = 80 μs. The vertical velocity of the slower A0-mode is in phase on the top and the bottom of the plate, while that of the faster mode is 180° out of phase. The opposite applies to the longitudinal velocity. The faster mode propagating in the plate is the one depicted in the lower-left corner in Figure 5. It has the phase velocity of 5435 m/s, and its nascent frequency lies between 160 and 170 kHz as seen from Figure 4.

Figure 6: Wave profiles at t = 80, 110, 140, and 170 μs.

Figure 7: Vertical displacements at the listening points before (left) and after (right) the defect.

Figure 8: Velocity profiles on the top (green) and the bottom (blue) of the plate at t = 80 μs.

Reference

1. H. Lamb, “On waves in an elastic plate,” Proc. R. Soc. Lond. A, vol. 93, issue 648, 1917.

Acoustics_Module/Ultrasound/lamb_waves_ndt_plate

Modeling Instructions

From the menu, choose .

New

In the window, click

Global Definitions

Parameters: Geometrical Parameters

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file lamb_waves_ndt_plate_geometry_parameters.txt.

In the text field, type Parameters: Geometrical Parameters.

Parameters: Model Parameters

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file lamb_waves_ndt_plate_model_parameters.txt.

In the text field, type Parameters: Model Parameters.

Steel

In the window, under right-click and choose .

In the window for , type Steel in the text field.

Locate the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rho_steel	kg/m³	Basic

Locate the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Pressure-wave speed	cp	cp_steel	m/s	Pressure-wave and shear-wave speeds
Shear-wave speed	cs	cs_steel	m/s	Pressure-wave and shear-wave speeds

Plastic

Right-click and choose .

In the window for , type Plastic in the text field.

Locate the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rho_plast	kg/m³	Basic

Locate the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Pressure-wave speed	cp	cp_plast	m/s	Pressure-wave and shear-wave speeds
Shear-wave speed	cs	cs_plast	m/s	Pressure-wave and shear-wave speeds