Lamb waves are a type of ultrasonic elastic waves that propagate guided between the two parallel surfaces of solid plates. Lamb waves have recently gained popularity from researchers and engineers in the non-destructive evaluation community for damage identification, due to their relatively low attenuation ratio, strong penetration capability, convenience of generation and collection, and high sensitivity to structural damage even of small size.
Lamb waves propagate as two infinite sets of modes: symmetric modes, in which the displacement is symmetrical about the midplane of the plate, and anti-symmetric modes, with displacements anti-symmetric about the midplane. Another important characteristic of Lamb waves is their dispersive behavior: Lamb wave velocity depends on both the excitation frequency (f) and the thickness (d) of the plate combined in a frequency–thickness (f·d) product.
This Python package presents tools to calculate and plot Lamb wave dispersion curves and particle displacement in traction-free, homogeneous and isotropic plates.
The code is tested with Python 3.7. Next section provides the prerequisites to run the program.
The code is dependant on the following external libraries: Numpy, Scipy, Matplotlib. These can be installed with Python's inbuilt package management system, pip. See Python's tutorial on installing packages for information about this issue. In short, the installation can be made as:
pip install numpy
pip install scipy
pip install matplotlib
In order to run validation.py, you will also need Pandas, which can be installed as:
pip install pandas
First, you need to import the Lamb class from the lamb module, and instanciate it. For this example, we are going to use a 10 mm Aluminum plate.
from lamb import Lamb
alum = Lamb(thickness=10,
nmodes_sym=5,
nmodes_antisym=5,
fd_max=10000,
vp_max=15000,
c_L=6420,
c_S=3040)thickness: Thickness of the plate, in mm.
nmodes_sym: Number of symmetric modes to calculate.
nmodes_antisym: Number of antisymmetric modes to calculate.
fd_max: Maximum value of frequency × thickness to calculate, in kHz × mm.
vp_max: Maximum value of phase velocity to calculate, in m/s.
c_L: Longitudinal wave velocity of the material, in m/s.
c_S: Shear wave velocity of the material, in m/s.
The following parameters are optional:
c_R: Rayleigh wave velocity of the material, in m/s. Defaults to None.
fd_points: Number of frequency × thickness points. Defaults to 100.
vp_step: Increment between phase velocity intervals. Defaults to 100.
material: Name of the material being analyzed. Defaults to "".
- Phase Velocity
To generate a plot of phase velocity as a function of frequency × thickness, you can use:
alum.plot_phase_velocity()This method produces the following plot:
You can use the following optional parameters with this method:
modes: Which family of modes to plot. Can be 'symmetric', 'antisymmetric' or 'both'. Defaults to 'both'.
cutoff_frequencies: Add cutoff frequencies to the plot. Defaults to True.
material_velocities: Add material velocities (longitudinal, shear and Rayleigh) to the plot. Defaults to True.
save_img: Save the result image as png in the results folder. Defaults to False.
sym_style: A dictionary with matplotlib kwargs to modify the symmetric curves (to change color, linewidth, linestyle, etc.).
antisym_style: A dictionary with matplotlib kwargs to modify the antisymmetric curves (to change color, linewidth, linestyle, etc.).
- Group Velocity
To generate a plot of group velocity as a function of frequency × thickness, you can use:
alum.plot_group_velocity()This method produces the following plot:
You can use the following optional parameters with this method:
modes: Which family of modes to plot. Can be 'symmetric', 'antisymmetric' or 'both'. Defaults to 'both'.
cutoff_frequencies: Add cutoff frequencies to the plot. Defaults to True.
save_img: Save the result image as png in the results folder. Defaults to False.
sym_style: A dictionary with matplotlib kwargs to modify the symmetric curves (to change color, linewidth, linestyle, etc.).
antisym_style: A dictionary with matplotlib kwargs to modify the antisymmetric curves (to change color, linewidth, linestyle, etc.).
- Wave Number
To generate a plot of wave number as a function of frequency × thickness, you can use:
alum.plot_wave_number()This method produces the following plot:
You can use the following optional parameters with this method:
modes: Which family of modes to plot. Can be 'symmetric', 'antisymmetric' or 'both'. Defaults to 'both'.
save_img: Save the result image as png in the results folder. Defaults to False.
sym_style: A dictionary with matplotlib kwargs to modify the symmetric curves (to change color, linewidth, linestyle, etc.).
antisym_style: A dictionary with matplotlib kwargs to modify the antisymmetric curves (to change color, linewidth, linestyle, etc.).
- Wave Structure
To generate a plot of the wave structure (i.e., the displacement profile across the thickness of the plate), you can use:
alum.plot_wave_structure(mode='A0', nrows=3, ncols=2, fd=[500, 1000, 1500, 2000, 2500, 3000])This method produces the following plot:
This method has to be used with the following parameters:
mode: Mode to be analyzed. Can be "A0", "A1", "A2", ..., "An" or "S0", "S1", "S2", ..., "Sn", with 'n' being the order of the corresponding mode.
nrows: Number of rows in the subplot.
ncols: Number of columns in the subplot.
fd: Array with the frequency × thickness values to analyze. The length of the array must be equal to nrows x ncols.
The following parameters are optional:
save_img: Save the result image as png in the results folder. Defaults to False.
inplane_style: A dictionary with matplotlib kwargs to modify the in-plane curves (to change color, linewidth, linestyle, etc.).
outofplane_style: A dictionary with matplotlib kwargs to modify the out-of-plane curves (to change color, linewidth, linestyle, etc.).
- Particle Displacement Field
To generate an animation of the particle displacement field, you can use:
alum.animate_displacement(mode='A0', fd=1000)This method produces the following animation:
This method has to be used with the following parameters:
mode: Mode to be animated. Can be "A0", "A1", "A2", ..., "An" or "S0", "S1", "S2", ..., "Sn", with 'n' being the order of the corresponding mode.
fd: Frequency × thickness value to animate.
The following parameters are optional:
speed: Delay between frames in milliseconds. It can be used to control the speed of the rotating vectors in the animation (a smaller value produces a faster animation). Defaults to 30.
save_gif: Set to True if you want to save the result animation as a gif in the results folder. Defaults to False.
save_video: Choose a video format if you want to save the result animation as a video in the results folder. Can be 'mp4', 'mov' or 'avi'. Defaults to False.
Note: If you want to save the animation as a gif, you should install ImageMagick and specify the full path to magick.exe like this before using the animate_displacement method:
Lamb.magick_path = 'C:/Program Files/ImageMagick-7.0.10-Q16/magick.exe'If you want to save the animation as .mp4, .avi or .mov, you should specify the full path to the ffmpeg executable in ImageMagick installation folder:
Lamb.ffmpeg_path = 'C:/Program Files/ImageMagick-7.0.10-Q16/ffmpeg.exe'If you are using some flavor of Unix, chances are ImageMagick is already installed on your computer.
- Save Results
To save all results in a txt file in the results folder, you can use:
alum.save_results()- Phase Velocity
You can use the attributes vp_sym and vp_antisym to find the phase velocity at a particular fd value or an array of fd values. They are dictionaries with interpolators at each mode, where the keys are "A0", "A1", "A2", ..., "An" (for vp_antisym) and "S0", "S1", "S2", ..., "Sn" (for vp_sym), with 'n' being the order of the corresponding mode.
For example, if you need the phase velocity for the S0 mode at 1000 kHz × mm, you can do:
alum.vp_sym['S0'](1000)And this should return 5265.14 m/s. Always make sure that the fd values are within the valid range for the corresponding mode (i. e., above the cutoff frequency and below the fd_max you chose). Also, make sure the mode selected is within the selected nmodes. For example, if you chose nmodes_sym = 5, you can use 'S0', 'S1', 'S2', 'S3' or 'S4'.
- Group Velocity
You can use the attributes vg_sym and vg_antisym to find the group velocity at a particular fd value or an array of fd values. They are dictionaries with interpolators at each mode, where the keys are "A0", "A1", "A2", ..., "An" (for vg_antisym) and "S0", "S1", "S2", ..., "Sn" (for vg_sym), with 'n' being the order of the corresponding mode.
For example, if you need the group velocity for the A1 mode at 2000, 3000, and 4000 kHz × mm, you can do:
alum.vg_antisym['A1']([2000,3000,4000])And this should return 3241.72, 3577.26, and 2486.33 m/s. Always make sure that the fd values are within the valid range for the corresponding mode (i. e., above the cutoff frequency and below the fd_max you chose). Also, make sure the mode selected is within the selected nmodes. For example, if you chose nmodes_antisym = 5, you can use 'A0', 'A1', 'A2', 'A3' or 'A4'.
- Wave Number
You can use the attributes k_sym and k_antisym to find the wave number at a particular fd value or an array of fd values. They are dictionaries with interpolators at each mode, where the keys are "A0", "A1", "A2", ..., "An" (for k_antisym) and "S0", "S1", "S2", ..., "Sn" (for k_sym), with 'n' being the order of the corresponding mode.
For example, if you need the wave number for the S3 mode at 8000 kHz × mm, you can do:
alum.k_sym['S0'](8000)And this should return 726.38 m-1. Always make sure that the fd values are within the valid range for the corresponding mode (i. e., below the fd_max you chose). Also, make sure the mode selected is within the selected nmodes. For example, if you chose nmodes_sym = 5, you can use 'S0', 'S1', 'S2', 'S3' or 'S4'.
In order to verify that the results obtained with this code are correct, a simple validation is performed using the Dispersion software included in the AGU-Vallen Wavelet tool, developed by in collaboration between Vallen Systeme GmbH, Aoyama Gakuin University (AGU), and University of Denver.
Run validation.py to perform the validation. The result should look like this:
-
Phase velocity validation:
-
Group velocity validation:
As can be seen, the results are almost identical. The only difference is that Dispersion has an extended velocity range.
For information about the equations implemented, please refer to:
[1] Rose, J. L., Ultrasonic Guided Waves in Solid Media, Chapter 6: Waves in Plates, Cambridge University Press, 2014.
[2] Graff, K. F., Wave Motion in Elastic Solids, Chapter 8: Wave Propagation in Plates and Rods, Dover Publications, 1975.
This project is licensed under the MIT License - see the LICENSE.md file for details.