computational modeling of ultrasound propagation using
TRANSCRIPT
2013 Simulia UK Regional User Meeting
1
Computational modeling of ultrasound propagation using
Abaqus explicit
Alison J. McMillan
Glyndwr University
2013 Simulia UK Regional User Meeting
2013 Simulia UK Regional User Meeting
2 Can Abaqus be used for wave and vibration problems?
• Absolutely! The beauty of finite element analysis is that there is no restriction to special cases.
• It is useful to revisit the special cases:
- Longitudinal waves
- Transverse waves
- Rayleigh waves
- Waves in wave guides
- Etc…
• Because comparison of finite element results with these can help with understanding and interpretation.
2013 Simulia UK Regional User Meeting
3
Ultra-sound as a method for Non Destructive Testing
• Ultra-sound is an acoustic wave with frequency above the range of human hearing, about 20,000 Hz.
• For the NDT of engineering materials, the useful range is 2 – 10×106 Hz, although for some lower density materials, the range 50 – 500×103 Hz is also used.
• Ultrasound NDT can be used to detect the presence of sub-surface defects and thickness profiles of features.
2013 Simulia UK Regional User Meeting
4
Waves in solids
Phase velocity of waves:
• Longitudinal
𝑐𝐿 =𝐸(1 − 𝜈)
𝜌(1 + 𝜈)(1 − 2𝜈)
• Transverse
𝑐𝑇 =𝐸
2𝜌(1 + 𝜈)
• Where a plane wave front impinges on a boundary, then typically both a longitudinal and transverse reflection and transmission will take place.
2013 Simulia UK Regional User Meeting
5
Waves in solids
Rayleigh waves
• Phase velocity
𝑐𝑅 = 𝑐𝑇0.862 + 1.14𝜈
1 + 𝜈
• This wave travels along the free surface, with motion perpendicular to the surface.
Hopkinson’s bar – 1D wave guide:
• Used for measuring the stress-strain properties of materials at high strain rate
𝑐𝐵 = 𝐸𝜌
2013 Simulia UK Regional User Meeting
6
Illustration – waves and vibrations in a beam
Flexural waves in a Bernoulli-Euler beam
• The governing equation
𝐸𝐼𝜕4𝑈3𝜕𝑥4
+ 𝜌𝐴𝜕2𝑈3𝜕𝑡2
= 𝑞(𝑥, 𝑡)
• Flexural rigidity, 𝐼 = 𝑤ℎ3 12
• Cross-sectional area, 𝐴 = 𝑤ℎ
• Time dependent forcing term, 𝑞 𝑥, 𝑡
• Phase velocity is dispersive (velocity depends on wavelength)
𝑐𝐹 =2𝜋
𝜆
𝐸𝐼
𝜌𝐴
2013 Simulia UK Regional User Meeting
7
Standing waves: Modal analysis
• Natural vibration modes are solutions of
𝐸𝐼𝜕4𝑈3𝜕𝑥4
+ 𝜌𝐴𝜕2𝑈3𝜕𝑡2
= 0
with boundary conditions encastre at one end and free at the other:
𝑈3 𝑥, 𝑡 = 𝑎𝑘𝑔𝑘 𝑥 sin(𝛼𝑘𝑘 𝑡)
𝑔𝑘 𝑥 = 𝐶𝑘 sin 𝑝𝑘𝑥 − sinh 𝑝𝑘𝑥 + 𝑅𝑘 cos 𝑝𝑘𝑥 − cosh 𝑝𝑘𝑥
𝑝𝑘 are the roots of cos 𝑝𝑘𝐿 + 1 cosh 𝑝𝑘𝐿 = 0
with 𝛼𝑘 = 𝐸𝐼 𝜌𝐴 (𝑝𝑘𝐿)2 and 𝑎𝑘 = 𝑔𝑘(𝐿) (𝜌𝐴𝛼𝑘)
2013 Simulia UK Regional User Meeting
8
Data used in this exercise
Aluminium properties
• Young’s modulus 70×103 MPa
• Poisson’s ratio 0.35
• Density 2.7×10-9 Tonnes/mm3
Beam geometry
• Length 100 mm
• Width 10 mm
• Thickness 2 mm
FEA model information
• Element type C3D8R, with
• Model comprising 40 by 10 by 10 elements
2013 Simulia UK Regional User Meeting
10
Low frequency forced motion
Quasi-static
• Applied forcing frequency << lower than the fundamental natural frequency
• The disturbance is transmitted through the component far quicker than the forced movement at the loading point
• Analysis run time is slow in Explicit analysis: more computationally efficient and no less rigorous to use an implicit analysis step
2013 Simulia UK Regional User Meeting
11
Intermediate frequency forced motion
• Selected forcing frequency 𝜔 = 50×103 rad/second
• This lies between the 4th and 5th flap modes
• At this frequency, 𝑐𝐹 =0.3834×106 mm/second and 𝜆𝐹 =48.18 mm
• Flexural wave takes 260.8×10-6 seconds to travel the length of the beam, 100 mm, cf the longitudinal and transverse waves, 15.50 and 32.27×10-6 seconds respectively
Deflection at measurement point
Time ×10-3 second 4 6 2 8 10 0
2013 Simulia UK Regional User Meeting
12
Wave pulse illustrated at 10×10-6 second intervals
Peak to peak 45-50 mm ≈ 𝜆𝐹
Wave travel distance ≈ 23 mm
10
50
20
30
40
60
300 ×10-6 seconds
2013 Simulia UK Regional User Meeting
13
Illustration of ultra-sound NDT of a 3D block
• Dimensions 20x20x10 mm
• Hole diameter 2 mm.
• Forcing frequency 5×106 Hz.
2013 Simulia UK Regional User Meeting
15
Wave profile
0 – 0.3×10-6 seconds
1.8 – 2.1×10-6 seconds
4.7 – 5.0×10-6 seconds
2013 Simulia UK Regional User Meeting
16
Signals received at measurement points
1
2
3
4 time
1 2 3 4 ×10-6 second
Wave arrival times: Longitudinal Rayleigh
Mea
sure
men
t poi
nts
2013 Simulia UK Regional User Meeting
17
Conclusions and recommendations
• Abaqus explicit can be effective in simulating wave pulses in components
• The post-processing of results requires interpretation; the development of post-processing add-on tools is an area for further development
• Abaqus explicit modelling can provide simulated data for demonstrating data processing techniques that could then be used on NDT data
• Simulated NDT data for components containing defects could then be used as reference catalogues, improving the interpretation capability of NDT