Computational modeling of ultrasound propagation using ...

2013 Simulia UK Regional User Meeting

1

Computational modeling of ultrasound propagation using

Abaqus explicit

Alison J. McMillan

Glyndwr University



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.


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.


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.


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


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


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)


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


9

Modal analysis solutions

165 Hz

1032 Hz

2885 Hz


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


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


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


13

Illustration of ultra-sound NDT of a 3D block

• Dimensions 20x20x10 mm

• Hole diameter 2 mm.

• Forcing frequency 5×106 Hz.


14

Surface deformation at 1×10-6 second intervals


15

Wave profile

0 – 0.3×10-6 seconds

1.8 – 2.1×10-6 seconds

4.7 – 5.0×10-6 seconds


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


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

Computational modeling of ultrasound propagation using ...

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