Computational modeling of ultrasound propagation using Abaqus

17 Pages·2013·3.09 MB·English

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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
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Can Abaqus be used for wave and vibration
problems?
Absolutely! The beauty of finite element analysis is that
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there is no restriction to special cases.
It is useful to revisit the special cases:
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Longitudinal waves
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Transverse waves
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Rayleigh waves
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Waves in wave guides
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Etc…
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Because comparison of finite element results with these
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can help with understanding and interpretation.
2013 Simulia UK Regional User Meeting
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Ultra-sound as a method for Non Destructive Testing
Ultra-sound is an acoustic wave with frequency above
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the range of human hearing, about 20,000 Hz.

For the NDT of engineering materials, the useful range
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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
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sub-surface defects and thickness profiles of features.
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Waves in solids
Phase velocity of waves:
Longitudinal
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𝐸(1 − 𝜈)
𝑐 =
𝐿
𝜌(1 + 𝜈)(1 − 2𝜈)
Transverse
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𝐸
𝑐 =
𝑇
2𝜌(1 + 𝜈)

Where a plane wave front impinges on a boundary, then
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typically both a longitudinal and transverse reflection
and transmission will take place.
2013 Simulia UK Regional User Meeting
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Waves in solids
Rayleigh waves
Phase velocity
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0.862 + 1.14𝜈
𝑐 = 𝑐
𝑅 𝑇
1 + 𝜈
This wave travels along the free surface, with motion
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perpendicular to the surface.

Hopkinson’s bar – 1D wave guide:
Used for measuring the stress-strain properties of
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materials at high strain rate
𝐸
𝑐 =
𝜌
𝐵
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Illustration – waves and vibrations in a beam
Flexural waves in a Bernoulli-Euler beam
The governing equation
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4 2
𝜕 𝑈 𝜕 𝑈
3 3
𝐸𝐼 + 𝜌𝐴 = 𝑞(𝑥, 𝑡)
𝜕𝑥4 𝜕𝑡2

Flexural rigidity, 𝐼 = 𝑤ℎ3 12
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Cross-sectional area, 𝐴 = 𝑤ℎ
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Time dependent forcing term, 𝑞 𝑥, 𝑡
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Phase velocity is dispersive (velocity depends on
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wavelength)
2𝜋 𝐸𝐼
𝑐 =
𝐹
𝜆 𝜌𝐴
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Standing waves: Modal analysis
Natural vibration modes are solutions of
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4 2
𝜕 𝑈 𝜕 𝑈
3 3
𝐸𝐼 + 𝜌𝐴 = 0
𝜕𝑥4 𝜕𝑡2

with boundary conditions encastre at one end and free at
the other:

𝑈 𝑥, 𝑡 = 𝑎 𝑔 𝑥 sin(𝛼 𝑡)
3 𝑘 𝑘 𝑘 𝑘

𝑔 𝑥 = 𝐶 sin 𝑝 𝑥 − sinh 𝑝 𝑥 + 𝑅 cos 𝑝 𝑥 − cosh 𝑝 𝑥
𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘

𝑝 are the roots of cos 𝑝 𝐿 + 1 cosh 𝑝 𝐿 = 0
𝑘 𝑘 𝑘

with 𝛼 = 𝐸𝐼 𝜌𝐴 (𝑝 𝐿)2 and 𝑎 = 𝑔 (𝐿) (𝜌𝐴𝛼 )
𝑘 𝑘 𝑘 𝑘 𝑘
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Data used in this exercise
Aluminium properties
Young’s modulus 70×103 MPa
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Poisson’s ratio 0.35
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Density 2.7×10-9 Tonnes/mm3
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Beam geometry
Length 100 mm
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Width 10 mm
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Thickness 2 mm
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FEA model information
Element type C3D8R, with
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Model comprising 40 by 10 by 10 elements
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Modal analysis solutions
165 Hz
1032 Hz
2885 Hz
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Low frequency forced motion
Quasi-static
Applied forcing frequency << lower than the
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fundamental natural frequency

The disturbance is transmitted through the component
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far quicker than the forced movement at the loading
point

Analysis run time is slow in Explicit analysis: more
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computationally efficient and no less rigorous to use an
implicit analysis step

2013 Simulia UK Regional User Meeting