Numerical and experimental analysis of a focused reflected wave in a multi-layered material based on a ray model

Ultrasonics 86 (2018) 41–48

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Ultrasonics journal homepage: www.elsevier.com/locate/ultras

Numerical and experimental analysis of a focused reflected wave in a multi-layered material based on a ray model Xiaoyu Yang a, Chengcheng Zhang b, Anyu Sun a,⇑, Xiaolong Bai a, Bing-feng Ju a, Qiang Shen b a b

The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, People’s Republic of China State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 9 October 2017 Received in revised form 21 December 2017 Accepted 22 January 2018 Available online 2 February 2018 Keywords: Multi-layered material Ray model Simultaneous measurement Multi-mode waves Phase difference Vðz; tÞ curve

a b s t r a c t A focal probe is used for the acoustic measurement of a thin layer of a material with unknown sound velocity. This is now possible, because an algorithm, based on the focused ray model, has been found. However, there are still several problems such as the assumption that the half-aperture angle equals the incident angle, the identification of the longitudinal-wave focus, and the composition of the signal. In this work, we study the multi-mode wave focus numerically and experimentally to identify the focused longitudinal waves. A theoretical multilayered focusing model has been introduced based on geometrical acoustics. In addition, a phase differentiation theory is proposed to find the incident angle for the focus of the tilted rays, which is referred to as maximum half-aperture angle in other studies. The Vðz; tÞ curve of a single layer, with a thickness of 1.5 mm and 2.0 mm, and a multi-layer are obtained using vertical translational movement. Both thickness and sound velocity are derived from the curve simultaneously. Our single layer experiments show that it is possible to focus multimode waves. The single and multi-layer experiments confirm the multi-layered focused ray model and phase differentiation theory. Furthermore, experiments are conducted to analyze the measured results. Ó 2018 Elsevier B.V. All rights reserved.

1. Introduction An acoustic thickness measurement of multi-layered material, whose sound velocity is unknown, is difficult to perform. Some authors use a focus probe that is typically used in acoustic microscopy to measure thickness and sound velocity simultaneously with the help of a ray model [1–3]. They used a probe with a very small aperture angle and samples with thin layers. This way, they can directly use the half-aperture angle of the probe as the angle of incidence. However, the angle of incidence at the focus point and the modes of the focused wave have not yet been studied. Other researchers obtained simultaneous measurements of the thinlayer material-parameters using an inversion algorithm based on ultrasound spectroscopy [4,5]. However, the estimated initial values for the iterative method are hard to find. Wavelet-based processing was developed to determine ultrasound velocity and material thickness simultaneously [6]. This measurement technology uses a one-shot transmission mode. However, it is not efficient, because it requires additional reference experiments- unlike other methods. More importantly, the samples in the studies above are

⇑ Corresponding author. E-mail address: [email protected] (A. Sun). https://doi.org/10.1016/j.ultras.2018.01.012 0041-624X/Ó 2018 Elsevier B.V. All rights reserved.

only thin single layers, while there are many practical problems with multi-layers. Some researchers already used the parametric model for ultrasonic waves propagating through a multi-layered material [7]. They obtained simultaneous measurements of the properties of multi-layered materials using a parameter-estimation algorithm based on the whole waveform, which increases complexity and reduces numerical stability when the number of parameters increases. The ultrasonic pitch-catch method is another method that can be used for simultaneous sound-velocity and thickness measurements for more than one layer [8]. This method requires both an angle probe and a normal probe. It can perform a relatively accurate measurement but at the cost of increasing complexity. In this paper, we use a focus probe for simultaneous thickness and sound-velocity measurements based on a ray model. The focus of the multi-mode waves is analyzed using a single layer model. Different wave modes were identified because the longitudinal bulk wave is most representative and used. The phase differences between waves with different angles of incidence and the existence of a critical angle affect the composition of the signal [9]. Thus, a theory about the phase difference of tilted rays was developed to find the incident angle for focused tilted rays. Experiments with single layer and multi-layered materials confirm the validity of the theoretical model and the new algorithm. This study

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complements the theory for reflective focus-probe waves in multilayered materials. The new analysis and experiments in this paper concentrate on practical implementation issues and contribute to the application of focus probes for simultaneous thickness and sound-velocity measurements in multi-layered materials. A theory for focused multi-mode waves is introduced in Section 2.1. Then, the theoretical ray model for single-mode focused waves in multi-layered material is discussed in Section 2.2 and the computation formulas are derived. Both signal composition and phase differences between the paraxial ray and the tilted ray during the vertical translation (movement) are analyzed numerically in Section 3. Experiments with single layer and multi-layered materials are conducted to verify the theory and the algorithm. Details of the experiments are shown in Section 4, and the results are shown in Section 5. The Vðz; tÞ curve is defined as the two-dimensional amplitude of a function for the movement distance z of the transducer and the time t in the step-time window. The ‘bottom left’ principle is proposed to identify the longitudinal-wave focus in the Vðz; tÞ curve, which contains unclear multi-mode peaks, in Section 5. Finally, measuring thickness efficiently as well as experimental problems is discussed in Section 6.

2. Theoretical model

surface-wave re-emits in the immersion liquid, while propagating along the sample surface [14]. This wave mode also contributes to the output signal. Thus, there are four fundamental wave modes to produce the output signal: longitudinal wave, transverse wave, transformed wave, and surface-wave mode. Because the velocity of the transverse wave is significantly lower than the longitudinal wave, three modes of bulk-waves focus on the back surface sequentially. When the focal probe moves down, the transverse wave focuses on the back surface first in Fig. 1(a). The transformed wave focuses then as shown in Fig. 1(b). The longitudinal wave focuses next as shown in Fig. 1(c). The time of flight (TOF) measurements of the three bulk waves suggest the opposite order. In addition, the surface wave does not focus at all. Thus the four modes obtain their characteristics in the Vðz; tÞ curve via vertical translation in the multi-layered material. The focus of the longitudinal wave can be distinguished from the bulk wave focuses based on the sequence in time and the displacement domain, which is defined as the ’bottom left’ principle. 2.2. Multi-layered model When the bulk wave with an initial incident angle of a focuses on the back surface of the layer in Fig. 2(b), the geometric formula of the dotted line and the solid lines, representing the bulk wave in the stratified material, can be written as

2.1. Multi-mode waves The analysis in this paper is based on two fundamental assumptions with respect to acoustic microscopy. Firstly, the sound velocity in crystals is higher than that in the immersion liquid; therefore, aberration in crystals can be ignored. Secondly, the dimensions of a lens are far larger than the wavelength in the liquid; hence, the wave in the model can be analyzed as geometric beams [10,11]. An incident ultrasonic wave in a single layer, immersed in the water with an incident angle of, is illustrated in Fig. 1. Longitudinal and transverse waves in solids have different refraction angles because of the difference in velocity. The waves return along a symmetrical route through reflection at the back surface of the layer. A modal transformation between the longitudinal and transverse waves occurs during the reflection at the back surface. The reflection angle of the longitudinal and transverse wave also obeys Snell’s refraction law [12]. The transformed wave, whose mode has been transformed after the reflection on the back surface, will also contribute the output signal- see Fig. 1(b). In addition, the surface wave will be simulated because of the critical angle [13]. The leaky

d  tanðaÞ ¼

n X hi  tanðbi Þ;

ð1Þ

i¼1

where a is the incident angle of the bulk wave between the liquid and the first layer, hi is the thickness of the layer i; bi is the refraction angle of the layer i; d is the vertical displacement for the focal position moving from the first surface to the back surface of layer n. Snell’s refraction law of multi-layered is

sinðaÞ sinðbi Þ ¼ ; c0 ci

ð2Þ

where c0 represents the sound velocity in liquid, and ci is the sound velocity in layer i. Time of flight (TOF) at the focal point for the liquid shown in Fig. 2(a) and the focal point in the back surface of the layer n in Fig. 2(b) is

2F ; c0 n X 2F 2d 2hi  Tn ¼ þ ; c0 c0 cosðaÞ i¼1 ci cosðbi Þ

T1 ¼

ð3Þ ð4Þ

Fig. 1. The theoretical model of the multi-mode waves propagating in a single layer and the focuses of (a) transverse wave, (b) transformed wave, and (c) longitudinal wave.

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Fig. 2. The theoretical model of (a) the longitudinal-wave focus in the surface and (b) the focused bulk wave propagating in a multi-layered material.

where F is the focal length in the immersion liquid. T 1 ; T n and d can be obtained directly from the echo signal in the Vðz; tÞ curve based on Eqs. (1)–(4). Furthermore, cn ; bn and hn can be formulated as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2hi ci u 2dc0 u  n1 i¼1 cos bi cos a cn ¼ t Pn1 2hi ; 2d T n  T 1 þ c0 cos i¼1 ci cos bi a   cn 1 bn ¼ sin ; sin a  c0 ! n1 d  c0 X hi  c i cos bn hn ¼  :  cos a i¼1 cos bi cn

ð6Þ

arriving from the center of the lens and a small angle with the acoustic axis [11]. There is no phase difference between rays arriving at the focal position from the central and peripheral regions of the lens in the immersion liquid. However, refraction will produce a phase difference between rays with different angles of incidence. The ray at normal incidence is used as reference. The TOF data of the central paraxial rays and the peripheral tilted rays with an incident angle a are

ð7Þ

tc ¼ 2

ð5Þ

! n X F d hi te ¼ 2  þ ; c0 c0  cos a i¼1 ci  cos bi

3. Numerical analysis 3.1. Focus in vertical translation When a single layer material, whose thickness and sound velocity are constants, is measured using a focused wave with a certain angle a, see Eqs. (1)–(4), the vertical displacement for the focus point moving from the front to the back surface is

d¼

h1  c1  cos a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; c2 2 c0  1  sin a  c12

! n Fd X hi ; þ c0 c i¼1 i

ð8Þ

0

where h1 and c1 are the thickness and sound velocity of the single layer, respectively, and a is smaller than the critical angle. This indicates that when the focal probe moves down, waves of all incident angles focus, based on their sequence rather than simultaneously because of the refraction angles. Larger angles of incidence require a larger vertical displacement. Therefore, after refraction, there is no position where all rays can focus in one point. 3.2. Signal phase Our signal analysis relies on the approximation hypothesis that the main contribution to the signal amplitude comes from the rays

ð9Þ ð10Þ

where tc is the TOF of the central paraxial wave, t e is the TOF of the peripheral tilted rays with an incident angle a. The peripheral tilted rays originate from the peripheral region of the lens. The maximum amplitude in the Vðz; tÞ curve occurs at the focal position in the solid layer where the maximum phase difference b satisfies the condition:

b ¼ 2pf 0  ðt c  t e Þ <

p 2

;

ð11Þ

where f 0 is the inherent frequency of the probe. The signal of the focused wave peak mainly consists of the central paraxial rays and the peripheral tilted rays. The rays are simplified to sinusoidal pulse-signals. Thus the phase difference between paraxial rays and tilted rays affects the amplitude of the composed signal. When the phase difference b reaches a certain value, the peripheral wave with the incident angle a reduces the peak voltage which is mainly determined by central paraxial rays. The incident angle a of the focused wave is not known for the numerical analysis. Thus, all possible angles within the maximum half-aperture angle need to be tried. Afterwards, the phase difference b is computed in order to find the incident angles of the tilted rays.

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4. Experimental 4.1. Method The peaks of the focused waves are identified in the Vðz; tÞ curve, which is a series of ultrasonic echo pulses collected from different transducer displacements z between an initial focal position and the material surface. Both vertical movement d and Dt can be obtained using the peaks in the Vðz; tÞ curve, where Dt i ¼ T i  T 1 ; T i represents the actual time-coordinate of peak i, and T 1 is the actual time-coordinate of the first peak with z ¼ 0. The time domain of the curves has been aligned for visualization by moving the time window. Thus the actual time coordinate of the peak is

Ti ¼ T 

2zinc  d ; f s  Dd

ð12Þ

where T is the time-coordinate in the curve, zinc is the increment for the time window, d is the vertical displacement, Dd is the step of the vertical displacement, and f s is the sampling frequency. It is clear that we cannot determine the angle of incidence of the tilted focused ray at the beginning. All possible angles of incidence within the maximum half-aperture angle are used to calculate phase b and height curves. According to the above analysis, which treats rays as sinusoidal pulse-signals, both estimated angle of incidence and measured thickness are found when the critical phasedifference of the focused wave(b ¼ p2 ) is satisfied in the curve. 4.2. Experimental setup The experimental system is essentially a reflective acoustic microscopy with the additional function of vertical translation. A V390 point-focusing transducer, provided by the Olympus Corporation, Japan, was used. The point-focusing probe has a nominal center frequency of about 50 MHz and a measured 3db bandwidth of about 41 MHz to 57 MHz. The focal length is 12.7 mm, and the half aperture angle is 14:5 . The sampling rate is 400 MHz and the Vðz; tÞ data were obtained using the vertical translation. The vertical position accuracy of the transducer is 0.5 lm. The transducer is initially moved to the focal position at the front surface of the first layer and before the automatic automatic vertical translation commences. The temperature was 27:3 for which the sound velocity in water is 1503 m/s. The single-layer experiments are designed to show evidence of multi-mode waves and wave mode transformation during the focusing process. The samples were two stainless-steel plates with thicknesses of 1.5 mm and 2.0 mm, respectively. The multi-layered experiments are designed to perform simultaneous thickness and sound-velocity measurements of multi-layered materials for longitudinal waves. The reference thickness of each layer was obtained by optical imaging. 5. Results 5.1. Single layers The Vðz; tÞ curve is obtained by vertical translation as shown in Fig. 3. The phase difference b and inversion thickness h, which correspond to a different assumed angle a for a tilted focused ray, are shown in Figs. 4 and 5. The estimated angle of incidence and measured thickness are found when the critical phase-difference of the focused wave (b ¼ p2 ) is satisfied. Repeated measurements were conducted to obtain both mean and variance. The inversion, reference thicknesses, and velocities of the single-layer sample are listed in Tables 1 and 2. There are three neighboring maxima in

Fig. 3. The Vðz; tÞ curve of materials, with thicknesses of (a) 1.5 mm and (b) 2.0 mm, with indication of multi-mode wave focuses.

the Vðz; tÞ curve for the single-layer material apart from the first echo. The measured thickness based on these three peaks is consistent with the thickness of the stainless-steel plate. The measured sound velocities of the longitudinal and transverse focused wave (first and third peak), and the reference velocities of stainless steel, are in full agreement with each other. Moreover, the measured velocity of the focused transformed wave (second peak) is about half the sum of the velocities of the longitudinal and transverse waves. The sequence of three peaks in the time and displacement domains satisfies the theoretical analysis of longitudinal wave, transverse wave, and transformed wave. Besides, the shape of the image of the first and third peaks in the curve are parallel to the first echo, while the second peak tends to bifurcate in a more complex way. The transformed waves undergo two kinds of transformation: longitudinal to transverse and transverse to longitudinal. Because the focal position of the transformed wave is not located on the acoustic axis, the shape of the second peak meets the characteristics of transformed wave. The single layer experiments confirm the occurrence of multi-mode waves. Furthermore, the focuses of the transverse wave and the transformed wave lag in the time domain but lead in the displacement domain compared

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Fig. 4. The phase difference b and calculated thickness h corresponding to different assumed angle a of tilted focused ray in a material with a thickness of 1.5 mm.

to the focused longitudinal waves. Thus the maximum in the Vðz; tÞ curve for longitudinal wave should be identified as ’bottom left’. The ’bottom left’ principle to identify the longitudinal wave is very useful for experiments with multi-layered material. 5.2. Multilayers The Vðz; tÞ curve for a multi-layer reveals many peaks- see Fig. 6. The focus of the transverse wave and transformed wave shows a short TOF and larger vertical displacement than longitudinal waves. As a result, the maximum of the longitudinal wave should be at the ’bottom left’ in the Vðz; tÞ curve. The focus of the longitudinal wave can be identified using the ’bottom left’ principle. The phase difference b and inversion thickness h correspond to a different assumed angle a of a tilted focused ray- see Fig. 7. The estimated angle of incidence and measured thickness are found when the critical phase difference for the focused wave (b ¼ p2 ) is met. Repeated tests were conducted to obtain both mean and variance. The inversion, reference thickness, and velocities of the multi-layered samples are listed in Table 3. 6. Discussion When the probe focuses on the front surface, the vertical movable range of the probe is F cos h, where h is the half-aperture angle.

Eq. (8) indicates that the necessary movable range required to change the focus through a layer is within the interval cos am ½h1cc1 ; h1cc1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , where am ¼ maxðh; aR Þ, and N ¼ cc1 ; aR is 2 0

0

1sin am N2

0

the critical angle. Thus the measurable thickness of the layer is limited by the focal length, maximum angle of incidence, and the critical angle. In the single layer experiments, the estimated angle of incidence a decreases when the plate thickness increases from 1.5 mm to 2.0 mm. This result clearly indicates coherence between the experimental and the corresponding phase differentiation theory. When the plate thickness increases, the TOF of the focused wave as well as the phase difference b increases according to Eqs. (9)–(11). Furthermore, the phase difference increases when a increases. Therefore, the estimated a will decrease to counteract the increase of the phase difference caused by the increased thickness. The signal-to-noise ratio (SNR) of the focused transverse wave is much lower. This has been shown in Fig. 3. Thus, it is not sufficient to measure transverse-wave velocity only. The transverse-wave focus experiments in this paper are performed only for confirmation. If the SNR of the focused transverse wave can be increased with additional research, such a method could be used for simultaneous measurement of many modal sound velocities. The measurement errors for the third layer are much larger in the multi-layer experiment, which is possibly due to the cumulative error.

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Fig. 5. The phase difference b and calculated thickness h corresponding to different assumed angle a of tilted focused ray in a material with a thickness of 2.0 mm.

Table 1 The inversion properties of a material with a thickness of 1.5 mm. Measured Properties (mean3r)

1 2 3

Reference Properties (common situation)

a ð Þ

ci (m/s)

hi ðlmÞ

Material

ct (m/s)

cl (m/s)

h ðlmÞ

7.75 ± 0.19 9.51 ± 0.68 12.8 ± 1.36

5584 ± 129 4261 ± 272 3073 ± 280

1454 ± 42 1592 ± 122 1489 ± 132

Stainless steel

3100

5790

1500

Peak

Peak 1,2,3 represent longitudinal-wave focus, transformed-wave focus, and transverse-wave focus.

Table 2 The inversion properties of a material with a thickness of 2.0 mm. Measured Properties(mean3r) Peak 1 2 3

Reference Properties(common situation)

a (°)

ci (m/s)

hi ðlmÞ

Material

ct (m/s)

cl (m/s)

h ðlmÞ

7.15 ± 0.23 8.83 ± 0.24 11.90 ± 0.71

5708 ± 298 4315 ± 204 3114 ± 149

1933 ± 85 2107 ± 137 1980 ± 129

Stainless steel

3100

5790

1970

Peak 1,2,3 represent longitudinal-wave focus, transformed-wave focus, and transverse-wave focus.

7. Conclusions Conventional focusing methods to measure sound velocity and thickness simultaneously use the half-aperture angle of the probe as angle of incidence. When the half-aperture angle is small

enough to ignore the phase difference of all rays and smaller than the critical angle, it can represent the maximum incident angle. When a larger aperture angle is used, the new algorithm presented in this paper is needed to find the estimated angle of incidence for the focused wave. Another problem preventing simultaneous

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X. Yang et al. / Ultrasonics 86 (2018) 41–48 Table 3 The inversion properties of a 3-layered material based on ray model. Measured Properties(mean3r) Peak 1 2 3

Refence Properties

a (°)

ci (m/s)

hi ðlmÞ

Material

cl (m/s)

h ðlmÞ

9.08 ± 0.66 8.19 ± 0.37 7.00 ± 0.36

6120 ± 428 4883 ± 329 6616 ± 654

462 ± 36 760 ± 44 632 ± 62

TC 4 OFC Mo

6171 4763 6250

476 777 575

Peak 1,2,3 represent longitudinal-wave focuses of first layer, second layer, and third layer. OFC represents Oxygen Free Copper which is 99:9% purity of copper, and TC 4 is a titanium alloy.

Fig. 6. The Vðz; tÞ curve of a multi-layered material with indication of longitudinalwave focuses in first three interfaces.

sound-velocity and thickness measurements in multi-layers is the multi-mode wave focus. The focal position of the longitudinal waves is hard to distinguish from other modal waves because

the captured signal is composed of all waves (longitudinal, transverse, transformed, and surface waves). Thus the neighboring maximum of the Vðz; tÞ curve are not all caused by a longitudinal wave. The numerical and experimental analysis have revealed the waveform and location features of different modal wave signals in the Vðz; tÞ curve, which is obtained from the vertical translation of the multi-layer using a reflective focal probe. The wave modes that produce the peaks in the Vðz; tÞ curve were identified. A theoretical model based on the geometrical acoustic and phase differentiation theory, was applied to enable the inverse calculation. The recovered values were compared using the reference values of the material parameters. Our results are consistent with the reference values within the permitted error. The experiments have validated the theoretical models and numerical analysis.

Fig. 7. The phase difference b and calculated thickness h corresponding to different assumed angle a of tilted focal ray in a multi-layered material.

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Acknowledgements This work is supported by National Natural Science Foundation of China (Grant Nos. 51425504, 51605429, 51575488, 51521001), Science Fund for Creative Research Groups of National Natural Science Foundation of China 51521064, Zhejiang Provincial Natural Science Foundation of China LZ13E050001, Project of Application Research on Public welfare Technology 2014C31086.

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