Propagation characteristic of laser-generated visco-elastic Rayleigh-like waves in stratified half-space
- Q. B. Han^{1}Email author,
- J. G. Shen^{2},
- X. P. Jiang^{1},
- C. Yin^{1},
- J. Jia^{1} and
- C. P. Zhu^{1}
https://doi.org/10.1186/s13638-016-0599-z
© Han et al. 2016
Received: 14 November 2015
Accepted: 4 April 2016
Published: 14 April 2016
Abstract
This paper reports on a study of the propagation characteristics of visco-elastic, Rayleigh-like waves in stratified half-space structures. Beginning with the Kelvin model, the characterization equation and the normal displacement of visco-elastic Rayleigh waves in stratified half-space structures are derived and the influence of the visco-elastic modulus on dispersion and attenuation is discussed. Theoretical calculations show that the attenuation-frequency curves perfectly match the phase-frequency curves. The effect of visco-elasticity on the attenuation of the Rayleigh-like wave is larger than its effect on dispersion. For “weak viscosity,” the attenuation is directly proportional to the viscosity modulus and the shear viscosity has a greater impact on the dispersion curves than does the bulk viscosity. The transient response of a visco-elastic Rayleigh wave is also simulated by means of Laplace and Hankel inversion transforms. The results are in good agreement with the theoretic predictions. It is believed the paper’s results and conclusions will provide insights and guidance for estimating visco-elastic parameters and for assessing adhesive quality.
Keywords
1 Introduction
Rayleigh waves, propagating on the free surface of an elastic half space, are well known. If the medium is homogeneous and isotropic, the velocity of the Rayeleigh wave depends on the elastic constants of the medium and not on the wavelength, i.e., the Rayleigh wave is non-dispersive, and the power density of the wave decays exponentially from the surface with a characteristic penetration depth of the order of a wavelength. However, in the presence of stratified half-space, the velocity will exhibit dispersion. Stratified half-space is a kind of common material structure [1–3]. Elastic wave fields in multilayered media are of considerable interest in a variety of applications, and have, therefore, been studied extensively over the years. After Rayleigh, Love, and Stoneley, Thomson and Haskell introduced the propagator matrix method that is later focused on by many authors. These works were carried out in the stratified elastic solid media model, and the propagator matrix technique is heuristic in many applications. The propagation of Rayleigh-like waves in a stratified half-space has been widely studied for use of nondestructive. Many years ago, Mason and Thurston [4] described surface acoustic waves (SAWs) in half substrate-coatings. Zininet et al. [5] pointed out that a pseudo-Rayleigh wave leaks energy into substrate. Lawr ultrasonics have been widely used to study the SAWs of multilayered adhesive structures [6–9], and its advantages include providing a non-contact, wide band, perfect source. Cheng et al. [10] have simulated laser-generated ultrasonic waves in a layered plate. All of the studies cited above assumed the adhesive layers are elastic solids because solidified adhesive layers closely resemble elastic solids. It should be remembered, however, that adhesive layers or coating have more attenuation than solids due to the presence of more “relaxation” or “creep.”
There are two attenuation mechanisms for sound waves in layered media. The first is due to leaking wave where energy is leaking from one type of material to the other. Wave leaking is common when the sound wave is traveling from a solid to a liquid [11–13]. The second mechanism is material damping. Media are never perfectly elastic but always show some degree of damping that absorbs the energy of mechanical waves. Material damping is often described in the research of the waves associated with earthquakes [14, 15]. Many researchers have applied the study of waves in a stratified half-space to nondestructive evaluation (NDE) of materials. For instance, Yew et al. [16] have assessed the bonging quality of for SH waves when the adhesive layers are considered visco-elastic layers. Deschmps et al. [17] have studied acoustic emission and reflection in an anisotropic plate. Chan and Cawley [18] have discussed the effect on Lame waves brought about by viscosity through changing imaginary part of phase velocity. Bernard and Lowe [19] have studied the velocity of energy propagation in a visco-elastic plate. However, in order to simplify the calculation, most of the researches adopt Kelvin-Voigt model which directly append an imaginary part (damping) to the phase velocity and the frequency dispersion caused by visco-elasticity is ignored.
In this paper, according to the fundamental visco-elastic theory, the characterization equations for the Rayleigh waves and Rayleigh-like waves in stratified half-space structures are found by means of the Laplace/Hankel transform method. Frequency dispersion and the attenuation related to the visco-elastic modulus in a half-space, substrate-coatings, and three-layer structures are analyzed, and the transient responses of visco-elastic Rayleigh wave are simulated.
In Section 1, the theory about attenuation modes is introduced, the governing equations are derived in the transformed domain, and the equivalent elastic force sources for laser ultrasonics are discussed. In Section 2, visco-elastic Rayleigh wave propagation characteristics including dispersion, attenuation, and transient response in half-space are simulated and analyzed. In Section 3, the procedure associated with the transfer matrix approach is presented and visco-elastic Rayleigh waves in two layered structures are considered. Finally, in Section 4, three-layer adhesive structures, e.g., a half infinite metal substrate—adhesive layer-metal film, are also studied. Studying the visco-elastic (or attenuation) characteristics of waves should help us to evaluate adhesive quality and material properties. The purpose of the paper is to quantitatively analyze attenuation and dispersion and the transient response properties of Rayleigh-like waves generated by a laser and provide a theoretical basic for the determination of visco-elastic characteristics of the coatings and adhesive layers.
2 Theory
2.1 Attenuation model
where \( {S}_{ij}={\sigma}_{ij}-\frac{1}{3}{\sigma}_{kk}{\delta}_{ij} \) is stress deflection tensor, \( {e}_{ij}={\varepsilon}_{ij}-\frac{1}{3}{\varepsilon}_{kk}{\delta}_{ij} \) is strain deflection tensor, σ and e are the stress symmetrical tensor and the strain symmetrical tensor, respectively; the strain is given by \( {\varepsilon}_{ij}=\frac{1}{2}\left({u}_{i,j}+{u}_{j,i}\right) \), u _{ i,j } = ∂u _{ i } /∂x _{ j }, and \( {P}^{\hbox{'}}={\displaystyle \sum_{k=0}^{m^{\hbox{'}}}{p}_k^{\hbox{'}}\frac{d^k}{d{t}^k}} \), \( {P}^{\hbox{'}\hbox{'}}={\displaystyle \sum_{k=0}^{m^{\hbox{'}\hbox{'}}}{p}_k^{\hbox{'}\hbox{'}}\frac{d^k}{d{t}^k}} \), \( {Q}^{\hbox{'}}={\displaystyle \sum_{k=0}^{m^{\hbox{'}}}{q}_k^{\hbox{'}}\frac{d^k}{d{t}^k}} \), \( {Q}^{\hbox{'}\hbox{'}}={\displaystyle \sum_{k=0}^{m^{\hbox{'}\hbox{'}}}{q}_k^{\hbox{'}\hbox{'}}\frac{d^k}{d{t}^k}} \), where \( {p}_k^{\hbox{'}} \), \( {p}_k^{\hbox{'}\hbox{'}} \), \( {q}_k^{\hbox{'}} \), and \( {q}_k^{\hbox{'}\hbox{'}} \) indicate the modulus in visco-elasticity mode, respectively, which are determined by material and visco-elastic mode.
2.2 Governing equations
In a cylindrical coordinate system, the displacement can be expressed by the potential functions φ and \( {\varOmega}_i\left(0,-\frac{\partial \psi }{\partial r},0\right) \) as u _{ i } = φ _{,i } + e _{ ijk } Ω _{ k,j } and Ω _{ i,i } = 0.
where the superscript H _{0} indicates a Hankel transform of order zero and p and s are the space frequency and time frequency, respectively; \( \alpha =\sqrt{p^2+\frac{s^2}{{c_l^{\#}}^2}} \), \( \beta =\sqrt{p^2+\frac{s^2}{{c_t^{\#}}^2}} \), \( {c}_l^{\#}=\sqrt{\frac{\lambda^{\#}(s)+2{\mu}^{\#}(s)}{\rho }} \), \( {c}_t^{\#}=\sqrt{\frac{\mu^{\#}(s)}{\rho }} \), \( {\lambda}^{\#}(s)=s\lambda (s)=\frac{1}{3}\left[\frac{Q^{\hbox{'}\hbox{'}}(s)}{P^{\hbox{'}\hbox{'}}(s)}-\frac{Q^{\hbox{'}}(s)}{P^{\hbox{'}}(s)}\right] \), and \( {\mu}^{\#}(s)=s\mu (s)=\frac{Q^{\hbox{'}}(s)}{2{P}^{\hbox{'}}(s)} \) represent complex Lame constants associated with frequency s; ρ is density; A, B, C, and D are constants in transform field.
where the superscript H _{1} indicates the Hankel transform of order one. The characterization frequency equations can be obtained by continuation of the displacement and the stress.
2.3 The equivalent elastic force sources
where \( \xi =\sqrt{\alpha^2+\frac{s}{\gamma }} \), γ is the thermal diffusion coefficient, and C _{0} is a constant related to the thermal and elastic properties; Q _{0} is the laser energy amplitude; Q(s) and Q(p) are the laser source function on time and space in the transform domain, respectively.
3 Visco-elastic Rayleigh waves in half-space
3.1 The characterization equation
where
Δ = 4p ^{2} αβ − (β ^{2} + p ^{2})^{2},
3.2 Dispersion and attenuation characteristics of the visco-elastic Rayleigh wave
From Eq. (8), the dispersion and attenuation characteristics of the visco-elasric Rayleigh wave can be obtained. The roots of the equation are complex related to frequency; with the imaginary part of the wave number is the attenuation factor in SI units of Np/m. The phase velocity of waves can be calculated from \( c=\frac{\omega }{\mathrm{real}(k)} \), where ω is angular frequency and k is wave number. In addition, s = jω and k = p.
where μ _{ k }, η _{ k }, and K are shear, viscous, and bulk modulus, respectively.
From the dispersion curves, we find that the visco-elastic Rayleigh wave is dispersive and is not the case in (non-dispersion) elastic materials, which is due to the influence brought by viscosity. The amount of dispersion is related to the magnitude of the viscosity, and the larger the magnitude of viscosity is, the stronger the dispersion. It is also shown that the viscosity has little effect on phase velocity. When K _{ η } = 1ηs, the phase velocity of the visco-elastic Rayleigh wave increases from 1.22 to 1.26 km/s in the frequency range 0–40 MHz and the change in phase velocity for the case of K _{ η } = 0.1ηs is negligible. Consequently, a sample can be regarded as non-dispersion in the case of weak viscosity, which is in correspondence with the results of Ping [23]. Accordingly, it is not very productive to study visco-elastic characteristics from the perspective of dispersion.
However, it is also shown that the attenuation of the visco-elastic Rayleigh wave increases with an increase of frequency and material viscosity. Compared with the dispersion curves, the attenuation curve variation with the frequency is more pronounced. It has been shown that the relative amplitude change in velocity versus frequency is 10 % less than that of the attenuation versus frequency, [24] and the attenuation for K _{ η } = 1ηs is higher by about one order of magnitude than for K _{ η } = 0.1ηs at the same frequency. Thus, the elastic constants could be determined by dispersion (velocity) and the viscous constants by an attenuation curve.
where \( {K}_B=\frac{\eta }{K} \), and η is the bulk viscosity modulus.
3.3 Transient response of visco-elastic Rayleigh
To observe attenuation effect produced by viscosity intuitively, the time domain transient response of visco-elastic Rayleigh wave was simulated by means of inverse Laplace and Hankel transform. Here, we use laser ultrasonic to simulate the transient response of visco-elastic Rayleigh wave, because pulsed laser sources provide a nondestructive, non-contact means of wide bandwidth acoustic wave generation, especially, its good repetition will be good for attenuation measurement; point excited source is adopted to focus easily on producing higher-frequency wave for attenuation measurement.
where t _{0} is the laser pulse rise time (in our calculation t _{0} = 10ηs), R is the laser pulse Gaussian radius (R = 0.1 mm), and Q _{0} is the absorbed laser energy. This source has been shown to accurately represent the stress field induced by a laser in a number of practical cases [21]. The representation is subject to the following assumptions: the heating is localized to the surface layer, the point of observation is outside of the volume defined by significant thermal diffusion, and the optical energy is converted to heat close to the irradiated boundary. The first and second assumptions hold if the thermal diffusion length is sufficiently less than the top layer thickness and the source to receiver distance, respectively. The third assumption holds as long as the top layer material is a strong absorber at the generation laser wavelength.
The profiles of the Rayleigh wave are similar to that in an elastic medium; there are three waveforms that can be identified by their arrival times: the first is the lateral (or Head) wave which propagates along surface with longitudinal velocity of medium, the second is shear lateral wave and very weak. The amplitude of Rayleigh wave is greatest and can be clearly identified. It is found that the transient response amplitude of the Rayleigh wave decreases with increase in the propagation distance for the same viscosity, which may attribute to the presence of geometric attenuation for cylindrical diffusion propagation. However, at the propagation distance, the amplitude of the Rayleigh wave decreases with increase of viscosity, from which the effect of viscosity on Rayleigh waves is clearly seen.
4 Substrate coating structure
4.1 The transfer matrix
where, T = N _{ n } M _{ n − 1} M _{ n − 2} ⋅ ⋅ ⋅ M _{ j } M _{ j − 1} M _{ j − 2} ⋅ ⋅ ⋅ M _{3} M _{2} M _{1} and \( N=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill -\beta \hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -\alpha \hfill & \hfill {p}^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -2\mu \alpha \hfill & \hfill \mu \left({p}^2+{\beta}^2\right)\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill \mu \left({p}^2+{\beta}^2\right)\hfill & \hfill -2\mu \beta {p}^2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right] \)
And, the normal displacement at top surface can also be obtained by means of inverting \( {\overline{u}}_z^{{}^{H_0}} \).
For visco-elastic layers, all the equations above remain true when λ, μ are replaced with λ ^{#}, μ ^{#} seen in Eqs. (11) and (12).
4.2 Slow on fast
The simulation parameters of two layers adhesive structures
Materials | c _{l}(km/s) | c _{t}(km/s) | Thickness (mm) | ρ (g/cm^{3}) |
---|---|---|---|---|
Al | 6.32 | 3.13 | ∞ | 2.72 |
Epoxy | 2.73 | 1.30 | 0.1 | 1.40 |
4.3 Fast on slow
For the case of fast on slow, the situation is more complicated by the presence of the cutoff velocity occurring at the transverse wave velocity of the substrate. This situation has been discussed by P. Zinin [5] et al. using V(z) curves. Here, we consider the phenomenon from the viewpoint of attenuation caused by viscosity of adhesive layers.
4.4 Transient response
5 Three-layer adhesive structures
Parameters of elastic epoxy and Al in three layers adhesive structures
Materials | c _{l} (km/s) | c _{t} (km/s) | Thickness (mm) | ρ (g/cm^{3}) |
---|---|---|---|---|
Al (upper) | 6.32 | 3.13 | 0.1 | 2.72 |
Al (Nether) | 6.32 | 3.13 | ∞ | 2.72 |
Epoxy | 2.73 | 1.30 | 0.05 | 1.40 |
6 Conclusions
In this paper, the propagation characteristics of visco-elastic, Rayleigh-like waves generated by a laser pulse are analyzed theoretically. Based on general visco-elastic theory and regarded the visco-elastic media as a Kelvin mode, the characterization frequency equations are found by means of the Laplace/Hankel transform, and transient displacement of the visco-elastic, Rayleigh-like wave is derived; the dispersion and attenuation curves due to viscosity are calculated numerically. It is shown that the dispersion of visco-elastic Rayleigh-like wave is associated with the magnitude of viscosity. In the presence of a “weak viscosity,” the viscosity has little influence on phase velocity, and the attenuation of the wave is approximately proportional to the viscosity modulus. The effect of shear viscosity on attenuation is much more than that of bulk viscosity. The transient responses of the visco-elastic, Rayleigh-like wave were also simulated by the Laplace and Hankel inverse transforms, from which the effect of viscosity on the Rayleigh-like wave is clearly shown; the simulated transient response results are also in good agreement with the dispersion and attenuation curves. The approach used here was the model we develop which may provide a useful tool for the determination of the visco-elastic parameters of the material.
Declarations
Acknowledgements
This work is supported by the Natural Science foundation of China Grant Nos. 11274091 and 11574072 and the Fundamental Research Funds for the Central Universities of Hohai University No: 2011B11014.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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