### The characterization equation

Let us study visco-elastic Rayleigh wave in half-space firstly; the concerned conclusion will help us to understand characteristics of visco-elastic Rayleigh-like waves. Considering the fact that the two upgoing bulk wave modes vanish such that two constants *B* and *D* are zero in Eq. (5) in the half-space *z* > 0. Substituting Eq. (5) into Eq. (6), the characterization equation of visco-elastic Rayleigh wave becomes

$$ 4{p}^2\alpha \beta -{\left({\beta}^2+{p}^2\right)}^2=0 $$

(8)

The normal displacement of the visco-elastic Rayleigh wave in transformed domain is obtained:

$$ {\overline{u}}_z^{{}^{H_0}}=-\alpha \frac{\varDelta_1}{\varDelta }+{p}^2\frac{\varDelta_2}{\varDelta } $$

(9)

where

*Δ* = 4*p*
^{2}
*αβ* − (*β*
^{2} + *p*
^{2})^{2},

and

$$ {\varDelta}_1=\left[\begin{array}{cc}\hfill -\frac{{\overline{\tau}}_{rz}^{{}^{H_1}}}{p{\mu}^{*}}\hfill & \hfill {\beta}^2+{p}^2\hfill \\ {}\hfill \frac{{\overline{\tau}}_{zz}^{{}^{H_0}}}{\mu^{*}}\hfill & \hfill -2{p}^2\beta \hfill \end{array}\right],{\varDelta}_2=\left[\begin{array}{cc}\hfill -2\alpha \hfill & \hfill -\frac{{\overline{\tau}}_{rz}^{{}^{H_1}}}{p{\mu}^{*}}\hfill \\ {}\hfill {\beta}^2+{p}^2\hfill & \hfill \frac{{\overline{\tau}}_{zz}^{{}^{H_0}}}{\mu^{*}}\hfill \end{array}\right] $$

The transient response under laser source illumination can now be obtained by using the inverse Laplace and Hankel transforms as

$$ {u}_z\left(r,t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\Big({\displaystyle \underset{\alpha -i\infty }{\overset{\alpha +i\infty }{\int }}{\overline{u}}_z^{{}^{H_0}}\left(p,s\right){e}^{st}}}ds\Big){J}_0(rp)pdp $$

(10)

### 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*.

Because its rheological property is mainly presented by shear deformation, when the shear deformation in the Kelvin model is only considered, we have

$$ \left\{\begin{array}{c}\hfill {S}_{ij}=2{\mu}_k{e}_{ij}+2{\eta}_k{\overset{.}{e}}_{ij}\hfill \\ {}\hfill \sigma =3Ke\hfill \end{array}\right. $$

(11)

where *μ*
_{
k
}, *η*
_{
k
}, and *K* are shear, viscous, and bulk modulus, respectively.

Combining Eq. (2) with Eq. (10) leads to

$$ \left\{\begin{array}{c}\hfill {\lambda}^{\#}(s)={\mu}_k\left(\frac{K}{\mu_k}-\frac{2}{3}-\frac{2}{3}{K}_{\eta }s\right)\hfill \\ {}\hfill {\mu}^{\#}(s)={\mu}_k\left(1+{K}_{\eta }s\right)\hfill \end{array}\right. $$

(12)

where \( {K}_{\eta }=\frac{\eta_k}{\mu_k} \) is the relaxation time of the visco-elastic medium (its SI unit is *s*), which indicates the time of strain lagging stress and *K*
_{
η
} = 0 represents an elastic medium, that to say, *K*
_{
η
} can describe the characteristics of viscous also. With epoxy as an example, the parameters are the bulk modulus *K* = 7.28 × 10^{9}kg/m. s^{2}, the shear modulus *u*
_{
k
} = 2.37 × 10^{9}kg/m. s^{2}, and the thermal diffusion coefficient *γ* = 0.001 cm^{2}/s. Since the coefficient *K*
_{
η
} for a solidified epoxy is very small, two cases of *K*
_{
η
} = 10^{−10}(0.1*ηs*) and 10^{−9}(1*ηs*) were chosen in our simulation to obtain the approximate attenuation [22]. By using Eqs. (8) and (12), the dispersion curves (phase velocity versus frequency) and attenuation curves (attenuation factor versus frequency) for two cases are plotted in Fig. 2a, b which presents *K*
_{
η
} = 0.1*ηs*, (c) and (d) indicate *K*
_{
η
} = 1*ηs*.

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
} = 1*ηs*, 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
} = 0.1*ηs* 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
} = 1*ηs* is higher by about one order of magnitude than for *K*
_{
η
} = 0.1*ηs* at the same frequency. Thus, the elastic constants could be determined by dispersion (velocity) and the viscous constants by an attenuation curve.

Above research, we assume that the only shear viscosity, if bulk viscosity is also taken into account in the Kelvin model, the constitutive equations can be written as

$$ \left\{\begin{array}{c}\hfill {S}_{ij}=2{\mu}_k{e}_{ij}+2{\eta}_k{\overset{.}{e}}_{ij}\hfill \\ {}\hfill \sigma =3Ke+3\eta \overset{.}{e}\hfill \end{array}\right. $$

(13)

$$ \left\{\begin{array}{c}\hfill {\lambda}^{\#}(s)=K\left(1+{K}_Bs-\frac{2{\mu}_k}{3K}-\frac{2{\eta}_k}{3K}s\right)\hfill \\ {}\hfill {\mu}^{\#}(s)={\mu}_k\left(1+{K}_{\eta }s\right)\hfill \end{array}\right. $$

(14)

where \( {K}_B=\frac{\eta }{K} \), and *η* is the bulk viscosity modulus.

In order to investigate the influence of bulk viscosity, we suppose *K*
_{
η
} = 0 firstly. Let us consider the case of *K*
_{
B
} = 1*ηs*, the calculated dispersion and attenuation curves are showed in Fig. 3. Compared these results with that of *K*
_{
η
} = 1*ηs* and *K*
_{
B
} = 0, it was found that the attenuation and dispersion caused by shear viscosity are much larger than that by bulk viscosity of the same frequency. In general, shear viscosity is greater than bulk viscosity for most of materials [1]. Therefore, bulk viscosity will be ignored, and only the shear viscosity will be taken in account in the following research.

### 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.

The time domain transient response of visco-elastic Rayleigh waves is related to excited source. Here, the laser source function is chosen to be

$$ Q\left(r,t\right)={Q}_0\left[\frac{2}{R^2} \exp \left(-2\frac{r^2}{R^2}\right)\right]\left[\frac{t}{t_0^3} \exp \left(-\frac{t}{t_0}\right)\right]\delta (z) $$

(15)

Its transform is given by

$$ {\overline{Q}}^{H_0}={Q}_0 \exp \left(-\frac{R^2{p}^2}{8}\right)\times \frac{1}{\left(1+{t}_0s\right)} $$

(16)

where *t*
_{0} is the laser pulse rise time (in our calculation *t*
_{0} = 10*ηs*), *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 Fourier-Bessel or Hankel transform is frequently used as a tool for solving numerous scientific problems and becomes very useful in the analysis of wave field. Equation (10) is used to calculate the transient response in stratified half-space. For a multilayered plate, there are an infinite number of singularities for particular frequency values in the integrand of the equations. These values correspond to an infinite number of poles associated with the zeroes of the Rayleigh-like frequency equation that relates frequency and wave number for guided waves in a layered plate. Since all of the poles are simple poles for a layered plate, the integral is carried out along a contour that is not on the imaginary axis so that the singularities can be avoided. Here, we apply the Secada method to inverse Laplace and Hankel transforms [25] to Eq. (9); this method uses an integral representation of Bessel functions for the transform as a weighted integral of Fourier components of the output function, by the means of computer and FFT technology, the inversing transient responses of the displacement can be obtained quickly. The transient responses of the Rayleigh wave at distance *r* = 5 mm and *r* = 7 mm for different *K*
_{
η
} are shown in Figs. 4 and 5. Again, *K*
_{
η
} = 0 implies the case of an elastic body.

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.