Abstract

For the non-destructive inspection of carbon fiber-reinforced plastic (CFRP), lasers can be used to generate ultrasonic waves. It is important to optimize the wavelength of the laser to ensure the intense excitation of a usable propagating mode. Real CFRP components used in the construction of airplanes and automobiles are often coated with several types of resin to protect against weathering. These resin layers change the excitation of the ultrasonic waves. Thus, the optimum laser wavelength may be changed by the coating resin. In this paper, we investigated the excitation of ultrasonic waves in a resin-coated CFRP plate using different laser wavelengths. We conducted experiments to convert the laser wavelength using periodically poled LiNbO3 (PPLN) devices. By injecting mid-infrared laser to a coated sample, we observed excited ultrasonic waves using a laser Doppler vibrometer. We found that transparent resins significantly increase the amplitude of the first-arriving longitudinal wave. Furthermore, when the laser was strongly absorbed in the surface layer, the excitation of longitudinal waves was suppressed. These results were clarified by a one-dimensional model of the thermal generation of ultrasonic waves. We concluded that a laser passing through a resin layer is a viable candidate for the effective inspection of coated CFRP by laser ultrasonic waves.

1 Introduction

Carbon fiber-reinforced plastic (CFRP) is often used in the construction of airplanes and automobiles. For example, CFRP is used for the body and main wing of Boeing 787 aircraft, accounting for approximately 50% of the entire plane [1]. The demand for CFRP continues to grow.

Following an accidental impact, defects such as inner delamination may occur in CFRP. As such damage decreases the strength of the CFRP, it is important to detect defects using non-destructive inspection [2] or structural health monitoring [3].

As an innovative method for non-destructive inspection, laser ultrasonics have received considerable attention [4]. In contrast to conventional inspection using ultrasonic probes, which requires ultrasonic couplants, the laser ultrasonic method can be performed in air because the laser generates ultrasonic waves without contacting the object. Furthermore, the laser-based method is easily applicable to curved sections because the oblique injection of the laser can generate ultrasonic waves. This is unlike the conventional method, in which probes must be moved along curved surfaces.

Various non-destructive inspection devices that use lasers to excite ultrasonic waves have been produced. One of them is a laser ultrasound inspection system (LUIS, iPhoton Solutions Corporation) [5] which detects defects by measuring the longitudinal waves that propagate in the thickness direction and are reflected by the back surface of the object. When delamination has occurred, the waves return earlier than expected. Another device is a laser ultrasonic visualizing inspector (LUVI, Tsukuba Technology Corporation) [68] which is based on changes in the generation and propagation of Lamb waves [9] by defects [10,11]. Abnormalities are discovered by combining results from discrete points in time.

To increase the inspection accuracy, it is important to clarify what wavelengths of laser are suitable for generating ultrasonic waves in CFRP. Currently, LUIS uses a CO2 laser (10.6 µm), whereas LUVI uses an Nd:YAG laser (1.064 µm). It has been reported that laser wavelengths in the mid-infrared (IR) range are suitable [1216], particularly 3.2 µm for CFRP components that have an epoxy resin layer on their surface. This argument is based on the relation between the penetration depth of the laser in an object and the intensity of the ultrasonic waves.

As CFRP components are often coated with resin to protect against weathering, we investigated the changes in the ultrasonic waves generated by different wavelengths of laser for a resin-coated CFRP. The absorption ratio of the laser can vary for different types of resins. Thus, it is possible that the most suitable wavelength will be changed by the coating. In this study, we investigated a urethane resin, which is used in the automobile industry. Using a CFRP plate coated with a urethane resin, we conducted a laser wavelength conversion experiment. In addition, we investigated the effect of the coating layer on the generation of ultrasonic waves using a one-dimensional model.

The remainder of this paper is organized as follows. In Sec. 2, we describe a laser wavelength conversion experiment for a resin-coated CFRP plate. In Sec. 3, to understand the experimental results, we present calculations based on the thermal generation of ultrasonic waves using a one-dimensional model. Finally, Sec. 4 summarizes the conclusions to this study.

2 Laser Wavelength Conversion Experiment

This section describes the results of an experimental investigation into the changes in ultrasonic waves generated by different wavelengths of laser.

Using a prepreg T700SC/2500 (TORAY, Japan), we molded CFRP plates ([0/45/90/−45]4s, 4.3 mm thick) in a hot press. One plate was coated with the urethane resin. First, an epoxy primer coat was applied, and then two urethane resin coats were added. The thickness of each layer was approximately 30, 35, and 45 µm, respectively. The coating layer was transparent to the human eye.

We first measured the transmittance spectra of the urethane resin in the IR region. We used calcium fluoride (CaF2, 25.4 mm diameter, 5 mm thick, WG51050, THOTLABS) as the base material. This was coated by the urethane resin in the same way as the CFRP sample. The transmittance and reflectance spectra of the base material and the coated sample were measured by a Fourier transform IR spectrophotometer (FT-IR-4200, JASCO). The results are shown in Fig. 1. The transmittance of the CaF2 sample was high (∼95%) in the range 1.5–5 µm, whereas that of the sample coated by the urethane resin had a significant dependence on the wavelength: it was 90% at 2 µm and 0% at 3.4 µm. As the reflectance was approximately constant (∼5%) in this region, as shown in Fig. 1(b), the IR rays must have been strongly absorbed by the coating layers at 3.4 µm.

We measured the transmittance spectrum for a thin CFRP plate (0.14 mm thick) formed using one layer of the prepreg. The transmittance was too small (less than 10−3) to be measured precisely in the range 1.5–5 µm.

Taking into account these resin spectra, we performed a laser wavelength conversion experiment. The experimental setup is shown in Fig. 2. The laser source (Quanta-Ray, Spectra-Physics) generated an Nd:YAG laser with a wavelength of 1.064 µm, a beam diameter of approximately 6 mm, a pulse width of 10 ns, and a repetition frequency of 30 Hz. The beam passed through a resonator consisting of a periodically poled LiNbO3 (PPLN) device and two mirrors. Under optical parametric oscillations, the Nd:YAG laser was converted into two different wavelengths. The longer is called the idler, and the shorter is the signal. To shut one of them off, we used a filter. The laser passing through the filter was injected to a test piece, whereby ultrasonic waves were generated due to thermal expansion. We detected out-of-plane displacements in the test piece at an epicenter using a laser Doppler vibrometer (OFV-505, OFV-2570, Polytec). To ensure detection, much of the laser from the Doppler vibrometer must have returned to the machine. Hence, we attached a reflection seal around the epicenter of the test pieces.

The converted wavelength of the laser can be changed using different crystals and mirrors [17]. Using several PPLN devices, we were able to obtain wavelengths ranging from 1.5 to 3.4 µm. The converted wavelength was also slightly affected by changes in temperature. Thus, the pedestal supporting the device was heated up to 50 °C. The effect of raising the temperature changed the wavelength by approximately a few tens of nanometers. Further information on the wavelength conversion to generate mid-IR light is reported in our previous report [17].

The energy per pulse of the converted laser was measured by a powermeter (head: QE12LP-S-MB, monitor: Maestro, Gentec-EO). We adjusted this by a few millijoules so as to generate sufficient displacement for the measurements without visibly damaging the samples.

Figure 3 shows the measured displacements of a bare CFRP plate and a urethane-coated CFRP plate for various wavelengths of laser. The horizontal axis is the time from the injection of the laser, and the vertical axis measures the out-of-plane displacement. Positive displacements indicate expansion, whereas negative ones represent contraction. The displacements were normalized with respect to the pulse energy.

The measured data may have contained errors; for example, the height of the first peak changed slightly in each experiment. Scattering was caused by changes in the beam direction after passing through a PPLN device, beam expansion with distance from the resonator, nonuniform spatial intensity distributions, deviations in the observation point given by the laser Doppler vibrometer from the epicenter, and nonuniform thickness of the resin layers. In the following, we investigate obvious changes that were considered to be greater than the margin of error.

Figure 3(a) shows the Nd:YAG (1.064 µm) results for the bare plate. The first-arriving mode was a longitudinal wave passing through the thickness. This is known as a precursor [18,19]. In Fig. 3(a), this corresponds to a very small bump in the positive direction at around 1.8 µs. After that, there was a depression. This could be caused by the in-plane forces associated with thermal expansion, as has been shown using a point-source [20]. After 2.5 µs, the shear wave reached the epicenter.

Figure 3(b) shows the results for the urethane-coated plate for 1.064 µm. We can see that the precursor was significantly enhanced. Furthermore, it was followed by several oscillations. Positive and negative displacements alternated while attenuating. Thus, the effect of the urethane coating was obvious. This enhancement of the precursor is desirable for non-destructive inspection. In addition, at around 5 µs, three enhanced longitudinal waves could be seen passing through the thickness of the plate.

Note that there was a time lag between the trigger and the injection of the laser in all cases of Fig. 3. Using Fig. 3(b), we can estimate the time lag as follows. We can see that the second longitudinal wave arrived at 5 µs. Since the first longitudinal wave appeared at 1.8 µs, it must have taken approximately 3.2 µs to pass through the double thickness of the plate. Therefore, 1.6 µs was required to propagate from one surface to the other. Hence, the laser was considered to be injected at 0.2 µs.

Figures 3(c)3(j) show the results for the converted laser wavelengths. For the bare CFRP plate, the precursor was very small for the various wavelengths. However, for the urethane-coated plate, the precursor was large, as shown in Figs. 3(d), 3(f), and 3(h). However, the precursor was significantly decreased at 3.4 µm. At this wavelength, the transmittance of the urethane layer was almost 0% (Fig. 1(a)), that is, the laser was strongly absorbed in the urethane layer. Thus, in a urethane-coated plate, the precursor was dependent on the wavelength of the laser. It was confirmed that, when we removed the coating resin by sanding, the excited precursor was similar to that of the bare CFRP plate.

We further investigated changes in the precursor at around 3.4 µm. By controlling the temperature of the crystal, we could slightly change the wavelength of the laser. The results are shown in Fig. 4. In Fig. 4(a), we can see that the amplitude of the precursor changed dramatically with the temperature of the crystal. As the other conditions except for temperature were the same, this difference in the precursor was considered to be mainly caused by changes in the wavelength of the laser. However, except for the longitudinal wave, the displacement was almost unchanged, as shown in Fig. 4(b). In this case, the wavelength of the laser had an effect on the longitudinal modes.

Figure 5 plots the change in amplitude of the precursor as a function of laser wavelength. The laser wavelengths were measured by an optical spectrum analyzer (AQ6376, YOKOGAWA). The converted laser had a spectrum bandwidth of order 0.01 µm around the center wavelength. Figure 5 also shows the transmittance from Fig. 1 for comparison. We can see that a significant reduction in the precursor occurred when there was 0% transmittance. Larger values of the transmittance corresponded to higher amplitudes in the precursor. In terms of the precursor, even very low values of the transmittance were significantly different from the case of 0% transmittance.

In summary, the following experimental facts were found.

1. In the case of a bare CFRP plate, the amplitude of the precursor was small as shown in left panels of Fig. 3.

2. On the other hand, in the case of a resin-coated CFRP plate, the amplitude of the precursor excited by the transparent laser was significantly increased (Figs. 3(b), 3(d), 3(f), and 3(h)).

3. The oscillation after the first positive displacement could be seen for the coated plate (for example, in Fig. 3(b)).

4. The amplitude of the precursor was decreased when the penetration depth of the laser into the coating resin layer was significantly decreased (Figs. 3(j) and 5).

In order to interpret the above experimental phenomena, we investigate longitudinal waves by using a one-dimensional model in Sec. 3.

3 Analysis of Longitudinal Waves Using the One-Dimensional Model

To clarify the effect of a resin layer on the generation of longitudinal waves, this section describes a theoretical investigation using an one-dimensional model. Furthermore, we examine the effect of the wavelength of the laser in terms of the penetration depth. The use of this one-dimensional model was inspired by the work of Dubois et al. [21].

3.1 Description of the One-Dimensional Model.

Figure 6 shows the coordinate system of the one-dimensional model. The Z-axis runs in the thickness direction of the plate. The origin is at the interface between the CFRP and the coating layer. The positive region is an infinite CFRP, and the negative region is the coating layer, which has a thickness of d.

The laser is injected from the negative direction to the plate. Part of the laser is absorbed by the coating layer, and the other part penetrates the CFRP. The intensity profile, I(z), can be expressed as
$I(z)={I0e−βc(z+d)(−d≤z<0)I0e−βcde−βz(0≤z)$
(1)
where I0 is the intensity of the injection laser at the surface of the coating layer, βc is the absorption ratio of the coating layer, and β is that of the CFRP. Here, we have neglected the reflection of the laser.
As in Ref. [21], time variations in the laser intensity, p(t), can be modeled by
$p(t)=tτ2e−(t/τ)$
(2)
where τ characterizes the pulse duration.
In absorbing the laser, the material becomes heated. The heating ratio, q, is obtained by
$q(z,t)=−dIdzp(t)={βcI0e−βc(z+d)p(t)(−d≤z<0)βI0e−βcde−βzp(t)(0≤z)$
(3)
As the resin and CFRP have significantly lower thermal conductivity than metals, this term can be neglected. In this situation, the change in temperature, T, is given by
$dTdt=qρc(orqρccc)$
(4)
where ρ is the density of CFRP, c is the specific heat of CFRP, ρc is the density of the coating layer, and cc is the specific heat of the coating layer. By integrating the above formula with time, we obtain the following expression:
$T(z,t)={I0e−βcdρcccβce−βcz∫0tp(t′)dt′(−d≤z<0)I0e−βcdρcβe−βz∫0tp(t′)dt′(0≤z)$
(5)
An increase in temperature causes thermal expansion, which is the source of ultrasonic waves. A general expression for the source term is given by
$−Cijklαkl∂T∂xj$
(6)
where Cijkl are the components of the fourth-order stiffness tensor and αkl are those of the second-order linear expansion coefficient tensor.
The fundamental equation of the CFRP is given by
$ρ∂2u(z,t)∂t2=C33∂2u(z,t)∂z2−(C31α1+C32α2+C33α3)∂T∂z(0≤z)$
(7)
where u denotes displacement in the z-direction, C3j(j = 1, 2, 3) are the elastic constants of CFRP, and αi(i = 1, 2, 3) are linear thermal expansion ratios. Similarly, the fundamental equation in the coating layer is
$ρc∂2uc(z,t)∂t2=(λc+2μc)∂2uc(z,t)∂z2−(3λc+2μc)αc∂T∂z(−d≤z<0)$
(8)
where uc denotes displacement in the z-direction, λc and μc are the Lamé constants of the coating layer, and αc is the linear thermal expansion ratio.
The boundary conditions are the continuities of the displacement and stress at the interface:
$uc(0,t)=u(0,t)$
(9)

$(λc+2μc)∂uc(0,t)∂z−(3λc+2μc)αcT(−0,t)=C33∂u(0,t)∂z−(C31α1+C32α2+C33α3)T(+0,t)$
(10)
In addition, as the stress is free at the surface of the coating layer
$(λc+2μc)∂uc(−d,t)∂z−(3λc+2μc)αcT(−d,t)=0$
(11)

The solution of the above equations under the boundary conditions is derived in the  Appendix.

3.2 Smallness of the Longitudinal Wave Excited in a Bare CFRP Plate.

We first investigated the effect of penetration depth on the generation of longitudinal waves. Penetration depth is the reciprocal of the absorption ratio, and its value varies with the laser wavelength.

We first considered the injection of laser to bare CFRP, as shown in Fig. 7(a). The calculated longitudinal waves at different penetration depths are presented in Fig. 7(b). These calculations used the elastic parameters of CFRP that were previously measured in our laboratory; the other thermal parameters were measured by a vendor. We can see that deeper penetration depths resulted in larger longitudinal waves [21]. As the penetration depth was very small (less than a few micrometers), the precursor of bare CFRP was very small as shown in the left panels of Fig. 3. Note that the displacement of the generated longitudinal wave was positive. No negative displacements were produced.

3.3 Enhancement of the Precursor by Coating.

Next, we considered the effect of the coating layer on the generation of ultrasonic waves. Here, we investigated a transparent coating layer. The laser was not absorbed by the coating layer but was completely absorbed in the CFRP. Changing the thickness of the coating layer allowed us to calculate the properties of the ultrasonic waves. In the following, calculations were performed using typical values of urethane resin parameters.

Figure 8(a) shows the infinite thickness case. When the laser was injected through the coating layer, the CFRP near the interface expanded due to the absorption of laser energy. As the CFRP was in contact with the coating layer, the CFRP received a reaction force from the coating layer at the interface. The reason of the enhancement of the longitudinal wave by coating was this reaction force. The CFRP tended to move in the positive direction from the interface, while the coating resin moved in the negative direction. The displacement propagated at the speed of the longitudinal waves. Therefore, at an observation point in the CFRP, a positive shift can be observed after the arrival time of the longitudinal waves. Similarly, a negative shift propagated at the speed of the longitudinal waves in the coating layer. The magnitude of the shift in the CFRP (x > 0) is
$−e−βxA+(v/vc)R1+(v/vc)RA$
(12)
where
$A=C31α1+C32α2+C33α3C33I0ρc$
(13)

$R=λc+2μcC33$
(14)

$v=C33ρ$
(15)
and
$vc=λc+2μcρc$
(16)

The final two equations give the magnitude of velocity in the CFRP and in the coating layer. The contribution of the first term of Eq. (12) is restricted to within the penetration depth of the laser. Hence, at points far from the interface, the shift is effectively given by the second term. Notably, this is independent of β. No matter how small the penetration depth 1/β is, a definite shift can occur.

3.4 Oscillation of the Arriving Wave Seen in a Coated Plate.

Figure 8(b) shows the results for a coating thickness of 1000 µm. Once the shift had propagated to the surface of the coating layer from the interface, it was reflected back toward the interface. A point in the inner region of the layer was further pulled toward the surface. When the shift arrived at the interface, it partly propagated into the CFRP. Thus, each point of the CFRP was displaced in the negative direction. The time interval between the first and second shifts corresponded to the time required for the longitudinal waves to propagate through the double thickness of the coating layer. The displacement is given by
$−e−βxA+(v/vc)R1+(v/vc)RA−2(v/vc)R(1+(v/vc)R)2A$
(17)

This quantity can be negative. Thus, although negative displacement could not arise in bare CFRP, as discussed above, in the presence of a coating layer, it may occur. This was consistent with the experimental measurements. Thus, it was clarified that the negative displacement is caused by the reflected wave at the surface of the resin coating.

Figure 8(c) shows the results for a thickness of 100 µm. The time intervals of the reflections were shorter because of the thinner coating layer, and several shifts could be seen at shorter intervals. In general, the displacement after the arrival of n shifts (n ≥ 2) is given by
$−e−βxA+(v/vc)R1+(v/vc)RA−2(v/vc)R(1+(v/vc)R)2{1−1−(v/vc)R1+vvc(v/vc)R+(1−(v/vc)R1+(v/vc)R)2−⋯+(−1)n−2(1−(v/vc)R1+(v/vc)R)n−2}A$
(18)

Thus, we observed many oscillations, which gradually decayed with time. This was in agreement with the experimental observation as shown in, for example, Fig. 3(b).

Figure 9 presents the results for thinner coating layers. As the intervals of reflection in these coating layers were small, the second shift canceled the first positive shift before the displacement reached its maximum value. Therefore, the amplitude became smaller as the coating layer became thinner. We experimentally observed that sanding the coating layer using a piece of sandpaper reduced the amplitude of the first-arriving longitudinal wave.

3.5 Dependence on the Penetration Depth of the Coating Layer.

In subsequent investigations, we considered absorption by the coating layer. As the laser was absorbed by both the coating resin and the CFRP, ultrasonic waves were generated by both materials. Changing the laser wavelength affected the ratio of absorption by the coating layer to that by the CFRP.

In Fig. 10, we show the changes in the first longitudinal wave. For smaller penetration depths, the amplitude of the longitudinal wave was smaller. This was consistent with experimental results (Fig. 5). In this calculation, we could see a delay in the arrival time for smaller penetration depths. In the experiments, the laser wavelength ranged within a band, including components that gave higher and lower penetration depths. Averaging the components obscured the delay in the rising time.

By changing the penetration depth of the coating layer, we could identify the decomposition of contributions to the first longitudinal wave (Fig. 11). For larger penetration depths, because the CFRP absorbed most of the laser, the majority of the longitudinal waves originated from the CFRP. For smaller penetration depths (<200 µm), the contribution from the coating layer was dominant. Because the two origins of the ultrasonic waves compensated each other, the generated ultrasonic wave was of approximately the same order for significantly different penetration depths ranging from 20 to 10,000 µm. When the penetration depth of the coating layer was very small (a few micrometers), the amplitude decreased. This tendency was consistent with experimental observations.

Our calculation showed that the contribution from the thermal expansion of CFRP was decreased around a few hundreds of micrometer. However, the experimental data showed that the amplitude of precursor was decreased around a few tens of micrometer. Therefore, we can conclude that the precursor receives the contribution from the thermal expansion of the coating layer.

4 Conclusion

We conducted laser wavelength conversion experiments and investigated ultrasonic wave generation in a resin-coated CFRP plate. To understand the experimental results, we investigated the generation of longitudinal ultrasonic waves using a one-dimensional model.

We found that CFRP coated by a transparent resin can generate a stronger longitudinal wave than bare CFRP. However, when the laser was strongly absorbed by the coating layer, the amplitude of the generated ultrasonic waves was significantly reduced. In particular, our urethane sample strongly absorbed 3.4 µm laser. Hence, wavelengths of 3.4 µm should be avoided. As the transmittance varies in different resins, the most suitable laser wavelength will be strongly dependent on the coating resin.

Other than the urethane resin coating used in the automobile industry, we performed experiments with other types of coatings, such as fluorine resin. The results indicated that different wavelengths of laser did not produce significant differences in the generated ultrasonic waves. This was because the penetration depth does not change much in the mid-infrared range.

The calculations in this study used a particular function describing the time variation of laser intensity and many resin parameters. Improving these factors through detailed measurements would enhance the reliability of our calculations.

During the experiments, we found that attaching several layers of scotch tape (thickness of approximately 40 µm) to the bare CFRP plate increased the first-arriving longitudinal wave. This was because the scotch tape played the role of a coating layer. This simple method for enhancing the ultrasonic waves may be applicable to non-destructive inspection.

Future work will investigate the effect of the laser wavelength on the Lamb waves. This is important for further development of laser ultrasonic visual inspection techniques based on Lamb waves.

Acknowledgment

We would like to thank Associate Professor Ashihara and his PhD student Morichika in the Institute of Industrial Science at the University of Tokyo for their IR measurements. We also thank Mr. Tamaki and Mr. Ohta at Maruyama Corporation for coating samples.

Funding Data

• This work is financially supported by the Consortium for Manufacturing Innovation and the cooperation research program of the Institute for Molecular Science.

Appendix: Solution of the One-Dimensional Model

We here introduce or reproduce the following notation:
$A≡C31α1+C32α2+C33α3C33I0ρc(A1)$
(19)

$Ac≡3λc+2μcλc+2μcαcI0ρccc(A2)$
(20)
and
$R≡λc+2μcC33(A3)$
(21)
where A is related to generation in the CFRP and Ac relates to the coating layer. In addition, the magnitude of the velocity of longitudinal waves in the CFRP is depicted as
$v=C33ρ(A4)$
(22)
and that in the coating layer is depicted as
$vc=λc+2μcρc(A5)$
(23)
The fundamental equations can be solved, for example, using a Laplace transformation. Using a unit step function θ(t), a solution in the region x ≥ 0 can be expressed as follows:
$u(x,t)=θ(t){{121(1+βvτ)2e−βx+βvt+121(1−βvτ)2e−βx−βvτ−(βvτ)βvt1−β2v2τ2e−βx−(t/τ)+β2v2τ2(β2v2τ2−3)(1−β2v2τ2)2e−βx−(t/τ)−e−βx}Ae−βcd$

$+θ(t−xv){−121(1+βvτ)2e−βx+βvτ+121−(v/vc)R1+(v/vc)R1(1−βvτ)2e−βv(t−(x/v))+(v/vc)R1+(v/vc)R(βvτ)βv1−β2v2τ2(t−xv)e−((t−(x/v))/τ)−11+(v/vc)Rβv1−β2v2τ2(t−xv)e−((t−(x/v))/τ)−(v/vc)R1+(v/vc)Rβ2v2τ2(β2v2τ2−3)(1−β2v2τ2)2e−((t−(x/v))/τ)−11+(v/vc)R2βvτ(1−β2v2τ2)2e−((t−(x/v))/τ)+(v/vc)R1+(v/vc)R}Ae−βcd$

$+θ(t−xv)(v/vc)R1+(v/vc)R{βcvc1+βcvcτ(t−xv)e−((t−(x/v))/τ)+βcvcτ(βcvcτ+2)(1+βcvcτ)2e−((t−(x/v))/τ)−1+1(1+βcvcτ)2eβcvc(t−(x/v))}Ace−βcd$

$+∑n=0∞θ(t−xv−(2n+1)dvc)(v/vc)R1+(v/vc)R(−1)n(1−(v/vc)R1+(v/vc)R)n×{−2βcvc1−βc2vc2τ2(t−xv−(2n+1)dvc)e−((t−(x/v)−(((2n+1)d)/vc))/τ)−4βcvcτ(1−βc2vc2τ2)2e−t−(x/v)−(((2n+1)d)/vc)τ+1(1−βcvcτ)2e−βcvc(t−(x/v)−(((2n+1)d)/vc))−1(1+βcvcτ)2eβcvc(t−(x/v)−(((2n+1)d)/vc))}Ac$

$+∑n=1∞θ(t−xv−2ndvc)2(v/vc)R(1+(v/vc)R)2(−1)n−1(1−(v/vc)R1+(v/vc)R)n−1{1(1−βvτ)2e−βv(t−(x/v)−(2nd/vc))−βv1−βvτ(t−xv−2ndvc)e−t−(x/v)−(2nd/vc)τ+βvτ(βvτ−2)(1−βvτ)2e−t−(x/v)−(2nd/vc)τ−1}Ae−βcd$

$+∑n=1∞θ(t−xv−2ndvc)(v/vc)R1+(v/vc)R(−1)n−1(1−(v/vc)R1+(v/vc)R)n−1×{−1−(v/vc)R1+(v/vc)Rβcvc1+βcvcτ(t−xv−2ndvc)e−t−(x/v)−(2nd/vc)τ+βcvc1−βcvcτ(t−xv−2ndvc)e−t−(x/v)−(2nd/vc)τ−21+(v/vc)Rβc2vc2τ2(βc2vc2τ2−3)(1−βc2vc2τ2)2e−t−(x/v)−(2nd/vc)τ+2(v/vc)R1+(v/vc)R2βcvcτ(1−βc2vc2τ2)2e−t−(x/v)−(2nd/vc)τ+21+(v/vc)R−1(1−βcvcτ)2e−βcvc(t−(x/v)−(2nd/vc))−1−(v/vc)R1+(v/vc)R1(1+βcvcτ)2eβcvc(t−(x/v)−(2nd/vc))}Ace−βcd(A6)$
(24)
Note that the above formula is finite at the limits βvτ → 1 and βcvcτ → 1. The term
$θ(t)121(1+βvτ)2e−βx+βvt(A7)$
(25)
which diverges for large t is canceled by
$θ(t−xv){−121(1+βvτ)2e−βx+βvτ}(A8)$
(26)
for $t≥xv$. In this way, the displacement is finite. Furthermore, the displacement can be shown to be continuous.
Similarly, a solution in the region x < 0 can be given as follows:
$uc(x,t)=θ(t){−(βcvcτ)βcvc1−βc2vc2τ2te−(t/τ)−βcx+βc2vc2τ2(βc2vc2τ2−3)(1−βc2vc2τ2)2e−(t/τ)−βcx−e−βcx+121(1+βcvcτ)2eβcvc(t−(x/vc))+121(1−βcvcτ)2e−βcvc(t+(x/vc))}Ace−βcd$

$+∑n=0∞θ(t+xvc−2ndvc)11+(v/vc)R(−1)n(1−(v/vc)R1+(v/vc)R)n{1(1−βvτ)2e−βv(t+(x/vc)−(2nd/vc))−βv1−βvτ(t+xvc−2ndvc)e−t+(x/vc)−(2nd/vc)τ+βvτ(βvτ−2)(1−βvτ)2e−t+(x/vc)−(2nd/vc)τ−1}Ae−βcd$

$+∑n=0∞θ(t+xvc−2ndvc)11+(v/vc)R(−1)n(1−(v/vc)R1+(v/vc)R)n×{(βcvcτ)βcvc1−βc2vc2τ2(t+xvc−2ndvc)e−t+(x/vc)−(2nd/vc)τ+vvcRβcvc1−βc2vc2τ2(t+xvc−2ndvc)e−t+(x/vc)−(2nd/vc)τ−βc2vc2τ2(βc2vc2τ2−3)(1−βc2vc2τ2)2e−t+(x/vc)−(2nd/vc)τ+vvcR2βcvcτ(1−βc2vc2τ2)2e−t+(x/vc)−(2nd/vc)τ+1+(1−vvcR)(−12)1(1+βcvcτ)2eβcvc(t+(x/vc)−(2nd/vc))+(1+vvcR)(−12)1(1−βcvcτ)2e−βcvc(t+(x/vc)−(2nd/vc))}Ace−βcd$

$+∑n=0∞θ(t+xvc−(2n+1)dvc)(−1)n(1−(v/vc)R1+(v/vc)R)n+1×{βcvc1−βc2vc2τ2(t+xvc−(2n+1)dvc)e−t+(x/vc)−(((2n+1)d)/vc)τ+2βcvcτ(1−βc2vc2τ2)e−t+(x/vc)−(((2n+1)d)/vc)τ+121(1+βcvcτ)2eβcvc(t+(x/vc)−(((2n+1)d)/vc))−121(1−βcvcτ)2e−βcvc(t+(x/vc)−(((2n+1)d)/vc))}Ac$

$+∑n=0∞θ(t−xvc−2(n+1)dvc)11+(v/vc)R(−1)n(1−(v/vc)R1+(v/vc)R)n×{1(1−βvτ)2e−βv(t−(x/vc)−((2(n+1)d)/vc))−βv1−βvτ(t−xvc−2(n+1)dvc)e−t−(x/vc)−((2(n+1)d)/vc)τ+βvτ(βvτ−2)(1−βvτ)2e−t−(x/vc)−((2(n+1)d)/vc)τ−1}Ae−βcd$

$+∑n=0∞θ(t−xvc−2(n+1)dvc)(−1)n(1−(v/vc)R1+(v/vc)R)n×{11+(v/vc)R(βcvcτ)βcvc1−βc2vc2τ2(t−xvc−2(n+1)dvc)e−t−(x/vc)−((2(n+1)d)/vc)τ+(v/vc)R1+(v/vc)Rβcvc1−βc2vc2τ2(t−xvc−2(n+1)dvc)e−t−(x/vc)−((2(n+1)d)/vc)τ−11+(v/vc)Rβc2vc2τ2(βc2vc2τ2−3)(1−βc2vc2τ2)2e−t−(x/vc)−((2(n+1)d)/vc)τ+(v/vc)R1+(v/vc)R2βcvcτ(1−βc2vc2τ2)2e−t−(x/vc)−((2(n+1)d)/vc)τ+11+(v/vc)R−121−(v/vc)R1+(v/vc)R1(1+βcvcτ)2eβcvc(t−(x/vc)−((2(n+1)d)/vc))−121(1−βcvcτ)2e−βcvc(t−(x/vc)−((2(n+1)d)/vc))}Ace−βcd$

$+∑n=0∞θ(t−xvc−(2n+1)dvc)(−1)n(1−(v/vc)R1+(v/vc)R)n×{−βcvc1−βc2vc2τ2(t−xvc−(2n+1)dvc)e−t−(x/vc)−(((2n+1)d)/vc)τ−2βcvcτ(1−βc2vc2τ2)2e−t−(x/vc)−(((2n+1)d)/vc)τ−121(1+βcvcτ)2eβcvc(t−(x/vc)−(((2n+1)d)/vc))+121(1−βcvcτ)2e−βcvc(t−(x/vc)−(((2n+1)d)/vc))}Ac(A9)$
(27)

It can be shown that uc(x, t) is finite at the limits βvτ → 1 and βcvcτ → 1. The continuity of uc(x, t) can also be shown.

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