## Abstract

Laser ultrasonics using guided waves excited by laser irradiation has attracted attention as an efficient nondestructive inspection method for carbon fiber-reinforced plastic (CFRP) composite laminates. In this article, to clarify the properties of laser-excited Lamb waves, we investigate the power flow of Lamb waves in an anisotropic CFRP laminate. The temperature rising caused by laser absorption is analytically calculated, and the derived thermal force is input to a finite element model to simulate the generation of Lamb waves. It is found that the power flow has an obvious directivity in a quasi-isotropic CFRP laminate. The power flow is smaller in the direction parallel to the carbon fibers of the surface layer and is larger in the perpendicular direction, but the maximum is in a slightly different direction; this result was validated by an experiment. Such directivity pattern is determined by two factors: the distribution of thermal force and the stacking sequence of CFRP laminates. Moreover, we investigate the different simulation models (metal-coated CFRP and cross-ply CFRP) to discuss the influence of each factor on the distribution of power flow separately.

## 1 Introduction

Carbon fiber-reinforced plastic (CFRP) is a composite material formed by embedding carbon fibers in a resin matrix as reinforcement. As an advanced composite material, CFRP is widely used in aerospace, civil, sports, and automotive and other industries due to its characteristics of high tensile strength, large modulus, and low density [1]. When CFRP structures are applied on transportation vehicles, these vehicles are likely to suffer from barely visible impact damages [2]. These damages can cause significant degradation in performance and serious safety risk in daily use.

Therefore, nondestructive inspection (NDI) is essential to detect these damages and assure the structure health [3]. Among all the NDI methods, laser ultrasonics has received more attention because it can be operated in a noncontact way and is capable of inspection on components with arbitrary curved surfaces [4]. When ultrasonic waves are excited by laser in a plate structure, a guided wave called Lamb wave is generated by the continuous reflection of longitudinal waves and shear waves from the upper and lower surfaces of the material [5]. Recently, a technique was developed to visualize the propagation of Lamb waves using waveform data collected by laser scanning [6,7]. The device based on the technique is portable and advantageous in cost. To improve the reliability of defect inspection by such laser-ultrasonic devices, the behavior of excited laser-ultrasonic waves must be carefully investigated.

Through many studies, the understanding of laser ultrasonics has been progressed. The generation of laser ultrasonics is based on the laser absorption by the surface of CFRP. By laser incidence, the irradiated area rapidly heats up. As a result, the rapid local thermal expansion appears and the thermal force thus generates ultrasonic waves, which is called a thermoelastic mechanism to excite ultrasonic waves. Within the plate structure, the laser-excited Lamb waves propagate in all directions from the irradiated area. Previous researchers [8–10] have used a numerical approach based on the finite element method (FEM) for the investigation of laser-generated Lamb waves and discussed the amplitude and frequency under different conditions.

In our previous study [11], a simulation method with a combination of analytical method and FEM for anisotropic CFRPs has been established. Our findings indicated that, even with a vertical incidence of laser to a quasi-isotropic CFRP, the peak-to-peak displacement of Lamb waves exhibited a difference in each propagation direction. Specifically, we observed larger displacements in the direction perpendicular to the carbon fiber of the surface layer (90 deg) compared to the direction parallel to the fiber (0 deg). However, the maximum value did not occur precisely at 90 deg. This phenomenon has implications for practical inspection because the detectability of the defects is dependent on the amplitude of Lamb waves. Thus, we investigate the factors that contribute to this directivity by considering the power flow of Lamb waves. Our research aims to identify the factors that influence the directivity pattern of Lamb waves generated in CFRP laminates.

This article is organized as follows. In Sec. 2, the power flow distribution of laser-excited Lamb waves is investigated by analysis and FEM. In Sec. 3, two factors contributing to the distribution are examined. In Sec. 4, the validity of both the simulation methodology and its results was verified through experiment. In Secs. 5 and 6, the influence of each factor on the power flow distribution is investigated by using models of metal-coated and cross-ply CFRPs, respectively. Section 7 is devoted to conclusions.

## 2 Simulation of Lamb Waves

In this section, we explain the simulation method constructed in the previous study [11]. The simulation has three steps. First, the temperature rising in CFRP due to the absorption of a laser is calculated based on a heat conduction equation. Second, the thermal forces derived from the temperature fields are calculated. Third, by introducing the thermal force into an FEM model, the generation and the propagation of Lamb waves are simulated. As a result, the directivity pattern is obtained.

### 2.1 Temperature Distribution of CFRP.

First, we need to specify the model of a pulsed laser. In this model, we set the *x*-axis of Cartesian coordinates along the carbon fiber orientation of the CFRP surface layer (Fig. 1). The material is in the region of *z* ≤ 0, and *z* = 0 is the surface of the material.

*f*(

*x*,

*y*) of the laser intensity on

*x*–

*y*plane is assumed as a Gaussian function with the radius of a laser beam

*d*:

*z*direction, the energy density of the laser attenuates exponentially:

*γ*is the absorption coefficient. When the penetration depth

*ɛ*is defined as the depth where the laser intensity decreases to 1/

*e*of the intensity on the surface,

*ɛ*is determined to be 1/

*γ*.

*p*(

*t*) is approximated by the following expression:

In Eq. (3), *τ* indicates the rise time of the laser pulse.

*I*

_{0}(W/m

^{2}). The heat distribution

*q*(W/m

^{3}) due to the absorption of the laser is expressed as

*z*because the laser energy is reduced by $g(z)\u2212g(z+dz)=\u2212(dg(z)dz)dz$. The total absorbed pulse energy

*E*is determined by integrating Eq. (5):

*f*(

*x*,

*y*),

*g*(

*z*), and

*p*(

*t*) specified above:

### 2.2 Temperature Distribution of CFRP.

Table 1 shows the parameters of the laser beam used in temperature calculation. The CFRP considered in this research is T700SC/2500 (Toray). The physical properties of a single layer of the CFRP are listed in Table 2. Here, axis 1 is the direction of carbon fibers.

Absorbed pulse energy E | Rise time | Beam radius d |
---|---|---|

6.0 mJ | 3.5 ns | 2.0 mm |

Absorbed pulse energy E | Rise time | Beam radius d |
---|---|---|

6.0 mJ | 3.5 ns | 2.0 mm |

Density ρ (kg/m^{3}) | 1540 | |

Stiffness coefficient C (_{IJ}I, J = 1, 2, …, 6) (GPa) | C_{11} | 133 |

C_{12}, C_{13} | 4.98 | |

C_{23} | 5.34 | |

C_{22}, C_{33} | 10.7 | |

C_{44} | 2.7 | |

C_{55}, C_{66} | 4.8 | |

Specific heat c (J/kg/K) | 923 | |

Heat conduction coefficient K (_{i}i = 1, 2, 3) (W/m/K) | K_{1} | 60 |

K_{2}, K_{3} | 0.59 | |

Thermal expansion coefficient α (_{i}i = 1, 2, 3) (1/K) | α_{1} | 4.8 × 10^{−7} |

α_{2}, α_{3} | 4 × 10^{−5} |

Density ρ (kg/m^{3}) | 1540 | |

Stiffness coefficient C (_{IJ}I, J = 1, 2, …, 6) (GPa) | C_{11} | 133 |

C_{12}, C_{13} | 4.98 | |

C_{23} | 5.34 | |

C_{22}, C_{33} | 10.7 | |

C_{44} | 2.7 | |

C_{55}, C_{66} | 4.8 | |

Specific heat c (J/kg/K) | 923 | |

Heat conduction coefficient K (_{i}i = 1, 2, 3) (W/m/K) | K_{1} | 60 |

K_{2}, K_{3} | 0.59 | |

Thermal expansion coefficient α (_{i}i = 1, 2, 3) (1/K) | α_{1} | 4.8 × 10^{−7} |

α_{2}, α_{3} | 4 × 10^{−5} |

When the laser illuminates the CFRP surface, the laser will pass through the very thin layer of epoxy resin on the surface to reach the carbon fibers directly, and the first layer of carbon fibers closest to the surface will absorb most of the laser energy (part of it will be reflected) [12]. Therefore, the penetration depth of laser into CFRP can be regarded as the diameter of a carbon fiber. In general, the diameter of a carbon fiber of CFRP is 5–10 *μ*m, and the diameter of carbon fiber used in this research is 7 *μ*m.

Figure 2 shows the temperature distribution on the surface plane (*z* = 0) based on Eq. (7). The temperature reaches its maximum at the center of the laser beam (*x* = 0, *y* = 0) and then decreases exponentially in all directions, and the region of temperature rising is only in the vicinity of the laser absorption region.

### 2.3 Thermal Force Distribution in CFRP.

Temperature rising causes thermal force, which generates ultrasonic waves. In this subsection, we calculate the thermal force from the temperature distribution.

*b*(

_{i}*i*= 1, 2, and 3) is expressed as

*C*(

_{ijkl}*i*,

*j*,

*k*,

*l*= 1, 2, and 3) is the stiffness coefficient tensor, and $ui$ is the displacement.

*C*and

_{ijkl}*α*. The surface force caused by temperature rising is given by

_{kl}*n*(

_{j}*j*= 1, 2, 3) is the normal vector to the surface. As discussed in the previous study [11], the total force along

*z*direction can be neglected due to the cancelation between the surface force (Eq. (10)) and the body force (Eq. (9)). Thus, we need to calculate only the in-plane force. We define an integrated in-plane force $bi\u2032(i=1,2,3)$ as the integration of

*b*along the

_{i}*z*direction. Since the depth of the temperature rising is much smaller than the mesh size Δ

*z*of FEM model, the integral along the depth direction within one grid is equal to the integral to infinity:

Figure 3(a) shows the magnitude of the integrated in-plane force $(b\u2032x)2+(b\u2032y)2$ around the laser irradiation point. Notably, the force is not axially symmetrical around the laser center. Instead, it exhibits larger forces in the *y* direction and smaller forces in the *x* direction. For instance, consider two points where the surface force varies with time: one located on the *x*-axis (*x* = 1.5 mm, *y* = 0 mm) and another on the *y*-axis (*x* = 0 mm, *y* = 1.5 mm). As depicted in Fig. 3(b), it can be observed that the point on the *y*-axis exhibits a larger magnitude at all time instances. This directional characteristic of the force could potentially induce directional properties in the waves it excites.

### 2.4 Results of Simulation.

The FEM model of the CFRP laminate has dimensions of 200 × 200 × 1.104 mm^{3} and is stacked up by eight plies of unidirectional CFRP prepreg with a sequence of [0/45/90/−45]_{s}. We use the commercial FEM software comwave (ITOCHU Techno-Solutions Corporation, Tokyo, Japan) to calculate the wave propagation from the thermal force. Cuboid elements are used with the mesh size of Δ*x* = Δ*y* = 0.15 mm and Δ*z* = 0.138 mm (equal to the thickness of a single layer). To ensure sufficient accuracy of a wave propagation analysis, the wavelength should be larger than 10 nodes, and the Courant number is set to be 0.8.

To solve Eq. (8) with FEM, we input the thermal force (Eqs. (11) and (12)) to the elements (the pink points at the center of the plate in Fig. 4).

As shown in Fig. 4, the circular arrayed points are set for receiving ultrasonic waves. The receiving points are located on the top surface at the same interval of 15 deg on the circle with a radius of 5 cm, whose center is the center of the laser irradiation point. *η* is the angle of the receiving points from the *x*-axis.

To investigate the power flow of the waves passing through the receiving point, we examine the velocity waveform at each point. We focus on the out-of-plane velocity, because, in our experiment, we utilized a single laser Doppler vibrometer, which measures the out-of-plane velocity. Figure 5 shows the out-of-plane velocity (*z*-velocity) of the waves propagating along the *x*-axis (*η* = 0 deg) and the *y*-axis (*η* = 90 deg). The first waves arriving at approximately 7 *μ*s are S0 modes, and the successive waves arriving at approximately 30 *μ*s are A0 modes. As the main vibration direction of the S0 mode is in-plane, the strength of the S0 mode in the out-of-plane direction is much smaller than that of the A0 mode.

### 2.5 Directivity of Power Flow.

*η*= 0 deg and

*η*= 90 deg), indicating different power flows for different directions. First, the sum of squared velocity in the

*z*direction (SSV) is defined as

*V*(

*n*) represents the out-of-plane velocity at each sampling time point. The physical quantity

*V*(

*n*) is chosen because, in the experiment, the vibration velocity along the

*z*direction is measured. Waveforms from 0 to 100

*μ*s are calculated in this simulation, where the sampling frequency is 84.45 MHz (

*N*= 8445). SSVs at different propagation directions calculated by ComWAVE are plotted in Fig. 7 (the maximum value of SSVs is taken as 1 for normalization). We can find the general trend that the SSV is smaller in the directions parallel to the carbon fiber of the surface layer (

*η*= 0 deg) and larger in the directions perpendicular to the carbon fiber (

*η*= 90 deg). It is noticed that the maximum value does not appear exactly in the

*y*direction (

*η*= 90 deg) but is slightly shifted toward a lower angle (

*η*= 75 deg), thereby breaking the symmetry of the power flow distribution with respect to the

*y*-axis (symmetry about 90 deg in Fig. 7).

*i*th direction is

*σ*(

_{ij}*i*,

*j*= 1, 2, 3) is the elastic stress. Using dispersion calculator (an open-source software developed by Dr. Armin Huber [13] for the calculation of dispersion curves and mode shapes of guided waves), we can calculate the distribution of the power flow

*P*

_{1},

*P*

_{2}, and

*P*

_{3}along the thickness direction. (According to the waveform shown in Figs. 5 and 9, because the energy of the stronger A0 mode is concentrating around 10 kHz, the calculation here is based on A0 mode at 10 kHz.) The directions of

*P*

_{1},

*P*

_{2}, and

*P*

_{3}are shown in Fig. 6. Note that axes 1, 2, and 3 are taken differently according to the propagation directions (

*η*).

The values of *P*_{2} and *P*_{3} are near zero, with only the in-plane *P*_{1} along the wavefront direction being significant. Considering that the magnitude of *P*_{1} varies in the thickness direction, we integrate *P*_{1} along the thickness direction and obtain *P*. As shown in Fig. 7, the distribution trend of power flow *P* of A0 mode is consistent with that of SSV. Therefore, the in-plane power flow can be estimated by SSV, which is experimentally measurable. Since it is difficult to measure the magnitude of power flow in actual inspection, it is beneficial that the distribution of power flow can be obtained directly by measuring the out-of-plane velocity.

## 3 Determining Factors of Directivity

In this section, we will discuss the factors influencing the pattern of the angular distribution of SSV. Given that the excitation of Lamb wave originates from the thermal force near the plate surface as calculated in Sec. 2, it is logical to deduce a direct bearing of the scale of thermal force on the amplitude of Lamb waves. As illustrated in Fig. 8, in accordance with Eqs. (11) and (12), the thermal force in the *y* direction (near 90 deg) considerably outweighs that in the *x* direction (near 0 deg and 180 deg). A comparison of Figs. 7 and 8 further reveals that the overall trend in SSV distribution also significantly favors the *y* direction over the *x* direction. This substantiates a strong correlation between these two quantities, suggesting that a greater thermal force indeed corresponds to a larger power of the Lamb waves.

If the thermal force is the sole determiner impacting the distribution of power flow, there should be a perfect alignment between the direction of maximum force and that of the maximum power flow. However, Fig. 8 shows that the direction of the maximum force is at the *y* direction, while Fig. 7 shows that the direction of the maximum power flow deviates from the *y* direction. The inconsistency between them shows that there are additional factors affecting the power flow distribution besides the thermal force distribution.

The conceivable second factor is the way of stacking up the CFRP laminate because the stacking sequence of CFRP laminates can also influence power propagation, potentially leading to directional power focusing. Previously, the focusing of Lamb waves in anisotropic plates was studied by Chapuis et al. [14] in the case of lead zirconate titanate (PZT) actuation.

**k**, and the magnitude is the reciprocal of the magnitude of phase velocity

*V*. Thus, the definition is

_{p}*ω*is the wave frequency and

*k*is the magnitude of

**k**. The power flow direction of Lamb waves in the laminate can be determined by drawing the slowness curve as a function of the propagation direction

*η*. The direction of the group velocity

**V**

*, in other words, the direction of the power flow, is normal to the slowness curve [15]:*

_{g}Thus, the power flow can be inconsistent with the direction of wave number in anisotropic materials.

To draw the slowness curve, the mode and frequency of the wave must first be determined. To understand the dominant component of waves (vibrating in the out-of-plane direction) traveling along *η* = 0 deg in Fig. 5, we use the fast Fourier transform (FFT) to draw a spectrum and find out in which frequency band the signal amplitude is mainly concentrated. As shown in Fig. 9, it is found that the dominant frequency components range from 0 to 100 kHz.

Since 10 kHz is the frequency at the highest point in the spectrum (Fig. 9), we draw the slowness curve of A0 mode at 10 kHz on an eight-ply CFRP laminate of [0/45/90/−45]_{s}. The software dispersion calculator is used to draw the phase velocity and the slowness curves for a certain mode at a certain frequency in CFRP laminates. Figure 10 shows the results.

As shown in Fig. 10(b), if a wave propagates at 90 deg (the wavefront is orthogonal to the propagation direction), the region of maximum amplitude is gradually skewed to the right side as the wave propagates, which means that the power flow will be slightly concentrated to the right side, and therefore, the signal will be stronger around 75 deg. This is why the maximum power of out-of-plane signals appears around 75 deg instead of 90 deg as shown in Fig. 7.

## 4 Experiment on CFRP Laminate

To verify the accuracy of the simulation, an experiment to propagate laser-ultrasonic waves in a CFRP laminate was conducted. The sample of the CFRP laminate has dimensions of 200 × 200 × 1.1 mm^{3} and is stacked up by eight plies of unidirectional CFRP prepreg with a sequence of [0/45/90/−45]_{s}. Figure 11 shows the experimental setup. The ultrasonic waves were generated using a Q-switched Nd:YAG laser (Ultra 100, Quantel), which has a wavelength of 1064 nm, a pulse power of up to 96 mJ, a pulse duration of 8.5 ns, and a beam radius of 2 mm. A laser Doppler vibrometer (LDV) system equipped with a sensor head (OFV-505, Polytec) and a decoder (OFV2570, Polytec) was used for receiving the out-of-plane velocity. The signals from the LDV were then filtered by a 1.5-MHz low-pass filter (3628, NF) before being recorded in an oscilloscope (TBS2102, Tektronix). The trigger signals from the Nd:YAG laser generator were also transmitted to the oscilloscope to obtain the laser emission time. The data were averaged multiple times in the oscilloscope to reduce noise.

In the experiment, considering the attenuation in the propagation path, we set the laser energy as 20 mJ to obtain recognizable signals and illuminated the laser beam vertically on the center of the CFRP laminate. Then, we observed waveforms at the same locations as the simulation model.

Waveforms of out-of-plane velocity at two points of *η* = 0 deg and *η* = 90 deg are shown in Fig. 12(a). We can see that the wave signal is always accompanied by the noise signal; thus, the S0 mode is not easy to distinguish, while the A0 mode is easy to be seen because of the large amplitude. This result agrees with the simulation result in Fig. 5. Furthermore, the directivity pattern shown in Fig. 12(b) is also similar to the simulation result shown in Fig. 7. From this agreement between the simulation and the experiment, the accuracy of the simulation is proved.

## 5 Power Flow Distribution in Metal-Coated CFRP

We can examine the changes in the power flow distribution by varying the two factors affecting the directivity pattern separately.

In this section, we change the anisotropic distribution of the thermal forces to an isotropic one by forming a metal coating on the surface of CFRP. A technology of covering the surface of CFRP wings and fuselages of airplanes with a metal coating has been developed to resist lightning strikes [16].

The thermal force distribution in metal-coated CFRP is analyzed by the following calculation. We take copper as an example, which is a material with excellent electrical conductivity.

### 5.1 Temperature Distribution of Metal.

There are two differences in physical properties between metal and CFRP. One is that the heat conduction coefficient of metal is generally larger to reach several hundred (W/m/K), while that of CFRP is only about 60 (W/m/K) even for the largest direction parallel to fibers (Tables 2 and 3). Another difference is that the depth of laser penetration into the metal is extremely low and close to zero (for laser with a wavelength of 1060 nm, the penetration depth in copper is 10 nm [17]).

Density ρ (g/cm^{3}) | 8.96 |

Young's modulus E (GPa) | 110–128 |

Poisson ratio | 0.34 |

Specific heat c (J/mol/K) | 24.4 |

Heat conduction coefficient K (W/m/K) | 401 |

Thermal expansion coefficient α (1/K) | 16.5 × 10^{−6} |

Density ρ (g/cm^{3}) | 8.96 |

Young's modulus E (GPa) | 110–128 |

Poisson ratio | 0.34 |

Specific heat c (J/mol/K) | 24.4 |

Heat conduction coefficient K (W/m/K) | 401 |

Thermal expansion coefficient α (1/K) | 16.5 × 10^{−6} |

*K*=

*K*

_{1}=

*K*

_{2}=

*K*

_{3}). Thus, we can obtain the following expression:

Note that the depth of temperature rise in the metal is determined by the heat conduction coefficient *K*, while that of CFRP is mainly determined by *γ*.

Since the large portion of the laser is reflected on the metal surface, we assume that the absorbed intensity of the laser is reduced to 1/3 of that on bare CFRP, i.e., the absorbed energy of a pulsed laser *E* is set to 2.0 mJ, to be closer to the actual situation.

*λ*and

*μ*are the Lame constants of the metal. Out-of-plane force is canceled by the surface force as in the case of bare CFRP.

Figure 13 shows the magnitude of the integrated in-plane force in the case of copper coating. In contrast to Fig. 3(a), we can see that the force distribution is isotropic, i.e., axisymmetric with respect to the laser center.

A layer of copper coating with a thickness of 50 *µ*m is added to the former FEM model (Fig. 4). The thickness of 50 *µ*m is thin enough not to affect the power flow characteristics of the entire plate, while at the same time ensuring that the laser heat is not transmitted to the CFRP and that the thermal force occurs only within the metal coating. The mesh size in the *z* direction is 0.05 mm within the coating. The positions of the signal receiving points and the input points of single forces in the *x*–*y* plane are set on the surface of the metal coating.

Performing the simulation similar to the previous one, we compare the change in the directivity of SSV when the CFRP surface is coated with copper. It is obvious that the directivity is greatly reduced as shown in Fig. 14. This is because the directivity of the thermal force disappears. However, the SSV is not exactly the same in all directions and the maximum value also appears around 75 deg.

To draw the slowness curve, we clarify the main frequency component of the signal as before. As shown in Fig. 15, the frequency spectrum of the out-of-plane signal is concentrated around 20 kHz. (Note that the dominant mode is A0 mode.)

Then, we draw the slowness curves of A0 mode at 20 kHz. It can be seen from Fig. 16 that the slowness curves of bare and copper-coated CFRP have approximately the similar shape. This indicates that the thin copper coating has little effect on the focusing of Lamb waves. Figure 14 shows that even though the thermal forces are isotropic, the directivity is caused by the stacking sequence. The distribution shows that the power flow in the *y* direction is slightly larger than that in the *x* direction, and the maximum value is around 75 deg.

We performed the same experiment on copper-coated CFRP. The copper coating was formed by Yoshino Denka Kogyo, Inc., using a wet plating method. One side of the laminate was coated with a thickness of 50 *μ*m.

As indicated by the experimental outcomes displayed in Fig. 17, two points are worthy of attention: First, in contrast to bare CFRP, the overall SSV of copper-coated CFRP is relatively weaker. This is largely due to the high reflectivity of metallic materials to the laser; a clean, smooth copper surface reflects more than 90% of the Nd:YAG laser (1064 nm) [17]. Thus, the copper coating absorbs considerably less laser energy compared to bare CFRP. Second, the directivity of the SSV of the copper-coated CFRP virtually disappeared, indicating that the experimental results align with the simulation results (Fig. 13). However, because the slowness curve of the copper-coated CFRP is slightly rounder, the power concentration is not as obvious as that of the bare CFRP. Coupled with the fact that the overall Lamb wave excitation of the copper-coated CFRP is weaker, the direction of power concentration is difficult to observe in the experiment.

## 6 Power Flow Distribution in a Cross-Ply CFRP

In this section, we alter the stacking sequence of CFRP laminate by considering a cross-ply laminate ([0/0/90/90]_{s}). Here, we conduct the simulation using the similar method as before and calculate the SSV in each direction. The distribution pattern of the SSV in the cross-ply laminate shows the symmetry with respect to the *y*-axis (Fig. 18). The difference between the maximum and minimum values of power flow increases and the maximum value occurs at 90 deg.

Extracting the dominant components in the same way, we find that the main frequency component of the out-of-plane signal is concentrated around 10 kHz as shown in Fig. 19. (Note that the dominant mode is A0 mode.)

From the slowness curve shown in Fig. 20, we can see that the slowness curve has a symmetry with respect to the *y*-axis. Therefore, the direction of the wave propagating along the *y*-axis has the same direction of the power flow. The power flow also maintains the symmetry with respect to the *y*-axis. As a result, it can be seen that as long as the force and slowness curves are symmetric with respect to a certain direction, the power flow distribution of Lamb waves also shows symmetry with respect to that direction.

## 7 Conclusions

In this research, we investigated the power flow of the ultrasonic waves generated in CFRP laminates exposed to laser irradiation. We initially presented our simulation methodology: temperature field derivation due to laser absorption, calculation of thermal forces based on temperature gradients, and analysis of ultrasonic wave propagation using FEM software. Through our simulations, we identified distinct directivity in the power flow of the excited waves in CFRP laminates ([0/45/90-45]_{s}). The SSV of the ultrasonic waves is larger in the direction perpendicular to the carbon fibers of the surface layer and smaller in the parallel direction. However, the distribution of power flow did not present symmetry with respect to 90 deg; the maximum power direction deviated from 90 deg and was manifested at 75 deg. This computational finding was then experimentally verified.

In our analysis, we tried to show that the directivity of the power flow is determined by two key factors: the anisotropic distribution of thermal forces and the stacking sequence of CFRP laminates dictating the shape of the slowness curves. Accordingly, we designed two different analyses to discuss the implications of these factors on the power distribution independently. In the first analysis, we eliminated the influence of the anisotropic distribution of thermal forces by applying an isotropic metal coating on the CFRP surface, thereby retaining the influence of the stacking sequence. The resultant anisotropy of power flow was significantly weakened, emphasizing the influence of thermal forces. In the second analysis, we changed the stacking sequence to cross-ply. By this alteration, the shape of the slowness curve is symmetrized with respect to 90 deg. The subsequent result showed that the directivity pattern of power flow exhibited the symmetry with respect to 90 deg, bolstering the validity of our hypothesis.

Upon further analysis, we discovered a primary–secondary relationship between the two factors affecting the power flow distribution. The directivity of the thermal force distribution plays a primary role; this was evident when we eliminated the anisotropy of the thermal force via a metallic coating, and most of the directivity disappeared. Conversely, the influence of the stacking sequence only makes slight shifts of power flow distribution which break the symmetry with respect to 90 deg. For the vertical incidence of laser beam in this study, since the distribution of laser-induced thermal forces relies on the carbon fiber orientation of the surface CFRP layer, knowing the carbon fiber orientation of the surface layer allows us to predict the general spread of power flow. However, in the case of oblique laser incidence, the distribution of thermal forces depends not only on the fiber direction of the surface layer but also on the incident angle and direction [11]. Therefore, more factors need to be considered when predicting the distribution of power flow.

The discussion of power flow distribution is meaningful for practical inspection. As shown in Fig. 21, during the NDI process, we detect internal defects in CFRP laminates by exciting Lamb waves through laser irradiation. If these waves encounter defects during propagation, the reflected waves are generated. The detection of any reflection waves implies the presence of internal defects. Moreover, stronger excited waves produce stronger reflection waves, thereby easing defect detection. It is thus possible to enhance detection efficiency through the selection of the receiving direction during practical inspection.

This work provides a reference for selecting the propagation path of ultrasonic waves in practical inspections.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.