Reducing the Rebound Effect in Microscale Laser Dynamic Forming through Multi-Pulse Laser Shock Loading

Abstract

Microscale laser dynamic forming, as a novel high-speed microforming technique, can overcome the shortcomings of traditional microforming methods. However, in practical applications, laser dynamic microforming technology is often affected by the rebound behavior of the workpiece, limiting the further improvement of processing quality and efficiency. This paper aims to reduce the rebound effect in laser dynamic forming by using multi-pulse laser shock loading. The forming results of workpieces under different laser energies and laser impact numbers were studied using experimental and numerical simulation methods. After multiple laser shocks, numerical simulations of the forming results were conducted using ANSYS/LS-DYNA software. These numerical simulation results were then experimentally validated and compared. The surface morphology of the workpieces was characterized using a confocal microscope and a scanning electron microscope (SEM). Energy-dispersive spectroscopy (EDS) was used to analyze the chemical element content changes in the collision regions at the bottoms of the workpieces after multi-pulse loading. The SEM and EDS results revealed the collision behavior patterns during the forming process. Finally, the forming laws of workpieces under multiple laser shocks were summarized.

1. Introduction

Microscale laser dynamic forming (LDF) [1] technology is a novel high velocity microforming process that does not require a traditional rigid punch. It has broad application prospects in industrial components [2], medical devices [3], power electronics [4], aerospace [5], and military fields [6]. However, in practical applications, the LDF process is often affected by the rebound behavior of workpieces, particularly when forming complex micro-features. The rebound effect can lead to the following problems: (1) local thinning or even fracture of the workpiece; (2) undesirable deformation; or (3) residual stresses. The causes of these problems arise from the physical processes that take place during the laser shock loading, which include: (1) propagation of the shock waves created by the laser pulse; (2) the formation of a stress–strain field as a result of the non-uniform forming velocity field; and (3) residual stresses due to permanent deformations. Traditional single-pulse LDF tends to cause rebound effects due to the high forming velocity resulting from high laser energy, which severely restrict the development and application of LDF.

The rebound effect, a significant phenomenon exhibited by the target material when subjected to impact loading, has garnered significant attention from numerous scholars [7,8,9,10]. Li et al. [11] conducted multi-pulse LDF experiments on ultrathin aluminum foils in a rough vacuum and in air. By investigating the effects of multiple pulses on the deformation behavior of the workpiece, it has been proven that the feasibility of reducing the rebound effect of the workpiece can be achieved by applying multiple pulses. Shen et al. [12] investigated the rebound effect that occurs during the formation of deep micro-features in LDF. To improve the forming precision, he used the plasticine layer as a pressure-carrying medium to absorb the impact-induced reflection wave. The results showed that the plasticine medium can reduce the rebound effect. Zheng et al. [13] utilized finite element methods to study deformation non-uniformity during the laser shock forming process and found that rebound effects occur when the laser spot diameter is too small. Gu et al. [14] analyzed and discussed the size effects in laser shock hydraulic microforming technology in terms of maximum protrusion height, surface morphology, micro-thickness, and elastic recovery. Wang et al. [15] employed the three-dimensional multi-particle finite element method (MPFEM) to investigate the dynamic response, stress–strain evolution mechanism, and rebound behavior of stamping blanks under laser loading characteristics, as well as the densification process of laser shock dynamic compaction of Al₂O₃/Al composite powder. They explored the effects of Em, hard particle Al₂O₃ content, and friction coefficient on the rebound and relative density of compacted blanks. The results showed that with the increase in laser energy and Al₂O₃ particle content, the rebound phenomenon of the compacted blanks became more pronounced; the smaller the friction coefficient and the higher the laser energy, the more evident the rebound phenomenon. Yan et al. [16] analyzed the rebound effect in electromagnetic forming when both a flat-bottom die and a wave-shape die were adopted. By adding liquid to this high-speed forming process, a strong shape control ability can be achieved.

To address the issue of rebound effects in LDF, various solutions have been proposed by researchers. Among them, the multi-pulse laser loading method, as an effective technical means, has attracted wide attention. By applying multiple laser pulses, multi-pulse laser loading can reduce the forming speed, effectively reducing the undesirable deformation caused by single high-speed laser shocks, and thus suppressing the rebound effect of the workpieces. However, current research on the suppression of rebound effects using multi-pulse laser loading is still relatively limited, particularly in terms of systematic numerical simulation and experimental verification.

This study proposes an effective method to suppress the rebound effect in LDF by decomposing the laser impact into several energetically smaller impacts. The results can provide a theoretical basis and technical support for improving the precision and stability of microforming technology. This method not only helps to improve the quality of LDF but also provides a reference for the optimization of other high-strain-rate forming processes [17].

2. Experimental Principles and Research Methods

2.1. Mechanism of Multi-Pulse Laser Dynamic Forming

LDF can be categorized into single-pulse laser dynamic forming (SPLDF) and multi-pulse laser dynamic forming (MPLDF). In SPLDF, successful material-to-die conformity can be achieved by adjusting the energy of the pulse laser. Lower laser energy may result in poor forming quality, while higher laser energy might introduce various defects. Therefore, finding an appropriate critical point [18] for the energy is crucial for experimental research.

In this study, the experimental setup for LDF consists of a blank holder, a confining medium (K9 glass), an ablative medium (black paint), rubber, a workpiece (copper foil of 30 μm thickness), and a micro-die, arranged from top to bottom. The blank holder ring is responsible for securing the entire experimental system platform, preventing displacement and ensuring tight contact between components during the forming process. K9 glass, known for its excellent optical properties and high resistance to laser-induced damage, is used as the confining medium, with a thickness of 3 mm to ensure experimental accuracy and to prevent plasma dispersion and loss after the laser impacts the ablative medium. The ablation layer is made of black paint, which generates a significant amount of plasma when irradiated by the laser. This plasma, constrained by the glass layer, acts downward on the soft film to form the workpiece. The rubber is made of 100 μm thick polyurethane material.

MPLDF is an advanced technology derived from SPLDF. Compared to traditional SPLDF, MPLDF increases the number of pulses, typically using multiple pulses to complete the forming process [11]. By dispersing energy, multi-pulse laser loading can effectively prevent the problem of excessive forming speed. This approach also avoids material damage defects and thermal deformation [19] caused by temperature influences, thereby ensuring the stability and quality of the forming process. The forming mechanism is illustrated in Figure 1.

As shown in Figure 1, it can be seen that MPLDF is very similar in principle to SPLDF, with the only difference being the number of laser pulses. Compared to the energy of a single pulse, each pulse in MPLDF has much lower energy. As multiple laser pulses act on the same position, the impact effect is more uniform and stable, resulting in better material forming outcomes, which helps enhance the strength and durability of the material surface. Additionally, the method of multi-pulse laser loading allows for more precise control of the surface forming of complex materials. Multiple low-energy laser pulses can effectively suppress the rebound effect induced by high-energy single-pulse laser forming, improving the forming accuracy. Overall, MPLDF leverages the cumulative effect of multiple pulses to achieve more uniform and stable processing of the workpiece material surface, thereby enhancing forming accuracy.

2.2. Micro-Die

The laser spot diameter is larger than the grooves and model features of the micro-die, resulting in a portion of the laser spot directly impacting the surface of the micro-die. Hence, the micro-die design needs to possess good impact resistance. In this study, ductile iron was selected as the material for the micro-die. The micro-die used in the experiment features two-level characteristics with trapezoidal and spherical features. The more complex the shape of the micro-die, the more uneven the velocity of the workpiece when it collides with the micro-die. This allows for more convenient study of the rebound behavior and collision behavior patterns during the workpiece forming process. The two-dimensional morphology and three-dimensional contour observed under a depth-of-field microscope are shown in Figure 2, where [A–B] represents the depth after forming the spherical groove and [C–D] represents the width of the bottom surface of the trapezoidal groove. The degree of fit and smoothness between the clamping ring, constraint layer, soft film, workpiece, and micro-die significantly influence the forming quality of the samples. The pressure applied by the 3 mm thick K9 glass constraint layer alone cannot ensure the fitting accuracy of all of the components. Therefore, in the experiment, a clamping ring [20] was designed to be placed above the constraint layer to ensure the forming precision of the workpiece, eliminating gaps between components and improving the forming result. The mechanical performance and material properties of the designed clamping ring are not demanding, as long as it can firmly press the lower constraint layer and the components below it. Ultimately, gray cast iron was used as the material for the clamping ring.

2.3. The Scheme Design of the Comparison Experiment

A comparative scheme was designed to investigate the numerical simulation and experimental comparison for suppressing rebound effects in MPLDF. We conducted numerical simulations utilizing ANSYS/LS-DYNA 2020 R2 software and recorded the dynamic responses of the workpieces. The effectiveness of multi-pulse loading in suppressing the rebound behavior and improving the forming quality of the workpieces was assessed. This ensured the consistency and reliability of the numerical simulations and experimental results, providing a scientific basis for enhancing the application of LDF.

As shown in Figure 3a, the confining medium and micro-die are molded as rigid parts in simulation. Constitutive models of workpiece and rubber are introduced in detail in the next section. Fabbro et al. [21] concluded that uniform laser shock pressure can be obtained in the confined ablation mode when the diameter of the laser beam is relatively large. In LDF experiments, the diameter of the laser beam (2 mm) is much larger than the width of the die cavity (600 μm). So, the laser beam acts on the whole area in the numerical simulation, as indicated in Figure 3a. And the shock pressure, $P$, is proportional to the square root of the absorbed laser intensity as follows:

$$P=B{I}^{1/2}$$

(1)

.

With $P$ in Kbar and $I$ in $\mathrm{G}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$, $B=21$ for glass-confined models. In the confined ablation mode, compared with the laser pulse duration, laser pressure duration can be prolonged 3 times. The laser pulse duration was 8 ns, so the corresponding shock pressure duration was set as 24 ns. Figure 3b shows the laser shock pressure history in the numerical simulation, where t represents the laser pulse duration.

2.4. Constitutive Model of Materials

(1)

Constitutive model of workpiece

In the numerical simulation, the workpiece (density: 8.96 g/cm³, elastic modulus: 110 Gpa) underwent dynamic responses to high strain and experiences plastic deformation. Poisson’s ratio [22] is one of the fundamental characteristics in the material models. In this work, its value is assumed to be constant in the elastic and plastic regions.

The Johnson–Cook model, an empirical constitutive model, is used to describe the plastic behavior of materials under high strain rates. Consequently, it is widely applied in simulating the behavior of metals in conditions such as impact, explosion, and high-speed deformation. This model is adopted as the constitutive model for the copper foil, expressed as follows [23]:

$$\sigma =(A+B{\epsilon }^n)(1+Cln{\displaystyle \frac{\dot{\epsilon }}{\dot{{\epsilon }_0}}})[1-{({\displaystyle \frac{T-T_r}{T_m-T_r}})}^m]$$

(2)

In the equation, A, B, n, ${\epsilon }_0$, and m are material constants and are the five major material characteristic constants of this model; $\sigma$ represents the equivalent yield strength in MPa; $\mathrm{T},{\mathrm{T}}_{\mathrm{m}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{T}}_{\mathrm{r}}$, respectively, denote the material’s temperature, room temperature, and the material’s melting point temperature; and C represents the strain rate constant. The constitutive model parameters for T2 copper used in this study are shown in

Table 1. The parameters of T2 copper in Johnson–Cook mode.

Material	Poisson’s Ratio	A	B	C	n	m	${\mathbf{T}}_{\mathbf{r}}$	${\mathbf{T}}_{\mathbf{m}}$	${\mathit{\epsilon }}_0$
T2 copper	0.34	89.63	291.64	0.025	0.31	1.09	27	12	1.0

[24,25]:

(2)

Constitutive model of rubber

In the process of LDF, polyurethane rubber (density: 1.22 g/cm³) with high elasticity was selected as the material for the soft film. Polyurethane rubber is a nonlinear material. In this paper, the Mooney–Rivlin constitutive model was used to characterize the rubber’s behavior. The stress component expression of the model is shown in Equation (3) [26]:

$${\sigma }_{ij}={\displaystyle \frac{\partial W}{\partial {\epsilon }_{ij}}}$$

(3)

The stress components of this model are obtained by differentiating the strain energy function with respect to the strain components, where W represents the strain energy function and ${\epsilon }_{ij}$ represents the strain components. For the strain energy function W, its expression is given by Equation (4):

$$W={\sum }_{k+m=1}^n{C}_{km}{(I_1-3)}^k{+(I_2-3)}^m+{\displaystyle \frac{k}{2}}{(I_3-3)}^2$$

(4)

where I₁, I₂, and I₃ are the invariants of the deformation tensor. As polyurethane rubber is an incompressible material, I₃ is set to 1. k is the bulk modulus of elasticity and Ckm are constants determining the material’s dynamic response. Typically, C₀₁ and C₁₀ are used to describe the deformation behavior in the Mooney–Rivlin model.

Table 2. Constitutive model parameters of polyurethane rubber.

Material	A (Shore Hardness)	C01	C10	Poisson’s Ratio
Urethanes	70	0.184	0.736	0.49997

provides the constitutive model parameters for the polyurethane rubber material [27].

2.5. Result Characterization

The KEYENCE VHX-1000C (Osaka, Japan) ultra-depth 3D microscope was employed to observe the surface morphology and 3D profiles of the deformed workpieces. Additionally, the scanning electron microscope (SEM), model S-3400N manufactured by Hitachi, Tokyo, Japan, was used for more detailed observations of the microstructure and morphology of the workpieces. Furthermore, X-ray energy-dispersive spectroscopy (EDS) was employed to detect and analyze the changes in chemical element content on the surface of the workpieces.

3. Results and Analysis

3.1. Rebound Phenomenon in Laser Dynamic Forming

During the LDF process, materials undergo plastic deformation due to laser shock loading. However, after the loading ceases, materials may partially recover from elastic deformation, leading to workpiece rebound. High material strength or hardness exacerbates the likelihood of rebound, constituting a significant factor in workpiece rebound. Additionally, the inappropriate selection of laser spot size and laser impact angle can result in uneven stress distribution during workpiece forming, further increasing the likelihood of rebound. Improper support or constraint during the impact process may destabilize workpiece deformation, significantly increasing rebound potential. Figure 4 illustrates the forming results after laser impact at different energy levels in numerical simulations, while Figure 5 shows the forming results of the workpiece after laser impact at different energy levels in experiments, with rebound points marked in both Figure 4 and Figure 5. Upon comparing the experimental and numerical simulation results, it was observed that areas with more severe rebound were primarily located at the junction of the bottom trapezoidal and spherical grooves of the workpiece. In Figure 4, we can observe a trend where the rebound amount of the workpiece in the experiment increases with the increase in laser energy, and this observation is consistent with the results obtained through numerical simulation in Figure 5. Numerical simulation can effectively predict both the location where rebound occurs and its variation trend. Subsequent detailed research on rebound suppression conducted through a combination of experimental and numerical simulation methods follows.

3.2. Influence of Laser Shock Times on Workpiece Forming Result

Figure 6 illustrates the forming process under two laser shocks. The first stage represents the initial phase. The second stage depicts the loading phase of the first laser impact, where the workpiece begins to deform. The third stage represents the contact stage between the workpiece and the micro-die, where collision occurs between the bottom of the workpiece and the micro-die, completing the forming of the spherical and trapezoidal grooves. The fourth stage represents the rebound phase of the soft membrane, marking the end of the first laser shock loading. The fifth stage is the second laser loading phase, where the two-level features at the bottom of the workpiece become more aligned with the micro-die, optimizing the forming result and suppressing the rebound effect caused by the high laser energy during the first laser loading. The sixth stage is the second rebound phase of the soft membrane, marking the end of the numerical simulation process for forming the workpiece with two laser impact loadings. When increasing the laser impact cycles, the process remains similar to the two-load scenario, with only a change in the number of laser loadings, thus repeating stages five and six. Next, a systematic numerical simulation study will be conducted on the forming results of the workpiece under different laser impact cycles.

Figure 7 compares the effects of one and two laser impacts at 1380 mJ laser energy. After a single laser impact at 1380 mJ energy, the workpiece exhibits significant rebound, particularly noticeable at the lower surface of the trapezoidal groove. However, with continued laser impact cycles, the rebound phenomenon disappears, and after two impacts, the bottom of the trapezoidal groove becomes smooth, indicating effective suppression of the rebound effect. Two points (points a and b) in the rebound area of the lower surface of the workpiece were selected to study their displacement histories. The displacement histories of these two points are shown in Figure 8. It can be observed from the graph that after the first laser loading, point a exhibits an initial peak displacement, followed by a sharp decrease, indicating the occurrence of rebound after the collision between the workpiece and the micro-die. Subsequently, the displacement of point a stabilizes at a reduced value, with a rebound displacement of approximately 0.01 mm, until the second laser loading, where the displacement returns to the peak height observed after the first laser loading. This indicates the suppression of the rebound phenomenon that occurred after the first laser loading, and the displacement finally stabilizes without further changes. The displacement history of point b is similar to that of point a, but the rebound displacement after the first laser loading is around 0.005 mm, indicating a less severe rebound effect compared to point a. This observation aligns with the experimental findings, where the rebound area of the workpiece is concentrated at the junction of the trapezoidal and spherical grooves. Therefore, point a was selected as the basic point for analyzing and suppressing the rebound effect in the subsequent research.

This study delved into the formation outcomes of the workpiece by progressively elevating the laser energy, specifically analyzing the effects at the distinct energy levels of 1420 mJ, 1550 mJ, 1690 mJ, and 1800 mJ. Figure 9 shows the comparison of the numerical simulation results for the workpiece under these four laser energy levels after one and two laser impacts. It can be observed that increasing the laser energy leads to higher von Mises stresses within the workpiece, resulting in more severe rebound effects. However, after increasing the number of laser impacts to two, the rebound phenomena induced by these four energy levels are all suppressed, and the rebound effect at the lower surface of the trapezoidal groove is significantly mitigated. Figure 10 depicts the displacement histories of four selected regions exhibiting rebound after laser impacts at energy levels of 1420 mJ, 1550 mJ, 1690 mJ, and 1800 mJ. From the graphs, it is evident that after the first laser impact, point a experiences an initial peak displacement, followed by an instantaneous decrease, and stabilizing at a lower level for a period. This phenomenon occurs due to the rebound of the workpiece after the first laser impact, and as the laser energy increases, the magnitude of the decrease after reaching the peak also increases, indicating a more severe rebound effect. However, after the second laser impact, point a experiences an upward displacement and eventually stabilizes near the peak height observed after the first impact. This is because the rebound effect of the workpiece is successfully suppressed after the second laser impact. Therefore, increasing the number of laser impacts can effectively mitigate the rebound effect.

Figure 11 presents a comparison of the three-dimensional profiles and surface morphologies of the workpiece after different numbers of laser impacts at high laser energies. By comparing the numerical simulations and experimental results presented in Figure 10 and Figure 11, the following findings can be observed: At a laser energy of 1420 mJ, the rebound suppression during the second impact, as simulated numerically, is approximately 5 μm, while the mean experimental rebound suppression is 4.21 μm (with a sample standard deviation of 0.88 μm). At 1550 mJ laser energy, the simulated rebound suppression during the second impact is around 10 μm, and the mean experimental suppression is 8.48 μm (with a sample standard deviation of 0.94 μm). When the laser energy reaches 1690 mJ, the simulated rebound suppression during the second impact is approximately 15 μm, and the mean experimental suppression is 14.01 μm (with a sample standard deviation of 1.05 μm). Lastly, at 1800 mJ laser energy, the simulated rebound suppression during the second impact is around 18 μm, and the mean experimental suppression is 15.14 μm (with a sample standard deviation of 1.08 μm). As the laser energy increases, the error between the numerical simulations and experimental measurements also tends to increase. This is attributed to the fact that in the numerical simulations, the micro-die is assumed to be a rigid body. Consequently, as the laser energy escalates, the rebound of the workpiece simulated numerically is greater than that observed in the experiments. Overall, the numerical simulations can effectively predict the suppression behavior of rebound in workpieces subjected to multi-pulse laser shock loading. It can be observed from Figure 10 and Figure 11 that as the laser energy gradually increases from 1420 mJ to 1800 mJ, the rebound effect of the workpiece is effectively suppressed after two laser impacts.

The process of the numerical simulation study on the workpiece subjected to three laser impacts at varying laser energies is presented above. The research indicates that after two impacts, the rebound effect of the workpiece is essentially suppressed. Next, we explore whether the rebound effect of the workpiece can be effectively suppressed after three impacts at low laser energies. Figure 12 shows the numerical simulation process of the workpiece subjected to three laser impacts at 1020 mJ laser energy. It can be observed that at low laser energies, the rebound effect of the workpiece is not completely suppressed after two impacts, and it is only fully suppressed after the third impact. Additionally, the displacement history of the rebound area of the workpiece after three laser impacts is characterized, as shown in Figure 13. It can be seen that, initially, the displacement history is similar to the displacement history after two impacts mentioned earlier, with an instantaneous drop after reaching the peak value, followed by a recovery to near peak height after the second impact. The difference lies in the fact that at low laser energies, the displacement of the workpiece after two impacts does not reach the peak value, indicating that a slight rebound effect is still present after the second impact. However, after the third impact, the displacement history returns to the vicinity of the peak height and stabilizes, indicating a better suppression effect on the rebound effect of the workpiece.

Figure 14a illustrates the surface morphology and three-dimensional profile of the workpiece after two laser impacts with a laser impact energy of 1020 mJ, while Figure 14b depicts the surface morphology and three-dimensional profile of the workpiece after three laser impacts. It can be observed that in Figure 14a, the rebound phenomenon is significantly suppressed, and the forming depth of the workpiece is increased, indicating a better forming result. After two impacts, the rebound phenomenon on the lower surface of the trapezoidal groove has not completely disappeared; however, after the third impact, the rebound phenomenon is essentially eliminated. By comparing the numerical simulations and experimental results presented in Figure 13 and Figure 14, it can be found that under a laser energy of 1020 mJ, the combined rebound suppression during the second and third impacts, as simulated numerically, is approximately 5 μm. The mean experimental suppression of rebound under these conditions is 4.63 μm (with a sample standard deviation of 0.93 μm). Therefore, the experiment serves as a good comparative verification of the numerical simulation results.

3.3. Variation Trend of Chemical Element Content on Workpiece Surface after Multi-Pulse Laser Loading

Next, we investigate the changes in surface chemical elements before and after the collision. The experimental instruments to be used for this purpose are the Hitachi S-3400N scanning electron microscope (Tokyo, Japan) and its associated energy-dispersive X-ray spectroscopy (EDS) system. To conduct a comparative analysis of the changes in chemical elements on the surface of the workpiece before and after collision, it is necessary to first measure the chemical element content on the surface of the raw material. Figure 15 illustrates the chemical element content on the surface of the raw material before collision. All experimental raw materials used in this study are T2 copper foil, as indicated in Figure 15, with a copper (Cu) element proportion of 98.42%. Due to inevitable contact with air, the surface undergoes slight oxidation, resulting in an oxygen (O) element proportion of 1.58%.

Figure 16 illustrates the proportion of surface chemical elements in three areas after three impacts with 835 mJ laser energy. The data reflect that the elements beneath the trapezoidal groove surface undergo relatively minor changes compared to before the collision. The proportion of carbon (C) elements on the wall of the spherical groove increases by approximately double compared to beneath the trapezoidal groove surface. However, upon measuring the proportion of carbon elements at the bottom of the spherical groove, the results show a proportion of 24.88%, indicating the most intense collision behavior at the bottom of the spherical groove and suggesting a significant increase in carbon atom content before and after the collision, indicating carbonization on the surface of the workpiece. Similarly, after three impacts with 1200 mJ laser energy, chemical element measurements in three different areas reveal similar patterns. The proportion of carbon elements in Area 1 exhibits minor changes compared to before the collision, while the proportion of carbon elements in Area 2 increases by approximately two-thirds. However, in the central Area 3, the proportion of carbon elements reaches 33.64%, higher than the proportion after three impacts with 835 mJ laser energy, indicating more intense collision behavior with increased laser energy and carbonization at the bottom. This trend is illustrated in Figure 17.

Figure 18 shows the velocity distribution at the bottom of the spherical groove of the workpiece in the numerical simulation. According to Figure 18, it can be observed that the velocity at the bottom of the spherical groove is relatively high during forming. The temperature rise due to plastic deformation at high strain rates can be calculated using Equation (5), where $\mathsf{\rho }$ is the material density ($\rho$ = 8.9 × 10³ kg/m³), C_v is the specific heat capacity (C_v = 394 J/kg·°C), t is time, T is temperature, $\mathsf{\mu }$ is the thermal conversion coefficient (usually taken as 0.9), $\dot{\mathsf{\epsilon }}$ is the strain rate, and $\mathsf{\delta }$ is the strain. During high strain rate deformation, the surface temperature of the workpiece increases rapidly, while the cooling rate is relatively slow, making it difficult for the heat to dissipate. Additionally, the increased contact with oxygen on the surface further exacerbates this effect. As a result, the surface temperature of the workpiece can become excessively high, intensifying the carbonization phenomenon. This is the fundamental reason why a significant increase in carbon atoms is observed at the bottom of the spherical groove during experimental measurements, as shown in Figure 19.

$$\rho C_v{\displaystyle \frac{dT}{dt}}=\mu \delta \dot{\epsilon }$$

(5)

4. Conclusions

This paper conducts numerical simulations and experimental studies on MPLDF. By comparing the results of numerical simulations and experimental research, this study investigates the suppression of rebound effects of workpieces after being impacted by different numbers of pulses at various laser energies. Numerical simulation can effectively predict both the location where rebound occurs and its variation trend. The multi-pulse action of the laser beam alters the propagation of the shock wave pulse and the distribution of the forming pressure field, resulting in a more uniform distribution of the strain field. The main conclusions are as follows:

(1)

The forming result of workpieces after being impacted once, twice, and three times at different laser energies was investigated. The results indicate that the workpiece exhibited a rebound effect after a single impact. However, by increasing the number of laser impacts, the rebound effect can be successfully suppressed. At high laser energy densities, impacting twice is sufficient to suppress the rebound effect, while at low laser energy densities, three impacts are required to effectively eliminate the rebound effect.

(2)

The surface morphology and three-dimensional contours of the formed workpieces were characterized, and the experimental results were consistent with the numerical simulations. The results of the comparative analysis indicated that appropriately increasing the number of laser impacts according to the laser energy density can effectively suppress the rebound effect. Through the combined methods of numerical simulation and experimental validation, it was found that the rapid temperature rise on the material surface due to high strain rate deformation, along with the difficulty in dissipating heat, exacerbated the occurrence of “carbonization”. This phenomenon is the fundamental reason for the significant increase in C atom content measured after the experiments.