Lagrangian Finite Element Model Formulation and Experimental Validation of the Laser Impact Weld Process for Ti/Brass Joining

Abstract

Information on the flyer deformation during laser impact welding (LIW) is an important aspect to consider when high reliability of the welded components is required. For this reason, accurate numerical models simulating thermal and mechanical aspects are needed. In the present work, the cross-section morphology during LIW of Ti/Brass joints at varying laser pulse energies is modeled by a 2D finite element (FE) model. A hydrodynamic plasma pressure model able to describe the evolution of the pressure load step by step, taking into account the progressive deformation of the flyer, was implemented. Hence, this paper proposes an alternative method to the conventional node concentrated forces or predefined velocity as flyer boundary conditions. The levels of the equivalent plastic strain (PEEQ), shear stress, and critical flyer velocity at the collision point were used as reference parameters to predict the success of the welding bond, distinguishing the welded area from the unwelded area. The model was validated by comparison with the experimental data, which showed the effectiveness of the proposed FE code in predicting the cross-section morphology of the welded materials. Moreover, practical industrial information such as variation in the flyer impact velocity, collision angle, and process temperatures was predicted by varying the process laser pulse energy according to the basic principle of the process.

1. Introduction

Technology, especially that related to bioengineering, automotive, aerospace, and pressure vessels, now requires the use of dissimilar materials to achieve high-performance products by leveraging their unique structures. However, welding these materials necessitates extreme processes, the understanding and optimization of which require powerful tools such as numerical simulation.

The joining of metals, which may differ in material properties, is a topic of great interest for both academic and industrial research activities because of the possibility of sharing the common advantages of different welded materials, maximizing the mechanical performance of the joints [1,2]. In this context, impact welding processes, within solid-state joining techniques, are recognized as one of the most promising approaches to achieve permanent connections with limited or no interface melting and a reduced processing time—in the order of microseconds [3,4].

More in detail, a driving force is used to generate a high energy rate collision between a moving part, defined as the “flyer”, and a stationary part, defined as the “target”, allowing for metallurgical bonding in a predetermined region, with impact speeds up to 10⁵ [mm/s]. Impact welding processes can involve several types of driving forces (e.g., explosives, electromagnetic, vaporizing foil actuator, laser-induced shock, etc.) able to join in a very short time, less than 1 [µs], dissimilar materials ranging from tens of micrometers to tens of millimeters thick.

Nowadays these techniques are considered convenient processes for welding metals with significantly different physical and thermal properties, overcoming the conventional fusion welding downside of the generation of brittle intermetallic compounds, residual stresses concentration, and uncontrolled distortions that are detrimental to the overall performance of the components [5,6]. Among the multiple impact welding strategies, the LIW technique represents a promising approach as it allows for the creation of spot welds to micro-scale components for a reduced action area. A schematic illustration of LIW is reported in Figure 1.

In detail, during LIW, the laser is focused on the ablative layer, allowing the formation of plasma. Under the pressure of the plasma, the flyer is accelerated up to 10⁵ [mm/sec] in a few nanoseconds towards the target. The high energy of the collision results in a jet that cleans both the surfaces of the flyer and the target, removing oxides and films and making a metallurgical bond possible when a critical value for shear stress, equivalent plastic strain (PEEQ), and flyer velocity is reached [7,8].

Although the impact welding technique dates back to the 1940s, the current experimental evidence concerning the welding mechanism and the deformation of the flyer is limited because of the very short collision time [9]. For this reason, in several studies, impact welding is numerically simulated in an attempt to better analyze and study the basic mechanisms underlying the process. Wang et al. [10], using a Smooth Particle Hydrodynamics (SPH) numerical approach, analyzed LIW between Al/Al and Al/Cu, simulating the load produced by the pressure of plasma by applying a predefined velocity to the flyer. The authors showed that the shear stress and effective plastic strain near the interface need to reach a threshold value to weld the materials successfully.

Gleason et al. [11] modeled the Al1100/SS304 LIW process by a plane strain purely Eulerian method considering the plasma pressure load as a distribution of node concentrated forces. Their study showed the reliability of the proposed numerical model in predicting the transient velocities, shear stresses, plastic strains, and temperatures during the dynamic process. While the Eulerian approach offers a simple nodal boundary condition application, avoiding mesh interference issues, it does not represent the spring back and cracking well or the jet phenomena.

Li et al. [12] proposed a new numerical approach by coupling the SPH method with the Lagrangian grid for the LIW of Ti/Ni, validating their results with experimental data. The authors compared their model with a conventional purely Eulerian and SPH formulation, assessing the efficiency of their methodology in applying the plasma pressure load directly on the top of the flyer, capturing the interface waveform, the center spring back, and the jet, representing the small vortex waves, and displaying the entire welding process temperatures evolution.

Other numerical methods were adopted to simulate LIW joint formation [4], showing the usefulness of the simulations in obtaining a deep knowledge of the process and filling the experimental information gap. Among the several combinations of dissimilar materials, in this study, the cross-section morphology during the LIW of Ti/Brass joints at varying laser pulse energies was investigated. Ti and Brass are both characterized by high strength and corrosion resistance (i.e., joints suitable for aerospace fuel systems and automotive exhaust systems application); however, the difference in their physical and chemical properties makes the welding of these performing materials challenging. In particular, the difference in melting temperature, thermal expansion, and conductivity may lead to unfavorable weld characteristics, such as non-regular heat flow during welding, resulting in residual stress concentration, hot cracks, and excessive component distortion. To better understand the thermo-mechanical aspects characterizing the evolution of this type of welding, a Lagrangian approach was adopted to simulate the LIW of Ti/Brass joints considering a hydrodynamic plasma model, based on the solution of Fabbro et al. [13], to impose pressure boundary conditions on the flyer, thus proposing an alternative method to the conventional node concentrated forces or predefined velocity as flyer boundary conditions. The numerical results were validated with experimental outcomes reported in the literature, demonstrating the reliability of the adopted technique in simulating the cross-sectional morphology. The collected evidence suggests the proposed method is a useful tool to optimize the welding of Ti and Brass, as it investigates complex aspects such as pressure distribution, bonding zone, effective plastic strain, shear stress, flyer velocity, collision angle, and process temperatures and delivers reliable results within 5–10 min depending on the pressure load boundary condition.

2. Numerical Model

The commercial FEA software Abaqus/Explicit (v. 2020), with a Lagrangian formulation, was used to simulate the LIW of Ti/Brass joints using a coupled thermo-mechanical analysis. Due to the axisymmetric loading conditions of the laser pulse pressure, a 2D axisymmetric model was developed taking into account the flyer, the spacer and the target, Figure 2.

In detail, the flyer and the target were modeled as a deformable body with 12,000 and 40,000 CAX4RT elements (four-node bilinear displacement and temperature, reduced integration with hourglass control), respectively, while the spacer was considered as a discrete rigid body. Considering that the thickness of the flyer was the thinner object of the numerical model (30 µm), an edge length of 5 µm was chosen for the size of the finite elements. The spacer was fixed as a boundary condition [14], and at the bottom side of the target plate, U2 = UR1 = UR2 = 0 was imposed, where U2 is the displacement along the y-axis, UR1 is the rotation on the x-axis, and UR2 is the rotation on the y-axis. The contact was modeled as surface-to-surface contact, setting a Coulomb friction coefficient equal to 0.25 [15]. The contact interaction property was considered a “hard contact”, allowing separation after contact. The fraction of plastic work converted into heat was considered assuming an inelastic heat fraction equal to 0.9 [16,17].

To simulate the material behavior of TA1 Titanium and H62 Brass the strain–strain rate–temperature-dependent Johnson–Cook (J-C) flow stress model was employed, as reported in Equation (1).

$$\sigma \left(\epsilon ,\dot{\epsilon },T\right)=\left[A+B{\left(\epsilon \right)}^n\right]\left[1+Cln\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\right]\left[1-{\left(\frac{T-T_0}{T_{melt}-T_0}\right)}^m\right]$$

(1)

where A [MPa] is the yield strength, B [MPa] and n, respectively, are the coefficient and the exponent of strain hardening, C is the coefficient of strain rate hardening, ε is the plastic strain, $\dot{\epsilon }$ and ${\dot{\epsilon }}_0$ are the plastic strain rate [s⁻¹] and the reference plastic strain rate, respectively, T [K] is the current temperature, T₀ [K] is the reference temperature, T_melt [K] is the melting temperature of the material, and m the thermal softening exponent. The material constants obtained from the literature are reported in

Table 1. TA1 Titanium and H62 Brass material constants for the J-C constitutive model.

Material	A [MPa]	B [MPa]	C	n	m	έ0	Tmelt [K]
Ti [18]	430	1430	0.047	1.05	1.2	1	1923
Brass [14]	112	505	0.009	0.42	1.68	1	1189

.

In this paper, the welding criterion proposed by Chizari et al. [7] was adopted to predict the region along the interface where bounding occurs. The authors reported that the value of the PEEQ at the collision point represents a reference parameter for welding quality. In detail, where the PEEQ reaches a threshold value of 0.7, good bonding occurs, while under this critical value, no welding is found. Moreover, the shear stress values and critical flyer velocity at the collision time were considered to confirm the joint was effectively welded. Hosseinzadeh et al. [17] demonstrated that when the shear stress at the interface exceeds the shear yield stress (based on the Von-Mises ratio), a perfect spot weld occurs, while [8] used an empirical model for the minimum flyer velocity (Equation (2)) resulting in LIW joint:

$$v_{fc}=\sqrt{\frac{1.154{\sigma }_u}{\rho }}$$

(2)

where $v_{fc}$is the critical flyer velocity, ${\sigma }_u$ is the flyer Ultimate Tensile Strength (UTS), and $\rho$ is the flyer material mass density. Using a ${\sigma }_u$ of 308 [MPa] and $\rho$ of 4527 [kg/m³] from [19], a threshold value of $v_{fc}$ = 280 [m/s] was calculated as the minimum welding velocity for the examined flyer.

A space- and time-varying plasma pressure load was imposed on the top surface of the flyer as a boundary condition. At each time step, the axisymmetric pressure behavior was obtained by scaling the pressure peak value at the spot center by a Gaussian distribution, based on the spatial energy distribution of the laser pulse [16]. According to previous works on LIW [11,12,20], for the temporal evolution of the loading curve, the piecewise analytical solutions of the one-dimensional confined ablation model proposed by Fabbro et al. [13] were adopted. These solutions, normalized by the pressure value at the laser pulse width, tp = 8 [ns], are shown in Figure 3 for heating, t ≤ tp, and cooling, t > tp, phases, respectively.

The peak value depends on both the constant pressure${ p}_0$ (Equation (3)) and the ratio between the initial plasma thickness, L₀ = 10 [μm], and the plasma thickness at the end of the heating phase, L(tp), evaluated by Fabbro’s model [13]:

$${ p}_0=0.1{\left(\frac{\alpha Z I_0}{3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(3)

where α = 0.1 is the fraction of laser energy converted to thermal energy, I₀ the laser power density (Equation (4)) and Z the equivalent acoustic impedance (Equation (5)), which combines shock impedances of the K9 glass transparent overlay, $Z_g$ = 1.14 × 10⁶ [g·cm⁻²·s⁻¹], and the Titanium flyer, $Z_{ti}$ = 2.73 × 10⁶ [g·cm⁻²·s⁻¹], calculated as follows:

$$I_o=\frac{4E}{\pi d^2{t}_p}$$

(4)

$$Z=2\frac{Z_g{Z}_{ti}}{Z_g+Z_{ti}}$$

(5)

According to the above parameters, the temporal pressure loads corresponding to E = 1420, 890, and 450 [mJ] laser energy, for a laser spot diameter d = 1.5 [mm], achieve peak values of 4.33, 2.91, and 1.57 [GPa], respectively.

3. FE Validation, Results, and Discussion

The FE model was validated by comparing the predicted results with those experimentally found [14]. The simulated process conditions were identical to the experimental setup, modeling a flyer size of 20 × 5 × 0.03 [mm], a target size of 20 × 20 × 0.1 [mm], a stand-off distance of 0.2 [mm], and a laser spot diameter of 1.5 [mm], and considering the effect of the ablative and confinement layer on the pressure load. The laser pulse energy consisted of three levels, including 450, 890, and 1420 [mJ], with a wavelength of 1064 [nm].

For each of the three process conditions, the numerical welded area was measured as the region, along the interface, where the three criteria (PEEQ, shear stress, and critical flyer velocity) simultaneously occurred (Figure 4). Then, the predicted data were compared with the experimental results to assess the reliability of the developed numerical model.

In Figure 5, the PEEQ, the shear stress, and the flyer velocity at the collision interface are reported to evaluate the cross-section welded area when the laser pulse energy is set to 890 [mJ].

Considering the flyer–target contact front, a PEEQ distribution higher than the established threshold value is reported in Figure 5a, underlying a good connection between the two materials. This result is further confirmed by the shear yield stress analysis at the interface. According to [18], welded bonding occurs when the shear stress in both the flyer and the target exceeds their respective threshold shear yield stress values. By considering the Von-Mises ratio (between the tensile and the shear equivalent stress), threshold shear yield stress values of 248 [MPa] for the flyer (Ti) and 65 [MPa] for the target (Brass) were estimated. Figure 5b demonstrates that the shear stress at the welding interface exceeds these threshold values for both the flyer and the target, confirming the good quality of the weld, as indicated by the PEEQ examination. Finally, the flyer velocity evolution, reported in Figure 5c, also endorses the presence of a permanent joint, showing a collision value of about 750 [m/s] that exceeds the threshold value previously found.

A detailed comparative analysis is reported in

Table 2. Comparison between the experimental [14] and numerical results along the cross-section.

Laser Pulse Energy [mJ]	Outer Diameter [µm]			Inner Diameter [µm]			Cross-Section Welded Area [mm2]
	Num	Exp	Err%	Num	Exp	Err%	Num	Exp	Err%
450	1238.32	1251.93	1.1%	948.12	933.15	1.6%	0.498	0.547	8.9%
890	1480.29	1440.36	2.8%	904.85	895.78	1.1%	1.077	0.999	7.9%
1420	1579.22	1505.02	4.9%	880.52	831.72	5.9%	1.349	1.236	9.2%

, showing a good agreement between the numerical and experimental results with an overall maximum error of less than 10% for all the investigated parameters and an average error of 8.6% when the cross-section welded area is considered. The acceptable difference between the predicted and experimental outcomes and the correct simulated interface morphology, when changing the laser set, highlights the robustness of the developed numerical strategy. The slight difference between the experimental and numerical results is related both to the accuracy of the experimental data and to the adopted numerical strategy, the Lagrangian approach with a one-dimensional hydrodynamic confined ablation model. In fact, while the mesh moves with the material and calculates plastic strain and stresses accurately, this FE model has several limitations in high-deformation scenarios due to severe mesh distortion, material jetting, and interface waveform. This aspect has a significant impact on the simulated results, leading to an adequate discrepancy, as reported in Table 2. Moreover, the trend in the welded area, inner diameter, and outer diameter of the welding interface, when varying the laser pulse energy, is respected; with the increase in the laser pulse energy from 450 to 1420 [mJ], an increase in the outer diameter and a decrease in the inner diameter are registered, with a corresponding enhancement in the welded area.

With the implemented numerical procedure, it is now possible to analyze the evolution of the process deeply by investigating several aspects characterizing this type of welding, with a particular interest in the flyer velocity and collision angle, as reported in Figure 6.

It is well shown that higher laser pulse energy induces higher flyer velocity, which increases the collision angle, allowing for a larger welding area. In fact, during LIW, the collision point is subjected to a rapid expansion of the air that needs a larger collision angle to completely get away, resulting in reduced spring back and enhanced welded area.

In particular, with the increase in the laser pulse energy, the flyer velocity at the collision time increased from 750 [m/sec] to 1370 [m/sec] with a variation in the collision angle from 18° to 27°and an enhancement in the cross-section welded area from 0.498 [mm²] to 1.349 [mm²], corresponding to an increase of 63%, as shown in Figure 7.

Moreover, the simulation predicts the maximum flyer impact velocity on the center of the pressure load distribution, thus denoting the accuracy of the model in simulating the annular-shaped weld, as shown in Figure 8.

This type of analysis and the numerical outcomes reported in Figure 5 and Figure 6 assess the reliability of the implemented numerical approach in predicting the influence of laser pulse energy during the LIW of Ti/Brass joints.

Finally, in Figure 9, the predicted process temperatures are shown. Because of the laser pulse energy increase, higher shear stresses and PEEQ resulted in a large amount of deformation heat that caused a maximum temperature rise in the areas where the welding occurred (Figure 9a). Taking into account the melting temperatures of the two materials (Table 1), and considering that, during the simulation, the temperature peak decreases in a few nanoseconds, not allowing evident thermal diffusion, the numerical code confirms a solid-state weld mainly mechanically induced with minimal or no melting volume (Figure 9b). In detail, with the increase in the laser pulse energy from 450 to 1420 [mJ], both the maximum temperatures of the flyer and target rise as previously discussed, but from the numerical results, it is evident that for a laser pulse energy lower than 890 [mJ], the welding is mainly mechanically induced since neither the flyer nor the target exceeds its melting temperature. Even if the predicted temperatures were not validated by an experimental comparison, this observation gives an idea about the possibility of studying the thermo-mechanical events taking place during the process.

As described above, the developed FE model has several restrictions when simulating high-deformation processes; nevertheless, compared with the Eulerian approach adopted in [11], this strategy demonstrates computational efficiency, delivering reliable results within 5–10 min depending on the pressure load boundary conditions.

4. Conclusions

In this work, a Lagrangian FE model, including a hydrodynamic plasma model to impose pressure boundary conditions on the top surface of the flyer, was developed to investigate the LIW of Ti/Brass joints at varying process laser pulse energies. The adopted pressure model resulted in an alternative method to the conventional node concentrated forces or predefined velocity as flyer boundary conditions.

The numerical approach was validated by comparison with experimental data, showing an acceptable difference between the predicted and experimental outcomes in terms of cross-section morphology. Moreover, the right trend in the process temperatures, flyer velocity, and collision angle were verified according to the basic principle of the process.

The obtained results indicate that the proposed numerical method is suitable for studying LIW characteristics with satisfactory accuracy. It can be used to predict the following at varying process laser pulse energy:

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The cross-section morphology (welded area and outer and inner diameters of the welding interface) according to experimental evidence;

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The variation in flyer impact velocity, collision angle, and process temperatures with an increasing trend when higher laser pulse energy values are used;

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The threshold values for PEEQ, shear stress, and critical flyer velocity that allow for a good connection between the two investigated materials;

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The thermo-mechanical events that mainly influence the welding spot.

The present model can be used to solve practical engineering problems since it allows for studying the complex aspects characterizing the collision time with a particular interest in pressure distribution, maximum instantaneous pressure, bonding zone, flyer velocity, collision angle evolution, temperature distribution, and shear and strain trends. The reliable results of the numerical simulation find practical industrial applications in evaluating the welding quality from several points of view, allowing us to show and compare point by point and step by step, during the entire evolution of the process, the basic thermal and mechanical phenomena characterizing the LIW connection. In particular, this method finds particular interest in aerospace, automotive, and medical products and microelectronics, where the joining of dissimilar metals, with higher mechanical performance and reduced weight, is a topic of great practical importance.

Thus, the developed tool can be used to obtain real information such as distortion, weld shape, and connection area, allowing us to plan an optimum operating process setup focused on component performance, tool life, and power consumption. Further investigation on this method will be focused on the possibility of studying different (i) joining materials (i.e., Al/Cu, Ti/Cu, Al/SS, Ti/Ni, etc.), defining the right welding window for each combination, (ii) experimental setups (i.e., single- and double-plate impact welding), and (iii) driving forces to accelerate the flyer, allowing us to generalize the developed method for different welding modalities.

Numerical simulation, now an essential part of the development of digital twins, continues to provide a significant competitive advantage for companies aiming to succeed in the global competition landscape.