**6. Multi-physics computation of the flyer in-flight behaviour**

In impact welding, the in-flight behaviour of the flyer determines the collision conditions. Generally, the flyer velocity prior to the impact governs the interfacial phenomena. This is the characteristic parameter that should be known depending on the process and adjustable parameters. Experimental measurements using laser velocimetry methods provide an accurate assessment of the flyer velocity but numerical computation offers a better description of the flyer velocity in terms of spatial and temporal distribution. This section presents a multiphysics computation of the MPW process behaviour. It covers the electromagnetic discharge through the coil and the coupled electromagnetic-mechanical computation of the flyer behaviour. A 3D model is described, including the physical interactions of the process, the governing equations, the resolution procedure, and both boundary and initial conditions. It is employed to show the capability of the model to compute the process behaviour and partic‐ ularly, the flyer kinematics and macroscopic deformation. Illustrations of spatially distributed impact velocity simulation are presented.

while the heat transfer take into account of the mechanical property changes due to the thermal

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261

**Figure 13.** Synoptic depiction of the multi-physics interactions involved in MPW process.

always smaller than the electromagnetic time step, Δ*t* ≪ Δ*T*.

magnetic field and eddy current can be performed using these equations.

E

¶<sup>B</sup> Ñ ´ ¶ = -

> <sup>E</sup> H j <sup>t</sup> e

¶ Ñ´ = +

**6.2. 3D coupled electromagnetic-mechanical model**

Moreover, solution time steps are important input parameters that govern the convergence of a simulation. In an electromagnetic-mechanical coupling, that requires for both electromag‐ netic and mechanical time steps during a simulation. In general the electromagnetic time step

characteristic diffusion time *D* is determined by, *D =* 1/*μσ*. The mechanical time step (Δ*t*) is

The electromagnetic problem is governed by the Maxwell equations (Eqs. 10–13) and the electrical and magnetic constitutive relations (Equations 14 and 15). The calculation of

t

¶

<sup>r</sup> r r uur (10)

<sup>r</sup> r r <sup>r</sup> (11)

/2*D*, where *p* and *D* are characteristic mesh size and characteristic diffusion time. The

effect.

Δ*T* ≤ *p*<sup>2</sup>

### **6.1. Governing physics and multi-physics interaction during MPW**

**Figure 13** describes the multi-physics phenomena involved in the MPW process. The in-flight behaviour of the flyer is mainly governed by the electromagnetic induction and the mechanical response of the material via the Lorentz force while the structural deformation modifies the distribution of magnetic field which in turn affects the electromagnetic interaction between the coil and flyer. This process produces the macroscopic deformation of the structure. Note that the skin depth effect and associated current confinement causes a Joule effect which heats the external part of flyer where the Lorentz force occurs. Generally, the eddy current intensity is high enough, up to several hundred of kA, to generate a strong heating that diffuses within the flyer. A conductive material is expected to involve a good heat transfer and vice versa for a material with low thermal conductivity. Metals such as steel suffer from strong heating due to this phenomenon whereas aluminium or copper seems to limit such heating effect. The consequence of which will make the variance in electromagnetic properties with the temper‐ ature that can change the Lorentz force. Therefore, a full physical description of the phenom‐ enon governing the flyer in-flight behaviour should include this electromagnetic-thermalmechanical interaction. However, under suitable conditions, an electromagnetic-mechanical coupling provides an accurate computation of the flyer kinematics.

The interfacial collision rather involves microscopic phenomenon that can be separately treated using the flyer kinematics given by the electromagnetic-mechanical macroscopic computation. The time-dependent flyer velocity distribution becomes the initial condition for the computation of the impact as previously described in section (Section 5.3). The structural changes involved by both interfacial dynamics and thermal kinetics imply the consideration of specific metallurgical phenomena that decide properties of the joint. However, the multiphysics simulation of the interface can be limited to the mechanical and thermal aspects to reproduce the morphological features of the interface. In the interface simulations (Section 5.3), the mechanical computation describes the interfacial kinematics with the plastic work heating while the heat transfer take into account of the mechanical property changes due to the thermal effect.

**Figure 13.** Synoptic depiction of the multi-physics interactions involved in MPW process.

Moreover, solution time steps are important input parameters that govern the convergence of a simulation. In an electromagnetic-mechanical coupling, that requires for both electromag‐ netic and mechanical time steps during a simulation. In general the electromagnetic time step Δ*T* ≤ *p*<sup>2</sup> /2*D*, where *p* and *D* are characteristic mesh size and characteristic diffusion time. The characteristic diffusion time *D* is determined by, *D =* 1/*μσ*. The mechanical time step (Δ*t*) is always smaller than the electromagnetic time step, Δ*t* ≪ Δ*T*.

#### **6.2. 3D coupled electromagnetic-mechanical model**

**6. Multi-physics computation of the flyer in-flight behaviour**

**6.1. Governing physics and multi-physics interaction during MPW**

coupling provides an accurate computation of the flyer kinematics.

impact velocity simulation are presented.

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In impact welding, the in-flight behaviour of the flyer determines the collision conditions. Generally, the flyer velocity prior to the impact governs the interfacial phenomena. This is the characteristic parameter that should be known depending on the process and adjustable parameters. Experimental measurements using laser velocimetry methods provide an accurate assessment of the flyer velocity but numerical computation offers a better description of the flyer velocity in terms of spatial and temporal distribution. This section presents a multiphysics computation of the MPW process behaviour. It covers the electromagnetic discharge through the coil and the coupled electromagnetic-mechanical computation of the flyer behaviour. A 3D model is described, including the physical interactions of the process, the governing equations, the resolution procedure, and both boundary and initial conditions. It is employed to show the capability of the model to compute the process behaviour and partic‐ ularly, the flyer kinematics and macroscopic deformation. Illustrations of spatially distributed

**Figure 13** describes the multi-physics phenomena involved in the MPW process. The in-flight behaviour of the flyer is mainly governed by the electromagnetic induction and the mechanical response of the material via the Lorentz force while the structural deformation modifies the distribution of magnetic field which in turn affects the electromagnetic interaction between the coil and flyer. This process produces the macroscopic deformation of the structure. Note that the skin depth effect and associated current confinement causes a Joule effect which heats the external part of flyer where the Lorentz force occurs. Generally, the eddy current intensity is high enough, up to several hundred of kA, to generate a strong heating that diffuses within the flyer. A conductive material is expected to involve a good heat transfer and vice versa for a material with low thermal conductivity. Metals such as steel suffer from strong heating due to this phenomenon whereas aluminium or copper seems to limit such heating effect. The consequence of which will make the variance in electromagnetic properties with the temper‐ ature that can change the Lorentz force. Therefore, a full physical description of the phenom‐ enon governing the flyer in-flight behaviour should include this electromagnetic-thermalmechanical interaction. However, under suitable conditions, an electromagnetic-mechanical

The interfacial collision rather involves microscopic phenomenon that can be separately treated using the flyer kinematics given by the electromagnetic-mechanical macroscopic computation. The time-dependent flyer velocity distribution becomes the initial condition for the computation of the impact as previously described in section (Section 5.3). The structural changes involved by both interfacial dynamics and thermal kinetics imply the consideration of specific metallurgical phenomena that decide properties of the joint. However, the multiphysics simulation of the interface can be limited to the mechanical and thermal aspects to reproduce the morphological features of the interface. In the interface simulations (Section 5.3), the mechanical computation describes the interfacial kinematics with the plastic work heating The electromagnetic problem is governed by the Maxwell equations (Eqs. 10–13) and the electrical and magnetic constitutive relations (Equations 14 and 15). The calculation of magnetic field and eddy current can be performed using these equations.

$$
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial \vec{t}}\tag{10}
$$

$$
\vec{\nabla} \times \vec{H} = \vec{\mathbf{j}} + \varepsilon \frac{\partial \vec{\mathbf{E}}}{\partial \mathbf{t}} \tag{11}
$$

$$
\vec{\nabla} \cdot \vec{B} = 0 \tag{12}
$$

Therefore, by solving these Equations 18–19, the two unknowns A

time step, the electromagnetic and mechanical computations are coupled.

to accurately capture the electromagnetic skin effect (Eq. 3).

calculations, Lorentz force is estimated.

**Figure 14.** 3D geometrical model.

Here *ε*¯ ˙

system can be solved. Finally, based on these solutions procedure and magnetic pressure

Magnetic Pulse Welding: An Innovative Joining Technology for Similar and Dissimilar Metal Pairs

In this study, the electromagnetic coupled numerical simulations were carried out using LS-DYNA® package with the solver version R8. The resolution scheme in the electromagneticmechanical solver uses both finite element method(FEM) and boundary element method (BEM) [58]. BEM is used toe valuate the surface current and electromagnetic field thus the magnetic field in the air is not required in LS-DYNA® simulations. FEM is used during the computation of eddy current and Lorentz force in the workpieces. At each electromagnetic

A typical situation of one turn coil with a separate field shaper model is considered as an illustrative 3D simulation case (**Figure 14**). The model consists of solid8 node elements for both workpieces and tools to handle the electromagnetic algorithm in LS-DYNA. The largest element size was selected based on the skin depth and ensured that the element size is sufficient

Material model was described using a simplified Johnson-Cook model (Eq. 20) in the numerical

0

& (20)

˙ is the strain rate.

e

&

é ù æ ö

<sup>0</sup> is the quasi-static threshold strain rate, treated as equal to *1/s. A, B, C* and *n* are

e

simulations to capture the high strain rate deformation behaviour of the work pieces.

1 *<sup>n</sup> A B Cln*

=+ + ê ú ç ÷ ê ú ë û è ø

( )

where, *σ*¯ and *σ*¯ are the von Mises equivalent stress and strain respectively, *ε*¯

 e

s

<sup>→</sup> and Φ in an electromagnetic

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263

$$
\vec{\nabla} \cdot \vec{E} = \frac{\mathcal{P}}{\mathfrak{E}} \tag{13}
$$

$$
\vec{\mathbf{J}} = \sigma \vec{\mathbf{E}} + \overline{\mathbf{J}\_s} \tag{14}
$$

$$
\vec{\mathbf{B}} = \mu \vec{\mathbf{H}} \tag{15}
$$

In these equations σ, μ and ε respectively represent electrical conductivity, magnetic perme‐ ability and electrical permittivity. E <sup>→</sup> , B <sup>→</sup> , H →, ρ, <sup>j</sup> <sup>→</sup> and Js → denote the electric field, magnetic flux density, magnetic field intensity, total charge density, total current density, and source current density respectively. In magnetic pulse forming and welding processes, there involves no charge accumulation and eddy current approximation follow a divergence free current density, those implies ρ = 0 and ε <sup>∂</sup> <sup>E</sup> → <sup>∂</sup> <sup>t</sup> =0.

Due to the divergence condition of Eq. (10) and Eq. (12), they should satisfy the following correlations written in Eq. (16) and Eq. (17) respectively.

$$
\vec{\mathbf{E}} = -\vec{\nabla}\Phi - \frac{\partial \vec{\mathbf{A}}}{\partial t} \tag{16}
$$

$$
\vec{\mathbf{B}} = \vec{\nabla} \times \vec{\mathbf{A}}\tag{17}
$$

where Φ and A → are respectively the electric scalar potential the magnetic vector potential. Since the mathematical degree of freedom satisfy the magnetic vector potential A <sup>→</sup> , a gauge equation is applicable. Using the aforementioned co-relations with the generalized Coulomb gauge condition, ∇(σA <sup>→</sup> )=0, one could separate the vector and scalar potentials as shown in Equation (18) and Equation (19) respectively.

$$\nabla \left( \sigma \vec{\nabla} \Phi \right) = 0 \tag{18}$$

$$
\sigma \frac{\partial \vec{\mathbf{A}}}{\partial \mathbf{t}} + \vec{\nabla} \times \left( \frac{1}{-} \vec{\nabla} \times \vec{\mathbf{A}} \right) + \sigma \vec{\nabla} \Phi = \overline{\mathbf{j}}\_{\text{s}} \tag{19}
$$

Therefore, by solving these Equations 18–19, the two unknowns A <sup>→</sup> and Φ in an electromagnetic system can be solved. Finally, based on these solutions procedure and magnetic pressure calculations, Lorentz force is estimated.

In this study, the electromagnetic coupled numerical simulations were carried out using LS-DYNA® package with the solver version R8. The resolution scheme in the electromagneticmechanical solver uses both finite element method(FEM) and boundary element method (BEM) [58]. BEM is used toe valuate the surface current and electromagnetic field thus the magnetic field in the air is not required in LS-DYNA® simulations. FEM is used during the computation of eddy current and Lorentz force in the workpieces. At each electromagnetic time step, the electromagnetic and mechanical computations are coupled.

A typical situation of one turn coil with a separate field shaper model is considered as an illustrative 3D simulation case (**Figure 14**). The model consists of solid8 node elements for both workpieces and tools to handle the electromagnetic algorithm in LS-DYNA. The largest element size was selected based on the skin depth and ensured that the element size is sufficient to accurately capture the electromagnetic skin effect (Eq. 3).

**Figure 14.** 3D geometrical model.

Ñ× = B 0 r r (12)

r r (13)

r uur <sup>r</sup> (14)

r r (15)

<sup>r</sup> r r (16)

r r <sup>r</sup> (17)

ÑÑ = (σΦ 0 ) <sup>r</sup> (18)

<sup>→</sup> , a gauge equation

→ denote the electric field, magnetic flux

<sup>ρ</sup> <sup>E</sup> ε Ñ× =

<sup>s</sup> J EJ = + s

B µH =

<sup>→</sup> , B <sup>→</sup> , H →, ρ, <sup>j</sup>

→ <sup>∂</sup> <sup>t</sup> =0.

correlations written in Eq. (16) and Eq. (17) respectively.

ability and electrical permittivity. E

density, those implies ρ = 0 and ε <sup>∂</sup> <sup>E</sup>

(18) and Equation (19) respectively.

where Φ and A

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condition, ∇(σA

In these equations σ, μ and ε respectively represent electrical conductivity, magnetic perme‐

density, magnetic field intensity, total charge density, total current density, and source current density respectively. In magnetic pulse forming and welding processes, there involves no charge accumulation and eddy current approximation follow a divergence free current

Due to the divergence condition of Eq. (10) and Eq. (12), they should satisfy the following

t

→ are respectively the electric scalar potential the magnetic vector potential. Since

<sup>→</sup> )=0, one could separate the vector and scalar potentials as shown in Equation

s

<sup>r</sup> rr r <sup>r</sup> uv (19)

¶

is applicable. Using the aforementioned co-relations with the generalized Coulomb gauge

<sup>A</sup> <sup>E</sup>

B A =Ñ´

the mathematical degree of freedom satisfy the magnetic vector potential A

A 1

<sup>σ</sup> A σΦ j tμ ¶ æ ö +Ñ´ Ñ´ + Ñ = ç ÷ ¶ è ø

¶ = -ÑF -

<sup>→</sup> and Js

Material model was described using a simplified Johnson-Cook model (Eq. 20) in the numerical simulations to capture the high strain rate deformation behaviour of the work pieces.

$$\overline{\sigma} = \left( A + B \overline{\varepsilon}'' \right) \left[ 1 + C \ln \left( \frac{\dot{\overline{\varepsilon}}}{\dot{\overline{\varepsilon}}\_0} \right) \right] \tag{20}$$

where, *σ*¯ and *σ*¯ are the von Mises equivalent stress and strain respectively, *ε*¯ ˙ is the strain rate. Here *ε*¯ ˙ <sup>0</sup> is the quasi-static threshold strain rate, treated as equal to *1/s. A, B, C* and *n* are constants, obtained from literatures, are listed in **Table 1**. Other mechanical, electromagnetic and thermal quantities used in the model are listed in **Table 2**.

The specification of the test case and the input current used in the numerical simulations are

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**Figure 15.** (a) Schematic illustration showing the main working area of the model (field shaper and workpieces) except

In these numerical simulations, a similar Al/Al and Al/Cu combinations were investigated for their in-flight behaviour in terms of impact velocity and the collision angle at the onset of impact. The properties used in these simulations are corresponding to aluminium alloy 2024 and a commercially pure copper respectively for Al and Cu. The collision angles were calculated estimated on the angle between the radial and longitudinal velocity components (respectively *Vr* and *Vz*, see **Figure 17** for detail) from the simulations. The impact velocity was calculated inside of the tube along the longitudinal direction. Sudden change in the resultant velocity was used to determine the onset of the impact and subsequent conditions. That is, immediately at the onset of the impact, the resultant velocity of the tube rapidly reduces. Based on the prediction of the corresponding onset time, resultant velocity and the angle of attack

Impact velocity and impact angle against the longitudinal distance from the top edge of the tube are provided in **Figure 16**. The highlighted regions in **Figure 16a** and **b** are able to come in contact during the simulation, which is in agreement with the ~9 mm contact distance observed in experimentally welded samples (**Figure 16c**) obtained under same assembly

These results suggest that the high speed dynamics for the particular welding case as shown in **Figure 15**. That is, for the particular configuration of the simulation (**Figure 15a**), the top edge of the tube is located slightly above the horizontal mid-plane of the field shaper. This condition causes the highest velocity and the first occurrence of impact slightly below the top edge of the tube. The angle measurements were followed the sign convention given in **Figure 17a**. A closer look of the in-flight collision dynamics shown in **Figure 17b** illustrates the

shown in **Figure 15**.

the coiland (b) the source current used in the model.

[*tan1 (Vz/Vr)*] were calculated.

configuration.

**6.4. Simulation results of the in-flight behaviour of the flyer**


**Table 1.** Johnson-Cook parameters used to prescribe the constitutive behavior of workpieces.


**Table 2.** Physical properties of the materials and theirs corresponding parts.

#### **6.3. Boundary and initial conditions and other specifications**

This section addresses a general procedure for the specification of boundary and initial conditions for an electromagnetic-mechanical model. In the current example, the bottom side of the tube and the top side of the rod are fixed during the simulation. Coil was considered as rigid and fixed during the simulations. The field shaper geometry was placed inside the coil and left free without any boundary conditions to well represent the experimental condition. Automatic surface to surface contact was prescribed between the rod and the tube to capture the contact behaviour during the collision. Electrical conductivity (σ) is defined by electro‐ magnetic card and the relative magnetic permeability (μr) was considered as one for all the materials, which indicates that the ferromagnetic effects of materials are neglected during the simulation, whose μr values are generally greater than one. The electric and magnetic fields are not only depending on the geometry, assembly configurations and input parameters but also they depend on the input and output surface of the current definitions in a coupled simulation. Therefore, in order to accurately represent the experimental conditions, the exact same connecter point areas are used in the numerical simulations.

In general, an electromagnetic simulation requires to be defined with at least one electrical circuit. In order to define an electrical circuit, a voltage or current curve is applied across the input and output surfaces. Alternatively, the circuit could be specified with R, L, C (resistance, load, and capacitance respectively) parameters of the exterior parts in a circuit without including these parameters of the coil and other components used in a particular simulation. The specification of the test case and the input current used in the numerical simulations are shown in **Figure 15**.

**Figure 15.** (a) Schematic illustration showing the main working area of the model (field shaper and workpieces) except the coiland (b) the source current used in the model.

#### **6.4. Simulation results of the in-flight behaviour of the flyer**

constants, obtained from literatures, are listed in **Table 1**. Other mechanical, electromagnetic

**Johnson-Cook parameters** *A* **(MPa)** *B* **(MPa)** *C n* Aluminium alloy 352 440 0.0083 0.42 Commercially pure copper 90 292 0.025 0.31

> **(***kg/m3* **)**

This section addresses a general procedure for the specification of boundary and initial conditions for an electromagnetic-mechanical model. In the current example, the bottom side of the tube and the top side of the rod are fixed during the simulation. Coil was considered as rigid and fixed during the simulations. The field shaper geometry was placed inside the coil and left free without any boundary conditions to well represent the experimental condition. Automatic surface to surface contact was prescribed between the rod and the tube to capture the contact behaviour during the collision. Electrical conductivity (σ) is defined by electro‐ magnetic card and the relative magnetic permeability (μr) was considered as one for all the materials, which indicates that the ferromagnetic effects of materials are neglected during the simulation, whose μr values are generally greater than one. The electric and magnetic fields are not only depending on the geometry, assembly configurations and input parameters but also they depend on the input and output surface of the current definitions in a coupled simulation. Therefore, in order to accurately represent the experimental conditions, the exact

In general, an electromagnetic simulation requires to be defined with at least one electrical circuit. In order to define an electrical circuit, a voltage or current curve is applied across the input and output surfaces. Alternatively, the circuit could be specified with R, L, C (resistance, load, and capacitance respectively) parameters of the exterior parts in a circuit without including these parameters of the coil and other components used in a particular simulation.

Commercially pure copper Rod 8900 124 0.34 3.48 × 107 Copper alloy Field shaper 7900 210 0.29 2.66 × 107 Steel Coil --------------------Rigid------------------- 4.06 × 106

**Young's modulus(***GPa***)** **Poisson's ratio**

2700 73 0.33 1.74 × 107

**Electrical**

**conductivity (***S/m***)**

and thermal quantities used in the model are listed in **Table 2**.

**Material Part Density**

Rod

**Table 2.** Physical properties of the materials and theirs corresponding parts.

**6.3. Boundary and initial conditions and other specifications**

same connecter point areas are used in the numerical simulations.

Aluminium alloy 2024 Tubeand

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**Table 1.** Johnson-Cook parameters used to prescribe the constitutive behavior of workpieces.

In these numerical simulations, a similar Al/Al and Al/Cu combinations were investigated for their in-flight behaviour in terms of impact velocity and the collision angle at the onset of impact. The properties used in these simulations are corresponding to aluminium alloy 2024 and a commercially pure copper respectively for Al and Cu. The collision angles were calculated estimated on the angle between the radial and longitudinal velocity components (respectively *Vr* and *Vz*, see **Figure 17** for detail) from the simulations. The impact velocity was calculated inside of the tube along the longitudinal direction. Sudden change in the resultant velocity was used to determine the onset of the impact and subsequent conditions. That is, immediately at the onset of the impact, the resultant velocity of the tube rapidly reduces. Based on the prediction of the corresponding onset time, resultant velocity and the angle of attack [*tan1 (Vz/Vr)*] were calculated.

Impact velocity and impact angle against the longitudinal distance from the top edge of the tube are provided in **Figure 16**. The highlighted regions in **Figure 16a** and **b** are able to come in contact during the simulation, which is in agreement with the ~9 mm contact distance observed in experimentally welded samples (**Figure 16c**) obtained under same assembly configuration.

These results suggest that the high speed dynamics for the particular welding case as shown in **Figure 15**. That is, for the particular configuration of the simulation (**Figure 15a**), the top edge of the tube is located slightly above the horizontal mid-plane of the field shaper. This condition causes the highest velocity and the first occurrence of impact slightly below the top edge of the tube. The angle measurements were followed the sign convention given in **Figure 17a**. A closer look of the in-flight collision dynamics shown in **Figure 17b** illustrates the potential collision condition for the particular assembly configuration used in this numerical study.

Although these investigations show that the variance in impact angle and impact velocity, they are not apparent for different inner rods, the influence is suggested as highly depended on the current frequency. That is, one could neglect the difference of the inner rod when predicting the in-flight behaviour at higher current frequencies than that of a critical frequency that can be estimated by equating the first collision time with the full diffusion time of magnetic field through thickness from the exterior to the inner surface of the flyer. In contrast, the difference in the impact condition is not negligible for various inner rods at lower frequencies than that of the critical one. At those lower frequencies, the impact can be influenced by the conductivity

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**Figure 18.** Time dependent radial component of the (a) Lorentz force and (b) velocity obtained from various points on

The multi-physics coupled simulations particularly have the benefits of predicting the time dependent the Lorentz body force (**Figure 18a**) and velocity (**Figure 18b**) during the process. Those macroscopic data determines the collision conditions which in turn decides the weld generation. They are crucial for the computation of the interface behaviour during the welding and thus serve as required input condition. Generally, the velocity distribution is appropriate for the interface behaviour simulation that can reproduce physically realistic results (Section

The magnetic pulse welding is identified as a promising alternative to produce multi material assemblies, while it provides the attractive benefits in terms of cost, reliability, ease of use, flexibility, rate of work, no requirement of consumable and environmental friendliness. The operational conditions were highlighted including the input voltage and initial gap between the flyer and rod were identified as important assembly parameters for a particular overlap configuration of the MPW process. Moreover, the MPW process is also highly influenced by

of the inner rod [59].

the flyer.

5.4).

**7. Conclusions**

**Figure 16.** Impact angles along the longitudinal distance from the top edge of the tube for the simulation in (a) and instantaneous resultant velocity at those corresponding points during the onset time in (b). The highlighted regions in (a) and (b) well represent the onset of impact. (c) MPW of Al/Cu sample and (d) final shape of the workpieces from a numerical simulation.

**Figure 17.** (a) Convention of the angle measurement and (b) a closer look of the in-flight dynamics of the flyer velocity and the impact angle variation.

Although these investigations show that the variance in impact angle and impact velocity, they are not apparent for different inner rods, the influence is suggested as highly depended on the current frequency. That is, one could neglect the difference of the inner rod when predicting the in-flight behaviour at higher current frequencies than that of a critical frequency that can be estimated by equating the first collision time with the full diffusion time of magnetic field through thickness from the exterior to the inner surface of the flyer. In contrast, the difference in the impact condition is not negligible for various inner rods at lower frequencies than that of the critical one. At those lower frequencies, the impact can be influenced by the conductivity of the inner rod [59].

**Figure 18.** Time dependent radial component of the (a) Lorentz force and (b) velocity obtained from various points on the flyer.

The multi-physics coupled simulations particularly have the benefits of predicting the time dependent the Lorentz body force (**Figure 18a**) and velocity (**Figure 18b**) during the process. Those macroscopic data determines the collision conditions which in turn decides the weld generation. They are crucial for the computation of the interface behaviour during the welding and thus serve as required input condition. Generally, the velocity distribution is appropriate for the interface behaviour simulation that can reproduce physically realistic results (Section 5.4).

## **7. Conclusions**

potential collision condition for the particular assembly configuration used in this numerical

**Figure 16.** Impact angles along the longitudinal distance from the top edge of the tube for the simulation in (a) and instantaneous resultant velocity at those corresponding points during the onset time in (b). The highlighted regions in (a) and (b) well represent the onset of impact. (c) MPW of Al/Cu sample and (d) final shape of the workpieces from a

**Figure 17.** (a) Convention of the angle measurement and (b) a closer look of the in-flight dynamics of the flyer velocity

study.

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numerical simulation.

and the impact angle variation.

The magnetic pulse welding is identified as a promising alternative to produce multi material assemblies, while it provides the attractive benefits in terms of cost, reliability, ease of use, flexibility, rate of work, no requirement of consumable and environmental friendliness. The operational conditions were highlighted including the input voltage and initial gap between the flyer and rod were identified as important assembly parameters for a particular overlap configuration of the MPW process. Moreover, the MPW process is also highly influenced by the current frequency and the electromagnetic skin depth effect. After that interfacial natures and weld variances were investigated for both similar and dissimilar material combinations, where it was identified that the weld variances develop under onset of bonding, wavy interface formation, irregular interfaces, jetting, occurrence of vortices, interfaces with defects and intermetallic formation. The vortex formation is highly apparent in the dissimilar assembly while this also forms the intermetallic phases at the interface as a clear distinct feature from the similar metal assembly. After that, numerical simulations were utilised to capture the interfacial features, this could serve as a potential method to identify the influencing param‐ eters during the weld formation. These simulations well capture the interfacial features including jetting and ejection phenomena. Moreover these simulations reveal the interfacial heating which closely resemble with the defects in terms of side occurrence and shape in those assemblies that indicates these simulations could be utilised to predict the optimum welda‐ bility window for various combinations. Finally an investigation of coupled electromagneticmechanical simulations provides insight understanding and in-flight kinematics of the flyer that predicts the collision conditions during the MPW process. In summary, all these results indicate the potential outlook and innovative nature of the process among other existing welding technologies. The multi-physics nurture with high speed dynamics and associated high strain deformation particularly makes the process more complex that requires extra attention while manipulating the process. However, promising benefits and the evidence of potentially permanent weld formation always pave the way and keep attracting the manu‐ facturing industries.

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## **Acknowledgements**

Authors acknowledge the funding for the MSIM project from "Région Picardie" and for COILTIM project financial support from "Région Picardie" and "Le fonds européen de développement économique régional (FEDER)". Authors also thank "PlateformeInnovaltech" for its collaboration. Moreover, authors greatly appreciate and acknowledge the permission from "PSTproducts GmbH" to reuse their images in this chapter.

## **Author details**

T. Sapanathan1 , R. N. Raoelison2\*, N. Buiron1 and M. Rachik1

\*Address all correspondence to: rija-nirina.raoelison@utbm.fr

1 Sorbonne universités, Université de Technologie de Compiègne, Compiègne cedex, France

2 Université de Bourgogne Franche Comté, Université de Technologie de Belfort Montbé‐ liard, IRTES EA7274, Belfort, France
