Effects of Discharge Current on Velocity of Moving Sheet and Joining Property in Welded Sheet Processed by Magnetic Pulse Welding — Influence of Circuit Inductance on Discharge Current —

Abstract

A systematic experiment was performed for four types of welding circuit with different circuit inductances and a common welding coil, where each circuit inductance is given by the sum of a different remaining inductance and a common effective inductance. It was demonstrated that increasing the circuit inductance causes adverse effects on the discharge current, resulting in a long first collision time and a low deformation velocity of the moving sheet. When the circuit inductance increases from 0.0587 µH to 0.2280 µH, the maximum current decreases from 223 kA to 132 kA at a discharge energy of 2.0 kJ. With the decrease in discharge current, the deformation velocity of the sheet decreases from 383 m·s⁻¹ to 164 m·s⁻¹. The higher the circuit inductance is, the lower the deformation velocity of the sheet is. With the lower velocity, the shearing load of the resulting welded sheet further decreases, ultimately leading to joining failure. However, in the circuit with a minimum inductance of 0.0587 µH, it is possible to weld an aluminum alloy sheet to a 1 GPa class high-strength steel sheet. It has been clarified that the decrease in circuit inductance improves the joining property of a welded sheet.

 

This Paper was Originally Published in Japanese in J. JSTP 63 (2022) 129–136.

Fig. 1 General outline of magnetic pulse welding circuit, and measurement of discharge current and collision time signal.

1. Introduction

Magnetic pulse welding is a solid-phase joining method for two sheets at a high energy rate. A moving sheet accelerated by an impulsive electromagnetic force collides with a fixed sheet placed in a gap, and the two sheets are joined with each other. Aluminum is the most suitable material for the moving sheet as it is highly conductive and light weight. A discharge circuit including an E-shaped flat one-turn coil L_c is generally used to generate the impulsive electromagnetic force.^1–3) When a large impulse current (a discharge current) from an energy-storing capacitor bank C passes through the coil L_c, an energy is supplied to the moving sheet connected by an electromagnetic coupling³⁾ and the sheet undergoes plastic deformation due to the electromagnetic force.^4–6) If the moving sheet collides with the fixed sheet at a high velocity⁶⁾ and the first collision occurs earlier than the maximum of the discharge current, the sheets join strongly.^4–6,8) The two conditions are satisfied by changing both the discharge energy of the bank C and the gap between the two sheets. With these conditions, a 6061-T6 aluminum alloy sheet could possibly weld to a 590 MPa high-strength steel sheet. The joined, welded sheet attains a wavy structure with continually changing amplitude and wavelength.^9–11) A high frequency discharge current is a principal factor in magnetic pulse welding. A waveform of the discharge current is shaped by three forming factors, and by charging voltage V of the bank C. The three forming factors are maximum current, oscillating period, and damping coefficient. The charging voltage V is equivalent to the discharge voltage. The forming factors and the charging voltage V affect the deformation velocity of the moving sheet and the joining strength of the welded sheet, but the degree of influence is still unclear. Generally, three circuit elements L, C, and R in commercial equipment of magnetic pulse welding are different¹²⁾ and a change in the three circuit elements is difficult. In this case, the waveform of the discharge current are different for the same discharge energy. In the case of distinct kinds of circuit elements, estimating the influence of the discharge current on the joining property of the welded sheet is difficult. If the equipment has an exchangeable circuit inductance and one welding coil, the forming factors of the discharge current and joining strength of the welded sheet can be investigated. Therefore, to estimate the influence of the discharge current, three kinds of inductance regulator boards were prepared, and an experiment was performed using four discharge circuits having different circuit inductances on only one welding coil.^13–15) The purpose of this study was to determine the influence of circuit inductance on the discharge current. Furthermore, the relationship between the discharge current and the joining strength of the welded sheet was clarified. Based on the experimental results, the relationship between an electric phenomenon and a deforming phenomenon is discussed in detail.

2. Principle of Welding and Measurement of Collision Time

2.1 Principle of welding

A general outline of magnetic pulse welding and a simultaneous measurement system for both discharge current I and collision time signal S_c are shown in Fig. 1. The experimental apparatus was composed of a capacitor bank C, an E-shaped flat one-turn coil L_c, an inductance regulator board L_R, a fixture, and two sheets with a gap (before welding). Electrical quantities were eddy currents i, discharge current I, and magnetic flux lines φ_I. The cross section of the coil L_c was a rectangle, and the central part of the coil L_c had a mirror symmetry at yz-plane. Magnetic pulse welding of the two sheets was performed using the discharge circuit, including the inductance regulator board L_R. When the gap switch G was closed, the current density of the central part of the coil L_c increased because the discharge current I was concentrated in the narrow central part () from both sides () of the coil L_c. A high-density magnetic flux φ_I appeared instantaneously around the central part of the coil L_c, and intersected the moving sheet (flyer sheet), which was connected through an electromagnetic coupling. The magnetic flux φ_I penetrated the moving sheet, and a voltage was induced by Faraday’s law; then, an eddy current i passed through the sheet in the opposite direction () to hinder its further penetration. However, the magnetic flux φ_i generated by the eddy current i is not shown in Fig. 1. If the magnetic flux density B in the moving sheet was clear, which was produced by both the flux φ_I and flux φ_i, the eddy current density i can be given by eq. (1), and an impulsive electromagnetic force f per unit volume can be obtained using eq. (2) by applying Faraday’s law, Ohm’s law, and Fleming’s left-hand rule.

  

\begin{equation} \textit{rot}\,\boldsymbol{i} = -\kappa(\partial\boldsymbol{B}/\partial t) \end{equation} (1)

  

\begin{equation} \boldsymbol{f} = \boldsymbol{i} \times \boldsymbol{B} \end{equation} (2)

where κ is electric conductivity and f is the vector product of i and B. The moving sheet undergoes a convex plastic deformation in xz-plane because an impulsive electromagnetic force f is generated in the sheet placed on the coil L_c.

Fig. 1

General outline of magnetic pulse welding circuit, and measurement of discharge current and collision time signal.

2.2 Measurement of collision time

In Fig. 1, the discharge current I is detected by a Rogowski coil and passes through an integration circuit, which is then recorded in an oscilloscope. The collision time signal S_c was simultaneously measured with the current I. The fixed sheet (parent sheet) also has a role of a pin detecting signal S_c. The signal S_c was detected by a pin-contact process, which uses a voltage induced in the moving sheet. The simultaneous measurement method does not need the pin or a power supply for the detection of the signal S_c. The signal S_c appears at a moment when the moving sheet top, which deforms into convex shape, contacts with the fixed sheet and passes from two copper foils to the measuring circuit shown in Fig. 1, and is then recorded in the oscilloscope.^5,6,16) However, in one experiment the detection of the signal S_c is only once because the state of the first contact electrically creates a holding circuit.

3. Relationship between Magnetic Pressure on a Moving Sheet and Discharge Current

3.1 Ratio of magnetic field to discharge current

The joining of two sheets was performed after the moving sheet deformed into a convex shape and collided with each other.^4,7,16) The deformation of the moving sheet was performed using the electromagnetic force f shown in Fig. 1, but the numerical analysis of magnetic pulse welding to use the force f was very difficult. A magnetic pressure was used for the numerical examination, and the pressure was equivalent to a component in the z direction of the electromagnetic force f, as shown in Fig. 1.¹⁷⁾ A magnetic flux density B of eq. (2) was used to calculate the magnetic pressure P, and the calculation of B started with the calculation of the magnetic flux density B_I generated by the discharge current I. The ratio H_I/I of the magnetic field H_I to the discharge current I on the underside of the moving sheet in Fig. 1 is shown in Fig. 2. The magnetic field intensities H_x and H_z were the components in the x and z directions, respectively. In Fig. 1, the ratio H_x/I were symmetric with respect to the vertical axis and had a mirror symmetry in the yz-plane. The ratio H_x/I was approximately constant at 90% and 58 m⁻¹ in the coil width of ±2.5 mm, which corresponds to a component generating an electromagnetic force f_z in the z direction with pressurizing action as guided by eq. (2).⁷⁾ The ratio H_z/I was symmetric with respect to the origin, which was 0 m⁻¹ at x = 0 mm and had the maximum value of 48 m⁻¹ at x = −2.5 mm. The ratio H_z/I corresponds to a component generating electromagnetic force f_x in the ±x direction with a stretching action. The ratio H_I/I has the unit m⁻¹, which represents a physical quantity that can be determined by the width or thickness of the coil and is not related to the discharge current I changing with time.

Fig. 2

Ratio of intensity of magnetic field to discharge current.

3.2 Magnetic pressure and magnetic flux density

The magnetic flux φ_I or φ_i was generated between the central part of the coil L_c and the moving sheet, as shown in Fig. 3. The magnetic flux φ_I was generated by the discharge current I and appeared around the central part of the coil L_c in the right-handed screw direction, where it intersected the moving sheet. While the flux φ_i was generated by the eddy currents i, the fluxes φ_I and φ_i overlapped each other because the currents I and i were in opposite directions. A composite magnetic flux can be expressed as the sum of the fluxes φ_I and φ_i and is denoted by φ_I + φ_i. The flux density B_I or B_i was equal to the surface density for the fluxes φ_I or φ_i, and had a component in the x and z directions together. The component of flux density B_i in the x direction was generated by the eddy currents i and was calculated based on the component of flux density B_I in the x direction.¹⁷⁾ In Fig. 3, the magnetic flux density B₁ on the surface of the moving sheet is the sum of the components of the flux density B_I and B_i in the x direction. The magnetic flux density B₂ on the back of the moving sheet can be calculated by eq. (3) because the magnetic flux density B₁ is gradually damping in the sheet.

  

\begin{equation} B_{2} = B_{1}\,e^{(-\tau/\delta)} \end{equation} (3)

where τ and δ are the thickness of the sheet and skin depth, respectively. The skin depth δ is expressed in eq. (4).

  

\begin{equation} \delta = (2/\omega\mu\kappa)^{1/2} = (T/\pi\mu\kappa)^{1/2} \end{equation} (4)

where ω and T are the angular frequency and oscillating period of the discharge current I, respectively, and μ is the permeability of the sheet. δ is proportional to the square root of the period T. A magnetic pressure P applied on the moving sheet was equivalent to the pressure difference between both sheet surfaces shown by the two bold lines, which can be given by eq. (5) by using both the flux densities B₁ and B₂.

  

\begin{equation} P = (B_{1})^{2}/2\mu - (B_{2})^{2}/2\mu = (B_{1})^{2} \times \{1 - e^{(-2\tau/\delta)}\}/2\mu \end{equation} (5)

If the skin depth δ and thickness τ of the moving sheet are given, the value of magnetic pressure P can be obtained.¹⁷⁾ When the thickness τ of the sheet is infinity, the magnetic pressure P would be P = B₁²/2μ and the largest. If the thickness τ is constant, the pressure P would be determined by the flux density B₁ and skin depth δ. The size |B_I| of the flux density B_I that composes the flux density B₁ is proportional to the discharge current I as per Ampere’s circuital law. Therefore, the flux density B₁ is proportional to the current I. The magnetic pressure P is greatly affected by the discharge current I because the pressure P is proportional to the square of the current I based on eq. (5).

Fig. 3

Magnetic flux generated between coil and moving sheet.

3.3 Magnetic pulse welding circuit and discharge current

An equivalent circuit corresponding to the magnetic pulse welding apparatus shown in Fig. 1 is called the welding circuit and is shown in Fig. 4. The equivalent circuit is composed of both the LCR discharge circuit and the moving sheet of the secondary side. C is the capacity of the capacitor bank C, V is a charging voltage, I is the discharge current with damping oscillation, and L_c and R_c are the inductance and resistance of the coil L_c, respectively. The circuits were of four types and the subscript n = 1, 2, 3, and 4 corresponds to circuits C₁, C₂, C₃, and C₄, respectively. L_rn and R_rn are the remaining inductance and resistance, respectively, and both were the sum of components except coil L_c in the LCR discharge circuit. The components of the LRC discharge circuit include the bank C, gap switch G, the coil L_c, wiring of the circuit, and inductance regulator board L_R. M is a mutual inductance concerning an electromagnetic coupling and was treated as a constant in this study. In Fig. 5, the welding circuit is expressed in a simplified equivalent circuit because the resistances R_c and R_m were low together, and the oscillation of the discharge current I was very strong. The current I was formulated using a simplified circuit.

Fig. 4

Equivalent circuit for magnetic pulse welding.

Fig. 5

Simplified equivalent circuit.

L_e and R_e in Fig. 5 represent effective inductance and resistance, respectively, which include the components of both the coil L_c and the moving sheet. L_e and R_e were common for all four welding circuits and can be expressed as eqs. (6) and (7), respectively, by using a coupling coefficient k, which can be expressed as k² = M²/(L_c L_m).

  

\begin{equation} L_{\text{e}} = L_{\text{c}}(1 - k^{2}) \end{equation} (6)

  

\begin{equation} R_{\text{e}} = R_{\text{c}} + k^{2}L_{\text{c}}\,R_{\text{m}}/L_{\text{m}} \end{equation} (7)

The discharge current I was calculated by applying a linear approximation as expressed in eq. (8).

  

\begin{equation} I = V/\omega L\,e^{-\alpha t}\,\mathit{sin}\,\omega t \end{equation} (8)

where α and ω are the damping coefficient and angular frequency, respectively, and are expressed in eq. (9).

  

\begin{equation} \alpha = R/2L,\quad \omega = 2\pi/T = \sqrt{1/LC - (R/2L)^{2}} \end{equation} (9)

where L and R are the circuit inductance and resistance, as expressed in eqs. (10) and (11), respectively.

  

\begin{equation} L = L_{\text{e}} + L_{\text{r}n} \end{equation} (10)

  

\begin{equation} R = R_{\text{e}} + R_{\text{r}n} \end{equation} (11)

When the welding circuit has a strong oscillation characteristic due to a very low circuit resistance R, the circuit inductance L is proportional to an approximate square of the oscillating period T, as shown in eq. (9). In the welding circuit shown in Fig. 4, the remaining inductance L_r1 and remaining resistance R_r1 correspond to the subscript n = 1 for the remaining inductance L_rn and remaining resistance R_rn, respectively, which are physical quantities in a standard circuit C₁. Circuit C₁ removes the inductance regulator board L_R from Fig. 4 and gets directly connected. L_r2, L_r3, and L_r4 are the remaining inductances, and R_r2, R_r3, and R_r4 are the remaining resistances for the circuits C₂, C₃, or C₄, respectively. In Fig. 5, the moving sheet is deformed using some of the energy stored in the effective inductance L_e but none of the energy stored in the remaining inductance L_rn.

4. Experimental Results and Consideration

The discharge current is a principal factor in magnetic pulse welding. When the discharge current passes through the coil L_c shown in Fig. 4, the moving sheet collides with the fixed sheet because the magnetic pressure is applied on the moving sheet. An explanation for this can be obtained by examining the discharge current, magnetic pressure, deformation velocity, and welded sheet, sequentially, and they are affected by the circuit inductance.

4.1 Relationship between the magnetic pulse welding circuit and discharge current

As shown in Fig. 1, an experiment is performed in conditions where the width of the central part of the coil L_c is 5 mm, capacity C of the bank C is 400 µF, discharge energy W is 2.0 kJ, and gap length is 1.0 mm. The moving and fixed sheets were composed of 1050-H24 pure aluminum and C1100-1/4H pure copper sheets, respectively, both having thicknesses of 1.0 mm. The inductance regulator board L_R, which was added to the standard circuit C₁, was a hollow rectangular prism with a thickness of 3 mm and heights of 30 mm at circuit C₂, 90 mm at circuit C₃, and 270 mm at circuit C₄. In Figs. 4 and 5, a regulator board L_R, which is equivalent to the increase in the remaining inductance L_rn, is inserted. The increase in the remaining resistance at the circuit C₄ was less than 3% compared with that of the standard circuit C₁ because of the influence of the skin depth. The discharge currents I flowing in the standard circuit C₁ and the circuits C₂, C₃, and C₄ are shown in Fig. 6. A measuring device from the PEM Corporation was used to determine the current I. In Fig. 6, the vertical, and horizontal axes show the current and time at an interval of 66.5 kA and 20 µs, respectively. All the four discharge currents I was damped oscillation currents and was calculated from eq. (8) by the assumption of the linear approximation. Although the discharge energy W and the gap length d were equal for all four discharge currents, three forming factors of the discharge current I, including the maximum current I_m, oscillating period T, and damping coefficient α, were different. The inductances of the four circuits were different owing to differences in remaining inductance L_rn. From Fig. 6, the circuit inductance L can be obtained for each I and can be expressed for each circuit inductances L₁, L₂, L₃, and L₄. The circuit inductances and the three forming factors of the four discharge currents are shown in Table 1. In Fig. 5, the effective inductance L_e of the four circuits was the same at 0.0336 µH. Based on the changes observed between the standard circuit C₁ and circuit C₄, the circuit inductance L increased from L₁ = 0.0587 µH to L₄ = 0.2280 µH of approximately 3.9 times. The remaining inductance L_rn was calculated by using eq. (10), for example, L_r1 of the standard circuit C₁ was 0.0251 µH. Furthermore, based on the increased L, the maximum current I_m decreased from 223 kA to 132 kA of approximately 0.59 times, and the oscillating period T increased from 30.5 µs to 60.0 µs of approximately 2 times. Consequently, T increased in proportion to the approximate square root of L. The duration of the current I increased from 100 µs to 360 µs because the damping coefficient α decreased from 27800 s⁻¹ to 7170 s⁻¹. Therefore, based on the increase in L, the maximum current I_m decreased, and both the oscillating period T and the duration of I elongated.

Fig. 6

Waveforms of discharge currents for four types of circuit at discharge energy of 2.0 kJ with gap of 1.0 mm (For two numerical values in each waveform, maximum current is vertically shown and oscillating period is horizontally shown).

Table 1 Inductances and elements of discharge currents in Fig. 6.

4.2 Influence of discharge energy on the discharge current

The maximum current I_m and the average slope I_m/t_m obtained by a standard circuit C₁ are shown in Fig. 7. The horizontal axis shows the discharge energy W. t_m is the elapsed time from a current start to the maximum current I_m as well as the time at the maximum current I_m. The maximum current I_m shown by an open symbol ○ is a reference value on |B| of B in eq. (2), and the average slope I_m/t_m shown by a solid symbol ● is a reference value about |∂B/∂t| of ∂B/∂t in eq. (1). The maximum current I_m or the average slope I_m/t_m is expressed in a curve of involution, which increases in proportion to the approximate square root of the discharge energy W. According to an increase in the energy W, t_m is almost equal regardless of W because an increase in the period T is limited to a few percent. Consequently, I_m/t_m increases on a curve similar to I_m. The higher the W, the larger the I_m and I_m/t_m. The influence of the discharge energy W on the maximum current I_m is shown in Fig. 8, where open symbols ○, □, △, and ◇ indicate the I_m of the circuits C₁, C₂, C₃, and C₄ respectively. A solid symbol ◆ indicates the average slope I_m/t_m of the circuit C₄. According to the increase in W, each I_m increased. In each W, I_m was the largest at standard circuit C₁, then I_m decreased from circuit C₂ to C₃, and became the smallest at circuit C₄. The slope of the tangent of each I_m near 2.5 kJ was the largest at the standard circuit C₁, and then the slopes of the other circuits gradually declined from circuits C₂ to C₃ and C₄. In the standard circuit C₁, and circuits C₂, and C₃, I_m was expressed as the curve in involution of W, which was proportioned to the approximate square root of W. In the circuit C₄, I_m was expressed as a linear function of W, which was equivalent to I_m = 32W + 68. In the circuit C₄, the average slope I_m/t_m of the solid symbol ◆ was expressed as a linear function at I_m/t_m = 2.25W + 4.9. However, when the discharge energy W was beyond 3.0 kJ at circuit C₄, an increase in both I_m and I_m/t_m was considered low because the slope of I_m and I_m/t_m was low at 32 and 2.25, respectively. I_m is the reference value for |B|, and I_m/t_m is the reference value for |∂B/∂t|. Based on eq. (1), eq. (2), and eq. (5), the larger the I_m and I_m/t_m, the larger the electromagnetic force f and the magnetic pressure P. Based on the examinations given in Table 1, Fig. 7, and Fig. 8, the circuit inductance L of the circuit C₄ has a maximum of 0.2280 µH, which makes it difficult to obtain a good effect with discharge energy W.

Fig. 7

Effects of discharge energy on maximum discharge current and average slope of discharge current.

Fig. 8

Relationship between maximum discharge current and discharge energy.

4.3 Discharge current and collision time signals

A collision time signal S_c was measured together with the current waveform I by a simultaneous measurement. An example of the current waveform I (upper line) and the collision time signal S_c (lower line) obtained from the standard circuit C₁ is shown in Fig. 9(a). Similarly, that of the circuit C₂ is shown in Fig. 9(b). In Figs. 9(a) and (b), the vertical axis shows current value at a scale of 133 kA/div., and the horizontal axis shows time at a scale of 2 µs/div. The discharge energy W is 2.0 kJ, and the gap length is 1.0 mm. From Fig. 9(a) of the standard circuit C₁, the maximum current I_m, the maximum time t_m, and the zero current time t₀ are 223 kA, 6.20 µs, and 15.04 µs respectively, and the average slope I_m/t_m is 35.9 kA·µs⁻¹. From Fig. 9(b) of the circuit C₂, the maximum current I_m, the maximum time t_m, and the zero current time t₀ are 189 kA, 8.12 µs, and 18.50 µs respectively, and the average slope I_m/t_m is 23.3 kA·µs⁻¹. Although the discharge energy W and the gap length d were equal for the two discharge currents, I_m, and t_m were different because of the differences in the circuit inductances L₁ and L₂. By subtracting the start time of the current I from that of the time signal S_c, the first collision time t_c can be obtained as 5.72 µs for the standard circuit C₁ and 7.28 µs for the circuit C₂.

Fig. 9

Waveforms of discharge current (upper) and collision time signal (lower) for two types of circuit.

4.4 Discharge current and the first collision time

The maximum current I_m and the first collision time t_c obtained by the standard circuit C₁ are shown in Fig. 10. The horizontal axis shows the discharge energy W, and the circuit C₁ does not have the regulator board L_R. An open symbol ○ indicates I_m and a solid symbol ● indicates t_c. With the increase in W, I_m increased, and t_c decreased. It can be considered that a decrease in the first collision time t_c is caused by the increase in the deformation velocity ν of the moving sheet because the gap length d remained constant. The influence of the discharge energy W on the first collision time t_c obtained for each circuit is shown in Fig. 11. t_c is indicated by ○ at the standard circuit C₁, by □ at the circuit C₂, by △ at C₃, and by ◇ at C₄. The higher the discharge energy W, the shorter the first collision time t_c. In each W, t_c was the smallest at the standard circuit C₁, increased from circuit C₂ to C₃ and was the largest at C₄.

Fig. 10

Effects of discharge energy on maximum discharge current and first collision time.

Fig. 11

Relationship between first collision time and discharge energy.

4.5 Relationship between magnetic pressure and circuit inductance

The influence of the circuit inductance L on the maximum current I_m of the discharge current I is shown in Fig. 12. The maximum current I_m was examined in the two discharge energies. The discharge energies W were 3.0 kJ and 2.0 kJ, which were differentiated by the large and small open symbols, respectively. The declining rates of the two curves were almost equivalent to each other. Four vertical lines on the horizontal axis indicate the circuit inductance L in the standard circuit C₁, and circuits C₂, C₃, and C₄. The maximum current I_m is indicated by ○ at a standard circuit C₁, by □ at the circuit C₂, by △ at a circuit C₃, and by ◇ at a circuit C₄. When the circuit inductance L increases from L₁ = 0.0587 µH to L₄ = 0.2280 µH of approximately 3.9 times, the maximum current I_m flowing in the circuit C₄ decreases to approximately 0.60 times compared with that of the standard circuit C₁. The magnetic pressure P was equivalent to the input of the moving sheet, which can be calculated by eq. (5). The pressure P is proportional to the approximate square of the current I and is affected by the skin depth δ, as shown in eq. (4). The maximum magnetic pressure P_m at the maximum current I_m in Fig. 12 is shown in Fig. 13. The maximum pressure P_m was calculated using the assumption that the moving sheet stays still. Two curves of the maximum pressure P_m are inversely proportional to the approximate four-fifths square of the circuit inductance L, and the declining rates of the two curves are almost equivalent to each other. According to the increase in the circuit inductance L, the maximum pressure P_m generated at the circuit C₄ decreased to approximately 0.31 times compared with that of the standard circuit C₁. The maximum current I_m flowing in the circuit C₄ decreased to approximately 0.60 times, and the square of I_m was 0.36 times. 0.31 times as shown in Fig. 13 is smaller than 0.36 times. Based on Table 1 and eq. (4), the skin depth δ is proportional to one-fourth of the square of the circuit inductance L and affects the magnetic pressure P as shown in eq. (5). The maximum pressure P_m declined to 0.31 times instead of 0.36 times because P_m is affected by the skin depth δ and the maximum current I_m.

Fig. 12

Effect of circuit inductance on maximum discharge current.

Fig. 13

Relationship between maximum magnetic pressure and circuit inductance.

4.6 Relationship between deformation velocity and magnetic pressure

In Fig. 1, the moving sheet undergoes plastic deformation with a convex shape after the discharge current I flow in the welding circuit. The deformation velocity ν of the moving sheet was equal to the moving velocity of the top of the sheet, which was equivalent to the first collision velocity at a gap length d. The deformation velocity ν at d = 1.0 mm was calculated by differentiating a height with respect to time based on an approximation curve drawn with the measured values. The approximation curve of any circuit can be obtained by using the first collision time t_c measured at each gap length 0.81, 1.0, and 1.17 mm.⁷⁾ The influence of the circuit inductance L on the deformation velocity ν is shown in Fig. 14, where the velocity ν is equivalent to the output of the moving sheet. Based on the increasing circuit inductance L, the deformation velocity ν decreased from 383 m·s⁻¹ at the standard circuit C₁ to 164 m·s⁻¹ at the circuit C₄ at 2.0 kJ and from 448 m·s⁻¹ at the standard circuit C₁ to 224 m·s⁻¹ at the circuit C₄ at 3.0 kJ. The rate of decrease was 0.43 times at 2.0 kJ and 0.50 times at 3.0 kJ. The deformation velocity ν is inversely proportional to the approximate square root of the circuit inductance L. The velocity ν is the largest at the standard circuit C₁, with the smallest inductance L₁ = 0.0587 µH and the smallest at circuit C₄, with the largest inductance L₄ = 0.2280 µH. The deformation velocity ν also varies depending on the discharge energy W because the material, and thickness of the moving sheet do not change. A relationship between the deformation velocity ν and the maximum P_m of the magnetic pressure P is shown in Fig. 15. The relationship between the output ν and input P_m in the moving sheet is expressed in a straight line and thereby increases proportionally.

Fig. 14

Effect of circuit inductance on deformation velocity of moving sheet.

Fig. 15

Relationship between deformation velocity of moving sheet and maximum magnetic pressure.

4.7 Deformation process of the moving sheet and collision velocity

The deformation process of the moving sheet is shown in Fig. 16.⁴⁾ The moment when the sheet collided with a fixed sheet is shown in Fig. 16(a), and the continuous oblique collision after Fig. 16(a) is shown in Fig. 16(b). In Fig. 16(a), the deforming shape of the moving sheet is in vertical axis symmetry, which is convex at the upper part and concave at both sides. The top of the convex portion first collides with the fixed sheet at a high velocity ν. Two sheets did not join at the moment of the first collision because an angle β between two contact surfaces on the two sheets was kept at 0°. After Fig. 16(a), the deformation toward the z direction of the moving sheet was stopped by the fixtures shown in Fig. 1, and then one collision point was divided into the two collision points shown in Fig. 16(b). After that, a non-collision part outside the two collision points on the moving sheet shown in Fig. 16(a) was deforming toward the direction of ν_p in the xz-plane and continuously colliding with the fixed sheet at a high velocity. Consequently, the collision angle β increased from 0° and simultaneously two collision points moved to the ±x direction at a high transfer velocity ν_c. The colliding surfaces on the two sheets were clarified because metal jets were emitted from the surfaces due to oblique collision. The clarified colliding surfaces were continuously pressurized by an impact force given by the oblique collision and electromagnetic force of eq. (2). It can be considered that the joining of the two sheets started from two positions satisfying a joining condition, and then finished away from the joining condition because the oblique collision moved away from the two positions toward both outside. During continuous collision, it was considered that the collision velocity ν_p was approximately equal to the velocity ν shown in Fig. 16(a) for a few brief moments, and then gradually decreased. The time elapsed in the process was slightly longer than the transfer time of the collision point, which was about 1.5 µs¹⁸⁾ at 5 mm coil width. It can be considered that the decrease in the first collision velocity ν of Fig. 16(a) led to a decline in the velocity ν_p of the oblique collision and in transfer velocity ν_c at the collision point.

Fig. 16

Schematic cross-sectional view of deformation process.

4.8 Joining strength of the welded sheet

The welded sheet joined by laps of the 1050-H24 aluminum sheet and C1100-1/4H copper sheet was 130 mm in width and 80 mm in length (hereafter, the welded sheet is designated as the A1050/C1100 sheet). The thickness of the two sheets was 1.0 mm each. A length of 80 mm of welded sheet was divided into seven pieces of approximately 10 mm width, and then the tensile strength of the welded sheet were examined in three pieces of the central part. The shearing load P_a of the welded sheet is regarded as the average value of the maximum loads obtained at the three pieces.²⁾ The relationship between the shearing load P_a of the welded sheet and the discharge energy W is shown in Fig. 17. The joining strength of the welded sheet was estimated using the shearing load P_a. Open symbols ○, □, △, and ◇ indicate the shearing load of the welded sheet collected by the standard circuit C₁, and circuits C₂, C₃, and C₄ respectively, and the three pieces of the central part are a joining in all and are not a separation on the joined surface. Solid symbols ●, ■, ▲, and ◆ indicate non-joining or separation generated due to the cutting of the welded sheet. Solid symbols , , and indicate the mixture of both the joining or separation of the three pieces. The welded sheet was ruptured in the aluminum sheet, and the minimum discharge energy W_d was 0.6 kJ at the standard circuit C₁, and 0.8, 1.3, and 2.2 kJ at the circuits C₂, C₃, and C₄, respectively. In other words, the higher the L, the higher the W_d. According to the velocity curve of 2.0 kJ shown in Fig. 14, with the increase in the inductance L, the first collision velocity ν shown in Fig. 16(a) decreases from 383 m·s⁻¹ at the standard circuit C₁ to 164 m·s⁻¹ at the circuit C₄. Similarly in A1050/A1050 sheet, the relationship between the shearing load P_a and the first collision velocity ν was determined, and the velocity ν at which the welded sheet ruptured in the base metal was more than 178 m·s⁻¹ at 2.0 kJ.⁶⁾ On the other hand, as shown in Fig. 17, the A1050/C1100 sheet collected from the circuit C₄ with 2.0 kJ was separated but not ruptured, when the deformation velocity of the moving sheet was 164 m·s⁻¹ based on Fig. 14. Therefore, it can be considered that the A1050/C1100 sheet is expected to rupture when the first collision velocity ν was higher than 178 m·s⁻¹ at least. In the continuous collision process shown in Fig. 16(b), the decline in the velocity ν with the increase in the inductance L led to a decrease in the oblique impact force as well as the velocity ν_p of the oblique collision and the transfer velocity ν_c at the collision point. Consequently, the two sheets had difficulty joining. Based on Figs. 14 and 17, the decrease in the first collision velocity ν is considered to have a bad influence on the joining property, such as the joining strength of the welded sheet. In other words, if the first collision velocity ν is high enough, there is always an area where the two sheets can be joined to each other because the collision angle β continuously increases from 0°.

Fig. 17

Shearing load of welded sheets recovered from four types of circuit.

4.9 Joined state and deformation shape of the moving sheet

The joined state of the welded sheet by each circuit is shown in Table 2. The discharge energies W were 2.0 and 3.0 kJ, and the gap length d was 1.0 mm. An open symbol ○ indicates rupture of the moving sheet composing the welded sheet, and a symbol indicates separation. The welded sheet collected at 3.0 kJ was joined by all four circuits, but the welded sheet at 2.0 kJ cannot be joined with circuit C₄. The joining capability of magnetic pulse welding increased with the increase in the discharge energy W.⁷⁾ According to the discrete data given in Table 2 and Fig. 14, the first collision velocity ν, which ruptured the A1050/C1100 sheet was greater than approximately 224 m·s⁻¹. Therefore, it may be considered that an influence of the increase in energy W appears on an opposite deforming part as well as the joined surface of the welded sheet. A sectional schematic view of the welded sheet is shown in Fig. 18. Two sheets can be joined in the shape of two lines in a vertical axis symmetry. The deforming part of the moving sheet was measured along a bold line. The deformation picture of the moving sheet and calibrated deformation curve are shown in Fig. 19. Figure 19(a) shows a deformation on the surface of the moving sheet welded by the standard circuit C₁ at 2.0 kJ. The deformation in x and z directions was observed along the y direction. The amplitude of the deformation was measured at intervals of 5 µm along the white arrow in the middle. Reference points of all deformations cannot be provided in the measurement, but a reference point at the x-axis only was provided by the mirror symmetry in the yz-plane³⁾ of the magnetic pulse welding phenomenon. The electromagnetic force generated in the moving sheet had a mirror symmetry in the yz-plane based on eq. (2) and Fig. 2. Therefore, the deformation of the moving sheet was in mirror symmetry, and thus the two joining parts of the welded sheet also had mirror symmetry. A measured deformation curve was moving with a linear transformation and parallel displacement on the x-axis. This procedure was repeated, and the measured deformation curve was corrected to satisfy the vertical axis symmetry. An example of a calibration curve is shown in Fig. 19(b). The reference point for the x-axis is on the axis symmetry line of the calibration curve, and the numerical values on the vertical axis were temporary. Therefore, a reference point in the z direction cannot be provided, and the depth in the z direction of the welded sheets cannot be compared. However, if the reference point is placed on the deepest point of the calibration curve, a difference in displacement on the z-axis can be found for the welded sheet. The deepest portion shown by a thin ○ in Fig. 19(b) was enlarged, and the deepest portion of the welded sheets that collected circuits C₂, C₃, and C₄ were also enlarged. All enlarged deformation curves are shown in Fig. 20 for 2.0 kJ and Fig. 21 for 3.0 kJ. Two pulsating waves provided based on the vertical axis symmetry were confirmed on each of the four deformation curves of 2.0 kJ and 3.0 kJ. The two pulsating waves appeared outside the coil width of ±2.5 mm, and their positions, and wavelengths were different based on the circuit and discharge energy W. An interval l_xn on the horizontal axis of the two pulsating waves is shown in Table 3, where underlines show that the tops of the deepest portions are flat. The maximum magnetic pressure P_m at the circuit C₂ at 2.0 kJ was approximately 260 MPa based on Fig. 13. The appearance of the flat portion suggests the possibility of exceeding the yield stress of approximately 230 MPa of the C1100-1/4H fixed sheet. The distance l_xn was the largest at the standard circuit C₁, decreased from circuit C₂ to C₃, and was the smallest at circuit C₄. If the discharge energy W was high, l_xn would have increased at all four circuits. In the direction of the vertical axis of Figs. 20 and 21, the displacement difference l_zn between the deepest point and the minimum value of the pulsating wave can be determined. The displacement difference l_zn is shown in Table 4. Both the distance l_xn and the displacement difference l_zn were at their maximum at the standard circuit C₁, decreased from circuit C₂ to C₃, and did not appear at circuit C₄. The relationship between the sizes shown in the two tables may be based on the velocity ν shown in Fig. 14. The moving sheet had limited upward deformation in the z direction due to fixtures shown in Fig. 1. On the other hand, the outward deformation in the ±x direction from the coil L_c was large because two restricted positions were far away from a coil of 5 mm width. Based on relations among Fig. 17 and Tables 2, 3, and 4, it can be considered that the interval and displacement differences in the two pulsating waves are related to the joining strength of the welded sheet.

Table 2 Effect of type of circuit on joining of welded sheets.

Fig. 18

Schematic cross-sectional view of welded sheet.

Fig. 19

Deformation on back of moving sheet composing welded sheet obtained with standard circuit C.

Fig. 20

Deepest part of deformation curve on back of moving sheet for four types of circuits at discharge energy of 2.0 kJ.

Fig. 21

Deepest part of deformation curve on back of moving sheet for four types of circuits at discharge energy of 3.0 kJ.

Table 3 Pulse interval in deepest part of deformation curve.

Table 4 Displacement difference in z direction between deepest point and pulse wave.

4.10 Joined interface in a welded sheet

In magnetic pulse welding operating on a high-tensile steel sheet or a 6061-T6 aluminum alloy sheet, a high deformation velocity of the aluminum sheet is necessary.⁷⁾ In the standard circuit C₁, the circuit inductance L was the minimum, and the joining between the 6061-T6 sheet and the high-tensile steel sheet at 1 GPa was enabled¹⁹⁾ at an experimental condition having a discharge energy of 3.0 kJ and a gap length of 1.17 mm. The joined part of the welded sheet as observed by SEM is shown in Fig. 22. Two types of structures were observed on the joined part. One type was an intermediate layer shown by an upward arrow ↑ and the other type was an adjacent surface shown by an open symbol □. The intermediate layer was not equally distributed and appeared locally with different thicknesses. The impact and electromagnetic forces were found to be acting on the welded sheet. The two forces increased with the increase in the maximum I_m and the average slope I_m/t_m of the discharge current I. In Fig. 16(a), the first collision velocity ν is high at approximately 430 m·s⁻¹. In the continuous collision process shown in Fig. 16(b), the colliding portion included the colliding surface, which was pressurized by the large impact force that initiated a plastic flow in the colliding portion but had no the impact force after the transfer of the collision point was over. However, the colliding portion was strongly pressurized by the large electromagnetic force because the discharge current I was greater than 190 kA in a duration from the first collision to approximately 7 µs. The moving sheet heated up due to eddy currents i, and the colliding surface was affected by the heat of friction, which simultaneously increased the temperature of the colliding portion. Consequently, it can be considered that the plastic flow was continued in the colliding portion and the portion developed on the joined part. The discharge current I with damped oscillation was reduced to zero at approximately 100 µs, and then both the eddy currents and the electromagnetic force were reduced to zero with time. After that, the joined part was not affected by either the electromagnetic force or the heat. The joined part at high temperatures can be quenched by heat dissipation toward the adjacent region in the two sheets.²⁰⁾ Consequently, it is considered that the welded sheet collected between the aluminum alloy sheet and the high-tensile steel sheet can be joined strongly.

Fig. 22

Cross-sectional view of A6061-T6/SPFH sheet (1.0/1.0) welded at discharge energy of 3.0 kJ with gap of 1.17 mm using standard circuit C₁.

5. Conclusion

Since the discharge current is the principal factor in magnetic pulse welding, the experiments were performed using four welding circuits composed of different circuit inductances and one welding coil. The contents are the measurement of the discharge current or the collision time signal, and the tension test of the welded sheet or the deformation measurement on moving sheet. The experimental results can be summarized as follows:

1. (1)    According to the increase in the circuit inductance, the maximum value of the discharge current decreased, but the oscillating period and duration of the current became longer, when the magnetic pressure applied to the moving sheet decreased in proportion to the approximate square of the current. Consequently, the deformation velocity of the moving sheet top decreased proportionally with the magnetic pressure.
2. (2)    Raising the discharge energy raises the deformation velocity of the moving sheet because the amplitude of the discharge current increases. However, in the welding circuit, where the circuit inductance is very high, obtaining a good effect from the discharge energy is difficult because the maximum value and the average slope of the discharge current reduce remarkably.
3. (3)    Raising the circuit inductance causes adverse effects on the discharge current, resulting in the low deformation velocity of the moving sheet and a low joining capability for the welded sheet. In this case, the smallest discharge energy that the welded sheet is ruptured at the base material becomes high because the impact force pressurizing the collided portion during the oblique collision lowers.
4. (4)    The moving sheet’s collision surface and the opposite surface’s deepest region both exhibit the deformation of the welded sheet, together with two pulsating waves. The deformation of the deepest portion is largest in the welded sheet collected by the welding circuit, whose circuit inductance is the smallest. The width of the outward deformation in the ±x direction is related to the joining strength of the welded sheet.
5. (5)    If the maximum value and average slope of the discharge current are large together, the collection of the welded sheet with the strong joining is enabled. It is possible to weld the 6061-T6 aluminum alloy sheet to the 1 GPa high-tensile steel sheet using a welding circuit with the smallest circuit inductance. To collect the welded sheet which was strongly joined, it is important to raise the magnetic pressure and the deformation velocity of the moving sheet by decreasing the circuit inductance.

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