Dynamic analysis of electromagnetic compaction of Ag-Cu-Sn multivariate mixed metal powders for brazing

Abstract

Electromagnetic compaction is a preferable high energy rate forming process for brazing powders to improve the brazing performance. The dynamic analysis of electromagnetic compaction of Ag-Cu-Sn multivariate mixed metal powders was conducted by an electromagnetic-mechanical-thermal coupled model and electromagnetic compaction tests. The coupled model consists of electrical circuit model, electromagnetic field model, kinematic model of driving board, thermal field model and powder compaction model, it can be used to make a complete dynamic analysis of the electromagnetic compaction including discharge current, eddy current, mutual interactions of magnetic fields, kinematic motion of driving board, compaction densification behaviors and the energy transfer. The electromagnetic compaction tests of Ag-Cu-Sn multivariate mixed metal powders were carried out with a self-developed WG-IV electromagnetic forming machine. The electromagnetic-mechanical-thermal coupled model has good agreement with the electromagnetic compaction tests, and can provide a more accurate dynamic analysis and optimized process design of electromagnetic compaction. An optimization study for higher relative density was completed by the coupled model, and the brazing performance of the optimized brazing sheet with higher relative density was evaluated to be advantageous in the brazing performance tests.

Keywords

Electromagnetic compaction dynamic model optimization study brazing performance

1. Introduction

Brazing is the most feasible and economical joining technique. It has significant potential for structural and electronic applications, especially for high-performance microelectronic packaging. Ag-Cu-Sn alloys have been widely used as an active brazing alloy due to the advantageous properties, such as environmental friendliness, good mechanical properties, corrosion resistance, preferable wettability and low leak rates [1,2]. However, due to the existence of brittle compounds and the poor plastic workability evolved during the alloys smelting process, the Ag-Cu-Sn alloys are hard to be made into specific-shaped brazing sheet of minimum thickness in the following compaction process. Electromagnetic compaction is a high energy rate forming process, which is completed by the pulsed energy generated by the two opposite magnetic fields in the adjacent conductors in a very short time (less than tens of milliseconds). It has many advantages in terms of high forming precision, greater compaction densification, enhanced cost effectiveness [3]. Electromagnetic compaction is a preferable forming process for Ag-Cu-Sn powders to improve the brazing performance, the Ag-Cu-Sn powders are firstly compacted into thin brazing sheet with higher density and uniform density distribution, then sintered for alloys smelting. The higher relative density and uniform density distribution of the brazing sheet are beneficial to dimensional control during sintering process, as well as to enhance the brazing performance.

Electromagnetic technique was firstly employed for powder compaction in 1976 [4], and has been further developed in recent twenty years due to the great interests from automobile, electronic and aerospace industries. Intensive researches were carried out to study the electromagnetic compaction process [5–7], and the effects of electromagnetic parameters on the microstructure, hardness, density and compressive strength of metallic alloy powders were experimentally investigated [8–13]. The electromagnetic compaction system is a mutual induction system composed of discharge coil and induction part, and the coupled interaction between electromagnetic field and kinematic field of induction part plays the dominant role for compaction densification. In order to better understand the dynamic behaviors of the electromagnetic compaction, many efforts have been taken to solve the coupled problems [14–17]. Paese [18,19] presented a simplified mathematical model to evaluate the mutual inductance between magnetic field of a flat spiral coil and deformed sheet. Cao [20] revealed that the electromagnetic field and the mechanical behavior of induction part during electromagnetic forming process are strongly interrelated, and the full-coupling dynamic analysis of electromagnetic forming process was more accurate than loose coupling or sequential coupling analysis. The densification behavior of brazing powders under the pulse energy is extremely complex, the inter-particle pressures is more than 1 GPa, and the brazing powders have to make high speed movement of over 10 ∼100 m/s in a transient time. The full understanding of the macroscopic response (relative density, stress evolution, compaction pressure, springback or segregation) during electromagnetic compaction is lacking. A coupled electromagnetic-kinematic dynamic model for electromagnetic compaction process is necessary to analyze the densification behavior of brazing powders, and provide theoretical guidelines for process design of electromagnetic compaction in order to achieve the higher density and uniform density distribution.

In this study, a dynamic analysis of electromagnetic compaction of Ag-Cu-Sn brazing powders was made by the compaction test and an electromagnetic-mechanical-thermal coupled model. The compaction test was carried out with a self-developed WG-IV electromagnetic pulse forming machine. The electromagnetic-mechanical-thermal coupled model consists of electrical circuit model, electromagnetic field model, kinematic model of driving board, thermal field model and powder compaction model. Then the discharge current, eddy current, mutual interactions of magnetic fields, kinematic motion of driving board, compaction densification behaviors and the energy transfer during electromagnetic compaction were detailed analyzed. At last, an optimization study for higher relative density was conducted by the coupled model, and the brazing performance of the optimized brazing sheet was evaluated in the performance test.

2. Electromagnetic compaction test

2.1. Electromagnetic compaction process

The capacitor bank consists of a series of capacitors in parallel assembly, and is connected to commercial grid when charging, the electrical energy stored in the capacitor bank after charging is given by $\begin{eqnarray}\displaystyle E_{0}=0.5CU_{0}^{2}. & & \displaystyle\end{eqnarray}$ (1)

Where C, U ₀ are the equivalent capacitance and voltage of capacitor bank, respectively.

As shown in Fig. 1, a flat spiral discharging coil, connected to the capacitor bank, is placed above a circular driving board. As discharge occurs, the damped oscillating current is released from the capacitor bank and flows through the discharge coil, then a pulsed magnetic field is generated and penetrates the driving board where an eddy current is induced, the interactions between magnetic field and the eddy current results in Lorentz’s forces that is the main driving force for accelerating the driving board, the driving board together with the tapered amplifier and upper punch is accelerated up to a desired velocity for powder compaction. As shown in Fig. 2, the brazing powders are packed into a cylindrical die, then the high-velocity upper punch impacts with high pressure on the powders. The compaction rate ranges from 10² to 10³, and the final relative density of the brazing sheet after electromagnetic compaction may be as high as over 0.90.

Fig. 1.

Scheme of the discharge circuit and pulsed magnetic field.

Fig. 2.

Scheme of the electromagnetic compaction system.

2.2. Compaction test

The compaction test was carried out with a self-developed WG-IV electromagnetic pulse forming machine at Wuhan University of Technology (China), detailed parameters of the electromagnetic pulse compaction system are listed in Table 1. The tapered amplifier, die, upper punch and lower punch are made of tool steel (TC70 in ISO4957) with HRC of 65 after heat treatment, the density is 7750 kg/m³, the specific heat capacity is 477 J/kg ⋅ K, and the thermal conductivity is 53 W/m ⋅ K. The discharge coil and driving board are made of copper material, the density is 8900 Kg/m³, the specific heat capacity is 385 J/kg ⋅ K, the thermal conductivity is 401 W/m ⋅ K, the electric conductivity is 59000000 S/m, and relative magnetic permeability is 0.99990. The total mass of the driving board, tapered amplifier and upper punch is 1.2 kg.

As shown in Fig. 3, two temperature sensors are respectively placed at the bottom surface of the holes and connected to DH5922N signal processing system. The two holes are respectively at 1/3 and 1/4 height of the die, and the bottom surface of the hole is about 2 mm far from the internal surface of the die. The brazing powders in this compaction test are 60Ag-31Cu-9Sn (wt.%) powders. The powders were prepared by gas atomization process and selectively sieved, the particle sizes are average values from the Mastersizer 2000 laser particle analyzer, and the particle morphologies of each elemental powders are shown in Fig. 4. The electromagnetic compaction test was completed at room temperature of 23 °C.

Fig. 3.

Scheme of temperature measuring.

Table 1

Parameters of the WG-IV electromagnetic pulse compaction system

	Parameter	Value
Flat spiral coil	Number of wings	7
	Outer diameter	110 mm
	Pitch	7
	Cross section	40 mm²
	Self-inductance	1.94 μH
	Resistance	0.4 mΩ
Capacitor bank	Capacitance	1100 μF
	Maximum voltage	11000 V
	Maximum energy	60 kJ
Driving board	Diameter	130 mm
	Thickness	1 mm

Fig. 4.

SEM images of Ag, Cu and Sn powder (a is Ag powders with 53 μm average diameter, b is Cu powders with 47 μm average diameter, c is Sn powders with 32 μm average diameter).

3. Dynamic model of electromagnetic compaction process

3.1. Formulation of dynamic model

3.1.1. Electrical circuit model

The electrical circuit model consists of a discharge circuit model and an eddy current circuit model, as shown in Fig. 5. The RLC discharge circuit in electromagnetic system consists of capacitor bank (C), resistor (R ₀) and inductor (L ₀) of the connecting cables and activation switch, as well as an equivalent resistor (R ₁) and a self-inductor (L ₁) of discharge coil coupled with the driving board. The differential equation of discharge circuit can be given by applying Kirchhoff’s voltage law. $\begin{eqnarray}\displaystyle (R_{0}+R_{1})\overrightarrow{I_{1}}+\displaystyle \frac{1}{C}\int \overrightarrow{I_{1}}dt+\displaystyle \frac{d}{dt}(L_{0}\overrightarrow{I_{1}})+\displaystyle \frac{d}{dt}(L_{1}\overrightarrow{I_{1}})+\displaystyle \frac{d}{dt}(M\overrightarrow{I_{2}})=\overrightarrow{U_{0}}. & & \displaystyle\end{eqnarray}$ (2)

Where $\overrightarrow{I_{1}}$ is the discharge current, $\overrightarrow{I_{2}}$ is the eddy current induced in driving board, M is the mutual-inductance between discharge coil and driving board.

The differential equation of eddy current circuit in driving board is given as, $\begin{eqnarray}\displaystyle R_{2}\overrightarrow{I_{2}}+\displaystyle \frac{d}{dt}(L_{2}\overrightarrow{I_{2}})+\displaystyle \frac{d}{dt}(M\overrightarrow{I_{1}})=0. & & \displaystyle\end{eqnarray}$ (3)

Where R ₂, L ₂ are resistance and self-inductance of eddy current circuit in driving board, respectively.

Fig. 5.

Electrical circuit model.

The discharge current flows through the flat spiral coil with a same intensity, frequency and phase, so the self-inductance of discharge coil can be given as, $\begin{eqnarray}\displaystyle L_{1}=\mathop{\sum }_{i=1}^{N}\displaystyle \frac{{\Phi}_{i}}{I_{1}}=\mathop{\sum }_{i=1}^{N}\int _{S}\displaystyle \frac{\overrightarrow{B}}{I_{1}}d\overrightarrow{S_{i}}. & & \displaystyle\end{eqnarray}$ (4)

Where N is the loop number of discharge coil, $\overrightarrow{B}$ is the magnetic flux density in each point of the ith loop of discharge coil, $\overrightarrow{S_{i}}$ is the surface area of ith loop.

The mutual-inductance between discharge coil and ith individual ring conductor in driving board can be given as, $\begin{eqnarray}\displaystyle M_{1i}=\displaystyle \frac{{\Phi}_{1i}}{I_{1}}\quad (i=1,2,...,N). & & \displaystyle\end{eqnarray}$ (5)

Where 𝛷_1i is the magnetic flux produced by $\overrightarrow{I_{1}}$ that links to the ith individual ring conductor in driving board.

The mutual-inductance between ith individual ring conductor and jth individual ring conductor in driving board is given as, $\begin{eqnarray}\displaystyle M_{ij}=\displaystyle \frac{{\Phi}_{ij}}{I_{2i}}\quad (i=1,2,...,N,j=1,2,...,N). & & \displaystyle\end{eqnarray}$ (6)

Where 𝛷_ij is the magnetic flux produced by $\overrightarrow{I_{2i}}$ in ith ring conductor that links to the jth ring conductor in driving board. What’s more, when i = j, M _ii is the self-inductance L _2i of the ith individual ring conductor in driving board.

The mutual-inductance between discharge coil and driving board is not a constant and varies with the displacement of moving driving board. An approximate approach for accurate calculation of mutual-inductance is to make a coupled analysis by considering the influence of displacement of driving board on magnetic field.

3.1.2. Electromagnetic field model

A pulse magnetic field is generated and eddy current is induced in the driving board as soon as the time-varying discharge current flows through the flat spiral coil. The Maxwell equations of the pulse magnetic field are as follows, $\begin{eqnarray}\displaystyle {\nabla}\times \overrightarrow{E}+\frac{d\overrightarrow{B}}{dt}=0 & & \displaystyle\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle {\nabla}\times \overrightarrow{H}-\overrightarrow{J}=0 & & \displaystyle\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle {\nabla}\times \overrightarrow{B}=0. & & \displaystyle\end{eqnarray}$ (9)

Where $\overrightarrow{H}$ is magnetic intensity, $\overrightarrow{J}$ is current density, $\overrightarrow{E}$ is electric intensity.

The materials in the magnetic field are equivalent to be linear, thermal independent and of isotropic properties, according to Ohm’s law, $\begin{eqnarray}\displaystyle \overrightarrow{J}={\sigma}_{e}\overrightarrow{E}. & & \displaystyle\end{eqnarray}$ (10)

Where σ_e is electrical conductivity.

And it is known that, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{d\overrightarrow{B}}{dt}=\displaystyle \frac{\partial \overrightarrow{B}}{\partial t}-{\nabla}\times (\overrightarrow{v}\times \overrightarrow{B})\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times \overrightarrow{A}=\overrightarrow{B}.\end{eqnarray}$ (12)

Where $\overrightarrow{v}$ is the velocity of medium in magnetic field, $\overrightarrow{A}$ is magnetic vector potential [21].

From Eqs (7), (10), (11) and (12), it is deduced as follow, $\begin{eqnarray}\displaystyle \overrightarrow{J}=-{\sigma}_{e}{\nabla}{\varphi}-{\sigma}_{e}\displaystyle \frac{\partial \overrightarrow{A}}{\partial t}+{\sigma}_{e}\overrightarrow{v}\times ({\nabla}\times \overrightarrow{A}). & & \displaystyle\end{eqnarray}$ (13)

Where $\varphi$ is the scalar electric potential resulting from the voltage applied to each elementary loop, and the ${−}\sigma_e\mathrm{\nabla}\varphi$ can be regarded as the external current applied to the medium [22].

From Eqs (8) and (13), it is deduced as follow, $\begin{eqnarray}\displaystyle {\nabla}\times ({\mu}^{-1}{\nabla}\times \overrightarrow{A})=-{\sigma}_{e}{\nabla}{\varphi}-{\sigma}_{e}\displaystyle \frac{\partial \overrightarrow{A}}{\partial t}+{\sigma}_{e}\overrightarrow{v}\times ({\nabla}\times \overrightarrow{A}). & & \displaystyle\end{eqnarray}$ (14)

Where 𝜇 is the magnetic permeability.

Then the magnetic force density in driving board can be obtained, $\begin{eqnarray}\displaystyle \overrightarrow{F_{m}}=\left(-{\sigma}_{e}\displaystyle \frac{\partial \overrightarrow{A}}{\partial t}+{\sigma}_{e}\overrightarrow{v}\times \overrightarrow{B}\right)\times \overrightarrow{B}. & & \displaystyle\end{eqnarray}$ (15)

3.1.3. Kinematic model of driving board

Driven by the magnetic force between discharge coil and driving board, the driving board together with tapered amplifier and upper punch is accelerated from zero velocity to a desired velocity for powder compaction. The kinematic model of driving board is given by, $\begin{eqnarray}\displaystyle m\displaystyle \frac{\partial ^{2}\overrightarrow{u}}{\partial t^{2}}=\int _{V}\overrightarrow{F_{m}}dV. & & \displaystyle\end{eqnarray}$ (16)

Where $\overrightarrow{u}$ , m are displacement and the total mass of driving board, tapered amplifier and upper punch, respectively. V is the volume of driving board.

And the accumulated kinematic energy before impacting on powders is given as follow, $\begin{eqnarray}\displaystyle E=0.5m\left(\int _{0}^{t}\int _{V}\displaystyle \frac{\overrightarrow{F_{m}}}{m}dVdt\right)^{2}. & & \displaystyle\end{eqnarray}$ (17)

3.1.4. Powder compaction model

The densification behavior during powder compaction is governed by the inter-particle interactions and the plastic deformations of powder particles under compaction pressure from upper punch. Johnson and Cook plasticity model, a typical high strain-rate sensitive and temperature dependent model, is employed to describe the dynamic mechanical properties of Ag-Cu-Sn brazing powders during electromagnetic compaction. The flow stress is given as follow, $\begin{eqnarray}\displaystyle {\sigma}=(A+B{\varepsilon}^{n})(1+Cln\dot{{\varepsilon}}^{\ast })\left[1-\left(\displaystyle \frac{T-T_{room}}{T_{melt}-T_{room}}\right)^{m}\right]. & & \displaystyle\end{eqnarray}$ (18)

Where A, B, C, n and m are user-defined input constants that are acquired from a series of physical tests, ϵ is effective plastic strain for particle’s plastic deformation, $\dot{{\varepsilon}}^{\ast }=\dot{{\varepsilon}}/\dot{{\varepsilon}}_{0}$ is effective plastic strain rate for $\dot{{\varepsilon}}_{0}$ (in units of 1/time), T _melt is melting temperature, T _room is reference temperature. The detailed parameters of the Ag, Cu and Sn powder material are listed in Table 2 [23].

Table 2

Parameters of Ag, Cu and Sn powder material

	Johnson-Cook plasticity model					Thermodynamic model
	A MPa	B MPa	C	n	m	Density kg/m³	Specific heat J/kg ⋅ K	Melting temperature K
Ag	70	177	0.023	0.27	1.09	1050	237	1230
Cu	103	316	0.018	0.31	1.29	8900	386	1350
Sn	23	113	0.035	0.33	1.35	7280	228	502

The contact interactions are modeled by penalty contact algorithm. An interface force, that is proportional to the amount of penetration, is applied between the slave node and its contact point in master surface. $\begin{eqnarray}\displaystyle F_{n}=k_{i}{\Delta}_{i}. & & \displaystyle\end{eqnarray}$ (19)

Where k _i is the interface stiffness, and Δ_i is the penetration of the slave node through the master segment which contains its contact point. In case of unacceptable penetration as a result of large interface pressure, the time step size Δt is necessary to be scaled down.

Friction in penalty contact algorithm is based on the Coulomb formulation, which is equivalent to an elastic-plastic spring. The computing steps are as follows: (1) computing the yield force, F _y = 𝜇F _n, where 𝜇 is frictional coefficient; (2) computing the incremental movement of the slave node, ${\rm\Delta}e=r^{n+1}({\xi}_{C}^{n+1},{\eta}_{C}^{n+1})-r^{n+1}({\xi}_{C}^{n},{\eta}_{C}^{n})$ , where (𝜉_C, 𝜂_C) is contact point coordinates; (3) updating the interface force to a trial force f ^∗ = f ⁿ − k _iΔe; (4) accepting the value f ⁿ⁺¹ = f ^∗, if |f ^∗|≤ F _y, then continues a new calculation in next time step; and (5) scaling the trial force if it is too large, f ⁿ⁺¹ = F _y f ^∗∕|f ^∗|, if |f ^∗| > F _y.

The thermodynamics during electromagnetic compaction is usually completed in several milliseconds. The frictional work at contact interfaces and the plastic deformation work of powders are considered as two main heating generations for temperature rise of both powders and die body. The heating generation during electromagnetic compaction is given by, $\begin{eqnarray}\displaystyle Q=Q_{1}+Q_{2} & & \displaystyle\end{eqnarray}$ (20)

Where Q ₁ is the frictional heating at frictional interfaces, and Q ₁ = 𝜇F _n v, where v is the relative sliding velocity at contact interface, and v = Δe∕Δt. Q ₂ is the heating from plastic deformation of powders, and Q ₂ = 𝜂σϵ, where 𝜂 is Taylor-Quinney coefficient.

The frictional heating at frictional interfaces flows into contacting parts through each side of the interface, and the energy partition R _i of contacting part i at frictional interface is given by, $\begin{eqnarray}\displaystyle R_{i}=1/[1+\{(K{\rho}C_{p})_{j}/(K{\rho}C_{p})_{i}\}^{0.5}]. & & \displaystyle\end{eqnarray}$ (21)

Where K𝜌C _p is the geometric mean thermal property (GMTP) of the contacting parts i, then the frictional heating flowing into contacting part i is Q _1i = R _i Q ₁.

The electromagnetic compaction is assumed to be an adiabatic process with anisotropic linear thermo-elasticity and linear Fourier heat conduction. The differential equations of heat conduction in a continuum material is given by, $\begin{eqnarray}\displaystyle {\rho}C_{p}{\dot{T}}=K{\nabla}^{2}T+\dot{Q}. & & \displaystyle\end{eqnarray}$ (22)

Where, 𝜌 = 𝜌(x _i) , C _p = C _p(x _i) and K = K(x _i) are temperature, material density, specific heat capacity and heat conduction coefficient.

3.2. Solving procedure and method

The solving procedure of the electromagnetic-mechanical-thermal coupled model is shown in Fig. 6. Once the initial parameters of the electromagnetic compaction system is determined, the discharge current in coil and the eddy current in driving board can be calculated, also the mutual inductance is available to calculate the magnetic force density in the time-varying magnetic field, then the displacement of driving board can be calculated by solving the kinematic model, now that the first solving iteration is completed. Actually the mutual inductance as well as the magnetic fields changes with the varying displacement of driving board, and have to be updated in the next solving interaction until the discharge comes to the end. As soon as the upper punch comes into compacting on the brazing powders in the die, the kinematic energy of driving board, tapered amplifier and upper punch, which is accumulated during the discharge period, is treated as the pulse energy input for powder compaction model.

Fig. 6.

Solving procedure of the electromagnetic-mechanical-thermal coupled model.

A finite element method (FEM) by the use of LS-DYNA^® package (R7 version) was taken to numerically solve the electromagnetic-mechanical-thermal coupled model. It is hypothesized for simplification that all electrical resistance of the discharge coil and driving board are temperature independent. The FEM modeling of electromagnetic compaction is shown in Fig. 7. The flat spiral coil is firstly coupled with the discharge circuit, once the electromagnetic fields have been computed, the Lorentz force is evaluated at the nodes of driving board and added to the mechanical solver for computing the kinematic motion of driving board, also the Ohmic heating induced by discharge current and eddy current is added to the thermal solver for computing the temperature field in the coil and driving board. The plastic deformation of brazing powders and frictional interactions during powder compaction are solved by mechanical solver, also the temperature rise of brazing powders are solved by the thermal solver to update the mechanical properties of brazing powders materials according to John-cook plasticity model. It needs to be noted that each powder particle is normalized as sphere and uniformly discretized. An actual number of the brazing powders are randomly assembled in the die, the inter-particle frictional coefficient is 0.23, while the die wall/powder frictional coefficient is 0.16.

Fig. 7.

FEM modeling of the electromagnetic-mechanical-thermal coupled model.

4. Result and analysis

4.1. Analysis of electromagnetic process

As shown in Fig. 8, the discharge current in flat spiral discharge coil and the eddy current in driving board are sinusoidal and exponentially decaying. The simulated discharge current has reasonable agreement with the measured discharge current. The eddy current represents a quicker decaying than discharge current, which is due to the motion of driving board in the magnetic field. As the driving board moves away from the discharge coil, the magnetic coupling between discharge coil and driving board becomes looser, which results in a further reduction of the eddy current in driving board.

Fig. 8.

Discharge current and eddy current.

The magnetic field per ampere was defined for convenience in this study to analyze the magnetic field distribution. The magnetic field generated in driving board is shown in Fig. 9. The magnetic field is symmetrically distributed, the maximum can be found at 0.64 radius, and the minimum at center or external edge of the driving board. Also several peaks exist close to each loop of the discharge coil. In the time-varying magnetic field, the magnetic field per ampere is directly determined by the distance between discharge coil and driving board, the magnetic flux density is inversely proportional to the distance, so the peaks can be further pronounced with the decreasing distance between discharge coil and driving board.

Fig. 9.

Distribution of magnetic field in driving board.

4.2. Kinematic analysis of electromagnetic compaction

The kinematic motion of driving board relies on the inductive electromagnetic force applied on driving board. The magnetic force density in all of the nodes of driving board was calculated in the time-varying electromagnetic field, then total magnetic force versus time was computed, as shown in Fig. 10. It can be seen that considerable magnetic forces are developed in the first cycle, but it appears more convincing to account for the whole magnetic force cycles to make accurate studies of the electromagnetic compaction. As time goes on, the magnetic force on driving board decreases rapidly due to the gradual decaying of the transient currents and the increasing distance between discharge coil and driving board. Also an attracting force instead of repulsing force occurs at 0.47 ms causes a negative effect on accelerating motion of driving board, which is resulted from the phase difference between discharge current and eddy current.

Fig. 10.

Magnetic force on driving board.

The velocity of driving board during the electromagnetic compaction process is shown in Fig. 11. Firstly, the driving board was accelerated under the pulse magnetic force, and increased up to about 9.6 m/s during the first pulse of magnetic force, then followed by a little velocity decrease of 0.23 m/s due to the attracting magnetic force between discharge coil and driving board. What’s more, the second cycle of the magnetic force contributes a further velocity increase from 9.37 m/s to 11.6 m/s, which demonstrates the necessities of accounting for the whole magnetic force cycles to make accurate studies of electromagnetic compaction. At 1.6 ms, the upper punch came into impacting on brazing powders in the die and began to apply pulse compaction pressure on brazing powders, but the velocity of driving board indicates no obvious decrease from 1.6 ms to 2.3 ms, this is because the compaction pressure is relatively small due to the lower relative density of the brazing powders in the first stage of compaction densification, in which inter-particle friction, sliding and rotation are the dominators of densification behaviors. And after 2.3 ms, the velocity of driving board decreases more rapidly because the plastic deformation of powder particles turns to be the dominator of densification behaviors in the second stage, in which the kinematic energy of the driving board, tapered amplifier and upper punch is mostly consumed by the plastic deformation of powder particles. After the end of electromagnetic compaction at 3.7 ms, the upper punch together with the tapered amplifier and driving board has a weak rebound velocity, which will be eliminated by mechanical friction or anti-rebound device.

Fig. 11.

Velocity of driving board.

The stress field of brazing powders during electromagnetic compaction is a macroscopic representation of densification behaviors including the plastic deformations of powder particles and massive inter-particle interactions. As shown in Fig. 12, the Von Mises stress field of the brazing powders is non-uniform and varies from 162 MPa to 802 MPa. The high Von Mises stress domain locates in the top corner near the contact area with the upper punch, the Von Mises stress is mostly above 610 MPa which indicates significant plastic deformations of powder particles and drastic contact interactions. The lower Von Mises stress domain locates in the bottom corner near the contact area with the lower punch, some particles in this domain may not go through complete plastic deformation, and some small pores can be found in this domain. Also an axial stress gradient was evolved in the center part of the brazing powders, which is related to the inter-particle friction.

Fig. 12.

Stress contour of brazing powders at the end of compaction.

4.3. Energy analysis during electromagnetic compaction

As shown in Fig. 13, temperature rise at the same positions was obtained in compaction simulation to compared with the experimental results. Though the simulated curves are a little lower than experimental curves during the compacting process and a little higher after the end of compaction, the simulated temperature rise at each point shows reasonable agreement with the experimental temperature rise.

Fig. 13.

Curves of temperature rise at measuring points.

The transient temperature evolution of brazing powders during electromagnetic compaction is an important concern to densification mechanism. A higher temperature rise is beneficial to compaction densification because more plastic deformations could be completed at high temperature. The temperature rise of powders is caused by the accumulated heating energy resulting from the irreversible energy conversion from plastic deformation works as well as the frictional work at the inter-particle interfaces and die wall/powder interfaces. As shown in Fig. 14(a), the temperature field at 3.7 ms is non-uniform and has significant temperature gradient both in axial or radial direction. The temperature rise in the top corner near upper punch keeps the highest throughout the compaction process, lower temperature rises at the central part and even lower close to the lower punch. Several particles in the top corner have the maximum temperature rise as high as 226 K, which is higher than the melting point of Sn alloy. The comparatively larger temperature gradient in top corner indicates the significant local densification and limited inter-particle thermal conductivity in the transient time. The lowest temperature rise in lower corner is about 21 K, which is due to the poor densification related to the lowest Von Mises stress. Due to the additional frictional heat at die wall/powder interfaces, the temperature rise of the brazing powders in direct contact with the die walls appears to be higher than that at the same height.

The temperature rise of discharge coil is the result of Joule heat caused by the pulse discharge current. As shown in Fig. 14(b), at 1.3 ms, about the end time of first discharge cycle, the temperature field of the discharge coil is relatively uniform, and the average temperature rise is 4.7 K. By contrast, the driving board has a non-uniform temperature field at the same time. The temperature peak is 6.59 K at 0.64 radius of the driving board, where is the maximum of the magnetic flux density as well as the induced eddy current. And approximately no temperature rise occurs in the center part due to less magnetic flux density and the limited heat conduction within driving board in the transient time. Most of the reinforcement or insulation materials for discharge coil are poor thermal conducting, the accumulated heating energy in the long-term discharge sequence may lead to thermally induced failure of the coil, so it is necessary to consider the thermal aspects during coil design and process optimization.

Fig. 14.

Distribution of temperature rise.

Energy transfer is of great importance to evaluate the feasibility of a specific electromagnetic compaction process. From the viewpoint of electromagnetic compaction system, the electrical energy initially stored in the capacitor bank is firstly transformed into a magnetic pressure on driving board, and the magnetic pressure is transferred into kinematic energy of driving board, tapered amplifier and upper punch, then the kinematic energy is used to drive the compaction densification of brazing powders. The energy transfer of electromagnetic compaction is completed as soon as the compaction pressure from upper punch decreases to zero. Actually, only parts of the electrical energy are transferred through this energy transfer chain to the brazing powders during electromagnetic compaction, others are lost due to electrical resistance of supply lines, discharge coil or switches, the magnetic field leak as well as the kinematic frictions. Most of the lost energy is transferred into heating energy, which contributes to the temperature rise of discharge coil, driving board, supply lines and brazing powders. As shown in Fig. 15, the plastic deformation work of powder particles represents an approximately quadratic increasing with time, which reveals that plastic deformation of powder particles is the dominating densification behavior in the second stage of powder compaction and the main contributor to temperature rise of brazing powders. Frictional work at inter-particle interfaces and die wall/powder interfaces is comparatively small throughout the compaction process, and approximately follows a linear relationship with time.

Fig. 15.

Energy transfer during electromagnetic compaction.

5. Application

5.1. Optimization study of electromagnetic compaction process

Relative density promotion is a paramount concern during electromagnetic compaction of brazing powders. Higher relative density can effectively lower the sintering temperature for target density, and results in less shrinkage and less distortion during sintering process, which are beneficial to better dimensional control with enhanced properties of brazing sheet. As analyzed in Section 4, there are many influential factors on densification behaviors during electromagnetic compaction of brazing powders. Due to the convenient adaptation in industrial application, discharge voltage, discharge capacitance and the size distribution of powders are usually the core concerns in order to achieve a higher relative density. A series of calculations with electromagnetic-mechanical-thermal coupled model were made in the optimization study for higher relative density. As shown in Fig. 16, the relative density increases with the increase of discharge voltage, an approximately linear relation is found between relative density and discharge voltage when the discharge voltage is lower than 3300 V, but a slight increase when the discharge voltage is over 3500 V. The relative density for 900 μF capacitance is lower than that for 1100 μF capacitance at the same discharge voltage due to the less impacting energy on powders, and the two curves gradually converge at higher discharge voltage.

Fig. 16.

Relation curves between discharge voltage and relative density.

Ag particles are the majority contents for 60Ag-31Cu-9Sn brazing powders. In condition of a 53 μm defined size of Ag particles, the sizes of Cu, Sn powders can be chosen as independent variables, second-degree polynomial model was applied in the response surface study for a promoted compaction densification of brazing powders in order to get a higher relative density. As shown in Fig. 17, under 3800 V discharge voltage and 1100 μF discharge capacitance, the relative density grossly decreases with the decrease of particle sizes of Cu, Sn powders. A drastic decrease of relative density occurs as soon as the particle size of Cu powders is lower than 30 μm, the observed effects of particle size on relative density can be attributed to the plastic strain gradients in particles and more frictional contacts of smaller particles. Two peak values of relative density are illustrated in the response surface in Fig. 17. The first is at 50 μm of Cu powders and 10 μm of Sn powders, which can be explained by the enhanced flowability resulted from the small Sn particles. The small Sn particles can effectively fill the adjacent pores of the large particles under the external compaction pressure, which is beneficial to the local densification during electromagnetic compaction. The second is at 27.5 μm of Cu powders and 45 μm of Sn powders, which can be explained by the reasonable size ratio between Cu powders and Ag powders. A reasonable size ratio can reduce the additional energy consumed by particle rearrangement, frictional sliding and strain hardening rate during electromagnetic compaction.

Fig. 17.

Response surface of size distribution of Cu, Sn powders.

5.2. Brazing performance test

Three types of brazing sheets were prepared under 3800 V discharge voltage and 1100 uF discharge capacitance of WG-IV electromagnetic pulse forming machine. The first was made of 53 μm Ag, 47 μm Cu and 50 μm Sn powders, the second was made of 53 μm Ag, 25.4 μm Cu and 43 μ m Sn powders, the third was made of 53 μm Ag, 47 μm Cu and 27 μm Sn powders. The three brazing sheets were sintered at 500 K for 20 minutes in a vacuum environment. The relative densities of the three brazing sheets were all promoted during this liquid-phase sintering, the first sheet was increased from 90.7% to 92.1%, the second sheet was increased from 92.4% to 93.8%, and the third sheet was increased from 92.6% to 94.1%. The evolved microstructures of the sintered sheets are shown in Fig. 18, and it can be concluded that the Cu-Sn alloying is the main contributor for enhancing the relative density during sintering. The third sintered sheet has less residual pores around the particles than the first sintered sheet, and the size of the pores are a little relatively smaller. This is because that the smaller pores in the brazing sheet with higher relative density can be more easily eliminated by the liquid phase of Sn material during sintering process.

Fig. 18.

Microstructures of sintered brazing sheets (a is the first sheet, b is the second sheet, c is the third sheet, and the scaling factor is 300).

The higher relative density of brazing sheet is a dominating factor for brazing performance. As shown in Fig. 19, the brazing process was completed at 650 K for 4 minutes in vacuum environment. The brazing performance test was carried out according to the Regulations of ISO5187-1985 Welding and Allied Processes. The tensile strength of the brazed structures is 175.7 MPa for the first brazing sheet, 186.4 MPa for the second brazing sheet and 192.2 MPa for the third brazing sheet. It can be concluded that higher relative density of brazing sheet is vital higher brazing strength.

Fig. 19.

Scheme of brazing test (a is fixture tool for brazing, b is the brazed structure).

As shown in Fig. 20, the irregular interfacial morphology observed at the brazing joint reveals a significant element diffusion into the base metal occurred during brazing process, also small amount of Cu elements of base metal diffused into the brazing materials. The uniform eutectic structures and fine grains in Fig. 20(c) are the main mechanisms of reliable metallurgical bonding, which is greatly conductive to the preferable mechanical performance. The coarse eutectic structures in Fig. 20(a) indicates weak metallurgical diffusion during brazing process. What’s more, more small pores, that are strongly related to stress concentrations and mechanical failures, can be found in the brazing microstructure for the first brazing sheet.

Fig. 20.

SEM images of brazing microstructure (a is for the first brazing sheet, b is for the second brazing sheet, c is for the third brazing sheet, and the red square is amplified in the black square in the right top corner).

6. Conclusion

The electromagnetic compaction of brazing powders was carried out with a self-developed WG-IV electromagnetic forming machine. An electromagnetic-mechanical-thermal coupled model was presented to make the dynamic analysis of electromagnetic compaction of brazing powders. The coupled model consists of electrical circuit model, electromagnetic field model, kinematic model of driving board, thermal field model and powder compaction model, it can be used to make a complete dynamic analysis of the discharge current, eddy current, mutual interactions of magnetic fields, kinematic motion of driving board, compaction densification behaviors and the energy transfer during electromagnetic compaction. The full coupled model has been validated by experimental results that it could provide a more accurate method for dynamic analysis and optimum design of electromagnetic compaction process. The optimization studies were completed by the coupled model to successfully promote the relative density of brazing sheet, and the brazing performance of the optimized brazing sheet with the higher density was verified to be advantageous in the performance test.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51475345) and the Open Fund Project of State Key Laboratory of Materials Processing and Die & Mould Technology (P2015-01).

Conflicts of Interest

The authors declare no conflict of interest.

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