Simulation and instrumentation of electromagnetic compression of steel tubes

Journal of Materials Processing Technology 211 (2011) 840–850

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Simulation and instrumentation of electromagnetic compression of steel tubes A. Vivek a , K.-H. Kim b , G.S. Daehn a,∗ a b

Department of Material Science and Engineering, The Ohio State University, 2041, College Road, Columbus, OH 43201, USA Automotive Steel Applications Research Group, POSCO, Daechi4-dong, Gangnam-gu, Seoul 135-777, Republic of Korea

a r t i c l e

i n f o

Article history: Received 5 November 2009 Received in revised form 9 August 2010 Accepted 23 August 2010 Available online 26 September 2010 Keywords: Steel Forming Photon Doppler Velocimetry Aluminum Buckling

a b s t r a c t The near-axisymmetric compression of high strength steel tubes by electromagnetic forming is studied both experimentally and analytically. Tubes with a nominal tensile strength of 440 MPa, outer diameters of 75 mm and wall thicknesses up to 2.3 mm were compressed using a 9 turn helical actuator and discharge energies up to 24 kJ. This can produce reduction in diameters beyond 15%. Experiments were instrumented for measurement of velocity, as well as primary and induced current during the experiments. These records were compared to those of a simple numerical model and found to be in close agreement both in final tube diameter as well as temporal behavior of current and velocity. The model is also exercised to consider system design outside the space examined experimentally. The present experiments also clearly demonstrated that by reducing the system capacitance, the rise time of the pressure pulse can be decreased and this reduces the tendency for Euler bucking in the tubes. This results in rounder tubes, which may be more useful as a manufactured product. © 2010 Elsevier B.V. All rights reserved.

1. Introduction There are two main goals of this work: demonstrating steel tube compression with an electromagnetic forming process and demonstrating that this process can be simulated with a relatively simple model that has good correspondence to experimental data. The verified model can be used to optimize energy and material use in the industry. Also, the validated model should be able to predict deformation at energy levels that cannot be achieved within safe operating limits of the equipment used presently, but can serve as a guide for next generation equipment. The diameters of high strength steel tubes are reduced in a nearly axisymmetric manner using standard electromagnetic forming. Velocity, primary current, secondary current and temperature change were measured in the experiment. Photon Doppler Velocimetry was used to measure velocity, while Rogowski coils were used to measure currents. A high-speed infrared thermometer was setup to monitor change in temperature. The data with high temporal resolution was compared with a numerical model. In this work it was also verified that reducing the pressure rise times inhibits buckling in steel tubes and results in better roundness of the compressed tubes. Capacitance of the capacitor bank was changed to change the rise time. The simulation part of this work is mainly based on a code that solves a series of differential equations to output the current, veloc-

∗ Corresponding author. Tel.: +1 614 292 6779; fax: +1 614 292 1537. E-mail address: [email protected] (G.S. Daehn). 0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2010.08.023

ity, temperature, charge, pressure, heat and strain as a function of time. The approach is an adaptation of the work by Gourdin (1989) and Pon (1997) with equations for mutual inductances for tube geometries largely following Jabonski and Winkler (1978). The code considers the electromagnetic interaction of a conductive tube and a solenoid coil. The simple model is shown to provide rapid and relatively accurate predictions of average strains as a function of process parameters. And while a variety of capacitance and charge voltage combinations can produce the same nominal strain, at lower capacitance values, the current discharges more rapidly. The present experiments show that more rapid imposition of the pressure pulse leads to reduced wrinkling and rounder final tubes. These effects are not predicted by the present modeling, but are consistent with other investigations of this effect. 1.1. Background Electromagnetic forming process is based on the Faraday’s law of electromagnetic induction resulting in repulsive Lorentz body forces between two conductive bodies carrying opposed currents. In this process, a brief intense current pulse is delivered from a capacitor bank into an actuator (typically a solenoid coil made of copper windings). If there is a metallic object nearby then an opposing current is induced in it by the transient magnetic field developed by the current in the actuator. The actuator is generally stationary while the workpiece is repelled at a very high speed. A variety of geometric arrangements can be used in electromagnetic forming including tube expansion or compression and a variety

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of sheet forming methods. This is broadly reviewed by Daehn (2006). One arrangement described by Kamal and Daehn (2007) can permit a nearly planar launch of sheet metal. Upon impact it is possible to produce a solid-state weld with another piece of metal. Commonly axisymmetric compression is used to join a tube onto a rod or another tube by means of a crimp or weld. Electromagnetically driven conformal interference fits can provide robust joining between dissimilar metal pairs. High velocity deformation driven by electromagnetic forming can also produce a significant increase in the formability of metals that rupture through instability. This has been mentioned in the works of Balanethiram and Daehn (1994) who first showed the concept, Imbert et al. (2005) studied this in the presence of impact and Thomas et al. (2007) considered the effect changing constitutive behavior at high strain rate. Moreover, this technique has potential advantage over conventional forming methods because of low cost one-sided tooling, lightweight forming equipment and high speed and process control. EMF represents an elegant and efficient way of reducing the diameter of metal tubes, pipes or extrusions. Deformation can be much more stable because of inertial stabilization of buckling. There is no contact between the tool and workpiece, in one example a stainless steel fuel element was processed while it is inside a plastic bag as reported by James and Philpott (1970). Unlike explosive forming, such as that studied by Florence and Vaughan (1968), there is instantaneously same pressure over the length and circumference of the tube. The precise control of pressure is a major advantage of this technique and hence small to large tubes can be compressed using the same basic equipment. This technique has the unique ability to cause variable reduction of tube diameter at multiple locations in a single step. It has been reported by Kleiner et al. (2005) that roundness of electromagnetically compressed aluminum tubes improves with shorter rise times. Al Hassani (1974) claimed, based on his computational analysis and experiments that geometrically similar tubes of the same material produce similar numbers of wrinkles and when deformed to the same maximum fractional reduction reveal similar final buckled shapes. He also showed that the number of wrinkles (for same length/diameter ratio, and reduction in diameter) increases with the a/h ratio, where a is the diameter and h is the wall thickness of the tube. This has been supported in the work by Psyk et al. (2004) who, based on thorough experimental work also stated that with the same reduction in diameter, a/h and length/diameter ratios, a softer material produces fewer wrinkles on compression than a harder material. Lindberg (1964) proved, mathematically and experimentally that in thin cylinders, out of many harmonics produced, high frequency and high stress modes dominate. All these studies used electromagnetic compression of aluminum tubes as their basic experiment to study this phenomenon. Aluminum alloys have likely been the most commonly used experimental material in electromagnetic forming because of high electrical conductivity, low density. Also its formability by traditional processes is less than that of steel. Steel is more difficult to form by EMF because it has lower electrical conductivity, higher density and commonly higher yield strength than aluminum alloys. Steel is a major structural material and there is need for optimization of process parameters for the electromagnetic compression of steel tubes. The same numerical or mathematical procedures can be used for steel or aluminum, provided proper account is made for the materials properties in each case. There has also been a continued effort to numerically model electromagnetic forming. Fenton and Daehn (1998) modeled electromagnetic sheet metal forming process using a two-dimensional (2D) arbitrary Lagrangian Eulerian (ALE) finite-difference computer code known as CALE. Their results were verified against the experimental data from the work by Takatsu et al. (1988), in which

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Aluminum 1050 sheet was bulged using a flat spiral coil. Kleiner et al. (2004) used two approaches: solving the mechanical and electromagnetic problems using separate programs, and solving the two fields simultaneously in the same software environments. Again the numerical results were verified against experiments. Thomas et al. (2007) theoretically predicted forming limit diagrams (FLDs) for electromagnetic expansion of Aluminum 6063 tubes and compared them with the necking strains observed while expanding thin aluminum tubes by passing current through a coaxially places helical brass coil. Simulations and experiments were found to be in reasonable agreement with each other and there was a significant increase in forming limits in comparison to the quasistatic case. Recent FEM codes such as LS-DYNA include electromagnetism routines for coupled mechanical-thermal-electromagnetic simulations as described by L’Eplattenier et al. (2006). Henchi et al. (2008) built on these new codes and proposed a method to use electromagnetic ring expansion coupled with LS-DYNA and LS-OPT to determine material constitutive behavior at high strain rates. The model optimized constitutive parameters based on experimental data that compared well with those of cold worked copper. This present study uses simpler, essentially, one-dimensional numerical modeling. 2. Methods A 9-turn helical solenoid coil in conjunction with a capacitor bank was used to compress high strength steel tubes of different diameters and thicknesses. Experiments were done at different energy levels. The effect of varying rise time was studied using a capacitor bank with variable capacitance. Current, temperature change and velocity were recorded over the time scale of the experiment using advanced instrumentation. After the experiments, the final outer diameters of the compressed tubes were measured. A numerical code was used to predict currents, temperature changes, strains and velocities at the energies used in the experiments. Subsequently, the simulation and experimental results were compared. 2.1. Experimental approach 2.1.1. Capacitor banks The capacitor banks used in the experiments are both capable of discharging within tens of microseconds. The first is a Magneform capacitor bank with a nominal maximum capacity of 16 kJ at its maximum charging voltage of 8.66 kV. This unit has a total capacitance of 426 ␮F, and an internal inductance of about 100 nH and is switched by 8 ignitron switches that are located on each of the bank’s 8 capacitors. The bank’s total capacitance can be decreased by removing capacitors from the circuit, but in this study all 8 capacitors were engaged in the circuit for each experiment. The second is also a Magneform capacitor bank but with a maximum capacity of 48 kJ at its maximum charging voltage of 10 kV. This unit has 8 capacitors each of which can be switched to change the capacitance of the circuit. Each capacitor has a capacitance of 120 ␮F and internal inductance of approximately 280 nH. In this study 8 or 4 capacitors were engaged in the circuit for a given experiment. This permitted variation in the current rise time. 2.1.2. Solenoid coil A custom solenoid coil was fabricated by Poynting, GmbH, of Dortmund Germany. The coil is based on a 9-turn helix of a copper alloy that is potted in a filament wound composite casing. It is designed to withstand a maximum electromagnetic pressure of 300 MPa. The coil has an inner diameter of 76.2 mm and a working height of 73 mm. An engineering cross-section of the coil and photo of the primary coil are shown in Fig. 1. A Mylar sheet was

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Fig. 1. (A) Photograph and (B) engineering cross-section of the 9-turn solenoid coil built by Poynting, GmbH. The coil has a rated magnetic pressure up to 300 MPa. All dimensions are in millimeters.

used between the coil and steel tube to be compressed as extra level of protection against arcing. For compression of tubes having a reduced diameter significantly less than the solenoid coil diameter, a field shaper made of aluminum alloy 7075 was used. Kapton tape was applied around and inside the field shaper to prevent arcing. An engineering drawing of the field shaper and the set up for using it is shown in Fig. 2. 2.1.3. Instrumentation Calibrated Rogowski coils purchased from Rocoil of North Yorkshire, England provided direct measurements of the primary current. A second coil looped through the bore of the tube, measured the current that flows through it. This includes the primary current through each turn and the secondary current. The induced current in the tube is measured by subtracting the measured current from the known amp-turns of the primary coil. The effective number of solenoid coil turns was determined by low energy discharges through the solenoid coil while measuring the currents from the capacitor bank terminals and through the bore of the solenoid coil without any tube in it. The ratio of the two currents is the effective number of solenoid coil turns. Tube velocity is measured by Photon Doppler Velocimetry (PDV). This is

a technique recently introduced by Strand et al. (2006) and is based on the Doppler shift between a reference laser beam and a reflected velocity-shifted beam, and the OSU implementation has been recently published, Daehn et al. (2008). The PDV is based on a 1 W 1550 nm erbium fiber laser with optical probes. The PDV uses the interference of unshifted and shifted light off the surface of a target. The components of shifted and unshifted light interfere at the detector. Due to changes in the frequency of the two light components, interference is produced and a resulting pulsating light intensity is created as the target moves. The PDV system produces a waveform as illustrated in Fig. 3. Every half wavelength of intensity represents a half wavelength or 0.775 ␮m displacement of the target surface. The PDV and Rogowski data acquisition was all triggered at a common time based on the measured initial rise of the primary current and captured on a LeCroy oscilloscope with a 2 MHz bandwidth and 5 Gs/s acquisition rates. This system can capture velocities up to 800 m/s for a period up to 2 ms and for up to 4 channels. A simple fast-Fourier technique is used to translate the raw oscilloscope traces of the intensity–time plot into tube velocity. In order to capture the velocity on the inside surface of the tube an angled optical path provided by a simple mirror is used implementing a novel periscopic lens and mirror system, as shown in Fig. 4.

Fig. 2. Field shaper: shown as a (A) CAD model and (B) in the experimental set up.

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Fig. 3. (A) Basic principle of the Photon Doppler Velocimeter (PDV) and (B) an example trace from the initial portion of ring expansion experiment in copper. Shifted light off the moving target interferes with unshifted light reflected off the lens to produce ‘beats’ in the mixed signal, as shown in the actual data at the bottom.

2.1.4. Materials Experiments have been performed on seam welded steel tubes provided by POSCO with outer diameters (OD) of 76.2, 75 and 50 mm with thicknesses of 2.3, 2.7 and 1.5 mm respectively. All had a nominal tensile strength of 440 MPa. Most of the experiments were carried out on tubes of 76.2 mm OD and 2.3 mm thickness. Problems were encountered in properly coupling energy to the tubes of smaller 50 mm OD to the pulse energy employing the field shaper shown in Fig. 2 as detailed later. Launch energies of 8, 12.8, 15.8, 18, 21 and 24 kJ have been used. In each case two PDV probes were focused at the middle of the bore on opposite sides of the tube. While placing the tube inside the solenoid coil it was made sure

that the weld line was facing the capacitor bank terminals. In this way the consistency of the experimental set up was maintained. Final tube temperature was measured after 2 s of completion of the experiment by replacing the PDV probes by an optical thermometer. The overall setup for these experiments is shown in Figs. 5–7. 2.2. Simulation: theory and background A simulation code was written in the Mathematica programming language to simulate these experiments. The program solves a series of differential equations to output the current, velocity, temperature, charge, pressure, heat and strain at different times.

Fig. 4. (A) Periscopic probe system for measuring velocity from the inside of a tube. (B) The probe shines up from the bottom and bounces off the 45◦ mirror and follows the same path back to the lens.

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Fig. 5. Side view of the typical experimental configuration.

Fig. 6. Top view of the typical experimental configuration.

2.2.1. Governing equations The system is modeled as two LRC circuits coupled through their mutual inductance, M. The primary circuit is given a subscript 1 and its governing equation is given by: d Q1 (L1 I1 + MI2 ) + R1 I1 + =0 C1 dt

(1)

Here L is inductance, I is current, R is resistance, Q is charge and C is capacitance. The tube is also modeled as a resistive and inductive circuit (without capacitance). Its governing equation is: d (L2 I2 + MI1 ) + R2 I2 = 0 dt

(2)

Currents running in opposite directions in the tube and solenoid coil give rise to an electromagnetic repulsion between the two elements. For a simple thin cylinder under pressure, P, the membrane hoop stress in the wall, , can be calculated as: Fig. 7. Experimental configuration showing the looped second Rogowski coil and infrared temperature sensor.

=P

r w

(3)

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Table 1 Summary of experimental results for same capacitance. Energy level (kJ)

C1 peak velocity (m/s)

8 8 8

23.5 – –

Average

C2 peak velocity (m/s)

Peak primary current (kA)

Rise time (␮s)

Final OD (mm)

% reduction in OD

Predicted % reduction in OD

– – –

83.44 83.46 –

28 29.1 –

74.8 75 75.1

1.8 1.5 1.4

3 3 3

23.5

–

83.45

28.55

75

1.5

3

12.8 12.8 12.8 12.8 12.8

50.7 – – – –

54.5 – – – –

101.29 – 103 – –

25.9 – 28.5 – –

71.7 72 72.7 72.2 72.3

5.9 5.5 4.6 5.2 5.1

6 6 6 6 6

Average

50.7

54.5

102.45

27.2

72.2

5.2

6

15.8 15.8

68.1 –

89.3 –

– 112.4

– 27.9

69.5 70.1

8.8 8.0

8 8

Average

68.1

89.3

112.4

27.9

69.8

8.4

6

Here w is the wall thickness and r is the radius of the cylinder. When the magnetic pressure, Pm exceeds the yield stress, the material will accelerate in accord with Newton’s 1st law. From this the workpiece acceleration, a, can be calculated as: a=

Pm − (wf /r) dV = w dt

(4)

Here  f represents the current flow stress of the material,  is its density and w is the thickness of the workpiece. Minor axial tension can also be produced in deformation. This stress should be small relative to the hoop compression, and much more sophisticated calculations would be needed to provide estimates. Now, the outstanding problem is calculating the magnetic pressure (or force) as a function of time during the electromagnetic forming experiment. There are two components to magnetic pressure. The first is due to the interaction between the tube and the inductor and is given by: Pm

1 dM I1 I2 = A dr

(5)

Here A is the surface area of the workpiece adjacent to the actuator and the tube radius is represented by r. The other component of magnetic pressure is the self-force that results from the current running in the tube. This force always tends to expand the tube and can be represented as: Pm,s =

1 1 dL2 2 I 2 A dr 2

(6)

This pressure will aid tube expansion and hinder tube contraction. P(m, s) is included for completeness, but under the conditions studied here is small. The material properties required in modeling are density, resistivity and the constitutive behavior for the material. It is assumed that the material is rate independent and its plastic constitutive law can be expressed as:  = kεN

(7)

In the code k is taken as 800 MPa and N is 0.4. Tensile tests conducted on the material yielded these values. The next class of inputs needed consist of values or relationships for inductances. These are based on the effective radii of the solenoid coil and tube rs , rt ; and the length (which is constrained to be the same for tube and solenoid coil) and the number of electrical windings or turns in the solenoid coil, n. For a solenoid coil or tube of finite length, l, where the solenoid coil’s length is greater than

the diameter the inductance can be estimated as: Ls =

100 rs2 n2 10l + 9rs

(8)

Here, 0 is magnetic permeability of air and its value is 4 × 10−7 and l is the length of the solenoid coil or the tube. Ls can be the inductance of tube, L1 or solenoid coil, L2 depending on the value of n. The value of n is 1 for a tube; while for a solenoid coil n is the number of turns of the coil. Mutual inductance is simply expressed as and can be bounded by the expression: M=k



L1 L2

(9)

where k ranges between 1 and zero for tight and no coupling, respectively. For tube compression, the mutual inductance has the form: 0 rs rt (t)n M= l



1−

rs2 − rt2 (t) rs2



(10)

Eq. (10) was used for computation of mutual inductance in the model. The effective radius of the solenoid coil is represented as rs , and tube’s effective radius is represented as rt (t). Next, it is important to develop proper values for the effective radii for the solenoid coil and tube. The effective radius should represent the mid-point for the current path in the solenoid coil or tube. Thus, if the current runs uniformly through each body it would represent the mid-line between the inner and outer radius. At high frequency, because of the proximity effect, the current will tend to ride on the surface nearest the other conductor. The depth of the current path is taken as the skin depth, ı:



ı=

2 o ω

(11)

where  represents material resistivity. The excitation frequency, ω, is given as:



ω=

1 LC

(12)

where L is the effective system inductance (which usually increases as the workpiece moves) and C is the bank capacitance. The system inductance L takes on two limiting values. At high frequency the value is low because the field cannot penetrate the workpiece. The other limiting value is if the field can completely penetrate the workpiece. In this case it takes on the value of L1 . The high frequency data is typically more appropriate when estimating ı, because most

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Fig. 8. (A) Downward axial view on a tube compressed at 15.8 kJ discharge energy. The focal plane of the image is approximately at the most constricted region of the tube. The dark line at the bottom of the tube is the weld seam, which represents a slightly stronger zone than the rest of the tube. Note that while the tube radius reduces in all locations, there is some onset of buckling. (B) The three images at the bottom show tubes compressed at 15.8 and 12.8 kJ where they were sectioned at the center of the compressed area. Percentage compression is 7.7%, 7.2% and 5.4%, respectively. The original diameter of each tube is 76.2 mm. The seam is pointed out by the arrows.

of the mechanical impulse is given early in the process, and further compression occurs by the inertia of the workpiece. Lastly, system resistance and bank (parasitic) inductance must be evaluated for the capacitor bank system. Both have been estimated by ringing the capacitor bank without an inductive coil in place. System resistance and inductance can be obtained by simple summations. The system resistance includes both that for the bank and solenoid coil (including possible skin depth effects) and the parasitic resistance needed in the code is separate from the solenoid coil’s inductance. 3. Results 3.1. Experimental data The results of all the physical experiments conducted on 16 kJ capacitor bank for compression of 76.2 mm diameter tubes are shown in Table 1. C1 is the velocity of the seam weld region while C2 is the velocity of a point opposite to it. In each case there is a lower peak velocity at the seam region because it is harder than

Table 2 Temperature change upon EM expansion. Discharge energy (kJ)

Temperature change (◦ C)

8 12.8 15.8

21 33 41

virgin material. The final OD for a non-round tube is estimated by taking the average of the maximum and minimum OD measured using conventional callipers. The current rise time was determined as the time required for the primary current to reach the maximum value. Temperature measurement results are shown in Table 2. An axial view with the focal plane of the image at the minimum bore of a tube compressed at 15.8 kJ is shown in Fig. 8(A). Also Fig. 8(B) shows sections cut at the center of the compressed zone for both trials at 15.8 kJ and the trial at 12.8 kJ. There is some significant out of roundness that can be seen as a precursor to buckling. Results from the experiments conducted on the 48 kJ capacitor bank are shown in Table 3. The PDV results are designated as C1 and C2 for the two respective PDV channels.

Table 3 Summary of experimental results for varying capacitance. Energy (kJ)

Capacitors

C1 peak velocity (m/s)

18 18 21 21 24 24

4 8 4 8 4 8

53 – 69 53 88 60

C2 peak velocity (m/s) 70 – 80 71 111 81

Peak primary current (kA)

Rise time (␮s)

No. of wrinkles

Final OD (mm)

% Reduction in OD

Predicted % Reduction in OD

101 – 110 100 122 107

26 – 25 35 25 34

8 8 9 9 11 11

70.1 69.9 67.2 67.2 64.1 64.5

8.1 8.3 11.8 11.8 15.9 15.4

9.5 10 11.5 12 13 13.5

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Fig. 9. Tubes compressed at 24 kJ and sectioned at the center of the compressed area. Tube roundness is checked by drawing tangent circles around the transverse sections.

Fig. 10. Experiments done on 16 kJ capacitor bank. (a) Velocity and (b) currents.

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Fig. 11. Experiments done on 48 kJ capacitor bank with variable capacitance. (a) Velocity plots for the case of 4 capacitors in the circuit. (b) Velocity plots for the case of 8 capacitors in the circuit.

The number of wrinkles on the formed tubes was counted as the number of depressions by feeling the surface of each tube with hand. These tubes also showed similar wrinkles to those at lower energies. The tubes compressed using 24 kJ were sectioned at the center of the compressed area and the pictures were taken of the cross-section. Both tubes had very similar average reductions in diameter. The pictures are shown in Fig. 9. Imaginary circles were drawn around both the tube, so that they were tangential to the tip of most of the buckles. It can be seen that tubes compressed with lesser capacitance of the discharge circuit, hence shorter rise times, conform better to the circular shape. Also, because of less circularity of the tube on the right it was not possible to overlay the circle on the tube while touching all the tips of the wrinkles. Electrical coupling between the primary actuator and secondary tube is quite important and is sensitively affected by the gap between the solenoid coil and tube as well as by the use of field shapers. In ideal coupling, the peak current in the secondary (tube) will be equal to the peak amp-turns in the primary. In the case of experiments done compressing a 76.2 mm OD tube, it is found that the peak current in the tube is very nearly 8.3 times the peak primary current. In the experiments compressing the 75 mm diameter tubes with the field shaper, much poorer coupling was encountered. For 75 mm OD diameter tubes the peak tube current is only 75% of that in the 76.2 mm tubes. This reduces the electromagnetic pressure also by a factor of 0.75, reducing the process efficiency. In attempts to compress the 50 mm diameter tubes using the aluminum field shaper, experiments were conducted at 16, 18 and 21 kJ energy levels. There was insignificant reduction in diameters at all the energy levels. Peak primary and secondary currents at 16 kJ were observed to be 90 and 633 kA respectively with rise time of 28 ␮s, with considerable tube heating. More refined approaches are required to produce good forming efficiency using a field shaper of this type. 3.2. Comparison of model and experiment Velocity–time traces are shown for the simulated and measured data for channel 1 and 2 (denoted as C1 and C2 ) in Fig. 10(a). Over-

all the measured and simulated velocity–time behavior agrees well both in trend and magnitude. At 8 kJ the velocity and tube displacement is under-predicted and at 15.8 kJ discharge, the velocity trace on unaltered metal agrees very well while the one at the weld is lower than predicted. This is likely related to the minor buckling of the tubes and stronger weld region. Similar, velocity–time traces for the experiments done on 48 kJ capacitor bank are shown in Fig. 11. In most cases the trend of the trace in virgin metal agreeing well with prediction and the one at the weld being under the predicted value is repeated. Errors can also be attributed to the varied local velocity due to buckling, which increases with higher percent reduction in diameter. The respective graph for compression of 75 mm OD tube, where there was poor electrical coupling, is also shown in Fig. 10. The current–time traces for experiments 1, 2 and 3 at 8, 12.8 and 15.8 kJ are shown in Fig. 10(b). Again there is generally very good agreement between simulated and measured currents. There is faster reduction in simulated current values after peak. This could be because of simplistic modeling of the mutual inductance, which can have greater errors as the tube moves away from the coil. Secondary currents were not measured at 18, 21 and 24 kJ energies. However, the simulated rise time and peak primary currents match closely the measured values. 4. Extrapolation from simulation The overall match between experiment and the modeling shows the leading factors in the system are captured, but a much more detailed study could be justified. In this study, little effort was expended at matching the constitutive behavior of the deforming steel. This is a difficult problem that is worthy of further detailed study. The heating and strain rates in the process are very high and split-Hopkinson bar test data were not available for the material being used. Also, the formula used for inductance of the solenoid coil assumes that coil’s length is greater than diameter. Here the length of the coil is slightly smaller than its diameter. And presently the tube and coil were assumed to have the same length and

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Fig. 12. Predicted tube reduction in diameter as a function of discharge energy (obtained by keeping all experimental values the same with the exception of charging voltage) and a comparison to experimental measurement is included. The experiments at energies above 16 kJ were conducted on the 48 kJ bank. Strains obtained from 16 kJ bank are plotted by square markers while those obtained from 48 kJ bank are shown as round markers.

the extra area that is deformed outside the actuator is ignored. Despite these approximations and assumptions, the comparisons between simulations and measured data shown in Figs. 10 and 11 indicate that the simulation compares very well to experimental measurement. At higher discharge energies, buckling becomes more pronounced and a more sophisticated 3D numerical model would be needed to predict the final strains reliably and accurately. However, for the preliminary purposes, the model can be expected to predict tube reduction as a function of energy for rough design purposes. Such a prediction is compared to experimental data in Fig. 12. If the strains measured after higher energy experiments conducted on the 48 kJ capacitor bank were plotted, then also, predicted and obtained values agree well with each other as seen in Fig. 12. This prediction was done by simply varying the charge voltage in the capacitor banks. This shows that there is generally good agreement between experiment and the model, with some overprediction at the 8 kJ energy. Also, at accessible energies of about 40 kJ nearly 20% decrease in diameter is predicted for the 73 mm tube length. One of the practical goals of this study is to be able to impart significant compressive strains to high strength steel tubes by electromagnetic discharge. The most promising ways to maximize efficiency are to minimize the solenoid coil to tube spacing, optimize the number of turns and to reduce the compressed length of tube. All three of these approaches have been studied with the model. The number of turns also seems to be near optimal. In going beyond 9 turns there is little increase in solenoid coil efficiency (as seen in Fig. 13) and its robustness may be sacrificed by using finer

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Fig. 14. Predicted levels of tube compression at a 16 kJ discharge and a 9-turn solenoid coil as a function of tube and coil height. This shows that significant increases in tube reduction are possible by reducing coil length from the current value of 73 mm.

conductors and thinner insulation. Curves of this type fall after a maximum as system inductance becomes too high. Also, the length of the tube can also be reduced, concentrating the energy over a shorter length. The results of this are shown in Fig. 14, where again, all simulation parameters, except for the solenoid coil length are held constant. Again, this indicates that significant constrictions in the tube can be developed by reducing the formed length. 5. Conclusions Complementary instrumented experiments and simulations were carried out on the electromagnetic compression of high strength (440 MPa tensile strength) 76.2 mm diameter, 2.3 mm wall thickness tubes. Significant final deformations over a length of about 75 mm were obtained with a fixed 9-turn helical compression coil. About 8% and 15% reductions in diameter were developed with 16 kJ and 24 kJ discharges from separate commercial capacitor banks. Comparisons between simulations and experimental data gave confidence that a simple numerical model can reliably predict system performance. With these observations, some practical issues in reducing tube diameters were addressed. First, either increasing the forming energy or reducing the compressed length can effectively and predictably increase the extent of diameter reduction. Second, the tubes will experience Euler buckling upon compression and this can be suppressed by the high-speed deformation. The present work shows that, in accord with previous literature, that if the rise time for the pressure pulse is decreased, buckling is suppressed. This reduction in rise time is easily accomplished by reducing capacitance. The effect of reduced capacitance on nominal deformation is easily captured with the numerical model, but this simple model does not capture buckling. The prediction of buckling can be the focus of further investigation. References

Fig. 13. Predicted tube compression at a 16 kJ discharge as a function of number of turns in the primary solenoid coil with all other simulation parameters fixed at those used in the experiments. Experimental average maximum strain is 8.4% as shown by the cross symbol.

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