Response of Materials Subjected to Magnetic Fields

This research program examined the interaction of conducting materials with magnetic fields through a series of experiments, the development of analytical models and the development and testing of numerical models. The experiments were a series of drop tests of conducting spheres through strong magnetic field gradients and were used for the development and validation of the analytical and numerical models under low speed conditions. Large decelerations were measured and shown to agree with predictions. These codes were then extended to higher speed conditions and used to examine magnetic compression and the generation of stresses in the high speed materials. Comparisons with the ALEGRA code were made. Scaling models were proposed for high speed interactions where the penetration of the magnetic field into the material is very small during the interaction time. Applications of the magnetic induced stress were explored for changing the impact characteristics of high speed objects against a protective shield, improving survivability. Introduction The most significant parameter indicating the effect of a magnetic field on a conducting material is the magnetic Reynolds number, Rm = μσvl, where μ is the magnetic permittivity, σ is the electrical conductivity, v is the velocity and l is a characteristic scale length. This can be also written as the magnetic diffusion time divided by the interaction time, so for very fast interactions, the magnetic Reynolds number is greater than 1. In this case the induced magnetic field in the material is significant and leads to strong forces opposing the penetration of the external magnetic field into the material. The objective of this project was to examine the effects of conducting materials moving in a magnetic field under conditions where the magnetic Reynolds number is significant. This objective had two foci: 1) To determine if a conducting material moving through a gradient magnetic field at high velocity would significantly slow down, and/or be affected by stresses associated with electrical currents produced by inductive effects. 2) To determine if magnetic flux compression could be utilized to slow down or deflect high velocity conducting materials. To address the first focus, an analytical approximation was derived for the case of a low velocity (~5 m/s) conducting sphere (~2.54 cm diameter) moving in a gradient (~0.2 m) magnetic field (4 Tesla max), followed by a simulation (modified commercial software) and experiment designed to confirm the approximation. The key result of this was to show that thermal effects were minimal yet the Lorentz forces acting on the sphere were significantly influenced by the radius of the sphere (R) and to a lesser extent, the length of the gradient (1/L) and finally the velocity (v). These forces did slow down the sphere significantly. The next step was to examine the case of a high velocity (>1000 m/s) conducting sphere. For this case, we used the modified commercial simulation software, verified by the low velocity case. The results showed almost no change in velocity due to the lack of momentum transfer, i.e. too little time in the gradient magnetic field. However, what it did show is that the Lorentz forces were extremely significant. Also, since these forces were asymmetric, the induced stress warped and flattened the sphere as it traveled in the magnetic field gradient. This was confirmed by Sandia National Laboratory using the ALEGRA code. Thus, while not changing speed, the pressure of impact could be very significantly reduced. A particular application for this was found for the protection of space platforms from space debris. In fact, it was noted that the greatest effect was generated where conventional protection was weakest. To address the second focus, a well-known analytical approximation of the magnetic flux compression generated by two, 1-D plates (in parallel) moving toward each other (perpendicularly) was verified by the simulation. This was used to show by simulation, in 2-D, how layers could be applied that significantly slowed down an initial mass due to the repulsion force that was produced by the magnetic flux compression. It must be noted that the volume between the plates must be near zero for the effect to be significant. A limitation of this concept had been that if the plates simply folded under pressure and did not collapse completely against each other, the effect would be negligible and therefore not useful for the purpose proposed. This is remedied by using very small plates in conjunction with each other thereby ensuring complete collapse of the volume and generating significant repulsion force. An analytical approximation of the magnetic force produced was developed. Key Publication: A. Giffin, M. Shneider, and R. B. Miles, "Potential Micrometeoroid and Orbital Debris Protection System Using a Gradient Magnetic Field and Magnetic Flux Compression", Applied Physics Letters, 97, 5, 2010. Detailed progress and results: Results of this program can be broken into four areas: 1) Experimental tests conducted in high magnetic fields for the development of benchmark measurements for validation of both analytical and computational models. 2) Development of analytical models for comparison with computational results for conditions that cannot be duplicated in the laboratory 3) Implementation of 3-D commercial code for modeling of dynamic interactions in conducting materials moving though high magnetic fields. 4) Primary analysis of effects on a conducting sphere moving at high velocity with use of modified 3-D commercial code. 5) Results using the ALEGRA code for the high velocity conducting sphere including deformation. 6) Primary analysis of effects of multiple plates utilizing magnetic flux compression. 7) Possible application implementing results and current technologies. 8) Additional analysis of various models in literature and off shoots of current ideas. 9) Current and future directions Areas 1) and 2) Experiments and Analytical Model Development Figure 1 shows the laboratory set up for the benchmark experiments. The magnet is a superconducting Helmholtz coil capable of operating at up to 6 Tesla. Access to the high magnetic field at the center of the magnet is by three orthogonal passages, including a vertical passage through which spheres are dropped and a horizontal passage through which a laser beam is passed for timing the arrival of the spheres at the center. For the experiment, various solid, hollow or thin shell metallic spheres were dropped into the magnetic field and the fall time from release to the arrival at the center of the field is recorded using laser beams, as shown diagrammatically in the pictures. The interaction of the magnetic field with these spherical objects leads to the development of eddy currents that produce an induced magnetic field that generates a force opposing the gravitational force, thus reducing the fall velocity and increasing the fall time. Figure 1: Geometry of the Helmholtz magnetic coils and photographs of the laboratory set up. Spheres are dropped through a vertical hole (yellow line shows the fall trajectory) and pass into the high field region at the center of the magnet coils. The fall time is recorded by a pair of laser beams as shown in green. Figure 2 shows the induced force on a falling solid sphere. The force is produced by the induced eddy currents. As the sphere moves through the external B field (Bext) that is static but increasing along the x-direction, a current (Iind) is induced in the z,x-plane (the E field). The view of the left plane is rotated on the right side so that the external B field is coming out of the plane while the induced B field (Bind) is directed into the plane. In both cases notice that the direction of the force is radially inward, with a net force opposing the motion. As the velocity increases, the induced currents become more confined to the outer perimeter of the sphere, and at high velocity the forces that are generated in a thin shell sphere and the forces generated in a solid sphere at equal velocity are almost identical. Figure 2: Induced current and induced forces associated with a sphere dropping through a positive magnetic field gradient. The left diagram is a view of the sphere seen from a position orthogonal to the magnetic field lines and the right diagram shows a cross section of the sphere seen from a position along the magnetic field lines. The directions of the induced currents and force vectors are indicated. Figure 3 shows the analytical model approximation. The actual magnetic field reverses sign as shown by the red curve on the right diagram. The analytical model uses a piecewise linear approximation shown in blue. The most important feature of the field is the gradient and the linear approximation accurately captures that slope. The relevant quantity is Bmax/L, the maximum magnetic field divided by the liner dimension over which that field changes from 0 to the maximum value. Figure 3: Modeled and actual magnetic field gradient. The actual gradient is calculated from the specifications given for the magnet. The modeled field is a piecewise linear approximation. Figure 4 shows the results of the drop tests and the analytical predictions. Both aluminum and copper alloy spheres were tested. The 1⁄2 “ diameter (6.35 mm radius) copper sphere had a conductivity that was 93% of the International Annealed Copper Standard (IACS) and had a fall time that was very close to the predicted time and close to the ideal copper fall time. As the formula above the figure indicates, the force that is generated is proportional to the product of conductivity, σ, and the velocity, v, times the 5 power of the sphere radius, so the 3⁄4” and 1” diameter (9.52 mm and 12.7 mm radius) aluminum spheres are much more strongly affected. The aluminum alloy in those spheres is less conducting than the ideal aluminum alloy, which has 60% the conductivity of pure copper. In these cases the conductivity was only 32% and 38% of IACS for the 3⁄4” and 1” aluminum spheres. The significant difference in the fall times associated with the same size aluminum and copper spheres is because of the different density of the two objects: the aluminum alloys are much lighter than copper. Note that the fall time for the solid 1” aluminum sphere is almost three times that of free fall. The analytical model captures the fall times to within 4% of the measured value. These experiments have allowed for the development and testing of an analytical model that forms the foundation for the validation of computational results. The falling spheres are large enough and move fast enough under gravitational acceleration to produce significant magnetic Reynolds number effects. The close agreement of the analytical model with the measured results indicates that the model is capturing those effects accurately and thus can be expected to be reliable for higher velocity and more complex high magnetic Reynolds number simulations. Figure 4: Results for drop tests of 1⁄2” diameter copper and 3⁄4” and 1” diameter aluminum solid spheres. The left hand diagram shows the recorded and predicted fall times. The right hand diagram shows the recorded and predicted percentage increase in fall times. Note that the aluminum alloys had conductivities that were significantly lower that the ideal conductivity of aluminum which is 60% of the International Annealed Copper Standard (IACS). Area 3) Modified Commercial Software In order to provide for further validation of the code that is under development, we purchased a versatile commercial software package, MagNet by Infolytica Corporation. Following an extended series of demonstration tests conducted by the company in order to determine the utility of the code for our purposes, the first component of the software was acquired on 10/27/08. This software uses finite element analysis to simulate electromagnetic effects on materials. The code can simulate dynamic interactions of metals with magnetic fields, but it does not have the capability of following any deformation of the metal objects and needs to have material properties specified before the run. The code is very useful for simulating high Reynolds number effects, magnetic field compression and forces on objects. It has a full 3-D capability. Training for use of the software was completed on 11/21/08. Figure 5 shows the comparison between the analytical approximation and the simulated model results using our modified 3-D software. Comparing these results further validates our computational software. This increases our confidence that the simulated results at high velocities (high magnetic Reynolds numbers) are accurate for real situations. Further, it also validates our analytical approximation that shows the force as F~σG2vr5 where σ is the conductivity, v is the velocity, r is the radius of the sphere and the field gradient, G=Bmax/L is the maximum magnetic field, Bmax divided by the length, L of the gradient. Figure 5: Comparison of analytical approximation and simulated numerical results using modified commercial code. Figure 6 shows the acceleration and velocity changes in a sphere moving at low velocity (as above). Notice that the acceleration changes sharply at around 450 msec. At this time the sphere enters the first significant change in the field; the beginning of the field gradient. It almost instantly reaches a terminal velocity at this point as can be seen in the second graph. The sphere stays at this velocity until it reaches the end of the gradient. As can also been seen, toward the end, the sphere begins to increase in speed again. This is due to the fact that the gradient is not linear and tapers off at the end. We will not concern ourselves with this as our primary focus is the initial change. Figure 6 These two graphs show the changes in the acceleration and velocity as a conducting sphere moves through the magnetic field gradient (0 to 4 Tesla over ~20cm) at low velocity (under the influence of gravity for 1 meter). Area 4) High Velocity Conducting Sphere Figure 7 shows the deceleration of a copper sphere of the same size and magnetic field as above, but it is traveling at high velocity (1 km/s). Notice the much larger deceleration, about 50 times greater than the low velocity case. However, in this case the reduction in velocity is minimal over the distance of the gradient (less than 1 m/s), even though there is a large deceleration as shown. This is due to low momentum exchange over the time of the travel. Figure 7 These two graphs show the changes in the acceleration and velocity as a conducting sphere moves through the magnetic field gradient (0 to 4 Tesla over ~20cm) at high velocity (under the influence of gravity for 1 meter). Figure 8 shows a 3-D surface picture of the induced force on a falling solid copper sphere with a radius of 12.7 mm (1" diameter) at 1 km/s, putting it into the high Reynolds number regime. The magnetic field goes from 0 T to 4 T over a span of ~20 cm in the x direction. As the sphere travels through the gradient, the color and arrows indicate the strength (red = strongest) and direction (approximately radial) of the pressure on the sphere. The largest forces are at the bottom of the sphere which has the effect of slowing down the sphere. At high velocities, this reduction in velocity is minimal over the distance of the gradient, again due to low momentum exchange. However, the forces become very large as the velocity or gradient increases due to the increased strength of the eddy currents. Therefore, the key aspect to note is the differential of the forces acting on the sphere. These forces will flatten and extrude the sphere into a scalene ellipsoid shape. Thus the pressure of the impact of the sphere against another object will be diminished. -10000 -8000 -6000 -4000 -2000