4.1 Introduction

Most sheet metal forming processes are inherently complex involving three dimensional plastic flows with changing material properties and boundary conditions and have resisted efforts of many years aimed at producing computer models accurate and timely enough to be integrated into the process design loop. The fairly recent development of powerful workstation computers and efficient Finite Element Method (FEM) codes capable of modeling plastic deformations, are now giving designers some ability to predict process performance and thus attempt optimum tooling design. Previously, design was based on a few gross analytic methods and heavy reliance on empirical relations. Arriving at the correct process and specific tool surface geometry for sheet metal stamping has consequently consisted of more art than science. There has historically been a heavy reliance on the skills of the tool and die craftsmen to complete the

development of metal forming process applications so that acceptable parts are produced. The early FEM codes were linear elastic useful only for predicting the initiation of plastic yielding. Not until codes capable of modeling large plastic strains, strain rates and intermittent tool-sheet contact with friction could any but the simplest sheet forming applications be successfully modeled. To realize the full benefit of the more sophisticated software, a better understanding of the plastic behavior of the metal under stress and tribology of the tool-sheet contact is required. Improvements in both areas have been made. However, the accurate assessment of frictional forces at the contact points remains the Achilles heel of modeling complex matched tool, sheet forming processes. None the less, the state of the art in FEM for matched tool forming of sheet metal at this time is such that, when applied by users with sufficient knowledge of the real forming process, the tool designs generated produce parts that are close to model prediction [42]. Matched tools designed with computer modeling require much less time in press trials. The ever increasing competitive pressures in the manufacturing arena require that ever more thorough and accurate process design methods be implemented thus insuring the continued refinement of these FEM applications. Unfortunately, the best match tool process modeling software can only be applied to the pre-form half of the MT-EM process. The problem is not the high deformation velocities. The better software packages are fully dynamic and can model the inertial effects of the high deformation velocities of the EM half of the process. However, the electromagnetic pulse forming half of the process requires software which include Maxwellís equations for the calculation of the time varying eddy currents in the work sheet. Along with the resultant body forces which come out of the Maxwellís equations, the thermal effects of the eddy current are important since they feed back into the eddy current level though changes in resistivity. In addition, the localized current heating can effect the local strain distribution in MT-EM forming (see Chapter 3) which in turn points to the general lack of constitutive data for the material state and velocity regime of the EM pulse forming event .
 

4.2 Quasi-static process model requirements

Projecting current trends in the stamping industry, development of a reliable method of modeling the complete MT-EM process will be needed for wide spread acceptance and implementation. The level of model resolution that will make industrial users comfortable is unknown at this time. This problem is recursive in that the metal forming industry of today is unlikely to make much use of a process that can not be modeled for design purposes. On the other hand the industry will not spend the resources to develop high resolution modeling of a process that is not used. Since commercial codes such as Pam Stamp and LS Dyna-3D [ ] are available for modeling the matched tool forming, the effort of this research was directed toward identification and qualification of a code which could accurately model the EM pulse forming part of MT-EM process. The integration of results of the pre-form stage model as geometry and material condition input to the EM stage will also be left as future work. For the near term, average values of the strain state of the pre-form panel can be used as input material condition to the electromagnetic- dynamic model software package.

Kinematically static modeling of the transient electromagnetic pulse event can serve as a useful tool in the design of a MT-EM process. Good use can be made of data predicting the magnetic fields, eddy currents and resulting force-pressure distribution in EM forming area without considering workpiece deformation or tool deflection. A motionless model of the transient electromagnetics is, in effect, predicting the state of the EM form area at time t=0+. Motionless model analysis can be used to optimize the coil geometry based on a preferred forming pressure distribution, identify the sources of inefficiency due to flux leakage and excessive resistance heating. Also motionless EM modeling can determine the maximum forces experienced by the coil and their distribution for structural design purposes.

To investigate the actual usefulness of the commercially available electromagnetic codes, a simple axisymmetric model of a 70 mm diameter, single rectangular turn, copper pancake coil covered by 0.5mm insulation and 0.5mm aluminum work sheet was built using the. Opera-2d software package produced by Vector Graphics which could predict transient electric and magnetic fields, eddy currents, resistance heat generation and solid body motion (if desired). In this model, the coil current was assumed uniform and the 0.5mm thick work sheet was motionless. The objective of the study was to investigate the effect of the coil mounting structure and coil-holder gap on the forming pressure for a given current pulse. Three coil holder materials, copper, stainless steel and laminated steel and three gap (ideal insulator) distances, 1mm, 3mm and 6mm were investigated. Copper coil holder represented a worse case for a coil holder that could reduce forming efficiency by supporting large parasitic eddy currents. The laminated steel represented a best case holder since it conducts magnetic fields but supports very little eddy current. Stainless steel was chosen as a realistic high strength holder material that would limit parasitic eddy current losses due to high resistivity. Gap material was a fiber filled composite with the same permeability of air. The modeling and analysis was done by engineers in the Energy Technology division of Argonne National Laboratory using coil, structure, workpiece, current-time data and analysis parameters supplied by the Hybrid forming program. Selected output graphs and data from the Opera 2d study are presented in Figures 4.1 to 4.4 and Table 4.1.
 
 

b) Resultant pressure distribution for copper coil holder

Figure 4.1b indicates that Opera-2d predicts the second half of the first current cycle generates a larger pressure on the work sheet than the first although its amplitude is lower. The resultant difference in force impulse between the first and second half current cycles is consistent for all cases as seen in the data of table 4.1. A reason for the predicted positive pressure difference is that part of the energy of the first half cycle goes into establishing the initial magnetic field around the coil. Such small differences are interesting be not consequential to the EM phase of the hybrid process design.

Figure 4.3: Opera-2d model, magnetic field for pancake coil

in a copper holder; 0.5 mm aluminum work piece, 3mm gap; input; Fig. 4.1a

Figure 4.4: Opera-2d model, magnetic field for pancake coil

in a laminated steel holder; 0.5 mm aluminum

work piece, 3mm gap; input; Fig. 4.1a

In Figures 4.2 to 4.4 the wide variation in predicted magnetic fields for the three holder materials separated from the coil by a 3mm insulation gap are shown. A copper holder supports large eddy currents which confine the magnetic field to the insulation volume. This results in a lower effective inductance for the coil and hence lower field energy for a given current which in turn lowers the available forming pressure. Since the current is a fixed input to this model, the largest effective inductance produces the strongest magnetic field and the greatest pressure on the work sheet. In Chapter 5 the relation between magnetic pressure and field energy next to the work sheet will be discussed in detail.
 

Review of table 4.1 generally leads to the conclusion that a laminated steel housing is best from a forming energy efficiency standpoint. The more accurate conclusion is that the coil holder should be designed to minimize the capacity to support parasitic eddy currents. This goal can be achieved by other means. In particular the use of specialized composite material for the main tool body in a MT-EM process is shown in Chapter 5 to have the same effect as laminated steel but allows for a less costly, cast construction .
 

4.3 Full dynamic model for MT-EM forming ; the GEM code.

GEM is a special type numerical hydrodynamic analysis code (Hydro-code) generally refer to as Smoothed Particle Hydrodynamics (SPH). This numerical method was initially developed in 1977 to model non-axisymmetric phenomena in astrophysics and was found to be easily adaptable to model other problems with complex physics. SPH has become quite popular among astro-physicist because itís easier to code application than other hydo-code methods such as Arbitrary Lagrange-Eulerian (ALE). Consequently a fairly large body of literature exists concerning the formulation and use of the SPH method. Two good detailed reviews of the method are found in Monaghn (one of the inventors)[55] and Benz [15]. A major advantage of SPH is that it uses a cloud of points rather than a grid work of finite elements to define an object. Without the grid, the required derivatives of a problem can be numerically calculated without the errors attendant with element distortions. The lack of grid and elements also greatly simplify the code generation so that 3D codes are not significantly more complex than a 1D code [15]. The core mechanism of SPH is an interpolation method which allow any function to be redefined in terms of its value at a set of disordered points. The value of the function at any arbitrary point in the problem space is then constructed by summing the influence of the function at every known point on the point in question. The amount of effect that a known point value contributes to the point of interest is determined by a chosen interpolation kernel. The kernel is required to be differentiable and have an unity integral value over its region of influence. Although formally, the influence summation must be taken over all known points, apriori knowledge of the physics being modeled allows the interpolation kernel to be chosen so that only an appropriate number of nearest neighbors are included. A interpolation kernel based on the Gaussian distribution function is commonly used. The major short coming of SPH is that it has inherent difficulty handling boundaries where one runs out of neighbors. One solution to the boundary difficulty includes the use of phantom particles. A more recent development is the adaptation of moving-least-square interpolants to SPH which is also claimed to alleviate other pathological problems such as erroneous strain and rotation rates and instability in tension that in SPH, occur under certain circumstances [19]. GEM employs both of these refinements.

Douglas Everhart principally developed GEM over the past few years at ARA Inc., Columbus, Ohio for application to problems of more commercial interest, including metal forming. GEM is currently an in-house 3-d analysis code used almost exclusively by ARA engineers and scientists. ARA in considering a commercialized, user friendly version of the code, provided a copy of it for the purpose of modeling the coupon tests reported in Chapter 3 . This arrangement was mutually beneficial since no other fully dynamic 3-d codes with electromagnetics was accessible for this study and ARA was in need of hard bench mark data in metal forming for their commercialization effort. The particular physical phenomena that GEM can currently model are:

Figure 4.5: Schematic of SPH scheme used in GEM after [27]

SPH approximation:

Where  is the mass density number (volume)

Interpolation kernel (3D B-spline):

Gradient SPH (to approximate derivatives):

(4.5)

(4.6)

Moving-Least-Squares SPH enhancement (to approximate functions);

Basic SPH time integration loop for mass flow (common to all hydro-codes)

SPH codes are explicit methods so that the numerical stability of the scheme is dependent on the size of the time step. The time step can be optimized internally subject to some maximum which depends on the physics being modeled. Since SPH principally model mass flows. the time step is usually linked to the speed of sound in the material. Electromagnetics, which has a much different time step requirement, is formulated as an implicit loop in GEM and is solved for each deformation time step. In order to efficiently model thin sheet deformations, Belytscho-Tsay shell elements (default shell element in DYNA3D) were integrated into GEM as specialized particles. GEM currently carries internally, six material strength models including the standard Von Mises and Johnson-Cook. There is also a single phase transformation solidification model developed by ARA. Four equations of state are available; Linear Polynomial, JWL Explosives, Ideal Gas Law and Crushable Material. The GEM code is compact and can be run effectively on 32 bit personal and mid range work station computers with adequate disk space (500MB). The source code easily fits on a 3.5 " floppy diskette i.e., it is less than 1.0 Mbytes in length.

Before the GEM software was fully operational at OSU, engineers at ARA Inc. built a model of the coupon and coil system that was used in the experiments reported in Chapter 3. Figures 4.6 to 4.8 summarize the results of the GEM simulation of a specific 6111-T4 coupon test (F15) and provides a comparison with the experimental data provided to ARA by the Hybrid Forming Project. ARA Inc extended GEM specifically for modeling the EM phase of the MT-EM hybrid forming process. A RLC circuit module was included in the extensions. Previously the current wave form had to be specified as an input to a GEM model for EM simulations. This requirement could be met for the benchmark simulations since the data from the coupon experiments was available. However, the coil current is strongly coupled to the other system variables and is definitely dependent. To be a useful design

tool for the MT-EM process, GEM required a RLC circuit module so that the true independent variable, capacitor charge energy, could be used in simulation studies of proposed systems.

A 3D rendering of the cloud of particles which define the deformed F15 coupon and the coordinate system of the GEM model are shown in Fig. 4.6. Reasonable agreement in the deformed shape of the major axis is found in a comparison with Fig. 3.6. More specific and detailed comparisons are made in model results from the runs made by ARA Inc. given in Figs. 4.7 to 4.9 .

The experimental current data shown in Fig 4.7 was measured at the coupon coil input (center) lead and is closely approximated by the special RLC module in GEM. In Figure 4.8 the plastic deformation of the coupon in the x-z plane, predicted by GEM from actual coil current data and from capacitor energy, RLC module is compared to actual deflection as measured from the high speed array camera image of the test. Figure 4.9 is the coupon deformation in the y-z plane calculated by GEM. Figures 4.10 to 4.12 contain more detailed results from the F15 model run at OSU facilities. Note that for all the following GEM output figures, the center plane of coil main center bar coincides with x=0.0 and the side, return legs are at +/- 20. mm Figure 4.10 shows the calculated evolution of the x-z profile shape at 50 microsecond intervals. The predicted profile evolution is generally quite similar to the shape evolution of the coupons in general of which 32D, shown in Fig. 3.5, is representative. Close inspection of Fig. 4.10 will reveal that GEM predicts that the coupons experience an amount of elastic recovery. Note that the final profile at 595 m sec. is about 1mm lower and a bit wider than the tallest profile which occurs at 295 m sec. Elastic recovery would be difficult to measure with the array camera system used. However, the basic physics of the coupon test event makes elastic recovery nearly certain at some level.

The z velocity of the sheet in the x-z plane, calculated by GEM, is shown in Fig. 4.11. Direct measurement of deformation velocity was not available for the coupon test. However, it can be inferred from the array camera pictures which indicated general agreement with the predicted distribution history of velocity in the x-z plane. The coupon final vibrations indicated by the small velocity oscillations at 595 m sec. is also consistent with the physics of impulsive loads on elastic bodies.

The predicted y current densities of the coupon eddy current during the deformation event shown in Fig. 4.12 are, like the velocities, not measured. Unfortunately, there is no data form the tests like the array camera images from which the eddy currents can be directly inferred. However, the shape evolution recorded by the array camera does indirectly indicate general eddy current strength and distribution. The highest forces are generated by the largest currents which results in the largest deflections in the shortest time. The initial large central current peaks seen in Fig. 4.12 is corroborated by the first narrow high bump seen in Fig. 3.5 and in a self consistent way by the first profile in Fig. 3.10.

Actual profiles of the deformation heights along the x axis of the F15 coupon and others appear in Fig. 4.13 along with the final x-z profile calculated by GEM. The final GEM profile is seen to be significantly lower than the measured F15 profile. The GEM profile also lower than those shown which were deformed at lower energies. The comparison made in Figure 4. 13 is, in reality, not bad since many model simplifications could account for a sizable portion of it. In particular the constitutive relation used in this model may not be accurate in the velocity regime of the test. The comparison of x-z plane deformation produced by ARA and given in Fig. 4.8 is a bit misleading in that the actual x-z plane profile is higher than that measured from the array camera image of a free edge. ARA engineers did not have the actual coupon and from the array image it is very difficult to see the dome of the coupon behind the highly reflective edge. The front edge down turn accounts for most of the difference between the F15 profile in Fig 4.9 and 4.13. Fortunately the coupon edge which faces the camera is not down turned to the extent as a back edge, as in Fig. 3.6, so the difference is not considered to have a significant effect on calculation of the average deformation velocity.

The asymmetry in the y-z plane is predicted by GEM ( see Fig. 4.9) but in the wrong direction. Returning to Fig. 4.9, a comparison with Fig 3.6 and the measured y-z profiles shown in Fig. 4.14, indicates a that there is a fundamental inaccuracy in the way that the version of GEM used for this simulation is calculating the deflection of the free edges of the coupon. GEM predicts a saddle shape while the actual profiles in Fig. 4.14 clearly shows the coupon free edges are down turned. The y-z profile discrepancy is certainly a manifestation of the pathological problem that SPH has with boundaries. In discussions with ARA engineers concerning this problem it was decided that the cause was related to how the phantom particles were dealt with in calculating the eddy currents of the coupon [27]. The y-z profile generated is consistent with eddy current being carried by the phantom particles, which lie beyond the free edge of the coupon. The solution to this problem is the stricter enforcement of the conservation of charge within the GEM code. The special shell element particles that are used to model thin work sheets will be defined as super elements consisting of four sub-elements. This arrangement is required for the approximation of the second order conduction equations. The program flow will, in effect, solve for the induction currents from the predicted B fields and use these results as input to the conduction equations for the work sheet. Implementation of the code corrections is in currently in process at ARA Inc. However, verification of the corrected code will need to be included in future work.

GEM has, in general, performed rather well for a numerical method invented for problems far removed from the present application. The only major predictive problem in GEM discovered during this study which has been discussed above, is considered solvable and is presently being corrected. Presently there is no evidence that the accuracy of GEM models will not be quite sufficient, after corrections are made to prevent currents in phantom particles. With additional development and some refinements to the user interface, GEM could very well be the heart of the design method for MT-EM as well as other static- dynamic hybrid sheet forming processes.