Volume 49, Number 3, 8 359 Numerical Analysis of Transient Phenomena in Electromagnetic Forming Processes Sorin PASCA, Tiberiu VESSELENYI, Virgiliu FIRETEANU Pavel MUDURA, Tiberiu TUDORACHE, Marin TOMSE and Sorin MURESAN he paper deals with numerical models for electromagnetic forming processes applied on thin metallic sheets. The complexity of such numerical analysis is due to a coupling of different phenomena and, as a consequence, to an interdependence of system parameters. The process consists in high rate deformation of the workpiece by the action of electromagnetic forces. The analysis of transient electromagnetic field in magnetoforming applications is based on numerical models developed under FLUX professional software, while the simulation of mechanical deformations is performed using ANSYS. The paper presents some partial results of a research project. Keywords: electromagnetic forming, transient magnetic and structural analysis, finite element models 1. INTRODUCTION Because of multiple advantages, the use of workpieces made by aluminium and its alloys has extended in many sectors e.g. in automotive industry. As a consequence, taking into account the particularities in processing such materials, some classical technologies are improved but, because these processes are not satisfactory, new unconventional technologies are initiated. High rate forming technologies such as electromagnetic forming, hydro-forming etc., used only in some special application in 6-7 and rarely used later, especially due to an insufficient control, becomes actually. Electromagnetic forming (EMF), or magnetoforming, is a high speed forming process where a pulsed magnetic field is used to processing metals with high electrical conductivity (aluminium, copper or their alloys) within a few microseconds. Usually, thin wall metallic pieces (sheets, tubes) are deformed at room temperature by intense impulsive forces generated by a rapidly time varying magnetic field. The main components of EMF facility for forming metallic sheets are presented in Fig. 1. By discharging the capacitor through the coil, a time dependent magnetic field is produced and eddy currents are induced in the workpiece. Between the coil and workpiece the repulsive forces occur. If the stress generated in the metallic piece exceeds the yield point of the material, plastic deformation is produced with high velocity and for a very short time [1]. The magnetoforming involves a high frequency process, high power and strongcoupled phenomena, in which the electrical and geometric parameters are time dependent. A rigorous analysis of the magnetoforming process must take into 8 Mediamira Science Publisher. All rights reserved.
36 ACTA ELECTROTEHNICA account the non-uniform distribution of the current density in conductive regions, respectively the transient skin effect [1]. In addition, the deformation of the workpiece modifies the magnetic field distribution and the motion determines supplementary eddy currents, which characterize a strong coupling between electromagnetic and mechanical/structural phenomena. References give several approaches for the study and modeling of magnetoforming systems in condition of some assumptions [1]. In [3] we have analyzed the transient electromagnetic regime of magnetoforming only for thin wall metallic tubes. Next step, this paper presents one proposed numerical approach of the magnetoforming process of metallic sheets as a transient magnetic application and analyzes the mechanical displacement and deformation of the workpiece under action of these impulsive forces.. MODELS AND FORMULATIONS We study the magnetoforming device for metallic sheets as an axisymmetric system, in the cylindrical coordinate system (r,ϕ, z) Fig.. The current density in the forming coil, the source of electromagnetic field, has azimuthally orientation, J ex = J ex ϕ and the time dependent magnetic vector potential A(r,z,t) has only azimuthally component. The governing equation in terms of A is: ν A DA ( r A ) + ν + σ = J r r r z z Dt ex (1) where ν is magnetic reluctivity and σ = 1/ρ is electric conductivity. The total derivative in DA A (1) has the expression = v rot A, Dt t with v the velocity. If the motion of workpiece is not considered (v=) the equation of the modified vector potential r A is: ν ν ( r A) + ( r A) + r r r z r z () 1 + σ ( r A) = J r t ex The second term in the left part of the equation represents the density of induced currents, non-null in the conductive regions of the study domain. This equation of transient magnetic field is solved by the method step by step in time domain. The study domain for a typical flat sheet magnetoforming application is shown in Fig.. The source J ex, non-null only in coil region, is an unknown value as it follows: if we consider the forming coil is of thin wire type the current density J ex is constant in whole the coil region and if the forming coil is of solid conductor type, the current density is not constant over the coil cross-section. At any time step, the second relation between A and J ex are given by the model of electrical circuit, considering magnetic field electric circuit coupling. Having the force vector in every element of the piece (i.e. volume density of electromagnetic force), at every time step, we can find the final position of each element after the action of forces. Electromagnetic forming, from mechanical point of view, is characterized by a large plastic deformation of the processed material, which occurs in a very short time. The mathematical theory of plasticity
Volume 49, Number 3, 8 361 adequately describes the time-independent aspect of the behavior of materials but this is inadequate for the analysis of time-dependent behavior. An approach to achieving a satisfactory formulation for time-dependent behavior has been to generalize plasticity to cases within the strain-rate-sensitive range []. One such generalization has been provided by the theory of viscoplasticity. Last decades, various forms of the theory of viscoplasticity have been provided (Perzyna, Cristescu and Peirce). An approach to the construction of equations for viscoplastic materials can be made using the extremum principle. This also can be used as a basis for the finite-element formulation of viscoplastic analysis. Considering the work of Perzyna, a viscoplastic material can be described by applying the infinitesimal theory for each incremental deformation. Perzyna introduces a function F (σ ij ) []: 1/ 1/ { σij ' σij '} F( σ ij ) = 1 (3) k where k is the static yield stress in shear. Considering F(σ ij ) as the function similar to the plastic potential in the theory of plasticity, the constitutive equation is expressed as: F = γ " Φ( F) (4) σ ij where γ is the viscosity constant of the material and Φ (F) is a function of F such that: Φ( F ), F =. Φ( F), F > Then, = γ ' σ σij ' 1 σ Φ σ (5) where 1/ ( 3/ ) { σ ij ' σij '} σ = so that σ is identical to the yield stress in uniaxial tension, and σ = 3 k is the static yield stress in tension. Squaring both sides of (5): σ σ ' σ ' ij ij Φ 1 σ σ = γ ' (6) and using ε = 1/ ( / 3) { ε ij }, we obtain: σ ε = γ ' Φ 1 3 σ (7) Then from (5) and (7), the constitutive eqn is: 3 ε = σ ij ' σ (8) with σ and ε as defined before. The effective stress σ in equation (8), depends on the strain-rate dependent function Φ, which is to be determined by the properties of the material. If we choose the 1/ m function Φ = (( σ / σ ) 1), then from equation (7): m ε σ = σ 1 + γ (9) Equation (9) is a familiar rate-dependence law and the exponent m is the strain-rate sensitivity index. The last equation is also used to model viscoplastic material behavior in the ANSYS finite element analysis software [4]. As it is stated in [4], this kind of material models the best impact like deformation processes. 3. APPLICATIONS AND RESULTS The application which is subject of numerical analysis consists of electromagnetic forming of 1mm thin aluminium alloy sheet, the geometry of the system being illustrated in Fig.. The forming coil has 1 to 6 turns made by copper, with different width of turns but with the same overall dimensions, like in Fig. 3. Within numerical model, the coil region is considered solid conductor, tacking into account the non-uniform distribution of current density in cross-section. The geometry of the system (3D variant) for 1 and 4 turns coil is presented in Fig. 4. For transient magnetic application, we have
36 ACTA ELECTROTEHNICA R = 35 e R =1 i symmetry axis 1 5 1 6 6 6 4 4 4 4.5.5.5.5.5.5 5 (1) () (3) (4) (5) sheet are modeled using ANSYS software, as a transient application. As an intermediate step in solving this complex magneto-mechanical coupled problem, in this work we obtain the forces from a motionless structure. Fig. 5 gives the time variation of coil current and of the axial component of electromagnetic force on metallic sheet and Fig. 6 presents the distribution of volume density of electromagnetic force vectors within workpiece region at pick current time, for some studied cases. 1,499 1 3 4 6 (E6) A Fig. 3. The geometry of the coil region in five studied cases, with different number of turns. 1 499,999E-3 -,5-1 (E-6) s a) b) (E6) N 1 3 4 5 a) 1 3 4 6 4 3 c) d) Fig. 4. The 3D geometry of computation domain, for 4 turns (a, c) or 1 turn (b, d) forming coil. analyzed both D and 3D structure and because the differences are not significant concerning of the values of electromagnetic forces, we have studied the deformations only in D geometry. Transient regime of electromagnetic field in the computation domain is solved using FLUX software. At each time step, we obtain the volume density of electromagnetic force within each element of metallic sheet region, which represents the loads in structural application. The deformation/displacement of 1 1 3 4 5 (E-6) s b) Fig. 5. The time variation of the coil current (a) and of the axial component of electromagnetic force acting on metal sheet (b), for 1,, 3, 4 or 6 turns forming coil, at the same stored energy. Material properties of metallic sheet are follows: Young s modulus E = 7 GPa, yield stress σ = 4 MPa, tangent modulus E t = 1 MPa, Poisson ratio ν =.33, strain rate
Volume 49, Number 3, 8 363,1E6 Ampere Current / Current Time coil 6 turns,5e6 sheet gap coil -49,999E3 s. a),5e6,e6 New ton 5E-6 1E-6 15E-6 Force / Axial component Time SHEET,15E6,1E6 b) Fig. 6. The vectorial representation of volume density of electromagnetic force acting on sheet, for 1-turn (a) and 6-turns (b) solid conductor coil. hardening parameter m =.5, viscosity parameter γ = 1, density γ m = 7 kg/m 3, electrical resistivity ρ =.7 1-8 Ωm. Trough numerical modeling we find that for free expansion of thin aluminium sheet with dimensions and properties above described it is not necessary a big amount of a stored energy. For example, with 1 J (C = μf, U = 1kV) in capacitor bank and with 6 turns solid conductor coil, we obtained at t p = 11 μs the first pick of coil current of 11.54 ka and the corresponding total axial electromagnetic force acting on sheet region of 8.8 kn. The exterior diameter of coil was 7 mm and of the free deformable zone of sheet 9 mm. The gap between sheet and coil is 1 mm. The damped oscillation process is illustrated in Fig. 7 by time variation of coil current and of the axial component of the total force in sheet region, from computation domain. Several steps of sheet forming process are presented at the end of the paper in Fig. 8.,5E6 5E-6 1E-6 15E-6 Fig. 7. Time variation of coil current and of axial component of electromagn. force in sheet region. electromagnetic forces, resulting at each time step from magnetic computation. In a future work, we will take into account the influence of sheet deformation on electromagnetic field distribution in computation domain at each time step and as a consequence on electromagnetic forces. Fig. 8. The deformation of metallic sheet at different time steps (next views). s. 4. CONCLUSIONS The paper presents a method for simulation of electromagnetic forming of an aluminium sheet, based on numerical model of transient electromagnetic field in computation domain and on transient analysis of viscoplastic forming of sheet under action of
364 ACTA ELECTROTEHNICA 5. ACKOWLEDGMENTS This work was supported by Romanian Ministry of Education and Research, Grant CNCSIS A 1319/7, 34GR/1.5. REFERENCES 1. C. Fluerasu, Equivalent schemes of electromagnetic forming installations, Rev. Roum. Sci. Techn. Electrotechn. et Energ., 16, 4, pp. 593 69, Bucharest, 1971.
Volume 49, Number 3, 8 365. Shiro Kobayashi, Soo-ik Oh, Taylan Altan, Metal forming and the finite element method, New York, Oxford, Oxford University Press, 1989. 3. S. Pasca, V. Fireteanu, Transient Magnetic Finite Element Model of Magnetoforming, Rev. Acta Electrotehnica, Vol. 45, No. 6/4, pp. 575-58. 4. *** Ansys, Inc. Theory Reference, Ansys release 1., 184, 5. Sorin PASCA Marin TOMSE University of Oradea Faculty of Electrical Engineering and Information Technology E-mail: spasca@uoradea.ro Tiberiu VESSELENYI Pavel MUDURA University of Oradea Faculty of Management and Technological Engineering Virgiliu FIRETEANU Tiberiu TUDORACHE POLITEHNICA University of Bucharest Electrical Engineering Dept. EPM_NM Laboratory Bucarest, Romania E-mail: firetean@amotion.pub.ro Sorin MURESAN University of Oradea Faculty of Science Oradea, Romania