Electromagnetic forming by distributed forces inmagnetic and nonmagnetic materialsMotoasca, T.E.; Blok, H.; Verweij, M.D.; Berg, van den, P.M.

Published in:IEEE Transactions on Magnetics

DOI:10.1109/TMAG.2004.834041

Published: 01/01/2004

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Citation for published version (APA):Motoasca, T. E., Blok, H., Verweij, M. D., & Berg, van den, P. M. (2004). Electromagnetic forming by distributedforces in magnetic and nonmagnetic materials. IEEE Transactions on Magnetics, 40(5), 3319-3330. DOI:10.1109/TMAG.2004.834041

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004 3319

Electromagnetic Forming by Distributed Forces inMagnetic and Nonmagnetic Materials

T. Emilia Motoasca, Hans Blok, Martin D. Verweij, and Peter M. van den Berg

Abstract—In this paper, we discuss the electromechanical forcedensities associated with pulsed electromagnetic fields in inho-mogeneous, linear media with conductive losses, in the contextof a process of shaping metal objects. We show that the con-ductivity and the gradients in permittivity and in permeabilitylead to volume forces, while jump discontinuities in permittivityand permeability lead to surface forces. These electromagneticforces are assumed to act as volume (body) source densities in theelastodynamic equations and as surface source densities in thecorresponding boundary conditions that govern the elastic motionof deformable matter. As an example, we apply the theory to thecalculation of the elastic field in a hollow cylindrical object made ofa conducting magnetic or nonmagnetic material. We compare thenumerical results with those for the classical theory of elasticitywith concentrated forces on the boundaries of the material as thesource of the elastodynamic field.

Index Terms—Conducting magnetic materials, elastic field,electromechanical forces.

I. INTRODUCTION

THE literature pertaining to forces in electromagneticbodies is vast. An extensive review on more fundamental

aspects has been given by Penfield and Haus [1] while theexpressions of the force distribution in magnetized materialhave been discussed by many other authors like Carpenter [2],[3], Byrne [4], Carter [5], Reyne et al. [6], and Bobbio [12].In their book, Penfield and Haus [1] consider the completephysical system that describes the motion of a continuumunder the influence of electromagnetic field. It consists ofthree mutually coupled subsystems: a mechanical subsystemdescribing the mechanics of the moving material masses;an electromagnetic subsystem describing the dynamics ofthe electromagnetic fields; and a thermodynamic subsystemtaking into account the internal energy and the generation ofheat and its flow. However, for the engineering application ofelectromagnetic forming where materials bodies are shapedin intense, pulsed electromagnetic fields, a number of obser-vations and assumptions related to forces in electromagnetic

Manuscript received December 24, 2003; revised May 12, 2004.T. E. Motoasca is with the Faculty of Electrical Engineering, Division of

Telecommunications Technology and Electromagnetics, Eindhoven Universityof Technology, 5600 MB, Eindhoven, The Netherlands (e-mail: t.e.mo-toasca@tue.nl).

H. Blok and M. D. Verweij are with the Faculty of Electrical Engineering,Mathematics and Computer Science, Department of Electrical Engineering,Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail:H.Blok@EWI.TUDelft.nl; M.D.Verweij@EWI.TUDelft.nl).

P. M. van den Berg is with the Faculty of Applied Sciences, Delft Universityof Technology, 2628 CJ Delft, The Netherlands (e-mail: P.M.vanden-Berg@TNW.TUDelft.nl).

Digital Object Identifier 10.1109/TMAG.2004.834041

bodies can be made. The force expression that follows fromthe general theory contains certain terms that are too small tobe of engineering importance. In particular, we assume thatpiezoelectric, magnetoelectric, and magnetoelastic effects arehigher order effects and in the first instance can be neglected.The deformation velocities of all points in the materialbody are supposed to be small in comparison with the velocityof light . In electromagnetic forming the particle deformationvelocities are typically in the order of – m/s. Thus,terms of first and higher order in may be neglected.

With these assumptions and the theory of Penfield and Haus[1] in mind, the simplified physical system for deriving forcedensities in electromagnetic forming devices consists of anelectromagnetic system and a mechanical system only, cou-pled through electromechanical force densities. Within a linearapproximation of the model of electromagnetic forming, theelectromagnetic force densities are assumed to act as volume(body) source densities in the elastodynamic equations and assurface source densities in the corresponding boundary con-ditions that govern the elastic motion of deformable matter.For the derivation of the distributed electromechanical forcein this simplified system, we start with a macroscopic modelapproach, where the force that impressed, external currents andcharges exert on the electromagnetic field in the configurationfollows from a balance of electromagnetic momentum. Most ofthe standard considerations on the subject focus on the relevantforces in static or quasi-static electric and magnetic fields (see,e.g., Stratton [7], Moon [8], Fano [9], Penfield and Haus [1]), oron the case of continuously differentiable spatial variations in theconstitutive properties (see, e.g., Stratton [7] and Landau [10]).

In the present paper, we follow the macroscopic approachalong the line of the analysis presented by Stratton [7]. Hisanalysis will be extended to the case of piecewise inhomo-geneous, isotropic media, thus allowing for interfaces acrosswhich the constitutive parameters jump by finite amounts. Wewill show that finite gradients in permittivity and permeabilitylead to distributed volume forces and interface conditionslead to distributed surface forces. The pulsed field behaviorintroduces a distributed volume force that is associated withthe time derivative of the electromagnetic momentum of thefield. Further, the distributed volume and surface forces leadto an elastodynamic wave field in the relevant medium. Themechanical stress associated with the elastic wave motion thendetermines the amount of mechanical deformation that themedium undergoes. In many papers [11], [13]–[15], [20], [17],[18] on electromagnetic forming, the present problem is dealtwith by the use of equivalent surface source accounting for the

0018-9464/04$20.00 © 2004 IEEE

3320 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004

Fig. 1. A pulsed current loopD with JJJ and � in a vacuum domainDirradiates an inhomogeneous, isotropic object D . D is exterior to the sourcedomain D and the object domain D . The domain D = D [ D [ D hasthe boundary @D.

total electromagnetic force exerted on the workpiece duringthe deformation process. We investigate for a simple formingproblem the differences between this approach and the presentmethod in which both the volume force and surface forcedensities as local source distributions generating the elasticwave motion are taken into account appropriately.

II. ELECTROMECHANICAL VOLUME AND SURFACE FORCES

We consider a configuration consisting of a pulsed currentloop with external current density and externalcharge density that irradiates a smooth inhomoge-neous, isotropic object with constitutive parameterslocated in vacuum as shown in Fig. 1. The electromagnetic fieldequations are given by

(1)

The compatibility relations are

(2)

Further, we introduce

(3)

as volume density of external electric charge. Across a source-free interface of jump discontinuity in , and , the field quan-tities satisfy the continuity conditions

continuous across interfacecontinuous across interface

(4)

andcontinuous across interface

continuous across interface(5)

where is the normal vector of the interface.

Fig. 2. Detail of the boundary of the material object D in Fig. 1.

We take as physical interpretation that the volume source den-sities of external currents and electric charges in domainexert on the field the volume force, cf. Stratton [7]

(6)

To arrive at an expression that shows how the electromagneticfield transmits this force to other parts of the configuration (e.g.,outside in Fig. 1), we use (1)–(3), to end up with

(7)

In (7), we have

(8)

where

(9)

which is the volume density of induced electric current. Further

(10)

is the Maxwell stress tensor with the unit tensor of rank two.Using Gauss’ divergence theorem, the last term of (7) can berewritten as

(11)

where

(12)

with as in Fig. 2. The results presented in (7) and (11)can be rewritten as the balance of electromagnetic momentumfor the configuration, viz.,

(13)

MOTOASCA et al.: ELECTROMAGNETIC FORMING BY DISTRIBUTED FORCES 3321

In this expression

(14)

is the volume force that the electromagnetic field exerts on theinduced volume density of electric current

(15)

is the volume force that the field exerts on gradients of permit-tivity and permeability. Further

(16)

is a force due to a jump in the Maxwell stress tensor at theboundary of . This jump is conjectured to act as a surfaceforce at for the elastodynamic wavefield in the configura-tion. At the boundary of , a force is exerted that equals

(17)

For its interpretation, take for the ballof radius and center at the origin. In the far-field

region, the behavior of the field radiated by the sources inguarantees that exists as . Similarly, alsoexists as . The corresponding term can be in-terpreted as the “radiation pressure” that the field, by carryingits momentum from the exciting sources to , exerts on the“sphere at infinity.” Note that exerting this pressure is compat-ible with the property that this sphere absorbs the power radiatedto it. This radiation pressure is irreversibly lost to the electro-magnetic momentum. Finally

(18)

is the electromagnetic momentum carried by the field.With the interpretation of the radiation pressure on the sphere

at infinity in mind, also the terms and have the struc-ture of an irreversible loss of momentum. It is noted that, whenno matter is present in , both terms will vanish. As a conse-quence, in

(19)

and

(20)

can be conjectured to be the driving volume source densities ofbody force in the elastodynamic field equations that govern thebehavior of dynamic stress and particle velocity in mechanicallydeformable matter. In conclusion, we have found that in an in-homogeneous medium, the total volume force consists of twoforces; see (19)–(20). Later, we present some numerical results

for these forces in a piecewise homogeneous material wherethese two forces still exist due to the jump in permittivity andpermeability.

III. FORCES IN A HOMOGENEOUS MAGNETIC OBJECT

From the discussion in Section II we observe that in a homo-geneous, highly conducting magnetic object with medium pa-rameters located in vacuum the volume density of inducedcharge vanishes and in we find

(21)

(22)

Since in the electromechanical system for electromagneticforming, the electromagnetic field may be considered transientdiffusive, the derivative with respect to time of the electromag-netic momentum is always negligibly small in comparison toother terms in the balance of electromagnetic momentum. Toconclude, the force exerted on is in the present case

(23)

In this case, the volume density of force inside is givenby

(24)

while the surface density of force at follows immediatelyfrom (11) and (12).

IV. ELASTODYNAMIC WAVE MOTION GENERATED BY VOLUME

FORCES AND SURFACE FORCES

Here, the subscript notation and summation convention willbe used. The elastodynamic wavefield is characterized by its ten-sorial dynamic stress , together with its vecto-rial particle velocity . These quantities satisfy theequation of motion

(25)

in which is the volume density of mass of themedium, and the constitutive equation

(26)

in which is the elastic stiffness tensor andis the strain. The latter quantity satisfies the strain-displacementrelation

(27)

in which is the particle displacement. The relation betweenparticle velocity and particle displacement is

(28)

In our isotropic object (see Fig. 3), we have

(29)

3322 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004

Fig. 3. Isotropic object D with elastic properties � ; � ; � .

Fig. 4. Cross section of the cylindrical configuration under consideration.

in which and are the Lamé coefficients of the medium.Combination of (26)–(29) results in the deformation rate

equation

(30)

Across the surface discontinuity in elastodynamic properties, the boundary conditions are

(31)

The surface density of force in (31) follows from (11) and(12).

V. APPLICATION TO A SIMPLE CONFIGURATION

The configuration in which the electromagnetic field and thenthe elastic field will be calculated is presented in Fig. 4. Thisconfiguration models the case of electromagnetic compressionof hollow cylindrical objects, where the object to be deformed(the workpiece) is placed inside a forming coil. The configura-tion can be divided into four cylindrical subdomains, namely,the inner space of the workpiece with , the workpiecedomain with , the air gap in between the workpieceand the forming coil with , and the space outside theforming coil with .

A. Electromagnetic Solution

The medium of the workpiece is homogeneous, linear, timeinvariant, locally and instantaneously reacting, and isotropic inits electromagnetic behavior, with permeability and electricalconductivity . The other media are assumed to be vacuum withpermeability and zero conductivity. The configuration is ex-cited by a single sheet antenna, located at radial position ,

carrying an electric current in the positive direction. This in-finitesimally thin sheet antenna models the forming coil in thereal electromagnetic forming system. The configuration underinvestigation is assumed to have infinite length in the direc-tion. This fact, combined with the axial symmetry of the con-figuration, gives that all quantities related to this configurationdepend on the radius and time , only. In the cylindrical sheetat , an electric current per unit length along the di-rection is present. The electromagnetic field is causally relatedto the action of this electric-current source. The given sourcegenerates a one-dimensional, -dependent field of which only

and differ from zero. We assumethat the time variation is such that the displacement current canbe neglected and the field components satisfy the diffusive elec-tromagnetic field equations

= 0, for (32)

The simplest way to construct solutions that satisfy these equa-tions together with the boundary and excitation conditions andensure causality is to use the Laplace transformation with re-spect to time. To illustrate the notation, let

(33)

where it has been assumed that the electric current source starts toact at the instant . The complex transform parameter (alsodenoted as the complex frequency) is taken in the right half of thecomplex -plane.TheLaplacetransformedquantitiesaredenotedwith a hat symbol and we omit the explicit -dependence in ournotation. We take the limit , so that we end up with theFourier transformed quantities, where j is the imaginary unit and

is the radial frequency, while denotes the frequencyof operation. In our numerical work, we use the fast Fourier trans-form (FFT) to compute the pertaining Fourier transforms.

In the conductive domain of the workpiece, the compo-nents of the electromagnetic diffusive field are obtained as

(34)

where and are the modified Bessel function of the firstand second kind, while the quantities

(35)

are the diffusion coefficient and the impedance of the diffusiveelectromagnetic field, respectively. In the vacuum domains, wehave zero conductivity and the following quasi-static equationshold:

(36)

The first equation indicates that in a vacuum domain the mag-netic field is constant, while the second equation shows that theelectric field is a linear combination of and . The constants

and are determined from the continuity conditions for the

MOTOASCA et al.: ELECTROMAGNETIC FORMING BY DISTRIBUTED FORCES 3323

magnetic and electric fields at , the excitation conditionthat the magnetic field at is equal to , and the solutionsin the vacuum domains. In effect, in the domain , wearrive at

(37)

where

(38)

(39)

with .Within the homogeneous isotropic domain with ,

thus in the cylindrical domain representing the workpiece, theapplication of (24) gives

(40)

In view of the continuity of the magnetic field strength at theboundaries at and of the workpiece, the electro-magnetic surface force density on the outer boundary is ob-tained from (12) as

(41)

while on the inner boundary of the workpiece we have

(42)

After an inverse Fourier transform of the field quantities givenby (37), all the quantities in the right-hand side of (40)–(42) areknown for the whole time interval of investigation and within thewhole configuration. The electromagnetic volume force densityand the electromagnetic surface force density can then be cal-culated very easily. For nonmagnetic workpieces, the surfaceforce densities are zero, while for magnetic workpieces, theyhave nonzero values.

B. Elastodynamic Solution

When we turn to the elastodynamic problem, we observe thatthe sources, exciting the elastic field, are the electromagneticforce densities. In our configuration at hand, only the radialcomponents of these force densities are nonzero, viz., the elec-tromagnetic force density in the workpiece and theelectromagnetic surface force densities at the twoboundaries of the workpiece. In a cylindrical coordinate system,the components of the stress, strain, and particle displacementare ,and , respectively. With these components, theequations of motion are

(43)

(44)

(45)

Since the workpiece has been assumed homogeneous andisotropic, the constitutive relations are given by

(46)

In the same coordinate system, the components of the strain aredefined as

(47)

There exists no radial stress within the air, so the boundary con-ditions to be applied at and are

(48)

(49)

respectively. In the modeled cylindrical configuration with infi-nite length, we will assume that the stresses and strains are uni-form along the length of the workpiece. This assumption, com-bined with the rotational symmetry of the configuration givesthat all elements of the stress and strain are functions only of theradius and time , and all and operators result in zero.Moreover, . In our cylindrical configuration with infi-nite length, two particular cases related to the elastic field maybe distinguished: the plane stress case and the plane strain case.The plane stress case models a tube with free ends, thus the lon-gitudinal displacement is allowed and the longitudinal stress

is assumed to be zero. Although there is a longitudinal dis-placement , the infinite length of the configuration does notallow the calculation of the longitudinal displacement . Theplane strain case models a tube with fixed ends, thus the longitu-dinal displacement is zero, and further the longitudinal strain

is also zero. The infinite length of our model can deal withthis case. Moreover, the case characterizes the practicalsituation in which the finite workpiece is clamped at the ends.Therefore, we will solve (43) for the case of plane strain.

The simplest way to construct solutions of the present elas-todynamic equations is again to use the Laplace transforma-tion with respect to time. As for the calculation of the electro-magnetic field, we take , where is the ra-dial frequency, while denotes the frequency of operation. Inthe Laplace transform domain, the nonzero components of thestrain tensor and of the stress tensor, in the case of plane strain

, are written as

(50)

and

(51)

3324 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004

After substitution of and in the Fourier transformedcounterpart of (43), it becomes

(52)

where is the compressional wave number andis the compressional or P-wave speed.

When the solution of the above equation is found (see the Ap-pendix), all nonzero components of the strain and stress tensormay be calculated in the frequency domain, in accordance to(50)–(51). The results may then be transformed back to the timedomain using an inverse FFT.

VI. METHOD OF EQUIVALENT SURFACE FORCES

In the literature related to electromagnetic forming (see[13]–[15], [17]), the electromagnetic forming problem is dealtwith the use of an equivalent surface force (pressure) accountingfor the total electromagnetic force exerted on the workpieceduring the deformation process. Although in the literaturerelated to electromagnetic forming, the elastic field is not dealtwith in detail, it seems to suggest a solution of the presentelastodynamic problem with the help of these equivalent sur-face forces as external sources. Thus, the electromagnetic forcedensity in the right-hand side of (43) is assumed to be zeroand we have to solve the following equation of motion:

(53)

supplemented with boundary conditions. Due to the existence oftheequivalentsurfaceforcesthatactat theboundariesof thework-piece, theboundaryconditions tobeappliedaredifferent fromtheonesin(48).Thenewboundaryconditionstobeappliedresultfromthefactthatthetotalelectromagneticforceactingontheworkpieceis replaced by an equivalent electromagnetic surface force, cal-culated with the Maxwell stress-tensor formula. Therefore, twoequivalent surface forces act on the workpiece, and the boundaryconditions in (48) and (49) become

(54)

(55)

In the next section we will use the method of equivalent surfaceforces as presented in (53)–(54) to compare the numerical re-sults with the ones obtained with our theory.

VII. NUMERICAL RESULTS

In this section, we present some numerical results for a typ-ical electromagnetic forming system designed for compressionof hollow circular cylindrical workpieces. The workpiece sub-jected to electromagnetic compression has an inner radius

mm and an outer radius mm. With this ge-ometry, two types of materials have been chosen: one nonmag-netic and one hypothetical linear magnetic material

. In both cases, the electrical conductivity of theworkpiece is S/m and the cylindrical current sheetaccounting for the forming coil is located at mm.

Fig. 5. Current per unit length in the sheet antenna.

Fig. 6. Spatial distribution of the real part (left) and the imaginary part(right) of the magnetic field in the frequency domain at f = 500 Hz, forthe nonmagnetic (� = � ) workpiece (dashed lines) and for the magnetic(� = 100 � ) workpiece (solid lines).

Further, in both cases, the workpiece has the same elasticproperties and it has a linear elastic behavior within the wholerange of stresses and strains. The workpiece has the Lamé co-efficients of elasticity N/mN/m , and a mass density kg/m . We firstcompare the results of the electromagnetic problem and subse-quently the results of the elastodynamic problem. In particular,we will present results for the radial displacement , the radialstrain , the tangential strain , the radial stress

, the tangential stress , and the longitudinalstress .

A. Electromagnetic Results

In our examples, we have considered that the current per unitlength flowing in the sheet antenna is a damped pulsetypically used in electromagnetic forming processes. In the leftplot of Fig. 5, the current per unit length in the sheet antenna inthe time domain is presented, while its frequency domaincounterpart is presented in the right plot of this figure.As we will observe later, the electromagnetic volume force den-sity in the frequency domain reaches its maximum value at about100–500 Hz. Therefore, at 500 Hz, we present the spatial dis-tribution of the complex magnetic field inside and outside theworkpiece (see Fig. 6). In the magnetic workpiece, we observea much larger decay of the magnetic field than in the nonmag-netic workpiece.

In Fig. 7, the electromagnetic volume force density at the outerinterface of the cylindrical workpiece has been presentedin the time domain and in the frequency domain. The results arenormalized with , since from their expressions itdirectly follows that the absolute values of the electromagneticforce density in a magnetic material are times largerthan the ones obtained in a nonmagnetic material. We observethat the time domain results are roughly the same, although in themagnetic workpiece, the electromagnetic volume force densitydecays tozeromore rapidly than inanonmagneticone. In the timedomain, the electromagnetic volume force density decays

MOTOASCA et al.: ELECTROMAGNETIC FORMING BY DISTRIBUTED FORCES 3325

Fig. 7. Normalized electromagnetic volume force density at r = b in the timedomain (left) and in the frequency domain (right) for the nonmagnetic (� = � )workpiece (dashed lines) and for the magnetic (� = 100 � ) workpiece (solidlines).

Fig. 8. Normalized electromagnetic volume force density (�=� ) fin space-time domain for the nonmagnetic (� = � ) workpiece (a) and for themagnetic (� = 100 � ) workpiece (b).

more rapidly than the current per unit length in the sheetantenna. In fact, it contains higher frequency components, whichcan also be observed from its frequency spectrum. Althoughthe dominant part of the electromagnetic force density isnegative, there is an extended time interval where the electromag-netic volume force density is positive. The positive values of theelectromagnetic volume force density are negligible as comparedwith the maximum value of the electromagnetic volume forcedensity of about 2 10 N/m .

In Fig. 8, the evolution of the electromagnetic force densityin the space-time domain both for a nonmagnetic and for a mag-netic workpiece is presented. For the magnetic workpiece, sig-nificant (nonzero) results of the electromagnetic volume forcedensity are expected only for a small part of the workpiece, i.e.,

Fig. 9. Electromagnetic surface force density f at r = a (left) and at r = b(right) in the time domain for the magnetic (� = 100 � ) workpiece.

only for a cylindrical domain located near the outer boundaryfacing the electric current sheet. Therefore, we have

presented the space-time evolution of the electromagnetic forcedensity only for a small part (21.8 mm 22 mm) of theworkpiece. Comparing the results for the magnetic and the non-magnetic workpieces, we observe indeed the large decay of theelectromagnetic force density in the negative radial direction ofthe magnetic workpiece, together with a different decay in time.

In case of a magnetic workpiece, there exists also an electro-magnetic surface force density on both sides of the workpiece.The time-domain results for the electromagnetic surface forcedensity are presented in Fig. 9, for and . As ex-pected, on the outer boundary closest to the sheet an-tenna, the absolute values of the electromagnetic surface forcedensity are larger than the ones of the electromagnetic surfaceforce density on the inner boundary .

Comparing Figs. 7 and 9, we notice that the electromagneticvolume force density (Fig. 7) decays to zero more rapidly thanthe electromagnetic surface force density (Fig. 9). This is due tothe fact the volume force density depends linearly on the mag-netic field [see (40)], while the surface force density is relatedto the square of the magnetic field [see (41) and (42)].

B. Elastodynamic Results

In Fig. 10, the components of the elastic field at inthe nonmagnetic and in the magneticworkpiece are presented. As the elastic problem is almost quasi-static, the radial displacement is most of the time negative andits shape is very similar to that of the dominant electromagneticvolume force densities. With the chosen elastic properties of thematerial, the values of the radial displacement are very small,and these values yield very small radial and tangential strains,

and , respectively. The tangential stress has the largestvalues from all the stress components, for both nonmagneticand magnetic workpieces. In the nonmagnetic workpiece, theradial stress is very small and can be neglected, while forthe magnetic workpiece it has nonzero values. The longitudinalstress has negative values that are about three times smallerthan the values of the tangential stress .

From our simulations, we have observed that the radial dis-placement at is slightly smaller than the radial displace-ment at . The results confirm the fact that during the elec-tromagnetic compression the workpiece becomes thicker.

C. Comparison With Method of Equivalent Surface Sources

Thespace-timeevolutionoftheradialdisplacement hasbeencalculated using our theory presented in Section V and with themethod of equivalent surface forces presented in Section VI.

3326 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004

Fig. 10. Elastic field components at r = a in the time domain for thenonmagnetic (� = � ) workpiece (dashed lines) and for the magnetic(� = 100 � ) workpiece (solid lines).

For the nonmagnetic workpiece, the radial displacementscomputed with both methods are roughly the same (see Fig. 11).A similar behavior is noticed for the pertaining stresses. There-fore, we do not present these results.

In order to have a better picture of the real differences be-tween the results obtained with the two methods, in Fig. 12 wepresent the temporal evolution of the radial displacement and ofthe tangential stress at the inner boundary and at the outerboundary of the workpiece. We notice that both methodsyield almost similar results.

Subsequently, for the magnetic workpiece, after using the twomethods, in Fig. 13 we present the temporal evolution of the ra-dial displacement and of the tangential stress at the innerboundary and at the outer boundary of the work-piece. We notice that the radial displacement has negativevalues when calculated with both methods, though the valuescalculated with our method are about four times larger than withthe method of equivalent surface forces. For the tangential stress

at the inner boundary of the workpiece, the same ob-servation is valid, while at the outer boundary of the work-piece the two methods give results that differ very much fromeach other.

The method of equivalent surface forces gives at the outerboundary a tangential stress that has the same behavioras the one at the inner boundary. Our method gives a tangentialstress that has a different behavior than the similar tangen-tial stress calculated with the method of equivalent surfaceforces, due to the existence of surface forces that are taking intoaccount the magnetic nature of the workpiece. Because thesesurface forces are negligible at the boundary , their in-fluence on the tangential stress behavior is very small, while atthe outer boundary , the surface forces are very large andit has a considerable influence on the behavior of the tangentialstress .

Fig. 11. Radial displacement u in space-time domain, calculated with thepresent theory (a) and with the method of equivalent surface forces (b), for thenonmagnetic (� = � ) workpiece.

Fig. 12. Radial displacement u and tangential stress � in time domain atr = a (upper graphs) and r = b (lower graphs), calculated with the presenttheory (solid lines) and with the method of equivalent surface forces (dashedlines), for the nonmagnetic (� = � ) workpiece.

Further, from Fig. 13 we observe that the tangential stress atthe outer boundary , in the case of a magnetic workpiecepresents more important sign variations when compared withthe results given in Fig. 12 for a nonmagnetic workpiece. Thereason is that near the surface force density decays very

MOTOASCA et al.: ELECTROMAGNETIC FORMING BY DISTRIBUTED FORCES 3327

Fig. 13. Radial displacement u and tangential stress � in time domain atr = a (upper graphs) and r = b (lower graphs), calculated with the presenttheory (solid lines) and with the method of equivalent surface forces (dashedlines), for the magnetic (� = 100 � ) workpiece.

Fig. 14. Radial displacement u in space-time domain, calculated with thepresent theory (a) and with the method of equivalent surface forces (b), for themagnetic (� = 100 � ) workpiece.

rapidly, while the volume force density, having an opposite sign,decays much slower (compare also Figs. 7 and 9).

In Figs. 14 and 15, we present also the full space-time evo-lution of the radial displacement and of the tangential stress

Fig. 15. Tangential stress� in space-time domain, calculated with the presenttheory (a) and with the method of equivalent surface forces (b), for the magnetic(� = 100 � ) workpiece.

, when calculated with the presented theory and with themethod of equivalent surface forces. We observe that the max-imum values of the radial displacement , when calculatedwith our method, are about four times larger than the corre-sponding values calculated with the method of equivalent sur-face forces, and their space-time evolution is very different. Thesame observation is valid for the tangential stress .

VIII. CONCLUSION

In this paper, the electromechanical force densities associatedwith pulsed electromagnetic fields in piecewise homogeneous,isotropic, linear media with conductive losses have been dis-cussed, in the context of their application in a process of shapingmetal objects. It has been shown that the conductivity and thegradients in permittivity and in permeability lead to volumeforce densities, while jump discontinuities in permittivity andpermeability lead to surface force densities. These electromag-netic force densities are assumed to act as volume (body) sourcedensities in the elastodynamic equations and as surface sourcedensities in the corresponding boundary conditions that governthe elastic and anelastic motion of deformable matter.

3328 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004

The practical configuration wherein the developed theory hasbeen applied consisted of a hollow cylindrical domain with ahigh electrical conductivity (representing the workpiece) placedinside a cylindrical sheet antenna (representing the forming coil)carrying a given electric current per unit length. The configu-ration has been assumed to have infinite length and to be axi-ally symmetric. Is has also been assumed that the displacementcurrent may be neglected and the diffusive field equations maybe applied in the conducting cylindrical domain. Further, in thedomains containing air with zero electrical conductivity , thequasi-static field equations have been applied. Our theory canbe applied for nonaxially symmetric geometries as well. How-ever, for these more general geometries the pertinent differentialequations, both the electromagnetic and elastic ones, have to besolved by numerical methods, e.g., a finite-element method de-veloped by Lee et al. [19], [20] and Besbes et al. [21].

The electromagnetic force, assumed to be the source of theelastic field, has been computed in the cylindrical domain withhigh electrical conductivity for two types of materials: one non-magnetic and one linear magnetic . Thevalues of the electromagnetic volume force density are muchlarger in a magnetic material than in a nonmagnetic one. Thenumerical results showed also that, in both cases, the electro-magnetic force density decays rapidly in time and space. Forthe magnetic material, the electromagnetic surface force den-sity has also been calculated. Its values are much smaller thatthe integrated value of the electromagnetic volume force den-sity over the thickness of the highly conducting domain.

In the literature related to electromagnetic forming, an equiv-alent pressure calculated with the use of the Maxwell stresstensor is used for the calculation of radial displacements. Thisformula uses only the values of the magnetic field at the innerand outer boundaries of the cylindrical domain, as they havebeen calculated with our model. Obviously, the theory of equiv-alent surface forces does not take correctly into account the mag-netic nature of the object (workpiece). As a result, the theory ofequivalent surface sources yields elastic deformations that arealso assumed to be incorrect. We finally remark that our compar-ison has been made within the linear approximation. For non-linear media, we anticipate that the method of the equivalentsurface sources will not be useful.

APPENDIX

SOLUTION OF THE ELASTODYNAMIC EQUATIONS

In order to arrive at the solution of (52), we write it as

(56)

where denotes the particular solution and de-notes the general solution of the homogeneous form of (52).

A. Particular Solution

We may construct the particular solution using theGreen’s function as follows:

(57)

where is the solution of equation

(58)

In an unbounded domain with a cylindrical source placed at, the solution of (58) is

for

for(59)

In our further analysis, we need also the radial derivative of theparticular solution. This is obtained as

(60)

where

(61)

A special case for the calculation of the elastodynamic field is. Then, (59) and (61) become

(62)

and

(63)

respectively. Until here, since all quantities in (57) and (60) areknown, we have calculated the particular solution andits radial derivative as functions of the radius . Thus,we also have the values of the particular solution and its radialderivative at the boundaries and of the cylindricaldomain representing the workpiece.

B. General Solution

The general solution of the homogeneous form of(52) is given by

(64)

where the coefficients and are obtained from the boundaryconditions

(65)

(66)

MOTOASCA et al.: ELECTROMAGNETIC FORMING BY DISTRIBUTED FORCES 3329

Similar to (56), we write as

(67)

where

(68)

and

(69)

with

(70)

(71)

From (65)–(66), we obtain the following system of equations inmatrix form:

(72)

where the elements of the known vector are given by

(73)

(74)

Thus, the coefficients and may be calculated as

(75)

(76)

When , the general solution in (64) may be written as

(77)

Similar with the procedure presented above, we get

(78)

The coefficients and are obtained from the application ofthe boundary conditions in (65)–(66). This procedure yields thefollowing system of equations:

(79)

(80)

where the functions and have been defined in(73)–(74). The coefficients and are

(81)

(82)

ACKNOWLEDGMENT

The authors would like to thank Prof. A.T. de Hoop for hisfundamental contributions to the theory of Section II.

REFERENCES

[1] P. Penfield and H. A. Haus, Electrodynamics of Moving Media. Cam-bridge, MA: MIT Press, 1967.

[2] C. J. Carpenter, “Surface-integral methods of calculating forces on mag-netized iron parts,” The Institution of Electrical Engineers, pp. 19–28,Monograph no. 342, Aug. 1959.

[3] , “Distribution of mechanical forces in magnetized material,” Proc.Inst. Elect. Eng., vol. 113, pp. 719–720, Apr. 1966.

[4] J. V. Byrne, “Ferrofluid hydrostatics according to classical and recenttheories of the stresses,” Proc. Inst. Elect. Eng., vol. 124, pp. 1089–1097,Nov. 1977.

[5] G. W. Carter, “Distribution of mechanical forces in magnetized mate-rial,” Proc. Inst. Elect. Eng., vol. 112, pp. 1771–1777, Sept. 1965.

[6] G. Reyne, J. C. Sabonnadiere, J. L. Coulomb, and P. Brisonneau, “Asurvey of the main aspects of magnetic forces and mechanical behaviorof ferromagnetic materials under magnetization,” IEEE Trans. Magn.,vol. MAG-23, pp. 3765–3767, Sept. 1987.

[7] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill,1941.

[8] F. C. Moon, Magneto-Solid Mechanics. New York: Wiley, 1984.[9] R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy

and Forces. New York: Wiley, 1960.[10] L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous

Media. Oxford, U.K.: Pergamon, 1960.[11] K. Baines, J. L. Duncan, and W. Johnson, “Electromagnetic metal

forming,” Proc. Inst. Mech. Eng., pt. 1, vol. 180, no. 4, pp. 93–104,1965–1966.

[12] S. Bobbio, Electrodynamics of Materials. Forces, Stresses, and Energiesin Solids and Fluids. San Diego, CA: Academic, 2000.

[13] G. K. Lal and M. J. Hillier, “The electrodynamics of electromagneticforming,” Int. J. Mech. Sci., vol. 10, pp. 491–498, 1968.

[14] C. Fluerasu, “Electromagnetic forming of a tubular conductor,” RevueRoumaine des Sciences Techniques, Serie Electrotechnique et Energe-tique, vol. 15, pp. 457–488, 1970.

[15] S. T. S. Al-Hassani, J. L. Duncan, and W. Johnson, “On the parametersof magnetic forming process,” J. Mech. Eng. Sci., vol. 16, pp. 1–9, 1974.

[16] S. H. Lee and D. N. Lee, “Estimation of the magnetic pressure in tubeexpansion by electromagnetic forming,” J. Mater. Process. Technol., vol.57, pp. 311–315, 1996.

[17] R. Winkler, Hochgeschwindigkeitsbearbeitung. Berlin, Germany:VEB Verlag Technik, 1973, pp. 294–367.

[18] H. Zhang, M. Murata, and H. Suzuki, “Effects of various working con-ditions on tube bulging by electromagnetic forming,” J. Mater. Process.Technol., vol. 48, pp. 113–121, 1995.

[19] S. H. Lee and D. N. Lee, “Finite element analysis of electromagneticforming for tube expansion,” Trans. ASME, J. Eng. Mater. Technol., vol.116, pp. 250–254, Apr. 1994.

[20] , “Estimation of the magnetic pressure in tube expansion by elec-tromagnetic forming,” J. Mater. Process. Technol., vol. 57, pp. 311–315,1996.

[21] M. Besbes, Z. Ren, and A. Razek, “A generalized finite element modelof magnetostriction phenomena,” IEEE Trans. Magn., vol. 37, pp.3324–3328, Sept. 2001.

T. Emilia Motoasca was born in Brasov, Romania, on November 3, 1971. Shereceived the B.S. and M.S. degrees in electrical engineering in 1996 and 1997,and the B.S. degree in economic sciences in 1997, all from the TransilvaniaUniversity of Brasov, and the Ph.D. degree in technical sciences from the DelftUniversity of Technology, Delft, The Netherlands, in 2003. Her doctoral re-search was focused on electrodynamics in deformable media for electromag-netic forming.

From 1997 to 1998, she was a Research Assistant at the Transilvania Uni-versity of Brasov and from 1999 to 2003 she was employed at the Laboratoryof Electromagnetic Research, Department of Electrical Engineering, Delft Uni-versity of Technology. Since November 2003, she has been a Postdoctoral Re-searcher at the Faculty of Electrical Engineering, Eindhoven University of Tech-nology, Eindhoven, The Netherlands. Her current research interests include an-alytical and numerical aspects of the theory of electromagnetic waves with ap-plications in medical devices.

3330 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004

Hans Blok was born in Rotterdam, the Netherlands, on April 14, 1935. He re-ceived the degree in electrical engineering from the Polytechnical School ofRotterdam in 1956, and the M.Sc. degree in electrical engineering (cum laude)and the Ph.D. degree in technical sciences (cum laude) from the Delft Univer-sity of Technology, Delft, The Netherlands, in 1963 and 1970, respectively.

Since 1968, he has been a Member of the Scientific Staff of the Laboratoryof Electromagnetic Research, Delft University of Technology. He has done re-search and lectured in the areas of signal processing, wave propagation, andscattering problems. In 1972 he was appointed Associate Professor (“reader”)at the Delft University of Technology, and in 1980 he became a Full Professor.From 1980 to 1982, he was Dean of the Faculty of Electrical Engineering.During the academic year 1983–1984, he was a Visiting Scientist at Schlum-berger-Doll Research, Ridgefield, CT, where he was involved in modeling ofelectromagnetic prospecting problems. Since then, he has been Visiting Scien-tist at Schlumberger-Doll Research on a regular basis. On and off, he was and isacting head of the Laboratory of Electromagnetic Research. He has been Emer-itus Professor since 2000. His current research interests are inverse scatteringproblems and near-field optics.

Martin D. Verweij was born in Alphen aan den Rijn, The Netherlands, in1961. He received the B.Sc. degree in electrical engineering from the Munic-ipal Polytechnical School, The Hague, The Netherlands, in 1983, and the M.Sc.degree in electrical engineering and the Ph.D. degree in technical sciences fromthe Delft University of Technology, Delft, The Netherlands, in 1988 and 1992,respectively.

From 1993 to 1997, he was a Research Fellow of the Royal NetherlandsAcademy of Arts and Sciences. For several months in 1995 and 1997, he wasa Visiting Scientist at Schlumberger Cambridge Research, Cambridge, U.K. In1998 he became an Assistant Professor, and in that same year he became anAssociate Professor, both in the Laboratory of Electromagnetic Research, DelftUniversity of Technology, where he is currently involved in the research and theteaching of the fundamentals of electromagnetic fields and waves. His presentresearch interests are the computational modeling of electromagnetic deforma-tion, nonlinear waves, and nano-optics. He has authored or coauthored morethan 30 papers in international journals and international conference proceed-ings.

Dr. Verweij is a national member of Commission B (Fields and Waves) of theURSI and a member of the Acoustical Society of America.

Peter M. van den Berg was born in Rotterdam, The Netherlands, on November11, 1943. He received the degree in electrical engineering from the PolytechnicalSchool of Rotterdam in 1964, and the M.Sc. degree in electrical engineering andthe Ph.D. degree in technical sciences from the Delft University of Technology,Delft, The Netherlands, in 1968, and 1971, respectively.

From 1967 to 1968, he was a Research Engineer at the Dutch Patent Office.Since 1968, he has been a member of the Scientific Staff of the Electromag-netic Research Group, Delft University of Technology. During the academicyear 1973–1974, he was Visiting Lecturer in the Department of Mathematics,University of Dundee, Scotland, financed by an award from the Niels StensenStichting, The Netherlands. During a three-month period in 1980–1981, he wasa Visiting Scientist at the Institute of Theoretical Physics, Goteborg, Sweden.He was appointed as a Full Professor at the Delft University of Technology in1981. During the years of 1988–1994, he also carried out research at the Centerof Mathematics of Waves, University of Delaware; these visits have been fi-nanced by a NATO award. During the summer periods of 1993–1995, he was aVisiting Scientist at Shell Research B.V., Rijswijk, The Netherlands. At present,his main research interest is the efficient computation of field problems usingiterative techniques based on error minimization, the computation of fields instrongly inhomogeneous media, and the use of wave phenomena in seismic dataprocessing. His major interest is in an efficient solution of the nonlinear inversescattering problem.