Three-dimensional finite element analysis of the magnetic tape slitting process

Journal of Materials Processing Technology 170 (2005) 71–88

Three-dimensional finite element analysis of themagnetic tape slitting process

Shashank Aggarwal, Bharat Bhushan∗, Noriko KatsubeNanotribology Laboratory for Information Storage and MEMS/NEMS, The Ohio State University,

650 Ackerman Road, Suite 255, Columbus, OH 43202, USA

Received 16 September 2004; received in revised form 4 March 2005; accepted 29 March 2005

Abstract

The quality of factory-slit tape edges plays an important role in determining the performance of tape drives and is a controlling factorfor data track density. A theoretical study of the slitting process and parameters involved could assist greatly in the optimization of theprocess and development of better quality slit edges. Various finite element studies exist in the area of sheet metal blanking and cutting buta three-dimensional study in the area of polymer slitting is lacking. In this study, a three-dimensional model of the slitting process of thepolymeric tape substrate is developed using the finite element method and the effect of various blade and web parameters on the quality of

study. The. A criterionted. Verticalffect on edgeelating edge

inge caned ine andmage

beencrack

tape

d by

r inhear

irs ofittingusedming

the edge is analyzed. A modification of a two-dimensional model developed earlier is used to compliment the three-dimensionalmaterial of a typical substrate has been accurately modeled and a failure criterion incorporated to model the material separationfor comparison of the quality of slit edge is proposed. Parametric study of the effect of blade and web parameters has been conducengagement, edge radius, blade speed, blade radius, longitudinal tension in the web and web thickness are varied and their equality is analyzed. An analytical model is developed to study the effect of blade and web parameters and a damage parameter rquality to these parameters is proposed.© 2005 Elsevier B.V. All rights reserved.

Keywords: Magnetic tape; Edge quality; Finite element analysis

1. Introduction

To increase storage capacity, future high-performance lin-ear tape systems will require the use of thinner magnetictapes, higher track densities, higher tape speeds and lowerhead tape spacing[1,2]. The need for higher track densi-ties necessitates the use of narrower data tracks that will beplaced closer to the edge. Quality of virgin tape edge is oneof the parameters, which determine the performance of a tapeguiding system[3]. During normal drive operation, the tapeedge-guide flange interaction results in damage to the tapeedges, leading to problems in tracking and generation of loosedebris. It has been established that the factory-slit tape edge isimperfect[4]. An imperfect tape edge contains tears, crackededges, or chunks of material removed or protruding from the

∗ Corresponding author. Tel.: +1 614 292 0651; fax: +1 614 292 0325.E-mail address: [email protected] (B. Bhushan).

edge (Fig. 1a). This would prevent data tracks from beplaced close to the edge of the tape. Tears in the edgalso lead to the removal of a chunk of material when usa drive. The debris may then enter head–tape interfaccause temporary or permanent data loss or physical dato the interface. If a crack in the material has alreadycreated, the amount of energy needed to propagate theis far less. Crack growth might easily occur when therubs against the tape guides[3–5].

Commercially available magnetic tapes are produceslitting a much wider web of tape into thin strips[1,2]. Thisslitting process is thus the single most important factodetermining the quality of the tape edges produced. Scut slitting is used to perform this operation.Fig. 1b showsthe process where the web of tape is fed between pacircular rotating blades separated by a tape width. This slprocess is essentially similar to the cutting action that iswith a pair of scissors; there are two sharp edges co

0924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2005.03.032

72 S. Aggarwal et al. / Journal of Materials Processing Technology 170 (2005) 71–88

Fig. 1. (a) Optical micrograph of a MP tape edge, (b) schematic showing tape slitting process, (c) blade geometries used in the industry and (d) schematicdiagram of a MP tape.

together at a converging angle. The speed at which the webenters the shearing interface is variable in the slitting process.Even more important than the web feed speed is the angle towhich the two rotating blades converge. When the scissorsare closed to just the right angle, they can cut through a pieceof paper by simply being pushed along; no closing motion ofthe scissors is needed. If the angle of the converging scissorblades is slightly off, the paper will tear; this applies to theprocess of slitting magnetic tape as well. Having the correctangle of convergence enables the feed speed of the web togenerally be slower, as fast, or faster than the linear speed ofthe rotating blades[4,6].

The slitting parameters, which may affect the quality ofthe slit edge, include the following: web feed speed, rotatingblade speeds, sharpness of rotating blades, vertical engage-ment between the two blades, uniformity of blade sharpness,angle of convergence of the pair of rotating blades, proper-ties of the web materials and axial tension of the web. Thus,improving the quality of the factory-slit tape edge wouldrequire a thorough analysis of the effect of these factors andtheir subsequent optimization.

Different kinds of blade geometries and blade assem-blies are used in the industry for slitting of magnetic tapes.The three most commonly used blade geometries, shown in

S. Aggarwal et al. / Journal of Materials Processing Technology 170 (2005) 71–88 73

Fig. 1c, are:

(1) Disk knife: the male knife is in the form of a cylindri-cal disk. The male knife can be side loaded in eitherdirection; the female knife is fixed and cannot be movedaxially. This design enables the upper male knife to cuton both sides. When the upper knife is side loaded thiscreates a negative cutting angle, however because of theminimal knife overlap and 0.500 mm knife thickness tapecan be slit with very little deformation of the media.

(2) Beveled knife: the male knife has a chamfer of around 60◦on the outside diameter. This knife configuration is usedwith a disk spring with each knife to provide constantforce when the knife is side loaded. Again, when thisupper knife is loaded it creates a negative cutting angle.

(3) Dished knife: the dished knife geometry and assembly areshown in the figure. This knife configuration is also usedwith a spring, unlike other configurations noted above theknife is dished 1–3.5◦, creating a positive cutting angle.

A typical commercial magnetic tape used for data record-ing consists of a base film with a thin magnetic particulatecoating as shown inFig. 1d. The base film is typically madeof poly(ethylene terephthalate) (PET) and the magnetic coat-ing is composed of acicular magnetic particles, polymericbinders, lubricants, a cross-linker or curing agent and sol-vents[2]. The base film, also called the substrate, constitutesm slit-t fileo ttingp sighti r, toa eredm bot-t h isb thatb

erics finitee ccessf lud-i thed intos ate-r eredo n val-u froma raintsT imateb itee ty ofp siredo

rm-i od[ ank-i d

slitting processes employ shearing as the mode of materialseparation. Slitting differs from the operations of blankingand punching as it involves the rotation of blades and hencethe material is cut in two directions simultaneously insteadof one. A thorough understanding of the process of slittingwould thus require that a three-dimensional analysis of theprocess be conducted. A three-dimensional analysis wouldallow a complete study of the process enabling the consider-ation of parameters such as blade engagement, blade radius,web tension, etc., which cannot be studied using a simpli-fied two-dimensional model. Limited work has been donein the area of three-dimensional modeling of the slitting pro-cess. A three-dimensional analysis of sheet metal slitting wasconducted by Wisselink and Huetink[11,12] using a self-developed finite element code. The failure of the materialduring shear slitting was modeled using a ductile fracture cri-terion. An assumed crack front was modeled into the initialmesh and a damage parameter was calculated to investigatethe correctness of the assumed crack front. However, a cou-pling between the crack front and damage parameter was notestablished.

The modeling of the slitting process of polymeric films dif-fers from that for metals. The response of polymeric materialsis considerably different from that of metals and it requires theuse of hydrostatic pressure dependent yield criteria. Analysisof the slitting process of polymeric films has been limited.B iono weba s wasf fect.M ticm ionals -d tapes eledu l sep-a n. At torsi y ofa peed,e ree-d etict thusn eredr ands aree beg pli-m

1 l foralyzelity.

2 ent

ore than 90% of the tape and hence its behavior duringing will have the single most prominent effect on the prof the edge generated from slitting the tape. A study of slirocess of the substrate could thus provide valuable in

nto the mechanics of slitting of magnetic tapes. Howeveccurately model the slitting process of tapes, a multi-layodel of the tape will be required including the top and

om coats. The modeling of the magnetic top coat, whicrittle and undergoes micro-cracking, would also requirerittle fracture be incorporated into the model.

The slitting process of the magnetic tape or its polymubstrate can be studied by simulating the process usinglement analysis. Finite element analysis has been su

ully used to model various manufacturing processes incng metal cutting and metal forming. The method involvesiscretization of a complex region defining a continuumimple geometric shapes called finite elements. The mial properties and the governing relationships are considver these elements and expressed in terms of unknowes at element nodes. A set of equations then resultsn assembly process considering the loading and consthe solution to these sets of equations gives the approxehavior of the continuum. All commercially available finlement packages use this basic principle with a varierocedures for solving the equations to generate the deutput[7].

Much work has been done in the area of metal fong and metal cutting utilizing the finite element meth8,9]. Of relevance are the simulations of the sheet blng and sheet punching operations[9,10]. Sheet blanking an

-

.

ollen and Denier[13] conducted a microscopic evaluatf the sheared edges of PET film cut under a variety ofnd blade parameters. A laminated structure of the edge

ound and fiber formation was identified as a major deeehan and Burns[14] studied slitting using photoelasicrographs and made observations of the two-dimens

tress distributions. Aggarwal et al.[15] conducted a twoimensional finite element analysis of the magneticlitting process. The polymeric tape substrate was modsing a pressure dependent yield function and materiaration was incorporated using the shear failure criterio

wo-dimensional model, while addressing a number of facnfluencing edge quality, does not allow a complete studll the parameters such as vertical engagement, blade stc., which can only be considered in a 3-D model. A thimensional analysis of the process of slitting of magn

apes or the polymer films, which constitute these iseeded. However, the 2-D analysis would not be rendedundant by the 3-D model. Rather, since the stresstrain distributions through the cross-section of the webxtremely difficult to plot using the 3-D analysis, they canenerated using a modification of the 2-D model to coment the 3-D.The objectives of the present study are:

. To develop a three-dimensional finite element modethe slitting process of magnetic tape substrate and anthe effect of blade and web parameters on edge qua

. Modify an existing two-dimensional model to complimthe 3-D study.


2. Finite element analysis

2.1. Three-dimensional model

A three-dimensional model for the slitting process is pro-posed. ABAQUS/Explicit software package is used for thefinite element formulation. ABAQUS/Explicit provides toolsto work with large deformation problems and the flexibilityto incorporate user-defined material models.

2.1.1. Geometry of the modelThe geometry of the model is shown inFig. 2a. The

web is modeled as a film of 8�m thickness, 90�m widthand 900�m length. Aggarwal et al.[15] conducted a two-dimensional analysis of the tape slitting process using equiv-alent plastic strain as the damage criterion. It was observedthat the damage values are limited to a thin region near theedge. Thus, with the assumption that damage will be lim-ited to a maximum distance of 50�m from the edge, the webwidth is chosen as 90�m. The blade width for the male bladeused in the industry is around 0.5 mm. However, since the webbeing considered has a width of 100�m, a blade width thatjust exceeds the web on either side is needed. Hence, the maleand the female blades on either side of the web are modeledas cylindrical discs of uniform width of 50�m and a radiusof 10 mm. The blades used in the industry have radii around5 lings 8t medt ffectt ouldn

2ed

f tivem rob-l akesi t ana nfig-u atios.A thet ech-n is (inw aly-s rialfl g iso LE)aw inearb con-t n oft tudyw cifiedm . Theb slit-

Fig. 2. (a) Three-dimensional and side views of the 3-D model of the slittingassembly showing the web and blade parameters and (b) schematic showingthe boundary conditions for the 3-D model.

ting process. Thus, the blades are modeled as rigid bodies tomake the analysis computationally efficient.

In ABAQUS, contact is defined between two surfacesby choosing one surface as the master and the other as theslave. The surfaces of the material with the higher elasticmodulus (the rigid blades) are chosen as the master and theother (web surfaces) as slave. Surfaces can be defined byeither specifying the elements, which constitute the surface(element-based surfaces), or the nodes which are includedin the surface (node-based surfaces). The element-based

0 mm. However, due to limitations posed by the modeoftware on the size of the blade as compared to the�mhick web, a blade radius of 10 mm was chosen. It is assuhat the choice of a smaller blade radius would only ahe absolute value of the results and the general trends wot be affected.

.1.2. MeshingThe finite element package ABAQUS/Explicit is us

or the analysis. ABAQUS/Explicit provides the adapeshing tool necessary for handling large deformation p

ems like the one being analyzed. Adaptive meshing mt possible to maintain a high-quality mesh throughounalysis, even when large deformations occur, by recoring the mesh to maintain the correct element aspect rdaptive meshing in ABAQUS/Explicit does not alter

opology (elements and connectivity) of the mesh. The tique combines the features of pure Lagrangian analyshich the mesh follows the material) and Eulerian anis (in which the mesh is fixed spatially and the mateows through the mesh). This type of adaptive meshinften referred to as Arbitrary Lagrangian–Eulerian (Analysis (ABAQUS/Explicit User’s Manual, 2002)[16]. Theeb is meshed using three-dimensional eight-node lrick elements with reduced integration and hourglass

rol. Adaptive meshing is applied over the entire regiohe web. The mesh density of the web used for this sas tested for mesh dependence of the results. The speesh is fine enough to rule out any mesh dependencelades undergo negligible or no deformation during the


surfaces are unable to maintain the correct contact proper-ties (such as no penetration) if any of the elements con-stituting the surface satisfies the specified failure criterionand fails. However, node-based surfaces allow elements nextto the surface to fail and still maintain the correct con-tact properties. Hence, the surfaces of the web are definedas node-based surfaces and those of the blades as elementbased.

2.1.3. Boundary conditionsFig. 2b shows the boundary conditions used for the model.

The current portion of web being analyzed is held symmet-rically between two sets of blades separated by a tape width.Hence, symmetry boundary conditions are applied to theends of the web, which are parallel to axis 1 restricting theirmovement in the direction of axis 3 (seeAppendix A). Thefeeding of the web through the rotating blades is simulatedby restricting the movement of the web in direction of axis1 and imparting an angular as well as linear velocity to theblades. The blades, which are mounted on shafts and hencecannot move axially, have their movement restricted in thedirection of axis 3.

2.2. Two-dimensional model

A two-dimensional model for the slitting process of thetfi sistso twob bi nsid-e pperh Theg andt bladei

thati uldn –A,F

del.T et-r tapew d tot n 1.T menti 2 ist

2

sen-s ialt suit-a rials

Fig. 3. (a) Two-dimensional model of the slitting assembly showing the weband blade parameters and mesh in the region with adaptive meshing and (b)schematic showing the boundary conditions for the 2-D model.

2.3.1. Material propertiesBhushan et al.[17] conducted uniaxial tension tests on T-

PET to estimate its tensile and dynamic properties. Tests wereconducted on 6.1�m thick sample at strain rate of 0.5 min−1.The engineering stress–strain curve from the study shown inFig. 4a and the material properties are listed inTable 1 [18].

2.3.2. Constitutive equationsFor metals, Von Mises (based on the critical value of

the second invariant of the stress tensor) and Tresca (basedon the critical value of maximum shear stress) criteria arecommonly used as the yield criteria. Polymeric materials

Table 1Material properties of T-PET at strain of 0.083 s−1

Modulus of elasticity (GPa) 6.30Poisson’s ratio 0.3Strain at yield (%) 3.10Yield stress (MPa) 150Breaking strength (MPa) 350Strain at break (%) 44

ape substrate was developed by Aggarwal et al.[15]. A modi-cation of that model is considered here. The model conf the web modeled as a rectangular section with thelades on either side of it (Fig. 3a). The geometry of the we

s defined by its thickness and the width. The blade is cored as a two-dimensional cross-section of the bottom/ualf of the actual three-dimensional cylindrical blade.eometry of the blade is defined by the blade thickness

he edge radius. To prevent stress singularities the sharps modeled with a 0.4�m edge radius.

Adaptive meshing is used for the region of the webs likely to undergo high deformation and hence woot be able to maintain a high-quality mesh (section Aig. 3a).

Fig. 3b shows the boundary conditions used for the mohe current portion of web being analyzed is held symmically between the two sets of blades separated by aidth. Hence, symmetry boundary conditions are applie

he ends of the web restricting the movement in directiohe blades are mounted on shafts and hence their move

n direction 1 is restricted. The movement in directionhe applied blade penetration.

.3. Material model

The substrate material is modeled using a pressureitive yield criterion. Material properties from a uniaxension test are used to define the material behavior. Able failure criterion is incorporated to model the mateeparation during slitting.


Fig. 4. (a) Engineering stress–strain curve for T-PET, (b) Coulomb–Mohr and Drucker–Prager yield surfaces in the principal stress space and (c) graphicalmethod for determination of hardening constants using experimental data from simple tension test.

differ from metals in their yield behavior. The yield stressunder uniaxial tension for a polymer is different from thatin uniaxial compression[19,20]. The yielding is hydrostaticpressure sensitive. Thus, the yield criterion for these materialsshould include the influence of the hydrostatic pressure. Thetwo classical yield criteria for hydrostatic pressure-dependentmaterials are the Coulomb–Mohr and Drucker–Prager cri-teria [21,22]. Fig. 4b shows the two criteria in the prin-cipal stress space and their loci on the deviatoric plane.The locus of the Coulomb–Mohr criterion is an irregularhexagon, whereas that of the Drucker–Prager criterion is acircle.

The Drucker–Prager criterion is easier to implement fromthe standpoint of numerical formulation because of its con-tinuously varying normal. Moreover, it can be extended insuch a way that with proper selection of different param-eters it closely approximates the Mohr–Coulomb criterion,if needed[16]. Chiang and Chai[23] used the extendedDrucker–Prager criterion to model the yield behavior of poly-meric adhesive layers. Lee and Ghosh[24] used a modifiedDrucker–Prager criterion to predict shear bands in variouspolymers and verified them with experimental data. Hence,the extended Drucker–Prager (EDP) criterion available inABAQUS/Explicit is used here to describe the response ofthe polymeric PET tape substrate.

The EDP yield function,F, is given as

F = q

2

[1 + 1

K−

(1 − 1

K

) (r

q

)3]

+1

3(σ1 + σ2 + σ3) tanβ −

(1

K+ 1

3tanβ

)σ0

t = 0

(1)

where

q ≡√

(3J2)

r3 ≡ 27

2J3

andσ1, σ2, σ3 are the principal stresses,J2 andJ3 the secondand third invariant of the deviatoric stress tensor, respectively,σ0

t the yield stress in uniaxial tension,β the friction angle ofthe material andK is the ratio of the yield stress in triaxialtension to the yield stress in triaxial compression are materialparameters. Defining

λ ≡ σ0c

σ0t,


whereσ0c is the yield stress in uniaxial compression test it has

been shown[23] thatK andβ related toλ through

K = λ+ 2

2λ+ 1(2)

tanβ = 3(λ− 1)

λ+ 2(3)

To completely describe the EDP model, in addition toσ0c

andλ, the parameterψ is needed, which is the angle betweenthe normal to the yield surface and the increment of plas-tic strain vector, dεp. Associated flow is achieved forψ =βand non-associated forψ �=β. Forψ = 0, the material is non-dilatational (no volume change in the plastic deformationregime).

The values ofψ andλ are not known for T-PET. However,experiments on various polymer systems have shownλ tovary between 1.2 and 1.5[19,25]. In this study, a value of 1.4was chosen. Experiments have shown that polymers sufferonly minor volume changes in the yielding or post-yieldingdeformation regimes[26,27]. Thus,ψ was set as zero in theEDP model for this study. Forλ= 1.4,K is calculated to be0.895 andβ to be19.44◦ from Eqs.(2) and(3).

The stress–strain behavior of T-PET is non-linear and itexhibits strain hardening. This strain hardening behavior wasincorporated using an isotropic hardening law. The rate ofct

R

wm umc ea traind froms thod( ET,pa

1 n-

le

2 ple

est

3 re

4. The parameterb is determined by fitting the resultsobtained (R and Q), using the least squares method, tofollowing relation, which is the integration of Eq.(4).

R = Q(1 − e−bεp) (7)

Parameterb was found to be 3.5. Now with the knowledgeof the parametersQ andb, the hardening rule representedby Eq.(4) can be implemented in the material model.

2.3.3. Failure criteriaThe material separation that causes the slitting of the tape

depends upon the damage caused to the material. The materialseparates when the damage has reached a critical value. Thecriterion, which defines the damage to the material and itscritical value, is called the failure criterion for that material.

In the case of shearing of metal sheets, various failurecriteria have been used. The Cockroft and Latham criterionis used for ductile fracture[29,30]. The criterion is based onthe void theory and assumes that fracture develops in a shearband when tensile strain energy reaches a critical value in theshearing process. The criterion addresses ductile fracture inmetals and is not suitable for polymers.

The shear failure criterion, used in sheet blanking opera-tions, is assumed to be relevant for polymer slitting and hasbeen used here[9]. The criterion is based on the value oft ts. Ita to them

ε

d

d

w ain.A

ω

w in,�

a entsi magep nowl-e lledt deda

r-p del.H lud-i eD tten

hange of the size of the yield surface,R, corresponding tohe isotropic hardening rule can be expressed as follows[28]:

˙ = b[Q− R]εeq (4)

here εeq is the equivalent plastic strain andb and Q areaterial constants.Q is the saturation stress or the maxim

hange in the size of the yield surface andb defines the ratt which the size of the yield surface changes as plastic sevelops. These material constants can be identifiedimple tension or compression tests using a graphical meFig. 4c). In this study, tensile stress–strain data for T-Presented inFig. 4a andTable 1, was used to determineQndb in the following manner:

. The plastic strainεp was determined from the experimetal data as,

εp = ε− εe = ε− σ

E(5)

whereε is the total strain from uniaxial test,εe =σ/E theelastic strain andE is the Young’s modulus from simptension test (Table 1).

. The isotropic hardening value is obtained from simtension as (Fig. 4c)

R = σ − σ0t (6)

whereσ0t is the initial yield stress from simple tension t

(Table 1).. Q is determined as an asymptotic value of the measuR

asεp increases, and is found to be 300 MPa (Fig. 4c).

he equivalent plastic strain at element integration poinddresses the failure due to the plastic damage causedaterial. Equivalent plastic strain,εp, is defined as

p =∫

dεp (8)

εp is given by

εp = 2√3

[(dεp1 − dεp

2)2 + (dεp

2 − dεp3)

2 + (dεp3 − dεp

1)2]1/2

(9)

hereεp1, ε

p2 andεp

3 are the components of the plastic strdamage parameter,ω, is defined as

= εp0 + ∑

�εp

εpf

(10)

hereεp0 is any initial value of the equivalent plastic stra

εp the increment of the equivalent plastic strain,εpf the strain

t failure and the summation is performed over all incremn the analysis. Failure is assumed to occur when the daarameter exceeds one. The model thus requires the kdge of only the equivalent plastic strain at failure, also ca

he critical equivalent plastic strain, which can be provis the input.

ABAQUS/Explicit does not provide the option of incoorating a failure criterion with the Drucker–Prager moowever, a failure criterion can be incorporated by inc

ng it in a user-subroutine (VUMAT) written for thrucker–Prager model. Hence, a VUMAT had to be wri


for Drucker–Prager yielding and a failure criterion was incor-porated by including it in the subroutine.

In previous works, crack propagation[9] and the elementdeletion method[31] have been used to achieve material sepa-ration. Both methods employ a failure criterion, which causesthe initiation and propagation of cracks or element deletionas the case may be. In this study, element deletion is used toremove elements from the mesh after they have failed. Whenthe shear failure criterion is met at an integration point, allstress components are set to zero and that material point fails.If all of the material points at any one section of an elementfail, the element is removed from the mesh.

3. Results and discussion

3.1. Generation of slit edges and evolution of slitting in3-D model

Fig. 5a shows the web in a partially slit position. As theweb goes through the rotating blades, the elements at the slit-ting interface reach the critical equivalent plastic strain valueand fail. These failed elements, which are unable to carry anyloads, assume arbitrary shapes and are ignored by the analysisprocedure. The failed elements are then removed manuallyusing the ABAQUS/CAE post-processor (the graphical userit gess lede inter-f

3

σ -p ichi occurb s thewc to bei res-s tiono s. Ina int es int nesso thes essesa the3

h thet train2t sesa s.

As is expected, bothσ22 andσ33 are highly compressive inthe region of the web directly under the blades as was inthe case of the 3-D study. The shear stress distribution showsextremely high values in the region where the web is expectedto slit and the values drop down as we move away from theregion. The high shear stress is caused by the shearing actionof the two blades on either side of the web.

The contours of equivalent plastic strain, which is used todefine the shear failure criterion used in this study, are shownin Fig. 7a and b for the 3-D and the 2-D studies, respectively.For both cases, the equivalent plastic strain is maximum atthe slitting interface where the web is expected to split anddiminishes with increasing distance from the edge.

3.3. Measure of edge quality

The quality of the edge produced from the slitting of theweb would depend upon the damage caused to the edge andthe extent of the damage around the edge. Since a criticalvalue of equivalent plastic strain is assumed as the failurecriterion in this study, its magnitude near the edge would bea measure of the damage. The higher the equivalent plasticstrain in a region, the more damaged it would be.

Fig. 8a shows the equivalent plastic strain contours forthe slit web. Observing closely the area near the edge (areaABCD) it can be seen that the maximum value of equivalentp easesw lowc stics erialf s. Am magev imumv r thed ualityo magea

1vingThele oftrainues

2 ell asrded.

3 ng theions

4 aver-rsus.

litieso theirs

nterface) to generate the slit edges of the web.Fig. 5b showshe evolution of the slitting process at four different statarting from the unslit web to the fully slit web. The failements have been removed using the graphical user

ace in each case.

.2. Stress contours

Fig. 6a shows the contours for the normal stressesσ11,22 andσ33 for the 3-D study.σ11 andσ22 show high comressive values in the region of the partially slit web, wh

s in contact with the blades. The compressive stressesecause of the blade surface pushing down on the web. Aeb slits, it moves tangentially to the blade (Fig. 2a). Thisauses the area of contact with the blade on the webnclined to the horizontal at an angle. The normal compive stress is thus split into two components in the direcf the axes 1 and 2 contributing to the two normal stresseddition, it is observed thatσ33 is also highly compressive

he same region of the partially slit web. The shear stresshe 3-D model are expected to be high through the thickf the web, in the region that is undergoing slitting byhearing action of the two blades. However, these strre not easily observable at the slitting interface using-D model.

The contours of the normal and shear stresses throughickness of the web can be obtained from the plane s-D study.Fig. 6b shows the normal stressesσ22 andσ33 and

he shear stressσ23 obtained from the 2-D study. The stresre plotted for a blade penetration of∼20% of web thicknes

lastic strain occurs at the edge and then gradually decrith increasing distance from the edge until it reaches aonstant value. The maximum value of equivalent platrain corresponds to the specified critical value for matailure and hence will be the same for all parametric studie

easure of edge quality would thus be based upon the daalues in the region around the edge rather than the maxalue of equivalent plastic strain at the edge. The loweamage in the region around the edge, the better the qf the edge. In order to calculate and compare the daround the edge, the following methodology was used:

. A path comprising several consecutive nodes (Fig. 8a) ischosen starting from the edge of the slit web and moin a direction perpendicular and away from the edge.length of the path is selected such that the last coupnodes fall in the region of constant equivalent plastic sso that all variations in equivalent plastic strain valwere captured.

. The equivalent plastic strain values at the nodes as wtheir distances from the edge are measured and reco

. The above steps are repeated for three locations aloedge; namely the center of the slit web and two locaton either side of the center at distances of 100�m from it.

. The values obtained from the three different paths areaged and plotted to get the ‘equivalent plastic strain vedistance from edge’ graph for a particular case study

These graphs are used for comparing the edge quabtained by using various blade and web parameters andubsequent optimization.


Fig. 5. (a) Partially slit web showing mesh state and failed elements and (b) mesh states at four consecutive stages during slitting.


Fig. 6. (a)σ11, σ22 andσ33 contours in a partially slit web for the 3-D study and (b)σ22, σ33 andσ23 contours for a blade penetration of∼20% of web thicknessfor the 2-D study.

Fig. 8b shows the equivalent plastic strain distribution forthe 2-D study and parameters used to compare edge quality inthe 2-D study. It is observed that the equivalent plastic strainforms a narrow band stretching between the two blade tips.

The damage zone in the middle of the tape would dictate theseparation of the bulk of the material. To obtain a good qualityedge during shear slitting, it would be required that the max-imum damage be limited to a thin and uniform band in this


Fig. 7. (a) Equivalent plastic strain contour in a partially slit web for the3-D study and (b) equivalent plastic strain contour for a blade penetration of∼20% of web thickness for the 2-D study.

zone. The thickness of this band would then be a measure ofthe extent of the damage to the tape away from the edge. Thesecond damage zone is near the blade surfaces in contact withthe tape. The area of this damage zone would be a measureof the damage caused to the top and back coat of the tape.

The quality of the tape edge will thus be dependent uponthe two parameters discussed above and defined inFig. 8b,namely

1. Thickness of damage zone in the middle of the tape.2. Area of damage zone around the blade surfaces in contact

with the tape.

The criterion for the goodness of the edge can then bestated as, “the maximum equivalent plastic strain should belimited to a thin region stretching through the tape cross-section and reach the critical value causing minimum damagearound the blade surfaces”. This is used as the criterion foroptimization of the slitting process and the parameters asso-ciated with it for the 2-D study.

3.4. Parametric study for 3-D model

The parameters whose effect was analyzed in this study arevertical engagement, edge radius, blade speed, longitudinaltension in the web and web thickness. The above parame-

Fig. 8. (a) Schematic showing equivalent plastic strain contours near the slitedge and criterion for comparison of edge quality for the 3-D study and (b)schematic showing parameters for comparison of edge quality for the 2-Dstudy.

ters are defined inFig. 2a, with respect to the slitting modelbeing considered. The test matrix for the study is given inTable 2.

3.4.1. Effect of vertical engagement in 3-D modelAs shown inFig. 1c vertical engagement for the blade

assemblies used in the industry vary from 0.1 mm to 0.25 mm.For this study, a nominal value of 0.1 mm is chosen.Fig. 9ashows the equivalent plastic strain graphs for the vari-ous blade engagements considered. An increase in bladeengagement from 0.05 mm to 0.2 mm leads to an increasein damage values the edge. The best quality of edge isobtained for an engagement of 0.05 mm. It can be inferredthat lowering the engagement leads to a better qualityedge. However, for a very low vertical engagement, the


Table 2Test matrix for parametric study using 3-D model

Model parameters Range of values

Vertical engagement(mm)

0.05 0.1a 0.15 0.2

Blade radius (mm) 5 10a 20Edge radius (�m) 0a 2 4 6Blade speed (m/s) 0.2a 4 6Longitudinal tension

per unit thickness(MPa)

1 10a 50 100

Web thickness (�m) 4 6 8a

a Nominal values.

web might not separate fully and hence an optimum valuewould have to be reached considering the above-mentionedfactors.

3.4.2. Effect of blade radius in 3-D modelAs shown inFig. 1c, the blade radii for the slitter blades are

around 50 mm. However, due to limitations posed by the soft-ware on the size of the blades as compared to the 8�m webthickness a nominal value of 10 mm is chosen for this studyand the radius is varied from 5 mm up to 20 mm.Fig. 9b showsthe equivalent plastic strain graphs for the various blade radiiconsidered. An increase in blade radius from 5 mm to 20 mmleads to a decrease in damage values near the edge. The bestquality of edge is obtained for a blade radius of 20 mm. It canbe inferred that increasing the radius leads to a better qualityof the slit edge.

3.4.3. Effect of blade radius and vertical engagementFrom the parametric studies of vertical engagement and

blade radius it is observed that the edge quality is directlyproportional to the blade radius and inversely proportional tothe vertical engagement. Thus, intuitively it can be proposedthat if the ratio of the vertical engagement to the blade radiusis kept constant, the edge quality should not be affected. Totest this assumption, three cases of different blade radii andv io oft es mainsc lader

3and

h oulddf f thes ger edge.T se of2 es to4 tinuei e

Fig. 9. (a) Effect of vertical engagement on edge quality for the 3-D study,(b) effect of blade radius on edge quality for the 3-D study and (c) combinedeffect of vertical engagement and blade radius on edge quality.

blades would need to be reground before they wear to an edgeradius of 4�m.

3.4.5. Effect of blade speed in 3-D modelStress–strain behavior of tape substrate is strain rate

dependent. T-PET shows stiffening with an increase in strainrate. In order to study the effect of blade speed on the edgequality, stress–strain data at a higher strain rate of 0.83 s−1

was assumed.Fig. 11a shows the stress–strain graphs for

ertical engagement were considered for which the rathe two was the same (0.01).Fig. 9c shows the results for thtudy. As can be observed, the damage near the edge reonstant for the same ratio of vertical engagement to badius.

.4.4. Effect of edge radius in 3-D modelAn ideal blade would have a perfectly sharp edge

ence no edge radius. However, with use the blades wevelop an edge radius due to wear. A nominal value of 0�m

or the edge radius is chosen for this study. The results otudy are shown inFig. 10. As expected, an increase in edadius leads to an increase in the damage values near thehe quality of the edge does not deteriorate for an increa�m from the nominal value. As the edge radius increas�m the damage values increase appreciably and con

ncreasing for edge radius of 6�m. It can be inferred that th


Fig. 10. Effect of edge radius on edge quality for the 3-D study.

T-PET at two strain rates of 0.083 s−1 and 0.83 s−1. For thisstudy, blade speeds of 2–6 m/s were considered. The resultsare shown inFig. 11b. As the blade speed increases the dam-age values near the edge increase. Thus, a lower blade speedwould produce a better quality edge because of the higherstrain hardening effect of the tape substrate material at higherstrain rates.

Fres

Fig. 12. (a) Effect of web thickness on edge quality for the 3-D study and(b) effect of vertical engagement on web thicknesses of 4�m and 8�m forthe 3-D study.

3.4.6. Effect of web thickness in 3-D modelTo study the effect of web thickness on the edge quality,

three tape thicknesses of 8�m, 6�m and 4�m are consideredbeing slit by blades of identical geometry. The results of thestudy are shown inFig. 12a.

As the web thickness is decreased from 8�m to 4�m thedamage values near the edge increase. The damage valuesare not significantly higher for the 6�m web. However, the4�m web shows appreciable damage when compared to theother two thicknesses. Thus, for the same blade assembly, the4�m web would have a poorer edge as compared to the 8�mweb.

A further study of blade parameters is conducted on the4�m web to understand the modifications needed to improvethe edge. It is known from the parametric studies conductedearlier that decreasing the vertical engagement between the

ig. 11. (a) Measured and assumed stress–strain curves for T-PET at strainates of 0.083 s−1 and 0.83 s−1, respectively, and (b) effect of blade speed ondge quality for 3-D study with T-PET material data defined at two differenttrain rates.

blades leads to an improvement in edge quality. The engage-ment was reduced to 0.05 mm from 0.1 mm for the 4�mweb. The results of this study are shown inFig. 12b. Thedamage values for the 4�m web with 0.05 mm engagementare significantly lower than those for 4�m blade and 0.1 mm


Fig. 13. Effect of longitudinal tension on edge quality for the 3-D study.

engagement. Therefore, if the tape thickness is reduced, theblade engagement would have to be reduced in order to obtaina similar quality of slit edge.

3.4.7. Effect of longitudinal tension in 3-D modelTo maintain tautness of the web of tape substrate to be

slit, it is wound on winders and re-winders prior to and afterbeing slit at the blade interface. The action of winding andrewinding the tape gives rise to a longitudinal tension in theweb. The value of longitudinal tension used in the industryis around 90 N/m, which corresponds to a stress of∼10 MPafor the cross-section of the web being considered for thisstudy (8�m by 90�m). This value of stress is specified asthe initialσ11 value throughout the web. For this study, stressvalues ranging from 1 MPa to 100 MPa are used. The resultsare shown inFig. 13. An increase or a decrease in the initialstress value from the nominal value of 10 MPa does not alterthe damage values near the edge. Hence, it can be observedthat a change in longitudinal tension in the web within thespecified range does not affect the quality of the edge basedon the criterion proposed in this study.

3.4.8. Effect of dish angle in dished knife in 3-D modelThe geometry of the dished knife is different from the disk

and the bevel knife geometries as shown inFig. 1c. The dishk ◦ ◦aco o thed es nop

TB

M

K

Fig. 14. Effect of dish angle for the dished knife geometry on edge qualityfor the 3-D study.

3.5. Parametric study for 2-D model

The parameters whose effect was analyzed in this 2-Dstudy are edge radius, web thickness and the dish angle fordished knife. The above parameters are defined inFig. 3a,with respect to the slitting model being considered. The testmatrix for the study is given inTable 4.

Fig. 15a shows the results for the study of the effect of edgeradius on edge quality. The equivalent plastic strain contoursare observed at a point prior to the initiation of material failureor separation at a blade penetration of∼20% of web thick-ness. It is observed from the figure, that a change in the edgeradius from the nominal value affects the equivalent plasticstrain contours. For a lower value of edge radius (0.2�m),the damage value in the middle of the tape is slightly higherwith the same damage area around the blade surfaces. Thus, alower value of edge radius would lead to a better quality edge.For the edge radius value of 0.6�m, the damage in the middleis lower and also the damage band is thicker than the one for0.4�m edge radius. Thus, it can be inferred that increasingthe edge radius leads to a degradation of edge quality.

To study the effect of tape thickness on the edge quality,three tape thicknesses of 8�m, 6�m and 4�m were consid-ered being slit by blades of identical geometry. The resultsof the study are shown inFig. 15b. The equivalent plasticstrain contours are plotted at a blade penetration of∼20% oft knessd ought f thed Thus,f orerq

TT

M

EW

nife has a taper with angles between 1and 3.5 to createpositive cutting angle. For this study, angles of 1–6◦ were

hosen (Table 3). The results are shown inFig. 14. The anglef the dished knife does not seem to make a difference tamage values near the edge and hence probably dolay an important part in determining edge quality.

able 3lade angle values for dished knife study

odel parameter Range of values

nife angle (◦) 1.534.56

t

ape thickness. It can be observed that as the web thicecreases, the value of the equivalent plastic strain thr

he thickness of the web reduces. Also, the thickness oamage zone increases with decreasing web thickness.

or a thinner web, same blade dimensions would give a pouality of edge.

able 4est matrix for parametric study using 2-D model

odel parameters Range of values

dge radius (�m) 0.2 0.4a 0.6eb thickness (�m) 4 6 8a

a Nominal values.


Fig. 15. (a) Effect of edge radius on edge quality for the 2-D study, (b) effect of web thickness on edge quality for the 2-D study and (c) effect of dish anglefor the dished knife geometry on edge quality for the 2-D study.

Fig. 15c shows the results for the study of the effect ofdish angle in a dished knife. The knife angles are varied from0◦ to 4.5◦. There is no observable change in the distributionof equivalent plastic strain for the three angles. Hence, it canbe inferred that dish angle does not affect the quality of theedge for the specified values.

3.6. Analytical study

The observations made from earlier analysis revealed thatthe damage near the slit edge remains constant for a fixedratio of the vertical engagement between the blades and theblade radius. It was also observed that the decreasing thethickness of the web lead to more damage near the slit edgewhich could be offset by lowering the vertical engagement. Tounderstand the relationship between these three parameters,namely vertical engagement, blade radius and web thickness,an analytical model is proposed.

Fig. 16a shows the cross-section of the blade assembly. IfR is the blade radius andh is the vertical engagement then itcan be seen that,

h = R− R cosθ = R(1 − cosθ)

or

cosθ = 1 − h

R

using Taylor’s expansion of cosθ and neglecting higher orderterms we get,

1 − θ2

2= 1 − h

R(sinceθ is very small)

or

θ =√

2h

R(11)

Fig. 16b shows the cross-section of the blade and webassembly. IfR is the blade radius andt is the web thicknessthen it can be seen that,

t = R+ R cosθ − 2R cosα = R(1 + cosθ − 2 cosα)

using Taylor’s expansion of cosα and cosθ and neglectinghigher order terms we get,

t

R= 1 + 1 − θ2

2− 2

(1 − α2

2

)


Fig. 16. (a) Illustration showing the calculation of angleθ for a specific bladeengagement, (b) illustration showing the calculation of angleα at the contactof top blade and web and (c) illustration showing calculation of parametera for a partially slit web.

Substituting forθ from Eq.(11)we get,

α =√t + h

R(12)

Fig. 16c shows the cross-section of the web in the processof slitting. Let us assume that the web is completely separatedat the point A. Ifa is the distance along the partially slit web

Fig. 17. Damage values for test cases with different web thickness and bladeengagements but the samea/θ.

then,

a = t

2 sinα, α ≈ 0

or

a = t

2α(13)

Table 5lists the various combinations of web thickness,vertical engagement and blade radii considered in this study.The corresponding values ofa and θ are calculated usingthe formulae derived above. The cases are listed in the orderof increasing edge damage around the slit edge or degradingedge slit edge quality as observed from the parametric studies.It can be observed that the damage around the slit edge isdirectly proportional to the ratio ofa andθ. That is, as theratio of a andθ increases, the damage around the slit edgedecreases or the edge quality improves. As an illustrativeexample,Fig. 17shows the damage values of two cases withdifferent web thicknesses and vertical engagements but thesamea/θ value of 22.09. As can be observed the damagevalues are very similar for the two cases.

Let us propose a damage parameter,k, such that higherthe value ofk, better the slit edge quality. Thus, the damageparameterk will be directly proportional to the ratio ofa andθ. This can be expressed mathematically as,

k

w

k

k

∝ a

θor k = c

a

θ

herec is an arbitrary constant.Now substituting from Eqs.(11)and(13)we have,

= ct

2αθ= c

t

2√

t+hR

√2hR

Thus,

= cRt

2√

2√h(t + h)


Table 5Values ofa andθ for different values oft, h andR

Web thickness,t (�m)

Vertical engagement,h (mm)

Blade radius,R (mm) a θ a/θ

8 0.10 20 6.28 0.10 62.858 0.20 20 6.24 0.14 44.178 0.05 10 4.46 0.10 44.588 0.10 10 4.44 0.14 31.434 0.05 10 3.14 0.10 31.434 0.10 10 3.12 0.14 22.098 0.20 10 4.42 0.20 22.09

4. Conclusions

A three-dimensional finite element model for the slittingof tape substrate has been developed. Material propertiesof the polymeric substrate have been accurately modeledusing the extended Drucker–Prager yield criterion and thestress–strain data available from literature. The shear failurecriterion has been incorporated into the material model bywriting a VUMAT user-subroutine.

A criterion for comparison of edge quality of the slit edgehas been developed making use of the equivalent plastic strainas the damage parameter and a parametric study of blade andweb parameters has been conducted. A modification of a two-dimensional model developed by Aggarwal et al.[15] hasbeen analyzed to compliment the three-dimensional analysisand both models show identical trends of edge quality for thecommon parameters considered.

An analytical model for the slitting of the tape substratehas been developed. A damage parameter relating the bladeand web parameters to the quality of the slit edge has beenproposed.

The following conclusions may be drawn from this study:

1. Of the blade parameters, vertical engagement, edge radiusand blade speed have a prominent effect on the quality ofthe slit edge. A decrease in vertical engagement or an

thefor

age-useddge.

s theofeedr thef the

2 ain aTheievee

3 of a

the slit edge quality.k is related to the blade radius,R, thevertical blade engagement,h, and the web thickness,t, inthe following manner:

k = cRt

2√

2√h(t + h)

For accurate modeling of the tape slitting process, a multi-layered model of the tape will have to be considered whichwould include the top and bottom coats. The modeling ofthe magnetic top coat, which is brittle and undergoes micro-cracking, would also require that brittle fracture be incor-porated into the model. Various other parameters also mightneed consideration such as the vibrations of the web, the gen-eration of thermal stresses at the slitting interface, etc.

In addition, experimental studies such as observing the slitedge quality for different values of the blade and web param-eters would be beneficial in order to verify and complimentthe theoretical analysis conducted in this study.

Acknowledgements

Financial support for this study was provided in parts bythe membership of the Nanotribology Laboratory for Infor-mation Storage and MEMS/NEMS (NLIM) and ImationC ger,T Thea tionC shidK elpw r.T ntf

Ac

tudyi width( na-l daryc e toa ndary

increase in blade radius leads to an improvement inedge quality. The quality of the edge remains sameidentical ratios of the blade radius and the vertical engment. An increase in edge radius, which may be caby wear of the blades, leads to a poorer quality slit eOf the values tested, edge radius of 2�m did not show adifference from the perfectly sharp blade. However, aedge radius is increased to 4�m and above, the qualitythe slit edge drops significantly. Increasing blade spalso leads to a poorer edge with higher damage neaedge. This is caused by the higher strain hardening opolymeric tape substrate at increased strain rates.

. Blade assemblies would need to be modified to obtgood edge quality when thinner webs of tape are slit.4�m web requires a lower vertical engagement to achthe same quality of edge as the 8�m web with the samset of blades.

. The quality of the slit edge can be expressed in termsdamage parameter,k, such that higher the value ofk, better

orp.-Advanced Technology Program (Program Manaed Schwarz, Peregrine Technology, St. Paul, MN).uthors thank Richard E. Jewett and Todd L. Ethen of Imaorp. for fruitful discussions throughout the study. Dr. Ra. Abu Al-Rub of Louisiana State University provided hith writing the VUMAT. The authors are also thankful to Mony Lewis of Industrial Tools, Inc., for providing importaeedback on the design and assembly of slitter knives.

ppendix A. Effect of using symmetry boundaryondition on result values

The portion of the web being analyzed in the present ss held between two pairs of blades separated by a tape12.7 mm). However, since the width of the web being ayzed is only 0.09 mm, the assumption of symmetry bounonditions is not fully correct. The other extreme would bssume that the edges, where currently symmetry bou


Fig. A.1. Effect of using two extreme boundary conditions on the slit edgequality.

condition is being applied, are stress free.Fig. A.1shows thecomparison between the two extreme cases. It can be seenthat changing the boundary condition from that of symmetryto a stress free condition does not alter the result values much.Hence, it can be safely assumed that data presented in thisstudy is independent of the symmetry boundary conditionbeing used.

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Three-dimensional finite element analysis of the magnetic tape slitting process

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