a simulation with finite elements to model steel...

UPTEC F 19023

Examensarbete 30 hp10 Juni 2019

A SIMULATION WITH FINITE ELEMENTS TO MODEL STEEL SHEET SLITTING A Master Thesis in Engineering Physics

Adam Ahlgren Peters

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

A SIMULATION WITH FINITE ELEMENTS TOMODEL STEEL SHEET SLITTING

Adam Ahlgren Peters

A steel slitting process is simulated using FEM (Finite Element Method) in order to seepotential defects along the edge in a steel sheet after it has been cut. The model'sresults were compared to microscope images of the steel sheet in order to verifyaccuracy. The purpose is conceptual and to find a model that successfully simulates asteel cutting process and (hopefully) how the edge depends on different parameters.The model developed seems to achieve this task, and a more thorough calibration ofthe model could result in (more) optimal parameters for the machine to use.

ISSN: 1401-5757, UPTEC F 19023Examinator: Tomas NybergÄmnesgranskare: Per IsakssonHandledare: Saed Mousavi

Populärvetenskaplig sammanfattning

Det ökande kravet p̊a allt mer högh̊allfasta st̊al (st̊al som t̊al attdeformeras mycket) har lett till att st̊alen utvecklas till att bli allt merh̊allbara och t̊aliga. Detta kan tära p̊a utrustningen vid klippning, men vadsom framförallt undersöks i denna rapport är huruvida en modell kanutvecklas för en specifik klipp-process (spaltning), där fokus ligger p̊a hurklippkanten ser ut när st̊alet klipps. Kantens skada p̊averkar st̊aletsh̊allfasthet mycket. För att inse detta kan man jämföra ett papper med ettlitet hack i kanten med ett papper som har hel kant; papperet med skadankommer rivas isär mycket enklare när man börjar dra i papperet längs medkanten. Det är därför av intresse att veta mer om kantskador.De simulerade resultaten har jämförts med mikroskopbilder p̊a klippkantenav st̊alet hur det ser ut när det klippts. En komplett kalibrerad modellskulle kunna användas för att optimera parametrar för klippningen.Förhopnningsvis skulle d̊a klippskadan minimeras, vilket skulle innebäraökad h̊allfasthet för st̊alet eftersom sprickor lättare propagerar därifr̊an detredan finns sm̊a sprickor eller ojämnheter i materialet.

Acknowledgements

I would like to thank my official supervisor Dr Saed Mousavi for all thehelp with material-related topics, my unofficial supervisor Dr LarsOlovsson (IMPETUS’s CEO) for all the help with simulation-relatedquestions. Also a thanks to Prof. Per Isaksson for the valuable feedback onmy report.

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Table of contents

1 Introduction 51.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 62.1 Parameter convention and definition . . . . . . . . . . . . . . 62.2 Slitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Stress-strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Damage model . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Residual stresses . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . 12

2.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 132.7.2 Finite difference scheme . . . . . . . . . . . . . . . . . 15

2.8 FEM software . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8.1 Time stepping . . . . . . . . . . . . . . . . . . . . . . . 162.8.2 Mass scaling . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Method 183.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Measurements and parameters . . . . . . . . . . . . . . . . . . 18

3.2.1 Stress-strain . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Slitting machine . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Residual stress . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Material model adaptation . . . . . . . . . . . . . . . . . . . . 203.3.1 Parameter calibration . . . . . . . . . . . . . . . . . . . 203.3.2 Choice of damage model . . . . . . . . . . . . . . . . . 20

3.4 Software setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.1 Element type . . . . . . . . . . . . . . . . . . . . . . . 213.4.2 Element order . . . . . . . . . . . . . . . . . . . . . . . 213.4.3 Element density . . . . . . . . . . . . . . . . . . . . . . 213.4.4 Time stepping and mass scaling . . . . . . . . . . . . . 233.4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . 233.4.6 Initial conditions . . . . . . . . . . . . . . . . . . . . . 23

4 Results 244.1 Stress-Strain curve . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Scanning Electron Microscope . . . . . . . . . . . . . . . . . . 24

4.2.1 Rollover zone . . . . . . . . . . . . . . . . . . . . . . . 25

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4.2.2 Shear zone (bright zone) . . . . . . . . . . . . . . . . . 254.2.3 Secondary shear zone (fracture zone) . . . . . . . . . . 264.2.4 Burr zone . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.5 Plastic deformation zone . . . . . . . . . . . . . . . . . 27

4.3 Two-dimensional model . . . . . . . . . . . . . . . . . . . . . 294.3.1 Energy balance (2D) . . . . . . . . . . . . . . . . . . . 30

4.4 Three-dimensional model . . . . . . . . . . . . . . . . . . . . . 314.4.1 Energy balance (3D) . . . . . . . . . . . . . . . . . . . 324.4.2 Different sharpness . . . . . . . . . . . . . . . . . . . . 334.4.3 Different gap . . . . . . . . . . . . . . . . . . . . . . . 344.4.4 Different damage tolerance . . . . . . . . . . . . . . . . 35

5 Discussion 365.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3 Stress-Strain curve . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Scanning Electron Microscope . . . . . . . . . . . . . . . . . . 375.5 Two-dimensional model . . . . . . . . . . . . . . . . . . . . . 385.6 Three-dimensional model . . . . . . . . . . . . . . . . . . . . . 385.7 Future improvements . . . . . . . . . . . . . . . . . . . . . . . 395.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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1 Introduction

Metallurgy has until recently depended solely upon trial and error. Therecent development in computer science -and computational powerspecifically- has led to a new complement to the more traditional approach.By successfully implementing a Finite Element Model (FEM) [1], one cannarrow down the necessary experiments needed for development and hencespeed up the development process. Ultimately, the strength along withother material properties of the steel alloy is what truly matters. If thesecan be precisely estimated numerically, the trial and error process wouldbecome far more effective.

1.1 Background

Due to the increased demand in high-strength steels (mainly in theautomotive industry), the need to precisely simulate the manufacturingprocess has increased as well. Steel plates are among the most commontypes of steel elements produced in the world. During the last two decadesthe yield strength of the hardest steel grades has had an extraordinaryimprovement. Materials with yield strength of 1.5 GPa and higher, are nowcommonly used, especially in automotive industry. As a result, theequipment used to cut such steel has seen a significant increase in wear.

1.2 Purpose

In short, the purpose is to construct a model based on observation of anindustrial process that involves steel slitting. The simulations are to becompared to images of the steel’s edge in order to verify accuracy. Thegoal is to give a hint of the optimal parameters for the machine involved(e.g slitting gap and slitting overlap).The purpose of this thesis is to lay a foundation for further materialstudies and process improvements. The study should be seen asconceptual; the method matters more than the precision of the calibrationthat depends upon experimental data. Ultimately, a well-developedmethod would improve the pace at which improvements can be made, sinceit narrows down the experiments considered necessary.As a first initial step towards minimising the damage on the producedsteel, the goal is to simulate the slitting process as precisely as possible.Once such a model exists, it should provide an easier way to minimise thewaste (and wear) through simulation experiments, in union with real-lifeexperimental controls.

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2 Theory

Since this project has roots in both the material and the computationalsciences, the relevant theory will be presented for both of these.

2.1 Parameter convention and definition

For simplicity the variable, abbreviation and parameter names are definedas in Table 2.1 unless specified otherwise. SI-units are used. Text in boldsymbolises a matrix of any size. Some variables may be reused from sectionto section and might not refer to the same value.

Symbol Meaning Unit∆ Difference Operational� ∆l/L Numericρ Density kg/m3

ν Poisson’s ratio Numericσ Stress Paω Ang. frequency rad/sa Acceleration m/s2

c Wave velocity m/sE Young’s modulus PaF Force NL Length mr Radius mm Mass kgt Time sv Velocity m/s

2.2 Slitting

Slitting process (knowledge) is of high importance, as it is an essential partof the manufacturing process. Since customers’ need are in focus anddifferent costumers have different needs, the material that is sold mighthave to be of different sizes. The slitting machine ensures the width iscorrect and according to the customers’ specifications. The slitting processis shown in Figure 1 schematically and in Figure 2 in reality, where one cansee how a steel sheet is split into several slimmer sheets.

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Figure 1: Slitting process shown schematically. The metal sheet (or mate-rial to be slitted) is the grey, horizontal line and exits out of the plane ifone considers this paper to define a 2D plane. The red, vertical lines repre-sent the circular cutters.

Figure 2: Slitting process. The material is slitted into slimmer chunks bythe circular cutters.

The process is important to understand, since the process in fact inflictssome damage to the edges of the material. Edge damage is kept at aminimum (for obvious reasons), but still has a high impact on thedurability of the material. For instance, the burr as seen in Figure 3 shouldbe kept inward if the material is to be folded. Otherwise the small burrmay fracture and severely impact the durability of the material, since alarger fracture then more easily may propagate deeper into the material.

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Figure 3: Characteristic edge of a slitted material.

2.3 Stress-strain

One of the most, if not the most essential parts of material characterisationis the stress-strain test. This test shows the tensile strength, yield strength,elasticity, plasticity, and failure for the material tested. A typicalstress-strain curve and setup can be seen in Figure 4.

Figure 4: Typical engineering stress (green) plotted together with the truestress (blue). As can be seen, they are identical up until the necking occurs.At 1 we have the yield strength and at 2 the maximum tensile strength. Redcross symbolises fracture. E is the slope of the elastic zone. Figure based onstress-strain figures in [2] and should be seen as purely schematic.

The material is attached in opposing ends. A linearly increasing force isapplied to drag the material apart. Initially, this will result in an elasticresponse of the material (much like a spring) and will follow Hooke’s Law[3]. The maximum value when the material is still elastic is called yield

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strength. Typically, high performance steel (such as SSAB’s Docol series[4]) have a very high yield strength. Needless to say, this property is verydesirable in products that needs to withstand high degrees of deformationwithout failure or plastic deformation. At some point however, the materialwill no longer be elastic but will instead become plastic. This means thematerial will start to (non-reversibly) deform as more force is applied, untilit ruptures. An ideally plastic material will have a plastic part of thestress-strain curve that is parallel to the x-axis. Steel however, continues toincrease in stress until it reaches a certain maximum (ultimate tensilestrength). At failure the material ruptures.For calibration of damage models, true stress-strain curve has to be used.The most common stress-strain curve however is the engineeringstress-strain curve [2] [3].

2.4 Material model

Over the years different material models have been developed with onepurpose; describe the material(s) properties mathematically. In this projectwe consider a model with parameters E, ρ, ν, ξ, tresca, and �0. Parametersnot considered are parameters that refers to temperature dependencies,such as thermal softening, thermal volymetric expansion, and adiabaticheat development. The material model used is a linear combination ofHooke’s law and Ludwik’s empirical formula [5]. It can be expressed as

σ(�) =

{E�, if � ≤ �pσ(�p) + C0(�− �p)C1 , if � > �p

(1)

E, �p, C0, C1 ∈ IR and constant(s).

where �p is the value of � where the elastic zone ends. �p and C0 arecalculated based on empirical data, C1 is an arbitrary constant that ischosen by trial and error, given the empirical data. Other parameters andvariables as defined in section 2.1.

2.5 Damage model

In this project it was decided to work with a so called uncoupled damagemodel. Damage growth is driven by plastic straining of the material(damage grows faster in tension than in compression). There is no gradualmaterial softening/degradation due to a increasing damage. However, asdamage reaches 100% (D=1) the material is suddenly failing and the

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element housing the failed integration point will be removed (eroded) fromthe model.The Cockcroft- Latham model was used as damage model because of itshigh suitability for ductile solids [6]. The model can be expressed1 as

D =1

Wc

∫ �pff0

max(0, σ1)d�pff +

1

ts

∫ t0

(σ̄1σs

)αsdt (2)

where D is the damage level of the element, σ1 is maximum principalstress, and in this case σ1 = σ̄1, �

pff the effective plastic strain, d�

pff the

effective plastic strain increment, αs an exponent controlling the time forspall fracture, ts is time to develop spall fracture at threshold stress, and σsis the spall strength. The element is eroded (deleted) when D = 1. Themost important parameter to calibrate here is the normalisation constantWc, since this determines how much the element may deform before it iseroded.

2.6 Residual stresses

Residual stresses2 are stresses that remain in a solid material. Such stressesare created in almost every manufacturing process; from casting to welding.They can occur without any external loads and have high impact on thestrength of the material. Overlooked and neglected these stresses maycause major failures in constructions (for instance, the famous crack in theLiberty Bell as seen in Figure 5 is the result of neglected and escalatedresidual stresses). Accounted for, these stresses may be used to increase thedurability of the material, which makes the subject highly relevant tostudy for the manufacturers.

1This notation might upset mathematicians since it looks like we integrate from zeroto the same variable as the function depends on (ex

∫ t0f(t)dt), however it is used for

convenience. The more correct way to express it is to change variables (so that∫ t0f(t)dt

becomes∫ t0f(s)ds).

2This section is a short resume of the book [7].

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Figure 5: As a famous example of residual stresses’ potential harm, thecrack in the Liberty Bell occurred as a result of residual stresses.

The stresses are self-equilibrating (hence the name residual). This basicallymeans that local areas of tensile and compressive stresses sum to create aforce and moment component that sums to zero, much like a conservationalmechanical system3. This leads to that the tensile stresses in the centralregion of the material is cancelled/countered by equal (after summing)compressive stresses at the surfaces. If the stresses are large enough, thematerial will deform elastically in order to preserve any plasticdeformation’s effect on the residual stresses. During the lifetime of thematerial external loads such as bending, temperature changes,transformation stresses, surface corrosion, and intergranular stresses allplay a role in how the residual stress might change over time and mightcause a failure after some time instead of directly after manufacturing,albeit some residual stresses are so large initially that the cause thematerial to fail almost immediately.All force components must always sum to zero when no external loads areapplied. Because of this, the stresses might not be very apparent and henceoverlooked or neglected. In material design, these stresses may be used toenhance the performance of the material. In brittle materials for example

3A classic example of this is planets that orbit stars thanks to the conservation ofenergy and angular momentum; forces sum to zero.

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(glass), a residual stress that compresses the surface at the expense oftensile stresses inside it will increase the performance since brittle materialsusually can withstand high compressive forces and are sensitive to cracks.By hardening (compression hardens the material) the brittle material’ssurface, the resistance to any cracks will increase at the surface. Apractical example of this is a car’s door glass. The glass is tempered tosuch a high degree that it will withstand a high degree of external violencewithout any cracks to be seen. However, once a crack do form, the entireglass panel will shatter. In a car this i very desirable, since it basicallyeliminates the risk of fatal bleeding injuries as a result of cutting on theshattered glass, since all the shattered pieces are so small.The residual stress components can be expressed on matrix form,

S =

σxx σxy σxzσyx σyy σyzσzx σzy σzz

(3)where each component is some function of space and time that symbolisesthe distribution in that corresponding direction. Since the stresses areequal and opposite, we have in (3) that ST = S, i.e S is symmetric(σij = σji, where i, j = {x, y, z}).

2.7 Finite Element Method

As a foundation of the Finite Element Software (see section 2.8) lies thefinite element method (FEM) [1]. Basically, this method uses the fact thatany geometry can be approximated with a linear combination of a finitenumber of elements that are composed of so-called basis functions, as isshown in Figure 6.

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Figure 6: The blue line is the exact solution to some diffusion-problem on[0,1] in 1D. The red line consists of two linear elements (three nodes) andapproximates the solution. The green line consists of three elements, andthe yellow line four. As can be seen, more elements is better when it comesto accuracy. Note that the elements have for the sake of simplicity beenchosen to have the same interval lengths.

2.7.1 Derivation

A minor derivation of the FEM the software uses is necessary in order tounderstand the advantages and drawbacks of different element types.Given that one wants to approximate the function u(x) with the linearcombination of some set of basis functions

u(x) =n∑i=1

ciφi, u ∈ [0, 1] (4)

where i is number of grid points located at xi and ci some constant definedfor node i. Transforming this into a local coordinate system so that theelements are defined for x = [0, 1] yields

ξ =x− xixi+1 − xi

, (5)

hence u is locally approximated by ξ, i.e for the 1D linear case

u(ξ) = c1 + c2ξ, ξ1,2 = [0, 1] (6)

which in turn leads to

c1 = u1c2 = −u1 + u2 (7)

that can be expressed on matrix form as

c = Au, where in this case (8)

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A =

(1 0−1 1

).

The approximate function with node values then becomes

u(ξ) = u1 + (−u1 + u2)ξ = u1(1− ξ) + u2ξ = u1N1(ξ) +N2(ξ)ξ (9)

where N1, N2 are linear basis functions for 1D elements. This derivationcan be performed for higher order elements as well, in higher dimensions.For second order 1D one makes the quadratic ansatz

u(ξ) = c1 + c2ξ + c3ξ2, ξ1,2,3 = [0, 1] (10)

and a cubic ansatz for third order elements. For 2D and 3D the derivationbecomes a bit more tricky, but the same method still applies (coordinatetransformation, followed by an ansatz, which leads to some basis functionsN1, N2, ... ,Ni. Some basis functions for the 2D second order is shown inFigure 8, and basis functions for the 1D first, second, and third order isshown in Figure 7.

Figure 7: The first, second, and third order basis functions for 1D.

Figure 8: Two of the quadratic basis functions for rectangular elements. N1and N2 are derived in section 2.7.1.

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2.7.2 Finite difference scheme

In order to solve time-dependent problems, the equations need to be solvedfor each timestep. This is derived in [8]. IMPETUS Afea’s Solver uses asecond order explicit scheme. For example, given the differential equation(harmonic oscillator) in [8]

∂2x

∂2t+ ω2x = 0 (11)

with ω ∈ IR, x0 = 1 and ∂x0∂t = 0, the integration scheme can be written onmatrix form as

xn+1 = Axn (12)

where

xn+1 =

(xn+1∂xn+1∂t

),xn =

(xn∂xn∂t

), and A =

(1− 1

2∆t2ω2 ∆t

−∆tω2 + 12γ∆t3ω4 1− γ∆t2ω2

).

The integration scheme (11) is stable if the eigenvalues of A λ1, λ2 satisfy

|λ1|, |λ2| ≤ 1. (13)

Solving (11) with (13) as constraints for ∆t, one gets that the criticaltimestep ∆tcr is

∆tcr =1

ω

√2

γ(14)

If the time step would be larger than this value, the solution would start tooscillate more and more until it diverges completely. This may lead to longsimulation times, if the grid is very fine (i.e a lot of elements requiressmaller time step).

2.8 FEM software

The FEM software used in this project is IMPETUS Afea’s solver. TheIMPETUS Afea Solver is a non-linear explicit finite element solver,specialised to model material deformations under load. The the solver usesa J2 (von Mises) yield criterion and associated plastic flow rule [9] [10], andhas slightly modified (optimised4) quadratic and cubic functions than whatis presented in Section 2.7.1. It is designed for a high level of robustness(double precision), capable of simulating advanced material distortion,

4As the exact source code is not available to the public, these will and can not bepresented. The derivation is there to create a basic understanding of the project.

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hydrodynamics, welding, explosives and collisions [9]. IMPETUS AfeaSolver works with Green-Naghdi stress rate, where material rotations aredefined through R in the relation (see also [8] [9])

F = RU (15)

where F is the deformation gradient, U is a symmetric stretch tensor(describes deformation), and R is a rotation tensor (describes rigid bodyrotation). For contact between objects, a penalty based contact algorithmand a Coulomb model of friction is used. A penalty based contactalgorithm allows for small geometric errors (penetrations between thebodies in contact). The applied contact pressure5 is a direct and linearfunction of the penetration error.The solver also includes an advanced post-processing visualisation tool andrelies entirely on Lagrangian equations of motion. The IMPETUS AFEASolver is GPU accelerated and is reliant on NVIDIA’s CUDA architecture.Thanks to most parts of the problem being parallelisable, the benefit of theGPU’s many cores is huge and speeds up calculations a whole lot. Thesolver was developed to simulate (amongst other things) large mechanicaldeformations and suits this project very well [9].

2.8.1 Time stepping

As stated in section 2.7.2, the time stepping scheme is of second orderaccuracy. However, since it is an explicit scheme, this is only true if thetime step size is less than a certain threshold. The time step size in anexplicit model is dependent on the size of the element and the wave speed.Using too large a time step means the speed of sound is large enough topropagate through an entire element in a single time step, which will resultin a unstable solution since this produces shockwaves.The critical time step is limited by the maximum eigenfrequency of thesystem6. Given that γ = 2 in IMPETUS’s software, (14) becomes

∆tcr =2

ωmax. (16)

5The contact stiffness in this work was set to 5.0 · 1015 Pa/m. That is, a penetrationerror of 1 µm will generate a contact pressure of 5.0 · 1015 Pa/m · 1.0 · 10−6 m = 5 · 109Pa = 5 GPa.

6This in turn can be derived from the central difference scheme, https://ftp.lstc.com/anonymous/outgoing/jday/olovsson2005_selective_mass_scaling.pdf in turnreferencing ”The Finite Element Method: Linear Static and Dynamic Finite ElementAnalysis”.

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https://ftp.lstc.com/anonymous/outgoing/jday/olovsson2005_selective_mass_scaling.pdfhttps://ftp.lstc.com/anonymous/outgoing/jday/olovsson2005_selective_mass_scaling.pdf

However, an exact calculation of ωmax is expensive. The Impetus AfeaSolver instead estimates the critical time step ∆tc as

∆tcr = min(∆te,∆tcont,∆tblast,∆tSPH) (17)

where ∆te is the critical element time step, ∆tcont is critical contact timestep, ∆tblast is the critical particle blast time step size, ∆tSPH is the criticalSPH (Smoothed-particle hydrodynamics) time step size.For a linear model ∆tcr is approximately equal to the Courant equation

C = ∆tn∑i=1

uxi∆xi

≤ Cmax, (18)

where uxi is the magnitude of velocity in dimension xi, ∆t is the time stepsize, and ∆xi is the step size in dimension xi. This will yield a roughestimate for more complicated elements where it is unclear how theelement length should be measured. If one lets c be the dilatational wavespeed through the material, Lc be the smallest element size, then one getsthat

∆t� ≈Lcc

(19)

c =

√E

ρ(20)

where E is Young’s modulus and ρ is the mass density. This implies that amaterial with a higher density and equal modulus E will have a lower wavespeed, leading to a larger stability limit [11] [12].

∆tstable = Lc

√ρ

E(21)

2.8.2 Mass scaling

In order to avoid simulation times that are excruciatingly long, thesoftware exploits the fact that the critical time step is related to the speedof sound (c) in the material [9] [8]. Equation (21) basically states that ifone adds artificial mass to the system, increasing the density ρ whilstleaving E unchanged, ∆tstable will increase. This effectively reduces c andhence increases the critical timestep size. The approximation leads toshorter simulation times, however this may come at the cost of increasedenergy in the system that derives from the added mass. If the added massis small enough this is negligible, however if the energies from the addedmass becomes large some results might not be very accurate.

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3 Method

Overall, the method presented should be seen as conceptual. This meansthat the precise choice of material and material parameters should matterless and the choice of method and proposed experiments matter more, asthe method should be applicable to some arbitrary, similar process.First, the process was observed. The process was then implemented inIMPETUS Afea’s IDE, along with all measurements. After the simulationwas complete, the results were objectively compared with material samplesof the chosen material. The edge of the slitted material (both from thesimulation and the samples) was also compared to well-establishedcharacterisation images to further measure the accuracy of the model.

3.1 Material

High strength steel for the sheet was SSAB’s Docol 930S [4]. The materialwas observed in the slitting machine, and test pieces were obtained formicroscopic study.

3.2 Measurements and parameters

3.2.1 Stress-strain

Stress-strain data was achieved by a classic stress-strain test of thematerial. However the cross-section area was not measured and hence theachieved curves could not be transformed into true stress-strain curves.Instead, the material model (1) was fitted to the part of the plastic zonebefore any real difference had occurred between the engineeringstress-strain and the true stress-strain. That is, the curve was optimised tofit experimental data until onset of diffuse necking.

3.2.2 Slitting machine

The slitting machine considered is depicted in Figure 9, where it is drawnschematically with each part scaled down by an equal amount. In Figure10 we see a zoomed-out version of the entire slitting process.

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Figure 9: The slitting machine at SSAB shown schematically from the side.

Figure 10: The slitting process at SSAB shown schematically from the side.

3.2.3 Residual stress

Residual stresses were applied arbitrary to a model of a large sheet to seewhat effect a certain distribution had on the material. The functionsapplied and the size of the residual stresses where chosen by studying [2]and [7].

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3.3 Material model adaptation

Since this is a conceptual study parameter calibration is not of the essence.However, some calibrations where necessary to perform in order to be ableto be verify the accuracy of the model.

3.3.1 Parameter calibration

By obtaining stress-strain curves for the steel, parameters C0, C1, and �pcould be calculated7 in MATLAB [13] by first transforming the non-linearequation (1) into a linear regression problem on matrix form

Ax = b (22)

where A =

�C10 1

�C11 1

�C12 1...

...

�C1N−1 1

, x =(

C0σ(�p)

), and b =

b0b1b2...

bN−1

.Solving for x (minimisation problem for each guess of C1), one arrives atan optimum rather quickly, given that the size of b is not too large8.

3.3.2 Choice of damage model

The Cockcroft- Latham (CL) damage model was chosen for this study. TheCL damage model is not very precise, however it is generally the go-tooption when there is limited amounts of material data. CL damage modelis appropriate for ductile materials and suits the needs for this project well[9].

3.4 Software setup

IMPETUS uses a command structure very similar to LS Dyna [14] to setupthe problem. The source code for this may be found in the Appendix. Theparameters used are stated in Table 1 unless otherwise specified. Thematerial model used is the function resulting from the calibration. SeeSection 2.4 and Figure 13 for these.

7See appendix for source code.8In practise this typically means that N < 104, although this is more a rule of

thumb and may vary a lot.

20

Parameter Value UnitE 210 · 109 Pagap 5 · 10−5 mheight 5 · 10−4 moverlap 3 · 10−3 mwidth 2 · 10−2 mR 0.1 mr 0.2 · 10−3 mv 3.33 m/sρ 7800 kg/m3

ν 0.3 NumericWc 1 · 109 Pa

∆tcr 4 · 10−8 s

Table 1: Gap corresponds to the slit space between cutters (x-direction),height the sheet thickness (y-direction), overlap the maximum overlap ofthe cutters in 3D (y-direction), width is cutters’ contact side width, R iscutter radius, r is radius of the edge that cuts the sheet, v is the velocity atwhich the sheet enters the cutters. Other variables defined as before. Allparameters based on real-life measurements.

3.4.1 Element type

In higher dimensions, such as 2D or 3D, the elements may be based onrectangles, triangles, tetrahedrons, hexahedrons, and (in more rare cases)pentahedrons (basically prisms in their appearance). In Figure 8, examplesof second order (quadratic) rectangular elements are shown.

3.4.2 Element order

Most common by far is the first order, which basically consists ofpiece-wise linear polynomial basis, i.e the functions described in section2.7.1 of the first order. These are less computationally expensive thansecond or third order elements, but does not approximate non-linear shapesor plastic deformation in general as well as the higher order types do.

3.4.3 Element density

The density of the elements is ultimately what limits the ability to seedetails in the simulated material. The more elements per volume, thehigher the detail (see Figure 6 for an example of this). However, thenumber of elements used is limited by the time it takes to simulate them.

21

By putting more elements where the fracture will occur and not as manywhere no fractures will occur, the resources are used more efficiently. Thisdistribution can be seen for the 2D-case in 11 and in Figure 12 for the3D-case. For the 2D model at most 28590 elements were used. For the 3Dmodel the element count was 907698 at most.

Figure 11: 2D model of the process. Note the denser distribution of ele-ments around the centre.

22

Figure 12: 3D model of the process. Note the denser distribution of ele-ments around the centre.

3.4.4 Time stepping and mass scaling

The time step size was set to be no smaller than 40 ns. If the criticaltimestep was smaller than this, mass-scaling was to be used. This led tosimulation times of 2-7 days for the 3D model and a few hours for the 2Dmodel. If no mass-scaling had been used, the 3D-model with the mostelements would have taken more than one month to complete, maybemore.

3.4.5 Boundary conditions

As boundary condition for the 3D model, the edges of the sheet (the edgeswith highest |x|-coordinate values) were chosen to only be able to movealong the z-axis. The cutters had boundary conditions that prevented anytranslational motion and were set to rotate around their centre with aconstant angular velocity |w|, such that the relative velocity with respectto the sheet was 0 at the point of contact (|w× r| = 0), i.e the cuttersrolled on(to) the sheet.

3.4.6 Initial conditions

Due to the effects of mass-scaling, it is better for accuracy and simulationtime to initialise an object that has very high density in the mid-section

23

(such as the simulated sheet) at a certain speed, instead of applying anacceleration. The sheet was set to an initial velocity v = vz = 3.3 m/s.

4 Results

4.1 Stress-Strain curve

The material model was fitted to the stress-strain curve for Docol 930Ssuccessfully. The resulting function can be seen in Figure 13.

Figure 13: Fitted curve with least squared error to the data.

4.2 Scanning Electron Microscope

In these SEM images, the direction ”up” is where the shearing tool firstimpacted the steel, and ”bottom” where the tool last was in contact withthe steel. This corresponds to the y-axis in the model if one considers thepart of the sheet that is on the negative side of the x-axis. The differentzones depicted here corresponds to what is shown in Figure 3, where thecutted zone is referred to as the bright zone. Note that some SEM imagesare rotated for convenience. Figure 14 depicts the slitted edge from theside.

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Figure 14: Overview of the slitted edge. The bright zone is found in the up-per part, with a distinct line separating it from the fracture zone.

4.2.1 Rollover zone

The rollover zone is characterised by its more or less rounded/compressededge at the top and a vertically straight, smeared-out zone beneath this.An effect of the initial compression of the material. See the upper part ofFigure 18 for a clear view of this. The rollover zone measured 173 µm inwidth and 57 µm in height.

4.2.2 Shear zone (bright zone)

The primary shear zone9 is the result of a compression in such a way thatthe bright zone becomes smooth and very even. The bright zone is seenzoomed in in Figure 15, and Figure 14 depicts the slitted edge from theside, where the bright zone is found in the upper part.

9Known as bright zone, due to its very shiny and smooth structure.

25

Figure 15: A zoomed-in image of the bright zone. Note the smoothness.

4.2.3 Secondary shear zone (fracture zone)

Secondary shear zone is located beneath the primary shear zone, see Figure14. Here, the material has suffered (almost) exclusively tensile forces andhas hence been ripped apart. The zone is characterised by its moon-likelandscape with mountain-like pikes and high variation in height comparedto the primary shear zone. A zoomed-in image of the fracture zone is seenin Figure 16

Figure 16: Fracture zone. Note how the material has been ripped apart andcreated the very non-smooth surface.

26

4.2.4 Burr zone

At the very bottom the burr zone is located. This is the result of tensileforces that has dragged the material out to such an extent that it hasplastically deformed and formed a sharp edge. The burr zone may be seenin Figure 18 at the very bottom left, as well as in Figure 17. The burr zonemeasured 40 µm in height.

Figure 17: Overview of the slitted edge a bit from the side. The burr zoneis the entire, outermost part of the edge.

4.2.5 Plastic deformation zone

The plastic deformation zone is defined as the zone that contains partsthat has been plastically deformed as a result of the slitting process. It istypically a semi-circle seen from the positive direction of the z-axis thatextends some distance into the negative x-direction of the sheet. In Figure18 the left sample has been slitted with a gap of 5 · 10−5 m between thetwo cutters, whereas the right sample depicts the exact same material, cutwith more gap ( 5 · 10−4 m) between the cutters.

27

Figure 18: Profile of the slitted material, achieved with EBSD. Note thehigh impact of the slitting gap; the right one has suffered a much higherdeformation as a result of this.

Figure 19: Profile of the slitted material, achieved with EBSD with plasticdeformation as overlay. The plastic deformation colour depicts the degreeof deformation (i.e damage), where red is highest and blue lowest. Notethat the overlay is normalised to its picture; that is, it not necessarily truethat the red zone to the left has the same plastic deformation degree as thethe red zone on the right.

28

Figure 20: Profile of the slitted material, achieved with EBSD. The plasticdeformation is depicted and lets one see the direction of the deformation.

Figure 19 depicts the samples from Figure 18, but with an overlay thatshows the degree of plastic deformation. Both Figure 18 and 19 wereachieved with EBSD (Electron BackScatter Diffraction) [16].

4.3 Two-dimensional model

The 2D model was performed in another project, see [17] for a completeproject description. Residual stresses were not considered in this model.The result for the 2D model may be seen in Figure 21. The 2D modelacted as an uncalibrated simplification in order to see if the simulation waspossible or not.

29

Figure 21: The damage zone is shown graphically. Note that the fracturenow follows the path of maximum damage thanks to the altered mesh. Tosee how the mesh was altered, please see [17].

4.3.1 Energy balance (2D)

Energy levels for the added mass in 2D are very small, as may be seen inFigure 22.

30

Figure 22: Here some different types of energies can be seen as a functionof time in the simulation. For accuracy, it is important (when mass scal-ing is used) that the kinetic energy created by added mass (Ekinetic−added)does not increase much. It is especially important if one wants to ne-glect the added mass completely that Ekinetic−added

Figure 23: Simulation with Wc = 1 GPa, r = 0.2 mm, overlap = 3 mm,and gap = 0.1 mm.

4.4.1 Energy balance (3D)

Energy levels for the added mass in 3D are large, as may be seen in Figure24. This means that the added mass may not be negligible in the sameway as for 2D, and measurements made in the model needs to take intoaccount the added mass.

32

Figure 24: Here some different types of energies can be seen as a functionof time in the simulation. For accuracy, it is important (when mass scal-ing is used) that the kinetic energy created by added mass (Ekinetic−added)does not increase much. It is especially important if one wants to ne-glect the added mass completely that Ekinetic−added

Figure 25: The edges of the tools have a sharpness radius r = 0.1 mm.

Figure 26: The edges of the tools have a sharpness radius r = 0.35 mm.

4.4.3 Different gap

A simulation with a smaller (infinitesimal) gap between the cutters isfound in Figure 27. Worth to note is that the edge appears to be moredamaged than when a small gap is used as space.

34

Figure 27: The edges of the tools have a sharpness radius r = 0.2 mm. Thegap between the two cutter tools is infinitesimal.

4.4.4 Different damage tolerance

Figure 28 depicts the case when the damage normalisation parameter Wc(see section 2.4 for definition) was set to half the value of what wasinitially simulated (Wc = 0.5 · 109 Pa). Figure 29 when the value was set todouble the initial value (Wc = 2.0 · 109 Pa). As can be seen, the elementsare allowed to deform to a much higher extent with higher values of Wc.

Figure 28: Wc = 0.5 GPa. The edges of the tools have a sharpness radiusr = 0.1 mm.

35

Figure 29: Wc = 2.0 GPa. The edges of the tools have a sharpness radiusr = 0.1 mm.

5 Discussion

Achieved results seem promising for future development. The 3D modelappears to simulate the slitting process very well, and might be able toaccurately predict new optimal parameters with the right calibration.

5.1 Software

IMPETUS Afea has proven itself to be extremely accurate and fast to workwith. The software’s robustness opens up for future, larger simulations onclusters. The mass-scaling’s effect on the 3D model are a bit unclear, theonly noticeable difference appears to be an increased pressure on thecutters, although this remains to be investigated. The time-step is quitehard to decrease due to the increase in simulation time, and by comparingthe SEM and 2D results with the 3D model one may assume that themodel is accurate despite the sinister energy plot seen in Figure 24. Thelargest problem by far with this study has been the fact that there arealmost no research or material models that handle fracture very well.Typically these (FEM) studies instead simulates the physics up untilfracture point. As follows from Section 2.7.1, one may come to realise thatthe higher order elements simulate plastic deformation much better thanthe linear elements do [8]. However, since we are more interested in howthe fracture propagates and not so much in how the plastic deformation

36

depicts itself, the trade-off was made to increase the element density at thecost of the elements’ polynomial order.

5.2 Material model

As of now, the only material model considered has been the one specifiedin Section 2.4. A development of a new material model12 (along withdamage model) would probably enhance the model further, however thecurrent material model must be seen as a very well-working compromise-especially since it may be calibrated further. The current material anddamage models are, after all, based on excessive experimental results andmay be expected to be very accurate.

5.3 Stress-Strain curve

Calibration of the material model may be seen in Figure 13. The fit wasmade due to the lack of true stress-strain data (which is hard to come by)and is an approximation of the true curve. The approximation relies on thefact that the true stress-strain relationship continues to grow for ductilematerials as seen in Figure 4.

5.4 Scanning Electron Microscope

SEM images were beyond expectations in accuracy and showed veryhigh-detailed images of the test pieces. This opened up for a more preciseand objective verification process of the simulated results, especially due tothe EBSD images seen in Figures 18, 19, and 20. The EBSD images showsthat the plastic deformation and damage simulated in the model resembleswhat actually occurs in the real-life process. Figure 14 shows that therollover zone and bright zone together fluctuates around 130 to 200 µm indepth, whereas the rollover zone itself measures around 50-60 µm. Thisverifies the results from the profile image in Figure 18, where the rolloverzone was measured to be 57 µm. The variation in bright zone depth maybe expected due to the fact that the steel in real life is a bit heterogeneous,as well as the fact that the cutters may vary a bit as well.There are major differences between the bright zone and the fracture zone,as can be seen in Figures 15 and 16. The difference is due to the fact thatthe material (on a microscopic case) has been compressed in the former

12A new material model is not an easy task to develop, and might even exceed a PhDin span.

37

case, and ripped apart in the latter. It is a bit unclear how much thisaffects the propagation of micro-cracks in the material, but the hypothesisis that the fracture zone is weaker than the bright zone due to the fact thatforce applied is more likely to be focused into one of the already existingcavities of the fracture zone.

5.5 Two-dimensional model

The two-dimensional model was developed in the project specified here[17]. During this time there was no material data available to calibrate themodel and the constants C0, C1 and �p were chosen arbitrarily. As a resultof this, the produced model results are not to be considered trustworthy,but rather a first, simple, proof of concept that the simulations indeed canbe made. The mesh in Figure 21 is a result of first simulating the processwith straight mesh, and then, the timestep before any elements are eroded,the most damaged elements were chosen and their coordinates extractedinto another, new simulation. This resulted in a very accurate prediction ofwhere the fracture will propagate, and is an effective way to reduce themesh bias when the element density is high. See [17] for a completedescription of this process. The 2D model became the basis for the 3Dmodel.

5.6 Three-dimensional model

Compared to the 2D version, the 3D model is far more complex since itincludes depth dimension along with rotation of cutters and otherparameters that correspond to those used in the real process. Albeit thecalibration made was kept to a minimum, the results does indeed seemvery promising for further studies. The influence of residual stresses seemsto be negligible compared to the models that doesn’t take residual stressesinto account. Also, since residual stresses typically appear in themagnitude of 200 MPa at most (which is on the surface) [2] [7], the sheetwould have to be larger in surface with equal thickness for stresses to haveany noticable effect. All tests performed in Section 4.4 were simulated bothwith and without residual stresses with no noticeable difference.Hypothetically, the model would have to include a larger portion of thesheet for these effects to appear. Simulations of such large models wereskipped mainly due to lack of memory on GPU, but also because theresults would be irrelevant due to the uncalibrated Wc.

38

5.7 Future improvements

The two most necessary experiments needed for an increase in accuracy area true stress strain curve for the desired material, along with a calibrationof Wc. A true stress strain curve would make one able to skip the datafitting step in Section 3.3, since the raw data then can be imported directlyinto IMPETUS’s IDE. For better model verification the pressure may bemeasured in real life on the cutters and then compared to those in themodel. Slitting is a continuous crack propagation process, whileCockcroft-Latham is a criterion for ductile crack initiation. Hence, it mightnot be the ideal model for this process. However, shifting to somethingmore accurate is not easy without access to results from a materialcharacterisation program. A future model might also consider the wear onthe cutters (currently they are considered to be rigid bodies) and be ableto predict when they should be replaced for better performance withregard to cost. A better calibrated model might also be able to predictoptimal parameters (cutter overlap, cutter gap etc.) for damageminimisation. CL damage model may cause mesh bias and should thereforebe analysed in future development. An alternative might be to use anenergy based crack propagation criteria (energy per unit crack). Thiswould likely reduce mesh dependency greatly while also ensure convergence(see parameter GI in [9]).A fully calibrated model should be able to estimate the spread ofmicro-cracks in the material by looking at the damage zone and theplastically deformed zone in the model, as these have proven themselves tobe accurate (compare, for instance, Figure 28 with Figures 18 and 19).Given that the micro-cracks may be estimated by the model, they could intheory be minimised based on the model and the model could then act as abasis for real-life experiments. The rapid development of GPU technologygives hope of an even more fine mesh, even on desktops.

5.8 Conclusions

Overall the model appears to successfully simulate the real-world process.Further calibrations may enhance the accuracy and perhaps make thesimulation able to predict a new, better optimal set of parameters for theprocess. The most accurate parameters, based on the results, appear to bewith Wc = 0.5 GPa and r = 0.1 mm. This is objectively achieved bycomparing the EBSD results in Figure 18 with the model results in Figure28. It appears to be (based on the results) that a less sharp cutting tooldeforms the steel more (Figure 26 versus 25), too small a gap damages the

39

steel more (Figure 27 versus 23), and it also seems like Wc should lie in theneighbourhood of 0.5 GPa in size. The results should be verified to ensurethis, although based on the EBSD images (Figures 18, 19, and 20) therollover zone does indeed worsen with a more round tool and much largergap.One may say, based on these results, that the simulation as of now needssome more calibration before it can be seen as a complete success, but asuccess nevertheless. Due to the uncertainty in the simulation and the needfor more essential parameters to be calibrated first, the effect of residualstress cannot be trusted as the model stands now.The 2D model may be seen as a successful simplification of the 3D model,as their results deviate very little from one another. This basically meansthat for better crack propagation and prediction, a finer mesh may be usedin the 2D model and used for comparison with the 3D model.Furthermore, the method the model was developed with may be applied toother industrial processes. The applications are uncountable, and albeitimprovements may not be guaranteed with this method, they are certainlyfar more likely to occur.

References

[1] Mathematics HandbookBy Lennart R̊ade and Bertil Westergren.Published 2004 by Studentlitteratur AB.

[2] Mechanical Behaviour of MATERIALSSecond editionMarc Meyers and Krishan ChawlaCambridge University PressUniversity Printing House, Cambridge CB2 8BS, UK

[3] Teknisk h̊allfasthetslära by Tore Dahlberg. Published 2001 byStudentlitteratur AB.

[4] SSAB’s dual-phase high-performance steel. Fetched 12/5 2019.https://www.ssab.se/produkter/varumarken/docol

[5] Ludwik’s empirical relation. Fetched 12/2 2019https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4980855/

40

https://www.ssab.se/produkter/varumarken/docolhttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC4980855/

[6] Cockcroft-Latham damage model.https://www.dynalook.com/conferences/

12th-international-ls-dyna-conference/keynotep-a.pdf

[7] Practical Residual Stress Measurement MethodsBy Gary S. SchajerPublished 2013 by WILEY

[8] On the Arbitrary Lagrangian- Eulerian Finite Element MethodBy Lars OlovssonDivision of Solid MechanicsDepartement of Mechanical EngineeringLinköpings universitet, SE-58183 Linköping, Sweden

[9] IMPETUS Afea’s homepage.https://www.impetus-afea.com/

IMPETUS’s manual, Cockcroft Latham:https:

//www.impetus-afea.com/support/manual/?command=PROP_DAMAGE_CL

IMPETUS’s manual, Material properties:https:

//www.impetus-afea.com/support/manual/?command=MAT_METAL

[10] Von Mises yield criterion.https://www.engineersedge.com/material_science/von_mises.htm

[11] On solid mechanics.https://classes.engineering.wustl.edu/2009/spring/mase5513/

abaqus/docs/v6.6/books/gsa/default.htm?startat=ch09s03.html

[12] Physics HandbookBy Carl Nordling and Jonny Österman.Published 2004 by Studentlitteratur AB.

[13] MATLAB’s homepage.https://se.mathworks.com/products/matlab.html

[14] LS Dyna’s homepage. Fetched 13/4 2019http://www.lstc.com/products/ls-dyna

[15] Example images of how sheared steel usually look. Fetched on 9/12019.https://www.thefabricator.com/article/stamping/

blanking-questions-have-you-on-the-edger

41

https://www.dynalook.com/conferences/12th-international-ls-dyna-conference/keynotep-a.pdfhttps://www.dynalook.com/conferences/12th-international-ls-dyna-conference/keynotep-a.pdfhttps://www.impetus-afea.com/https://www.impetus-afea.com/support/manual/?command=PROP_DAMAGE_CLhttps://www.impetus-afea.com/support/manual/?command=PROP_DAMAGE_CLhttps://www.impetus-afea.com/support/manual/?command=MAT_METALhttps://www.impetus-afea.com/support/manual/?command=MAT_METALhttps://www.engineersedge.com/material_science/von_mises.htmhttps://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/gsa/default.htm?startat=ch09s03.htmlhttps://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/gsa/default.htm?startat=ch09s03.htmlhttps://se.mathworks.com/products/matlab.htmlhttp://www.lstc.com/products/ls-dynahttps://www.thefabricator.com/article/stamping/blanking-questions-have-you-on-the-edgerhttps://www.thefabricator.com/article/stamping/blanking-questions-have-you-on-the-edger

[16] The EBSD technique. Fetched 12/5 2019.https:

//en.wikipedia.org/wiki/Electron_backscatter_diffraction

[17] Project report of the previous study. Finished 19/2 2019https:

//drive.google.com/open?id=164wfgkZi_fncVFxsJgDVpoEPWscY_Zq5

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https://en.wikipedia.org/wiki/Electron_backscatter_diffractionhttps://en.wikipedia.org/wiki/Electron_backscatter_diffractionhttps://drive.google.com/open?id=164wfgkZi_fncVFxsJgDVpoEPWscY_Zq5https://drive.google.com/open?id=164wfgkZi_fncVFxsJgDVpoEPWscY_Zq5

APPENDIX3D model script.

1 ∗UNIT SYSTEM2 SI3 ∗PARAMETER4 L = 0.15 D = 0.086 h = 0 .5 e−37 o f f = 0.00958 gap = 0.2∗%h9 R = 0.1

10 width = 0.0211 over lap = 0 .3 e−312 v = 3.3313 w = %v/%R14 tend = 0.5∗(%D+%o f f )/%v15 ∗TIME16 [%tend ] , 0 , 4 . 0 e−817 #18 # −−− MESH −−−19 #20 ∗COMPONENTBOX21 ” Sheet − cente r f i n e ”22 3 , 3 , 40 , 30 , 75023 [−%gap−1.5∗%h ] , [−%h / 2 ] , [−%D/4−%o f f ] , [%gap+1.5∗%h ] , [%h / 2 ] ,

[−% o f f ]24 ∗COMPONENTBOX25 ” Sheet − cente r coa r s e ”26 4 , 4 , 1 , 1 , 1827 [−%gap−1.5∗%h ] , [−%h / 2 ] , [−%D−%o f f ] , [%gap+1.5∗%h ] , [%h / 2 ] , [−%

D/4−%o f f ]28 ∗COMPONENTBOX29 ” Sheet − l e f t ”30 5 , 4 , 10 , 1 , 2431 [−%L/ 2 ] , [−%h / 2 ] , [−%D−%o f f ] , [−%gap−1.5∗%h ] , [%h / 2 ] , [−% o f f ]32 ∗COMPONENTBOX33 ” Sheet − r i g h t ”34 6 , 4 , 10 , 1 , 2435 [%gap+1.5∗%h ] , [−%h / 2 ] , [−%D−%o f f ] , [%L/ 2 ] , [%h / 2 ] , [−% o f f ]36 ∗MERGE DUPLICATED NODES37 P, 4 , P, 4 , [%h /10 ]38 ∗INCLUDE39 ” Cutter 1”40 . . / c u t t e r r 0 . 2 . k41 1 , 1 , 142 0 , 0 , 0 , [%gap / 2 ] , [%R−%over lap / 2 ] , 043 ∗INCLUDE44 ” Cutter 2”

43

45 . . / c u t t e r r 0 . 2 . k46 1 , 1 , 1 , 10000 , 10000 , 147 0 , 0 , 0 , [−%gap / 2 ] , [−%R+%over lap / 2 ] , 048 −1, 0 , 049 ∗CHANGE P−ORDER50 PS , 124 , 251 ∗SMOOTH MESH52 PS , 124 , 60 .053 ∗SET PART54 12455 1 , 2 , 456 ∗MERGE57 P, 3 , P, 4 , [%h /100 ]58 ∗REDISTRIBUTE MESH CARTESIAN59 160 P, 3 , X, 0 , 161 0 , 0 , [−% o f f ]62 ∗TRANSFORM MESH CARTESIAN63 1 , P, 3 , 0 , 444464 ∗FUNCTION65 444466 0 .4∗ y ∗ (1 − abs ( x )/(%gap+1.5∗%h) )67 ∗GEOMETRYBOX68 12369 [−%L/ 4 ] , [−%h / 2 ] , [−%D−%o f f ] , [%L/ 4 ] , [%h / 2 ] , [−% o f f ]70 #71 # −−− MATERIALS −−−72 #73 ∗MAT METAL74 1 , 7800 .0 , 210 .0 e9 , 0 . 3 , 175 176 ∗FUNCTION77 178 647 .81 e6 + 1438.5 e6∗ epsp ˆ0 .379 ∗PROP DAMAGE CL80 1 , 181 1 .0 e982 ∗MAT RIGID83 ” Cutter ”84 2 , 7800 .085 #86 # −−− PARTS −−−87 #88 ∗PART89 ” Cutter 1”90 1 , 291 ” Cutter 2”92 2 , 293 ” Sheet − cente r ”

44

94 3 , 195 ” Sheet − edges ”96 4 , 197 #98 # −−− BOUNDARY CONDITION and CONTACT −−−99 #

100 ∗CONTACT101 ” gene ra l contact ”102 1 , 1103 PS , 34 , PS , 12 , 0 . 1 , 5 . 0 e15104 0 , 0 , 333105 83 , 83 , 0 , 0 , 1106 ∗FUNCTION107 83108 p∗vtang109 ∗GEOMETRYBOX110 333111 [−%width ] , [−%R] , [−%D−%o f f ] , [%width ] , [%R] , 0112 ∗SET PART113 34114 3 , 4115 ∗SET PART116 12117 1 , 2118 ∗BC MOTION119 ” Cutter 1”120 1121 P, 1 , XYZ, YZ122 V, RX, 1001123 ∗BC MOTION124 ” Cutter 2”125 2126 P, 2 , XYZ, YZ127 V, RX, 1001 , −1.0128 ∗FUNCTION129 1001130 −%w131 ∗INITIAL VELOCITY132 PS , 34 , 0 , 0 , [%v ]133 ∗BC MOTION134 ” Sheet l e f t ”135 3136 G, 1 , Y137 V, Z , 1002138 ∗BC MOTION139 ” Sheet l e f t ”140 4141 G, 2 , Y142 V, Z , 1002

45

143 ∗FUNCTION144 1002145 %v146 ∗GEOMETRY SEED COORDINATE147 ” Sheet l e f t ”148 1149 [−%L/ 2 ] , 0 , [−%D/2 ]150 ∗GEOMETRY SEED COORDINATE151 ” Sheet r i g h t ”152 2153 [%L/ 2 ] , 0 , [−%D/2 ]154 ∗END

MATLAB code for curvefit:

1 c l o s e a l l2 c l e a r a l l3 data = x l s r e a d ( ’ doco l930s . x l sx ’ ) ;4 data2 = x l s r e a d ( ’ docol1400m . x l sx ’ ) ;5 e p s i l o n 2 = data2 ( 1 : end , 1 ) ;6 sigma2 = data2 ( 1 : end , 2 ) ∗1 e +6;789 e p s i l o n = data ( 1 : end , 1 ) ;

10 sigma = data ( 1 : end , 2 ) ∗1 e +6;11 y = sigma−sigma (1) ; %set x (0 )=01213 p = [ e p s i l o n . ˆ ( 0 . 3 ) ones ( s i z e ( y ) ) ]\ sigma ; % Estimate

Parameters14 % i f y = k∗x .ˆ0.5+C, then p (1)= k and p (2)=C15 z = e p s i l o n . ˆ 0 . 3∗ p (1)+p (2) ;16 o r i g = e p s i l o n .ˆ0 .3∗0 .1∗10ˆ9+1 .5∗10ˆ9 ;1718 d i sp ( ’ k = ’ )19 d i sp (p (1 ) )2021 d i sp ( ’C = ’ )22 d i sp (p (2 ) )2324 f i g u r e25 hold on26 p l o t ( ep s i l on , sigma )27 % p lo t ( eps i l on2 , sigma2 )28 p l o t ( ep s i l on , z )29 % p lo t ( ep s i l on , o r i g )30 % legend ( ’ Docol 930S ’ , ’ Docol 1400M’ , ’ F i t t ed curve y=k∗ eps ˆ{0.3}+

C ’ , . . .31 % ’ Or i g i na l mate r i a l model ’ )32 legend ( ’ Docol 930S ’ , ’ F i t t ed curve y=k∗ eps ˆ{0.3}+C ’ )33 x l a b e l ( ’ e p s i l o n ’ )

46

34 y l a b e l ( ’ sigma ’ )

MATLAB code for mesh alteration:

1 c l e a r a l l2 format long3 data=load ( ’ damage l i s t41 . txt ’ ) ;4 x = data ( : , 1 ) ;5 y = data ( : , 2 ) ;6 z = data ( : , 3 ) ;7 dmg = data ( : , 4 ) ;89 %k = length ( y ) ;

1011 % for k = 1 : l ength ( y )12 % i f z ( k )˜=0 && y˜=013 % y ( k ) =100;14 % x ( k ) =100;15 % end16 %17 % end1819 i =1;20 j =1;21 % Take away elements where z !=022 while i

45 i f (maxD( k ) == 0)46 maxD( k ) = DMG( i ) ;47 end48 %else49 i f k==150 i f (maxD( k )1)56 i f (maxD( k )

a simulation with finite elements to model steel...

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