Biaxial Testing of Sheet Metal: An Experimental-Numerical Analysis
Gerard Quaak MT 08.10
TU/e Master Thesis May, 2008
Engineering thesis committee prof.dr.ir. M.G.D. Geers (Chairman) dr.ir. J.P.M. Hoefnagels (Coach) ir. C. Tasan (Coach) dr.ir. P.J.G. Schreurs dr.ir. H. Vegter
Eindhoven University of Technology Department of Mechanical Engineering Computational and Experimental Mechanics
Abstract
With the recent increase in the popularity of advanced high strength steels (e.g. dual phase, TRIP) in automotive industry, new challenges have arisen. Conventional continuum models are found not to capture the reported pre- mature ductile failures in such steels, which are governed by damage evolution. Another challenge is understanding and predicting metal behaviour under com- plex strain paths. The ability to precisely capture these eects in continuum models is important for the sheet metal forming industry, in order to carry out these processes as ecient as possible. However, improvements of the numerical tools highly depend on the development of accurate and practical experimental techniques. A testing device for biaxial deformation of sheet metal is such an experimental tool that has been studied by many researchers before. Although several experimental set-ups have been proposed in the literature, most of these designs are not capable of providing information up to the point of fracture. Additionally, these set-ups are not usable for real-time, in-situ examination of the deforming structure with advanced microscopic techniques such as SEM, AFM, surface prolometry, or digital image correlation, because a miniaturized form is not available or was never investigated. The main goal of this project therefore is to nd a practical and accurate set- up that can deform sheet metal specimens under varying complex strain paths, while allowing for real-time, in-situ microscopic examination. For this pur- pose, several experimental set-ups (e.g. bulge, punch, Marciniak and cruciform tests) have been studied and compared both experimentally and numerically. For testing cruciform samples a simplied in-plane biaxial-loading set-up was designed and build, while Marciniak tests were carried out at Corus RD&T. The corresponding experimental results were used to verify and compare these tests in terms of practical aspects (e.g. specimen preparation). The computa- tional results are used to analyze stress and strain distributions and for better understanding of the eects of miniaturization. Combining literature, numerical and experimental test results, it was concluded that the cruciform and Marciniak test are the most promising set-ups for minia- turized biaxial testing of sheet metal with in-situ microscopic examination. Both test have their limits, but when taken these into account can provide valuable data.
i Samenvatting
Met de recente populariteit van 'advanced high strength' staalsoorten (dual phase, TRIP) in de automotive industrie, zijn nieuwe vraagstukken ontstaan. Conventionele continuum modellen blijken niet in staat om de aanwezige taaie breuk te beschrijven, waardoor het voorspellen van het materiaal gedrag tijdens omvormen van deze metalen niet mogelijk blijkt. Wanner niet lineaire rek paden een rol spelen blijken de huidige continuum modellen zelfs nog minder accuraat. Voor het verbeteren van de numerieke gereedschappen zijn echter betrouwbare experimentele technieken nodig, zodat het materiaal gedrag voor deze groep metalen is te bepalen. Een voorbeeld van zo een opstelling, is er een om biaxiale deformatie van plaat staal te bestuderen. Hiernaar is reeds veel onderzoek aan besteed, maar hoewel verschillende gereedschappen onderzocht zijn, bestaat er tot op heden geen opstelling om tot breuk te deformeren zonder externe invloeden. Daar komt nog bij dat om de micro structuur van het materiaal te onderzoeken geavanceerde technieken als SEM en AFM of digital image correlation gebruikt moeten worden, wat inhoudt dat een miniatuur opstelling nodig is. Mogelijkheden voor een dergelijke opstelling zijn nog niet eerder onderzocht. Het belangrijkste doel van dit project is dan ook het vinden van een praktische manier om plaat staal onder veranderende rek paden, met de mogelijkheid real- time metingen te doen. Hiervoor zijn verschillende experimentele opstellingen (bulge, punch, Marciniak en cruciform test) numeriek en experimenteel onder- zocht en vergeleken. De numerieke resultaten geven o.a. inzicht in spannings- en rek velden en de eecten van miniaturisatie, de experimentele resultaten worden gebruikt voor vericatie en vergelijk op praktische gebied, zoals het maken van test samples. Voor het testem van kruisvormige trekstaven is een opstelling ontworpen en gebouwd, als onderdeel van dit onderzoek. Voor de Marciniak testen is gebruik gemaakt van een opstelling beschikbaar gesteld door Corus RD&T. De literatuur studie, numerieke en experimentele resultaten laten zien dat zowel de kruisvormige trekstaven als de Marciniak test bruikbaar zijn in een geminia- turiseerde vorm. Beide tests hebben beperkingen, maar wanneer hier rekening mee wordt gehouden kunnen zij waardevolle informatie opleveren.
ii Acknowledgement
This Master project was mainly carried out at Eindhoven University of Tech- nology, in the group of professor Marc Geers. Experimental work has been done at Corus RD&T in IJmuiden and in the University Multi-scale Labora- tory. Special thanks goes to Sjef Garenfeld for his time and patience in the design process of the biaxial testing set-up and specimens. I also want to thank Marc van Maris, supervisor of the Multi-scale Laboratory, for his guidance and advice during experimental work and Tom Engels for his help with the tensile testing machine. From I want to thank Corus RD&T Menno de Bruine, oper- ator of the Marciniak test set-up, Carel ten Horn and Louisa Carless for their time and eort with the experimental work on the Marciniak test. At Philips Drachten a lot of insight was provided in possible material removal techniques, for which I want to express my gratitude to Gerrit Klaseboer, Harmen Altena and Willem Hoogsteen for receiving us and a good discussion of the material removal problem. I also want to thank Johan Hoefnagels and Cem Tasan for their input in the project, the many evenings discussing and the help with the nal report. Gerard Quaak Eindhoven, May 2008
iii Contents
Abstract i
Samenvatting ii
1 Introduction 1 1.1 Formability and strain path dependency ...... 2 1.2 Objective ...... 4 1.3 Strategy ...... 4
2 Literature Survey 5 2.1 Introduction ...... 5 2.2 Bulge test ...... 8 2.3 Punch test ...... 11 2.4 Marciniak test ...... 14 2.5 In-plane loading with cruciform geometry ...... 17 2.6 Summary ...... 22
3 Numerical methodology 23 3.1 Material model ...... 23 3.2 Bulge test ...... 24 3.3 Punch test ...... 25 3.4 Marciniak test ...... 27 3.5 In-plane loading with cruciform geometry ...... 28
4 Experimental methodology 29 4.1 Marciniak test ...... 29 4.1.1 In-plane testing with the Marciniak set-up ...... 29 4.1.2 Specimen manufacturing ...... 30 4.2 In-plane loading with cruciform geometry ...... 30 4.2.1 Design of a test set-up ...... 30 4.2.2 Tests with in-plane cruciform geometry ...... 34 4.2.3 Specimen manufacturing and characterization ...... 36
5 Results 39 5.1 Bulge test ...... 40 5.1.1 Miniaturization ...... 40 5.1.2 Summary ...... 42 5.2 Punch test ...... 43
iv CONTENTS
5.2.1 Miniaturization ...... 44 5.2.2 Summary ...... 46 5.3 Marciniak test ...... 47 5.3.1 Working principle of the Marciniak test ...... 47 5.3.2 Numerical - experimental study of the Marciniak test . . 49 5.3.3 Minaturization ...... 53 5.3.4 Summary ...... 55 5.4 In-plane loading with cruciform geometry ...... 56 5.4.1 Optimization of the cruciform design ...... 56 5.4.2 Proof of principle ...... 57 5.4.3 Specimen manufacturing and characterization ...... 60 5.4.4 Miniaturization ...... 66 5.4.5 Summary ...... 67 5.5 Comparative evaluation of the set-ups ...... 68
6 Conclusions and recommendations 72 6.1 Conclusions ...... 72 6.2 Recommendations for future work ...... 73
Bibliography 75
A Electrical Discharge Machining 79
B Electrical Chemical Machining (ECM) 83
C Specimen Preparation: TegraPol or Target System 86
v Chapter 1
Introduction
Metals, and in particular sheet metals, are used in a wide variety of appli- cations in industry, with the main elds of application being packaging (food containers, beverage cans), automotive and aerospace industry. As material costs are a substantial part of the costs of manufactured products and most of the products are produced in large numbers, large cost reductions can be achieved by lowering the amount of material used. Moreover, several other reasons that can be thought of for lowering the amount of used material, e.g. lowering weight and lowering impact on the environment by polluting. All of these reasons result in eorts to achieve material use reduction without quality loss in the product.
Figure 1.1: Advantages and disadvantages of using aluminium instead of steel for a standard medium size car [44]
In the automobile industry a rst attempt to achieve weight reduction was done by using low density materials like aluminium, magnesium and plastics, but recent studies show a promising future for steels instead. The International Iron and Steel Institute (IISI) computed how the use of aluminium makes the car body lighter, but does not have a signicant eect on the total weight of the car and causing more environmental impact because of higher equivalent
CO2-emissions (see gure 1.1). The development of Advanced High Strength Steels that can replace the existing steels is therefore closely followed by the automotive industry. [16, 44]
1 CHAPTER 1. INTRODUCTION
Figure 1.2: Overview of steel grades as used in the automotive industry [44]
Figure 1.2 shows an overview of regularly used types of steel for automotive applications. In this gure the materials furthest to the left are most suitable for forming virtually any desired shape, while consuming relative low amounts of energy. The steels on the right, however, can withstand much larger forces, which makes manufacturing more dicult. An ideal material would have prop- erties of both steel types, being highly formable, yet very strong. Under certain circumstances, a good combination of these properties can be achieved e.g. by suitable phase transformations.[44] Advanced High Strength Steels (AHSS) have such properties, e.g. a relatively high yield strength and high hardening rate compared to conventional steels. In the past decades signicant amounts of eorts are put in setting up strategies for improving FE models to capture these failures. However, to make optimal use of FE-modelling, a good description of the materials behaviour is necessary, which relies on the accuracy of the used constitutive laws for describing the material behaviour. The deformation-induced evolution of metal micro structure for which, as will be explained in more detailed later, new experimental tools are necessary. [44]
1.1 Formability and strain path dependency
In the previous section the problems with AHSS were shown, which will be explained in more detail with the help of the Forming Limit Diagram (FLD) that will be introduced now. The FLD is based on the assumption that for forming purposes, the maximum deformation is limited by the initiation of unstable deformation, e.g. necking. When forming metal sheets the material is subjected to dierent strains and strain paths, which have been found to have dierent maximum allowable deformations. Therefore in industry the FLD, as shown in gure 1.3 (a), is used to show these limiting deformations. On the axes are the strains in the two principle directions in the plane, with the line giving the point of necking for the combination of strains at that point. The numbers in the gure show the strain paths pure shear (1), simple tension
2 CHAPTER 1. INTRODUCTION
Figure 1.3: a) Schematic forming limit diagram; b) Stress limit diagram (From Ba- nabic [3])
(2), plane strain tension (3) and biaxial tension (4). These maxima are only true for linear strain paths to that point, and are therefore not a simple to use as it seems. The limits of the FLD are becoming clear when testing a sheet metal under a changing strain path. The rst gure, 1.4(a), shows the strain paths up to neck- ing for linear strain paths on an undeformed sheet of metal. The second gure, 1.4(b), shows where necking starts for a sheet metal that is rst deformed under uniaxial tension, and then by biaxial tension. A large increase in formability is found, that was not predicted by the original FLD. When starting with biaxial tension followed by uniaxial tension, a large decrease in formability is found, as shown in gure 1.4(c). This eect is stronger for AHSS then for conventional steels, which makes the need for understanding what happens necessary to be able to use the new steels up to full potential. [3, 24]
Figure 1.4: a) Linear strain paths on an undeformed sheet; b) Forming with uniaxial tensile state followed by biaxial; c) Forming with biaxial tensile stage followed by uniaxial (From Banabic [3])
A quite similar concept, but not less sensitive to strain path changes and thus the strain history of the material, is the stress forming limit diagram, as shown in gure 1.3(b). A disadvantage of the stress based forming limit is uncer- tainty of the computed stresses, which in practice can only be determined from measured or computed strain elds. A FE model could be used to determine these forming stresses, but therefore the used material model should accurately describe the material behaviour.[24]
3 CHAPTER 1. INTRODUCTION
1.2 Objective
The challenges that have grown due to the increased use of AHSS in new de- signs, lead to the goal of this project. This is the development of an experimen- tal methodology to deform a sheet metal specimen under biaxial tension. The need for such a methodology to analyze microstructural changes under biaxial loading is obvious, as no such set-up exists. The data obtained with such a test method can then be used to predict damage evolution and thus in the future make better FE-modelling possible for (re)designing products. An experimental set-up that can be used to study the microstructural changes will have to t in or under standard microscope systems and has to be usable with digital image correlation systems for in-situ examination of the deforming material. Such a set-up can then be used in future to study and characterize new materials, as developed by the industry. The most important properties that will be considered are:
• The existence of a homogeneous stress- and strain distribution in the studied area of the specimen, which is not inuenced by contact, friction or other eects introduced by the test equipment.
• The possibility to deform under complex strain paths, preferably with the option to change the strain path during a test.
• The initial point of fracture and the crack itself should be free to form. Inuences from specimen or set-up must be minimized so the obtained material data is as undistorted as possible.
1.3 Strategy
Working towards a suitable test method, various known methods for testing under biaxial loading are studied. The literature survey in chapter 2 is meant to provide better understanding of the problem and dierent set-ups, so a choice can be made which testing methods will be used. The validation and further studying of the most promising set-ups was done both numerically and experimentally. The models and the assumptions made to simplify the computations are being discussed in the rst part of chapter 3. The second part of chapter 3 contains the experimental set-ups used to validate the numerical work. The results of both numerical and experimental work are discussed in chapter 4. The numerical and experimental results are compared in order to nd lim- itations and possible future improvements for both, resulting in an extensive overview of all the studied set-ups.
4 Chapter 2
Literature Survey
In the last decades several scientists have studied methods to deform sheet metal under complex strain paths, including punch tests [3, 36, 39], bulge pressure tests [49, 50], viscous pressure forming tests [37], biaxial compression tests [30] or cruciform tests [11, 19, 32]. Currently the most used method for determining FLCs in the industry is by using punch tests, which are known to overestimate the maximum allowable strains [3]. As the exact amount of the overestimation of can not be exactly determined, an unknown error in the resulting FLD makes a relatively large safety margin (up to 10 %) is applied to the maximum allowable strains when using the material in a forming process. This means more material will be used to make a safe structure or product, leading to higher costs. [3, 45, 47] This chapter will rst describe several properties of the biaxial testing set-ups, that will be used in the following sections to compare the dierent set-ups. A denition of biaxial stress is giving, followed by denitions for the working plane, geometrical constraining and properties to measure. The set-ups to be discussed are the punch test, the bulge test, the cruciform tensile test and the Marciniak test, with a study of the workings of each set-up and recent developments. The last section gives an overview of the studied set-ups, for easy comparing of each set-up.
2.1 Introduction
Biaxial loading
In the biaxial stress state forces are working in two directions on an innitesimal small volume, the third direction is the out of plane direction that is related to the two in plane directions, just like an uniaxial stress state as shown in gure 2.1 on the left. The stresses working on the volume under biaxial stress can be visualized, as shown in gure 2.1 on the right: forces are acting on the four areas perpendicular on the plane, from which the stresses can be computed
5 CHAPTER 2. LITERATURE SURVEY dividing the force by the area it is acting on. Strains in a biaxial deformation can than be computed via equations 2.1 to 2.3. Often it is more convenient to measure strains, equations 2.4 and 2.5 are given for calculating stresses from known strains. σ3 = 0 as there is no force acting on the plane. These equations are only valid in the elastic regime, whereas in the plastic regime pure biaxial loading only takes place up to localization. [13]
Figure 2.1: Uniaxial and biaxial stress states [13]
1 ε = (σ − νσ ) (2.1) 1 E 1 2
1 ε = (σ − νσ ) (2.2) 2 E 2 1
ν ε = − (σ + σ ) (2.3) 3 E 1 2
E σ = (ε + νε ) (2.4) 1 (1 − ν2) 1 2
E σ = (ε + νε ) (2.5) 2 (1 − ν2) 2 1
A complicating factor in the biaxial case is to determine the area that the forces are acting on, which makes determining stresses σ1 and σ2 more dicult than for the uniaxial case. Furthermore, during an actual manufacturing process the biggest problem is determining the plastic response. This cannot be described with a set of equations as given above. An important observation is that real biaxial loading only occurs up to lo- calization. Due to damage, necking and failure in a material, asymmetry is introduced and the simplied approaches as used in the elasticity regime are not correct anymore. Still the elastic behaviour is important, as this is where the nal failure mode might be determined.
6 CHAPTER 2. LITERATURE SURVEY
Working plane
Some experimental set-ups test a material in-plane, others out-of-plane, de- pending mostly on tooling. As out-of-plane testing gives rise to bending, it is preferred to test in-plane. This means stresses and strains are constant over the thickness of the sheet, which makes computing of the stresses and measuring the strains less complicated. Some studies also show an inuence to the forming limit while comparing in- plan and out-of-plane testing. Forming limits up to 6 % higher where found with out-of-plane testing of the same material. [47]
Geometrical constraints
The geometry of a tested specimen or the set-up itself can inuence the data measured during an experiment. Possible inuences are areas of contact where friction plays a role or a geometry that is sensitive to a certain mode of failure. These so-called geometrical constraints can be introduced by contact or friction with the used tool set in the region of interest, by asymmetry of the tool set, by non-isotropic material behaviour or by a geometry that leads to stress concentrations. A well known example is anisotropy in sheet metals, which gives rise to the need to test a material in more then one orientation relative to the rolling direction of the sheet. When deforming biaxially, the anisotropy will introduce a weaker direction, which is more likely to fail. [47]
Measuring stresses, strains or forces
Not every experiment has the same possibilities for measuring stresses, strains or forces. As stated before, directly measuring stresses would be the most ideal solution in most cases, but this is hardly ever possible. Stresses are normally calculated from either a strain eld or forces and the area they are working on. Measuring strain elds can be done with a digital image correlation set-up that for sheet metal can measure strains on top or bottom surface. Stresses can be calculated with the use of a FE-model, but therefore depends on an accurate material model. The other solution is measuring forces and the area they work on, as is done for uniaxial tensile tests. This is only possible when both the force on an area and the area itself can be measured. This works ne for a simple tension test, but for more complex stress states this is often not possible.
7 CHAPTER 2. LITERATURE SURVEY
2.2 Bulge test
The bulge test is a well described experimental set-up for biaxial loading, where pressure is used to deform a specimen. The set-up consists only of a pressure chamber and clamping mechanism. The bulge test is mostly used for testing thin lms, as bending stresses can be neglected for that case. The pressure can be build up by a gas, a uid or even a owing polymer [21, 37].
Figure 2.2: Pure biaxial bulge test with rounded die [21]
In gure 2.2 a simple bulge test set-up is shown, with the most important properties visualized being t0 and td, the initial and nal thickness of the sheet, dsheet the diameter of the sheet, dc the diameter of the die cavity, hd the height of the dome and RC the radius of the die edge. Rd is the radius of the bulge in a circular set-up. Rd is divided in two values R1 and R2 for an elliptic bulge, with the two radii relating to the bulge radius in the principle directions. For large apertures, the membrane theory can be used to compute stresses, strains and pressures, as will be discussed in the next section.
Membrane theory
σ σ p 1 + 2 = (2.6) R1 R2 t where σ1 and σ2 are the principle stresses on the sheet surface, R1 and R2 the radii, perpendicular to each other, p the pressure applied to the sheet and t the thickness of the sheet. In the pure biaxial case where the bulge is a perfect bowl, R1 = R2 = Rd and σ = σ1 = σ2, so the equations can be simplied to
pR σ = d (2.7) 2td with td the thickness at the top of the dome. The eective stress can be written as
8 CHAPTER 2. LITERATURE SURVEY
µ ¶ p R σ = d + 1 (2.8) 2 td
In the above relations the dome radius Rd is dened as
2 2 ((dc/2) + Rc) + hd − 2Rchd Rd = (2.9) 2hd and the thickness at the top of the dome as
µ ¶2 1 (2.10) td = t0 2 1 + (2hd/dc) with dc, Rc and hd dened as shown in gure 2.2. This model includes the plastic deformations by using a correction for hardening, although in a simpli- ed way. The hardening component can be adjusted by replacing the exponent 2 in equation 2.10 by (2 − m), where m is the hardening power law exponent. [21] Slota et al [51] state that a die aperture of at least one hundred times the thick- ness of the sheet is needed to be able to neglect bending inuences. Especially for determining reliable strain elds this is important, because the strains vary with the thickness of the sheet due to bending.
Advantages and Disadvantages
An advantage of the bulge test is the absence of contact (and therefore fric- tion) in the area of interest, which makes the analytical solution less complex. There are no geometrical constraints due to the tooling or the geometry of the specimen.1 Some disadvantages of the bulge test include the large height dierence be- tween the deformed and undeformed specimen, making it dicult to use lens systems for online and in-situ measurements (e.g. imaging correlation analy- sis). Moreover, only strains in the ε1 > 0 and ε2 > 0 region can be determined, as the sheet needs to be clamped over the whole outer region to prevent the pressurizing air or uid from escaping. The high pressure also leads to uncontrollable neck and crack propagation, be- cause of the force controlled nature of the experiment. Necking and fracture might occur in a split second, with no tools available to measure the phe- nomenon. The high pressure needed in a miniature set-up might even prove to be a problem to reach in a conventional set-up without the use of a large hydraulic system or polymer as pressure body. [37]
1In a bulge test the thickness of the sheet varies, with the thinnest part of the sheet forming in the centre. This can be considered a geometrical constraint, as it forces fracture at this point.
9 CHAPTER 2. LITERATURE SURVEY
Further literature
The most important properties described extensively in literature including out-of-plane bending, especially for small apertures [15, 51], uncertainty of the exact shape and thickness of the bulge [15, 49] and uncertainty of the moment of fracture [50]. A big disadvantage is found when changing strain distributions, as this leads to building new die shapes for every wanted distribution [3, 9, 50].
10 CHAPTER 2. LITERATURE SURVEY
2.3 Punch test
A second, somewhat similar approach, is the use of a punch to deform sheet metal under many strain paths, including biaxial tension. Several standardized tests are available (e.g Keeler, Nakazima and Hasek tests) as described by Banabic [3]. Although all these tests are used to determine the same material properties, there are several dierences. The biggest disadvantage of the punch test is the presence of contact, as this both gives rise to geometrical constraints and adds friction to the problem. An advantage of the punch test is its ability to undergo various strain paths, all of them up to necking and fracture. Many ideas to achieve dierent strain paths have been proposed and will be briey discussed. Changing the strain path during a test is practically impossible for a punch test, as tooling geometry and specimen are xed in most cases.
Punch set-ups
The Keeler test uses punches of dierent radii to vary the strain path of the tested sheet metal specimen, introducing dierent strain paths due to geome- try and friction variations. The specimens are the same for every test, which makes the test easy to prepare. The dierent punch shapes, as shown in gure 2.3, make the test more time consuming if a larger part of the FLC is to be determined. The test can only determine the positive part of the FLC, ε1 > 0 and ε2 > 0. An alternative where the same specimen, but only one type of punch are used, is the Hecker test. In this case the amount or type of lubricant is varied, which gives dierent strain paths. For this test again only the positive part of the FLC can be found. [3]
Figure 2.3: Punch shapes as used in the Keeler test [3]
In the industry the Nakazima test (or sometimes the similar Hasek test) is most often used to determine material properties. For both tests a simple hemispherical punch and a circular die are used, while the shape of the specimen determines the strain path. Especially for the Nakazima test both tooling and specimen are relatively simple. The Nakazima specimen, as shown in gure
2.4 on the left side, only dier in width W . Strain paths for ε1 > 0 can be
11 CHAPTER 2. LITERATURE SURVEY found with this test. The main disadvantages apart from those due to friction are possible wrinkling and measurement errors caused by the curvature of the punch. Specimens proposed by Hasek (gure 2.4) can be used if wrinkling is a problem. The advantages and disadvantages are the same as for the Nakazima test. The advantage of avoiding wrinkling is countered by the extra work needed to manufacture the specimen.
Figure 2.4: Specimen geometries for Nakazima and Hasek punch tests [3]
Tooling inuence
The test methods as shown up to here have dierent regimes they can be used for, as shown in gure 2.5. This clearly shows the limits of the uniaxial tension test, the bulge test and Keelers test. It also shows how dierent tests can lead to dierent results, mainly because of dierences in tooling and deformations because of that.
Figure 2.5: FLCs established using dierent testing methods: 1. Hasek; 2. Nakaz- ima; 3. Uniaxial tension; 4. Keeler; 5. Hydraulic bulge [3]
12 CHAPTER 2. LITERATURE SURVEY
Sheet thickness
Another observation is sheet thickness inuencing the results in all set-ups, caused by dierences in bending stresses. Both Raghavan and Banabic show a rising FLC for thicker sheets, showing how a thicker sheet, with more material to ow, is leading to higher forming limits. This is observed for both in-plane and out-of-plane testing and can therefore not be described as a pure bending eect. More likely is the presence of an edge eect, leading to stiening of the surface of the sheet and leading to earlier fracture in thin sheets as there is less material in the centre to distribute the stresses introduced by deformation. [3, 47]
Overestimation of fracture strains
A second disadvantage of the punch test is the overestimation of acceptable strains, mainly because of tooling introduced geometrical constraints on necking behaviour. [3, 45, 47] An eect found in punch tests, mainly due to friction, is localizing of the neck away from the centre of the specimen. A second problem is that the punch test does not allow diuse necking of the material, leading to larger formability [38]. This happens as the material on top of the punch sticks to the punch, resulting in lower strains. Depending on the shape of the punch, the test method and the lubrication this determines where the material fails and under what strain path. For most punch tests this behaviour is unwanted, but tests like Keelers are partly based on this principle. A test with a hemispherical punch and varying lubrication states can be used to determine failure from pure biaxial strain paths (in the centre) to almost pure stretching. [3, 8]
Measuring
The punch test can be used with an image correlation system, as the top area is free from obstacles, but the strain eld can only be measured at the outer layer of the sheet. As the strain eld will not be homogeneous through the thickness of the sheet and therefore the measured strains might not represent the actual strain eld. The eect of friction on the surface of the punch might also introduce an error that has to be compensated for when wanting to measure the real material properties instead of the properties under the given set of restrictions.
13 CHAPTER 2. LITERATURE SURVEY
2.4 Marciniak test
An alternative punch test was proposed by Marciniak and Kuczy«ski [39], re- sulting from their theory on loss of material stability under biaxial tension, which manifests itself by a groove running perpendicular to the largest princi- ple stress. They showed how their hypothesis about local strains concentrating in this groove could experimentally be veried with a set-up as shown in gure 2.6. The idea behind the Marciniak test is it simply converting a vertical force into a biaxial force in the horizontal plane. This is done by a at punch deforming a test specimen indirectly via a washer sheet with a central hole. The hole expands radially as the punch moves in and because of friction the tested sheet of metal expands with the washer. The radial friction forces in the contact region between washer and sheet also prevent the sheet from fracturing near the rounded edge of the punch, with the largest strains found in the at central part of the specimen. The central part is now uniformly balanced, biaxially loaded, with no contact in the area, allowing failure to occur anywhere in this region.
History
In 1977 Tadros and Mellor [53] expanded the theory of Marciniak and Kucz«ski by adding dierent tooling shapes. They proposed using elliptical shaped tool- ing, resulting in various biaxial loads from pure biaxial to aspect ratios of 1:7. They give results for several materials, for some the test set-up works, for oth- ers like brass 70/30 it does not. Further research by Mellor showed dierent damage behaviour up to fracture for brass. [46, 47]
Figure 2.6: Schematic diagram of the Marciniak test tooling set-up for in-plane test- ing of sheet metal [47]
As dierent punch geometries are a costly method for testing, other options to achieve dierent strain paths have been investigated. One method in particular seems to have potential and is described by Raghavan [47] as a simple technique to generate in-plane forming limits. His proposal, based on earlier work by Gronostajski and Dolny [20], diers from the others by the use of dierent
14 CHAPTER 2. LITERATURE SURVEY specimen and washer geometries. With this combination, compared to earlier methods, a wide range of strain paths can be prescribed.
Strain paths
The Marciniak set-up has been used to study the role of material defects under balanced biaxial stretching conditions, but the Raghavan proposal makes it useable for failure under dierent strain paths. Any strain path from uniaxial to balanced biaxial can be achieved with the right washer and sheet geometry. The used sheet and washer geometries are shown in gure 2.7 and divided in several types, depending on strainpaths that can be generated with them. [47]
Figure 2.7: Typical specimen (top) and washer (bottom) congurations used for drawing and stretching strain states in the in-plane Marciniak test fol- lowing from the Raghavan proposal [47]
In tests with dierent types of steel and aluminium both the elliptical tool- ing suggested by Tadros and Mellor, and the the varying washer geometries suggested by Raghavan were capable of reaching strains of up to 40 %. The largest dierence is the fact that Raghavans method can go into negative minor strain paths, i.e. in his tests he spans a range from -25 % to 40 %, while the elliptical punch only reaches positive strains. The type I Raghavan geometry spans minor strains from -25 % to -10 % for determining forming limits in the draw region, the type II geometry can be used for determining the plane strain region, with minor strains from -10 % to 10 %. Geometries III and IV give strain paths in the stretching region, with the latter equal to the classic biaxial balanced Marciniak test. Both Tadros and Mellor, and Raghavan found positive minor strains from 15% to 40% with these geometries. [47, 53]
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Tooling geometry constraints
The mentioned articles not only show high strains can be reached, they even deform up to fracture and succeed in that. In the classic Marciniak biaxial balanced case, the type IV geometry, the central part of the specimen is under uniformly biaxial tension and failure can therefore initiate anywhere in the central region. This results in several nearly biaxially loaded necking areas. With all other geometries, both for Raghavan and the elliptical tooling, the strain distribution is not perfectly uniform, resulting in failure near the centre of the specimen. This again is considered a geometrical constraint, although the observed fracture paths suggest at least some defect sensitivity, as the fracture paths vary between similar tests. [47]
Further literature
Some other properties of the Marciniak test discussed in literature include the ability to see the inuence of anisotropy (r-value) [2, 47] and the inuence of sheet thickness [47, 53]. Also the simple set-up that can be build on a conven- tional tensile tester with only the specimen geometry to vary for dierent strain paths [14, 47]. set-ups in several sizes have been used ranging from diameter of 75mm [53] to large enough plates to cut out tensile specimen for uniaxial test- ing [14]. Strain measurements can be done with an image correlation system, which is easy because of the at nature of the area of interest [14].
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2.5 In-plane loading with cruciform geometry
The set-ups shown earlier mostly test the sheet metal out-of-plane, introducing bending stresses. As this is undesired when producing "clean" material data to determine the properties of a material, in-plane alternatives have been studied for a long time to replace the out-of-plane set-ups.
Figure 2.8: Flat cruciform shape with necking widths for the arms (1) and the centre (2) shown
A lot of research is focusing on cruciform shaped specimen to overcome the out-of-plane problem. The basic idea of a cruciform specimen is based on a standard tensile test, but with a second direction of loading added. The four arms of a cruciform specimen can be given a displacement, thus introducing tensile forces in two directions perpendicular to each other in the centre of the specimen, as shown in gure 2.8.
History
The idea of using at, cruciform shaped specimen has been researched since the sixties, by Shiratori and Ikegami (1967), Hayhurst (1973), Kelly (1976), Makinde (1989) and several others [11, 30]. The methods described by them full the requirements as mentioned in the introduction, by generating an ho- mogeneous strain distribution in the thickness direction, yielding in the central part of the specimen and being capable of describing dierent strain paths. Not all methods are useful for reaching necking or fracture conditions though, for dierent reasons. Several authors studied the possibilities of cruciform specimen for determining yield loci or hardening, which has the advantage of only going into the yield region and no further. Promising results for determining yield-locus where found by Müller and Pöhlandt [42] by using a specimen as shown in gure 2.9(a). For this geometry, high stress localization is found near the notches, but for deformation up to yield the geometry is useable. A similar goal, but with a dierent geometry, was achieved by Hoferlin et al.
17 CHAPTER 2. LITERATURE SURVEY
[26], who used a square sheet with multiple small clamps to prevent introducing an in-plane bending moment. The method was used to experimentally deter- mine the yield locus of ve dierent materials and comparing it with nite element simulations. Several authors used cruciforms related to the geometry as shown in gure 2.9(b). [19, 31, 32, 57] These all have slits in the direction parallel to the tensile forces in common, with the idea behind it being the avoidance of bending forces in the plane of interest. Kuwabara et al. [31] claims the slits to make the strain distribution in the biaxially loaded zone almost uniform. Some of these studies use curved arms, others use straight arms, some even use both to compare. Kuwabara [31, 32] tested low-carbon sheet metal and determined experimen- tally the plastic work for a strain range up to ε < 0.03 in the biaxially loaded zone under load ratios of 4:2 and 4:4. In a second work they determine the yield surface, with the use of an abrupt strain path change. A similar specimen is used by Wu et al. [57] for testing a biaxial tensile set-up capable of realiz- ing complex loading paths. No local strain measurements where done for this set-up though, making it dicult to compare the useability. An optimization of the Kuwabara specimen was performed by Gozzi et al. [19], in order to study the mechanical behaviour of extra high strength steel. They had a problem with reaching the desired amount stress, as failure occurred before reaching that stress in the biaxially loaded region. A geometry as shown in gure 2.9 b) was used and optimized, where the notches where changed to keep the stress in the corners low enough to prevent failure there. Dierent lengths of slits where found to be preferable in some situations. A dierent geometry is proposed by Yu et al. [58] in a study on forming limits for sheets under complex strain paths. Using a nite element model to optimize, they come up with a cruciform shape as shown in gure 2.9(c). The centre of the cruciform has been thinned, with a cross-shape thinned area surrounding a bowl shaped area that is even further thinned. The general idea behind this shape is obtaining the most uniform stress distribution in the central region. According to the authors, complex strain paths can be achieved by adjusting the velocity ratios imposed on the specimen arms. Demmerle and Boehler [11] describe several cruciform geometries in their arti- cle, ranging from uniform thickness specimen with trapezoidal arms to plates clamped by three or more limbs from each side. The geometry they investigate most is the one proposed by Kelly, as shown in gure 2.9(d). From their they look into two alternatives based on the original specimen of Kelly, namely 2.9(e) and 2.9(f) where the centre area is thinned in a square form or round form re- spectively. They nd a uniform stress distribution in the centre, but with stress localizations in the corners. A last design proposed by them is shown in gure 2.10 and is found the best as it has the lowest stress localization out of the centre. The design was found to expensive to realize though and never tried. A problem described by Demmerle and Boehler [11] is loading of anisotropic materials, which will result in a distortion of the loading axes. Some solutions
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Figure 2.9: Several cruciform geometries as used in dierent studies; a) Müller and Pöhlandt [42]; b) Kuwabara and Gozzi et al. [19, 32] ; c) Yu et al. [58] ; d) Kelly [27] ; e) Square modied Kelly [11]; f) Round modied Kelly [11]
19 CHAPTER 2. LITERATURE SURVEY
Figure 2.10: Optimal design of a biaxial specimen by Demmerle [11] for this problem that are proposed in literature are making slots in the arms [19, 31, 57] or giving the tensile set-up that is used more degrees of freedom [7, 11]. No studies where found on using the cruciform geometry up to fracture under biaxial loading, other then for composites [52].
Alternative approaches
An alternative to weakening the centre part of the cruciform is strengthening the arms. This method is used in composite testing Smits et al. [52], where it is relatively easy to shape the test specimen in the desired form. For sheet metals no literature was found on strengthening the arms, although some al- ternative solutions to be though of are to glue extra material to the arms or to change material properties in the arms, for instance by changing the mate- rials microstructure by case- or surface hardening (e.g. carburizing, nitriding, boriding or titanium-carbon diusion).
Manufacturing
The main problem mentioned for a thickness reduced cruciform specimen is the change in material properties due to manufacturing and the change in sheet properties due the removal of the outer layer. The former problem can only be minimized, by studying removal methods that introduce only minimal damage to the original material, including Electro Discharge Machining as is used in other studies and said to have little impact on the tested material. [56]
Test set-ups
The set-up as proposed by Kuwabara et al. [31] and also by Smits et al. [52] uses hydraulics to drive the tensile tests. This has the advantage of being able to directly connect the two opposing hydraulic cylinders via to a common hydraulic reservoir and thus keeping the pressures (and therefore forces) exactly the same. With a servo controlled system it is possible to change the strain path.
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A second approach found in literature is the use of a mechanism, xed in a conventional tensile testing machine [26, 43]. set-ups like this can be divided in two main groups - vertical and horizontal loading - as is shown in gure 2.11. All mechanisms need to be adjusted for diering strain paths, which make a path change during an experiment impossible.
Figure 2.11: Possible mechanisms in conventional tensile stage. Top pictures: verti- cal under stage; Bottom pictures: horizontal under compression [43]
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2.6 Summary
The four methods discussed in this chapter, the bulge test, the hemispherical punch test, the in-plane cruciform tensile test and the Marcianiak test are the most practical tests for creating biaxial stress elds. Each of them has its advantages and disadvantages that will be analyzed in this report with more details using both numerical and experimental tools. For the bulge test, the biggest advantage is its simplicity and the existence of an analytical model. The useability of the analytical model and the typical dimensions for a miniaturized set-up is analyzed. In the punch test, the most important property inuencing the results is friction. Therefore the inuence of friction when miniaturizing is analyzed. The Marciniak test, advertised as having perfect in-plane, biaxial loading will be both numerically and experi- mentally veried. The main properties that will be investigated are friction, tooling geometry and tooling forces, for nding the possibilities for a miniatur- ized set-up. The last set-up to be analyzed is the in-plane cruciform set-up, for which the main challenge will be designing and building the set-up and producing the specimens. Combining the results of the four methods, the most suitable method for biaxial testing of sheet metal to fracture will be suggested, with recommendations for possible further investigation.
22 Chapter 3
Numerical methodology
The literature survey describes several experimental set-ups that are used for biaxial testing of sheet metal in the industry. Each of these set-ups has its advantages and disadvantages, some of which can be investigated with the help of a numerical model. The models used in this study are presented here, after in the rst section the used material model is described, as this is the same for all numerical models.
3.1 Material model
All numerical and experimental work is done with IF-steel, which is provided by Corus RD&T. In MSC.Marc the elasto-plastic material model is used to describe the IF-steel, which is an isotropic model, using Young's Modulus (E), Poisson's Ratio (ν) and a plasticity criterium as input parameters.
Table 3.1: Material properties for MSC.Marc isotropic elasto-plastic material model
Property Value Young's Modulus [GPa] 45 Poisson's Ratio [-] 0.29 Initial Von Mises Yield Stress [MPa] 130
Plasticity is modelled with the piecewise linear method, using a table of equiv- alent plastic strain and Von Mises stress. The used values for these properties are given in table 3.1, with the plasticity curve that is given in gure 3.1. The stress in this gure is determined in a standard tensile test, which per de- nition is equal to σVM (the Von Mises stress) as used by MSC. Marc. From the obtained stress-strain distribution only the part up to necking is loaded in MSC. Marc, as the elastic-plastic material model is dened as such.
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Figure 3.1: Plasticity curve as used in the MSC.Marc material model
Necking, damage and localization
When interpreting the numerical results, it is important to understand how plasticity is implemented, as this can otherwise lead to wrong assumptions. The plasticity curve as used here only describes the stress evolution up to necking of the material. The reason for this is that local strains become much higher and result in necking, which is not described by the elasto-plastic material model. Furthermore the numerical model cannot accurately describe localiza- tion, which is found when the material starts to neck and fail. To avoid this problem a failure criterium or local damage model is needed, which is beyond the scope of this project as such a damage model can only be determined with the obtained experimental data from the biaxial set-ups.
3.2 Bulge test
The model for the bulge test is based on the analytical model given by Gutscher [21] as shown in gure 2.2 in the literature survey. The most important prop- erties to study are the resulting stress- and strain elds and possibilities to miniaturize.
The FE model
The model was build using straight axisymmetric thick shell elements of type 1, which are suitable for large displacements and large deformations. Shell elements are chosen for their suitable computational behaviour and bending incorporation. The degrees of freedom (axial, radial and right hand rotation) are sucient to describe the problem, and a pressure boundary condition can be used as well.
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Figure 3.2: FE Model of the bulge test (expanded view is used for the shell elements) (MSC.MARC Mentat)
The element only has one integration point for stiness and two integration points for mass and pressure determination, which means small time steps are necessary when describing large plastic deformations. Stresses over the thickness of the element are integrated with Simpson's rule, over 5 layers and 11 integration points (default settings that are suitable for complex plastic deformation). To take into account the thickness variation during the bulge test, the updated lagrange method is used.
Table 3.2: Properties of the bulge test model
Property Value
Sheet thickness t0 [mm] 0.7 Cavity diameter dc [mm] 20 to 150 Clamp rounding Rd [mm] 5.25
To determine the needed pressure and maximum dome height at the point of necking, it is assumed that failure occurs when the Von Mises strains reach the maximum of 380MP a as determined in a uniaxial tensile test.
3.3 Punch test
The punch test set-up that will be studied is the hemispherical punch, as this is the most commonly used punch type in industry, e.g. in Nakazima tests. The main goal of the analysis will be to study the eect of friction on deformation, as
25 CHAPTER 3. NUMERICAL METHODOLOGY friction is believed to inuence the measured material parameters. The model is build up from solid elements, as they are more suitable for visualizing stress- and strain gradients. The set-up is modelled axisymmetric to keep the number of elements as low as possible.
Figure 3.3: FE Model of the punch test (MSC.MARC Mentat)
FE model
The axisymmetric model has four bodies in it, namely the sheet that is modelled using solid axisymmetric elements of type 10, a four node quadrilateral element with bilinear interpolation1, the punch and the two clamps that are modelled as rigid body curves, see gure 3.3.
Table 3.3: Properties of the computational punch model
Property Value Punch diameter [mm] 50 Inner diameter die [mm] 56 Radius die edge [mm] 2 Clamp force [kN] 200 Number of elements 2000 Element Type Axisymmetric solid (quad), bilinear (Type 10)
To study the eect of friction the coulomb model is used, which is used in MSC.Marc to describe any friction except for shear friction. The coecient of friction, Mps will be varied between 0 and 1. (For more information on friction and friction coecients, see Giancoli [18]). To investigate the miniaturization problem the same model is used with the geometry scaled down to the dierent punch sizes.
1Higher order elements can not be eectively used, as in contact MSC.MARC only uses the bilinear approximation
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3.4 Marciniak test
Similar to the bulge and punch models, the Marciniak FE model is described here, followed by the most important properties that will be studied. The model can be used to gain more insight in the general workings of the set-up and eects of several design parameters, including punch edge geometry, punch size and washer geometry.
FE model
The model was build using element type 10, axisymmetric solids, which were also used in the punch test analysis2. Tests showed 3D shell element can be used as an alternative, but for faster computation the axisymmetric solids approach was chosen. In table 3.4 the used set-up and geometries are described. The model describes the experimental set-up that is available at Corus RD&T and will also be used for experimental work.
Figure 3.4: FE Model of the Marciniak test (MSC.MARC Mentat)
The simulations to investigate the inuence of changing parameters are per- formed with dimensions of the experimental set-up that is available at Corus RD&T. The inuence of friction is studied by varying friction coecients be- tween the washer and sheet (Mws), and between washer and punch (Mwp) are 22D axisymmetric shell elements are available and can be used to describe the model geometry. These shell elements do not work well in contact though when using more then one deformable body
27 CHAPTER 3. NUMERICAL METHODOLOGY
Table 3.4: Dimensions of the Marciniak test set-up for the FE Model
Property Value Punch diameter [mm] 50 Punch edge radius [mm] 10 Inner diameter die [mm] 53.78 Radius die edge [mm] 3 Clamp force [kN] 200 Number of elements 2000 Element Type Axisymm. solid (quad), bilinear (Type 10) varied between 0 and 1. The friction is modelled using the Coulomb Friction model in MSC. Marc.
3.5 In-plane loading with cruciform geometry
The cruciform model is computed with 3D elements to describe the thickness reduced geometry, with only one-eight of the cruciform being modelled when possible.
Table 3.5: Dimensions and properties of the cruciform model [54]
Property Value Width (w) [mm] 10.0
Radius of rounding (Rw) [mm] 3.0 Radius of bowl (Rreduced) [mm] 6.5 Thickness sheet [mm] 0.7 Number of elements (one-eight) 5000 Element Type 3D solid (hex), trilinear (Type 7)
Boundary conditions in the cruciform test are symmetry planes and displace- ments of the arms only, as they are there is no contact in the set-up outside of the clamping area. To compare the numerical results with the experimental re- sults, the numerical strain elds have been calculated for the same strain types as can be found experimentally. This is total strain, major strain and minor strain. The resulting stress eld is used to nd where the maximum tensile stress is reached, which is assumed to be the starting point for necking. Parameters that have been varied to optimize the numerical model are the thickness reduction and the radius between the arms. All models have a bowl shaped thickness reduction, where the radius in the plane of the sheet is kept equal at all times.
28 Chapter 4
Experimental methodology
To verify the numerical results from the tests as mentioned in the previous section, several experiments have been carried out. The experimental work can be divided in two main parts, the Marciniak test and the cruciform geometry.
4.1 Marciniak test
To compare the numerical results with experimental results, IF steel specimens were tested on a set-up that was made available by Corus RD&T. The set-up used is an existing tooling set, normally used for testing deep drawing prop- erties. This means the used geometry is mostly pre-determined by the tooling available. The properties that were varied were washer hole size and the amount of friction between washer and sheet.
4.1.1 In-plane testing with the Marciniak set-up
The test set-up is a hydraulic punch with a 100mm die attached. The punch measures 50mm in diameter and has a rounding of 10mm. The die used to clamp the specimen has an outer diameter of 100mm and an inner diameter of 53.78mm. The edge of the die is rounded with a radius of 4.5mm, which is considered large enough not to cause fracture on the die. The clamping force used in the tests is set to 300kN, so no material can ow out of the die. The speed of the punch can be adjusted for each test, but is normally set to 10mm/min. To make clear photos for strain measurements, some of the tests are carried out with stops and the last test is carried out at a slow punch speed without stops to make photos. Both a simple digital photo camera and an advanced photo camera set-up as used by Corus RD&T where used. Several specimen have been painted with a high contrast pattern for use with the Aramis system, using standard black and white paint. This system works by following the distortion of a pattern that is sprayed on top of the specimen of interest, comparing each
29 CHAPTER 4. EXPERIMENTAL METHODOLOGY picture that is taken to the previous. Lightening for the test was provided by ambient light and a clip-on diuse LED light. Furthermore a pattern of regular dots was applied to each of the specimens for use with the Argus strain measurement system.
4.1.2 Specimen manufacturing
Manufacturing of specimen for the Marciniak test is done by stamping and wire EDM. The 100mm sheets to be tested are stamped from a steel strip at the Corus workshop. The 100mm washer sheets are cut by wire EDM, after which the hole in the centre is removed with the same process. As the width of the specimen is much larger then the EDM or stamping aected zone, no inuence of the manufacturing process is found in the test specimen.
A set of washer was produced with holes of 5mm, 7.5mm, 10mm, 10.5mm, 11mm, 11.5mm and 12mm, as 11mm was numerically found to be the best choice.
4.2 In-plane loading with cruciform geometry
The cruciform geometry will be experimentally tested, to verify the numerical results and to study possible diculties that occur in an experimental environ- ment. The experimental set-up that was used to carry out the biaxial tests was designed rst, and therefore the design of this set-up is explained rst.
4.2.1 Design of a test set-up
A set-up to test the cruciform specimen at the Eindhoven University of Technol- ogy was designed keeping several guidelines in mind. First of all the set-up had to go all the way to fracture, resulting in a total elongation of approximately 4mm. The maximum forces to be expected are under 2kN in two directions. The set-up has to pull the cruciform with equal force from all sides, with all the forces acting in one plane. The possible use of the Aramis strain measuring equipment is preferred, and so is the possibility to change the strain path for a test. Table 4.1: Properties for the experimental set-up
Property Size Displacement [mm] 4-6 Force [kN] 2-5 Other Strain path changeable Strains measurable
Several designs have been discussed, including the use of a Kammrath und Weiss tensile stage with additional parts to pull in the perpendicular direction.
30 CHAPTER 4. EXPERIMENTAL METHODOLOGY
The compactness of such a set-up makes this impossible to achieve in the plane of the force acting on the cruciform though. Other designs ended up with the similar problems, leaving a mechanism to be placed in a standard tensile machine as the best option. Several of these mechanisms have been analyzed, with the cruciform either horizontal or vertical position in relation to the mechanism. The vertical solu- tion has the obvious advantage that the deformation of the cruciform is easily followed, as it is clearly visible from the sides, with no obstructions by either the mechanism itself of the tensile machine. Practical design of a set-up with the cruciform clamped vertically was found to be a big challenge, as a simple mechanism cannot elongate the cruciform in four directions at the same time. Slider mechanisms or double hinges would be necessary, which lead to friction problems that can make the test rig instable. [17]
Figure 4.1: Principle of the biaxial tester and non-linear movement of the joint
The nal design was therefore made with the cruciform in a horizontal position, where a compression movement of the Zwick 1474 tensile machine is used to push four arms outwards and thus stretching the cruciform biaxially. The set-up was build at the university workshop, with the use of standard parts mostly. As a result of the use of ball joints in the mechanism, the vertical movement of the tensile testing equipment is not perfectly transferred in horizontal direction. As shown in gure 4.1, the joint directly responsible for the horizontal movement of the test specimen is following an arch-like path. For small displacements the displacement is approximately linear, which means linear displacements can be accurately described. The photo of the set-up (gure 4.2) shows the total set-up. This set-up can be placed in a conventional tensile testing machine, placing the bottom cylinder (2) in a centre ring on the tensile testing machine and screwing a standard cylindrical connector to the top cylinder (3) to attach the set-up to the moving part of the machinery. When the tensile tester then moves in the direction of the blue arrows, the joints of the arms move out in the direction of the red arrows. In the set-up available at the Technical University of Eindhoven the bottom cylinder is xed, with the top cylinder being pressed down by the tensile tester.
31 CHAPTER 4. EXPERIMENTAL METHODOLOGY
Figure 4.2: Experimental set-up for testing cruciform specimen
32 CHAPTER 4. EXPERIMENTAL METHODOLOGY
The eight 125mm arms form four mechanisms, with three joints each to convert the vertical displacement into a horizontal displacement. To change the load distribution, the arm length can be changed by using longer or shorter arms, so dierent strain paths can be tested. Small adjustments can be made by elongating the arms itself. The arms have ball joints as shown in gure 4.2 (5), that can freely rotate in all directions to prevent a clamped cruciform from being suppressed in more directions then there are degrees of freedom. The clamping mechanism has been made by pressing a clamp onto the clamping area of the cruciform, as shown in gure 4.2 (6) and (7). The four clamps can move in vertical direction and have a thread that is used to pre-stress the clamped cruciform. This makes it possible to adjust the position of the cruciform and get it aligned. The test set-up is used in a standard tensile machine as stated, which gives two problems. The rst is the fact that the clamped specimen, although loaded in-plane, will move in vertical direction as the joints that are connected to the clamps move in vertical direction. The second problem is the fact that the cruciform is clamped horizontally, with the tensile machine present above and beneath, which makes it more dicult to have a clear view of the cruciform. The rst problem is easily solved, as preliminary tests showed the camera is able to keep focus over the distance the cruciform moves. The other problem was solved by adding a cavity in the lower cylinder of the biaxial test set-up (3) where a mirror is placed under 45◦. This mirror gives a clear view (4) on the deformed specimen, with the possibility to add a camera set-up for online strain measurements in front of the tensile tester. The camera can be pointed at the mirror in the bottom cylinder through a hole in the cylinder, as shown in gure 4.2.
Misalignment
Misalignment of the forces in a cruciform set-up can be introduced by clamping the specimen in under an angle in one of the clamps. In gure 4.3 stress distributions are shown in cruciform specimen under dierent misalignments, to analyze the resulting behaviour. The four test are all exactly the same, with the cruciform being pulled 2mm in each direction, so not reaching the maximal tensile stress. In all three misaligned cases, a movement of 1mm away from the principle direction is added. Each of these possible misalignments results in a highest stress in the centre of the cruciform, with hardly any distortion of the stress and strain elds. This observation is an important one, as it shows how the test is insensitive to misalignment due to faulty or skew clamping. The biggest inuence of misalignment as predicted by the simulations is the development of a preferred necking direction, as bands of higher stress develop. These bands develop in places where the material is likely to fail without this extra high stress as well, which makes the impact less signicant.
33 CHAPTER 4. EXPERIMENTAL METHODOLOGY
Figure 4.3: Stress distribution in a cruciform specimen and the inuence of mis- alignment in the cruciform set-up; a) Symmetric tensile test; b) One arm misaligned; c) Two arms with opposite misalignment displacement; d) Two arms with equal misalignment
4.2.2 Tests with in-plane cruciform geometry
To run a biaxial tensile test, the following steps need to be carried out. First the tensile set-up is fastened in the specially designed holder, in which the arms can be positioned in a 45◦angle. For the current set-up, the holder is xed in the exact position to achieve this. The mechanism needs to be position in the holder in the right direction, which can be checked by aligning the black marks on the holder and mechanism for both the top and bottom cylinder. With the mechanism in the holder a cruciform specimen can be fastened in the set-up. It is important to make sure the clamps are on the device before attaching the cruciform, as the cylindrical part of the clamps cannot be added with a cruciform in place. It is recommended to x the cruciform in two opposite clamps rst, while making sure the clamps are aligned. With the rst two clamps tightened, the other two clamps can be fastened, where it is important again to align the clamps before tightening. When the four clamps are fastened and checked, the set-up is ready to be placed in the tensile testing machine. To move the mechanism without deforming the clamped specimen, two steel bars are screwed to the mechanism. While attaching the bars, it is important that the bottom cylinder is kept in contact with the holder. The alignment markers have to stay in touch as well. If the bars are xed, the mechanism can
34 CHAPTER 4. EXPERIMENTAL METHODOLOGY be taken from the holder and placed in the tensile machine. To attach the mechanism to the Zwick tensile machine, a metal connector is screwed to the top cylinder. The hole in this connector has to be aligned with the mirror hole in the bottom cylinder. If the connector is in place, the mechanism can be placed on the metal centre pad in the tensile machine and the compression rig can be lowered into the connector. Positioning the compression rig low enough for the steel xation bar to be put in place requires some caution, the rig may not exert any force yet. With the mechanism fastened, the compression rig can be directed to the zero position by balancing the force to zero. To start the test, the Zwick software is used and set-up for a standard compression test.
Force displacement measurements
As the experimental set-up as mentioned above does not measure forces or displacements on the specimen itself, these have to be computed from data obtained from the Zwick tensile testing equipment. The force-displacement curves from the tensile testing equipment can be converted are obtained via the controlling software and give time, displacement and force.
As the angle of the arms of the test rig are approximately 45◦, the force and displacement of the tensile tester are simple to determine. The movement of the joint to which the clamps are xed is approximately equal for the horizontal and vertical displacement. The force on the cruciform in one direction is ap- proximately half the force measured by the tensile machine. This does not take into account any losses in the testing rig though, which cannot be determined without adding load cells on the separate arms.
Aramis strain measurements
Using the mirror set-up as described above, strain measurements are made with the Aramis system. When the biaxial tensile set-up is placed and fastened in the tensile tester, the Aramis system can be installed. The camera is pointed at the 45◦mirror and adjusted so it shows the cruciform specimen in the Aramis software. A (diuse) light source is then installed, after which the shutter time of the camera can be set. If needed the needed shutter time can be changed by adjusting the diaphragm of the camera1. After the camera is set-up and focused, the Aramis software is set-up to take a photo every second, which is the minimum time step. For slower tests this might be adjusted, a number of 50 to 100 photos are needed for the strain eld calculations.
1For shutter times close to 50Hz the background lighting inuences the photos, which makes using the diaphragm needed to be able to increase or decrease the shutter time without over- or underlightening
35 CHAPTER 4. EXPERIMENTAL METHODOLOGY
4.2.3 Specimen manufacturing and characterization
During the manufacturing process of the cruciform specimen several manufac- turing steps are used, of which some might lead to damage or errors in the geometry. The steps will therefore be explained here, and several test to inves- tigate the inuence of the main production step, EDM, are introduced.
3
Figure 4.4: Shape and size of the cruciform specimen
The rst step in manufacturing the cruciform specimen is cutting the cruciform shape, with dimensions as given in gure 4.4. The cutting is done by wire EDM, this will not have a huge inuence on the material, as the thickness of the aected material is small compared to the thickness and width of the specimen. The second step for manufacturing the cruciform specimen is die sink EDM, to thin the centre of the cruciform. EDM is used as it can achieve the wanted precision and can cut the wanted geometry, without a big distortion of the original material, according to several studies (see Appendix A). The inuence of the EDM process is characterized by several experimental tests. These include height prolometry, SEM, grain size measurements, nano indentation and tensile tests, which are briey discussed.
Surface prolometry
The EDM process copies the geometry of the electrode into the workpiece material, removing material from both the workpiece and the electrode. The nal product should be a perfect copy, which will be veried using the Sensofar Optical microscope. The microscope is used to make a height prole.
36 CHAPTER 4. EXPERIMENTAL METHODOLOGY
The settings used for the height prole are a resolution of 4 with the 5x mag- nication. A height of 1000µm is scanned to capture the cruciform and the removed bowl shape.
Scanning Electron Microscopy
SEM images of the cross-section of the as received steel sheet and the thickness reduced specimen are prepared by grinding and polishing with the Struers tar- get machine. The specimen where cut mechanically from the sheet and grinded for more then 2 mm to remove any deformed material. The results are com- pared with the results for the as received material. To make the grains visible, the cross-section was etched with the same method as the microscopy samples.
Grain size measurements
Using the Zeiss Axioplan 2 grain size measurements of thickness reduced spec- imen are compared to specimen from as received material. The preparation of the specimen is the same as for the tests with as received material. The samples are cut from as received material and then a cross-section is grinded and polished to make the surface at enough for microscopy. After the nest polishing step, the surface is etched using a 30 second nittal bath (5 % solution) and a 20 second step with Marshall's Reagent. The combination of these two attacks all the grain boundaries and makes the grains clearly visible. The recipe of the Marshall´s reagent is as follows, constituted by two parts. Part A consists of 5 ml sulfuric acid (concentrated), 8 g oxalic acid and 100 ml of water. Part B is a 30% solution hydrogen peroxide solution. Before use, mix part A and B in equal parts and use the mixed solution fresh. For better results, 1ml of hydrouoric acid per 100ml of solution can be added. Part A can be stored, the mixture cannot.
Nano indentation
Nano indentation test are carried out on thickness reduced specimen and spec- imen of the as received material. For each position 21 indentations where done over the thickness of the specimen with a Berkovitch tip, with a 50 µm spacing over the thickness and a nal displacement of 500µm into the material. The sets of measurements perpendicular to the thickness have been made 100 µm from each other. The location of the indents is shown in gure 4.5. The hard- ness that is used to compare the microstructure over the thickness of the sheet and to compare the original material with the thickness reduced material is computed by the indentor software. Preparation of the specimen was carried out with the Struers Target System.
37 CHAPTER 4. EXPERIMENTAL METHODOLOGY
Figure 4.5: Positions where nano indentation was done to determine local hardness.
Tensile tests
The tensile tests as done to characterize the eect of EDM are carried out with the same specimens as the earlier mentioned tests. The specimens for this test have been thickness reduced using EDM or wire EDM though, to study the eect of these processes.
Thickness treduced is 200µm, the transition from the clamping area is a smooth one due to rounding by the EDM process.
treduced t
6,5 12
10 4
31
Figure 4.6: Tensile bar specimen with thickness reduction for tensile testing
38 Chapter 5
Results
For the out-of-plane set-ups, the most important challenge about the three studied set-ups is the possibility to build a miniaturized version for use with microscopy. This is analyzed for the dierent set-ups, and it will play an important role in the end to give an answer to the question which test is the best choice for biaxial testing with microscopy tools. For the cruciform specimen however, miniaturization is not the most important property to analyze, as the biggest challenge here lies not in miniaturization but in deformation up to fracture with as little distortions as possible.
39 CHAPTER 5. RESULTS
5.1 Bulge test
The advantage of the bulge test is the availability of an analytical model, as was shown in the literature review. This model is valid for large set-ups, but for smaller set-ups it might not be as usable as bending will become more inuencing. For the bulge test the stress distribution in a specimen near fracture is shown in gure 5.1, as predicted by the FE model. The distribution shows only a small gradient over the thickness of the sheet in the top of the dome. For smaller set-ups however, the role of bending stresses increases, as is analyzed in the next section.
Figure 5.1: Computed Von Mises stress eld for the bulge test with an aperture diameter of 50mm close to fracture
A second disadvantage of the bulge test originating in the concept itself, is to keep top of the bulge, where fracture occurs, in focus for use with microscope techniques. This means the distance between microscope and specimen has to be kept constant by moving the microscope along with the growing bulge. A last problem is the explosive burst that is to be expected when the material fractures, resulting in both dangerous situations and deformation of the fracture zone.
5.1.1 Miniaturization
Miniaturization of a bulge set-up means the aperture which allows the sheet to deform, will be decreased in diameter. For apertures of 20 to 200mm the results are given in table 5.1. The decreasing height for a miniaturized set-up is a good sign, as for microscopy only limited height is available in most set-ups. The rising pressures on the other hand are a big challenge, as this means the set-up needs to be reinforced to cope with the forces acting on it. High pressure is dangerous under microscopes as well, especially in a SEM vacuum chamber where an sudden increase of the internal pressure will damage the microscope. The numerical computations made with MSC.Marc have been compared with
40 CHAPTER 5. RESULTS
Table 5.1: Numerical results at fracture of bulge test simulations with MSC.Marc
for varying cavity diameters for stresses close to σfracture Cavity diameter Height Pressure Normalized height1
dc [mm] hdmax [mm] Pmax [MPa] hd/dc [-] 20 6.60 30.00 0.330 25 7.85 25.00 0.314 30 9.05 21.10 0.302 40 11.60 16.40 0.290 50 13.95 13.25 0.279 60 16.55 11.25 0.276 80 21.30 8.50 0.266 100 26.30 6.90 0.263 150 38.65 4.65 0.258 200 50.75 3.50 0.254 the analytical solution, as shown in gure 5.2. The analytically determined pressure at the point of necking can be calculated by rearranging equation 2.8:
2σ Pmax = (5.1) ( Rd + 1) td
Figure 5.2: Comparison of numerical and analytical solution for bulge pressure ex- periment near fracture
The analytical model follows the numerical model for large apertures, but for small set-ups the analytical model predicts a lower pressure then is found nu- merically. The reason for this change originates from the neglection of bending stresses in the analytical model, which makes the analytical model overestimate the thickness reduction. The advantage of the bulge test having an analytical model to determine stresses therefore does not hold for a miniaturized set-up, which means numerical computing is still needed.
41 CHAPTER 5. RESULTS
For the results as shown above a miniaturized version of the bulge test small enough to t under a microscope would need to have cavity diameter with a maximum of 50mm, considering the total set-up with clamps will then easily measure 100mm in diameter. The height dierence of approximately 15mm is relatively large, especially for use with microscopes.
5.1.2 Summary
• Pressure for testing up to fracture is predicted to reach 12MP a or more, which can lead to dangerous situations in case of fracture. The explosive character of the test also can lead to alteration of the fracture surface, which is unwanted.
• The increasing pressure results in a second disadvantage, being the need for a robust set-up. Miniaturizing now leads to the need for a stronger set- up, to cope with the higher pressures, and for very small bulge diameters this results in an increased size of the test apparatus.
• For use with microscopes the area where fracture will occur has to be kept in the same plane, which is dicult with a bulge test.
• One of the biggest advantages of the bulge test normally found, is the good description of the forces and stresses by the analytical model, but for a miniaturized set-up the analytical model is found not to be correct anymore.
42 CHAPTER 5. RESULTS
5.2 Punch test
The punch test, in several forms, is the most used material characterization test in industry. The reason for this is simple, as the test can be used over a large range of strain paths and test specimens are easy to make. A disadvantage that is taken for granted though is that friction inuences the measured material data. When looking at the stress distribution as shown in gure 5.3, the punch test and the bulge test look to behave similar, but this is not entirely true. This originates in the fact that for a sheet on a punch, localization is postponed due to restriction of material ow. This is not the case in the bulge test, where the maximum allowed strains thus are found to be larger.
Figure 5.3: Numerically found stress eld for a 50mm spherical punch test at fracture
Data obtained from the punch test simulations is given in gures 5.4 where the inuence of friction is shown. In table 5.2 the location of necking relative to the centre is given in relation to friction, where higher friction results in fracture away from the centre. Experimental work conrms this problem, e.g. at Corus RD&T punch tests frequently have to be repeated several times before fracture at the centre of the contact area of punch and specimen is found. [40] This inuence of friction leads to a fundamental problem of the punch test: Friction is known to inuence the measured material data, but the friction coecient itself is often unknown. This results in measured material data that does not represent the actual material behaviour, but material behaviour in a certain set-up. This eect was also found in literature, as was shown in gure 2.5.
43 CHAPTER 5. RESULTS
Table 5.2: Properties of the miniaturization study FE model
Friction coecient Necking Distance Sheet Thickness [-] from centre [mm] in centre [mm] 0 0 0.490 0.01 0.05 0.492 0.025 0.1 0.495 0.05 0.4 0.499 0.10 1.0 0.504 0.20 3.5 0.507 0.30 5.0 0.511 0.40 8.6 0.525 0.50 10.7 0.546 0.75 15.4 0.581 1.00 16.5 0.616
Figure 5.4: The inuence of friction between punch and sheet on the stress distribu- tion in a 50mm punch near fracture
5.2.1 Miniaturization
A logical result for miniaturizing a punch set-up is the growth of the inuence of bending stresses. Therefore a normal industrial set-up with a punch diameter of 100mm is compared with a miniaturized version with a punch diameter of 30mm. In the standard set-up, as shown in gure 5.5 b), the gradient of the stresses and strains over the thickness of the element is small. The stresses on the outer layer are only 1.5 % higher then on the inner layer. The strains vary even more, measuring approximately 7 % increase from outer to inner layer. For the same region in a miniaturized set-up we nd much steeper gradients, as shown in gure 5.5 a), with variations up to 20 % for the plastic strain. Figure 5.5 clearly shows how a miniaturized punch set-up gives diculties in determining material properties, as the assumption that neglecting bending
44 CHAPTER 5. RESULTS
Figure 5.5: Comparison of large punch (D = 100mm) (a) and small punch (D = 30mm) (b) stress- and strain distributions (Von Mises) in a radius of 2.5mm round the centre at fracture; Note that the eect is exaggerated due to scaling. stresses is not inuencing the results, does not hold anymore. A useable punch size of 50mm gives strain gradients of 8 % in the failure region, which is still unwanted.
Table 5.3: Maximum punch forces at necking, obtained via FE-modelling in MSC.MARC for dierent set-up sizes
Punch diameter [mm] Punch Force [kN] 10 3.8 20 8.3 30 12.7 40 17.3 50 21.8 60 26.3 80 35.1 100 43.9
A last property determined via the FE-modelling of the punch test is the nec- essary force for a given size of the set-up. Table 5.3 shows the punch forces obtained from the MSC.MARC model. Punch forces are directly related to the size of the set-up, decreasing with miniaturization. For a set-up of 50mm forces of approximately 22kN are needed, which is still a very high force to be achieved by a small set-up. Smaller set-ups are not desirable, as bending starts playing a larger role, inuencing the results even more.
45 CHAPTER 5. RESULTS
5.2.2 Summary
• Fracture in the punch test is not similar to fracture in a free surface under biaxial tension, due to friction and bending. This results in an error in the material behaviour that is measured.
• An increase or decrease in friction results in a change in the measured data. As the inuence of friction cannot be measured, the eect of it on the measured material data is unknown.
• Miniaturizing results in an increased bending stress, due to the decrease in the punch radius. Therefore the material data measured in a miniaturized set-up is even more inuenced by external factors then in a large set-up.
• Forces needed for a miniaturized set-up with a diameter of 50mm are found to be around 22kN.
46 CHAPTER 5. RESULTS
5.3 Marciniak test
The Marciniak test makes it possible to transfer a vertical displacement in to a horizontal load, which can be used to deform a sheet biaxially in the horizontal plane. The modication proposed by Raghavan enables a sample to be loaded not only biaxially, but in multiple strain paths, as has been described in the literature study. This makes the Marciniak test an interesting set-up to use, if it can be miniaturized for use with microscope equipment.
5.3.1 Working principle of the Marciniak test
Figure 5.6: Overview of the stress-eld in the Marciniak test under loading up to necking
The numerical model is rst used to verify the claims that no stress gradient is to be found in the biaxially loaded region of the specimen, which can be seen in gure 5.6. The stress eld is found to be similar as found in literature. When look at the stress distribution away from the centre, a problem occurs. The stress (σcentre) is found to be lower then the stress on the outer radius of the punch (σedge) and for all numerically tested set-ups it was found that:
σ edge ≥ 1 (5.2) σcentre This indicates failure is expected to always occur away from the centre of the sheet. However, when looking at the test results of the set-up that was made available by Corus RD&T (see 4.1.1) three modes of failure are found. An often observed failure mode is cutting behaviour, where the edge of the washer hole cuts into the tested sheet, thereby initiating a crack and guiding it in a circular direction. A second failure mode is a typical deep drawing failure, where the sheet and washer fracture in the sidewall after stretching. The wanted failure mode is fracture in the centre under biaxial loading, which is found in some of the tests. In gure 5.7 (a) cutting, (b) deep drawing and (c) random crack mode are shown.
47 CHAPTER 5. RESULTS
Figure 5.7: Three modes of failure found when testing sheet metal with the Marciniak test; a) Cutting; b) Deep drawing; c) Random crack
The experimental results show that the set-up does result in fracture in the centre, under the right circumstances. The numerical computations and exper- imental results are found to do show similarities, as both seem to be critical for reaching fracture in the centre. Therefore the experimental results have been analyzed in more detail to nd a possible explanation. To determine whether the stresses occurring in the experimental Marciniak test reach higher values outside of the biaxially loaded region as well, the local strains were measured using a regular pattern on the undeformed sheet and comparing this to the re- sulting pattern on the tested sheet. This system, known as Argus, needs every dot to be visible in at least 3 and preferably 5 photos taken from dierent an- gles, but this was found to be an impossible task with the small test specimen. A photo and a set of microscope pictures of the sample as shown in 5.8 have been used instead to approximate the strains manually instead.
Figure 5.8: Marciniak test with regular pattern on top to determine the strain eld
By comparing the increase in distance between the undeformed pattern and the pattern after the test, local strains even can be computed as ε = (L − L0)/L0. For several fractured specimen, this results in strains of (28 +/- 2%) and (29 +/- 2%) for the respectively the centre and edge of the test specimen. This results in a measured value of approximately 1.04 for σedge/σcentre, again larger then one. The numerical model is veried with another geometry as well, as used in the article by Raghavan [47].
It seems that with the quotient of σedge/σcentre is larger then one experimentally
48 CHAPTER 5. RESULTS as well,but the specimen can still fracture in the centre. An explanation for this is that interaction between washer and sheet restricts the tested sheet from localizing in the contact area. This result is in agrement with experimental results, where necking is only found in the free centre of the specimen and not on the washer. The critical balance that is observed is likely to originate from this phenomenon as well, as σedge can not be a lot larger then σcentre to still get fracture in the centre.
Figure 5.9: Close-up of the random fractured sheet with necks stopping at the washer-sheet interface for the 7.5mm washer hole
An interesting observation is made on the randomly fractured sheets and shown in gure 5.9). It can be seen that several necks have formed, that all stop at the point where the washer is in contact with the sheet. This leads to the hypothesis that the deformation away from the centre can indeed be larger then in the centre, but due to restriction of localization the sheet cannot fail in the edge and therefore eventually fails in the centre. From the above it can be concluded that the restriction of localization is a key factor in the Marciniak test. It is therefore important to get a better understanding how several design parameters inuence the stresses. This infor- mation can then be used when miniaturizing. The numerical model is found to be useable for this, as the simulations seem to predict the (absolute) very well. The point where localization starts can not be predicted yet, and therefore the location of the nal crack can not be predicted.
5.3.2 Numerical - experimental study of the Marciniak test
For better understanding of possibilities to optimize the Marciniak test for miniaturizing, the important design parameters will be analyzed numerically. The parameters that are studied are the friction between punch and washer
(Mpw), the friction between washer and sheet (Mws), the punch edge radius, the washer hole radius, inner die radius and die edge radius. These numerical computations will be complemented with a set of experimental tests, to verify the numerical results. These test have been carried out with a set-up available at the Corus RD&T. The results are shown in table 5.4. In table 5.4 the washer hole radius is the size of the hole in the washer, punch
49 CHAPTER 5. RESULTS
Table 5.4: Marciak Tests at Corus - Results
Nr. Washer hole Punchspeed Stops Roughened Load Displ. Failure mode radius [mm] [kN] [mm] 1 11 10 mm/min No No 65.0 14.9 Cutting 2 11 20 mm/min No No 64.0 14.4 Cutting 3 11 2 mm/min No No 63.4 14.0 Cutting 4 10 10 mm/min No No 65.3 15.8 Cutting 5 7.5 10 mm/min No No 73.0 16.8 Random 6 5 10 mm/min No No 78.1 12.1 Deepdraw 7 10 10 mm/min No Yes 67.4 14.8 Random 8 10 10 mm/min No Yes 66.1 15.7 Random 9 10 10 mm/min Yes Yes − − Cutting 2 10 10 1 mm/min Yes Yes 70.0 14.7 Random 11 7.5 1 mm/min No Yes 85.2 15.7 Deepdraw 12 10.5 1 mm/min No Yes 72.7 14.6 Random 13 7.5 10 mm/min No Yes 79.4 14.7 Deepdraw speed is the speed of the punch moving up. Stops were used in some test to take photos during the test, for use with the Aramis strain measurement system. Roughening was done by grinding of the washer, to increase the amount of friction between washer and sheet. All tests but the rst one were done with lubrication of the punch, as it was not possible to vary the lubrication conditions.
Friction
The eect of friction was studied by varying the friction coecient in the sheet- washer (Msw( and punch-washer contact (Mpw). It is important to know that friction coecients for unlubricated steel on steel contact go as high as 0.65, while lubricated contact friction coecients can be as low as 0.04, in the case of perfect lubrication with teon for instance [18].
Figure 5.10 shows the inuence of (Msw). The results show a rather small but clear inuence, where low friction between the sheets results in higher stresses on the outside of the tested sheet as the washer can not transfer its deformation on to the tested sheet. For high friction coecients the stresses in washer and sheet grow to become of the same size, resulting a relatively higher stress in the centre.
Figure 5.10: Inuence of friction between the washer and the sheet; Dpunch = 50mm; Rpunchedge = 10mm; Rwasherhole = 11mm; Mwp = 0.05;
50 CHAPTER 5. RESULTS
Experimentally the eect of increasing friction between washer and sheet can clearly be seen that for a washer with a hole radius of 10mm. This results in a cutting mode type of failure for low friction (test nr. 4), while for increased friction a random crack forms in the biaxially loaded region (tests nr. 7, 8 and 10). The increased friction is does not result in a huge improvement of the
σedge/σcentre value, but due to high sensitivity to this value a small change in friction can be of inuence. This delicate balance is also found when comparing tests nr. 5 and 11, where a 7.5mm washer hole was used.
The second set of models shows the inuence Mpw, as shown in gure 5.11. The friction coecient in the washer-sheet contact is set to 0.8, as this high value was found to be needed. As is expected, a low friction results in a better stress distribution, but again the inuence seems to be small when only checking the stress quotient. From gure 5.11 it can not be seen how the stress distribution changes for friction coecients higher then approximately 0.5, where the biaxial loaded region never reaches the maximum tensile stress due to failure in a deep drawing mode.
Figure 5.11: Inuence of friction between the washer and the punch; Dpunch = 50mm; Rpunchedge = 10mm; Rwasherhole = 11mm; Msw = 0.8;
The experimental results have not been used to verify the inuence of Mwp, as changing lubrication conditions in a controlled manner was not possible.
Washer hole size
The radius of the washer hole is a simple property to adjust, so for an experi- mental set-up it is important to know its inuence to take advantage of it. In gure 5.12 it can be seen how the washer hole diameter has an optimum at around 10mm. The innitesimal small washer hole equals a deep drawing test of a sheet that is twice the thickness of the tested material.
Figure 5.12: Inuence of the washer hole radius; Dpunch = 50mm; Rpunchedge = 10mm; Mwp = 0.05; Msw = 0.8;
51 CHAPTER 5. RESULTS
The experiments conrmed that for small washer holes the failure mode indeed becomes similar to a deep drawing test (e.g. test nr. 6). Tests nr. 11 and 13 show failure in a deep drawing mode as well for the 7.5mm washer hole size. This shows both friction and washer geometry can cause this failure mode, as the 7.5mm washer hole of test nr. 5 fails in a random fracture due to lower friction between washer and sheet. A larger washer hole can result in the washer sliding of the punch and thereby initiating a cutting type of failure. This is found with the 11mm washer hole in tests nr. 1, 2 and 3. The fact that for 10mm and 10.5mm washer holes random failure is possible shows once more the sensitivity of the Marciniak test.
Punch edge radius
The radius of the punch edge is important when miniaturizing, as it inuences the stress eld and can thus be used to lower the value of σedge/σcentre. To analyze the inuence, dierent radii edge radii have been numerically tested. The washer hole size was adjusted to be approximately , Rpunch −Rpunchedge ·0.9 so the washer sheet stays on top of the punch during the test.
Figure 5.13: Inuence of the punch edge radius; Dpunch = 50mm; Rwasherhole = 11mm; Mwp = 0.05; Msw = 0.8;
As is shown in gure 5.13 the maximum stress in the central region of the tested sheet cannot reach its maximum for small radii of the punch edge. This can easily be explained, as for an innitesimal small radius the punch has become a cutting tool, where the tested sheet will fail due to shear.
Other design parameters
Numerical computations have been done to study the inuence of the inner die radius and die edge radius as well, showing no inuence on the stress distribu- tion.
Conclusions
The numerical and experimental results show similar trends, which means the model can be used for a qualitative analysis. The small dierences between the
52 CHAPTER 5. RESULTS centre and edge stresses show that it is not easy to lower the stress quotient, which means optimizing the set-up for miniaturization will not be easy.
5.3.3 Minaturization
All the earlier studied properties work together in a miniaturized set-up. When scaling down the set-up in total, the curve in gure 5.14 is the result. This shows how the quotient of the stresses raises when miniaturizing, which is to be expected due to increasing bending stresses. Also it can not predict a minimum size that will still work, but it does show a clear trend.
Figure 5.14: Inuence of miniaturizing the Marciniak set-up; Rwasherhole = 11mm; Mwp = 0.05; Msw = 0.8;
To nd the best set-up, several design parameters have been analyzed and from the results obtained it is possible to construct a numerical model of a miniature Marciniak test choosing all the best values. This perfect set-up has large friction on the sheet-washer interface and low friction on the punch interface and a washer with a hole scaled to the optimal size found for a 50mm punch. Table 5.5: Geometry for the miniaturized Marciniak set-up
Property Value Punch diameter [mm] 40 Punch edge diameter [mm] 8 Washer hole size [mm] 15 Inner die radius [mm] 42 Die edge radius [mm] 3.5
The result for this optimized set-up was found to be marginally worse (σedge/σcentre = 1.015) then for the simply scaled set-up as shown in gure 5.14 (σedge/σcentre = 1.012). This again shows how optimizing the Marciniak test set-up is not as trivial as for other set-ups. With a more extensive optimization it is thought to be possible to nd a better geometry, and several other design parameters can be considered as well.
Additional washer on top
The use of an additional washer on top of the standard set-up is thought to result in better restriction of localization, and more force transferring between the washers and sheet. The expected decrease of σedge/σcentre is not found though, as the computational model results in worse behaviour then the original test. Experimentally adding an extra washer might still work due due to an
53 CHAPTER 5. RESULTS increase in restriction of localization that the washer and additional washer have on the tested sheet, but at the expense of increased forces.
Washer material
An alternative approach for changing the stress eld is to change the material of which the washer is made. Finding a suitable material might prove dicult though, as the following requirements have to be met:
• The material needs to have high friction with the sheet material.
• The material needs to have low friction in the contact with the punch.
• The material must be roughly as formable as the tested sheet, so the washer can drag the sheet outward.
• The material must be capable of reaching stresses comparable to the stresses in the tested sheet so it can transfer forces and restrict localiza- tion.
Some materials that come to mind are polymers, rubbers and other metals. The last group will result in higher forces, as another metal has to be as strong or stronger then the tested material. Polymers and rubbers can easily undergo the large deformations, but due to their generally lower strength might not able to transfer enough stress to and from the tested sheet. As friction and the area of contact play an important role in the transfer of stresses as well, optimization of a set-up with another material proves to be a complicated task, but there is room for improvement.
Other design restrictions
When the miniaturizing both the needed forces and maximum height of the set-up are important, therefore these values where computed for ve dierent sizes of the punch. When comparing the values in 5.6 with those found for the punch test, it immediately becomes clear that the forces in the Marciniak test are a lot higher. The maximum height is found to be relatively low, as 11mm is not expected to cause problems with microscopy set-ups. The forces can prove to be a problem though, as a set-up strong enough to cope with forces of approximately 70kN might be impossible to be build on a small scale. For high strength steel or thicker sheets the forces will even be higher. The computed punch forces as shown in table 5.6 were compared with the forces found in the experimental set-up. For a washer with a 11mm hole radius it was found that the computed maximum force equals 62kN, the measured force lays between 63 and 65kN. For a smaller washer of 7.5mm the computed force is 70kN, which is again a little lower then the measured punch force of 73kN.
54 CHAPTER 5. RESULTS
Table 5.6: Maximum punch forces at necking for a Marciniak test set-up;
Rwasherhole = 11mm; Mwp = 0.05; Msw = 0.8; Punch diameter [mm] Force [kN] Maximum height [mm] 25 28 6.2 35 41 8.2 50 62 11.1 75 92 15.6 100 125 20.3
An idea to decrease the needed force for the Marciniak set-up is to use a thinner washer sheet. It was numerically veried that reducing the thickness of the washer up to 50% is possible without a signicant inuence on the stress eld. This results in a force that is approximately 20% lower, depending on washer hole radius as well.
Conclusion
The results for miniaturization show that a challenge lies ahead when wanting to miniaturize the Marciniak set-up. There are several possibilities to improve the used set-up though, which leaad to opportunities to miniaturize.
5.3.4 Summary
The experimental results clearly show the Marciniak test works, although an exact prediction of the failure mode is dicult. Friction between washer and sheet is found to be important, as is the washer size. The numerical results obtained with the FE model showed lower stresses in the centre of the cup then on the edge. This is conrmed experimentally and leads to the hypothesis that necking or localization is restricted on the washer-sheet interface.
• The Marciniak test is found to experimentally work and can be qualita- tively described with a numerical model up to necking.
• The Marciniak test shows a high sensitivity to friction and tool geometry, resulting in failure modes other then fracture under biaxial tension. For miniaturization purposes optimization is needed to nd the best tooling geometry, which probably varies with material properties of the tested specimen.
• Building a Marciniak test of dimensions small enough to t under a mi- croscope is found to be possible. However, a disadvantage is the need for high forces up to 50kN or more, which ask for a clever design when miniaturizing.
55 CHAPTER 5. RESULTS
5.4 In-plane loading with cruciform geometry
Many researches studied the use of cruciform specimen over the last decades, with most of the studies focussing on yielding properties. One step further is studying necking and fracture with cruciform test specimen, which leads to the new challenge of having to raise the stresses in the centre of the cruciform so failure occurs there. A study performed by Roel Vos [54] showed the cruciform specimen can be used up to fracture, but a thickness reduction is needed in the central part to make it fail before one of the arms does. After analyzing several geometries a bowl shaped thickness reduction was suggested, which will be the geometry used for further numerical and experimental work. The study by Vos predicts a reduced thickness of 0.2mm should be sucient to reach fracture in the centre of a cruciform with 10mm width arms.
5.4.1 Optimization of the cruciform design
To verify the earlier results and analyze the resulting stress and strain elds, new simulations have been run. The resulting strain distributions for Von Mises, major and minor strains are shown in gure 5.15. The two important properties that are found from the numerical analysis are the biaxial straining of the centre and the strain band developing from the centre outwards. The rst indicates failure can occur in the centre, the shows geometrical constraining. The diagonal strain band shows how a crack can only develop in this direction, which means the orientation of the crack is not dened by the material but by the geometry. This results in the need for testing with specimen of dierent orientation to fully characterize a material that is anisotropic.
Figure 5.15: Strain elds as determined numerically showing Von Mises strains, ma- jor strains and minor strains
For an experimental set-up the clamping area of the specimen is important to prevent slip in the arms. As the width of the clamps in the biaxial testing set-up is 12mm, this is the maximum width the specimen can have. To reduce the forces on the set-up and reduce the change of clamp slip, the width of the arms for experimental testing has been reduced to 6mm. This results in the dimensions as shown in table 5.7. The thickness of the centre is adjusted to 180µm to increase the chance of fracture in the centre.
56 CHAPTER 5. RESULTS
Table 5.7: Dimensions of the cruciform specimen as shown in gure 4.4
Property Value Width (w) [mm] 6.0 Radius of rounding (Rw) [mm] 3.0 Radius of bowl (Rreduced) [mm] 4.75 Thickness sheet [mm] 0.7 Thickness of centre [µ m] 180
5.4.2 Proof of principle
Using the experimental set-up as described in section 4.2.1, a set of cruciform specimen with dimensions as given in table 5.7 was biaxially loaded up to failure. From six specimen tested, only one fractured in the centre, showing either the thickness reduction predicted by the numerical analysis is not enough or the material behaviour in the thickness reduction is not exactly the same as the global material behaviour of the IF-steel.
A new set of specimen with a thickness reduced area of 160µm was manufac- tured and tested, resulting in ve successful tests out of ve tested specimen. Fracture in all specimen occurred as predicted numerically, as is shown in gure 5.16. The crack clearly develops diagonal, and video footage show the crack starts in the exact centre of the specimen.
Figure 5.16: Cruciform specimen fractured under biaxial deformation, with and without an Aramis pattern on the surface
For comparison of the experimental results with the numerical results, the Aramis system is used to measure the local strains on the tested specimen. All ve experiments show similar strain distributions, and similar strain values at fracture, as is shown in gure 5.17. These measured strains are similar to the computed strains as shown in gure 5.15. The reason for the dierence between measured and computed strains, espe- cially in the centre of the specimens, is the fact that local strains are measured, which are not numerically computed. The problem can also be originating in the manufacturing of the cruciform specimens, which could lead to damage or alteration of the material properties. This will be investigated in the next section of this chapter. With the computed and measured strain elds being so similar, the next step is to analyze the displacements and forces acting on the cruciform. This is done by
57 CHAPTER 5. RESULTS
Figure 5.17: Strain elds measured with Aramis for three dierent cruciform speci- mens, showing Von Mises strains, major strains and minor strains measuring the displacement and force at the tensile machine, the Zwick 1474, as no load cells are present in the biaxial tester. As the set-up is only for small displacements, the displacements at the clamps can easily be determined from this as 1/2 of the total displacement. The total elongation of the cruciform in one direction therefore is equal to the displacement of the tensile machine. For the forces a similar computation can be made, as due to the 45◦angles the total vertical force is equal to the total horizontal force, which means 1/4th of the total force is acting on each clamp.
Figure 5.18: Elongation-Force diagrams for the cruciform specimens deformed up to fracture (Total elongation of the cruciform in one direction)
The stress acting on the arms of the cruciform can now be calculated from these results, by dividing the force acting on the arms by the area of the cross-section.
58 CHAPTER 5. RESULTS
This results in a stress of σ = F/A = 1200/(6 · 10−3 ∗ 0.7 · 10−3 = 286MP a, which is clearly under the maximal value of 325MP a as was found as the maximum allowed (engineering) stress for IF-steel. To determine the stresses in the biaxially loaded centre of the cruciform numerical computations are needed for which the material model has to be used. A problem that arises here is to choose the right parameters for the material model, as the material behaviour of the cruciform is not the same as the material behaviour of the sheet metal. This originates from the fact that sheet metals are not homogenous over the thickness of the sheet, as is assumed in simple material models. An eect that cannot be numerically modelled and was not experimentally studied is the inuence of size eects. As the grains in the used IF steel are approximately 10 − 15µm, a thickness of 150µm is expected to not be sensible to size eects, but for material with larger grains this might become an issue that has to be dealt with.
59 CHAPTER 5. RESULTS
5.4.3 Specimen manufacturing and characterization
As was mentioned earlier for the punch test, material data as measured in a test is not always due to deformation of the material. In case of the punch a clear eect is introduced by friction, in the case of the cruciform an eect might be found due to specimen preparation. In this section the eects of manufacturing cruciform specimen with EDM will be analyzed, while more information about EDM in general can be found in Appendix A. It is known from literature that the EDM process alters the surface of a metal, as vaporized material falls back on the surface en solidies. This so-called white layer, or recast zone, exists as a hard layer on top of the original material, with micro cracks in it due to thermal expansion. To get a better understanding of the inuences of manufacturing the cruciform specimen, several characterization techniques including surface prolometry, microscopy, grain size measurement, tensile testing and nano indentation have been used.
Surface prolometry
To get a better idea of the inuence of EDM on the surface of the material and on the microstructure, the rst step in characterizing the EDM process for manufacturing of cruciform specimen is to analyze both the surface and cross-section of the specimen. Using the Sensofar confocal microscope, height proles where made of the cru- ciform specimen, as shown in gure 5.19. This image shows the surface texture of the thickness reduced area to dier from the original material, as it looks more bumpy. A second observation is asymmetry found in the specimen, mean- ing the top and bottom sides of the cruciform are not exactly similar. This was found for all the tested cruciform specimen.
Figure 5.19: Height prole as measured with the Sensofar confocal microscope, showing top and bottom side of a cruciform specimen with measure- ments on the same scale
A last observation is the existence of variation in the thickness, although this is found to be less then 15µm for the measured specimen. This is an increase of more then 9% for a 160µm thick specimen.
60 CHAPTER 5. RESULTS
Microscopy analysis
A rst hint that reveals the EDM process might alter the material behaviour, is found by examining the surface of the thickness reduced specimen. Figure 5.20 shows how a layer of resolidied material is left behind by the process. When comparing with the as received material, it is clear that EDM does change the surface of the sheet, even though the measured global roughness is the same as for the as received material.
Figure 5.20: Microscope pictures of IF and by EDM thickness reduced IF showing resolidied material from the EDM process
Grain size measurements
In gure 5.21 the small grains on the edge and larger grains in the centre of the original sheet are visible. This shows that the assumption of a homogeneous material is not correct a representation of reality.
Figure 5.21: Cross-section of IF-steel sample after etching with Nittal / Marshall Reagent
A similar study of the cruciform sample shows slightly larger grains near the edge of the cross-section, in the same order as the grains that where originally
61 CHAPTER 5. RESULTS originated at those positions before the outer layer was removed. Due to the random distribution in the grain size and the limited resources to compute the average grain size over a large area of the cross-section, no qualitative compar- ison can be made, but comparing several smaller areas show no indication of grain size increasing or decreasing due to the EDM process.
Figure 5.22: Microscopic pictures comparing the thickness reduced area with the original sheet
Scanning Electron Microscopy
When analyzing the cross-section of a sheet metal with a SEM, the original material shows a clean surface with no irregularities while the cross-section that was subjected to the EDM process shows deposited foreign material on the outside of the sheet, as can be seen in gure 5.23. The size of these resolidied droplets range from 10 to 30 µm in width and 3 to 6 µm in height. This so- called white layer (or recast layer) has been investigated more, to nd out the inuence on the biaxial tensile tests.
Figure 5.23: Close-ups of the cross-section the edge of both the unaltered sheet and the edge after EDM processing, showing an irregular edge with resolidied material attached to the surface
62 CHAPTER 5. RESULTS
A rst observation when analyzing the cruciform specimen in SEM after frac- ture, is the existence of micro cracks in the mentioned recast layer. In gure 5.24 an example of such a crack in the recast layer is shown, resulting in a notch in the under lying steel. The existence of a micro crack on the surface can lead to fracture starting at the surface instead of failure from the middle of the sheet, as is found normally in ductile fracture. To verify fracture did not start at the surface, the crack tip of the cruciform specimen is compared to the crack tip of the Marciniak test specimen. The reason to use the Marciniak specimen as a reference is that for this specimen no external inuences are present that alter the biaxial fracture mode.
Figure 5.24: SEM image of the fractured recast layer in a biaxially tested cruciform specimen
When comparing the two crack tips, a similar type of fracture is observed. In gure 5.25 the crack tips for the Marciniak specimen and the cruciform speci- men are shown, which do not show a dierence that can lead to the conclusion that the fracture mechanisms dier. Both the Marciniak and cruciform speci- men are found to have no visual evidence of a crack starting from the surface of the sheet.
Figure 5.25: SEM images of the crack tip of biaxially fractured material in the Marciniak test and the cruciform test showing many similarities
63 CHAPTER 5. RESULTS
Tensile tests
Microscopy showed possible eects of the EDM process are to be found in the altering of the surface roughness, as the microstructure of the material is not changed. To analyze the inuence of this change on the biaxial measurements, the following section shows the results of mechanical testing of the material. The results of the tensile tests (gure 5.26) show a clear inuence of the EDM and wire EDM processes, especially for the fracture strain and hardening shows a minor decrease as well. The huge decrease of the fracture strain can be explained by the micro cracks that have formed. For biaxial loading these do not show a clear inuence, as the measured maximum stresses are found to be roughly the same for all tests. For the uniaxial tensile tests the cracks are of more inuence though, due to easier localization at the notch that was formed. This is the result of the equal thickness of the tensile bar.
450
400
350
300
250
200 True Stress [MPa] 150
100 Uniform Thickness Specimen 50 Wire EDM Thickness Reduced Sink EDM Thickness Reduced 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 True Strain [−]
Figure 5.26: Results of the tensile tests for thickness reduced IF-steel
The dierences between die sinking EDM and wire EDM shown in the stress- strain curves, showing an increase of the fracture strain for wire EDM. This shows that wire EDM and die sinking EDM can not considered as resulting in the exact same damage to the material, as was suggested by the EDM workshop. [10] For hardening it is more logical to increases, due to the addition of the recast layer, but the response of the total specimen shows a decrease. This can be explained by the heterogeneity of the sheet through the thickness, of which only the centre part is tested due to the thickness reduction. From the nano indentation results it can be found that the centre of the sheet has a lower hardness, resulting in less hardening of the thickness reduced specimen.
64 CHAPTER 5. RESULTS
Nano indentation
The hardness as determined by nano indention is visually represented in gure 5.27, where the results for both an unaltered sheet (in red) and a thickness reduced sheet (in black) are given. The results show an increase in hardness from the centre of the material outwards, which for the unaltered material increases approximately 20%, from a hardness of around 1.4 GPa to almost 1.7 GPa near the edges. The measurements on the thickness reduced sample show a similar result, with the hardness increasing outwards from the centre. The average hardness of the thickness reduced samples increases slightly faster then the hardness of the original material, but in the centre part of the cross-section no inuences are found at all. The measured lower hardening in the tensile tests is thus found to originate from the fact that the part of the sheet that is measured has a lower hardness, due to the removal of the harder outer layer.
Figure 5.27: Hardness distribution determined by nano indentation
In the gure the error bars show an uncertainty of up to 5% for most of the data, which is caused by the inuences of the microstructure of the material, like grain boundaries. A larger uncertainty is usually found near the edges, mainly due to specimen preparation. In appendix C an example of increased uncertainty is given, caused by the polishing step used.
Eects of thickness reduction on measured material behaviour
An eect of material removal is an inuence on the global behaviour of the sheet metal, e.g. hardening, as was shown in gure 5.27. This change of measured sheet properties originates in the heterogenous structure of a typical metal sheet as shown in gure 5.28. For sheets containing dierent phases like dual phase steels, this eect is even stronger as the microstructure vary greatly over the thickness then for the tested IF-steel. This eect introduces opportunities when determining material properties. Re- moving a part of the sheet by thinning may be carried out in such a way that in- stead of testing the sheet, testing a small part of the material or microstructure in the sheet becomes possible by removing the other layers. Some possibilities include material removal from one side only, as shown in 5.28 in the bottom picture. This approach would test the material near the surface, instead of the
65 CHAPTER 5. RESULTS
Figure 5.28: Inuence of microstructural inhomogeneities over the thickness of a sheet metal material in the centre of the sheet, and might provide insight in the layered structure of some sheet metals. Combining material properties found through the thickness of the sheet can then give more insight in the deformation behaviour of the sheet as a whole. This combined material model should give the same result as the a model determined for the sheet metal itself, as both average the materials behaviour over the thickness. A challenge that remains is to remove the material without inuencing the measured material data. For the removal of the inuencing recast layer, the use of ECM to polish or using ECM as only step instead are to be considered. Research into optimizing the manufacturing process has been started, but has not been nished during this project. There are strong indications though that the recast layer can be fully removed, as was conrmed during a meeting with Philips. [41]
5.4.4 Miniaturization
With the earlier mentioned set-ups, the possibility for building a miniature version of an existing set-up is the main problem. For cruciform specimens this is not the case, as the designed set-up already is build at small scale.
The current set-up for 60x60mm specimen can be decreased in size by shorten- ing the arms. From the numerical computations it can be found that the strain eld is only showing a gradient up to 12mm from the centre of the specimen. This means a specimen of 24mm, not including the clamping area, should be possible. When reducing the width of the arms, an even smaller specimen can be used. The limiting factor is only the material used, as large grains or other microstructural properties can lead to size eects when miniaturizing into a small length scale.
Tensile forces and displacements
For a cruciform with arms of 6mm width, the needed forces were found to be less then 3kN. By decreasing the width of the arms this force can be lowered even more. For building a set-up, the low forces are not expected to be a problem. Experimental and numerical results show the cruciform specimens have a max-
66 CHAPTER 5. RESULTS
Table 5.8: Numerical results for force and displacement for a cruciform with a centre thickness of 160µm
Property Value Maximum Elongation [mm] 4.5 (2 x 2.25) Maximum Force [kN] 2.95 (2 x 1.47) imum displacement of 4 − 6mm. This value can be lowered by shortening the arms, which gives more space for clamping as well.
Table 5.9: Comparing numerical and experimental results for the strain eld in a cruciform with a centre thickness of 160µm
Property Numerical Experimental Maximum Strain in the Arms [-] 0.11 0.15-0.20 Maximum Strain in the centre [-] 0.41 0.46-0.65
When combining the results, no reason can be found that makes building a biaxial testing device for cruciform specimen that can be used with SEM and other microscopes.
5.4.5 Summary
The cruciform specimen is found to be an excellent method for testing under biaxial loads. Both size and forces can be kept small for this set-up, making it relatively easy to build as a miniature set-up.
• The cruciform test is found to be experimentally usable to deform metal sheets up to fracture under biaxial loading.
• The numerical and experimental data up to necking is found to be remark- ably similar for the used simple numerical model. This makes designing and optimizing the cruciform geometry a relatively easy task.
• Miniaturization is not an issue with the cruciform specimen, unless scaling down into the domain where size eects become important. Forces and displacements are small in comparison to other techniques.
• The manufacturing of cruciform specimen is the biggest challenge, al- though promising processes have been discovered and are currently being investigated.
67 CHAPTER 5. RESULTS
5.5 Comparative evaluation of the set-ups
The four tested set-ups have all got their advantages, which will be discussed in this section. The result of this is summarized in table 5.10.
Limitations of set-up and specimen
As is shown in literature, some set-ups provide better material data then others. This is mainly inuenced by in-plane or out-of-plane testing, contact in the area of the measurement and constraints that lead to force failure modes:
• The bulge test is an out-of-plane test, resulting in bending inuences, which make the measured data deviate signicantly from the actual ma- terial data.
• The punch test is an out-of-plane test, like the bulge test, making bending also a problem. Contact in the area of interest makes the measured data deviate even more for describing the material.
• The Marciniak test is an in-plane test, with no contact in the area of interest. The Marciniak test deforms the sheet under pure biaxial load, without any distortions from contact or bending.
• The in-plane cruciform test also has no contact or bending inuencing the measured data, but due to the specimen geometry a forced failure direction exists.
Specimen preparation
• The specimen for a bulge test is cut out of the original plate, to t in the clamping die. No special manufacturing steps are needed and because of that all materials, as long as they are impenetrable for uids, can be tested with the bulge test.
• The punch test specimen is exactly the same as for the bulge test. Also for the punch test there are no real limitations on what materials can be tested.
• For the Marciniak test a specimen like the one for the bulge and punch test is needed. A washer sheet is needed as well, which is a copy of the test sheet, with a hole cut out in the centre. Determining the exact shape of washer is a challenge, as the test is sensitive to small changes in the shape. For testing dierent materials, it is likely that the set-up needs to be optimized again, resulting in dierent set-ups or washer geometries for dierent materials. Strong heterogeneity over the thickness can result in early fracture at the punch edge due to bending stresses.
68 CHAPTER 5. RESULTS
• The in-plane cruciform test has the most challenging specimen design. Manufacturing of a thickness reduced test specimen involves EDM or ECM, which is both time consuming and can alter the material. In the- ory any material can be tested with the cruciform loading test, but the eect of EDM or ECM might not be equal for every material. Strongly heterogenous sheets can prove to be dicult to characterize, as the dif- ferent layers have to be taken into account.
Measuring opportunities
• With the bulge test image correlation or microscopy can be used, but the lens will have to move with the growing bulge. This results in a challenging set-up, that involves constant measuring of the bulge height. The measurements at the surface are not representative for the whole sheet, as the stress- and strain elds are not uniform.
• The punch test has the same problem with non uniform stress- and strain elds as the bulge test. Keeping the top of the bulge at the same distance from the lens is easier though, as the movement of the punch dictates the movement of the sheet.
• In the Marciniak test the area of interest stays in the same plane, which makes digital image correlation or microscopy easy. The stress eld can be computed, if the material model is known, which can be a challenge as it is the material model that needs to be determined.
• For the in-plane cruciform test a digital image correlation system or mi- croscope can be used as well, as the area of interest stays in-plane. The uncertainty in the exact geometry of the specimen makes determining the stress eld more dicult, and involves numerical computing again for the thickness reduced centre.
Strain paths and strain path changes
• The bulge test can only be used for strain paths with ε1 > 0 and ε2 > 0. To achieve dierent strain paths, elliptical dies with dierent aspect ratios are needed. There is no possibility to change the strain path during a test
• The punch test can be used for strain paths with ε1 ≥ 0, while ε2 can be both positive and negative. This can be achieved easiest by changing the geometry of the test specimen, for instance with a Nakazima test. Strain path changes can not be made during a test
• The Marciniak test has the same strain path range as the punch test, ε1 ≥ 0 while ε2 can be both positive and negative. This is done with dierent specimen and washer geometries [47]. Varying the strain path during the test is not possible
69 CHAPTER 5. RESULTS
• The in-plane cruciform test is the only test where the strain path can be changed during a tensile test. All strain paths under tensile loading can be achieved by changing the load on the dierent arms
Miniaturizing
• Miniaturizing the bulge test results in two problems that make it unde- sired to do so. The rst is the increased bending inuence for a small bulge test. Even more problematic is the pressure of over 10MP a that is needed to go up to fracture in a bulge test with 50mm diameter. This results in uncontrollable and dangerous situations at fracture.
• The punch test shows the same problems with bending as the bulge test, but decreasing the size of this set-up results in a lower force needed to go to fracture, but an increase in friction inuence due to the decrease of the area the forces are working. The needed forces still go as high as 20kN and to be able to neglect bending a large set-up is preferred.
• In the Marciniak test the bending problem is not found, which makes miniaturizing easier. The critical balance between the dierent parame- ters makes a miniature set-up dicult to optimize and to get it working. Also high forces are needed to deform up to fracture, as forces go as high as 60kN for a punch diameter of 50mm
• To deform a cruciform specimen up to fracture, forces as low as 2.5kN are sucient. The only fundamental limitation for miniaturizing is the thickness reduction, that still has to be large enough to not feel size eects
70 CHAPTER 5. RESULTS 2 ε and 1 ε kN 5 . Yes, directionture of is dictated frac- Thinned sheet,sion job preci- alteringmen speci- geometry in arms Inuence becausethinning of for thickness Load distribution of the machine, positivenegative and 2 , 2 1 ε ε kN simple washer geometry No, fracture isstart free anywhere for to biaxial case the Specimen andshapes, washer , positive No additional inuence, besides averagingterial ma- propertiesthickness over positive and negative 50 2 ε kN , positive and nega- 1 fracture starting point Punch shapes, specimen shapes, friction, positive ε tive Additional inuence be- cause of bending 20 Comparison of the dierent biaxial test methods di- mm 50 2 Table 5.10: ε for a and , corresponding 1 ε kN 20 MP a Bulge TestNoNoNo Punch TestUnaltered sheet NoStraineld, Yes pressure Unaltered Marciniak sheet TestElliptical dies, Yes, friction only Straineld, dictates pos- the punchitive force Straineld, punch force Unaltered Cruciform sheet StrainNo and eld, tensile a load YesAdditional No inuence be- cause of bending Bending10 Noto Yes No Bending No Bending near etches In the range of grainsize Yes ameter test Property In-plane Contact in region of interest Geometrical con- straints in region of interest Specimen prepa- ration Measuringtions op- Strain elds Strainchange Heterogeneous path materials Miniaturizing problems Expected forces
71 Chapter 6
Conclusions and recommendations
6.1 Conclusions
From the result it can be concluded that the best options for a miniature set-up to study biaxial deformation and strain path changes are in-plane loading using a cruciform geometry (ILCG) and the Marciniak test. Which of these is better is dictated by the results that are wanted, as each has its own advantages and disadvantages. For the cruciform set-up, the following can be concluded:
• The cruciform test set-up is best a better option to be used under a microscope, as it can be miniaturized relatively easily. Necessary forces and the specimen size are both small enough to make a miniaturized set- up possible. Forces as low as 2kN are found to deform IF steel up to fracture.
• A disadvantage of the cruciform set-up is the challenge that lies in manu- facturing test specimen. Manufacturing induced eects can lead to data not representative of the tested sheet, due to either damage introduced in the tested material or altered material properties over the thickness of the specimen. To obtain reliable data these eects need to be kept as small as possible.
• A second disadvantage of the thickness reduction is the need to deter- mine the needed thickness reduction to reach fracture. The ideal thick- ness therefore depends on material behaviour and therefore the optimal cruciform geometry changes with every material.
• A concurring advantage associated with the need for precise specimen preparation in order to achieve thickness reduction in the cruciform spec- imen, is the possibility to test dierent layers in a sheet metal. As most
72 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS
sheet metals are heterogeneous over their thickness, this can prove to be a welcome addition to standard sheet tests.
• A huge advantage of the cruciform test set-up, that is not found in any other test, is the possibility to change strain paths during a test. Com- plex strain paths can be described relatively easily, by prescribing the displacements of the clamps. The other set-ups that have been analyzed have no possibilities for doing this.
• The Marciniak test is an interesting alternative, as it undistorted biaxial loading results due to the nature of the test and no distortion of the measured data by specimen preparation is found. A miniaturized test set-up needs punch forces as high as 50kN for IF-steel, which might be challenging when designing a small set-up.
• When miniaturizing the Marciniak set-up many design parameters are inuencing a critical balance between the biaxial fracture mode and other failure modes. This results in a complex optimization problem, which cannot easily be solved. It is likely that a miniaturized set-up can be optimized to work, but no proof can be given at this moment. Also, optimizations are needed for dierent materials so a set-up can be used to test more then one material.
• If there is need for changing strain paths, the Marciniak test cannot be used. The only strain path changes that can be carried out involve using pre-strained material to cut test specimen from. This gives severe limi- tations to the variety of strain paths that can be assessed, and leads to a more lengthy test procedure. Also when testing under dierent strain paths dierent washer and specimen shapes are needed. This involves more optimizations, just as the original test set-up.
6.2 Recommendations for future work
The challenges that need to be solved to develop an ideal set-up that is useable under all circumstances are the following:
• The in-plane cruciform specimen has not been optimized yet, only the earlier found geometry of Vos has been used and reduced in thickness to achieve fracture. To make the cruciform test smaller, an optimization of the cruciform size, the thickness reduction and geometry of the thickness reduction is recommended.
• A study into EDM, ECM or another advanced method for material re- moval is recommended, in order to make the cruciform specimen with a less distorted microstructure. A method useable in a laboratory en- vironment is desirable, to make the specimen design cycle shorter. A study into optimizing the EDM process or using ECM has been started, in cooperation with Philips DAP [41], but results are pending.
73 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS
• To understand the Marciniak test, studying of the local necking be- haviour is necessary, so numerical results can be improved to describe the Marciniak test with the localizations that were found. The improved numerical model can then be used to analyze miniaturization further.
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78 APPENDIX A. ELECTRICAL DISCHARGE MACHINING
Appendix A
Electrical Discharge Machining
Theory
The EDM process is basically a cathode-anode set-up of two electrodes, sepa- rated by a liquid dielectric, with a tool and workpiece. During the machining process, a voltage is applied over the gap between anode and cathode, causing the dielectric to break down. During the 'on-time' of the electrode a plasma channel grows, surrounded by a vapour bubble. The dense liquid dielectric restricts the growth of the plasma channel width, which results in the energy being concentrated in a small volume. Local temperatures can reach as high as 20 000 to 40 000 K, with plasma pressures as high as 3 kbar. The shape of this plasma channel as seen in gure A.1.
Anode Melt Anode (+) Cavity Cathode Compressed Liquid
Shockwave Plasma Front Anode Ambient Liquid Erosion rate Dielectric
Cathode (-) Cathode Melt Cavity ‘On-time’ Figure A.1: Schematic diagram of the EDM process showing the circle heat sources, plasma conguration and melt cavities (left); The dierence between anode and cathode material removal rates in respect to 'on-times' (right) [12]
High energy levels in the plasma cause both electrodes to melt, but due to the high plasma pressures, vaporization is limited. When the pulse is turned on, the anode will be the rst to rapidly melt because of fast moving electrons, and melting of the cathode starts later during the process, due the lower mobility of the positive ions. This results in dierent material removal rates for the anode
79 APPENDIX A. ELECTRICAL DISCHARGE MACHINING and cathode side, as is shown in gure A.1. For die sinking machines the cathode is usually the workpiece, resulting in relatively large 'on-times' in the order of 10-100 µs to make sure material is removed from the workpiece, instead of the tool. For most wire machines, the cathode is the wire, cutting into the anodic workpiece. The pulse time for wire machines therefore is generally much shorter, in the order of less then 10 µs. At the end of the 'on-time', the current is terminated and a violent collapse of the plasma channel follows. This causes the superheated, molten liquid on the surface of the electrodes to explode into the dielectric uid. Part of the material is then carried away, the remainder resolidies on the surface. [12, 25]
White layer and Heat Aected Zone (HAZ)
Bleys et al [6] saw several dierent layers in the thermally inuenced zone in EDM. They report formation and thickness of these layers depends on the process conditions and work piece properties. The 'white layer', a molten and resolidied layer, also known as the recast layer is a result of EDM. In this layer micro cracks are found, that seldom go deeper then the layer itself. Also present are micro-holes and droplets of resolidied material. Below the recast layer, the heat aected zone (HAZ) is found. The material in this zone has not been melted, but did undergo thermal inuence. Several layers are present, although they are not easily distinguished. It is shown that it is possible to reduce the recast layer to a thickness in the order of micrometres, by nishing in several steps with decreasing pulse current. Hardness test on an EDM machined surface show an increase in hardness in the HAZ, because of either diused carbon or a ner grain structure due to rapid cooling after the EDM process. The study shows residual stresses up to 500 MP a, present from several µm to almost 10 µm into the material.
EDM parameters and their inuence
Four parameters determine the material removal rate and thickness of the heat aected zone and recast layer. These parameters are pulsed current, pulsed voltage, pulse 'on-time' and pulse 'o-time'. The rst three determine the en- ergy put into the process, the latter determines the cooling time in between pulses. According to dierent sources, pulsed current and pulse 'on-time' in- uence the surface roughness, recast layer thickness and induced stresses most [22, 23, 48]. Varying pulse 'on-time' and pulse current has shown similar results for several researches. Kiyak and Çakr [29] show a relation for pulse time and current on AISI P20 tool steel, where pulse time clearly has more inuence then pulse current. They conclude a better nish is obtained for low currents and short pulse 'on-time', in combination with a relatively high pulse 'o-time'. Guu et al
80 APPENDIX A. ELECTRICAL DISCHARGE MACHINING
[23] give an empirical model that describes how the surface roughness changes a b in relation to pulse current (Ip) and pulse 'on-time' (τon) Ra = A ∗ (Ip) (τ) that can be tted to experimental data.
Lee et al [34] investigated the inuence of pulse 'on-times' of 25 µs up to 600 µs, showing surface roughness, tool wear and material removal rate is almost constant for pulse 'on-times' higher then 50 µs. Keskin et al [28] show a similar result, but for pulse 'on-times' of 100 µs and higher, with dierent power settings and materials. Lee and Li [35] studied the eect of pulse current and pulse 'on-time' on the thickness and composition of the recast layer. They conclude the recast layer grows with higher pulse current or pulse 'on-time'. Denser materials where found up to 15 µs into the material for pulse 'on-times' of 12.5 µs. The compo- sition of the top layer was studied with energy dispersive X-ray method (EDX), showing higher levels of carbon in the layer. The study of Guu et al [23] gives a similar result, and gives an empirical model for the recast layer thickness, t a b t = A ∗ (Ip) (τon) . The experimentally found thickness varies from 7 to 31 µm, for relatively long pulse 'on-times' of 20 to 180 µs. A study of Guu et al [23] shows correlation between tensile strength and machin- ing parameters, where a lower tensile strength was found for EDM machined materials. In a later article, Guu [22] shows the depth of micro-cracks grows with increasing pulse current and pulse 'on-time' as well, measuring 1272 to 1873 nm in depth for currents up to 1.5 A and 'on-times' up to 6.4 µs. Re- moving the recast layer is named as a possible solution to remove this eect.
Figure A.2: Surface roughness for positive (left) and negative (right) polarity on tool [1]
A study by Amorim and Weingaertner [1] shows clear dierences for using positive or negative electrodes as tooling. The large dierence in removal rates is due to the plasma channel shape as explained earlier. An optimal 'on-time' of around 50 µs is found for a positively charged tool, while an 'on-time' of around 8-12 µs is found optimal for a negatively loaded tool. This eect was also predicted by [12]. Amorim and Weingaertner also studied the inuence of polarity on the surface roughness, as shown in gure A.2. This gure clearly shows how a negatively charged tool (tool as cathode) gives a smoother nish,
81 APPENDIX A. ELECTRICAL DISCHARGE MACHINING at the cost of a much lower material removal rate and greater tool wear. A study by Bleys et al [6] into surface roughness shows very low roughness values can only be obtained by using low currents and inverse polarity, so-called 'EDM polishing'. A surface roughness Ra of 0.09 to 0.26 µm is shown to be possible by EDM polishing.
Tool wear
A side eect that should not be forgotten, is tool wear, as wearing of the tool cannot be prevented. Wang et al [55] show increased tool wear for high and short pulse currents. For extremely long pulse times, the dierence in wear becomes less and eventually virtually negligible. As for a ne surface roughness it was found that the pulse current should be low, this will not cause any problems for EDM where a ne surface roughness is required.
82 APPENDIX B. ELECTRICAL CHEMICAL MACHINING (ECM)
Appendix B
Electrical Chemical Machining (ECM)
Theory
The concept of using chemical solutions for material dissipation has already been patented in 1929, but was not improved much till the late 50s and 60s, when aircraft industry started to using ECM. In the last decade the microma- chining step has evolved, allowing surfaces to be machined with micrometre resolution and polished with the same machine. This specialized use in the form of Electrochemical Micro Machining (EMM) show great potential as no tool wear exists and high removal rates are possible. Alternatives like laser cutting, EDM and other non conventional machining tools are mostly thermal oriented, therefore making high precision more dicult due to heating of the workpiece. [4, 5] The ECM EMM process physical background lies in anodic dissolution of workpiece material, where the workpiece and tool act as anode and cathode, separated by an electrolyte. By applying an electric current, the anode work- piece dissolves locally, so shaping it to become the mirrored image of the tool. The electrolyte, often a salt solution, is then used to bind the free ions from the dissolved material and removes it by owing through the machining gap. [4] The standard ECM process makes precise machining of vertical walls nearly impossible, but with a clever tooling geometry as shown in gure B.1 it can be done. A more complex dual pole set-up can be used to achieve even better results. [33] The ECM EMM process is still under development and likely to be more enhanced over the coming years, but is being used in several commercial man- ufacturing processes. Manufacturing steps like cutting slots in sheet metal, machining MEMS components and shaping of surgical equipment can all be done with ECM. [4, 33]
83 APPENDIX B. ELECTRICAL CHEMICAL MACHINING (ECM)
Figure B.1: Comparison of dierent anode-cathode set-ups for EMM. a) uninsulated, b) insulated, and c) dual pole tools [33]
ECM parameters and their inuences
The most important parameters in the ECM process are again pulse 'on- time', pulse 'o-time', pulsed current and pulsed voltage, like for the EDM process. The combination of these parameters determines the width of the inter-electrode gap, by resulting in equilibrium. As the material is not evapo- rated during the ECM process, heating up of the workpiece can be controlled a lot better, which results in a cleaner surface. [5] For high precision material removal low currents are preferred, so the removed material contaminating the die-electric can ow away before inuencing the local material removal rates. The inter-electrode gap can be as small as a few micrometre, but this limits the maximum material removal rate. Other properties inuencing the accuracy of the ECM process are the choice of the electrolyte and the tool, where the latter should be thermally and electrically conductive, corrosion resistant and sti enough to withstand the electrolyte pressure without vibrating. [5] The electrolyte used is a last process parameter that can be used to enhance the ECM process, as dierent materials may need dierent electrolytes. A dierent electrolyte can also be used to favour certain chemical reactions and thereby change the material removal process. Electrolytes can be divided in
84 APPENDIX B. ELECTRICAL CHEMICAL MACHINING (ECM) two main types: passivating electrolytes, that contain oxidizing anions, like sodium nitrate or sodium chlorate, and non-passivating electrolytes containing more aggressive anions such as sodium chloride. The rst are used for better machining precision, the latter for higher removal rates. [33]
Surface roughness
Surface nishing by ECM is better then for most other processes, but as gure B.2 shows this does depend on the length scale being considered. The reason for the worsening for larger length scales is the development of wave-like patterns on the surface because of hydrodynamic vortex phenomena (dierences in ow speed). Choosing the right ECM tooling helps to reduces this surface roughness problem. [5, 33]
Figure B.2: Comparison of surface roughness for conventional polishing and ECM [33]
Tool wear
The problem of EDM where the electrode shape changes is not found in ECM, as only hydrogen gas is evolving at the cathode, thus no material is removed from the electrode during the process.
85 Appendix C
Specimen Preparation: TegraPol or Target System
An important aspect to take into account when interpreting data, is the amount of scatter found. For data sets where the scatter is large, there might be a problem with the acquisition of the data. This might be due to a problem with the set-up or a problem with the used specimen.
Figure C.1: Maximum load in an indentor test over the cross-section of a metal sheet on a specimen prepared with the TegraPol system
While using the nano indentor to determine the hardness gradient of the IF steel sheet, a large scatter was found on specimen prepared with the TegraPol polishing system as can be seen in gure C.1. It was found that this was related to the atness of the specimen.
86 APPENDIX C. SPECIMEN PREPARATION: TEGRAPOL OR TARGET SYSTEM
Figure C.2: Maximum load in an indentor test over the cross-section of a metal sheet on a specimen prepared with the Target system, showing lower scatter then for the TegraPol prepared specimen
The data obtained from specimen prepared with the Struers Target system, where the surface was found to be better then for conventional polishing, showed a lot less scatter. Even though one side of the specimen still showed rounding of the edges, as can be seen in gure C.2 the scatter for the same measurement was decreasing.
87