Technische Universiteit Eindhoven Mechanical Engineering Mechanics of Materials Eindhoven, January, 2007

Bachelor Final Project Report

A computational study of biaxial sheet metal testing: effect of different cruciform shapes on strain localization

R. Vos MT 07.03

Contents

1 Introduction 21.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Finite element simulations and results 72.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Effect of arm width . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Effect of corner shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Effect of numbers of arms . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Effect of thickness reduction . . . . . . . . . . . . . . . . . . . . . . . . 132.2.5 Complex loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Discussion 21

4 Conclusions 22

1

Chapter 1

Introduction

Sheet metal forming processes are commonly used in the automotive, packaging and con-struction industries. Successful manufacturing of sheet parts requires careful experimentsand simulations to assess the forming behavior of the material. Some information can beobtained from uniaxial testing. But as sheet metals are subject to strain paths ranging fromshear to biaxial straining, more information on the behavior of the sheet material under dif-ferent loading conditions has to be obtained. The material behavior in these different strainpaths and the correlation between this behavior and the underlying microstructure could beexamined by the use of a biaxial testing stage. The goal of this project is to come up with asuitable specimen geometry for such a stage, that would allow the examination of sheet metalyielding, necking and fracture at different stress states, with biaxial being the most importantone.

1.1 Problem statement

One method proposed in the literature to test biaxial behavior of a material, is applying axialand/or torsion loads and internal pressure on thin-walled cylinder tubes [1]. Essential forthis method is that the material has to be a circular tube. This method can therefore not beapplied to sheet material, so a different method is needed. Many researchers proposed theuse of a cruciform shaped specimen [1-7]. Via a numerical analysis performed on a cruciformspecimen, it is possible to obtain the required information of the behavior under differentstrain paths. The problem of using a cruciform specimen is that strain localization usuallyoccurs in the arms [4]. This means that the yielding, necking and fracture do not occur in thearea which is subjected to biaxial loading, i.e. the central region of the cruciform. So biaxialfailure properties of the material cannot be obtained by using such geometries. Therefore theshape of the cruciform has to be altered in such a way that failure occurs in the biaxiallyloaded zone.

1.2 Literature survey

Muller et al.[2] proposed two new methods to determine the yield locus of the biaxially loadedzone of a sheet metal. These methods are the inclined tensile test and the cross tensile test.In the inclined tensile test a strip material is stretched under an angle. In the cross tensiletest longitudinal and transverse forces are applied on a cruciform specimen. For the crosstensile test they optimized the specimen as shown in Fig. 1. This specimen allows one toobtain high strains in the biaxially loaded zone before failure. High stresses are found in the

2

Fig. 1. Geometry of the cross specimen used byMuller et al.

Fig. 2. Geometry of the cross specimen used byHoferlin et al.

neighborhood of the notches. Experimentally only the cross tensile test has been carried out.

Hoferlin et al. [8] used a specimen as shown in Fig. 2 to design a biaxial tensile stage todetermine the yield locus of thin steel sheets. The used material is cold-rolled steel with fivedifferent carbon concentrations, measured offset stresses and r-values. They made severalfinite element simulations and determined the yield locus of each material. Subsequently theyield locus of the five materials was determined by experiments. The results were comparedto the yield locus which was theoretically determined.

Kuwabara et al. [1] used a cruciform specimen with slits in the arms (Fig. 3) to carryout biaxial tensile tests on cold-rolled steel sheet to determine its elastic-plastic deformationbehavior under biaxial tension. They claim that the slits make the strain distribution inthe biaxially loaded zone almost uniform. The material used in the tests was as-received

Fig. 3. Geometry of the cross specimen used by Kuwabara et al.

cold-rolled low-carbon steel with a thickness of 0.8 mm. They determined experimentally thedegree of strain scatter of the material in the biaxially loaded zone under load ratios of 4:2and 4:4. They also determined experimentally the contours of plastic work for a strain rangeof ε < 0.03 and compared their results with existing yield criteria.

Kuwabara et al. [3] used the same specimen as proposed in [1] (see Fig. 3) to determinethe yield surface in the vicinity of a current loading point by experimental testing on a cold-rolled steel sheet and an aluminum alloy sheet. They claim that they verified the potential of

3

a new method for determining the yield surface by experiments. This new method consist ofdetermining the yield surface by using an abrupt strain path change, which has been proposedearlier by Kuroda and Tvergaard [9].

Yu et al. [4] studied the sheet forming limit for complex strain paths. Using finite ele-ment models, they optimized a cruciform specimen with a reduced thickness area in the armsand in the central region, by changing the geometric parameters shown in Fig. 4. The optimal

Fig. 4. Quarter of the cross specimen used by Yuet al.

Fig. 5. Geometry of the cross specimen used byWu et al.

shape of the specimen has the most uniform stress distribution in the central region and isable to generate a large deformation during the stretching of the specimen. The deformationof the specimens was analyzed under uniaxial-tension and under biaxial-tension. The authorsclaim to have found that complex strain paths can be realized by adjusting the velocity ratiosimposed on the specimen arms.

Wu et al. [5] used the specimen shown in Fig. 5 to build a biaxial tensile test system that canrealize complex loading. They built a biaxial tensile test system which is capable of realizingcomplex loading paths with good accuracy, but no local strain measurements were carried out.

Gozzi et al. [6] developed a new specimen design to study the mechanical behavior of extrahigh strength steel. In earlier studies a specimen was used as can be seen in Fig. 6 (a). Theproblem with this specimen was that failure in the material occurred before the desired stressand strain were reached in the biaxially loaded zone. Therefore a new model had to be devel-oped. The biaxially loaded zone was reduced in area to reach higher levels of stress and strain

(a) Old specimen (b) New specimen 1 (c) New specimen 2

Fig. 6. Geometry of the different cross specimens used by Gozzi et al.

in this area. The notches were changed to keep the stress in the corners lower and to prevent

4

necking in this area. The slits in the arms were changed to obtain the best stress/straindistribution in the biaxially loaded zone. The authors came up with two specimens as shownin Fig. 6 (b) and Fig. (c). In specimen b, the slits all have the same length. In specimen c,the two outer slits have a reduced length by 3 mm. For different load cases, one of these newspecimen shapes is the best to use. The authors showed that these two new specimens allowone to study the mechanical behavior of extra high strength steel.

Smits et al. [7] performed finite element simulations and experiments on several cruciformspecimens to find an optimized specimen shape for biaxial testing of fibre reinforced compositelaminates. The material used in the models is glass fibre reinforced epoxy.

Fig. 7. Geometry and stress localization of the different cross specimens used by Smits et al.

Fig. 7 shows the examined geometries and the first principal strain in the material pre-dicted by the finite element program. In geometry A, failure occurs in the arms. To relocatethis failure to the center, first a thickness reduction in the center of the specimen was made,which is shown in geometry B. This resulted in an increased principal strain in the center.Subsequently the corner geometry was changed as shown in geometry C. This resulted inhigh strains and possible failure of the specimen in the center. Changing the geometry of thecorners was necessary, because the fibres at ±45 degrees carried the load from one arm to aperpendicular one, which resulted in unloading of the center of the specimen. In geometryD, a larger thickness reduction area was used. The strain variation over the most loadedaxis of the milled zone was examined for each geometry, by finite element simulations andexperiments. A higher strain was found in the experiments in the transition zone between thefull thickness and the thickness reduced area compared to the finite element predictions. Thisdifference may be eliminated in a more detailed model with a smooth thickness reduction.Geometry C showed the most uniform distribution of strains, both in the simulation and inthe experiments. To determine which geometry is the best, also experimental biaxial failuredata was obtained. The highest failure strains were found for geometry C, which indicatesthat this is the most optimal specimen. Failure started at the corners between the two armsfor all geometries, but for geometry B and C the complete biaxially loaded test zone wasdamaged.

5

Summary

In the papers discussed above, several different geometries were used. The used geometrieshave round corners, notched corners or slits in the arms. Some have a thickness reductionand some do not. Only two papers show the effect of a few different shapes of the specimen,but one is considering high strength steel [6] and one is considering a composite material [7].This means that the effect of different shapes on the location of the maximum straining havenot been investigated thoroughly. Most of the papers studied the specimen deformation up tothe point of yielding, but not up to the point of fracture and even not to the point of necking.To find an optimized specimen in which the yielding, necking and damage occur in the center,information on the effect of strain localization in different shapes is necessary. Therefore thisreport focuses on investigating the stress and strain distribution in the cruciform specimenwhen different parts of the geometry are changed.

6

Chapter 2

Finite element simulations andresults

2.1 Method

The finite element simulations were carried out with the finite element package Marc/Mentat2005. The element type used for two-dimensional analyses is the plane stress element quad 4(type 3), because the cruciform specimen is loaded in its plane by prescribed displacements.For the three-dimensional models, element type hex 8 (type 7) was used. Considering sym-metry and loading conditions, only a quarter of the cruciform has to be modeled. The usedmaterial is interstitial free (IF) steel with a thickness of 0.7 mm, a Young’s modulus of 45 GPaand a Poisson’s ratio of 0.29. A hardening curve up to the point of necking (see Fig. 8) wasused to describe the plastic yielding. All these values were determined from data obtainedfrom a uniaxial tensile test of the same material, as part of a NIMR project MC 2.05205a atTU/e. Mesh dependency has been investigated while simulating the geometries. It has beenfound that the mesh has no influence on the area of localization, only on the accuracy of thevalue of the stress.

Fig. 8. Hardening curve used in thesimulations

Fig. 9. Sketch of the biaxial stage proposed byKammrath and Weiss

Each of the simulated geometries has a total length of 60 mm, taking the dimensions ofthe proposed biaxial stage into consideration [10]. A sketch of this stage is shown in Fig. 9.On each of the four clamps a tensile load is being applied by an independent motor. Theused load cells have a maximum capacity of 5000 N. The travel range is maximum 8 mmfor each of the four clamping devices. To be able to place the stage under a microscope, thedimensions of the test stage should not exceed 260x260 mm2.

To find a suitable specimen geometry, the effect of different aspects of the geometry arestudied as follows: first the effect of arm width is examined; second the effect of different

7

corner shapes is studied; third the effect of the number of arms of the cruciform is examined,using sharp, round and notched corners. Because thickness reduction can contribute to therelocation of the localization from the arms to the center of the specimen [7], different shapesof this reduction are also examined. Finally the most suitable geometry is loaded in twostages, first equi-biaxially and then uniaxially as well as vice versa, to see if the failure stilloccurs in the center of the specimen.

For comparing the several geometries, the von Mises stress distribution in the materialis determined. This stress gives a good insight into the overall magnitude of the stress andpredicts the plastic deformation of the material under triaxial loading from results obtainedfrom a uniaxial test [11]. The values of the von Mises stress in the material are indicated bya color code. An example of combinations of colors and values is shown in Fig. 10. Note thatthe this legend does not apply to each von Mises stress figure.

Fig. 10. Values of the von Mises stress (MPa) indicated by colors

2.2 Results

2.2.1 Effect of arm width

To determine the effect of the width of the arms, geometries with sharp corners and an armwidth of respectively 10 mm, 20 mm and 30 mm have been simulated. Fig. 11 shows the

(a) Arm width: 10 mm (b) Arm width: 20 mm (c) Arm width:30 mm

Fig. 11. Von Mises stress at necking as a function of the arm width

stress in the specimens at the end of the simulation. For each width necking occurs in themiddle of the arms. This means that in a geometry with rectangular arms, necking alwaysoccurs in the arms, as was also concluded in [4].

Fig. 12 (a) shows the stress in the center, corner and in the middle of the arms during thestretching for the specimen with a 10 mm arm width. From this figure it is clear that initiallythe highest stress is reached in the corners. However, this stress decreases once the arms startto neck, because of elastic recovery. The same trend holds at the center of the specimen,albeit at a lower level of stress. In the neck, however, the stress continues to increase.

8

Fig. 12 (b) shows the stress-displacement curve for the specimen with 20 mm arm width.In this figure it is interesting to observe that the stress in the corner of the specimen wellexceeds the level which is needed in a uniaxial test to initiate necking ( i.e. the highest stressreached in Fig. 8) before necking finally occurs in the arms. The explanation for this is that

(a) Arm width: 10 mm (b) Arm width: 20 mm

Fig. 12. Stress-displacement curve for the specimens with 10 mm and 20 mm arm width

the observed high stress is very local in the corners, which means that there is a very largestress gradient going from the corner to the center of the specimen. The stress distributionin the arms is more uniform and the deformation is less constrained by surrounding material,which means that necking in this area is more favorable. The figure also shows that the stressreached in the center is almost the same for both specimens. The graph of the specimenwith 30 mm arm width is not shown, because the simulations show the same effect as for thespecimen with 20 mm arm width.

Table 1Maximum values of von Mises stress (MPa) reached in the corners

region arm width 10 mm arm width 20 mm arm width 30 mm

center 300 315 350

corner 450 540 660

Table 1 shows the maximum stress reached in the corner and in the center for the threespecimens. The maximum stress reached is higher as the arm width is increased. The expla-nation for the increased stress in the center and corners is that the arm length is decreasedand necking in the arms is therefore delayed compared to the narrower, more slender arms.As a result, a higher stress is reached in the center and corners of the specimen before it dropsas a result of the necking of the arms.

Although the higher stress in the center is desired, the higher stress in the corners isundesired. This is because in the ideal geometry localization should take place in the biaxiallyloaded zone. This means that the stress gradient from corner to center is decreased so thatnecking could possibly begin in the corners. To minimize this possibility in the most suitablespecimen, it is best to use the specimen in which the stress in the corners is the lowest: thespecimen with an arm width of 10 mm.

2.2.2 Effect of corner shape

The stress in the corners can be further lowered by changing the corner shape. To determinewhich corner shape decreases the stress the most, specimens with round and notched cornershave been simulated with arms of 10 mm width.

9

The examined geometries with round corners have corners with radii of 1, 3 and 5 mm.The results (Fig. 13) show that necking still occurs in the arms, which was to be expected,since the radius makes necking near the center less favourable. The only difference between

(a) Corner radius: 1 mm (b) Corner radius: 3 mm (c) Corner radius: 5 mm

Fig. 13. Von Mises stress at necking as a function of the corner radius

different corner radii is the maximum stress reached in the corners during the stretching ofthe arms. These maximum stress values are shown in table 2. The stress for the geometry

Table 2Maximum values of von Mises stress (MPa) reached in the corners

geometry stress (MPa)

sharp corner 450

round corner radius 1 mm 450

round corner radius 3 mm 360

round corner radius 5 mm 315

with a corner radius of 1 mm is the same as for the sharp corner geometry. This means that asmall corner radius does not decrease the stress in the corners. When the radius is increased,the stress in the corners is decreased. A corner radius of 3 mm decreases the stress in thecorners by about 20 %. The difference between a corner radius of 3 mm and 5 mm is small.

(a) Round corner, radius: 3 mm (b) Notch corner, radius: 3√

2− 3 mm

Fig. 14. Stress-displacement curve for the rounded corner specimen and for the notched corner specimen

Fig. 14 (a) shows the stress-displacement curve for the geometry with a corner radius of3 mm, in the center, the arm and the corner of the specimen. If these curves are comparedwith the curves of the specimen with sharp corners and arm width 10 mm (Fig. 12 (a)), itappears that the stress in the corners now remains below that in the arms. The radius of 3

10

mm has thus entirely removed the stress concentration in the corners. At the same time, thestress in the center of the specimen is approximately 10% lowered, which is rather unwanted.

Fig. 15. Distance between different corners is constant

The examined geometries with notched corners have notches with radii of respectively√2 − 1, 3

√2 − 3 and 5

√2 − 5 mm. The different radii were chosen in such a way, that the

distance between notch roots and the sharp corner is the same as the distance between theroot of the round corners with radii 1, 3 and 5 mm and the sharp corner (see Fig. 15). Thismakes it possible to compare this geometry to the geometry with round corners. Fig. 15shows the final results of the simulations of the specimens with notched corners.

(a) Radius notch:√

2− 1 mm (b) Radius notch: 3√

2− 3 mm (c) Radius notch: 5√

2− 5 mm

Fig. 16. Von Mises stress at necking as a function of the notch radius

From Fig. 16 (a) it is clear that a small notch still leads to necking in the arms. When theradius of the notches is increased, the necking starts in these notches (Fig. 16 (b), (c)). Thiseffect was also found in [2,6]. The explanation for these results is that the arms are reducedin width in the area where the notches are present. This area is therefore less strong andnecking is more likely in that cross section. It seems that the notches need to have a certainradius to decrease the width enough to move the localization from the arms to the notches.Fig. 14 (b) shows the stress-displacement curve for the specimen with notches with a radiusof 3√

2−3 mm. During the entire loading history, the stress is the highest in the area betweenthese notches. Furthermore, the stress in the center of the specimen is larger than the stressin the arms. If this figure is compared to Fig. 14 (a), one observes that the stress in thecenter of the specimen with notches is higher than in the specimen with round corners. If acomparison is made with the sharp corner geometry, it seems that the stress in the center isnow increased by approximately 10 %. So the notched geometry helps to increase the stressin the biaxially loaded zone, but it also weakens a certain area even more than the middle ofthe arms. This increases the danger of necking elsewhere than in the biaxially loaded zone.

11

2.2.3 Effect of numbers of arms

In the industry different strain path behavior of sheet metal material is examined by forminglimit tests on Nakazima strips as shown in Fig. 17. This test could possibly be simulatedby an in-plane model, which in the most ideal case consists of a circular specimen which isradially displaced, as shown in Fig. 18. The problem is that such a test stage is impossible to

Fig. 17. Nakazima strips used in bulge tests Fig. 18. Ideal in-plane biaxial test specimen

construct physically and therefore it cannot be tested experimentally. In order to be able toapply in-plane loads, the specimen needs to have arms. Because four arms may not representthe ideal case, more arms could be introduced. The effect of more than four arms is thereforeinvestigated by constructing specimens with 6 and 8 arms. To exclude the shape of the cornerhaving an effect on the location of localization in a specimen with more arms, simulationshave been made with all the three different corner shapes. The radii of the round corner andnotched corner are the same to make a comparison possible.

(a) Sharp corners (b) Round corners (c) Notched corners

Fig. 19. Von Mises stress at necking in specimens with 6 arms for each corner shape

(a) Sharp corners (b) Round corners (c) Notched corners

Fig. 20. Von Mises stress at necking in specimens with 8 arms for each corner shape

Figures 19 and 20 show results similar to those found in the simulations of the specimenwith four arms. For the specimen geometries with sharp corners first a high stress is found

12

in the corners of the specimen, followed by necking in the arms. For the specimens with theround corners necking occurs in the arms and for the specimens with notches the neckingoccurs between the notches. Hence, the increase in the number of arms has no effect on thelocalization.

Table 3 shows the maximum stress values reached in the center for specimens with 4, 6and 8 arms, for all three different corner shapes. As can be seen, the stress is slightly higherfor the specimens with sharp and round corners as the number of arms is increased. Theexplanation for this is that the length of the arms is decreased when more arms are used,which, as concluded earlier in this report, increases the stress reached in the corners and inthe center. For the specimens with notches, the stress even decreases when more arms areused. This can be explained by the fact that the necking occurs further away from the centeras the number of arms is increased.

Table 3Maximum values of von Mises stress (MPa) reached in the center

corner shape 4 arms 6 arms 8 arms

sharp 300 320 325

round 250 280 285

notch 360 325 315

Although the stress is increased in the center, the necking still occurs in the arms, or at thetransition from the center to the arms. Therefore more arms do not result in an improvementover four arms. This means that a specimen with more than four arms is not necessary forbiaxial testing.

2.2.4 Effect of thickness reduction

To make sure that necking of the material happens in the biaxially loaded zone, a thicknessreduction in the center of the specimen is necessary. Several different shapes of thicknessreductions are examined: one with the shape of a circle (specimen A), one with a shape ofa cruciform (specimen B), one with both shapes (specimen C), and one with the shape ofa bowl (specimen D). All geometries have round corners to decreases the danger of neckingelsewhere than in the biaxially loaded zone.

Specimen A

In geometry A the biaxially loaded zone is homogeneously reduced in thickness in the formof a circle. First the necessary thickness reduction is examined by a two-dimensional analysison specimens in which the thickness is reduced from 0.7 mm to respectively 0.5, 0.4 and0.3 mm. Note that the transition from one thickness to the other is abrupt, because it is atwo-dimensional analysis. The radius of the circle is 5 mm. This radius was chosen becausea bigger radius also reduces the thickness of parts of the arms and makes the arms weaker.This would increase the chance of failure in the arms, which is undesired.

From Fig. 21 it can be concluded that only in the specimen with a thickness of 0.3 mmthe localization occurs in the biaxially loaded zone. In this geometry there are two types oflocalization regions, region A and region B, as can be seen in Fig. 21 (c). Region A is in therounded corner between two arms and region B is in the top right area of the circle where thethickness has been reduced. Fig. 22 shows the stress-displacement curve for these two regionsas well as for the center of the specimen. Elastic recovery occurs in the center, whereas thestress in regions A and B continues to increase. This means that the final necking takes place

13

(a) Thickness center: 0.5 mm (b) Thickness center: 0.4 mm (c) Thickness center: 0.3 mm

Fig. 21. Von Mises stress at necking in the specimens as a function of the thickness reduction

along both these regions, but away from the center. Note that the stress at which neckingstarts is significantly higher than that is necessary for necking in the arms.

Fig. 22. Stress-displacement curve of the geometry with a thickness of 0.3 mm in the center

The localization in regions A and B can possibly be explained by the abrupt changein thickness. Therefore the effect of a more stepwise thickness reduction is also examined.Because the smooth thickness reduction starts from a circle with a radius of 5 mm, a smallercircle has the necessary thickness of 0.3 mm. Therefore the effect of a smaller circle radiusis examined first. This is done by a two-dimensional analysis on specimens with a thicknessreduction circle of 4 mm and 3 mm. The results are shown in Fig. 23.

(a) Circle radius: 4 mm (b) Circle radius: 3 mm

Fig. 23. Von Mises stress at necking in the specimens as a function of the circle radius

Fig. 23 (b) shows that a reduction circle with a radius of 3 mm results in necking in thearms. Hence, it appears that the minimum radius is approximately 4 mm.

14

For the more stepwise reduced specimen the thickness reduction therefore starts at a radiusof 5 mm and ends with a thickness of 0.3 mm at a circle of 4 mm (see Fig. 24 (a)).

(a) Geometry (b) Von Mises stress at necking

Fig. 24. Geometry and Von Mises stress for specimen A with a more stepwise reduced thickness

Fig. 24 (b) shows that there are still two localizations regions. So the amount of thicknessreduction has no effect on the localization. To examine if a smooth thickness reduction changesthe pattern, a three-dimensional finite element model has been made of this specimen. Thecross section of the area with a reduced thickness is shown in Fig. 25 (a).

(a) Cross section of the biaxially loaded zone.Units in mm. (b) Von Mises stress at necking

Fig. 25. Geometry and Von Mises stress at necking of the three-dimensional model for specimen A

Fig. 25 (b) shows almost the same results as obtained from the two-dimensional model:highest stress in the top right area of the reduced thickness circle and also high stress in thecorner of the specimen. Hence, a homogenous thickness in the center with a smooth thicknessreduction does not give the desired result.

Specimen B

Next a two-dimensional analysis is performed on geometry B. In this geometry the thicknessreduction has the form of a cruciform as shown in Fig. 26 (a). This shape enlarges the distancebetween the corner of the specimen and the thickness reduction area. The thickness of thereduction area is respectively 0.3 and 0.2 mm. The length of the arms of the inner cruciformis 5 mm, to prevent any reduction of the thickness in the arms of the specimen. Fig. 26 (b)and Fig. 26 (c) show that the thickness of the inner cruciform has to be below 0.3 mm, tohave necking in this area. Fig. 26 (c) shows also that there are again two stress localizations:one in the corner of the entire specimen (region A) and one in the corner of the reductionarea (region B).

15

(a) Geometry (b) Thickness center: 0.3 mm (c) Thickness center: 0.2 mm

Fig. 26. Geometry and Von Mises stress at necking for specimen B as a function of the thickness reduction

Fig. 27 shows the stress-displacement curve of these two areas and for the center of thespecimen. The stress in regions A and B continues to increase, when the center starts toelastically recover. The stress in region B is again the highest during the entire loadingprocess.

Fig. 27. Stress-displacement curve for specimen B

Specimen C

Specimen C has a thickness reduction in the center which is similar to the one studied in [4],i.e. a circle inside a cruciform (see Fig. 28 a) although high stresses are expected on the edgeof the inner circle. This inner circle has a thickness of 0.2 mm, because the results of specimenB showed that otherwise necking occurs in the arms. Note that this is also a two-dimensionalanalysis. Fig. 28 (b) shows the results of the simulation. The stress localizes in the innercircle. But the highest stress is still not reached in the exact center of the specimen, but inthe top right area of the inner circle, as expected. Although the stress localization is near thecenter, this is still not the ideal specimen.

Specimen D

It seems that the only way to have the failure starting in the specimen’s center is by makingthis area the weakest point of the specimen. Therefore a three-dimensional model is madein which the thickness is gradually reduced from both sides of the specimen to a minimumthickness in the center, as can be seen in Fig. 28 (a). The thickness reduction starts at abigger radius than 5 mm, so now the arms do have a certain area where the thickness isreduced. Although this can result in necking of the arms, it seems to be necessary, because

16

(a) Geometry (b) Von Mises stress at necking

Fig. 28. Geometry and Von Mises stress at necking for specimen C

otherwise the thickness reduction at a certain radius of the circle is to small. The thickness ofthe center point is 0.2 mm, to be sure that the thickness reduction suffices to force localizationthere. Although from earlier results a thickness reduction as used in specimen C is expectedbecause of the high stress in the neighborhood of the center, a spherical reduction is used.This is because such a shape is easier to manufacture, while it is still expected to give thewanted results.

(a) Cross section of biaxially loaded zone. Unitsin mm. (b) Von Mises stress at necking

Fig. 29. Geometry and Von Mises stress at necking for specimen D

Fig. 29 (b) shows that the stress localizes in the exact center for this geometry, withslightly lower stresses in bands towards teh corners. Fig. 30 (a) shows the stress-displacementcurve for both areas and for one arm of the specimen. At a displacement of 2.8 mm, the armsstart to elastically recover, but the stress in the center and in the corner continue to increase.This means that both areas start to neck, but because the stress in the center is the highest,failure occurs first in the center. Note that this stress is again higher than needed for neckingin the arms. When this stress is reached, the axial stress at the end of the arms is about 300MPa. This means that the biaxial test stage has to exert a load of about 2100 N.

The strain path followed by the material in the center is shown in Fig. 30 (b). The ratiobetween the major and minor strain is almost equal to one during the entire loading process,which means that the center is deformed under equi-biaxial tension. This specimen thereforeseems to be suitable for biaxial testing up to the point of failure.

17

(a) Stress-displacement curve (b) Strain path

Fig. 30. Stress-displacement curve and strain path of specimen D

2.2.5 Complex loading

Specimen type D has also been subjected to straining paths that are more general than equalbiaxial, namely first equi-biaxial strain followed by uniaxial strain as well as vice versa. Thefinal displacement ratio of x-arm to y-arm is 2:1. The results are shown in Fig. 31. As can beseen, the stress localization takes place in the exact center of the specimen for both loadings.Note that there is still relatively high stress in the corners, which is also seen in the results ofspecimen D under equi-biaxial loading. The only difference is that this stress in the cornersis a bit more to the right, which can be explained by the fact that the x-arm has a higherdisplacement.

(a) Uniaxial-biaxial loading (b) Biaxial-uniaxial loading

Fig. 31. Von Mises stress at necking of specimen D under complex loading

Fig. 32 shows the stress-displacement curves of both loadings. These figures show that thex-arm starts to elastically recover at almost the same displacement for both loadings. Thestress at which the center starts necking, is also almost the same. Note that the stress neededfor necking in this model is slightly higher than that is needed for the necking to occur in thearms of the specimen without a thickness reduction.

18

(a) Uniaxial-biaxial loading (b) Biaxial-uniaxial loading

Fig. 32. Stress-displacement curve of specimen D under complex loading

Fig. 33 shows the strain paths of the center of the specimen for both loadings. These resultsshow that the strain paths of the center can be controlled by adjusting the loadings of thearms of the specimen. This means that complex strain paths can be realized. Because failurestarts in the center of the specimen and complex strain paths can be realized, this specimenhas the most suitable geometry for examination of sheet metal yielding, necking and fracturefor different stress states.

(a) Uniaxial-biaxial (b) Biaxial-uniaxial

Fig. 33. Strain paths of the center of specimen D under complex loading

19

Chapter 3

Discussion

The results of the specimen with sharp corners show high stress concentrations in the corners,but no plastic flow is present in this area. The necking eventually occurs in the arms of thespecimen, independent of the width of the arms. The stress in the corners can be decreasedby using round corners. Using notches results in necking in the area where the notchesreduce the specimen arm width, but this area is still part of the arms which are only underuniaxial loading. Using more arms shows essentially the same results as the simulations of thespecimens with four arms. The specimen shapes with more than four arms do not performbetter than that with four arms. All results of these simulations show necking in the arms.This means that the arms are the weakest point of the specimen and therefore this simplecruciform specimen is not suited for biaxial loading up to the point of necking. Note thatusing slits, as proposed in [1, 3, 5, 6] to obtain a uniform stress distribution in the biaxiallyloaded zone up to the point of yielding, will not change this picture and it will weaken thearms even more.

A thickness reduction in the center does result in necking in the biaxially loaded zone.However, if the biaxially loaded zone has a homogenous thickness reduction, this necking doesnot start in the center but in a certain area of the reduction zone. For a circular thicknessreduction this occurs in the top right area of the circle. For a cruciform thickness reductionit occurs in the corner of this cruciform. Using a reduction of both a circle and a cruciformalso does not result in the desired outcome, although the highest stress does occur in theneighborhood of the center. This is promising, but still not good enough. The only way tohave the yielding and necking starting in the center of the specimen appears to be by makingthis center the thinnest part of the specimen, by applying a bowl shaped thickness reduction.Because this area has to ’compete’ with the weak arms, the center of the specimen needs tohave a very small thickness, compared to the thickness of the arms. Because the thickness ofthe specimen is already small, the thickness of the center must be very small and the questionremains if this is physically possible to make with the existing technologies like spark-erosion,without changing the microstructure of the material.

More investigation needs to be performed to find the exact needed dimensions of thethickness reduction of the specimen and also more investigation on the mesh dependency hasto be performed. Other investigations need to be performed on for example strengtheningthe arms, which will possibly result in a specimen which is physically easier to make.

20

Chapter 4

Conclusions

Finite element simulations have been carried out on different cruciform specimen shapes tostudy the effect of different geometries. The objective was to obtain a specimen in which theyielding, necking and fracture occur at the center of the specimen, where a truly biaxial stressstate exists. The results obtained in this study can be summarized as follows:

1. In specimens with sharp corners, necking occurs in the arms, independent of arm width.

2. Round corners decrease the stress in the corners. In a specimen with notch corners, thenecking occurs in area between the notches, if the notches are sufficiently deep.

3. The number of arms does not change the localization.

4. A homogenous thickness reduction in the center of the specimen results in high stressat the edge of this area.

5. A thickness reduction in the form a bowl results in yielding, necking and damage at thecenter of the specimen, even under complex loading.

21

References

[1] T. Kuwabara, S. Ikeda, K. Kuroda, Measurement and analysis of differential work harden-ing in cold-rolled steel sheet under biaxial tension, Journal of Materials Processing Technology80-81 (1998) 517-523

[2] W. Muller and K. Pohlandt, New experiments for determining yield loci of sheet metal,Journal of Materials Processing Technology 60 (1996) 643-648

[3] T. Kuwabara, M. Kuroda, V. Tvergaard and K. Nomura, Use of abrupt strain pathchange for determining subsequent yield surface: experimental study with metal sheets, Actamater (2000) 2071-2079

[4] Y. Yu, M. Wan, X.D. Wu, X.B. Zhou, Design of a cruciform biaxial tensile specimenfor limit strain analysis by FEM, Journal of Materials Processing Technology 123 (2002) 67-70

[5] X. Wu, M. Wan and X.B. Zhou, Biaxial tensile testing of cruciform specimen undercomplex loading, Journal of Materials Processing Technology 168 (2005) 181-183

[6] J. Gozzi, A. Olsson and O. Lagerqvist, Experimental Investigation of the Behavior ofExtra High Strength Steel, Society for Experimental Mechanics, Vol.45, No. 6, (2005) 533-540

[7] A. Smits, D. van Hemelrijck, T.P. Philippidis, A. Cardon, Design of a cruciform specimenfor biaxial testing of fibre reinforced composite laminates, Composites Science and Technol-ogy 66 (2006) 964-975

[8] E. Hoferlin, A. van Bael, P. van Houtte, G. Steyaert and C. de Mar, Biaxial tests oncruciform specimens for the validation of crystallographic yield loci, Journal of MaterialsProcessing Technology 80-81 (1998) 545-550

[9] M. Kuroda, V. Tvergaard, Acta mater., 1999, 47, 3879

[10] The ’BiAx’ Concept, Kammrath and Weiss, Special Developments Microscopy

[11] R.T. Fenner, Mechanics of Solids, Blackwell Scientific Publications, 1989, 517

22