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

Transcription

1 Bachelor Final Project Report A computational study of biaxial sheet metal testing: effect of different cruciform shapes on strain localization R. Vos MT Technische Universiteit Eindhoven Mechanical Engineering Mechanics of Materials Eindhoven, January, 2007

2 Contents 1 Introduction Problem statement Literature survey Finite element simulations and results Method Results Effect of arm width Effect of corner shape Effect of numbers of arms Effect of thickness reduction Complex loading Discussion 21 4 Conclusions 22 1

3 Chapter 1 Introduction Sheet metal forming processes are commonly used in the automotive, packaging and construction industries. Successful manufacturing of sheet parts requires careful experiments and simulations to assess the forming behavior of the material. Some information can be obtained from uniaxial testing. But as sheet metals are subject to strain paths ranging from shear to biaxial straining, more information on the behavior of the sheet material under different loading conditions has to be obtained. The material behavior in these different strain paths and the correlation between this behavior and the underlying microstructure could be examined by the use of a biaxial testing stage. The goal of this project is to come up with a suitable specimen geometry for such a stage, that would allow the examination of sheet metal yielding, necking and fracture at different stress states, with biaxial being the most important one. 1.1 Problem statement One method proposed in the literature to test biaxial behavior of a material, is applying axial and/or torsion loads and internal pressure on thin-walled cylinder tubes [1]. Essential for this method is that the material has to be a circular tube. This method can therefore not be applied to sheet material, so a different method is needed. Many researchers proposed the use of a cruciform shaped specimen [1-7]. Via a numerical analysis performed on a cruciform specimen, it is possible to obtain the required information of the behavior under different strain paths. The problem of using a cruciform specimen is that strain localization usually occurs in the arms [4]. This means that the yielding, necking and fracture do not occur in the area which is subjected to biaxial loading, i.e. the central region of the cruciform. So biaxial failure properties of the material cannot be obtained by using such geometries. Therefore the shape of the cruciform has to be altered in such a way that failure occurs in the biaxially loaded zone. 1.2 Literature survey Müller et al.[2] proposed two new methods to determine the yield locus of the biaxially loaded zone 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 tensile test longitudinal and transverse forces are applied on a cruciform specimen. For the cross tensile test they optimized the specimen as shown in Fig. 1. This specimen allows one to obtain high strains in the biaxially loaded zone before failure. High stresses are found in the 2

4 Fig. 1. Geometry of the cross specimen used by Müller et al. Fig. 2. Geometry of the cross specimen used by Hoferlin 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 to determine the yield locus of thin steel sheets. The used material is cold-rolled steel with five different carbon concentrations, measured offset stresses and r-values. They made several finite element simulations and determined the yield locus of each material. Subsequently the yield locus of the five materials was determined by experiments. The results were compared to the yield locus which was theoretically determined. Kuwabara et al. [1] used a cruciform specimen with slits in the arms (Fig. 3) to carry out biaxial tensile tests on cold-rolled steel sheet to determine its elastic-plastic deformation behavior under biaxial tension. They claim that the slits make the strain distribution in the 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 the degree of strain scatter of the material in the biaxially loaded zone under load ratios of 4:2 and 4:4. They also determined experimentally the contours of plastic work for a strain range of 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 determine the yield surface in the vicinity of a current loading point by experimental testing on a coldrolled steel sheet and an aluminum alloy sheet. They claim that they verified the potential of 3

5 a new method for determining the yield surface by experiments. This new method consist of determining the yield surface by using an abrupt strain path change, which has been proposed earlier by Kuroda and Tvergaard [9]. Yu et al. [4] studied the sheet forming limit for complex strain paths. Using finite element models, they optimized a cruciform specimen with a reduced thickness area in the arms and in the central region, by changing the geometric parameters shown in Fig. 4. The optimal Fig. 4. Quarter of the cross specimen used by Yu et al. Fig. 5. Geometry of the cross specimen used by Wu et al. shape of the specimen has the most uniform stress distribution in the central region and is able to generate a large deformation during the stretching of the specimen. The deformation of the specimens was analyzed under uniaxial-tension and under biaxial-tension. The authors claim to have found that complex strain paths can be realized by adjusting the velocity ratios imposed on the specimen arms. Wu et al. [5] used the specimen shown in Fig. 5 to build a biaxial tensile test system that can realize complex loading. They built a biaxial tensile test system which is capable of realizing complex 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 extra high strength steel. In earlier studies a specimen was used as can be seen in Fig. 6 (a). The problem with this specimen was that failure in the material occurred before the desired stress and strain were reached in the biaxially loaded zone. Therefore a new model had to be developed. 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

6 necking in this area. The slits in the arms were changed to obtain the best stress/strain distribution in the biaxially loaded zone. The authors came up with two specimens as shown in 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 new specimen shapes is the best to use. The authors showed that these two new specimens allow one to study the mechanical behavior of extra high strength steel. Smits et al. [7] performed finite element simulations and experiments on several cruciform specimens to find an optimized specimen shape for biaxial testing of fibre reinforced composite laminates. 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 predicted by the finite element program. In geometry A, failure occurs in the arms. To relocate this 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 in high strains and possible failure of the specimen in the center. Changing the geometry of the corners was necessary, because the fibres at ±45 degrees carried the load from one arm to a perpendicular one, which resulted in unloading of the center of the specimen. In geometry D, a larger thickness reduction area was used. The strain variation over the most loaded axis of the milled zone was examined for each geometry, by finite element simulations and experiments. A higher strain was found in the experiments in the transition zone between the full thickness and the thickness reduced area compared to the finite element predictions. This difference 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 in the experiments. To determine which geometry is the best, also experimental biaxial failure data was obtained. The highest failure strains were found for geometry C, which indicates that this is the most optimal specimen. Failure started at the corners between the two arms for all geometries, but for geometry B and C the complete biaxially loaded test zone was damaged. 5

7 Summary In the papers discussed above, several different geometries were used. The used geometries have round corners, notched corners or slits in the arms. Some have a thickness reduction and 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 have not been investigated thoroughly. Most of the papers studied the specimen deformation up to the 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 this report focuses on investigating the stress and strain distribution in the cruciform specimen when different parts of the geometry are changed. 6

8 Chapter 2 Finite element simulations and results 2.1 Method The finite element simulations were carried out with the finite element package Marc/Mentat 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 symmetry and loading conditions, only a quarter of the cruciform has to be modeled. The used material is interstitial free (IF) steel with a thickness of 0.7 mm, a Young s modulus of 45 GPa and a Poisson s ratio of A hardening curve up to the point of necking (see Fig. 8) was used to describe the plastic yielding. All these values were determined from data obtained from a uniaxial tensile test of the same material, as part of a NIMR project MC a at TU/e. Mesh dependency has been investigated while simulating the geometries. It has been found that the mesh has no influence on the area of localization, only on the accuracy of the value of the stress. Fig. 8. Hardening curve used in the simulations Fig. 9. Sketch of the biaxial stage proposed by Kammrath and Weiss Each of the simulated geometries has a total length of 60 mm, taking the dimensions of the 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. The used load cells have a maximum capacity of 5000 N. The travel range is maximum 8 mm for each of the four clamping devices. To be able to place the stage under a microscope, the dimensions of the test stage should not exceed 260x260 mm 2. To find a suitable specimen geometry, the effect of different aspects of the geometry are studied as follows: first the effect of arm width is examined; second the effect of different 7

9 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 the relocation of the localization from the arms to the center of the specimen [7], different shapes of this reduction are also examined. Finally the most suitable geometry is loaded in two stages, first equi-biaxially and then uniaxially as well as vice versa, to see if the failure still occurs in the center of the specimen. For comparing the several geometries, the von Mises stress distribution in the material is determined. This stress gives a good insight into the overall magnitude of the stress and predicts the plastic deformation of the material under triaxial loading from results obtained from a uniaxial test [11]. The values of the von Mises stress in the material are indicated by a color code. An example of combinations of colors and values is shown in Fig. 10. Note that the 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 Effect of arm width To determine the effect of the width of the arms, geometries with sharp corners and an arm width 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 the middle of the arms. This means that in a geometry with rectangular arms, necking always occurs 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 the stretching for the specimen with a 10 mm arm width. From this figure it is clear that initially the highest stress is reached in the corners. However, this stress decreases once the arms start to 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

10 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 well exceeds the level which is needed in a uniaxial test to initiate necking ( i.e. the highest stress reached 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 large stress gradient going from the corner to the center of the specimen. The stress distribution in 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 stress reached in the center is almost the same for both specimens. The graph of the specimen with 30 mm arm width is not shown, because the simulations show the same effect as for the specimen with 20 mm arm width. Table 1 Maximum values of von Mises stress (MPa) reached in the corners region arm width 10 mm arm width 20 mm arm width 30 mm center corner Table 1 shows the maximum stress reached in the corner and in the center for the three specimens. The maximum stress reached is higher as the arm width is increased. The explanation for the increased stress in the center and corners is that the arm length is decreased and 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 drops as a result of the necking of the arms. Although the higher stress in the center is desired, the higher stress in the corners is undesired. This is because in the ideal geometry localization should take place in the biaxially loaded zone. This means that the stress gradient from corner to center is decreased so that necking could possibly begin in the corners. To minimize this possibility in the most suitable specimen, it is best to use the specimen in which the stress in the corners is the lowest: the specimen with an arm width of 10 mm Effect of corner shape The stress in the corners can be further lowered by changing the corner shape. To determine which corner shape decreases the stress the most, specimens with round and notched corners have been simulated with arms of 10 mm width. 9

11 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 of the arms. These maximum stress values are shown in table 2. The stress for the geometry Table 2 Maximum 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 a small 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 the corners 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: 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 of 3 mm, in the center, the arm and the corner of the specimen. If these curves are compared with the curves of the specimen with sharp corners and arm width 10 mm (Fig. 12 (a)), it appears that the stress in the corners now remains below that in the arms. The radius of 3 10

12 mm has thus entirely removed the stress concentration in the corners. At the same time, the stress 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, and 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 the root of the round corners with radii 1, 3 and 5 mm and the sharp corner (see Fig. 15). This makes it possible to compare this geometry to the geometry with round corners. Fig. 15 shows the final results of the simulations of the specimens with notched corners. (a) Radius notch: 2 1 mm (b) Radius notch: mm (c) Radius notch: 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 the radius of the notches is increased, the necking starts in these notches (Fig. 16 (b), (c)). This effect was also found in [2,6]. The explanation for these results is that the arms are reduced in width in the area where the notches are present. This area is therefore less strong and necking is more likely in that cross section. It seems that the notches need to have a certain radius 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 radius of mm. During the entire loading history, the stress is the highest in the area between these notches. Furthermore, the stress in the center of the specimen is larger than the stress in the arms. If this figure is compared to Fig. 14 (a), one observes that the stress in the center of the specimen with notches is higher than in the specimen with round corners. If a comparison is made with the sharp corner geometry, it seems that the stress in the center is now increased by approximately 10 %. So the notched geometry helps to increase the stress in the biaxially loaded zone, but it also weakens a certain area even more than the middle of the arms. This increases the danger of necking elsewhere than in the biaxially loaded zone. 11

13 2.2.3 Effect of numbers of arms In the industry different strain path behavior of sheet metal material is examined by forming limit tests on Nakazima strips as shown in Fig. 17. This test could possibly be simulated by an in-plane model, which in the most ideal case consists of a circular specimen which is radially 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 to apply in-plane loads, the specimen needs to have arms. Because four arms may not represent the ideal case, more arms could be introduced. The effect of more than four arms is therefore investigated by constructing specimens with 6 and 8 arms. To exclude the shape of the corner having an effect on the location of localization in a specimen with more arms, simulations have been made with all the three different corner shapes. The radii of the round corner and notched 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 specimen with four arms. For the specimen geometries with sharp corners first a high stress is found 12

14 in the corners of the specimen, followed by necking in the arms. For the specimens with the round corners necking occurs in the arms and for the specimens with notches the necking occurs between the notches. Hence, the increase in the number of arms has no effect on the localization. Table 3 shows the maximum stress values reached in the center for specimens with 4, 6 and 8 arms, for all three different corner shapes. As can be seen, the stress is slightly higher for the specimens with sharp and round corners as the number of arms is increased. The explanation 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 in the center. For the specimens with notches, the stress even decreases when more arms are used. This can be explained by the fact that the necking occurs further away from the center as the number of arms is increased. Table 3 Maximum values of von Mises stress (MPa) reached in the center corner shape 4 arms 6 arms 8 arms sharp round notch Although the stress is increased in the center, the necking still occurs in the arms, or at the transition from the center to the arms. Therefore more arms do not result in an improvement over four arms. This means that a specimen with more than four arms is not necessary for biaxial testing Effect of thickness reduction To make sure that necking of the material happens in the biaxially loaded zone, a thickness reduction in the center of the specimen is necessary. Several different shapes of thickness reductions are examined: one with the shape of a circle (specimen A), one with a shape of a cruciform (specimen B), one with both shapes (specimen C), and one with the shape of a bowl (specimen D). All geometries have round corners to decreases the danger of necking elsewhere than in the biaxially loaded zone. Specimen A In geometry A the biaxially loaded zone is homogeneously reduced in thickness in the form of a circle. First the necessary thickness reduction is examined by a two-dimensional analysis on specimens in which the thickness is reduced from 0.7 mm to respectively 0.5, 0.4 and 0.3 mm. Note that the transition from one thickness to the other is abrupt, because it is a two-dimensional analysis. The radius of the circle is 5 mm. This radius was chosen because a 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 mm the localization occurs in the biaxially loaded zone. In this geometry there are two types of localization regions, region A and region B, as can be seen in Fig. 21 (c). Region A is in the rounded corner between two arms and region B is in the top right area of the circle where the thickness has been reduced. Fig. 22 shows the stress-displacement curve for these two regions as well as for the center of the specimen. Elastic recovery occurs in the center, whereas the stress in regions A and B continues to increase. This means that the final necking takes place 13

15 (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 necking starts 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 change in 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 smaller circle has the necessary thickness of 0.3 mm. Therefore the effect of a smaller circle radius is examined first. This is done by a two-dimensional analysis on specimens with a thickness reduction 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 the arms. Hence, it appears that the minimum radius is approximately 4 mm. 14

16 For the more stepwise reduced specimen the thickness reduction therefore starts at a radius of 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 thickness reduction has no effect on the localization. To examine if a smooth thickness reduction changes the pattern, a three-dimensional finite element model has been made of this specimen. The cross 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 the corner of the specimen. Hence, a homogenous thickness in the center with a smooth thickness reduction does not give the desired result. Specimen B Next a two-dimensional analysis is performed on geometry B. In this geometry the thickness reduction has the form of a cruciform as shown in Fig. 26 (a). This shape enlarges the distance between the corner of the specimen and the thickness reduction area. The thickness of the reduction area is respectively 0.3 and 0.2 mm. The length of the arms of the inner cruciform is 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, to have 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 reduction area (region B). 15

17 (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 the specimen. The stress in regions A and B continues to increase, when the center starts to elastically recover. The stress in region B is again the highest during the entire loading process. 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 edge of the inner circle. This inner circle has a thickness of 0.2 mm, because the results of specimen B showed that otherwise necking occurs in the arms. Note that this is also a two-dimensional analysis. Fig. 28 (b) shows the results of the simulation. The stress localizes in the inner circle. But the highest stress is still not reached in the exact center of the specimen, but in the top right area of the inner circle, as expected. Although the stress localization is near the center, 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 making this area the weakest point of the specimen. Therefore a three-dimensional model is made in which the thickness is gradually reduced from both sides of the specimen to a minimum thickness in the center, as can be seen in Fig. 28 (a). The thickness reduction starts at a bigger radius than 5 mm, so now the arms do have a certain area where the thickness is reduced. Although this can result in necking of the arms, it seems to be necessary, because 16

18 (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 of the center point is 0.2 mm, to be sure that the thickness reduction suffices to force localization there. Although from earlier results a thickness reduction as used in specimen C is expected because 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 the wanted results. (a) Cross section of biaxially loaded zone. Units in 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, with slightly lower stresses in bands towards teh corners. Fig. 30 (a) shows the stress-displacement curve for both areas and for one arm of the specimen. At a displacement of 2.8 mm, the arms start 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 necking in the arms. When this stress is reached, the axial stress at the end of the arms is about 300 MPa. 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 ratio between 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 therefore seems to be suitable for biaxial testing up to the point of failure. 17

19 (a) Stress-displacement curve (b) Strain path Fig. 30. Stress-displacement curve and strain path of specimen D Complex loading Specimen type D has also been subjected to straining paths that are more general than equal biaxial, namely first equi-biaxial strain followed by uniaxial strain as well as vice versa. The final displacement ratio of x-arm to y-arm is 2:1. The results are shown in Fig. 31. As can be seen, 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 of specimen D under equi-biaxial loading. The only difference is that this stress in the corners is a bit more to the right, which can be explained by the fact that the x-arm has a higher displacement. (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 the x-arm starts to elastically recover at almost the same displacement for both loadings. The stress at which the center starts necking, is also almost the same. Note that the stress needed for necking in this model is slightly higher than that is needed for the necking to occur in the arms of the specimen without a thickness reduction. 18

20 (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 results show that the strain paths of the center can be controlled by adjusting the loadings of the arms of the specimen. This means that complex strain paths can be realized. Because failure starts in the center of the specimen and complex strain paths can be realized, this specimen has the most suitable geometry for examination of sheet metal yielding, necking and fracture for different stress states. (a) Uniaxial-biaxial (b) Biaxial-uniaxial Fig. 33. Strain paths of the center of specimen D under complex loading 19

21 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 the specimen, independent of the width of the arms. The stress in the corners can be decreased by using round corners. Using notches results in necking in the area where the notches reduce the specimen arm width, but this area is still part of the arms which are only under uniaxial loading. Using more arms shows essentially the same results as the simulations of the specimens with four arms. The specimen shapes with more than four arms do not perform better 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 simple cruciform specimen is not suited for biaxial loading up to the point of necking. Note that using slits, as proposed in [1, 3, 5, 6] to obtain a uniform stress distribution in the biaxially loaded zone up to the point of yielding, will not change this picture and it will weaken the arms 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 does not start in the center but in a certain area of the reduction zone. For a circular thickness reduction this occurs in the top right area of the circle. For a cruciform thickness reduction it occurs in the corner of this cruciform. Using a reduction of both a circle and a cruciform also does not result in the desired outcome, although the highest stress does occur in the neighborhood of the center. This is promising, but still not good enough. The only way to have the yielding and necking starting in the center of the specimen appears to be by making this 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 to have a very small thickness, compared to the thickness of the arms. Because the thickness of the specimen is already small, the thickness of the center must be very small and the question remains 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 the thickness reduction of the specimen and also more investigation on the mesh dependency has to be performed. Other investigations need to be performed on for example strengthening the arms, which will possibly result in a specimen which is physically easier to make. 20

22 Chapter 4 Conclusions Finite element simulations have been carried out on different cruciform specimen shapes to study the effect of different geometries. The objective was to obtain a specimen in which the yielding, necking and fracture occur at the center of the specimen, where a truly biaxial stress state 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, the necking 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 stress at the edge of this area. 5. A thickness reduction in the form a bowl results in yielding, necking and damage at the center of the specimen, even under complex loading. 21

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