Pure Bending Determination of StressStrain Curves for an Aluminum Alloy


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1 Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. Pure Bending Deterination of StressStrain Curves for an Aluinu Alloy D. TorresFranco, G. UrriolagoitiaSosa, G. UrriolagoitiaCalderón, L. H. HernándezGóez, A. olinaballinas, C. TorresTorres, B. RoeroÁngeles Abstract.  The uniaxial tensile or copressive test provides a fundaental indication of strength and stiffness properties for a given aterial. The priary concept to characterize a aterial by axial testing is changing. Currently, there is a pure bending experiental ethod by which it is possible to obtain siultaneously tensile and copressive stressstrain curves. In this paper, it is presented a pure bending ethod to deterine the stressstrain behavior of a aterial. Results were obtained on final anufactured speciens fro Aluinu alloy 6063T5. The advantages of this ethod are related ainly with its versatility and easy ipleentation. Besides, it deonstrates feasibility to deterine previous loading history in a coponent. Additionally it does not need extensive echanical preparation of speciens. The ethod presented is based on calculations fro the value of the bending oent and the corresponding value of the strain produced in the upper and lower surfaces of the specien. Bending experiental results were copared against axial testing. Results and the affinity derived fro the balance in stressstrain curves are satisfactory. Index Ters Analytical ethods, pure bending test and stressstrain curves T I. INTRODUCTION HE stressstrain curves are the best way to describe the echanical behavior of aterials. The ost significant property of a solid is its echanical stability. In other words the aterial resistance is strongly related with shape and size, by the application of an external agent []. The echanical properties describe how a aterial is affected by internal as well as external agents, including forces in tension, copression, ipact, fatigue, etc []. The echanical properties are very iportant for the selection of aterials with requireents that arise fro a particular use. Very suitable aterials for certain applications can be copletely useless for others jobs. Therefore, before selecting a aterial, it is iportant to anuscript received arch 07, 0; revised arch 7, 0. This wor was supported in the National Polytechnic Institute (IPN) and the National Council for Science and Technology (CONACyT). D. TorresFranco, G. UrriolagoitiaSosa, G. UrriolagoitiaCalderon, L.H. HernándezGóez, A. olinaballinas, C. TorresTorres and B. RoeroÁngeles are with the National Polytechnic Institute in Section of Postgraduate Studies and Research, School of echanical and Electrical Engineering, building 5, nd floor, Adolfo Lopez ateos Professional Unit Zacatenco, Col. Lindavista, C.P , exico D.F., exico, phone: ; Ext.: (eail: hotail.co, ISBN: ISSN: (Print); ISSN: (Online) consider what will be expected fro the coponent at service. So, it is iportant to deterine which echanical properties are required at service, so that the aterial can present an appropriate response [3]. The aterials ay be characterized through tests that are standardized. Axial testing to obtain the echanical properties usually can be referred as tension and copression tests. However, in the engineering field it has been opted for a different alternative in order to deterine the aterial behavior. In this sense, bending tests provide ultiple benefits; three of the ost iportant aspects of pure bending tests are () the siultaneous deterination of the tensile and copressive curves; () the agnitude and effect of prior loading history into the aterial, and finally; (3) it is virtually unnecessary to produce a specific or standardized speciens to perfor the test [], [4]. The use of a bending test to deterine echanical properties was investigated for the first tie by Herbert in 90, who published a study showing results in ters of tensile and copressive stress strain curves of a cast Iron aterial [5]. A few years later, siilar experiental techniques were perfored by Nadai [6] and arin [3]. By 98, Laws [7] published a research on a ethod by which he calculated stressstrain curves through the data of a bending test for a coposite aterial. Results were copared against those obtained fro an axial tensile test. One year later ayville and Finnie [8] conducted a study on a bending ethod. They found the stress strain tensile and copressive behavior for three different aterials that were previously prestrained. In recent years, UrriolagoitiaSosa et al. [9] published a new bending ethod. It is proposed an equation to obtain siultaneously the tensile and copressive behavior in beas with strain hardening and Bauschinger effect. This procedure allows to deterine the effect on the induced residual stress field. Additionally, Schajer and An [] developed an inverse calculation for the siultaneous deterination of stress strain curves fro bending test data. They used the inverse calculation to reduce the possible error percentage found by other researchers. This paper presents an analytical ethod that is based on the bending test for providing stress strain curves related with both directions of loading (tension and copression). The coponents tested are pieces directly used by the autootive industry in exico. The results (stressstrain relationship) obtained through this new analytical approach are copared with experiental axial tensile data and copression test evidence, which validate the proposed ethod. WCE 0
2 Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. A. Theoretical Details [9] II. ETHODOLOGY This new proposed ethod is based on the assuption that the tensile and copressive stress strain curves of a aterial have an asyetric distribution. Furtherore, it is considered that the specien has a cross section with wide b and height h, which is subjected to pure bending oent. Also, the cross section plane reains flat before and after the bending [0]. The corresponding strains and stresses at the top and botto surfaces of the speciens are given by t, c, σ t and σ c respectively. The instantaneous location of the neutral axis of the bea for the hypothetical state is assued to be at a distance h t and h c fro the tensile and copression surfaces. The total depth h, is equal to the su of h t and h c. A linear strain distribution across the depth of the specien is considered following Tiosheno s assuption. Under pure bending condition, the neutral axis is located by iposing the equilibriu condition that the total axial force vanishes. The second condition required for the solution is the equilibriu of internal and external oents. Let the strain range t, be described by a series of strain increents, j, and let the corresponding atching strain increent for c be. The strains and strain increents for given oent levels are obtained fro test data. At any given instance of loading, and considering axial equilibriu and areas under the stress strain curve on the tensile and copressive sides, the equation for the total nuber of increents is []: dissipated on the tensile side as on the copressive sides over the increent of j. Therefore (4) can be written as: ( + ) t c [ σ ( + )] Assuing that the stresses and strain at increents to  have been deterined, the tensile stress at any increent can be obtained fro (6) as: σ t jσ / t where, / ( + ) t c [ ] and ( ). Alternatively we can write it as; c + t [ ( + )] (6) (7) (8) σ (9) and by considering any two consecutive increents  and, and generalizing for tensile and copression stresses, we can write (8) as; σ σ () σ α (0) α It is possible to write the equation for equilibriu of internal and external oents as: σ + σ () t t c c Fro the linearity of strain distribution across the depth it follows that: t c (3) t c + ( ) t Therefore () can be rewritten as: ( + ) t c c [ σ + σ ] Fro (), it can be deduced that for each increent j σ σ (5) (4) This iplies equality of total strain energy stored or ISBN: ISSN: (Print); ISSN: (Online) where α t for tension stress and α c for copression stress. The (9) gives the average stresses at any increent.. The tension and copression stresses in this wor were obtained using (0) [] [3]. B. Experiental Procedure and aterial The aterial used for the experients was an alloy of Aluinu 6063T5 [4]. Eight speciens of square cross section (.7 x.7 ) with a total length of 50 were used. Bending tests were developed on a four points configuration [0]. The specien is part of a bar used as loc lever for car s door and it was decided to siplify the proble by transforing the geoetry of the coponent into a bea. A heat treatent procedure of annealing was used to free any previous load history into the speciens. The speciens were placed into an oven and heated at 50 c for 5 inutes. Finally, the saples were slowly cooled down inside the oven. All the speciens were instruented with strain gauges (EA LZ0 [5]) at the top and botto surfaces (Figure ). The strain gauges are extreely helpful to obtain the strain data caused by the application of the oent into the specien. The bending oent produces a siultaneous tensile and copressive deforation into the bea. The WCE 0
3 Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. experiental bending procedure was perfored in order to collect inforation about deforations and to calculate the stress to characterize the aterial. The best anner to produce a pure bending oent by an experiental test is by the use of a four point bending rig (Figure ). the tie to apply the bending oent. Additionally, in order to reduce the possible friction between the eleents, a strip of silicone PTFE, thic was added between the seicircle indentation and the cylindrical bloc. C. Results and discussions Before bending tests were carried out, 3 tensile and 3 copressive speciens were prepared fro the sae batch of 6063T5 aluinu alloy aterial. The speciens were annealed, and tests were perfored by following standards [6] [7]. The average tensile result obtained fro a tensile test in a for of a stress strain curve is illustrated in Figure 3. It is iportant to state, that because the aterial was annealed, its behavior was siilar in tension as in copression, so it was decided do not present the copressive stress strain curve. Fig.. The position of the gauges on the bea in Aluinu alloy. Fig. 3. Tensile results for an annealed Aluinu alloy speciens Fig.. Syste of four points of loading In order to characterize the aterial, the theoretical developent for the bending ethod was based on the assuption that a pure bending oent configuration will not produce shear stress into the specien and the bending oent will be always constant at the central section of the bea. These considerations allow describe tensile and copressive stress properties fro strain data. Figure, shows the configuration of the fourpoint bending rig and the set up of the specien. It is iportant to stand out that the four point rig has several circular blocs ade fro steel, which are positioned in a seicircle indentation produced in the ain blocs of the rig. The principal purpose of these cylinders is to allow free bending oveent of the specien, so no pulling will be produced at ISBN: ISSN: (Print); ISSN: (Online) Fig. 4. Four point bending procedure, strain data against oent The bending experiental procedure was perfored in a hydraulic axial achine. For each test a bending oent of 55 N was applied. The strain data obtained by the oent WCE 0
4 Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. effect fro the bending procedure presented in this research can be observed in Figure 4. Fro the strain data collected in the bending test, it can be observed that the strain data fro the top and botto surfaces are alost identical. Fro this observation it can be concluded, that the bending procedure was perfored correctly. The annealing process erased the stress strain prior loading history in the aterial and that the stress strain curve of the coponent is virtually identical in tension and copression. It is iportant to ention that the results of these cases are the average of five bending tests. stresses) in both directions for deterining anisotropic behavior. In this sense, both tensile and copressive axial testing ought to be perfored to deterine any prior loading history, but in ost cases the cause cannot be deterined, because the preparation of the noralized speciens will add a loading history to the coponent. The new analytical ethod for deterination of the siultaneous behavior in tension and copression by a stressstrain curve can reduce the nuber of tests to be ade to fully characterize the aterial; oreover resources could be saved by the application of the bending procedure. For the bending procedure, it is also necessary to perfor at least three testing procedures for the characterization of the aterial. In any case, the entire tensile and copressive easureents ust be perfored. Additional speciens that have a flat geoetry can be tested without adding any anufacturing process. Fig. 5. Coparison of stress strain results for a 6063T5 Aluinu alloy Figure 5 presents the results of both tests (axial tension and four point bending), showing the relation between the stressstrain curves using the bending ethod presented in this paper against the axial tensile test. It is clear that the results are alost identical between both ethods. However, there is a sall difference in the yield region between stressstrain curves, being slightly greater the yield strength found by the bending procedure. UrriolagoitiasSosa et. al. [], [] have explained this effect as a consequence of the abrupt change in the stress strain behavior and the atheatical procedure used by the equation syste applied for this case. Nevertheless, it is clear that the bending procedure has provided an efficient ethodology to characterize the aterial, and several echanical properties can be obtained as the axial tensile test. Bending test is considered as a new ethod to deterine the echanical properties in a aterial, but it has not been noralized yet. However, it is a test that provides soe advantages and it has not been properly exploited, because ost of laboratories engaged in echanical testing consider that the tensile test is the ost efficient and correct one, which in ost of the circustances this is not so. Nonetheless, this new bending ethod can get siultaneously the properties of a aterial in tension and copression. Also it can be used to deterine possible prior loading history (hoogenous loading, strain hardening and Bauschinger effect) and non hoogeneous loading (residual III. CONCLUSION An analytical ethod for calculation tension and copression stressstrain curves exhibited during the characterization of annealed aterial bending is presented. The resulting stressstrain curves obtained fro the bending ethod applied to bea speciens closely agree when they are copared against the axial experiental procedure for all cases. This eans that the evaluation of the bending ethod with the proposed conditions was generally correctly. This ethod can be used in any case where the behavior of the stressstrain relation is an iportant issue. It is worth rearing that this technique could also be applied to a aterial with previous history. However, anticlastic effects could be a proble, since it is not possible to deterine the ultiate stress with the approach of the bending test. The ultiate stress ay be easily achieved in an axial test, due to the action of the load, where the specien starts to be narrowed, so this can be easured directly. It would be interesting to found how could be deterined the ultiate resistance by using the bending approach. ACKNOWLEDGENT The authors appreciate the econoic support and equipent provided by the National Council for Science and Technology (CONACYT) and of the National Polytechnic Institute (IPN) for this study. Also, the authors indly acnowledge the financial support fro the Institute of Science and Technology of Federal District in exico. REFERENCES [] G. S. Schajer, Y. An, Inverse calculation of uniaxial stress strain curves fro bending data, J. Eng. at. Tech, vol. 3, no. 4, pp , 009. [] G. UrriolagoitiaSosa, J. F. Durodola and N. A. Fellows, Deterination of tensile and copressive stress strain curves fro bend tests, Appl. ech. at., , pp , 004. [3] J. arin, echanical Behavior of Engineering aterials, Ed. London: PrenticeHall, 96, pp. 97. [4] G. UrriolagoitiaSosa, J. F. Durodola and N. A. Fellows, Deterination of residual stress in beas under Bauschinger effect using surface strain easureents, Strain, vol. 39, no. 4, pp , 003. ISBN: ISSN: (Print); ISSN: (Online) WCE 0
5 Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. [5] A. Herbert, Über den Zusaenhang der Biegungselastizatät des Gusseinsens it seiner Zugund Drucelastizatät (On the Connetion Between the Bending Deforation of Cast Iron With Its Tension and Copression Deforation), Z. Ver. Dtsch. Ing., vol. 54, pp. 398, 90. [6] A. Nadai, Plasticity, Ed. New Yor: c Graw Hill, 93, pp. 65. [7] V. Laws, Derivation of the Tensile StressStrain Curve fro Bending Data, J. at. Sci., vol. 6, pp , 98. [8] A. R. ayville and I. Finnie, Uniaxial StressStrain Curves fro a Bending Test, Exp. ech., vol., no. 6, pp. 970, 98. [9] G. UrriolagoitiaSosa, J. F. Durodola and N. A. Fellows, A ethod for the Siultaneous Derivation of Tensile and Copressive Behavior of aterials Under Bauschinger Effect Using Bend Tests, Proc. Int. ech. Eng. Part C; J. ech. Eng. Sci., vol. 0, pp , 006. [0] N. E. Dowling, echanical Behavior of aterials, Ed. New Jersey: PrenticeHall, 993, pp. 3. [] S. P. Tiosheno and J.. Gere, echanics of aterials, 3th ed., Ed. London: Chapan and Hall, 99, pp [] G. UrriolagoitiaSosa, Analysis of Prior Strain History Effect on echanical Properties and Residual Stresses in Beas, PhD Thesis Oxford Brooes University, 005, pp [3] D. TorresFranco, Design of a New echanical Test to Identify aterial Properties, Specialty Engineering Thesis, National Polytechnic Institute, exico, 007, pp [4] etals Depot, Steel, Aluinu, Stainless and Brass, etals Catalog, etals Depot, pp. 9, 99. [5] Student anual for Strain Gage Technology, easureent Group, Bolletin 309D, easureent Group, pp. 73, 99. [6] BSI 000, Tensile Testing of etallic aterials; part and 4, British Standards Institution, BS EN 000 and BS EN 0004, 990. [7] AST E989a, Standard Test ethods of Copression Testing of etallic aterials at Roo Teperature, AST International E989a, pp. 8, 000. ISBN: ISSN: (Print); ISSN: (Online) WCE 0
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