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2 Journal of Materials Processing Technology 209 (2009) Contents lists available at ScienceDirect Journal of Materials Processing Technology journal homepage: Development of simple shear test for the measurement of work hardening Y.G. An, H. Vegter, J. Heijne Corus Research, Development & Technology, P.O. Box , 1970 CA IJmuiden, The Netherlands article info abstract Article history: Received 11 June 2008 Received in revised form 31 October 2008 Accepted 7 November 2008 Keywords: Simple shear Work hardening Experimental technique FEM simulation The simple shear test achieves large deformation without plastic instability in comparison with the uniaxial tensile test or the plane strain tensile test. However, the strain measurement can be very time consuming if the appropriate apparatus is not available. Moreover, the effect of specimen geometry on the measured work hardening has not been well investigated. The present work describes a simple shear test and a new strain extensometer based on a rotary angle transducer. FEM simulations and experimental work were carried out to evaluate the effect of specimen geometry on shear strain distribution, to study the effect of material parameters on the measured work hardening, and to correct the edge effect based on the geometry of test specimen. The correction method proposed for the simple shear test was validated experimentally Elsevier B.V. All rights reserved. 1. Introduction The accuracy of sheet metal forming simulations is highly dependant upon the constitutive equations. Recently, Barlat et al. (2005) and Banabic et al. (2005) proposed yield functions where anisotropy is introduced by means of linear transformations of the stress tensor and multiple fitting parameters to produce the desired anisotropy. While Vegter and van den Boogaard (2006) took Bezier interpolation as yield function based on measured yield stress in different strain modes. These descriptions are flexible enough to describe different yield loci shapes; however, they usually involve many parameters that have to be determined experimentally. As a result, the plastic properties derived from the uniaxial tensile test are not enough, and further experiments such as biaxial stretching, plane strain tension and even the simple shear tests are necessary. An and Vegter (1998, 2005) developed the through-thickness compression test and analysed the effect of friction on the measured work hardening. An et al. (2004) validated a novel and simple test procedure for an accurate measurement of work hardening in the plane strain tension. Bouvier et al. (2006) used simple shear test to characterise the plastic anisotropy of sheet material. Based on the measured plastic behaviour in different strain modes, parameters of complex yield functions can be fitted and used in finite element codes for sheet metal forming simulations. In order to define the plastic behaviour of sheet metals in the shear mode, Marciniak and Kolodziejski (1972) proposed a plane torsion test for the assessment of failure sensitivity. Tekkaya and Corresponding author. Tel.: ; fax: address: yuguo.an@corusgroup.com (Y.G. An). Pöhlandt (1982) developed a procedure for the plane torsion test to determine stress and strain curves. This test method, together with the inclined tensile test, was further analysed by Tekkaya et al. (1982a, 1982b). Later Miyauchi (1984) proposed the simple shear test which can achieve large deformation without plastic instability in comparison with the uniaxial tensile test. Unfortunately this test was not studied thoroughly and not widely used for the characterisation of the plastic behaviour of sheet metals. One of the reasons was that the test was difficult to perform, and the strain measurement was different from the conventional method. For accurate strain measurement, it can be quite time consuming if the appropriate method is not available. Moreover, appropriate procedures to correct the edge effect for accurate work hardening based on measured data in the simple shear test have not been fully developed yet. In the present work, a simple and robust strain extensometer is developed based on a rotary angle transducer. FEM simulations and experimental work were carried out to evaluate the effect of specimen geometry on shear strain distribution; to study the effect of material parameters on the measured work hardening; and to correct the edge effect based on the geometry of the test specimen. Finally the correction method proposed for the simple shear test was validated experimentally. 2. Description of experimental technique The test device for the simple shear test developed in the lab has two symmetrical shear zones as shown in Fig. 1. The specimen is fully clamped by three sets of clamping blocks and is acted on by an axial load. This test approach is quite similar to that developed by Miyauchi (1984). The main advantage of this arrangement is that the lateral forces acting on the shear zones are balanced, while only /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.jmatprotec

3 Y.G. An et al. / Journal of Materials Processing Technology 209 (2009) Fig. 3. The distribution of shear displacement across the shear width, indicating that the displacement is linearly distributed in the shear zone centre. Fig. 1. The experimental device for the simple shear test. a: rotary angle transducer, b: clamping block, and c: specimen. the axial force is measured for the calculation of nominal shear stress. As a consequence, almost no friction is involved in the test compared with the test methods where one shear zone is employed in the test. For the simple shear test, different specimen geometries can be used. In the present study, two geometries have been investigated as illustrated in Fig. 2. The first geometry is simply a rectangular strip, and the second is a strip with slits as used by Miyauchi (1984). The first specimen geometry is simple and easy to prepare, however, there may be some premature failure due to stress concentration at the edges of the specimen. The second specimen geometry can avoid the problem to some extent, however, the effect of specimen geometry on the stress and strain distribution and the measured work hardening has to be studied. Shear strain measurement with high accuracy cannot be easily achieved with a conventional extensometer for displacement. In order to develop a robust strain measuring method, the displacement field in the shear zone was investigated. Fig. 3 shows the displacement in the shear direction along the shear zone width for low carbon steel. The shear length was 35 mm and the shear width was 5 mm in the test. The result indicates that the displacement along the shear zone width is linearly distributed. Based on the measured information, an angular extensometer was made using a rotary angle transducer as shown in Fig. 1 for the measurement of the shear angle in the centre of shear zone. The rotary angle transducer is conventionally used to measure the rotation angle in mechanical engineering. In the simple shear test the relative displacement of the two legs connected to the rotary angle transducer causes the rotary angle transducer to rotate, so that the shear angle can be measured and the shear strain be calculated. 3. Evaluation of specimen geometry for the simple shear test For the simple shear test, different specimen geometries have been used, such as a piece of strip, or a specimen with slits. The effect of specimen geometry on the strain and stress distribution was evaluated using FEM simulations. Furthermore, the strain distribution was investigated experimentally using 1 mm grids and strain measurement system PHAST for low carbon steel DX54DZ. The averaged mechanical properties and plastic anisotropy r-value of the material are listed in Table 1. For FEM simulations, the FEM code Dieka which is an implicit code developed by the University of Twente was used (Dieka 9.1, 3D discrete shear triangle elements with two integration points through the thickness were used in the simulations. In the present investigation, two geometries were studied; a rect- Fig. 2. Illustrations of the specimen geometry for simple shear test. L and W are specimen length and effective shear width, respectively. White areas are the shear zones, and the grey areas are clamping zones.

4 4250 Y.G. An et al. / Journal of Materials Processing Technology 209 (2009) Table 1 The average mechanical properties of the material. Material. DX54DZ. 0.2 [MPa] UTS [MPa] u [%] t [%] r n Fig. 4. Mesh used for FEM simulation for specimen slits. angular specimen and a specimen with slits. For the rectangular specimen, the shear gauge width was 5 mm, for the specimen with slits, the effective shear gauge width, or the slit width was 5 mm, and the corner radius was 1.5 mm. The specimen length was 55 mm for both geometries. In the numerical simulations, only one shear zone of the specimen was modelled because of the symmetry of the test geometry. Fine meshes were used close to the edges to account for the localised deformation as shown in Fig. 4. The Hill 48 yield function and the Swift work hardening model, = 527( ε) 0.219, were used. For a comparison with the simulations, the simple shear test with specimen length of 55 mm and 35 mm were carried out for the two specimen geometries and the strain distributions were measured. The strain distributions, ε xx, in the FEM simulations for the two specimens are plotted in Fig. 5. As observed in the experiments, stress and strain concentrations at the edges due to clamping lead to fracture at large shear strains. At similar shear displacement, the specimen with slits has less local high deformation than rectangular specimen. For most low carbon steels, this local high strain is not a problem to achieve large shear strains. However, for ultrahigh strength steels, such as dual-phase steels with tensile strength of 600 MPa or higher, premature failure due to the stress concentration may occur. So a specimen with slits may delay premature failure to some extent. The shear stress and shear strain distribution along the shear length in the FEM simulations are shown in Fig. 6 for the rectangular specimen. Along the edges of the specimen, the strain mode is uniaxial tension or compression, so that the shear stress and shear strains are zero. The shear strain and stress build up rapidly from the Fig. 6. The shear stress and shear strain distribution along the centre of shear length for a rectangular specimen. In the simulation, a length of 50 mm and shear width of 5 mm is used. Fig. 7. Strain distribution along the centre of shear zone for the rectangular specimens measured using 1 mm dot grids and PHAST strain measurement system. edges, and after approximately 5 mm from the edges, they reach a stable level. This strain distribution characteristic is not dependent on material parameters, and is mainly due to the geometry. The strain distribution is confirmed by the experimental data measured by PHAST for low carbon steel as shown in Fig. 7. The results indicate that through the majority of shear length of the specimens the Fig. 5. The normal strain distribution, ε xx, close to the edges of specimen. At similar shear displacement, there is less stretching at the edges for the specimen with slits.

5 Y.G. An et al. / Journal of Materials Processing Technology 209 (2009) strains for the whole specimen. So the specimen with slits, independent of the clamping method, is not recommended in the shear test for the measurement of work hardening. 4. The measurement of work hardening and its correction If the inhomogeneity of the shear stress distribution near the free edges is neglected, the shear stress can be evaluated by dividing the recorded shearing load by the area of the shear cross-section of the sample. Therefore, the average shear stress is given by av = F S (1) Fig. 8. The shear stress and shear strain distribution along the centre of shear length for specimen with slits. In the simulation, an effective shear length of 45 mm and shear width of 5 mm are used. shear strains are homogenously distributed except near the edges, provided that the specimen length is not too small. The shear stress and shear strain distribution in the FEM simulations along the shear length are shown in Fig. 8 for the specimen with slits. Similar to the rectangular specimen, the strain mode is uniaxial tension or compression along the edges, so the shear stress and shear strains are zero. The shear strain and stress build up rapidly from the edges, and after approximately 3 mm from the edges, they reach a maxima. After that the stress and strain start to decline with increasing distance towards the specimen centre. For a specimen with a shear width of 5 mm, at least 15 mm is needed before the shear strain reaches a stable value. Such a strain distribution characteristic is not dependent on material parameters, and is mainly due to the geometry like rectangular specimen. The strain distribution is observed experimentally in the PHAST measured data for low carbon steel as shown in Fig. 9. The results indicate that the shear strain magnitudes are quite different even they have similar shear displacement. Further, no stable strain distribution can be observed in the middle of the specimen with a specimen length of 35 mm. The strain distribution in the FEM simulations and experimentally measured results indicate that the shear strains are homogenously distributed through the majority of the shear length of the rectangular specimens except near the edges. While for the specimen with slits, the strain distribution is inhomogeneous and the strain measurement on one spot is not representative of the where F is the imposed force measured with the load cell of the testing machine, and S is the area of the shear cross-section of the rectangular specimen, respectively. As shown in the simulation results in the previous section, the stress and strain distributions are not homogeneous, so the geometry of the specimen will influence the accuracy of the measured work hardening. This effect needs to be investigated by FEM simulations. In the simple shear test, the stress tensor has the components in the test frame; ( ) xx av 0 [] = av yy 0 (2) the components xx and yy being due to the rotation of material axis relative to the shear direction. In order to estimate the error in approximating the shear stress by this average value, the shear stress ratio is defined as follows: K s = av (3) cm cm = ( xx yy 2 ) xy (4) where the cm refers to the maximum shear stress at the centre of the specimen. For the calculation of this stress, all the stress components in the FEM simulations, xx and yy, xy are taken into account. Based on our FEM simulations, the specimen geometry has a large effect on the measured shear stress. As a result, a geometric parameter has been defined as follows: G s = L (5) W where L and W refers to the specimen length and shear width of the rectangular specimen, respectively. This parameter is used in the following FEM simulations and experiments. Strain measurements in the modelled simple shear test were made using a virtual extensometer, simulated by the relative displacement of two nodes in the centre of the specimen. The deformation load was also calculated in the simulation, so that the averaged stress and strain data could be calculated in the same manner as in the test Effect of material parameters Fig. 9. Strain distribution along the centre of shear zone for specimens with slits measured using 1 mm dot grids and PHAST strain measurement system. Since the strain state varied between simple shear in the middle of the specimen and uniaxial tension along the edges, the thinning or thickening of material in the neighbourhood of the edges depends on the plastic anisotropy r-value. The relative strength of the uniaxial state and shear state is another factor that may influence the measured shear stress in the test. To study these effects, the relevant parameters were varied in the FEM simulations as shown in Table 2. The materials were assumed to be planar isotropic and the Hill 48 yield loci and the Von Mises were used in simulations. In

6 4252 Y.G. An et al. / Journal of Materials Processing Technology 209 (2009) Table 2 The stress factors for the yield loci used in the FEM simulations a. Model r fu fs Hill Von Mises VM r VM r a fu and fs refer to the stress factor in the uniaxial and shear state, respectively. addition, the Von Mises yield function was modified and described by the Vegter model using different r-values; VM r0.5 and VM r1.35 where r-values of 0.5 and 1.35 were used instead of 1, so that the effects can be easily investigated. The Swift work hardening model for low carbon steel, = 527( ε) 0.219, was used in the simulations. The effect of r-value on the measured shear stress was investigated using the Von Mises yield function and the Vegter model that describes two modified variants of the Von Mises yield function with different r-values. The VM r0.5 and VM r1.35 yield loci have the same uniaxial stress point, shear stress point, plane strain point and equi-bialxia point as the Von Mises yield function but different r-values, so the effect of r-value can be studied. In the FEM simulations two specimen geometries, G s = 10 and 5, were used as shown in Fig. 10. The results indicate that the effect of r-value is minor, while the effect of specimen geometry is significant. For specimen with a geometry ratio of G s = 10, the effect of r-value on the shear stress ratio is negligible. For a geometry ratio of G s = 5, the effect was slightly increased at large strains, but still too small to be measured in practice. At the same r-value, the difference in the shear stress ratio between the two geometries is approximately 2%. The effect of relative strength in the shear state and the uniaxial state on the shear stress ratio is shown in Fig. 11. Because the effect of r-value is negligible, the effect of relative shear strength on the shear stress ratio was studied using the Hill 48 yield function with an r-value of 2, and the Von Mises yield function. The shear stress factor for these two descriptions is and 0.548, respectively. The results indicate that the effect of geometry remains the same, approximately 2%, at the same shear stress factor. The effect of relative shear strength becomes significant for a small geometry ratio and increasing strain. At the geometry ratio of G s =10 and shear strain of 0.5, the difference is approximately 0.5%. At a small geometry ratio G s = 5, the difference remains small, and until a shear strain of 0.5 it is still within 1%. In short summary, the simulation results clearly show that the material parameters have only very limited influence on the measured shear stress in the test. The Fig. 11. The effect of shear point on the yield loci on the shear stress ratio for different geometries. The Hill 48 and Von Mises yield function have a shear stress factor of and 0.577, respectively. most important factor for the accuracy of measured shear stress is the geometry Effect of specimen geometry ratio on the stress ratio Relationship between shear stress ratio and geometry ratio The relationship between the shear stress ratio and the geometry ratio was analysed based on the strain distributions of the rectangular specimen that were numerically calculated and experimentally measured. If the specimen geometry ratio is large enough, then the strain distribution is homogeneous in the centre of the specimen. The total shear force, F t, is a summation of shear force in the centre and of the shear force over the shear length close to the edges, F t = F cm + 2F e (6) if the total shear length, l t, is long enough, consisting of a shear length with homogeneous shear stress, l c, and shear length with inhomogeneous shear stress distribution, l e, then l e is calculated as l e = l t l c (7) 2 because the shear length with inhomogeneous shear stress distribution is influenced by material properties and shear width, W, the shear length, l e, can be written as l e = f ew 2 = f el t (8) 2G s here f e is an unknown material parameter that has influence on the shear length with inhomogeneous shear stress distribution. the total shear force can be written as F t = cm l c + e 2l e (9) here e is the equivalent shear stress over the shear length l e. The l c and l e can be substituted by l t, the shear force is expressed as Fig. 10. The effect of r-value on the shear stress ratio for different geometries. In the simulations, the r-value was modified with the rest of yield loci not touched. F t = cm l t + f e ( e cm ) l t (10) G s The averaged shear stress is calculated by the total shear force over total shear length, and the shear stress ratio is the ratio of averaged shear stress over shear stress in the centre, then the relationship between shear stress ratio and inverse specimen geometry ratio is obtained as 1 K s = f e cm e cm /G s (11)

7 Y.G. An et al. / Journal of Materials Processing Technology 209 (2009) Fig. 12. The relationship of specimen geometry ratio and shear stress ratio obtained in FEM simulation. Fig. 14. The measured shear stress and strain for specimens with geometry ratio of 6, 8 and 10. Corrected shear stress and strain curves were plotted in the graph. here the slope of the linear relationship, f e ( cm e )/ cm, is a function of the material properties and unknown. It has to be solved by FEM simulations for the shear stress correction Effect of geometry ratio on shear stress ratio based on FEM The effect of specimen geometry on the shear stress was investigated using different specimen geometry ratios, G s = L/W, infem simulations. In the previous section, it has been shown that the material parameters do not play a significant role for the accuracy of the measured shear stress in the test. In this study of specimen geometry, only Von Mises yield function was used in simulations. As shown in Fig. 11, the shear stress ratio does not decline much with shear deformation, in particular for specimens with a large geometry ratio; the shear stress ratio at a shear strain of 0.20 was taken as an index to study the effect of geometry ratio on the shear stress ratio. The relationship between the geometry ratio and the shear stress ratio is plotted in Fig. 12. The result indicates that if the shear zone is infinitely long and narrow, the measured averaged shear stress is approaching the maximum shear stress. At a geometry ratio of 10, that is to say at a shear length 10 times larger than the shear zone width, the shear stress ratio is approximately 2.5% lower than the maximum shear stress. While a geometry ratio of 5 will lead to approximately 5% lower shear stress. If the effect of material parameters is taken into account, the geometry ratio of more than 10 is preferred based on the argument that most low carbon steels have a yield locus between the Von Mises and the Hill 48 with an averaged r-value of 2.0, and that the shear stress factor is less influ- enced by material variations and plastic deformation. The geometry ratio of 10 is realistic and can be easily realised in practice. Then the 2.5% difference can be used as a correction factor for the measured work hardening data in the simple shear test, independent of the materials tested. 5. Experimental validation of measured work hardening To validate the simple shear test, low carbon steel DX54DZ was tested. The averaged mechanical properties and plastic anisotropy r-values are listed in Table 1. The test was carried out on a MTS hydraulic machine that has a maximum load capacity of 300 kn. The crosshead speed was 0.10 mm/s, which gave a strain rate of approximately s 1. Rectangular specimens with different geometry ratios were used for the validation. The deformed specimen is shown in Fig. 13. Fig. 14 shows the measured shear stress and shear strain curves for the geometry ratio of 6, 8 and 10. There are some differences between different geometries. The stress and strain curve for G s =6 is the lowest amongst the three measured curves. Based on the correction curve shown in Fig. 12, corrections were made to the measured data and the corrected curves were also plotted in the graph. After the correction, the corrected curves overlap for G s =8 and 10, while the corrected curve for G s = 6 is still lower than those for the other two geometries. The results indicate that the correction gives approximately the right values for the shear stress if the geometry ratio is larger than Conclusions Fig. 13. The sample for the simple shear test. a: clamping zones, b: middle clamping zone and c: shear zones. (1) A simple shear test apparatus with new robust angular measurement has been established which gives reproducible shear stress and shear strain data. (2) Two different specimen geometries have been evaluated, based on numerical simulations and experimental data. For rectangular specimens, the strain distribution is quite homogeneous in the majority of shear length of specimen, while for specimens with slits, the strain distribution is not homogeneous over majority of shear length. It is recommended that the rectangular specimen, not the specimen with slits, be used for the measurement of strain and strain data. (3) Numerical simulation of the simple shear test indicates that material parameters, such as the plastic anisotropy r-value, the shear stress factor, have a minor effect on the measured shear stress, provided that the geometry ratio of the specimen is

8 4254 Y.G. An et al. / Journal of Materials Processing Technology 209 (2009) large enough. Otherwise the effect of material properties may become significant for specimen with a small geometry ratio. (4) Numerical simulation of the simple shear test indicates that the specimen geometry has a critical influence on the accuracy of the measured shear stress. Based on the simulations, a correction curve for the measured shear stress was derived. It is suggested that a geometry ratio of more than 8 should be used, so that the measured shear stress can be reliably corrected. References An, Y.G., Vegter, H., The difference in plastic behaviour between bulging test and compression test for sheet steels and aluminium alloys. In: Proceedings of the Working Groups Meeting of the IDDRG, Genval, Belgium, pp. M1 5. An, Y.G., Vegter, H., Elliott, L., A novel and simple method for the measurement of plane strain work hardening. J. Mater. Process. Technol , An, Y.G., Vegter, H., Analytical and experimental study of frictional behaviour in through-thickness compression test. J. Mater. Process. Technol. 160, Banabic, D., Aretz, H., Comsa, D.S., Paraianu, L., An improved analytical description of orthotropy in metallic sheets. Int. J. Plasticity 21 (3), Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., Linear transformation based anisotropic yield functions. Int. J. Plasticity 21, Bouvier, S., Haddadi, H., Levee, P., Teodosiu, C., Simple shear tests: experimental techniques and characterisation of the plastic anisotropy of rolled sheets at large strains. J. Mater. Process. Technol. 172, Marciniak, Z., Kolodziejski, J., Assessment of sheet metal failure sensitivity by method of torsioning the rings. In: Proceedings of the 7th Biannual Congr. of the IDDRG, Amsterdam, pp. s Miyauchi, K., A proposal of a planar simple shear test in sheet metals. Sci. Pap. RIKEN 78, Tekkaya, A., Pöhlandt, K., Determining stress strain curves of sheet metal in the plane torsion test. Ann. CIRP 31, Tekkaya, A., Pöhlandt, K., Dannenmann, E., 1982a. Methoden zur Bestimmung der Fließkurven von Blechwerkstoffen. Ein Überblick, Teil I. Blech Rohre Profile 29 (9), Tekkaya, A., Pöhlandt, K., Dannenmann, E., 1982b. Methoden zur Bestimmung der Fließkurven von Blechwerkstoffen. Ein Überblick, Teil II. Blech Rohre Profile 29 (10), Vegter, H., van den Boogaard, A.H., A plane stress yield function for anisotropic sheet material by interpolation of biaxial stress states. Int. J. Plasticity 22,