Manufacturing Letters 21 (2019) 50–55 Contents lists available at ScienceDirect Manufacturing Letters journal homepage: www.elsevier.com/locate/mfglet The effect of deformation gradient on necking and failure in hole expansion test Surajit Kumar Paul Department of Mechanical Engineering, Indian Institute of Technology Patna, Bihta, Bihar 801106, India a r t i c l e i n f o Article history: Received 30 May 2019 Received in revised form 25 June 2019 Accepted 11 August 2019 Available online 12 August 2019 Keywords: Uniaxial tensile test Hole expansion ratio Diffuse neck Localized neck Finite element simulation a b s t r a c t Hole expansion ratio (HER) is widely used to represent stretch-flangeability of sheet metal. The state of stress at the edge of central hole is uniaxial tensile in nature during hole expansion test (HET). The strain/deformation is uniform throughout the width of the sample prior to the commencement of necking in a tensile test specimen. However, finite element investigation confirms the presence of prominent strain/ deformation gradient in HET sample. Only one free edge i.e. central hole edge presents in HET sample. These two effects are responsible for the higher HER than the uniaxial tensile total elongation of the material. Ó 2019 Society of Manufacturing Engineers (SME). Published by Elsevier Ltd. All rights reserved. 1. Introduction Stretch flangeability of sheet metal can be described as an ability to resist an edge crack during edge stretching deformation of sheet metal. Hole expansion ratio (HER) calculated from hole expansion test (HET) is normally utilized as a measure to know the stretch flangeability of sheet metal. HER can be defined as HER ¼ 100 df di di ð1Þ where di and df are the initial and finial diameter of the central hole respectively. Normally after observation of thickness crack at the central hole edge of the sheet metal sample, the HET is stopped. Paul [1] and Yoon et al. [2] performed finite element (FE) simulation of HET and reported that uniaxial tensile deformation takes place at the central hole edge where cracking initiates. More accurately, the stress state in gauge length of uniaxial tensile and central hole edge of HET samples are alike. On the other hand, uniaxial tensile test cannot be used to predict HER directly. Many researchers have tried to correlate HER with various tensile properties like ultimate tensile stress [3,4], yield stress to ultimate tensile stress ratio [5], post uniform elongation [2,3], strain rate sensitivity [2], coefficient of normal anisotropy [4] etc. However, these correlations are found to be valid for limited and similar kind of materials. Researchers also reported various correlation between the HER and microstructural features of materials [6–8] but limited to similar type of materials. E-mail addresses: surajit@iitp.ac.in, paulsurajit@yahoo.co.in Researchers also pointed out that the fabrication of central hole by punching operation in the specimen for the HET creates dimples and micro-cracks at the central hole shear edge [9–11]. This fact motivated research groups to assume HER as a fracture based parameter, so as to correlate with fracture toughness [2,9,10,12] and notch mouth opening displacement [13]. Eq. (1) clearly indicates that, HER is purely an elongation based dimensionless parameter. In p contrast, the term fracture toughness having a unit MPa m, contains information regarding material’s strength. For that reason, establishing a correlation with a purely elongation based parameter (i.e. HER) and strength/energy based parameter (i.e. fracture toughness) is not physics based. Paul [14] determined local maximum diffuse strain in a uniaxial tensile specimen from digital image correlation based technique. He showed that value of HER considering defect free (i.e. EDM machined) central hole is roughly identical to the local maximum diffuse strain measured from the uniaxial tensile test. He also proposed that the presence of deformation gradient terminates/delayes the onset of diffuse necking in HET. Literature review suggests that the reason behind higher HER than the total tensile strain of the material is still not clear and need to be understood. Hence, an effort has been made in the current work to find out the reason behind it. A detailed FE simulation is done in present investigation to understand the effect of deformation gradient in necking/failure during HET. 2. Experiment and finite element simulation Cold rolled dual phase (DP) (an advance high strength steel (AHSS) which contains ferrite and martensite phases) steel is https://doi.org/10.1016/j.mfglet.2019.08.004 2213-8463/Ó 2019 Society of Manufacturing Engineers (SME). Published by Elsevier Ltd. All rights reserved. 51 S.K. Paul / Manufacturing Letters 21 (2019) 50–55 selected for the present study. The initial thickness of sheet is around 1.4 mm. Edge cracking during stretch flanging operation of soft mild steel is not a problem, however it is a major restriction for the case of AHSS in regard to their application. For this reason, DP steel is selected for this study. A 60° conical punch is used for HET. A 10 mm diameter center hole is present in a 100  100 mm2 square shape sheet specimen. The center hole is prepared by both punching and electrical discharge machining (EDM) process. The punching is done with a clearance of 12% [15,16]. A steady strain rate of 0.001/s is applied during uniaxial tensile test of 25 mm gauge length flat tensile specimens. Contact type 25 mm gauge length extensometer is utilized to measure the engineering strain during the uniaxial tensile experiment. Engineering stress–strain curve of DP steel is depicted in Fig. 1(a). Isotropic strain hardening law proposed by El-Magd [17] is utilized to fit the true stress–strain curve of DP steel. El-Magd [17] stress-strain law can be stated as r ¼ r0 þ Aep þ B 1 e b ep
ð2Þ where, A, B and b are the material constants for El-Magd law. For DP steel, the values of r0, A, B and b are 500 MPa, 410 MPa, 340 MPa, and 9.8 respectively [18]. Input stress–strain curve for elastoplastic FE simulation is illustrated in Fig. 1(b). Isotropic strain hardening model (Eq. (2)) and von Mises yield criterion based on continuum plasticity theory are assumed in this work. Uniaxial tensile and hole expansion simulations are conducted with commercial software package ABAQUS/Explicit. To simulate the damage and failure, the Marciniak and Kuczynski (M-K) model [19] is introduced in the finite element model. A small zone with slightly lower Fig. 1. (a) Engineering uniaxial tensile stress-strain curve of DP steel, and (b) True tensile stress-strain curve and fitting of El-Magd law for DP steel. (c) Schematic illustration of HER in classic forming limit diagram, Point A represents HER for punched hole and Point B represents HER for EDM machined hole; (d) for DP steel, comparison of HER for punched and EDM machined hole, (e) hole expansion test samples of DP steel after completion of hole expansion test for punched and EDM machined hole. 52 S.K. Paul / Manufacturing Letters 21 (2019) 50–55 thickness is considered in M-K model [19]. This damage zone does not actually present in the material. It can be described as a numerical concept to study the failure strain for non-damaged material. One-quarter of the specimen is modeled to simulate the HET. One damage zone with dimension of 1° is selected in 90° sheet sample with thickness of 1.4 mm. Thickness ratio in damage zone and perfect zone in sheet sample is 0.99. Yu et al. [20] have adopted similar assumption and procedure of simulation considering MK model. Shell elements with reduced integration (S4R) are employed to discretize the sheet specimen. Along the thickness direction, the number of integration points is selected as 5. Because of high strain gradient at the damage zone in forming, smaller element size is used in the probable zone of damage initiation to obtain more accurate strain as an output. Punch, upper and lower dies are modeled as rigid body. The punch movement is set at a constant velocity of 1 mm/s. Coefficient of Coulomb friction between blank and tool is selected as 0.2. Element deletion at equivalent failure strain of 0.9 [21] is on during finite element simulation of both uniaxial tensile and HETs. Different strain components (example: longitudinal, transverse, thickness and in-plane shear strains) are determined just before the fracture, at a location in the necked zone. Thereafter, von Mises equivalent strain is calculated from those strain components. To simulate the necking phenomenon in the tensile sample, MK model with same set of parameters are used (like thickness ration of 0.99, S4R element for meshing, and equivalent fracture strain of 0.9). Fig. 2. (a) necking in DP steel sample during uniaxial tensile test, (b) finite element simulation model with local reduced thickness, (c) variation of in-plane maximum principal strain along the width of the sample, (d) in-plane maximum principal strain and diffuse necking, (e) in-plane maximum principal strain and localize necking in the damage zone, and (f) in-plane maximum principal strain and cracking in the damage zone, (g) comparison of engineering stress-strain between experimental and finite element simulation, (h) evolution of local principal strains with uniaxial tensile test progression. S.K. Paul / Manufacturing Letters 21 (2019) 50–55 3. Results and discussions Uniaxial tensile deformation exists at the central hole edge of test sample in HET [1,2]. If the HER for two test conditions, punched and EDM machined central holes are schematically plotted in forming limit diagram (minor strain in x-axis and major strain in y-axis), then the two points A and B are located as shown in Fig. 1(c). Point A denotes HER for the punched hole, while point B denotes HER for EDM machined hole specimen. Point A is located far below the forming limit curve (FLC), however point B is located above the FLC. The location of point A in the FLC is due to the presence of pre-existing defects such as dimples and cracks at the punched hole’s edge. HER performance of DP steel for punched and EDM machined hole can be compared from Fig. 1(d). It can be noticed that the HER of EDM machined hole specimen is far higher than the HER of punched hole specimen. HER of EDM machined hole specimen is also far higher than the uniform and total elongation of the material. Similar trend is also reported in literature [2,22]. Through thickness cracks are located at the central hole’s edge after the HET, depicted in Fig. 1(e). Large numbers of 53 small cracks and one through thickness crack are visible for the punched hole, while only two through thickness cracks are visible for EDM machined hole. No sign of diffuse necking (i.e. undulation near crack) is visible for sample with EDM machined hole (Fig. 1 (e)). Fig. 2(a) shows diffuse necking during uniaxial tensile test of DP steel sample. Gauge length of uniaxial tensile sample and mesh details are depicted in Fig. 2(b). Fig. 2(c) illustrates uniform strain/deformation along the width of the sample before necking. Fig. 2(d) shows reduction of sample’s width in the diffuse neck zone. After completion of diffuse necking, the axial strain increases at the expense of thickness reduction i.e. width of the sample remain constant and this phenomenon is known as localized necking (Fig. 2(e)). Finally failure (cracking and separation) happens at the localized neck zone, when the equivalent plastic strain reaches failure strain (Fig. 2(f)). The comparison of engineering stressstrain between experimental and FE simulation is depicted in Fig. 2(g). The evolution of local principal strains with uniaxial tensile test progression is illustrated in Fig. 2(h). Magnitude of minimum in-plane and out-of-plane principal strains are almost same Fig. 3. (a) one quarter of sheet specimen for finite element simulation of hole expansion test, (b) in-plane maximum principal strain in hole expansion test sample, and (c) variation of in-plane maximum principal strain along the width of the sample (i.e. moving away from the central hole edge), (d) evolution of principal strains at the failure initiation location. 54 S.K. Paul / Manufacturing Letters 21 (2019) 50–55 Fig. 4. (a) out-of-plane maximum principal strain (i.e. thickness strain), (b) in-plane maximum principal strain and localize necking in the damage zone, and (c) in-plane maximum principal strain and cracking in the damage zone. before initiation of diffuse neck. After initiation of diffuse neck, increment rate in out-of-plane principal strain is higher to some extent than the minimum in-plane principal strain. After initiation of localized necking, maximum in-plane principal strain and outof-plane principal strains increase rapidly, while an increment of minimum in-plane principal strain is found to be insignificant. Fig. 3(a) shows one-quarter of sheet specimen used in HET. Comparatively, fine mesh is used in the damage zone (red color) as illustrated in Fig. 3(a). In the perfect zone mesh size is 1° and in the damage zone is 0.125°. Fig. 3(b) and (c) show in-plane maximum principal strain in HET sample and variation of in-plane maximum principal strain along the width of the sample (i.e. increasing distance from the central hole edge) respectively. Highest in-plane maximum principal strain is visible at the central hole edge and it gradually reduces as moved away from the central hole edge. Paul [23] also has observed maximum deformation and failure initaion at the central hole edge for conical punch, while litile bit away from the central hole edge for flat-bottom and hemispherical punches. The in-plane maximum principal strain becomes zero where upper and lower dies hold the sheet specimen. The difference between uniaxial tensile and HET results from simulation and experimental can be stated as: (i) the strain/deformation is uniform along the width of the sample before onset of necking for uniaxial tensile test, while strain/deformation gradually reduces with rising distance from the central hole edge for HET. (ii) uniaxial tensile test sample has two free edges along its width, while HET sample has only one free edge (i.e. central hole edge) because another edge is held between upper and lower dies, and deformation becomes zero at that area. (iii) crack is visible in necked zone only for an uniaxial tensile test, whereas multiple cracks may be visible anywhere at the central hole edge for the HET. Multiple cracks for the punched hole sample could be due to the hole edge preparation method. Evolution of principal strains at the failure initiation location during HET is illustrated in Fig. 3(d). FE simulation result shows formation of localized neck in Fig. 4 (a) during HET. Thickness strain in damage and perfect zones are also plotted in selected zoom area in Fig. 4(a). Before and after through thickness crack formation, in-plane maximum principal strain is plotted in Fig. 4(b) and (c) respectively. From the FE simulation of HET, it can be confirmed that diffuse neck is absent/delayed and direct formation of localized neck is observed. In experimentation also evidence of diffuse neck (i.e. wave on central hole edge) is not visible for EDM machined sample (Fig. 1(e)). Uniform elongation of DP steel is measured to be approximately 17.5% from both experiment and finite element simulation (Fig. 2 (d)). The diffuse neck initiates for DP steel after an elongation of 17.5% during uniaxial tensile test. After initiation of diffuse neck, the deformation is concentrated in the diffuse neck zone (Figs. 2(d)-3(e)), and as a consequence total elongation of only 25% is evident. However, Paul [14] showed through local strain analysis by digital image correlation technique that local axial strain can reach up to 95%, after the completion of diffuse neck. If diffuse neck is suppressed/delayed, then 95% uniform deformation is possible for DP steel. The suppression/delay of diffuse neck in HET logically explain HER of 110% for DP steel with EDM machined central hole edge. After uniform deformation, localized necking and through thickness crack formation also have a role in HER and this can explain the higher HER than the local axial strain after completion of diffuse neck. Reasons responsible for the suppression/delay of diffuse neck in HET are proposed in the present work, they are (i) presence of deformation gradient, and (ii) presence of only one free edge and constraint due to continuous presence of material on the other side. 4. Conclusions Author showed from in-depth finite element simulation and experimental evidence that diffuse neck followed by localized neck are visible for uniaxial tensile test, while no evidence of diffuse neck is evident and only formation of localized neck is observed for HET. The strain/deformation before necking is uniform throughout the width of the sample for normal uniaxial tensile S.K. Paul / Manufacturing Letters 21 (2019) 50–55 experiment. However presence of a prominent deformation gradient is noticed for HET. Central hole edge of the HET sample experienced highest strain/deformation and zero strain/deformation is evident where upper and lower dies hold the sample. Author explains the suppression/delay of diffuse neck in HET by the existence of deformation gradient and single free edge. References [1] Paul SK, Mukherjee M, Kundu S, Chandra S. Prediction of hole expansion ratio for automotive grade steels. Comput Mater Sci 2014;89:189–97. [2] Yoon JI, Jung J, Lee HH, Kim G-S, Kim HS. Factors governing hole expansion ratio of steel sheets with smooth sheared edge. Met Mater Int 2016;22:1009–14. [3] Paul SK. Non-linear correlation between uniaxial tensile properties and shearedge hole expansion ratio. J Mater Eng Perform 2014;23:3610–9. [4] Chatterjee S, Bhadeshia HKDH. Stretch-flangeability of strong multiphase steels. Mater Sci Technol 2007;23:606–9. 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