Strain Distribution Analyses of Sheared AHSS

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Villarreal-Cheretti, César A. Salazar-García, Miguel A. Quiñones, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7272703/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 31 Oct, 2025 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted 5 You are reading this latest preprint version Abstract Manufactured goods subjected to complex forming operations, such as drawing, piercing, or stamping, involve cutting and shearing into blanks; shearing promotes the development of strain gradients within the piece, which may result in cracks or fissures in the finished component, and would cause the rejection of the blank. The origin of such defects is under discussion, as they can be attributed to the material or the cutting process. This work proposes a visioplastic analysis based on the macroscopic flow lines developed in the material during shearing. The method is applied to samples cut from advanced high-strength steels that were sheared with different clearances between the shearing blades used. The transversal section of the samples was prepared and etched to reveal these flow lines; the coordinates along five lines were recorded and fed into an algorithm that fitted them to cubic splines to obtain the coordinates and derivative of a new set of data points from which the strain components were computed. The method was tested in two different Advanced High-Strength Steels (AHSS) of various thicknesses and microstructures. It revealed essential details on the shear mechanics that would allow for the improvement of shearing and blanking operations. Formability shearing microstructure visioplastic analysis strain distribution Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Introduction The global concern for the increase of greenhouse gas emissions has resulted in stringent worldwide regulations imposed on automakers that have responded by reducing vehicle weight and size and enhancing combustion engine efficiency [ 1 , 2 ]. Automakers and their suppliers have to fulfil conflicting requirements, such as improving the stiffness of thicker sections, increasing fuel consumption, or incorporating attractive and aesthetic designs into new models, which may compromise forming and joining thinner sections. Steelmakers have responded by developing Advanced High-Strength Steels (AHSS) characterised by multiple phases and microstructures to enhance strength and formability [ 3 – 6 ]. Multiple automotive components are manufactured following sheet metal forming operations performed on pieces and blanks sheared from rolled strips. The strain to which the sheared strip is subjected is affected by the geometry of the process and by the characteristics and properties of the material; recent reviews on the shearing process can be found elsewhere [ 7 , 8 ]. Figure 1 illustrates a typical profile and surface observed in sheared samples. The material deforms plastically due to the pressure exerted by the blade to cause the rollover; actual shearing produces a bright, burnished zone, which will develop into a rougher fractured zone. The fracture angle φ can be used to characterise the fracture. The burr shown towards the bottom region in Fig. 1 is attributed to either excessive wear or clearance set between the blades [ 9 , 10 ]. Shearing of AHSS may result in the generation of high strain gradients within the piece being cut, which may promote the occurrence of fissures or cracks that may be affected by geometrical parameters such as the thickness of the strip or the gap between the shearing blades of dies, by tool wear, by the mechanical properties of the material and by the different microstructural constituents present in it [ 7 – 19 ]. Attempts to evaluate the strain distribution during shearing have been conducted using finite element modelling [ 13 , 20 – 29 ] or hardness testing [ 30 , 31 ]. Local values of strain have been evaluated by analytical [ 32 – 35 ] or metallographic [ 36 – 40 ] visioplastic techniques; in the former case, a geometric pattern is inscribed either on the surface, or a meridian plane of the sample and the strains are deduced from the distorted pattern after deformation. Metallographic analyses are conducted on specimens that cannot be sectioned due to their geometric or processing conditions. The strain distribution is calculated assuming that the grain aspect-ratio varies with the change in shape of the specimen, or by the changes in trajectory of aligned characteristics such as dual-phase structures, sulphide stringers or macroscopic flow lines. This work aims to present the results of a model based on the visioplastic analysis of flow lines detected in sheared samples from AHSS strips. The coordinates of individual flow lines were fed into a model that evaluates normal and shear strains after fitting them into cubic splines. The model allowed for discerning different features attributed to the material or the shear geometry. Experimental procedure Two AHSS, which will be identified as A and B in this work, were selected for their study; both steels were produced by the thin slab casting route and were delivered in their hot-rolled condition. The chemical composition of the steels is shown in Table 1 . The thickness of the strips was 3.6 and 2.9 mm for steels A and B , respectively. The transformation of the steels after hot rolling and the resulting microstructural constituents expected were assessed by a commercial thermodynamic and kinetics computer package based on a modified Scheil-Gulliver model [ 41 ]. The selection of these steels was due to the difference in performance when manufacturing automotive components [ 16 , 19 ]. Table 1 Chemical composition of the steels (% mass). Steel C Mn P S Cu Ni Si Cr Al Ti Mo N A 0.065 1.721 0.011 0.002 0.086 0.165 0.543 0.796 0.039 0.015 0.318 0.0061 B 0.046 1.086 0.008 0.003 0.099 0.037 0.257 0.043 0.056 0.146 0.013 0.0081 Figure 2 shows the tooling set-up used in the shearing tests (a); bottom (b) and top (c) sides are also included. This design was used as it allowed for strips of different thicknesses to be sheared with different clearances or gaps between the shears. Figure 3 shows the blades made from a quenched and tempered D2 steel to a 59 Rockwell C hardness of 20 mm thick and 280 long. The top blade was symmetrically machined with an angle of 1.84° to vary the height from 30 mm at the edges to 35 mm in its centre; the bottom blade had a height of 25 mm (a); the position within a mechanical press is shown (b). The blade clearances were of 5, 10, 15 and 20%. Table 2 shows the gaps left between the blades. Table 2 Gaps between the blades in mm. Steel Thickness (mm) Clearence (%) 5 10 15 20 A 3.6 0.18 0.36 0.54 0.72 B 2.9 0.145 0.29 0.435 0.58 The sheared specimens were prepared for their metallographic examination in a normal plane that was polished and etched to reveal the flow lines and record the coordinates at different points along them. This was done following standard metallographic procedures using grinding papers of different grits starting with 220 and polishing it with diamond pastes of 9 and 3 µm and silica gel of 0.25 µm. The surfaces were etched with a 3% solution of nitric acid (HNO 3 ) in ethyl alcohol (CH 3 CH 2 OH). Observations were made using a digital microscope, Keyence VHX-7000 at the same magnification. Specimens from both steels were prepared for their metallographic inspection by etching them in a 1% aqueous solution of sodium metabisulfite (Na 2 S 2 O 3 ) and 4% picric acid (C 6 H 2 (NO 2 ) 3 OH) in ethyl alcohol [ 42 ]. Tensile tension tests conducted on specimens of 12.7 mm in width and 50 mm in gauge length cut along 0,45 and 90° with respect the rolling direction of the sheets in a computer driven mechanical following the ASTM E8/E8M standard [ 43 ]. The load-displacement data were converted into real stress-strain (σ-ε) curves assuming constancy of volume [ 44 ]. Five different flow lines were selected in each specimen, and the coordinates of various points along them were recorded to evaluate the strain distribution. These lines were located around 10% in thickness from either surface of the steel strip, identified as top ( TS ) and bottom ( BS ) surfaces; another two were at a quarter of the thickness from either surface, recognized as top ( TQ ) and bottom ( BQ ) quarters and at mid-thickness, identified as M . The algorithm used to evaluate local values of strains is based on fitting cubic polynomial spline curves within an interval defined by two consecutive points along the flow lines. This procedure assures the continuity of not only the curve, but also the continuity of its derivative [ 45 ]. Such an algorithm has determined the strain within heavily deformed specimens [ 37 – 39 ]. Figure 4 shows such a case for a line constructed from the coordinates of six data points measured at mid-thickness ( M ) of a sample of steel A sheared with a 5% clearance. The spline curve used is divided into a new set of coordinates from which the strain evaluation analysis is conducted. The components of strains are calculated from the coordinate axes shown in Figure 1. The strain due to bending, which will be defined as ε y , is caused by compressive forces, and it is calculated by: ε y = log (h i /h o ) where h i and h o are respectively the height of an individual new point and that of the initial point of the spline curve, in which it is assumed to be within the unstrained portion of the sample. It is assumed that plastic deformation does not produce a change in volume of the specimen, therefore ε x = -ε y and ε z = 0. Shearing will be conducted only on one plane, so the value for γ xy will be given as: γ xy = tan (α) where α is the angle made by the spline curve and the shear strain is its slope; γ yz and γ zx are null, as no deformation takes place in the z direction, see Figure 1 (b). The equivalent strain (ε) will be calculated as [44]: ε = 2/3 [(ε y 2 + 0.75 γ xy 2 )] 1/2 The values of ε y , γ xy and ε will be called in this work as bending, shear and shear strains, as indicated in Figure 4 (b). Results The stress-strain curves from both steels are shown in Fig. 5 that confirms that both types of steels can be rated to an ultimate tensile strength of 800 MPA [ 3 – 6 ], Table 3 . Although the shape of the curves of both steels differ, as those from steel A reflect those of multiphase AHSS, as they do not show an abrupt yield point phenomenon and exhibit a high strengthening rate after yielding, whereas the shape of the curves from steel B resembles that of a high strength low alloy (HSLA) steel in the sense that they exhibit the yield point or Lüders phenomenon and a lower strengthening rate towards the early stages of plastic deformation [ 3 , 4 ]. Table 3 Mechanical properties of the steel strips. Steel A B Inclination to rolling 0° 45° 90° 0° 45° 90° Strain at yielding 0.005 0.005 0.005 0.006 0.006 0.005 Lüders strain - - - 0.032 0.032 0.035 Yield stress 697.6 671.9 708.0 724.7 766.6 752.9 Uniform strain 0.090 0.085 0.068 0.141 0.139 0.123 Ultimate tensile stress 996.4 959.3 985.2 890.5 897.7 898.8 Strain to fracture 0.137 0.141 0.111 0.186 0.193 0.168 Yield and ultimate tensile stress in MPa Figure 6 shows the sheared surfaces of the specimens from steels A and B with the various clearances, in which the difference between the steels is notable, as no fissures are detected in the samples from steel A . Still, they occur in all the specimens from steel B , except the one sheared with the largest clearance, 20%. Figures 7 and 8 show the micrographs and points selected for drawing the five flow lines selected in each specimen for steels A and B , respectively. These images show the marked contrast between both steels, as was pointed out when Fig. 6 was presented. Table 4 shows the proportion of rollover, bright and fracture zones, and burr, see Fig. 1 , measured from images such as those shown in Figs. 6 to 8 . The occurrence of fissures in the samples is also presented in this table. The values for angle φ, which is made by the fracture zone with respect to the vertical, shearing, direction, see Fig. 1 (a), were measured from the broken lines drawn in Figs. 7 and 8 and are also reported in Table 4 . Table 4 Amount of various features (%) on the surfaces of the sheared samples; the occurrence of fissures and that of the fracture angle are reported. Steel A B Clearance (%) 5 10 15 20 5 10 15 20 Rollover 2.8 3.8 4.4 9.2 15.4 13.5 12.9 6.8 Burnish 6.3 12.6 11.3 13.9 27.5 37.1 28.2 23.1 Fracture 90.9 83.5 84.3 73.3 57.1 49.4 58.9 65.1 Burr 0.0 0.0 0.0 3.6 1.2 0.9 3.6 5.0 Fissures No No No No Yes Yes Yes No Angle φ (°) 10 -1 -6 -9 -26 -14 -28 3 The flow lines shown in Figs. 7 and 8 were selected to start within the unaffected material and end at the sheared surface. Their coordinates were then related to those of the micrographs for their analysis. Figures 9 and 10 show, respectively, the values of the different strain components calculated along the five different flow lines in samples from steel A sheared with 5 and 20% clearance. The values for the equivalent strains evaluated at the four clearances for steels A and B are shown in Figs. 11 and 12 . The values of equivalent strain in a three-dimensional plot for steels A and B sheared with 5 and 20% clearances are shown in Figs. 13 to 16 . The low lines and the points used to fit them are plotted at zero equivalent strain; the trace of sheared surface will be plotted towards the right-hand side. Figures 13 and 15 show the distributions for the 5% clearance, whereas those for a 20% clearance are shown in Figs. 14 and 16 . These diagrams confirm the difference in behaviour between steels A and B . The strain values in steel A are higher when the clearance increases from 5 to 20%, whereas those in steel B are reduced. Moreover, the fissures observed in the samples from steel B do not occur at the highest clearance. Discussion The computed values for the equivalent strain were used to construct the maps shown in Figs. 17 and 18 for steels A and B , respectively. The contour lines were drawn for strains of 0.05, 0.1, 0.2, 0.5 and 1.0. These contours follow a similar shape in the samples from steel A , Fig. 17 , in the sense that higher values of strain occur towards the sheared edge, and will exhibit high strain gradients, as the material would be undeformed at a distance of less than 1 mm from the sheared surface; higher values of equivalent are achieved as the clearance of the shearing tools increases. The distribution within the samples from steel B is somewhat different. Although higher strain values are encountered towards the sheared surface, the strain gradients will be shallower, as unstrained material will not be found below 1.5 mm, except for the case of the sample cut with a 20% clearance. The micrographs from Fig. 8 show the development of fissures within the sample when clearances of 5, 10, and 15% were used; these fissures are incipient in the sample with a 20% clearance. Burr is observed in all samples from material B ; its values are shown in Table 4 and can be observed in Figs. 8 and 18 . The difference in the strain distribution within the steels would be related to their difference in mechanical properties expressed in terms of their stress-strain curves shown in Fig. 5 . Both steels show a yield strength close to 700 MPa, 692.5 MPa for steel A and 748.1 MPa for steel B , which exhibits an average Lüders strain of 0.033; steel A shows a higher hardening rate after yielding than that of steel B . The tensile strength of steel A is of 980.3 MPa at a strain 0.081, and a strain to fracture of 0.130. Steel B exhibits a tensile strength of 895.7 MPa at a strain of 0.134, and a strain to fracture of 0.182. Figure 19 shows the strengthening rate vs. strain curves (dσ/dε-ε) for steels A and B calculated from the stress vs. strain curves (σ-ε) obtained from samples tested along the rolling direction (0° in Fig. 5 ). The width of the Lüders strain of steel B is indicated in the graph. Figure 19 shows that the hardening rate of steel A is higher than that of steel B at lower strains and particularly while this last steel is within the Lüders regime, which will allow for straining to proceed to a higher degree in this material and would result in the extended deformation observed to occur in the diagrams shown in Fig. 18 , while the higher hardening rate would be responsible for the higher strain gradients appreciated in the corresponding diagrams from steel A , Fig. 17 . The lower strength of steel B will be responsible for the burr observed to occur in its sheared samples as the softer material will be pushed into the gap left by the shears as has been documented experimentally and modelled by finite element analyses [ 7 , 8 , 10 – 14 , 17 , 20 – 29 ]. Previous work on steels with similar compositions of either type of steel had revealed that the microstructure and hardening behaviour of these steels are different. However, both are in their hot-rolled condition [ 16 , 19 ]. Steel A , due to their higher contents in Si and Mn, as well as the addition of Cr, Ni and Mo would transform after rolling into a higher proportion of acicular structures, martensite and bainite, than steel B , which relies in the strengthening by precipitation of TiN that results from the addition of Ti and the amount of N grabbed during melting [ 46 ]. Figure 20 displays the continuous cooling transformation diagrams (CCT) for steels A and B , which were computed at the centre and surface of 3.6 mm thick strips processed in the run-out table of a compact, thin slab line, assuming conditions to those documented for a similar line [ 47 ]. Diagrams a and b were predicted by the software mentioned above [ 41 ], assuming a finishing rolling temperature of 900ºC and a fully recrystallised austenite grain size of 30 µm [ 48 ], in which transformation in steel A will start into bainite as it cools down. In contrast, steel B will begin transforming into ferrite. The cooling curves indicated as S and C for surface and centre in the diagrams follow rates close to 10 C/s. The isothermal temperatures for A 1 , A 3 and M s are drawn as broken lines, the curves for the start of transformation to ferrite (F s ), pearlite (P s ) and banite (B s ), as well for the end of pearlite (P f ) and bainite (B f ) are drawn as full lines. Figure 20 also shows in diagrams b and c the volume fraction (X v ) predicted by the software [ 40 ] for ferrite (F), pearlite (P), bainite (B), martensite (M) both steels at various rates, austenite (A) is predicted to occur only a obove 20C/s in steel A. The shaded areas in these diagrams correspond to rates ranging from 5 to 20 C/s. The microstructures in Fig. 21 were observed in samples prepared from steels A and B that were etched with the tint reagent used for identification of various structures as martensite appears white, bainite black and ferrite tan [ 42 ]. It is confirmed from these images that the structure of steel A is made of ferrite, bainite and martensite, which are tinted tan, black and white; whereas that of steel B is mainly made of ferrite and bainite. Figure 22 shows the end of a fissure detected in the specimen from steel B sheared with a 10% clearance in which a series of cuboid particles are detected close or within the crack surface, and are suspected to cause the failure. These particles have been identified as TiN in steels of similar composition that exhibited cracks during processing [ 16 , 19 ]. It is noticeable that the angle φ in the samples sheared from steel A is affected by the clearance as it changes from positive to negative, Table 4 , implying that blades and toolings will be subjected to higher attrition and wear when the clearance is tightened, as has been reported in previous studies [ 7 – 11 , 17 , 23 – 25 , 28 – 31 ], therefore, it would be recommended the use of a 10 to 15% clearance when trimming or shearing this type of steel. The case of steel B is difficult to assess due to the occurrence of fissures and burr in almost all situations and would require a clearance of at least 20% to obtain a clear cut in which no fissures were developed and the burr is slight. Conclusions The procedure used to compute the local values of strain components show that the steels behave in a different way when sheared. Steel A , in which its microstructure contains bainite and martensite, tends to develop steeper strain gradients in the vicinity of the cut, whereas steel B , in which a predominance of ferrite and bainite is encountered exhibit shallower gradients. The difference in the development of such gradients can be explained in terms of the strengthening rate (dσ/dε) taking place in either steel after yielding. Steel B , which exhibit Lüders phenomenon, is not subjected to the higher strengthening rate that steel A shows, hence the shallower gradients in steel B . The strengthening in steel A affects the angle φ, and it would be the case when using small clearance, such a 5%, that the new surface will impinge and press the tools promoting the high wear rates observed to occur un such instances. Therefore, these steels require for a higher clearance, such as 15%. The case of steel B is not clear, as the samples exhibit burr in all cases and, with the exception of the 20% clearance, all of them exhibit fissuring associated with the presence of TiN precipitates used to enhance its strength. Declarations Support/Funding The authors thank the support provided by the Mexican Secretaría de Ciencia, Humanidades, Tecnología e Innovación (Ministry for Science, Humanities, Technology and Innovation) to OFVC and CASG. Conflicts of interest/Competing interests The authors declare that they have no conflict of interest related to this work. Authors' contributions O.F. Villarreal-Cheretti. Experimental work, data gathering and analysis, writing. C.A. Salazar-García. Experimental work, data gathering and analysis, writing. M.A. Quiñones. Industrial testing and analysis, writing. N.F. Garza-Montes-de-Oca. Data analysis, industrial processing, writing. R. Colás. Data analysis, writing, corresponding author. References Miller JD, Façanha C (2014). 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Eval., 13:28-38. https://doi.org/10.1520/JTE10757J Colás R, Sellars CM (1987) Strain distribution and temperature increase during plane strain compression testing. J. Test. Eval., 15:342-349. https://doi.org/10.1520/JTE11032J Rodríguez A, Colás R, Olvera G, Fodor P (2000) Strain distribution analysis of hot forged seamless pipe fittings. Mat. Sc. Technol., 16:171-174. https://doi.org/10.1179/026708300101507668 Rodríguez-Rodríguez MG, Valdés-Covarrubias E, Guerrero-Mata MP, Colás R (2001) Visioplastic analysis of experimental rolling of steel. J. Mat.: Des. Appl., 215:155-163. https://doi.org/10.1243/1464420011545003 Unckel H (1937) A study of the deformation of the macrostructure of some two-phase alloys by cold rolling. J. Inst. Met., 61:171-196. Colás R (1988) A note on the deformation of strain gradients within deformed specimens. J. Mech. Working Technol., 16:335-340. https://doi.org/10.1016/0378-3804(88)90060-5 Colás R, Grinberg A (1999) A study of strain localization by means of reduced gage compression testing. J. Mat. Proc. Techn., 88:276-283. https://doi.org/10.1016/S0924-0136(98)00406-3 Cárdenas CA, Guerrero-Mata MP, Colás R (2003) Forming analysis of a tubular connecting bolt. J. Mat. Proc. Techn., 134:53-58. https://doi.org/10.1016/S0924-0136(02)00918-4 Gusel L, Rudolf R, Kosec B (2009) Analysis of a strain rate field in cold formed material using the visioplasticity method. Metalugija, 48:103-107. Saunders N, Guo Z, Li X, Miodownik AP, Schillé J-Ph (2003) Using JMatPro to model materials properties and behavior. JOM, 55 :60-65. https://doi.org/10.1007/s11837-003-0013-2 Lepera FS (1979) Improved etching technique for the determination of percent martensite in high-strength dual-phase steels. Metallography, 12:263-268. https://doi.org/10.1016/0026-0800(79)90041-7 ASTM E8/E8M-21 (2021) Standard test methods for tension testing of metallic materials, ASTM Int., West Conshohocken, PA. www.astm.org Dieter GE, Bacon D (1988) Mechanical Metallurgy. SI Metric Edition Advanced McGraw-Hill Book Co., London. Fowler AH, Wilson CW (1966). Cubic spline: A curve fitting routine (No. Y-1400 (Rev. 1)). Union Carbide Corp., Oak Ridge, Tenn. Y-12 Plant. https://www.osti.gov/servlets/purl/4524486 Turkdogan ET (1989) Causes and effects of nitride and carbonitride precipitation during continuous casting. Iron Steelmaker, 16:61-75. Hernández L, Guerrero-Mata MP, Leduc LA, Colás R (2004). A model for the run out table cooling in a compact rolling mill. J. Phys. IV, 120:513-518. https://doi.org/10.1051/jp4:2004120059 Zambrano PC, Delgado AL, Guerrero-Mata MP, Colás R, Leduc LA (2003) Hot rolling of light gauge steel strip. ISIJ Int., 43:1030-1035. https://doi.org/10.2355/isijinternational.43.1030 Cite Share Download PDF Status: Published Journal Publication published 31 Oct, 2025 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted Editorial decision: Major Revisions Needed 14 Oct, 2025 Reviewers agreed at journal 29 Aug, 2025 Reviewers invited by journal 06 Aug, 2025 Editor assigned by journal 05 Aug, 2025 First submitted to journal 01 Aug, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7272703","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":496877792,"identity":"83abc997-bbe5-4782-8386-9e6c1056b653","order_by":0,"name":"Oscar F. Villarreal-Cheretti","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Oscar","middleName":"F.","lastName":"Villarreal-Cheretti","suffix":""},{"id":496877793,"identity":"b82dffd6-4c2f-4f9e-aef9-e09f1f600ae3","order_by":1,"name":"César A. Salazar-García","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"César","middleName":"A.","lastName":"Salazar-García","suffix":""},{"id":496877794,"identity":"c459930f-732a-41dd-862f-3ada73b9a654","order_by":2,"name":"Miguel A. Quiñones","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Miguel","middleName":"A.","lastName":"Quiñones","suffix":""},{"id":496877795,"identity":"490d385b-cf6d-468d-a4b3-a73685851e04","order_by":3,"name":"Nelson F. Garza-Montes de Oca","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Nelson","middleName":"F. Garza-Montes","lastName":"de Oca","suffix":""},{"id":496877796,"identity":"a0700347-9a6f-4292-b73b-f15b3964dc6d","order_by":4,"name":"Rafael Colás","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVRIie3PMW7CMBSA4WdZIotLVlcMuUIYK0BcxVEkpjB16WhkyVkMM0zcgpVEXrlAZakqCyvJ5lYZmqYMLCRlq1T/g+3Bn+wH4HL9yTBvNo8Dyor60PM6CfohJAPI198E30M0aZ7tAH66WJQW3gLi6UyPP3dBHwMqyuQ2oYdcDAg8DxWZMT1fmaHEgB83u9skpBEfADC0hyTUc2VQTXr4oZ2IDwtsqvxzoZ+Umf6GSEqARYomoMGaqJPUs8gRCVms6CnMl9zEEiPROoufCv1qX9hE+fHx3VZmsk1FXpQt5PK9y45ks/Ku+9dV91x2uVyu/9IXSbRQ05+c5nMAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0001-6017-7244","institution":"Universidad Autónoma de Nuevo León","correspondingAuthor":true,"prefix":"","firstName":"Rafael","middleName":"","lastName":"Colás","suffix":""}],"badges":[],"createdAt":"2025-08-01 15:30:54","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7272703/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7272703/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00170-025-16883-z","type":"published","date":"2025-10-31T15:56:56+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":89047253,"identity":"86ec72c9-7e32-408e-a399-14242fc294f1","added_by":"auto","created_at":"2025-08-14 06:58:13","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":84013,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the cross-section (a) and the surface (b) of sheared samples showing the reference axes.\u003c/p\u003e","description":"","filename":"image1.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/fa1f5cbaa0c5417df8127e37.jpg"},{"id":89047662,"identity":"369b3ae1-b180-4ceb-b2bd-f9bc758b039c","added_by":"auto","created_at":"2025-08-14 07:06:13","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":351513,"visible":true,"origin":"","legend":"\u003cp\u003eTooling used for the shearing test (a) and details of the bottom (b) and top (c) components.\u003c/p\u003e","description":"","filename":"image2.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/8a059d6e89405fcb53af34a9.jpg"},{"id":89047262,"identity":"a1afafa9-c791-4cb1-a61f-b156acb71bb0","added_by":"auto","created_at":"2025-08-14 06:58:13","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":306653,"visible":true,"origin":"","legend":"\u003cp\u003eGeometry of the blades used for shearing (a), dimensions in mm; setup of the tooling in a mechanical press (b).\u003c/p\u003e","description":"","filename":"image3.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/7e00bead938e5982ac2e012d.jpg"},{"id":89047660,"identity":"ec7f50c3-611e-4a2f-a982-08afd8c0d169","added_by":"auto","created_at":"2025-08-14 07:06:13","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":201650,"visible":true,"origin":"","legend":"\u003cp\u003eFitting of cubic splines through the coordinates of the data points of a flow line (a) from which the strain components are evaluated and plotted as a function of length (b).\u003c/p\u003e","description":"","filename":"image4.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/c0158e9902c3d851f7947584.jpg"},{"id":89047250,"identity":"4864dc3d-477e-4458-8971-3d6a473e1ff0","added_by":"auto","created_at":"2025-08-14 06:58:13","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":174571,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain curves for steels \u003cstrong\u003eA\u003c/strong\u003e and \u003cstrong\u003eB\u003c/strong\u003e tested at 0, 45 and 90° with respect to the rolling direction.\u003c/p\u003e","description":"","filename":"image5.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/8c21f927b0f2a20cc9cad5fc.jpg"},{"id":89047657,"identity":"6e046a80-d00a-4b81-b020-608966cdf168","added_by":"auto","created_at":"2025-08-14 07:06:13","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":511828,"visible":true,"origin":"","legend":"\u003cp\u003eSheared surfaces of steels \u003cstrong\u003eA\u003c/strong\u003e and \u003cstrong\u003eB\u003c/strong\u003e at different values of clearance.\u003c/p\u003e","description":"","filename":"image6.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/ecc2871b9fe344d13f877eac.jpg"},{"id":89047667,"identity":"16cfe02c-a3c0-4c08-bc89-139db4bccbf2","added_by":"auto","created_at":"2025-08-14 07:06:14","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":594917,"visible":true,"origin":"","legend":"\u003cp\u003eMicrographs of the samples sheared from steel \u003cstrong\u003eA\u003c/strong\u003e showing the points selected along five flow lines; the angle φ is measured from the deviation from the vertical direction (y) of the broken line drawn at the bottom-end of the fracture zone, see Figure 1 (a).\u003c/p\u003e","description":"","filename":"image7.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/17986410f234bf51e6f88c84.jpg"},{"id":89048427,"identity":"7b3143f7-0c90-4154-b915-cf0d6bde1ad9","added_by":"auto","created_at":"2025-08-14 07:14:13","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":823413,"visible":true,"origin":"","legend":"\u003cp\u003eMicrographs of the samples sheared from steel \u003cstrong\u003eB\u003c/strong\u003e; similar features to those in Figure 6 are indicated.\u003c/p\u003e","description":"","filename":"image8.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/33c4e86ca50cea0617a6626b.jpg"},{"id":89047259,"identity":"f98890e8-a5c8-4fa8-9f6c-6f99d3ea9a2b","added_by":"auto","created_at":"2025-08-14 06:58:13","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":196346,"visible":true,"origin":"","legend":"\u003cp\u003eStrain components evaluated along the five flow lines in the sample from steel \u003cstrong\u003eA\u003c/strong\u003e sheared with a 5% clearance.\u003c/p\u003e","description":"","filename":"image9.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/e0ff999af6c0669ac277e49f.jpg"},{"id":89047268,"identity":"666971cc-a285-421d-add4-917c9f4eaeb0","added_by":"auto","created_at":"2025-08-14 06:58:14","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":203751,"visible":true,"origin":"","legend":"\u003cp\u003eStrain components evaluated along the five flow lines in the sample from steel \u003cstrong\u003eA\u003c/strong\u003esheared with a 20% clearance.\u003c/p\u003e","description":"","filename":"image10.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/66c3ee118f7630bfb5a55759.jpg"},{"id":89048430,"identity":"709efa0d-4f5d-4d03-814f-9aae8d996c43","added_by":"auto","created_at":"2025-08-14 07:14:13","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":206029,"visible":true,"origin":"","legend":"\u003cp\u003eValues of the equivalent strain computed from the flow lines at the various clearances in steel \u003cstrong\u003eA\u003c/strong\u003e.\u003c/p\u003e","description":"","filename":"image11.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/4ffe1f578dca41b992b60641.jpg"},{"id":89048801,"identity":"11777c3a-eb2d-4592-afe7-b4e5d83746b6","added_by":"auto","created_at":"2025-08-14 07:22:14","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":210364,"visible":true,"origin":"","legend":"\u003cp\u003eValues of the equivalent strain computed from the flow lines at the various clearances in steel \u003cstrong\u003eB\u003c/strong\u003e.\u003c/p\u003e","description":"","filename":"image12.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/5e6bb850cf2404c49125dcdd.jpg"},{"id":89047273,"identity":"c85f57c3-90e9-4e27-8b4c-c5957640fa21","added_by":"auto","created_at":"2025-08-14 06:58:14","extension":"jpg","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":193273,"visible":true,"origin":"","legend":"\u003cp\u003eValues of the equivalent strain developed by shearing steel \u003cstrong\u003eA\u003c/strong\u003e with a 5% clearance; the flow lines and the points used to construct them are plotted at zero strain.\u003c/p\u003e","description":"","filename":"image13.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/acae0cd1390207033fcc5c2f.jpg"},{"id":89048435,"identity":"de88b7c6-1730-4c64-aa10-00212bb25147","added_by":"auto","created_at":"2025-08-14 07:14:14","extension":"jpg","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":210650,"visible":true,"origin":"","legend":"\u003cp\u003eValues of the equivalent strain developed by shearing steel \u003cstrong\u003eA\u003c/strong\u003e with a 20% clearance; the flow lines and the points used to construct them are plotted at zero strain.\u003c/p\u003e","description":"","filename":"image14.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/4fc83c2acda7be7af021fc01.jpg"},{"id":89047283,"identity":"5c350003-2e38-453f-bad5-dd37ed225a7d","added_by":"auto","created_at":"2025-08-14 06:58:14","extension":"jpg","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":210677,"visible":true,"origin":"","legend":"\u003cp\u003eValues of the equivalent strain developed by shearing steel \u003cstrong\u003eB\u003c/strong\u003e with a 5% clearance; the flow lines and the points used to construct them are plotted at zero strain.\u003c/p\u003e","description":"","filename":"image15.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/a9080b9cb699f4e927895b03.jpg"},{"id":89047678,"identity":"0bd2c880-0e0b-45c6-a76c-a3f19502d75d","added_by":"auto","created_at":"2025-08-14 07:06:14","extension":"jpg","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":195481,"visible":true,"origin":"","legend":"\u003cp\u003eValues of the equivalent strain developed by shearing steel \u003cstrong\u003eB\u003c/strong\u003e with a 20% clearance; the flow lines and the points used to construct them are plotted at zero strain.\u003c/p\u003e","description":"","filename":"image16.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/00e342140573dfa0d8b534d8.jpg"},{"id":89047275,"identity":"3e58cbea-d823-4d01-9d72-7cc21bf2d7da","added_by":"auto","created_at":"2025-08-14 06:58:14","extension":"jpg","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":264771,"visible":true,"origin":"","legend":"\u003cp\u003eStrain distribution maps for the sheared samples from steel \u003cstrong\u003eA\u003c/strong\u003e.\u003c/p\u003e","description":"","filename":"image17.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/cfdf65c0af868ad459d4038a.jpg"},{"id":89047272,"identity":"e3927b54-16e1-4496-9000-2db4b53fe9b4","added_by":"auto","created_at":"2025-08-14 06:58:14","extension":"jpg","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":153739,"visible":true,"origin":"","legend":"\u003cp\u003eStrain distribution maps for the sheared samples from steel \u003cstrong\u003eB\u003c/strong\u003e.\u003c/p\u003e","description":"","filename":"image18.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/82d436598de63c80a35af462.jpg"},{"id":89047669,"identity":"784cc059-fe78-436a-a49c-541632bd44bb","added_by":"auto","created_at":"2025-08-14 07:06:14","extension":"png","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":102212,"visible":true,"origin":"","legend":"\u003cp\u003eHardening rate (dσ/dε-ε, \u003cstrong\u003ea\u003c/strong\u003e and \u003cstrong\u003eb\u003c/strong\u003e) and stress (σ-ε, \u003cstrong\u003ec\u003c/strong\u003e and \u003cstrong\u003ed\u003c/strong\u003e) vs. strain curves for steels \u003cstrong\u003eA\u003c/strong\u003e and \u003cstrong\u003eB\u003c/strong\u003e; the Lüders strain, ε\u003csub\u003el\u003c/sub\u003e, of steel \u003cstrong\u003eB\u003c/strong\u003e is indicated. The tests were conducted in specimens machined along the rolling direction.\u003c/p\u003e","description":"","filename":"image19.png","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/f47fdf132ddd30f85d1e3cb9.png"},{"id":89047688,"identity":"fb62a157-0617-4669-be63-0d6bcc6e25f7","added_by":"auto","created_at":"2025-08-14 07:06:15","extension":"jpg","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":310438,"visible":true,"origin":"","legend":"\u003cp\u003eCCT diagrams computed from the chemical compositions of steels \u003cstrong\u003eA\u003c/strong\u003e and \u003cstrong\u003eB\u003c/strong\u003e (a and b, respectively) and the volume fraction predicted for the various microstructural components (c and d respectively).\u003c/p\u003e","description":"","filename":"image20.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/e054081a3abed5f16bc0f7ca.jpg"},{"id":89048433,"identity":"53797666-3e9e-469b-8f19-48be2ac91380","added_by":"auto","created_at":"2025-08-14 07:14:14","extension":"jpg","order_by":21,"title":"Figure 21","display":"","copyAsset":false,"role":"figure","size":705620,"visible":true,"origin":"","legend":"\u003cp\u003eMicrographs of steels \u003cstrong\u003eA\u003c/strong\u003e at surface (\u003cstrong\u003ea\u003c/strong\u003e) and centre (\u003cstrong\u003ec\u003c/strong\u003e) and of steel \u003cstrong\u003eB\u003c/strong\u003e at surface (\u003cstrong\u003eb\u003c/strong\u003e) and centre (\u003cstrong\u003ed\u003c/strong\u003e). The rolling direction is perpendicular to the observed plane.\u003c/p\u003e","description":"","filename":"image21.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/9ebe12e6bb29c77efd4075a5.jpg"},{"id":89048432,"identity":"d4ff20de-b0cc-49da-8f62-adcd51e94108","added_by":"auto","created_at":"2025-08-14 07:14:14","extension":"jpg","order_by":22,"title":"Figure 22","display":"","copyAsset":false,"role":"figure","size":405076,"visible":true,"origin":"","legend":"\u003cp\u003eExamination of a fissure of the sample from steel \u003cstrong\u003eB\u003c/strong\u003e sheared with a 10% clearance; TiN particles are detected along and close to it and were surrounded by circumferences.\u003c/p\u003e","description":"","filename":"image22.tiff.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/c0ad203c36503e29b70abf27.jpg"},{"id":95040844,"identity":"42bd1c3a-8d4e-4cad-bcb6-498e6454744b","added_by":"auto","created_at":"2025-11-03 16:10:39","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7337201,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7272703/v1/84658379-8875-4487-a18a-8e9403fece5b.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eStrain Distribution Analyses of Sheared AHSS\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe global concern for the increase of greenhouse gas emissions has resulted in stringent worldwide regulations imposed on automakers that have responded by reducing vehicle weight and size and enhancing combustion engine efficiency [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Automakers and their suppliers have to fulfil conflicting requirements, such as improving the stiffness of thicker sections, increasing fuel consumption, or incorporating attractive and aesthetic designs into new models, which may compromise forming and joining thinner sections. Steelmakers have responded by developing Advanced High-Strength Steels (AHSS) characterised by multiple phases and microstructures to enhance strength and formability [\u003cspan additionalcitationids=\"CR4 CR5\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e–\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eMultiple automotive components are manufactured following sheet metal forming operations performed on pieces and blanks sheared from rolled strips. The strain to which the sheared strip is subjected is affected by the geometry of the process and by the characteristics and properties of the material; recent reviews on the shearing process can be found elsewhere [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates a typical profile and surface observed in sheared samples. The material deforms plastically due to the pressure exerted by the blade to cause the rollover; actual shearing produces a bright, burnished zone, which will develop into a rougher fractured zone. The fracture angle φ can be used to characterise the fracture. The burr shown towards the bottom region in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e1\u003c/span\u003e is attributed to either excessive wear or clearance set between the blades [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eShearing of AHSS may result in the generation of high strain gradients within the piece being cut, which may promote the occurrence of fissures or cracks that may be affected by geometrical parameters such as the thickness of the strip or the gap between the shearing blades of dies, by tool wear, by the mechanical properties of the material and by the different microstructural constituents present in it [\u003cspan additionalcitationids=\"CR8 CR9 CR10 CR11 CR12 CR13 CR14 CR15 CR16 CR17 CR18\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e–\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. Attempts to evaluate the strain distribution during shearing have been conducted using finite element modelling [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan additionalcitationids=\"CR21 CR22 CR23 CR24 CR25 CR26 CR27 CR28\" citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e–\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e] or hardness testing [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eLocal values of strain have been evaluated by analytical [\u003cspan additionalcitationids=\"CR33 CR34\" citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e–\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] or metallographic [\u003cspan additionalcitationids=\"CR37 CR38 CR39\" citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e–\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e] visioplastic techniques; in the former case, a geometric pattern is inscribed either on the surface, or a meridian plane of the sample and the strains are deduced from the distorted pattern after deformation. Metallographic analyses are conducted on specimens that cannot be sectioned due to their geometric or processing conditions. The strain distribution is calculated assuming that the grain aspect-ratio varies with the change in shape of the specimen, or by the changes in trajectory of aligned characteristics such as dual-phase structures, sulphide stringers or macroscopic flow lines.\u003c/p\u003e\u003cp\u003eThis work aims to present the results of a model based on the visioplastic analysis of flow lines detected in sheared samples from AHSS strips. The coordinates of individual flow lines were fed into a model that evaluates normal and shear strains after fitting them into cubic splines. The model allowed for discerning different features attributed to the material or the shear geometry.\u003c/p\u003e"},{"header":"Experimental procedure","content":"\u003cp\u003eTwo AHSS, which will be identified as \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e in this work, were selected for their study; both steels were produced by the thin slab casting route and were delivered in their hot-rolled condition. The chemical composition of the steels is shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The thickness of the strips was 3.6 and 2.9 mm for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e, respectively. The transformation of the steels after hot rolling and the resulting microstructural constituents expected were assessed by a commercial thermodynamic and kinetics computer package based on a modified Scheil-Gulliver model [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]. The selection of these steels was due to the difference in performance when manufacturing automotive components [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c12\" colnum=\"12\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c13\" colnum=\"13\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eChemical composition of the steels (% mass).\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"13\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSteel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eC\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMn\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eP\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eS\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003eCu\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003eNi\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e\u003cp\u003eSi\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c9\"\u003e\u003cp\u003eCr\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c10\"\u003e\u003cp\u003eAl\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c11\"\u003e\u003cp\u003eTi\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c12\"\u003e\u003cp\u003eMo\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c13\"\u003e\u003cp\u003eN\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eA\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.065\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1.721\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.011\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.002\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.086\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.165\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e0.543\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e0.796\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e0.039\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e\u003cp\u003e0.015\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e0.318\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e0.0061\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eB\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.046\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1.086\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.008\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.003\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.099\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.037\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e0.257\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e0.043\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e0.056\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e\u003cp\u003e0.146\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e0.013\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e0.0081\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the tooling set-up used in the shearing tests (a); bottom (b) and top (c) sides are also included. This design was used as it allowed for strips of different thicknesses to be sheared with different clearances or gaps between the shears. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the blades made from a quenched and tempered D2 steel to a 59 Rockwell C hardness of 20 mm thick and 280 long. The top blade was symmetrically machined with an angle of 1.84° to vary the height from 30 mm at the edges to 35 mm in its centre; the bottom blade had a height of 25 mm (a); the position within a mechanical press is shown (b). The blade clearances were of 5, 10, 15 and 20%. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the gaps left between the blades.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eGaps between the blades in mm.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"6\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eSteel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eThickness (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"4\" nameend=\"c6\" namest=\"c3\"\u003e\u003cp\u003eClearence (%)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eA\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e3.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.18\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.54\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.72\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eB\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e2.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.145\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.435\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.58\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eThe sheared specimens were prepared for their metallographic examination in a normal plane that was polished and etched to reveal the flow lines and record the coordinates at different points along them. This was done following standard metallographic procedures using grinding papers of different grits starting with 220 and polishing it with diamond pastes of 9 and 3 µm and silica gel of 0.25 µm. The surfaces were etched with a 3% solution of nitric acid (HNO\u003csub\u003e3\u003c/sub\u003e) in ethyl alcohol (CH\u003csub\u003e3\u003c/sub\u003eCH\u003csub\u003e2\u003c/sub\u003eOH). Observations were made using a digital microscope, Keyence VHX-7000 at the same magnification. Specimens from both steels were prepared for their metallographic inspection by etching them in a 1% aqueous solution of sodium metabisulfite (Na\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e) and 4% picric acid (C\u003csub\u003e6\u003c/sub\u003eH\u003csub\u003e2\u003c/sub\u003e(NO\u003csub\u003e2\u003c/sub\u003e)\u003csub\u003e3\u003c/sub\u003eOH) in ethyl alcohol [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eTensile tension tests conducted on specimens of 12.7 mm in width and 50 mm in gauge length cut along 0,45 and 90° with respect the rolling direction of the sheets in a computer driven mechanical following the ASTM E8/E8M standard [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e]. The load-displacement data were converted into real stress-strain (σ-ε) curves assuming constancy of volume [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eFive different flow lines were selected in each specimen, and the coordinates of various points along them were recorded to evaluate the strain distribution. These lines were located around 10% in thickness from either surface of the steel strip, identified as top (\u003cb\u003eTS\u003c/b\u003e) and bottom (\u003cb\u003eBS\u003c/b\u003e) surfaces; another two were at a quarter of the thickness from either surface, recognized as top (\u003cb\u003eTQ\u003c/b\u003e) and bottom (\u003cb\u003eBQ\u003c/b\u003e) quarters and at mid-thickness, identified as \u003cb\u003eM\u003c/b\u003e.\u003c/p\u003e\u003cp\u003eThe algorithm used to evaluate local values of strains is based on fitting cubic polynomial spline curves within an interval defined by two consecutive points along the flow lines. This procedure assures the continuity of not only the curve, but also the continuity of its derivative [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e]. Such an algorithm has determined the strain within heavily deformed specimens [\u003cspan additionalcitationids=\"CR38\" citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e–\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eFigure 4 shows such a case for a line constructed from the coordinates of six data points measured at mid-thickness (\u003cstrong\u003eM\u003c/strong\u003e) of a sample of steel \u003cstrong\u003eA\u003c/strong\u003e sheared with a 5% clearance. The spline curve used is divided into a new set of coordinates from which the strain evaluation analysis is conducted. The components of strains are calculated from the coordinate axes shown in Figure 1. The strain due to bending, which will be defined as ε\u003csub\u003ey\u003c/sub\u003e, is caused by compressive forces, and it is calculated by:\u003c/p\u003e\u003cp\u003e\u0026nbsp;ε\u003csub\u003ey\u003c/sub\u003e = log (h\u003csub\u003ei\u003c/sub\u003e/h\u003csub\u003eo\u003c/sub\u003e)\u003c/p\u003e\u003cp\u003ewhere h\u003csub\u003ei\u0026nbsp;\u003c/sub\u003eand h\u003csub\u003eo\u003c/sub\u003e are respectively the height of an individual new point and that of the initial point of the spline curve, in which it is assumed to be within the unstrained portion of the sample. It is assumed that plastic deformation does not produce a change in volume of the specimen, therefore ε\u003csub\u003ex\u003c/sub\u003e = -ε\u003csub\u003ey\u003c/sub\u003e and ε\u003csub\u003ez\u003c/sub\u003e = 0.\u003c/p\u003e\u003cp\u003eShearing will be conducted only on one plane, so the value for γ\u003csub\u003exy\u003c/sub\u003e will be given as:\u003c/p\u003e\u003cp\u003eγ\u003csub\u003exy\u003c/sub\u003e = tan (α)\u003c/p\u003e\u003cp\u003ewhere α is the angle made by the spline curve and the shear strain is its slope; γ\u003csub\u003eyz\u003c/sub\u003e and γ\u003csub\u003ezx\u003c/sub\u003e are null, as no deformation takes place in the z direction, see Figure 1 (b).\u003c/p\u003e\u003cp\u003eThe equivalent strain (ε) will be calculated as [44]:\u003c/p\u003e\u003cp\u003eε = 2/3 [(ε\u003csub\u003ey\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e + 0.75 γ\u003csub\u003exy\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e)]\u003csup\u003e1/2\u003c/sup\u003e\u0026nbsp;\u003c/p\u003e\u003cp\u003eThe values of ε\u003csub\u003ey\u003c/sub\u003e, γ\u003csub\u003exy\u003c/sub\u003e and ε will be called in this work as bending, shear and shear strains, as indicated in Figure 4 (b).\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eThe stress-strain curves from both steels are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e5\u003c/span\u003e that confirms that both types of steels can be rated to an ultimate tensile strength of 800 MPA [\u003cspan additionalcitationids=\"CR4 CR5\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Although the shape of the curves of both steels differ, as those from steel \u003cb\u003eA\u003c/b\u003e reflect those of multiphase AHSS, as they do not show an abrupt yield point phenomenon and exhibit a high strengthening rate after yielding, whereas the shape of the curves from steel \u003cb\u003eB\u003c/b\u003e resembles that of a high strength low alloy (HSLA) steel in the sense that they exhibit the yield point or L\u0026uuml;ders phenomenon and a lower strengthening rate towards the early stages of plastic deformation [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eMechanical properties of the steel strips.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSteel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e\u003cp\u003eA\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"3\" nameend=\"c7\" namest=\"c5\"\u003e\u003cp\u003eB\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eInclination to rolling\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e45\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e90\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e45\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e90\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eStrain at yielding\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.005\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.005\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.005\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.006\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.006\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.005\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eL\u0026uuml;ders strain\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.032\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.032\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.035\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eYield stress\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e697.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e671.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e708.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e724.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e766.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e752.9\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUniform strain\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.090\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.085\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.068\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.141\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.139\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.123\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUltimate tensile stress\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e996.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e959.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e985.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e890.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e897.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e898.8\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eStrain to fracture\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.137\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.141\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.111\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.186\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.193\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.168\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e\u003cp\u003eYield and ultimate tensile stress in MPa\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the sheared surfaces of the specimens from steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e with the various clearances, in which the difference between the steels is notable, as no fissures are detected in the samples from steel \u003cb\u003eA\u003c/b\u003e. Still, they occur in all the specimens from steel \u003cb\u003eB\u003c/b\u003e, except the one sheared with the largest clearance, 20%. Figures\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e7\u003c/span\u003e and \u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e8\u003c/span\u003e show the micrographs and points selected for drawing the five flow lines selected in each specimen for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e, respectively. These images show the marked contrast between both steels, as was pointed out when Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e6\u003c/span\u003e was presented. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows the proportion of rollover, bright and fracture zones, and burr, see Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e1\u003c/span\u003e, measured from images such as those shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e6\u003c/span\u003e to \u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The occurrence of fissures in the samples is also presented in this table. The values for angle φ, which is made by the fracture zone with respect to the vertical, shearing, direction, see Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e1\u003c/span\u003e (a), were measured from the broken lines drawn in Figs.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e7\u003c/span\u003e and \u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e8\u003c/span\u003e and are also reported in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eAmount of various features (%) on the surfaces of the sheared samples; the occurrence of fissures and that of the fracture angle are reported.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"9\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSteel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"4\" nameend=\"c5\" namest=\"c2\"\u003e\u003cp\u003eA\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"4\" nameend=\"c9\" namest=\"c6\"\u003e\u003cp\u003eB\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eClearance (%)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRollover\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e3.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e4.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e9.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e15.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e13.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e12.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e6.8\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBurnish\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e6.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e12.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e11.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e13.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e27.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e37.1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e28.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e23.1\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eFracture\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e90.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e83.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e84.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e73.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e57.1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e49.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e58.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e65.1\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBurr\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e3.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e1.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e3.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e5.0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eFissures\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eNo\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNo\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003eNo\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003eNo\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003eYes\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003eYes\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003eYes\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003eNo\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAngle φ (\u0026deg;)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-26\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-14\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-28\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e3\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe flow lines shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e7\u003c/span\u003e and \u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e8\u003c/span\u003e were selected to start within the unaffected material and end at the sheared surface. Their coordinates were then related to those of the micrographs for their analysis. Figures\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e9\u003c/span\u003e and \u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e10\u003c/span\u003e show, respectively, the values of the different strain components calculated along the five different flow lines in samples from steel \u003cb\u003eA\u003c/b\u003e sheared with 5 and 20% clearance. The values for the equivalent strains evaluated at the four clearances for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e are shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e11\u003c/span\u003e and \u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e12\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eThe values of equivalent strain in a three-dimensional plot for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e sheared with 5 and 20% clearances are shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e13\u003c/span\u003e to \u003cspan refid=\"Fig19\" class=\"InternalRef\"\u003e16\u003c/span\u003e. The low lines and the points used to fit them are plotted at zero equivalent strain; the trace of sheared surface will be plotted towards the right-hand side. Figures\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e13\u003c/span\u003e and \u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e15\u003c/span\u003e show the distributions for the 5% clearance, whereas those for a 20% clearance are shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e14\u003c/span\u003e and \u003cspan refid=\"Fig19\" class=\"InternalRef\"\u003e16\u003c/span\u003e. These diagrams confirm the difference in behaviour between steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e. The strain values in steel \u003cb\u003eA\u003c/b\u003e are higher when the clearance increases from 5 to 20%, whereas those in steel \u003cb\u003eB\u003c/b\u003e are reduced. Moreover, the fissures observed in the samples from steel \u003cb\u003eB\u003c/b\u003e do not occur at the highest clearance.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe computed values for the equivalent strain were used to construct the maps shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e17\u003c/span\u003e and \u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e18\u003c/span\u003e for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e, respectively. The contour lines were drawn for strains of 0.05, 0.1, 0.2, 0.5 and 1.0. These contours follow a similar shape in the samples from steel \u003cb\u003eA\u003c/b\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e17\u003c/span\u003e, in the sense that higher values of strain occur towards the sheared edge, and will exhibit high strain gradients, as the material would be undeformed at a distance of less than 1 mm from the sheared surface; higher values of equivalent are achieved as the clearance of the shearing tools increases. The distribution within the samples from steel \u003cb\u003eB\u003c/b\u003e is somewhat different. Although higher strain values are encountered towards the sheared surface, the strain gradients will be shallower, as unstrained material will not be found below 1.5 mm, except for the case of the sample cut with a 20% clearance. The micrographs from Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e8\u003c/span\u003e show the development of fissures within the sample when clearances of 5, 10, and 15% were used; these fissures are incipient in the sample with a 20% clearance. Burr is observed in all samples from material \u003cb\u003eB\u003c/b\u003e; its values are shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and can be observed in Figs.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e8\u003c/span\u003e and \u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e18\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eThe difference in the strain distribution within the steels would be related to their difference in mechanical properties expressed in terms of their stress-strain curves shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Both steels show a yield strength close to 700 MPa, 692.5 MPa for steel \u003cb\u003eA\u003c/b\u003e and 748.1 MPa for steel \u003cb\u003eB\u003c/b\u003e, which exhibits an average L\u0026uuml;ders strain of 0.033; steel \u003cb\u003eA\u003c/b\u003e shows a higher hardening rate after yielding than that of steel \u003cb\u003eB\u003c/b\u003e. The tensile strength of steel \u003cb\u003eA\u003c/b\u003e is of 980.3 MPa at a strain 0.081, and a strain to fracture of 0.130. Steel \u003cb\u003eB\u003c/b\u003e exhibits a tensile strength of 895.7 MPa at a strain of 0.134, and a strain to fracture of 0.182. Figure\u0026nbsp;\u003cspan refid=\"Fig22\" class=\"InternalRef\"\u003e19\u003c/span\u003e shows the strengthening rate vs. strain curves (dσ/dε-ε) for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e calculated from the stress vs. strain curves (σ-ε) obtained from samples tested along the rolling direction (0\u0026deg; in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e5\u003c/span\u003e). The width of the L\u0026uuml;ders strain of steel \u003cb\u003eB\u003c/b\u003e is indicated in the graph. Figure\u0026nbsp;\u003cspan refid=\"Fig22\" class=\"InternalRef\"\u003e19\u003c/span\u003e shows that the hardening rate of steel \u003cb\u003eA\u003c/b\u003e is higher than that of steel \u003cb\u003eB\u003c/b\u003e at lower strains and particularly while this last steel is within the L\u0026uuml;ders regime, which will allow for straining to proceed to a higher degree in this material and would result in the extended deformation observed to occur in the diagrams shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e18\u003c/span\u003e, while the higher hardening rate would be responsible for the higher strain gradients appreciated in the corresponding diagrams from steel \u003cb\u003eA\u003c/b\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e17\u003c/span\u003e. The lower strength of steel \u003cb\u003eB\u003c/b\u003e will be responsible for the burr observed to occur in its sheared samples as the softer material will be pushed into the gap left by the shears as has been documented experimentally and modelled by finite element analyses [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan additionalcitationids=\"CR11 CR12 CR13\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan additionalcitationids=\"CR21 CR22 CR23 CR24 CR25 CR26 CR27 CR28\" citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e].\u003c/p\u003e\u003cp\u003ePrevious work on steels with similar compositions of either type of steel had revealed that the microstructure and hardening behaviour of these steels are different. However, both are in their hot-rolled condition [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. Steel \u003cb\u003eA\u003c/b\u003e, due to their higher contents in Si and Mn, as well as the addition of Cr, Ni and Mo would transform after rolling into a higher proportion of acicular structures, martensite and bainite, than steel \u003cb\u003eB\u003c/b\u003e, which relies in the strengthening by precipitation of TiN that results from the addition of Ti and the amount of N grabbed during melting [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e]. Figure\u0026nbsp;\u003cspan refid=\"Fig23\" class=\"InternalRef\"\u003e20\u003c/span\u003e displays the continuous cooling transformation diagrams (CCT) for steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e, which were computed at the centre and surface of 3.6 mm thick strips processed in the run-out table of a compact, thin slab line, assuming conditions to those documented for a similar line [\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e]. Diagrams a and b were predicted by the software mentioned above [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e], assuming a finishing rolling temperature of 900\u0026ordm;C and a fully recrystallised austenite grain size of 30 \u0026micro;m [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e], in which transformation in steel \u003cb\u003eA\u003c/b\u003e will start into bainite as it cools down. In contrast, steel \u003cb\u003eB\u003c/b\u003e will begin transforming into ferrite. The cooling curves indicated as S and C for surface and centre in the diagrams follow rates close to 10 C/s. The isothermal temperatures for A\u003csub\u003e1\u003c/sub\u003e, A\u003csub\u003e3\u003c/sub\u003e and M\u003csub\u003es\u003c/sub\u003e are drawn as broken lines, the curves for the start of transformation to ferrite (F\u003csub\u003es\u003c/sub\u003e), pearlite (P\u003csub\u003es\u003c/sub\u003e) and banite (B\u003csub\u003es\u003c/sub\u003e), as well for the end of pearlite (P\u003csub\u003ef\u003c/sub\u003e) and bainite (B\u003csub\u003ef\u003c/sub\u003e) are drawn as full lines. Figure\u0026nbsp;\u003cspan refid=\"Fig23\" class=\"InternalRef\"\u003e20\u003c/span\u003e also shows in diagrams b and c the volume fraction (X\u003csub\u003ev\u003c/sub\u003e) predicted by the software [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e] for ferrite (F), pearlite (P), bainite (B), martensite (M) both steels at various rates, austenite (A) is predicted to occur only a obove 20C/s in steel A. The shaded areas in these diagrams correspond to rates ranging from 5 to 20 C/s. The microstructures in Fig.\u0026nbsp;\u003cspan refid=\"Fig24\" class=\"InternalRef\"\u003e21\u003c/span\u003e were observed in samples prepared from steels \u003cb\u003eA\u003c/b\u003e and \u003cb\u003eB\u003c/b\u003e that were etched with the tint reagent used for identification of various structures as martensite appears white, bainite black and ferrite tan [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. It is confirmed from these images that the structure of steel \u003cb\u003eA\u003c/b\u003e is made of ferrite, bainite and martensite, which are tinted tan, black and white; whereas that of steel \u003cb\u003eB\u003c/b\u003e is mainly made of ferrite and bainite. Figure\u0026nbsp;\u003cspan refid=\"Fig25\" class=\"InternalRef\"\u003e22\u003c/span\u003e shows the end of a fissure detected in the specimen from steel \u003cb\u003eB\u003c/b\u003e sheared with a 10% clearance in which a series of cuboid particles are detected close or within the crack surface, and are suspected to cause the failure. These particles have been identified as TiN in steels of similar composition that exhibited cracks during processing [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eIt is noticeable that the angle φ in the samples sheared from steel \u003cb\u003eA\u003c/b\u003e is affected by the clearance as it changes from positive to negative, Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, implying that blades and toolings will be subjected to higher attrition and wear when the clearance is tightened, as has been reported in previous studies [\u003cspan additionalcitationids=\"CR8 CR9 CR10\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan additionalcitationids=\"CR24\" citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e, \u003cspan additionalcitationids=\"CR29 CR30\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], therefore, it would be recommended the use of a 10 to 15% clearance when trimming or shearing this type of steel. The case of steel \u003cb\u003eB\u003c/b\u003e is difficult to assess due to the occurrence of fissures and burr in almost all situations and would require a clearance of at least 20% to obtain a clear cut in which no fissures were developed and the burr is slight.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eThe procedure used to compute the local values of strain components show that the steels behave in a different way when sheared. Steel \u003cb\u003eA\u003c/b\u003e, in which its microstructure contains bainite and martensite, tends to develop steeper strain gradients in the vicinity of the cut, whereas steel \u003cb\u003eB\u003c/b\u003e, in which a predominance of ferrite and bainite is encountered exhibit shallower gradients.\u003c/p\u003e\u003cp\u003eThe difference in the development of such gradients can be explained in terms of the strengthening rate (dσ/dε) taking place in either steel after yielding. Steel \u003cb\u003eB\u003c/b\u003e, which exhibit L\u0026uuml;ders phenomenon, is not subjected to the higher strengthening rate that steel \u003cb\u003eA\u003c/b\u003e shows, hence the shallower gradients in steel \u003cb\u003eB\u003c/b\u003e.\u003c/p\u003e\u003cp\u003eThe strengthening in steel \u003cb\u003eA\u003c/b\u003e affects the angle φ, and it would be the case when using small clearance, such a 5%, that the new surface will impinge and press the tools promoting the high wear rates observed to occur un such instances. Therefore, these steels require for a higher clearance, such as 15%. The case of steel \u003cb\u003eB\u003c/b\u003e is not clear, as the samples exhibit burr in all cases and, with the exception of the 20% clearance, all of them exhibit fissuring associated with the presence of TiN precipitates used to enhance its strength.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eSupport/Funding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors thank the support provided by the Mexican \u003cem\u003eSecretaría de Ciencia, Humanidades, Tecnología e Innovación\u003c/em\u003e (Ministry for Science, Humanities, Technology and Innovation) to OFVC and CASG.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of interest/Competing interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no conflict of interest related to this work.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors' contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eO.F. Villarreal-Cheretti. Experimental work, data gathering and analysis, writing.\u003c/p\u003e\n\u003cp\u003eC.A. Salazar-García. Experimental work, data gathering and analysis, writing.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eM.A. Quiñones. Industrial testing and analysis, writing.\u003c/p\u003e\n\u003cp\u003eN.F. Garza-Montes-de-Oca. Data analysis, industrial processing, writing.\u003c/p\u003e\n\u003cp\u003eR. Colás. Data analysis, writing, corresponding author.\u003c/p\u003e"},{"header":"References","content":"\u003col start=\"1\" type=\"1\"\u003e\n\u003cli\u003eMiller JD, Fa\u0026ccedil;anha C (2014). The state of clean transport policy. A 2014 synthesis of vehicle and fuel policy developments. International Council on Clean Transportation, New York. https://theicct.org/wp-content/uploads/2021/06/ICCT_StateOfCleanTransportPolicy_2014.pdf \u003c/li\u003e\n\u003cli\u003eModi S. Baron J (2017) EPA Mass Reduction Analysis - Observations and Recommendations. 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IV, 120:513-518. https://doi.org/10.1051/jp4:2004120059 \u003c/li\u003e\n\u003cli\u003eZambrano PC, Delgado AL, Guerrero-Mata MP, Col\u0026aacute;s R, Leduc LA (2003) Hot rolling of light gauge steel strip. ISIJ Int., 43:1030-1035. https://doi.org/10.2355/isijinternational.43.1030\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Formability, shearing, microstructure, visioplastic analysis, strain distribution","lastPublishedDoi":"10.21203/rs.3.rs-7272703/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7272703/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eManufactured goods subjected to complex forming operations, such as drawing, piercing, or stamping, involve cutting and shearing into blanks; shearing promotes the development of strain gradients within the piece, which may result in cracks or fissures in the finished component, and would cause the rejection of the blank. The origin of such defects is under discussion, as they can be attributed to the material or the cutting process. This work proposes a visioplastic analysis based on the macroscopic flow lines developed in the material during shearing. The method is applied to samples cut from advanced high-strength steels that were sheared with different clearances between the shearing blades used. The transversal section of the samples was prepared and etched to reveal these flow lines; the coordinates along five lines were recorded and fed into an algorithm that fitted them to cubic splines to obtain the coordinates and derivative of a new set of data points from which the strain components were computed. The method was tested in two different Advanced High-Strength Steels (AHSS) of various thicknesses and microstructures. It revealed essential details on the shear mechanics that would allow for the improvement of shearing and blanking operations.\u003c/p\u003e","manuscriptTitle":"Strain Distribution Analyses of Sheared AHSS","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-14 06:58:08","doi":"10.21203/rs.3.rs-7272703/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revisions Needed","date":"2025-10-14T08:41:55+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2025-08-29T08:24:03+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-07T02:42:17+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-05T12:34:27+00:00","index":"","fulltext":""},{"type":"submitted","content":"The International Journal of Advanced Manufacturing Technology","date":"2025-08-01T11:30:27+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"81c6442f-d5ef-4f54-a30c-292a1cbd310e","owner":[],"postedDate":"August 14th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-11-03T16:08:00+00:00","versionOfRecord":{"articleIdentity":"rs-7272703","link":"https://doi.org/10.1007/s00170-025-16883-z","journal":{"identity":"the-international-journal-of-advanced-manufacturing-technology","isVorOnly":false,"title":"The International Journal of Advanced Manufacturing Technology"},"publishedOn":"2025-10-31 15:56:56","publishedOnDateReadable":"October 31st, 2025"},"versionCreatedAt":"2025-08-14 06:58:08","video":"","vorDoi":"10.1007/s00170-025-16883-z","vorDoiUrl":"https://doi.org/10.1007/s00170-025-16883-z","workflowStages":[]},"version":"v1","identity":"rs-7272703","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7272703","identity":"rs-7272703","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}