- Email: [email protected]
Evaluation of neck shape of notched round-bar specimens during tensile test
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Procedia Manufacturing 00 (2018) 000–000 Available online atatwww.sciencedirect.com Available www.sciencedirect.com Procediaonline Manufacturing 00 (2018) 000–000
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www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Manufacturing 15 (2018) 1762–1768 Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia
17th International Conference on Metal Forming, Metal Forming 2018, 16-19 September 2018, 17th International Toyohashi, Japan Conference on Metal Forming, Metal Forming 2018, 16-19 September 2018, Toyohashi, Japan
Evaluation of neck shape of notched round-bar specimens during
EvaluationEngineering of neck shape of notched round-bar during Manufacturing Society International Conference 2017,specimens MESIC 2017, 28-30 June tensile test 2017, Vigo (Pontevedra), Spain tensile test Takeshi Nishiwakia, *, Masanobu Muratab, Yoshinori Yoshidac Costing models for capacity optimization Industry 4.0: Trade-off Takeshi Nishiwaki *, Masanobu Murata ,in Yoshinori Yoshida Dept. of Mechanical Engineering, Daido University, Nagoya, Aichi 457-8530, Japan used capacity and operational efficiency Dept.and of Mechanical Engineering, Daido University, Nagoya, Aichi 457-8530, Japan Dept.between of Production Systems, Nagoya Municipal Industrial Research Institute, Nagoya, Aichi 456-0058, Japan a,
b
c
a
b
a
c bDept.
of Production andEngineering Systems, Nagoya Municipal Industrial Research Institute, Nagoya, Aichi Center for Advanced Die and Technology (G-CADET), Gifu University, Gifu-City, Gifu456-0058, 501-1193,Japan Japan c Center for Advanced Die Engineering Gifu University, Gifu-City, Gifu a and Technology (G-CADET), a,* b b 501-1193, Japan
A. Santana , P. Afonso , A. Zanin , R. Wernke a
University of Minho, 4800-058 Guimarães, Portugal
Abstract b Unochapecó, 89809-000 Chapecó, SC, Brazil Abstract Notched round-bar tensile tests were conducted on three types of metals, and their flow stress curves were identified. Deformation Notched round-barwas tensile tests were three types andintheir stress curvesradius were identified. Deformation of the specimens recorded usingconducted two CCDoncameras and of themetals, changes theirflow cross-sectional were measured using a of the specimens was recorded CCDtensile cameras andwas thecorrected changes in their cross-sectional radiusand were digital image correlation method.using The two average stress through an inverse analysis, the measured numericalusing resultsa Abstract digital image method.results. The average tensile stress was corrected through analysis, andwere the numerical results coincided withcorrelation the experimental Measurements of the longitudinal profilesan of inverse a necked specimen also carried out. coincided the experimental results. Measurements the longitudinal profiles of a necked specimen were interconnected, also carried out. Numerical analysis results the identified flow stressofcurve showedwill a good experimental results. Under thewith concept of using "Industry 4.0", production processes be agreement pushed towith bethe increasingly Numerical analysis usingtime the identified flow stress curvemuch showed a good agreement the experimental results. information basedresults on a real basis and, necessarily, more efficient. In with this context, capacity optimization © 2018 The Authors. Published by Elsevier B.V. goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability and value. © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. © Peer-review under responsibility of the the scientific committee committee of the the 17th 17th International Conference onMetal Metal Forming. instead of Peer-review under responsibilityand of scientific of International Conference on Forming. Indeed, lean management continuous improvement approaches suggest capacity optimization Peer-review under responsibility of the scientific committee of the 17th International Conference on Metal Forming.
maximization. The study of capacity optimization and costing models is an important research topic that deserves Keywords: Tensile test; Notched specimen; Flow stress; Local necking; Degital image correlation contributions from both thespecimen; practical and theoretical perspectives. This paper presents and discusses a mathematical Keywords: Tensile test; Notched Flow stress; Local necking; Degital image correlation model for capacity management based on different costing models (ABC and TDABC). A generic model has been developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s 1. Introduction value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity 1. Introduction optimization might hideupoperational inefficiency. A flow stress curve to a large amount of strain is required for an accurate plastic forming simulation of forging © 2017 The Authors. Published by Elsevier B.V.of A flow stress curve up to a large amount strain is requiredthrough for an accurate plastic forming ofalthough forging and stamping processes. In general, the flow stress is measured tensile tests using flat barsimulation specimens, Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference and stamping processes. In general, the flow stress is measured through tensile tests using flat bar specimens, although a flow stress curve after a uniform elongation cannot be obtained. Therefore, the flow stress after local necking is 2017. aapproximated flow stress curve afterana extrapolation uniform elongation be obtained. Therefore, flow after necking through using acannot hardening law. However, in the order to stress improve thelocal accuracy of isa approximated through an extrapolation using a hardening law. However, in order to improve the accuracy of a Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency
1. Introduction
* Corresponding author. Tel.: +81-52-612-6111; fax: +81-52-6125623. * E-mail Corresponding Tel.: +81-52-612-6111; fax: +81-52-6125623. address:author. [email protected] The cost of idle capacity is a fundamental information for companies and their management of extreme importance E-mail address: [email protected]
in modern©production systems. In general, it isB.V. defined as unused capacity or production potential and can be measured 2351-9789 2018 The Authors. Published by Elsevier 2351-9789 2018 Authors. Published Elsevier B.V.hours of Peer-review underThe responsibility of theby scientific committee 17th International on Metal Forming. in several© ways: tons of production, available of the manufacturing, etc.Conference The management of the idle capacity Peer-review under responsibility thefax: scientific committee * Paulo Afonso. Tel.: +351 253 510of 761; +351 253 604 741 of the 17th International Conference on Metal Forming. E-mail address: [email protected]
2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 17th International Conference on Metal Forming. 10.1016/j.promfg.2018.07.248
2
Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
1763
simulation, it is necessary to measure experimentally the flow stress up to the large strain region. Therefore, some methods for obtaining the flow stress from the tensile test results after necking have been developed [1]. We have developed a method for obtaining a flow stress curve until a fracture occurs [2, 3]; with this method, a tensile test using notched round-bar specimens is conducted and a stress correction is applied to the experimentally measured curve. Stress correction factors are identified through an inverse analysis using the finite element method (FEM) of the tensile test and an optimization method to minimize the deviation between the experimental and numerical loads - cross sectional radius curve (P-a curve, see Fig. 1). Therefore, a FEM analysis using the flow stress curve identified through this method reproduces a cross sectional radial change at the bottom of the neck, however we have not confirmed whether the overall profiles of the neck shapes are consistent with the experimental results. In addition, in our previously proposed method, images taken using a CCD camera were binarized, and the diameters of the neck bottom were measured through an extraction of the contour line. Therefore, the measurement accuracy of the radius is insufficient with a material with a small elongation, such as medium carbon steel, because of the small amount of deformation prior to failure. Recently, the studies on identifying the parameters in Swift’s law and Voce’s law using a digital image correlation method (DIC) for strain measurements during material testing have been reported [4, 5]. DIC can be applied to not only strain measurements but also to three-dimensional shape measurements accurately using an image analysis of sub-pixel processing. Therefore, we tried to improve our proposed method using DIC for shape measurements. 2. Experimental procedure 2.1. Notched round-bar tensile test A notched round-bar tensile test was applied to two ductile metals, low carbon steel (SS400, in JIS) and an aluminum alloy (A5056-H34, in JIS), and a low ductile metal, middle carbon steel (S45C, in JIS). Their mechanical properties are shown in Table 1. The parameters of Swift’s law determined within the range of uniform elongation are also shown. The uniform elongation of S45C is extremely small, at 2%. Fig. 2 shows the dimensions of the specimens. The diameters of the notch bottom were 5mm, and four kinds of notch radius were prepared. It is possible to change the history of the triaxial stress state in the neck bottom by changing the initial notched radius R0. As the radius decreases, the triaxiality and the tensile load increase. The specimens were previously coated with a random pattern, and their deformations were recorded using two CCD cameras (GRAS-20S4M，Point Grey Research, see Fig. 1). The two cameras simultaneously captured the same side, covering for 90 degrees of the specimens. The tensile speed was 3mm/min, and the sampling speed of the images was 8 fps. This corresponds to an average strain increment of 0.0014 per frame in in the case of SS400.
Fig. 1. Schematic diagram of tensile test recorded using two cameras.
Fig. 2. Shapes of notched round-bar specimens.
Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
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Table 1. Mechanical properties and the parameters of Swift’s law. Swift’s law ; σ = F (ε0+εp)n
Tensile strength /MPa
Yield strength /MPa
Uniform elongation /%
F /MPa
n
ε0
S45C
776
575
2
1052
0.07
0.00028
SS400
473
537
19
788
0.19
0.002
A5056-H34
318
225
11
567
0.21
0.013
2.2. Measurement of cross-sectional radius of neck bottom using digital image correlation The surface displacement and strain distribution at the neck bottom of the specimens were analyzed using DIC software (Vic-3D, Correlated Solutions) with two pairs of deformed images. At the same time, three-dimensional profiles of the surface of the specimens were also measured. Fig. 3(a) shows an example of the three-dimensional profiles of the neck bottom analyzed using the DIC. A profile of the cross-section perpendicular to the tensile axis at the bottom is approximated as a circle using the least squares method. The minimum cross-sectional radii of the neck bottom a were continuously obtained from the circles (see Fig. 3(b)). a
b
Fig. 3. Measured profiles of notched round-bar specimen (SS400, R0 = 20 mm); (a) three-dimensional coordinates of the surface of specimen; (b) cross-sectional contour lines at neck bottoms from the initial state to fracture.
Fig. 4. Comparison of measurement methods of section radius.
Fig. 5. Corrected flow stress curve for inverse analyses.
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Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
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Fig. 4 shows a comparison of the measurement method between the DIC and a conventional image analysis [6]. Although the measurement error in the conventional method was about ± 30 μm, the DIC decreased the error more than one order of magnitude, and the neck radius curve is smooth. This DIC method can be applied to measure the small amounts of deformation. 2.3. Stress correction method of flow stress As a stress correction method of the flow stress from the average tensile stress, an inverse analysis method was used. In this method [2, 3], the strain range from uniform strain to fracture strain is divided into N sections (see Fig. 5), and the equivalent stress σflow I for each equivalent plastic strain εI is obtained by multiplying the average tensile stress σzave I using a correction coefficient xI, which is defined through the following equation. σflow I ＝ xI σzave I (0≦xI≦1, I=1,2,…,N) .
Fig. 6. Error between experimental and computed curves.
(1)
Fig. 7. FEM models of notched round-bar tensile tests.
The correction coefficient xI is obtained using an inverse analysis such that the mean-square error of the target tensile load Pi measured through the experiment and tensile load Fi(x) calculated from a FEM analysis is minimized (see Fig. 6). In the inverse analysis, an optimization method, namely, the successive response surface method (SRSM) [7], is adopted. LS-DYNA971 was used for the numerical simulations. Axial symmetry along the tensile axis and mirror symmetry at the neck bottom were assumed. All the models are shown in Fig. 7. The neck bottom is divided into 0.1 mm meshes to reproduce the shape of the neck. It is assumed that the material is elasto-plastic and follows the von-Mises yield criterion. 3. Experimental results 3.1. Tensile test results and corrected stress-strain curve Fig. 8(a) shows experimental curves of tensile load P versus the change in radius (a0-a). The tensile load of medium the carbon steel (S45C) was the highest, and the low carbon steel (SS400) and the aluminum alloy (A5056-H34) followed in order of tensile strength. With the same material, as the initial notch radius R0 was decreased, the tensile loads increased and the fracture occurred earlier. A specimen with an initial notch radius of 3 mm was the earliest to break, and that of S45C failed when the radius decreased by only 0.15 mm. Even if a material with less ductility has only small deformations prior to a fracture, the dimensional change can be measured with high accuracy using the DIC. Numerical results on the tensile load P versus the change in radius (a0-a) are also shown in Fig. 8(a). These are the final results of an inverse analysis, and are described in the section 2.3. The corrected flow stress was used in these FEM analyses. The experimental load curve and numerical analysis results are almost overlapped under all conditions.
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Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
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Using the corrected flow stress, the relation between the tensile load and change in radius up to a failure has been reproduced. In the deformed states from mark A to mark I in Fig. 8(a), the longitudinal profiles of the neck bottom were investigated in detail. These results will be described later in section 3.2. a
b
Fig. 8. (a) Experimental curves and inverse analysis results regarding tensile load P versus change in radius (a0-a) curve, and (b) corrected flow stress curves obtained through inverse analyses.
Fig. 8(b) shows the equivalent stress-equivalent plastic strain curves identified through a stress correction. Despite the different radii R0 of the notch bottoms, the identified curves overlap the same curve with the same material. The range in the stress-strain curve obtained is the most extensive in the case of the specimen (R0 = 20 mm). The stress-strain curve at up to the equivalent strain of 0.3 has been identified for S45C. The largest strain obtained through this method is more than 10 times that under uniform strain. 3.2. Comparisons of longitudinal profiles of deformed specimens Fig. 9(a) shows the longitudinal profiles of a notched specimen (SS400, R0 = 20 mm) during a tensile test. These four profile curves were taken during the initial state A, under maximum load B, after local necking C, and just prior to fracture D. Comparing A and B, the y coordinate of curve B decreases overall, which indicates that it is a deformation within the range of uniform elongation. Comparing B, C, and D, the y coordinate near the center of the notch bottom locally decreases, which indicates that local necking has occurred. In addition, the results of FEM analyses using the corrected stress-strain curve are shown. The numerical results coincide with the experimental results. The entire shape can be reproduced in spite of using only the cross-section radial change (a0-a) at the necked bottom under a stress correction. It is thought that the corrected flow stress is accurate, thereby proving the validity of the correction method. Fig. 9(b) shows the longitudinal shapes of the SS400 specimens, which have various initial notch radii. All curves were taken just prior to fracture. Except for G (R0 = 3 mm), the centers of the D, E, and F curves are nearly the same in term of radius and profile along the longitudinal direction just prior to fracture. This means that, even though the initial radius of the notch bottom was different, the state of stress at the break was about the same. The profile of the numerical calculation also represents the experimental results well. According to the calculation results of the FEM, the stress triaxiality just prior to failure of the D, E, and F specimens was almost the same value, 0.95. The notched specimen of SS400 with large ductility has not succeeded in providing different stress states until fracture. Fig. 9(c) shows the longitudinal shapes of notched specimens (R0 = 20 mm) of the three materials just prior to fracture. The specimen of S45C, which fractured early, showed the smallest change of only 0.25 mm at the bottom. The profile of the numerical calculation also represents the experimental results well.
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Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
a
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b
c
Fig. 9. Longitudinal profiles of specimens during tensile test: profiles of (a) various changes in radius (SS400, R0 = 20 mm), (b) various notch bottom radii (SS400, just prior to fracture) and (c) various materials (R0 = 20 mm, just prior to fracture).
3.3. History of surface strain of deformed specimens The surface strain was also measured using the DIC. Fig. 10 shows the experimental and numerical curves of the surface strain versus a change in radius (a0-a). The numerical longitudinal strain εz and the circumferential strain εθ of SS400 were consistent with the measurement results of the DIC. In the case of S45C, both results showed good agreement. However, in the case of A5056-H34, good agreement was observed until the change in radius reached about 0.3 mm, but a deviation began thereafter, and a deviation of about 7% was eventually observed in the longitudinal strain. To identify this cause, not only the strain at a single point on the surface of the specimen, but also the overall changes in the cross sectional shape were investigated. Fig. 10 shows the histories of circularity of the cross sections in the notch bottom. In the S45C and SS400 specimens, there were few changes from the initial circularity during the tensile test. On the contrary, the circularity of A5056-H34 increased from about a 0.3 mm change in radius, which means that the sectional shape changed from a circle. In other words, a non-axisymmetric deformation was observed, which deviates from an isotropic assumption. Therefore, it is thought that the FEM results are not consistent with the experimental curve. 4. Conclusions In this study, the notched round-bar tensile tests were conducted on a middle carbon steel (S45C), a low carbon steel (SS400), and an aluminum alloy (A5056-H34) using an image analysis with the DIC, and their flow stress curves were determined using a proposed stress correction method. The longitudinal profiles of the necked specimen were measured and compared with the FEM results. The following conclusions were obtained. 1) The flow stress of a low ductile material such as middle carbon steel up to the fracture point can be obtained using a digital image correlation method. 2) In the case of S45C and SS400, the load-radius curve, longitudinal profiles, and surface strain obtained through FEM is consistent with the experimental results. The validity of the stress correction method could be proved.
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3) In spite of the consistency in the load-radius curve and longitudinal profiles, a deviation was observed in the longitudinal strain of A5056-H34. A non-axisymmetric deformation occurred, and the isotropic assumption was not maintained until the fracture point. a
b
Fig. 10. Longitudinal and circumferential strain of specimens during tensile test: (a) SS400, R0 = 20 mm, and (b) A5056-H34, R0 = 20 mm.
Fig. 11. Changes in circularity of cross section at necked bottom.
References [1] G. Mirone, A new model for the elastoplastic characterization and the stress-strain determination on the necking section of a tensile specimen, International Journal of Solids and Structures, 41(2004) 3545–3564. [2] M. Murata, Y. Yoshida, T. Nishiwaki, Stress correction method for flow stress identification by tensile test using notched round bar, Journal of Materials Processing Technology, 251 (2018) 65–72. [3] M. Murata, Y. Yoshida, T. Nishiwaki, Identification of ductile fracture parameter with stress correction method using notched round-bar tensile test, Procedia Engineering, 207 (2017) 2060–2065. [4] S. Coppieters, S. Cooreman, H. Sol, P.V. Houtte, D. Debruyne, Identification of the post-necking hardening behavior of sheet metal by comparison of the internal and external work in the necking zone, Journal of Materials Processing Technology, 211 (2011) 545–552. [5] J.H. Kim, A. Serpantié, F. Barlat, F. Pierron, M.G. Lee, Characterization of the post-necking strain hardening behavior using the virtual fields method, International Journal of Solids and Structures, 50 (2013) 3829–3842. [6] Y. Yoshida, N. Yukawa, T. Ishikawa, Determination of ductile damage parameters by notched round bar tention test using image analysis, NUMIFORM’2004, AIP Conference Proceedings, 712 (2004) 1869–1874. [7] N. Stander, K.J. Craig, On the robustness of a simple domain reduction scheme for simulation based optimization., Engineering Computations, 19 (2002) 431–450.
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Procedia Manufacturing 00 (2018) 000–000 Available online atatwww.sciencedirect.com Available www.sciencedirect.com Procediaonline Manufacturing 00 (2018) 000–000
ScienceDirect ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Manufacturing 15 (2018) 1762–1768 Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia
17th International Conference on Metal Forming, Metal Forming 2018, 16-19 September 2018, 17th International Toyohashi, Japan Conference on Metal Forming, Metal Forming 2018, 16-19 September 2018, Toyohashi, Japan
Evaluation of neck shape of notched round-bar specimens during
EvaluationEngineering of neck shape of notched round-bar during Manufacturing Society International Conference 2017,specimens MESIC 2017, 28-30 June tensile test 2017, Vigo (Pontevedra), Spain tensile test Takeshi Nishiwakia, *, Masanobu Muratab, Yoshinori Yoshidac Costing models for capacity optimization Industry 4.0: Trade-off Takeshi Nishiwaki *, Masanobu Murata ,in Yoshinori Yoshida Dept. of Mechanical Engineering, Daido University, Nagoya, Aichi 457-8530, Japan used capacity and operational efficiency Dept.and of Mechanical Engineering, Daido University, Nagoya, Aichi 457-8530, Japan Dept.between of Production Systems, Nagoya Municipal Industrial Research Institute, Nagoya, Aichi 456-0058, Japan a,
b
c
a
b
a
c bDept.
of Production andEngineering Systems, Nagoya Municipal Industrial Research Institute, Nagoya, Aichi Center for Advanced Die and Technology (G-CADET), Gifu University, Gifu-City, Gifu456-0058, 501-1193,Japan Japan c Center for Advanced Die Engineering Gifu University, Gifu-City, Gifu a and Technology (G-CADET), a,* b b 501-1193, Japan
A. Santana , P. Afonso , A. Zanin , R. Wernke a
University of Minho, 4800-058 Guimarães, Portugal
Abstract b Unochapecó, 89809-000 Chapecó, SC, Brazil Abstract Notched round-bar tensile tests were conducted on three types of metals, and their flow stress curves were identified. Deformation Notched round-barwas tensile tests were three types andintheir stress curvesradius were identified. Deformation of the specimens recorded usingconducted two CCDoncameras and of themetals, changes theirflow cross-sectional were measured using a of the specimens was recorded CCDtensile cameras andwas thecorrected changes in their cross-sectional radiusand were digital image correlation method.using The two average stress through an inverse analysis, the measured numericalusing resultsa Abstract digital image method.results. The average tensile stress was corrected through analysis, andwere the numerical results coincided withcorrelation the experimental Measurements of the longitudinal profilesan of inverse a necked specimen also carried out. coincided the experimental results. Measurements the longitudinal profiles of a necked specimen were interconnected, also carried out. Numerical analysis results the identified flow stressofcurve showedwill a good experimental results. Under thewith concept of using "Industry 4.0", production processes be agreement pushed towith bethe increasingly Numerical analysis usingtime the identified flow stress curvemuch showed a good agreement the experimental results. information basedresults on a real basis and, necessarily, more efficient. In with this context, capacity optimization © 2018 The Authors. Published by Elsevier B.V. goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability and value. © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. © Peer-review under responsibility of the the scientific committee committee of the the 17th 17th International Conference onMetal Metal Forming. instead of Peer-review under responsibilityand of scientific of International Conference on Forming. Indeed, lean management continuous improvement approaches suggest capacity optimization Peer-review under responsibility of the scientific committee of the 17th International Conference on Metal Forming.
maximization. The study of capacity optimization and costing models is an important research topic that deserves Keywords: Tensile test; Notched specimen; Flow stress; Local necking; Degital image correlation contributions from both thespecimen; practical and theoretical perspectives. This paper presents and discusses a mathematical Keywords: Tensile test; Notched Flow stress; Local necking; Degital image correlation model for capacity management based on different costing models (ABC and TDABC). A generic model has been developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s 1. Introduction value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity 1. Introduction optimization might hideupoperational inefficiency. A flow stress curve to a large amount of strain is required for an accurate plastic forming simulation of forging © 2017 The Authors. Published by Elsevier B.V.of A flow stress curve up to a large amount strain is requiredthrough for an accurate plastic forming ofalthough forging and stamping processes. In general, the flow stress is measured tensile tests using flat barsimulation specimens, Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference and stamping processes. In general, the flow stress is measured through tensile tests using flat bar specimens, although a flow stress curve after a uniform elongation cannot be obtained. Therefore, the flow stress after local necking is 2017. aapproximated flow stress curve afterana extrapolation uniform elongation be obtained. Therefore, flow after necking through using acannot hardening law. However, in the order to stress improve thelocal accuracy of isa approximated through an extrapolation using a hardening law. However, in order to improve the accuracy of a Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency
1. Introduction
* Corresponding author. Tel.: +81-52-612-6111; fax: +81-52-6125623. * E-mail Corresponding Tel.: +81-52-612-6111; fax: +81-52-6125623. address:author. [email protected] The cost of idle capacity is a fundamental information for companies and their management of extreme importance E-mail address: [email protected]
in modern©production systems. In general, it isB.V. defined as unused capacity or production potential and can be measured 2351-9789 2018 The Authors. Published by Elsevier 2351-9789 2018 Authors. Published Elsevier B.V.hours of Peer-review underThe responsibility of theby scientific committee 17th International on Metal Forming. in several© ways: tons of production, available of the manufacturing, etc.Conference The management of the idle capacity Peer-review under responsibility thefax: scientific committee * Paulo Afonso. Tel.: +351 253 510of 761; +351 253 604 741 of the 17th International Conference on Metal Forming. E-mail address: [email protected]
2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 17th International Conference on Metal Forming. 10.1016/j.promfg.2018.07.248
2
Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
1763
simulation, it is necessary to measure experimentally the flow stress up to the large strain region. Therefore, some methods for obtaining the flow stress from the tensile test results after necking have been developed [1]. We have developed a method for obtaining a flow stress curve until a fracture occurs [2, 3]; with this method, a tensile test using notched round-bar specimens is conducted and a stress correction is applied to the experimentally measured curve. Stress correction factors are identified through an inverse analysis using the finite element method (FEM) of the tensile test and an optimization method to minimize the deviation between the experimental and numerical loads - cross sectional radius curve (P-a curve, see Fig. 1). Therefore, a FEM analysis using the flow stress curve identified through this method reproduces a cross sectional radial change at the bottom of the neck, however we have not confirmed whether the overall profiles of the neck shapes are consistent with the experimental results. In addition, in our previously proposed method, images taken using a CCD camera were binarized, and the diameters of the neck bottom were measured through an extraction of the contour line. Therefore, the measurement accuracy of the radius is insufficient with a material with a small elongation, such as medium carbon steel, because of the small amount of deformation prior to failure. Recently, the studies on identifying the parameters in Swift’s law and Voce’s law using a digital image correlation method (DIC) for strain measurements during material testing have been reported [4, 5]. DIC can be applied to not only strain measurements but also to three-dimensional shape measurements accurately using an image analysis of sub-pixel processing. Therefore, we tried to improve our proposed method using DIC for shape measurements. 2. Experimental procedure 2.1. Notched round-bar tensile test A notched round-bar tensile test was applied to two ductile metals, low carbon steel (SS400, in JIS) and an aluminum alloy (A5056-H34, in JIS), and a low ductile metal, middle carbon steel (S45C, in JIS). Their mechanical properties are shown in Table 1. The parameters of Swift’s law determined within the range of uniform elongation are also shown. The uniform elongation of S45C is extremely small, at 2%. Fig. 2 shows the dimensions of the specimens. The diameters of the notch bottom were 5mm, and four kinds of notch radius were prepared. It is possible to change the history of the triaxial stress state in the neck bottom by changing the initial notched radius R0. As the radius decreases, the triaxiality and the tensile load increase. The specimens were previously coated with a random pattern, and their deformations were recorded using two CCD cameras (GRAS-20S4M，Point Grey Research, see Fig. 1). The two cameras simultaneously captured the same side, covering for 90 degrees of the specimens. The tensile speed was 3mm/min, and the sampling speed of the images was 8 fps. This corresponds to an average strain increment of 0.0014 per frame in in the case of SS400.
Fig. 1. Schematic diagram of tensile test recorded using two cameras.
Fig. 2. Shapes of notched round-bar specimens.
Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
1764
3
Table 1. Mechanical properties and the parameters of Swift’s law. Swift’s law ; σ = F (ε0+εp)n
Tensile strength /MPa
Yield strength /MPa
Uniform elongation /%
F /MPa
n
ε0
S45C
776
575
2
1052
0.07
0.00028
SS400
473
537
19
788
0.19
0.002
A5056-H34
318
225
11
567
0.21
0.013
2.2. Measurement of cross-sectional radius of neck bottom using digital image correlation The surface displacement and strain distribution at the neck bottom of the specimens were analyzed using DIC software (Vic-3D, Correlated Solutions) with two pairs of deformed images. At the same time, three-dimensional profiles of the surface of the specimens were also measured. Fig. 3(a) shows an example of the three-dimensional profiles of the neck bottom analyzed using the DIC. A profile of the cross-section perpendicular to the tensile axis at the bottom is approximated as a circle using the least squares method. The minimum cross-sectional radii of the neck bottom a were continuously obtained from the circles (see Fig. 3(b)). a
b
Fig. 3. Measured profiles of notched round-bar specimen (SS400, R0 = 20 mm); (a) three-dimensional coordinates of the surface of specimen; (b) cross-sectional contour lines at neck bottoms from the initial state to fracture.
Fig. 4. Comparison of measurement methods of section radius.
Fig. 5. Corrected flow stress curve for inverse analyses.
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Fig. 4 shows a comparison of the measurement method between the DIC and a conventional image analysis [6]. Although the measurement error in the conventional method was about ± 30 μm, the DIC decreased the error more than one order of magnitude, and the neck radius curve is smooth. This DIC method can be applied to measure the small amounts of deformation. 2.3. Stress correction method of flow stress As a stress correction method of the flow stress from the average tensile stress, an inverse analysis method was used. In this method [2, 3], the strain range from uniform strain to fracture strain is divided into N sections (see Fig. 5), and the equivalent stress σflow I for each equivalent plastic strain εI is obtained by multiplying the average tensile stress σzave I using a correction coefficient xI, which is defined through the following equation. σflow I ＝ xI σzave I (0≦xI≦1, I=1,2,…,N) .
Fig. 6. Error between experimental and computed curves.
(1)
Fig. 7. FEM models of notched round-bar tensile tests.
The correction coefficient xI is obtained using an inverse analysis such that the mean-square error of the target tensile load Pi measured through the experiment and tensile load Fi(x) calculated from a FEM analysis is minimized (see Fig. 6). In the inverse analysis, an optimization method, namely, the successive response surface method (SRSM) [7], is adopted. LS-DYNA971 was used for the numerical simulations. Axial symmetry along the tensile axis and mirror symmetry at the neck bottom were assumed. All the models are shown in Fig. 7. The neck bottom is divided into 0.1 mm meshes to reproduce the shape of the neck. It is assumed that the material is elasto-plastic and follows the von-Mises yield criterion. 3. Experimental results 3.1. Tensile test results and corrected stress-strain curve Fig. 8(a) shows experimental curves of tensile load P versus the change in radius (a0-a). The tensile load of medium the carbon steel (S45C) was the highest, and the low carbon steel (SS400) and the aluminum alloy (A5056-H34) followed in order of tensile strength. With the same material, as the initial notch radius R0 was decreased, the tensile loads increased and the fracture occurred earlier. A specimen with an initial notch radius of 3 mm was the earliest to break, and that of S45C failed when the radius decreased by only 0.15 mm. Even if a material with less ductility has only small deformations prior to a fracture, the dimensional change can be measured with high accuracy using the DIC. Numerical results on the tensile load P versus the change in radius (a0-a) are also shown in Fig. 8(a). These are the final results of an inverse analysis, and are described in the section 2.3. The corrected flow stress was used in these FEM analyses. The experimental load curve and numerical analysis results are almost overlapped under all conditions.
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Using the corrected flow stress, the relation between the tensile load and change in radius up to a failure has been reproduced. In the deformed states from mark A to mark I in Fig. 8(a), the longitudinal profiles of the neck bottom were investigated in detail. These results will be described later in section 3.2. a
b
Fig. 8. (a) Experimental curves and inverse analysis results regarding tensile load P versus change in radius (a0-a) curve, and (b) corrected flow stress curves obtained through inverse analyses.
Fig. 8(b) shows the equivalent stress-equivalent plastic strain curves identified through a stress correction. Despite the different radii R0 of the notch bottoms, the identified curves overlap the same curve with the same material. The range in the stress-strain curve obtained is the most extensive in the case of the specimen (R0 = 20 mm). The stress-strain curve at up to the equivalent strain of 0.3 has been identified for S45C. The largest strain obtained through this method is more than 10 times that under uniform strain. 3.2. Comparisons of longitudinal profiles of deformed specimens Fig. 9(a) shows the longitudinal profiles of a notched specimen (SS400, R0 = 20 mm) during a tensile test. These four profile curves were taken during the initial state A, under maximum load B, after local necking C, and just prior to fracture D. Comparing A and B, the y coordinate of curve B decreases overall, which indicates that it is a deformation within the range of uniform elongation. Comparing B, C, and D, the y coordinate near the center of the notch bottom locally decreases, which indicates that local necking has occurred. In addition, the results of FEM analyses using the corrected stress-strain curve are shown. The numerical results coincide with the experimental results. The entire shape can be reproduced in spite of using only the cross-section radial change (a0-a) at the necked bottom under a stress correction. It is thought that the corrected flow stress is accurate, thereby proving the validity of the correction method. Fig. 9(b) shows the longitudinal shapes of the SS400 specimens, which have various initial notch radii. All curves were taken just prior to fracture. Except for G (R0 = 3 mm), the centers of the D, E, and F curves are nearly the same in term of radius and profile along the longitudinal direction just prior to fracture. This means that, even though the initial radius of the notch bottom was different, the state of stress at the break was about the same. The profile of the numerical calculation also represents the experimental results well. According to the calculation results of the FEM, the stress triaxiality just prior to failure of the D, E, and F specimens was almost the same value, 0.95. The notched specimen of SS400 with large ductility has not succeeded in providing different stress states until fracture. Fig. 9(c) shows the longitudinal shapes of notched specimens (R0 = 20 mm) of the three materials just prior to fracture. The specimen of S45C, which fractured early, showed the smallest change of only 0.25 mm at the bottom. The profile of the numerical calculation also represents the experimental results well.
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Takeshi Nishiwaki et al. / Procedia Manufacturing 15 (2018) 1762–1768 Author name / Procedia Manufacturing 00 (2018) 000–000
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b
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Fig. 9. Longitudinal profiles of specimens during tensile test: profiles of (a) various changes in radius (SS400, R0 = 20 mm), (b) various notch bottom radii (SS400, just prior to fracture) and (c) various materials (R0 = 20 mm, just prior to fracture).
3.3. History of surface strain of deformed specimens The surface strain was also measured using the DIC. Fig. 10 shows the experimental and numerical curves of the surface strain versus a change in radius (a0-a). The numerical longitudinal strain εz and the circumferential strain εθ of SS400 were consistent with the measurement results of the DIC. In the case of S45C, both results showed good agreement. However, in the case of A5056-H34, good agreement was observed until the change in radius reached about 0.3 mm, but a deviation began thereafter, and a deviation of about 7% was eventually observed in the longitudinal strain. To identify this cause, not only the strain at a single point on the surface of the specimen, but also the overall changes in the cross sectional shape were investigated. Fig. 10 shows the histories of circularity of the cross sections in the notch bottom. In the S45C and SS400 specimens, there were few changes from the initial circularity during the tensile test. On the contrary, the circularity of A5056-H34 increased from about a 0.3 mm change in radius, which means that the sectional shape changed from a circle. In other words, a non-axisymmetric deformation was observed, which deviates from an isotropic assumption. Therefore, it is thought that the FEM results are not consistent with the experimental curve. 4. Conclusions In this study, the notched round-bar tensile tests were conducted on a middle carbon steel (S45C), a low carbon steel (SS400), and an aluminum alloy (A5056-H34) using an image analysis with the DIC, and their flow stress curves were determined using a proposed stress correction method. The longitudinal profiles of the necked specimen were measured and compared with the FEM results. The following conclusions were obtained. 1) The flow stress of a low ductile material such as middle carbon steel up to the fracture point can be obtained using a digital image correlation method. 2) In the case of S45C and SS400, the load-radius curve, longitudinal profiles, and surface strain obtained through FEM is consistent with the experimental results. The validity of the stress correction method could be proved.
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3) In spite of the consistency in the load-radius curve and longitudinal profiles, a deviation was observed in the longitudinal strain of A5056-H34. A non-axisymmetric deformation occurred, and the isotropic assumption was not maintained until the fracture point. a
b
Fig. 10. Longitudinal and circumferential strain of specimens during tensile test: (a) SS400, R0 = 20 mm, and (b) A5056-H34, R0 = 20 mm.
Fig. 11. Changes in circularity of cross section at necked bottom.
References [1] G. Mirone, A new model for the elastoplastic characterization and the stress-strain determination on the necking section of a tensile specimen, International Journal of Solids and Structures, 41(2004) 3545–3564. [2] M. Murata, Y. Yoshida, T. Nishiwaki, Stress correction method for flow stress identification by tensile test using notched round bar, Journal of Materials Processing Technology, 251 (2018) 65–72. [3] M. Murata, Y. Yoshida, T. Nishiwaki, Identification of ductile fracture parameter with stress correction method using notched round-bar tensile test, Procedia Engineering, 207 (2017) 2060–2065. [4] S. Coppieters, S. Cooreman, H. Sol, P.V. Houtte, D. Debruyne, Identification of the post-necking hardening behavior of sheet metal by comparison of the internal and external work in the necking zone, Journal of Materials Processing Technology, 211 (2011) 545–552. [5] J.H. Kim, A. Serpantié, F. Barlat, F. Pierron, M.G. Lee, Characterization of the post-necking strain hardening behavior using the virtual fields method, International Journal of Solids and Structures, 50 (2013) 3829–3842. [6] Y. Yoshida, N. Yukawa, T. Ishikawa, Determination of ductile damage parameters by notched round bar tention test using image analysis, NUMIFORM’2004, AIP Conference Proceedings, 712 (2004) 1869–1874. [7] N. Stander, K.J. Craig, On the robustness of a simple domain reduction scheme for simulation based optimization., Engineering Computations, 19 (2002) 431–450.