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УДК 539.424
Forming limit diagram generation from in-plane uniaxial and notch tensile test with local strain measurement through digital
image correlation
S.K. Paul1, S. Roy2, S. Sivaprasad2, and S. Tarafder2
1 Department of Mechanical Engineering, Indian Institute of Technology Patna, Patna, Bihar, 801106, India 2 Fatigue & Fracture Group, CSIR-National Metallurgical Laboratory, Jamshedpur, 831007, India
Requirement of forming limit curve appears indispensible for property check during formable sheet metal development cycle and quality control of formable sheet metal at production shop floor. In the present work, left side of the forming limit curve (uniaxial tensile to plane strain tensile) is experimentally determined from in-plane uniaxial and notch tensile test with local strain measurement through digital image correlation technique. A novel procedure has been developed to generate forming limit curve with considerably reduced experimental effort. The proposed new procedure is based on a combination of in-plane uniaxial notch tensile test with local strain measurement and modeling. The forming limit curve generated using the new procedure has been compared with the standard procedure (Nakajima test) and a good correlation has been obtained.
Keywords: forming limit diagram, digital image correlation, in-plane uniaxial notch tensile test, Nakajima test, necking, local strain
DOI 10.24411/1683-805X-2018-14010
Построение диаграммы предельного формоизменения при испытании на одноосное растяжение в плоскости образцов с надрезом методом корреляции цифровых изображений
S.K. Paul1, S. Roy2, S. Sivaprasad2, and S. Tarafder2
1 Индийский технологический институт Патны, Патна, Бихар, 801106, Индия
2 Национальная металлургическая лаборатория, Джамшедпур, 831007, Индия
Применение диаграмм предельного формоизменения незаменимо для определения характеристик листового металла при подборе параметров процессов штамповки, а также для контроля качества формованного листа на производстве. В настоящей работе экспериментально определена левая часть кривой предельного формоизменения (одноосное растяжение до растяжения в условиях плоской деформации) при испытании на одноосное растяжение в плоскости образцов с надрезом с измерением локальной деформации методом корреляции цифровых изображений. Предложен новый метод построения диаграммы предельного формоизменения, который позволяет значительно сократить экспериментальные затраты. Данный метод сочетает в себе проведение испытаний на одноосное растяжение образцов в плоскости с измерением локальной деформации и моделирование. Проведено сравнение кривых предельного формоизменения, полученных с использованием предлагаемого метода и стандартной методики (тест Nakajima), и показано их хорошее соответствие.
Ключевые слова: диаграмма предельного формоизменения, корреляция цифровых изображений, испытание на одноосное растяжение в плоскости образцов с надрезом, тест Nakajima, образование шейки, локальная деформация
1. Introduction
Forming limit diagram (FLD) is a widely used failure criteria (necking limits) for sheet metals. The forming limit curve (FLC) is generally represented by a series of points within a plot of minor strain on the abscissa and major strain
on the ordinate for linear strain paths ranging from uniaxial tension to equibiaxial tension. The principal strains at each point on the FLC correspond to the limit strains associated with the onset of localized necking under specific linear strain paths. For safe sheet metal forming of an engineer-
© Paul S.K., Roy S., Sivaprasad S., Tarafder S., 2018
ing component, the strain levels should be less than the FLC of that sheet metal with which the component is made. Apart from necking, the stamped component can be discarded because of other various factors such as wrinkling, insufficient stretching, excessive thinning etc. [1—6].
Normally two standardized experimental methods Mar-ciniak [7] and the Nakajima [8] are used to determine FLC. Normally a flat-bottomed cylindrical punch is used to deform the sheet metal specimen for Marciniak test, while hemispherical punch is used for Nakajima test. The different strain paths can be achieved by varying widths of specimens in both the test methods. In industrial sheet metal forming process, various factors like geometry of forming press, nature of deformation, previous deformation history, lubrication, anisotropy of the material etc. could affect the formability of the final formed part [9-14]. Experimentally find out the effect of such factors in FLC is a tedious task and hence prediction of FLC based on theory of plasticity and instability conditions gained popularity [15-18]. Theoretical procedures could assist in quick determination of FLC, while it cannot be replace of experimental FLC, specifically for new grades of materials. Therefore, researchers paid attention in recent years to generation of theoretical FLC with calibration of single/multiple experimental data points [19]. Singh et al. [19] used CrachLab (Mar-ciniak-Kuczynski method) to generate FLC and calibrate it with experimental uniaxial tensile test data. They reduce number of experiment significantly.
Number of literature on analytical and numerical procedures to determine FLC are available. Initially Swift [20] and Hill [21] introduced analytical bifurcation method for plane stress condition to predict FLC. Subsequently Storen and Rice [22], and Hutchinson et al. [23, 24] also applied bifurcation method to find out FLC analytically. Marciniak and Kuczynski [7] determined FLC numerically by introduction of geometrical imperfection (i.e. thickness hetero-
geneity). Latter large number of research paper published on Marciniak-Kuczynski method with inclusion of many complexities like inhomogeneity ratio and angle, through thickness stress etc. [25-29]. Empirical formulas are also available to predict FLC, namely Keeler and Brazier [17], Raghavan et al. [18] and Paul [30]. Many damage based models also present to predict FLC [31-34]. But effectiveness of the empirical formula and model is questionable for new grades of materials. In the present work, a procedure is proposed to generate FLC from in-plane uniaxial tensile and notch tensile test with local strain measurement through digital image correlation. Also calibration procedure is discussed for existing empirical model with one experimental data point effectively.
2. Experimentation
Cold rolled 1.4 mm thick dual phase steel (contains ferrite and martensite phases) was selected for this investigation. All experiments are carried out at laboratory environment in a servo-electric test frame of 35 kN capacity. All tests are continued until fracture of the sample. Commercially available 2D digital image correlation system from LaVision [35] is used for local strain measurement. Speckle pattern foil, supplied by LaVision, is used on one side of the specimen. Normally, 100-200 images per experiment are stored for further processing of data. Figure 1 shows the geometry of in-plane notch tensile sample to simulate plane strain condition. Tardif and Kyriakides [36] also used similar notch geometry to achieve plane strain condition.
3. Results and discussion
Contours of local strain components for in-plane uniaxial tensile test, in the loading direction e ^ and transverse direction e^ are shown in Figs. 2, a and 2, b respectively. For investigation of evolution of local strain com-
Fig. 1. Notch tensile specimen before (a) and after test (b)
4 ' 6 ' 8 10 12 Position, mm
Fig. 2. Tensile test of dual phase steel: contour of local strain in loading direction eyy (a), contour of local strain in transverse direction exx (b), variation of strain in loading direction eyy for vertical middle position of the specimen (c), and variation of strain in transverse direction e^ (d). Overall strain: 1.2 (1), 6.3 (2), 12.3 (3), 22.3 (4), 23.3 % (5)
ponents at various overall tensile strain levels, a vertical line in the middle of the specimen gauge section is selected. Figures 2, c and 2, d illustrates the evolution of local strain components 8 and 8along the selected vertical line.
5 10 15 20 Position, mm
Fig. 3. Notch tensile test of dual phase steel: contour of local strain in loading direction e yy (a), contour of local strain in transverse direction e^ (b), variation of strain in loading direction e yy for horizontal position at the lowest cross section of the specimen ( c), and variation of strain in transverse direction e (d)
Evolution of local strain components within the gauge length of the specimen almost homogeneous up to uniform elongation of the material and deformation is concentrated in a local region after uniform elongation. Similarly Figs. 3, a and 3, b shows the contours of local strain components in the loading direction e and transverse direction e for in-plane uniaxial notch tensile test respectively. A horizontal line in the minimum cross sectional area of the specimen is selected for detailed examination of evolution of local strain components with increasing deformation. Along the selected horizontal line, evolution of local strain components eand eare depicted in Figs. 3, c and 3, d respectively.
Local strain in loading direction eand transverse direction e for in-plane uniaxial tensile test prior to fracture (necking) is plotted in Fig. 4, a. Similarly local strain in loading direction eand transverse direction efor in-plane uniaxial notch tensile test just before the fracture is shown in Fig. 4, b. At the middle of in-plane uniaxial notch tensile specimen proper plane strain condition is achieved (Fig. 4, b). For in-plane uniaxial notch tensile test, fracture is initiated from the middle of the specimen. If proper plane strain condition is not achieved at the middle of the specimen then fracture can be initiated from the notch edge and necking strains cannot be find out. The local strains in loading direction eand transverse direction efor in-plane uniaxial tensile and in-plane uniaxial notch tensile test in the necked zone are collected from Figs. 4, a and 4, b, and plotted in Fig. 4, c. Local strain in loading direction eand transverse direction ecan be designated as major and minor strains respectively. Figure 4, c shows comparison of FLC obtained from Nakajima test and in-plane tensile tests with digital image correlation. Therefore, from in-plane tensile tests with digital image correlation the left side of the FLC (uniaxial tensile to plane strain tensile) can be successfully generated experimentally.
But the complete FLC (uniaxial tensile to equi-biaxial tensile) cannot be generated from in-plane uniaxial tensile experiments with changing notch geometry. In the current work, a novel method is proposed to generate complete FLC with an empirical model calibrated by an experimental data point. Most crucial point on FLC is FLC0 (forming limit at plane strain condition). It is already illustrated that accurate determination FLC0 is possible by in-plane uniaxial notch tensile test with digital image correlation. For a known FLC0, complete generation of FLC is possible from Paul model [30]. According to the Paul model [30], left side of FLC can be determined from Eq. (1):
e! = FLCo -e 2, (1)
where e1 and e 2 are the major and minor strains respectively.
The right side of FLC can be determined from Eq. (2): ei = (1 + FLCo)(1 + e2)p -1, (2)
50
40
30
,20 c
0 -10
-20
Ta
—I-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-r
2 4 6 8 10 Position, mm
12
10 15 20 Position, mm
In-plane tensile test with DIC □ Nakajima test
0.0-0.2 -0.1
0.0 0.1 0.2 Minor strain
0.3 0.4 0.5
Fig. 4. Dual phase steel: local strain in loading direction e (1) and transverse direction e^ (2) for in-plane uniaxial tensile test (a) and notch tensile test (simulating plane strain condition) (b); and comparison of FLC obtained from Nakajima test and in-plane tensile test with digital image correlation (c)
wherep is a material constant, andp can be calculated from Eq. (3):
p = 1.0834 exp (- 1.4114FLC0) - 0.361. (3)
In the current procedure FLC0 is experimentally determined from in-plane uniaxial notch tensile test with digital image correlation and the complete FLC is calculated from
□
□
1 ! O Plane strain tensile test
with DIC
! □ Nakajima test
|-Paul model calibrated with
! one experimental data point
0.0 H—■—i—■—l—■—i—■—i—■—i—■—i—■— -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Minor strain
Fig. 5. Comparison of FLC obtained from Nakajima test and Paul model calibrated with one experimental data point
Eqs. (1)-(3). Figure 5 shows the comparison of FLC obtained from Nakajima test and Paul model calibrated with one experimental data point. The proposed method is able to successfully predict complete FLC which is matched reasonably well with the experimental FLC for dual phase steel.
4. Conclusions
In this article, an experimental procedure is proposed to determine the left side of FLC (uniaxial tensile to plane strain tensile) from in-plane uniaxial tensile and notch tensile test with local strain measurement by digital image correlation. Proposed procedure is validated by experimental FLC data of dual phase steel.
A novel procedure is also developed in the present work to generate complete FLC by an empirical model which is calibrated by an experimental data point. FLC0 (forming limit at plane strain condition), the most critical point on FLC, is determined experimentally. FLC0 is used to calibrate the empirical model. FLC constructed by new procedure shows reasonable well matching with experimental FLC of dual phase steel.
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Поступила в редакцию 13.06.2018 г.
Сведения об авторах
Surajit Kumar Paul, PhD, Assist. Prof., Indian Institute of Technology Patna, India, [email protected], [email protected] Satish Roy, B.Tech, Res. Assist., CSIR-National Metallurgical Laboratory, India, [email protected] S. Sivaprasad, PhD, Senior Principal Sci., CSIR-National Metallurgical Laboratory, India, [email protected] S. Tarafder, PhD, Chief Sci., Head of Division, CSIR-National Metallurgical Laboratory, India, [email protected]
