Forming Limit Diagram Generation from In-Plane Uniaxial and Notch Tensile Test with Local Strain Measurement through Digital Image Correlation

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.


INTRODUCTION
.orming limit diagram (.LD) is a widely used failure criteria (necking limits) for sheet metals. The forming limit curve (.LC) 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 .LC correspond to the limit strains associated with the onset of localized necking under specific linear strain paths. .or safe sheet metal forming of an engineering component, the strain levels should be less than the .LC 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. [16].
Normally two standardized experimental methods Marciniak [7] and the Nakajima [8] are used to determine .LC. Normally a flat-bottomed cylindrical punch is used to deform the sheet metal specimen for Marci-niak 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 [914]. Experimentally find out the effect of such factors in .LC is a tedious task and hence prediction of .LC based on theory of plasticity and instability conditions gained popularity [15 18]. Theoretical procedures could assist in quick determination of .LC, while it cannot be replace of experimental .LC, specifically for new grades of materials. Therefore, researchers paid attention in recent years to generation of theoretical .LC with calibration of single/multiple experimental data points [19]. Singh et al. [19] used CrachLab (MarciniakKuczynski method) to generate .LC 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 .LC are available. Initially Swift [20] and Hill [21] introduced analytical bifurcation method for plane stress condition to predict .LC. Subsequently Storen and Rice [22], and Hutchinson et al. [23,24] also applied bifurcation method to find out .LC analytically. Marciniak and Kuczynski [7] determined .LC numerically by introduction of geometrical imperfection (i.e. thickness heterogeneity). Latter large number of research paper published on Marci-niakKuczynski method with inclusion of many complexities like inhomogeneity ratio and angle, through thickness stress etc. [2529]. Empirical formulas are also available to predict .LC, namely Keeler and Brazier [17], Raghavan et al. [18] and Paul [30]. Many damage based models also present to predict .LC [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 .LC 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.

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, 100200 images per experiment are stored for further processing of data. .igure 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.

RESULTS AND DISCUSSION
Contours of local strain components for in-plane uniaxial tensile test, in the loading direction yy ε and transverse direction xx ε are shown in .igs. 2a and 2b respectively. .or investigation of evolution of local strain components at various overall tensile strain lev-.ig. 1. Notch tensile specimen before (a) and after test (b) (color online). (d) most homogeneous up to uniform elongation of the material and deformation is concentrated in a local region after uniform elongation. Similarly .igs. 3a and 3b shows the contours of local strain components in the loading direction yy ε and transverse direction xx ε 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 yy ε and xx ε are depicted in .igs. 3c and 3d respectively.
.ig. 4. Dual phase steel: local strain in loading direction yy ε (1) and transverse direction xx ε (2) for in-plane uniaxial tensile test (a) and notch tensile test (simulating plane strain condition) (b); and comparison of .LC obtained from Nakajima test and in-plane tensile test with digital image correlation (c).
Local strain in loading direction yy ε and transverse direction xx ε for in-plane uniaxial tensile test prior to fracture (necking) is plotted in .ig. 4a. Similarly local strain in loading direction yy ε and transverse direction xx ε for in-plane uniaxial notch tensile test just before the fracture is shown in .ig. 4b. At the middle of in-plane uniaxial notch tensile specimen proper plane strain condition is achieved (.ig. 4b). .or inplane 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 yy ε and transverse direction xx ε for in-plane uniaxial tensile and in-plane uniaxial notch tensile test in the necked zone are collected from .igs. 4a and 4b, and plotted in .ig. 4c. Local strain in loading direction yy ε and transverse direction xx ε can be designated as major and minor strains respectively. .igure 4c shows comparison of .LC 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 .LC (uniaxial tensile to plane strain tensile) can be successfully generated experimentally.
But the complete .LC (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 .LC with an empirical model calibrated by an experimental data point. Most crucial point on .LC is .LC 0 (forming limit at plane strain condition). It is already illustrated that accurate determination .LC 0 is possible by in-plane uniaxial notch tensile test with digital image correlation. .or a known .LC 0 , complete generation of .LC is possible from Paul model [30]. According to the Paul model [30],

CONCLUSIONS
In this article, an experimental procedure is proposed to determine the left side of .LC (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 .LC data of dual phase steel.
A novel procedure is also developed in the present work to generate complete .LC by an empirical model which is calibrated by an experimental data point. .LC 0 (forming limit at plane strain condition), the most critical point on .LC, is determined experimentally. .LC 0 is used to calibrate the empirical model. .LC constructed by new procedure shows reasonable well matching with experimental .LC of dual phase steel.