Uniaxial Near Plane Strain Tensile Tests Applied to the Determination of the FLC 0 Formabillity Parameter

Department of Metallurgical and Materials Engineering, Escola Politécnica – Poli, Universidade de São Paulo – USP, Av. Prof. Mello Morais, 2463, CEP 05508-900, São Paulo, SP, Brazil Escola SENAI “Nadir Dias de Figueiredo”, Av. Ari Barroso, 305, CEP 06216-901, Osasco, SP, Brazil Centro Estadual de Educação Tecnológica Paula Souza, Faculdade de Tecnologia de São Paulo – FATEC-SP, Av. Tiradentes, 615, CEP 01124-060, São Paulo, SP, Brazil


Introduction
The last decade saw an increasing interest in the understanding of the physical metallurgy associated with the evolution (during straining) of both microstructure (damage accumulation/microvoids evolution) and crystallographic texture, aiming at locating the yield point and, consequently, ductile fracture during sheet metal drawing 1,2 .This interest relies on the support of true stress -true strain curves, on using alternative flow criteria (quadratic and nonquadratic) and on using Forming Limit Curves FLCs, i.e., on characteristics of the material to be drawn.
These Forming Limit Curves (FLCs) introduced by Lankford (1947), Keeler and Backofen and Goodwin [3][4][5] , allow a comprehensive representation of sheet formability and have been widely used as a criterion in the optimization of the drawing process and as an aid in die designing 2,[5][6][7] .
The Nakazima simulation test (1968) has been commonly applied for the evaluation of the FLCs.This test is sensitive to the sheet thickness, surface conditions, lubricants, tool type and geometry 2,5 , besides influences inherent to the test itself, since the sample does not remain flat, but is increasingly curved during straining (i.e. the strain path is not entirely contained in the sheet plane) 8 .Further, it should be added that obtaining the FLC curves via Nakazima is time consuming and expensive, as it requires the preparation and testing of several samples of different geometries and dimensions.The minimum recommended number of samples from the industrial practice is 30 samples.A typical setup consists of three replicas of each The methodology used in determining the FLC curves is based on the analysis of the deformation of sheet samples, which contains a circle grid printed over its surface.The samples are deformed in different conditions, in order to simulate different strain paths to which an actual part would be submitted during forming.The results of all sorts of tests designed for such purpose, being either intrinsic or simulated (regardless of friction) , consists in measuring the ellipses (i.e. the deformed circles of the printed circle grid) near the fracture region, calculating the largest principal strain (ε 1 ) and smallest principal strain (ε 2 ) in the sheet plane 5,[9][10][11] .A plot of these points generates V-type curves, which allow defining the boundary of conformational limits of that sheet (ASTM E2218, ISO 12004-2:2008) 12,13 .This point is where local thinning starts (reduction of resisting section) and that, at the end, culminates with fracture (generating the fracture limit curves, FrLC).
The apparent transferability of the concept of the FLC is tempting, but it is known that the strain path (which is not always a straight line) in formed parts influences the position of the FLC 9,10,14,15 .This path can be described by the strain ratio β = ε 2 /ε 1 .A path corresponding to biaxial tension (stretching) occurs for β ~ 1.A path close to plane strain is associated with ε 2 ~0 (equivalent to β=0).A path corresponding to deep drawing situations find values in the region -1<β<-0.5 [7] .Studies conducted in several types of automobile parts 5,15,16 show that over 80% of formed pieces usually fail under conditions of near plane strain (β~0), which is also the minimum of the obtained FLCs.
This propensity for failure under near plane strain conditions and the previously mentioned disadvantages of the Nakazima test 17 brought out the intrinsic tensile test under the condition of plastic deformation near to the plane strain 5,15,16 condition.In this case, the full determination of the FLC is avoided and all analysis is based on the FLC 0 point, which corresponds to the minimum (lowest point) of the FLC curve under plane strain, i.e., for the condition for which the smallest principal strain in the plane of the sheet vanishes: ε 2 → 0.
The objective of this work is to evaluate the possibility of replacing Nakazima tests by a fast and safe determination of the FLC 0 value through tensile tests that will lead to near plane strain deformation, using a smaller number of samples.

Material
Two kinds of blanks were used in the present work: a 0.75 mm thick Interstitial Free (IF) steel sheet and a 1.48 mm thick spheroidized SAE 1050 carbon steel sheet.Chemical compositions, as furnished by the suppliers, and mechanical properties (according to ABNT NBR 16284; ASTM E 517) 18,19 are given, respectively, in Tables 1 and 2. The first steel is ductile and widely used in drawing industries, especially by automobile manufacturers and by home appliance industries, while the latter has higher mechanical strength, which usually impairs formability.The microstructure (ferrite matrix containing spheroidal cementite) somewhat decreases this drawback and the steel is mainly used in applications such as toecaps for safety boots.
Steel formability may be evaluated by a series of mechanical properties, derived from a conventional tensile test, these are: yield stress (σ y ), ultimate tensile stress (UTS), elongation for a gauge length of 80 mm (ε f ), plastic anisotropy ratio (r α , where α refers to the angle between rolling direction and tensile sample loading direction) and the parameters of Hollomon's equation, defined by The values of these parameters for both steels, provided by the supplier, are given in Table 2.

Samples and testing methods
The geometries of the samples used specifically for plane strain (tensile) tests are shown in Figure 1.The dimensions were based on Wagoner's previous studies 20 .
The technique called serigraphy was used for recording a set of circles on the metallic samples.It is a simple process which provides good sharpness for measurements of the circles.It relies on the transfer of drawings onto serigraphy chromes and from them onto the metal sheet surfaces, via ink tanks.
The sequential steps of serigraphy refer to the production of the chrome; choice of the serigraphy tissue; preparation of frames (degreasing, drying, emulsification, drying, additional emulsification, drying, exposure to ultraviolet light; development; drying) and applying ink onto the surface.
The methodology used for the tensile tests under condition of plastic near plane strain, was the following: • Pre-recording of a net of circles of d 0 = 2 mm on the sample surface; • Loading of the samples, leading them to necking/  rupture through uniaxial tension.Three samples were used for each condition, with the major axis parallel either to the rolling direction (RD) of the original sheet or to the transverse (TD) direction; • Measurement of the ellipses along the longitudinal axis of the plastically deformed samples, adjacent to the necking region, for the determination of the major and minor axes, d 1 and d 2 , respectively, using an image analysis system (CAMSYS).Six ellipses were measured, three to the left and three to the right of the center line in the necked region.This system allows the automated reading of major and minor axes (d 1 and d 2 ) in the ellipses close to the thinning area.From these values, the true principal strains are calculated through: The assessment of FLC 0 values resulting from the unidirectional near plane strain plastic deformation is obtained by comparing results of the same parameter arising from a full determination of the respective FLCs using the Nakazima test, performed in the present work.Results are reported as true (i.e., not engineering) strains, according to Equation 2.
The equipment used for determining the FLC 0 values in near plane strain (tensile test) are briefly described below: • Projector of vertical profile: Objective lenses 10, 20, 50 and 100x and digital reader with geometric processor; • Universal Testing Machine with maximum capacity 600 kN electromechanical drive and speed ranging from 0.01 to 300 mm/min.The tests were conducted under displacement control; • Wire electro-erosion machine: to obtain low roughness in the cut face, hence preventing crack nucleation at this site during tensile testing.The Nakazima simulation tests 21 were carried out in a Erichsen press, with a 100mm-diameter punch.Sample sizes for the IF steel were 220 × 50, 220 × 80, 220 × 100, 220 × 110, 220 × 120, 220 × 130, 220 × 140, 220 × 160, 220 × 175 and 220 × 220 mm and 0.75 mm thickness.In the case of the AISI 1050, the same sample sizes were tested, but the samples with width smaller than 140 mm invariably broke in the blank holder, therefore results for this steel will be limited to 220 × 140, 220 × 160, 220 × 175 and 220 × 220 mm samples, with thickness 1.48 mm.Both steels were investigated in the RD configuration only.

Results and Discussion
The critical strains for necking of the samples subject to Nakazima's test are presented in Table 3.Each value represents the average of a large number of circles (also given in the table) and the standard deviation of the measurements is represented in parenthesis (referring to the value's last digit).As expected, the IF steel presents superior drawability when compared to the AISI 1050 steel.Based on these results, the coordinates of FLC 0 can be derived following the ASTM E2218 standard 12 , corresponding, respectively, to (0.52, 0.00) and (0.42, 0.00) for the IF and the 1050 steels.Although the result is compatible with the lower formability of the 1050 steel, analysis of the base properties of both steels (Table 2) would imply a worse behavior, suggesting that the FLC 0 parameter, derived from Nakazima's test, is overestimated (at least for the 1050 steel).
The results of the near plane strain tests are shown in Figure 2, together with the ones obtained from the Nakazima's tests for both steels.As observed, the values of the near plane strain test are smaller than those expected from the traditional FLC curve.As discussed before, this outcome is expected, since Nakazima's test is affected by friction and geometric factors related with the interaction between punch and the specimen.
The issue of the influence of geometrical and friction factors in the determination of Forming Limit Curves has been addressed already by several authors 17,22 .Charpentier, for example, showed, that samples deformed under curvilinear deformation paths (off the stretching plane) presented higher limit strains as compared with true biaxial tests under the same conditions 22 .

Figure 1 .
Figure 1.Sample geometry, used in the uniaxial tensile tests.

Figure 2 .
Figure 2. Results from the Near Plane Strain tests (nps), solid symbols, compared with the traditional Nakazima's test (N) results (empty symbols).The estimated FLC 0 value for Nakazima's test is indicated for both steels.The lines are drawn just as a guide to the eye.

Table 1 .
Analyzed composition of the investigated steels.

Table 2 .
Base mechanical properties of the investigated steels.

Table 3 .
Results from Nakazima's test.Limit strains are reported as true plastic strains (Equation2) and were defined by measuring the ellipses adjacent to localized necking.