Optimization of initial blank shape to minimize earing in deep drawing using finite element method

Transcription

1 Optimization of initial blank shape to minimize earing in deep drawing using finite element method Naval Kishor, D. Ravi Kumar* Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi , India Abstract Deep drawing of highly anisotropic sheets causes earing in the formed cup (formation of wavy edge at the top). In the present work, an attempt has been made to study this earing problem in deep drawing of cylindrical cups using a flat bottom punch from extra-deep drawing steel sheets using a finite element method based software LSDYNA. The mechanical properties and the formability parameters of the sheets were determined from the simple tensile tests. These properties along with experimental design parameters were used as input for simulation of deep drawing. Significant earing was observed and it compared with experimentally determined earing (by measuring cup height at several points) with reasonably good accuracy. To optimize the initial blank shape to minimize earing, the flow of material was observed at various steps during the process and accordingly blank shape was modified. Simulation of blanks with modified initial shape showed significant reduction in earing. Experiments were carried out on a 401 hydraulic press using the modified blank shapes and it has been found that results agreed well with the analysis results. Keywords: Deep drawing; Earing; Anisotropy; Limiting draw ratio; Finite element method 1. Introduction The importance of the sheet metal working in modern technology is due to the ease with which metal may be formed into useful shapes by plastic deformation processes in which the volume and mass of metal are conserved and metal is displaced from one location to another. In addition to this, the ability to produce a variety of shapes from flat sheets of metal at high rate of production is one of the outstanding qualities of the sheet metal working processes. Deep drawing is one of the widely used sheet metal working process in the industries to produce cup shaped components at a very high rate. Cup drawing besides its importance as forming process also serves as a basic test for the sheet metal formability. In deep drawing a metal sheet is used to form cylindrical components by a process in which the central portion of the sheet is pressed into die opening to draw the metal into desired shape without folding of the corners [1]. In deep drawing the metal at the center of the blank is subjected to biaxial tensile stress due to the action of the punch and is thinned down. Metal in the outer portion of the blank is subjected to a compressive strain in the circumferential direction and a tensile strain in the radial direction. As a result of these two principal strains there is a continual increase in the thickness as the metal moves inward [2]. The efficiency of the deep drawing process depends upon many parameters and the choice of these parameters is very important to achieve the high drawability. The deformation of a sheet metal in deep drawing can be quantitatively estimated by draw ratio (DR), which is defined as ratio of initial blank diameter to the diameter of the cup drawn from the blank (approximately equal to the punch diameter). For a given material there is a limiting draw ratio (LDR), representing the largest blank that can be drawn through a die without tearing. The drawability of a metal depends on two factors [1]: 1. The ability of the material in the flange region to flow easily in the plane of the sheet under shear. 2. The ability of side wall material to resist deformation in the thickness direction. The mechanical properties which are considered to be important in sheet formability are average plastic strain ratio (r) and the strain hardening exponent (n). The strength of the final part as measured by yield strength must also be considered. Taking both of the above mentioned factors into account it is desirable in the drawing operation to maximize material flow in the plane of sheet and to

2 N. Kishor, D. Ravi Kumar/ Journal of Materials Processing Technology (2002) maximize resistance to material flow in a direction perpendicular to the plane of sheet [ 1 ]. The flow strength of the sheet metal in the thickness direction is difficult to measure, but the plastic strain ratio (r) compares strengths in the plane and thickness directions by determining true strains in these directions in a tension test. For a given metal strained in a particular direction, r is expressed as fit where e w is the true strain in the width direction and e t the true strain in the thickness direction [1]. Generally sheet metal is anisotropic, that is, the properties of the sheet are different in different directions. The average plastic strain ratio (?) gives the ratio of average flow strength in the plane of the sheet to average flow strength normal to the plane of the sheet. Higher r means higher resistance to thinning, i.e. deeper cup can be drawn [3-5]. Hence, a high value of r denotes that the material has a very good drawability, i.e. high LDR, which is a desirable property for deep drawing operation but high anisotropy also leads to earing in the drawn cup [2,4]. Earing is defined as the formation of waviness on the top of the drawn cup. Ears are formed due to uneven metal flow in different directions and is directly related with planar anisotropy (Ar). The variations of flow strength in the plane of the sheet are termed as planar anisotropy. In case of isotropic material where planar anisotropy (Ar) is zero, no ears will form. But sheets are manufactured by cold rolling and annealing and due to this preferred orientation is developed in the sheet, which results in anisotropy in the sheet metal. If Ar is positive, ears form at 0 and 90 to rolling direction, and ears form at +45 to rolling direction if Ar is negative. Ideally a sheet with very high r and zero Ar is good for deep drawing to draw deeper cups in single draw but it is almost impossible to manufacture a sheet with very high normal anisotropy and with zero planar anisotropy [2,4]. Mostly sheets, which possess high normal anisotropy, possess high planar anisotropy too, leading to significant earing. Earing is undesirable since it requires an additional processing step where uneven top edge of the cup is trimmed. This results in loss of material, production rate and increase in cost. El-Sebaie and Mellor [6] conducted various experiments with sheets having w-values varying between 0.2 and 0.5 and found out that the w-value has little effect on LDR while the r value plays a dominant role and that for materials having r value less than unity, the LDR should be approximately constant for all values of n or variation is very small. It was also observed that in many cases higher LDR would be achieved if the instability site could be transferred from the cup wall to the flange. Several theoretical analyses have been proposed for predicting the LDR [3,5]. There has been considerable effort in recent years towards the development and use of FEM (finite element method) models to solve the deep drawing problems in a better way. Kobayashi [7] successfully solved the deep drawing problem with hemispherical punch by rigid plastic FEM techniques and found that results were in excellent agreement with the experimental results. Several investigations have been conducted in the area of earing prediction and effect of material properties such as r, Ar and n on the ear height. In the last few years, researchers [3,8-15] used different techniques to optimize the initial blank shape to minimize the earing. Initial work was done by Chung and Richmond [9] to optimize the initial blank shape to minimize the earing. They proposed the design of ideal forming processes based on the ideal forming theories. Chung et al. [10] developed a sequential design procedure to optimize sheet-forming processes based on the ideal design theory, FEM analysis and experimental trials. They used this procedure to design a blank shape for a highly anisotropic aluminum alloy sheet that resulted in a deep draw circular cup with minimum earing. In the present work an attempt has been made to develop a procedure for optimization of initial blank in such a way that the proposed procedure can be implemented easily. 2. Methodology As mentioned earlier the occurrence of earing is pronounced in the drawn cup from a sheet of high planar anisotropy (Ar). The methodology used in the present work for the prediction of LDR, earing, and optimization of initial blank shape to minimize the earing is presented in the following sections FEM simulation About FEM software LSDYNA An FEM based software LSDYNA is used in the present work to simulate the deep drawing process. LSDYNA is a nonlinear dynamic simulation software which can simulate different types of sheet metal forming processes like deep drawing, stretch forming, bending, etc., to predict the stresses, strain, thickness distribution, punch load and effect of various design parameters of tooling on process efficiency and final product. The most important and crucial part of simulation in software is the selection of appropriate material model. LSDYNA contains various material models (for viscous plastic, elastic plastic and rigid plastic materials), and each model has different suitability, so selection of correct material model as per the requirement is the prime necessity to get the accurate output or simulated results. Most of the material models require detailed material properties such as mass density, young's modulus of elasticity, strain hardening exponent, r 0, r 45, and r 90, K, etc., as input to preprocessor before running the solver. In addition to material properties, preprocessor also requires input of detailed process parameters such as static and dynamic friction coefficient, punch velocity, blank holding force, sheet

3 22 N. Kishor, D. Ravi Kumar/Journal of Materials Processing Technology (2002) thickness, shell thickness, damping coefficient, etc. In the present work, B arlat's anisotropic yield criterion [ 16] model has been used Determination of LDR As mentioned earlier that LDR represents the largest blank that can be drawn through a die without tearing. LDR was predicted by the software using the blanks of different diameter while keeping the tooling and other input parameters constant. The process parameters used in simulation are as follows: Static friction coefficient Dynamic friction coefficient Type of elements used Number of elements used in blank Punch velocity used Blank holding force used Shell thickness Quadrilateral and triangular shell elements mm/s 20 kn 1 mm A meshed model was constructed in the preprocessor of the software and the same model was used with blanks of different diameter. It has been found that when the blank size exceeds a critical diameter, the complete flow of material from the flange does not occur due to buckling and stretching takes place in the sheet below the punch. This represents the failure in actual drawing operation. The tensile and anisotropic properties of the sheet required for simulation have been actually determined by experimentation (explained in Sections and 2.2.2) and these properties along with other process parameters used in deep drawing experiments have been used as input data for simulation Prediction of earing Simulations using the software were done for several blanks of different diameter by constructing the meshed model in the preprocessor and using the appropriate input parameters. The height of the simulated cup was found out by the following procedure: 1. Circular blank was divided into 24 uniform parts by drawing lines at 15 interval from center to the circumference of the blank (i.e. angle between two adjacent points is 15 ). 2. The node number on the circumference of the blank was found corresponding to the points where the above lines cut the circumference of the blank. 3. The displacement in Z direction was found out from the post processor of LSDYNA for a particular node number. 4. The displacement of the flat portion of the cup from the initial blank position was found out. 5. The difference between the displacement from step 3 and 4 gives the cup height at a particular node. Same procedure was repeated for all the nodes Optimization of initial blank shape to minimize the earing Several simulations were carried out to observe the earing behavior with respect to planar anisotropy (Ar) and blank diameter. Correlations were established between the coordinates of points, where 0 and 90 lines to rolling direction cut the circumference of circular blank and corresponding coordinates of the points on the modified blank. A procedure for optimization of blank shape was established, which consists of two stages: the first stage is the modification of initial circular blank by using Eqs. (10) and (11) (for details see Section 3.6) and the second stage is the further modification of the modified blank from the first stage (if required). Several optimized blanks, obtained from the first stage of modification, were simulated using LSDYNA to check the validity of the correlated equations. The effective diameter and effective DR were calculated for the corresponding modified blanks. Method of determination of effective diameter and effective DR is explained in detail in Section 3.6. Though prediction of earing and optimization of initial blank shape was done for several blanks (corresponding to different DRs) using the above procedure, results corresponding to only one DR (i.e. 2.17) are being reported in this paper Experimental procedures Determination of tensile properties of extra deep drawing (EDD) steel sheet As mentioned in the previous section the accurate input of material properties is very important for simulation using the software. In view of this, uniaxial tensile tests were conducted on a 101 Universal Tensile Machine (UTM). Six specimens, two each cut from 0, 90 and 45 to the rolling direction were tested. Common tensile properties like yield stress, ultimate tensile stress and percentage elongation have been obtained from the load-elongation data. Yield stress has been calculated by taking the load at 0.2% strain by offset method. The loadelongation data obtained from the uniaxial tensile tests were used to calculate the strain hardening exponent (n) and strength coefficient K by assuming that the true stress (cr)-true strain (e) curve for the sheet metal follows the power law of hardening in the uniform plastic deformation region [2]: a = Ks n (1) where K is the strength coefficient and n the strain hardening exponent. The slope of the log a vs log e plot gives the value of strain hardening exponent and intercept on y-axis gives K.

4 N. Kishor, D. Ravi Kumar/ Journal of Materials Processing Technology (2002) Determination of normal anisotropy (r) and planar anisotropy (Ar) As mentioned earlier, plastic strain ratio (r) is the ratio of true strain in width direction to true strain in thickness direction. To measure the r-value following ASTME8M method, six tensile specimens were made, two each in 0, 45 and 90 to rolling direction. These specimens were elongated in uniaxial tension up to a predetermined strain (18% elongation in the present case). The variations of width and length at five different points, before and after the elongation, were measured in the gauge portion of the specimens and average values have been used for calculation of r as given below: t -I In (w o /w) ln(wl/w o l o ) where e w, e t and i are the true strain in width, thickness and length directions, respectively, and WQ and l 0 are the initial width and length, while w and I are the final width and length (after elongation). Normal anisotropy or average plastic strain ratio was calculated by [4]: (2) 2r 45 ) (3) where r 0, r 90 and r 45 denote the plastic strain ratio in 0, 90 and 45 to rolling direction, respectively, determined using Eq. (2). The planar anisotropy (Ar) was calculated using [5]: Ar = \(r 0 + r 90-2r 45 ) (4) Deep drawing experiments The LDR obtained from the simulation using LSDYNA was verified by conducting deep drawing experiments on a 1001 hydraulic press. The setup used in the present work is similar to any commonly used setup for deep drawing. Blanks of different diameter (80-88 mm in steps of 0.5 mm) were deep drawn and the maximum blank diameter that could be successfully drawn was found out. The limit was reached when cup fractured at the point little above the punch profile radius. The LDR obtained from the experiments was compared with the LDR obtained from the simulations using LSDYNA. The dimensions of the tools used in experiments such as punch, blank holder, die, etc., are given in Table 1. Cylindrical punch with flat bottom was used in the present work. A blank holding force in the range of kn was used in the operation to prevent the wrinkling. Ear height formed on the simulated cups from LSDYNA is verified by conducting the experiments with the blanks of same diameter as used in the simulation. The experimentally drawn cup height was measured using a height gauge vernier caliper (with a least count of 0.02 mm). The height was measured at 24 points (the angle between two adjacent points is 15 ). Experiments were also conducted on the modified initial blanks obtained from the correlated equations (first stage of optimization). The cup height was measured at various points and % ear height was compared with that of the cups obtained by drawing before modification with the same effective DR. 3. Results and discussions 3.1. Chemical composition The material used in the present work is low carbon steel sheet of EDD grade of 1 mm thickness. The variation of thickness in the sheet is less than 0.05 mm. This material has been chosen because of its extensive usage in automobile and other industries for deep drawing applications. The chemical composition of sheets has been analyzed spectroscopically. The composition by weight in percentage is given in Table Tensile properties The tensile properties namely yield stress (YS), ultimate tensile stress (UTS) and % elongation obtained from the uniaxial tensile tests are given in Table 3. It is clear from Table 3 that % elongation is high in EDD steel sheet (45%) compared to most commonly used sheet materials such as low carbon steel, aluminum, brass, etc., indicating the high ductility. This high value of elongation is due to controlled cold rolling and annealing and very low % of carbon in the material as shown in Table Strain hardening exponent (n) and anisotropy parameter (?) As mentioned earlier normal anisotropy (r) and strain hardening exponent («) are very important for formability of Table 2 Chemical composition of EDD steel sheet C Si Mn S P Cr Al Fe Remaining Table 1 Tool dimensions for deep drawing experiments Die radius (mm) 20.7 Die entry/profile radius (mm) 6 Punch radius (mm) 19.5 Punch profile radius (mm) 4 Sheet thickness (mm) 1 (±0.05) Table 3 Tensile properties Material EDD steel sheet of EDD steel sheet Yield stress (N/mm 2 ) 155 Ultimate tensile stress (N/mm 2 ) 250 Percentage elongation (using 50 mm gauge length) 45

5 N. Kishor, D. Ravi Kumar/Journal of Materials Processing Technology (2002) Table 4 Strain hardening Material EDD steel sheet coefficient and anisotropy of EDD steel sheet n K (MPa) r ? r 1.43 Ar a sheet. The values of n, r, and Ar determined from uniaxial tensile tests (as per the procedure explained earlier) are given in Table 4. It is clear from Table 4 that normal anisotropy (r) is greater than 1 which is desirable for high drawability in deep drawing and cups of high depth can be drawn in single draw with sheets of high r values. It can be expected that the material will have high LDR due to its high r value. Also the material has very high planar anisotropy (Ar) which indicates the possibility of high ear formation in deep drawing [2]. The material possesses good stretchability also as indicated by reasonably high n value. A high n value is useful for uniform distribution of strain during stretch forming [4] Limiting draw ratio (LDR) LDR of this material, determined from the deep drawing experiments, is given in Table 5. LDR can also be analytically calculated as follows. Initially, Whiteley [3] proposed the following equation to predict LDR: LDR, (5) where/is the factor which accounts for influence of process conditions on drawing efficiency. It can be observed from this equation that LDR increases with increasing r but it ignores the effect of strain hardening exponent («). Later Leu [5] proposed the following method to calculate the LDR by incorporating the effect of both n and r. LDR, = (6) The values of LDR predicted by the above analytical methods and FEM simulation method were compared with experimental value and the % difference is given in Table 5. The theoretical LDR from both the methods (with Eqs. (5) and (6)) were calculated on the assumption that the drawing efficiency (/) is 0.9 [5]. The experimental value of LDR for EDD steel sheet is 2.22 which is significantly higher than the value of LDR for most commonly used low carbon steel sheet (generally LDR for most of the low carbon steel is around 2). The high value of LDR in EDD steel sheet is due to its high normal anisotropy (r). Since high normal anisotropy indicates high resistance to thinning in the cup wall during drawing, it leads to higher LDR. As mentioned, Eq. (5) proposed by Whiteley [3] takes into account the effect of normal anisotropy r on LDR but ignores the effect of strain hardening exponent (n) on LDR. Eq. (6) proposed by Leu [5] incorporates the combined effect of both normal anisotropy (r) and strain hardening exponent (n) on LDR. The value of LDR obtained from Eq. (6) is much closer to experimental value of LDR, which shows that LDR also depends significantly on the w-value. The explanation for this could be based on two different modes of mechanism which occur during deep drawing of annealed and cold worked (work hardened) materials. In the first mode, the fracture occurs in the cup wall under plane strain tension and is most likely to apply to annealed materials. The second mode is in the flange under uniaxial tension and this is most likely to apply to materials that have been previously cold worked. Since this material is very ductile, thinning due to plane strain stretching in the cup wall cannot be ignored. Hence n also plays an important role in determining LDR of the material. The LDR obtained from the FEM simulation, is 2.23, which is very close to experimental value Though a large number of comparisons have to be made to find out the exact level of accuracy of the software predictions, it appears that it can predict LDR with accuracy better than the analytical methods as obtained in this case Ear height measurement The material used in the present work has a very high value of Ar (planar anisotropy), which indicates that the earing problem in the deep drawn cups will be significant with this material. In view of this, earing behavior of this material has been studied by simulation on software and also experimentally. Simulated cup drawn for a blank of diameter mm is shown in Fig. 1. Fig. 2 shows the photograph of drawn cups experimentally for different DRs (including cup for a DR of 2.17). The cup height measured on experimentally drawn cup was compared with the simulated cup height as shown in Fig. 3 for a blank of DR From Fig. 2 it can be observed that four ears have formed, two each at 0 and 90 to rolling direction as expected from the positive Ar value of this material. This has been correctly predicted by the FEM also (shown in Fig. 1). However, the cup height Table 5 Comparison of LDR from different methods Theoretical LDR from Whitely formula (LDR0 Theoretical LDR from Leu formula (LDR 2 ) LDR from experiment (LDR 3 ) LDR from FEM simulation (LDR 4 ) % Difference between LDR 3 and LDR t % Difference between LDR 3 and LDR 2 % Difference between LDR 3 and LDR

6 N. Kishor, D. Ravi Kumar/ Journal of Materials Processing Technology (2002) STEP 21 TIME = 2.5OTD1 S5E-0O2 could be due to small eccentricity in the blank due to placing of the blank holder on the sheet before drawing Optimization of the initial blank shape to minimize earing Fig. 1. Simulated cup drawn from the blank of diameter mm (equivalent to DR 2.17) showing earing. predicted by the simulation at four points, where ears have formed, is lower when compared to the actual measured cup height. It is also observed from Fig. 3 that the ear height is non-uniform in case of experimentally drawn cups. This The procedure adopted for optimization of initial blank shape to minimize earing is as follows: 1 A model was constructed in the preprocessor of the LSDYNA using the same tooling dimensions as given in Table 1. Fig. 4 shows a typical arrangement of tools for simulation of deep drawing. The final cup height of drawn cup was calculated using Eq. (7), from which stroke length of the punch was determined. The final cup height was calculated by using / = d 1 + Adh (7) where D is the blank diameter, d the punch diameter and h the cup height. 2. Circular blank of mm diameter was simulated and the directions of ear formation were observed. In the Fig. 2. Experimentally drawn cups from blanks of different diameter showing earing. - cup height from experiment - cup height from simulation Angle in degree from rolling direction Fig. 3. Comparison of cup height for a DR of 2.17 (84.52 mm blank diameter).

7 26 N. Kishor, D. Ravi Kumar/Journal of Materials Processing Technology (2002) STEP 2 TIME =1.3187BQ3E-DD3 Fig. 4. Typical meshed model used in deep drawing, constructed in the preprocessor of LSDYNA. present work, ears were found at 0 and 90 to rolling direction due to positive Ar value. In general for BCC materials, blank can be modified either by removing the material from the 0 and 90 to rolling direction or by adding the material in 45 to rolling direction. It has been found out that the first method of optimization is better when compared to the second method. 3. The tsr\ and Ar 2 values were found out as follows: An = r 0 - r 45 (8) Ar 2 = r 90 - r For modifying the blank a circle of required diameter was drawn on a graph paper and was divided into eight equal parts by drawing eight lines from center at an interval of 45. The coordinates of the points where these lines meet the circumference were found out from the drawing shown in Fig. 5. These points are numbered as 1-8 in Fig To reduce the material in 0 and 90 to the rolling direction, coordinates of four new points were obtained (two each on x-axis and y-axis). These points were numbered as 1', 3', 5' and 7'. The x- and y- coordinates of these four points can be obtained as follows: Modified X-coordinate = R 3Arj Modified 7-coordinate = R 3Ari (9) (10) (11) where R is the radius of circular blank, Arj and Ar 2 are calculated from Eqs. (8) and (9). The above coordinates were obtained after observing the material flow in a large number of simulations. 6. Now, four arcs are drawn with four new centers Ci, C 2, C3 and C4 (as shown in Fig. 5). Each arc connects three points, two of which are intersection points of lines drawn at 45 to rolling direction and the third point is the newly identified point on x- or y-axis as the case may be. These four arcs are 2 1' 8, 8 7' 6, 6 5' 4 and 4 3' 2. Fig. 6 shows the modified blank obtained by the first stage modification for the DR of 2.17 and Fig. 7 shows the simulated cup for the same modified blank as shown in Fig. 6. Fig. 8 shows the experimentally drawn cups from modified blanks for several DRs. It can be observed from Fig. 8 that ear height was reduced significantly and at a few locations very small ears (wavy edge at the top) were formed. The results from the drawn cup after first stage of modification shows a significant reduction of earing but indicates the need for further modification of the blank. Procedure for the second stage of modification is described below. The material flow behavior at each time step (the whole punch stroke length is divided into time steps) was observed during simulation of deep drawing. It has been found that time step 16 gives the clear view of material flow behavior and shows where material is to be added or removed as the case may be. Fig. 9 shows the flow behavior of metal at 16th time step for a modified blank shown in Fig. 6. The material is added where metal flow rate is high to modify the blank further (i.e. second stage of modification). The coordinates corresponding to the points, where metal flow rate is high, were increased by half the depth of the valley at that location. Fig. 10 shows the simulated cup from the modified blank obtained from second stage of modification. It has been found out that the second stage of modification is required when either the blank diameter was very close to LDR diameter or when it was very small Experiments with modified initial blank shape The shape from first stage of modification was used to cut the initial blanks to conduct experiments or in other words the earing produced by the optimized blank shape from the simulation was verified experimentally. If this work is to be done experimentally, it would require a large number of trials and consume lot of time and energy. The blanks of optimized shapes were first cut to approximate dimensions on a shear machine and finished to exact dimensions by filing. Due to modification of the initial blank shape, the effective area of the initial blank changed and hence the actual DR. The procedure for calculation of effective diameter is described below: 1. The circular blank was made in AutoCAD of required diameter. 2. The circular blank was modified by drawing the four arcs (each through three successive points) as explained in Fig The effective area (A t ) of the modified blank was obtained directly by using AutoCAD. 4. Effective diameter was calculated by assuming that the final area is equal to the area of a circle (A{ = 3.14-df). 5. Effective DR was calculated as "cup

8 N. Kishor, D. Ravi Kumar/ Journal of Materials Processing Technology (2002) C: original center Ci, C 2, C 3 and C 4 : New centers on modified blank 1,2,3,4,5,6,7,8 : Points on circular blank 1', 2', 3', 4', 5', 6', and 7'8': Points on modified blank after first stage Initial circular blank Fig. 5. Method of drawing/sketching modified blank. Modified blank after first stage STEP 21 TIME = B7E-002 Y I Fig. 6. Modified blank after first stage of modification from circular blank of diameter mm (DR 2.17). Y X Fig. 7. Simulated cup drawn from the modified blank of Fig.

9 28 N. Kishor, D. Ravi Kumar/Journal of Materials Processing Technology (2002) Fig. 8. Cups drawn experimentally from modified blanks obtained after first stage of modification. The difference in the blank dimensions between the modified blank from the first stage and second stage is very small. Due to non-availability of accurate cutting facility to cut irregular blanks, only first stage of modification was considered for experimental work. Fig. 8 shows the experimentally drawn cup from modified blank of different diameters. It can be seen from Fig. 8 that ear height was reduced significantly. However, small projections at random locations were observed, which could be due to uneven cut profile of the blank. The uneven top edge of the cup could be due to practical problem with exact centering of the blank with respect to die and punch. The blank could have shifted slightly due to the pressure of the blank holder. Fig. 11 compares the cup height obtained experimentally as well as from simulation for DR 2.1. It can be observed from Fig. 11 that minimum cup height obtained from the experimentally drawn cup is 33 mm (before modification), while minimum cup height after modification is little less than 33 mm. A close observation of Fig. 11 shows some eccentricity (in the plot of cup height after modification) in the drawn cup. The minimum cup height after modification can be improved by the use of proper experimental setup to ensure the proper centering of blank. The minimum cup height is the height of the cup that will be obtained after trimming the uneven top edge of the cup. % ear height is calculated as follows: % ear height = (max. cup height min. cup height) min. cup height 100 Fig. 12 compares the % ear height before and after the first stage of modification for the DR 2.1 for the drawn cup by simulation. It can be observed that % of ear height above the minimum cup height before modification is 12-14%, which reduces to 3 4% after modification. This has been experimentally verified. Fig. 13 compares the % ear height before and after the first stage of modification for DR 2.1 STEP 16 TIME = E-002 STEP 21 TIME = 2.5D70220E-GD2 Fig. 9. Material flow at time step 16th in deep drawing of the blank shown in Fig. 6. Fig. 10. Simulated cup drawn from the blank obtained after the second stage of modification for the same blank diameter.

10 N. Kishor, D. Ravi Kumar/ Journal of Materials Processing Technology (2002) actual cup height before modification - actual cup hieght after modification - simulated cup height after modification Angle in degree from rolling direction Fig. 11. Comparison of cup height before and after modification of blank shape for an effective DR of % ear height (sim) before modification - % ear height (sim) after modification Angle in degree from rolling direction Fig. 12. Comparison of % ear height above the minimum cup height before and after modification of blank shape for a DR of 2.1 (simulation). for the experimentally drawn cup. It can be observed that % of ear height above the minimum cup height before modification is 13-15%, which reduced to 5-7% after first stage of modification. It is expected that % ear height above the minimum cup height further reduces by having the following: (a) use of modified blank from the second stage of modification, (b) proper cutting facility to cut irregular shapes accurately, and (c) new experimental setup in which blank can be placed accurately at the center. % ear height before modification % ear height after modification Angle in degree from rolling direction Fig. 13. Comparison of % ear height above the minimum cup height before and after modification of blank shape for a DR of 2.1 (experimental).

11 30 N. Kishor, D. Ravi Kumar/Journal of Materials Processing Technology (2002) Conclusions Based on the results and discussion presented in the preceding sections, the following conclusions can be drawn: FEM is a very powerful tool for the analysis of deep drawing problems. LDR of a low carbon steel sheet of EDD grade has been predicted using a FEM-based software and it has been found that predicted value LDR using the software LSDYNA is very close to the experimental value. It is also found that the predicted value from FEM is more accurate than values predicted by the analytical methods. It is possible to reduce the extent of earing in deep drawing of highly anisotropic sheets using a noncircular blank. This initial blank shape has been optimized by considering the material flow in different directions during deep drawing and planar anisotropy of the material. Simulated results showed that ear height reduced significantly in the drawn cups using the modified initial blank shape. The results of experiments conducted on the blanks of modified shape agreed closely with the simulated results. With second stage of modification of initial blank shape, it is expected that earing will reduce further. References [1] S.L. Semiatin, ASM Metals Handbook: Volume Forming and Forging, vol. 14, International Handbook Committee, 1988, pp [2] E. Dieter George, Mechanical Metallurgy, vol. 4, McGraw Hill Book Company, London, [3] R.L. Whiteley, The importance of directionality in drawing quality sheet steel, Trans. ASM 52 (1960) [4] S.P. Keeler, Understanding sheet metal formability, National Steel Corporation, USA 3 (1968) [5] D.-K. Leu, Prediction of the limiting drawing ratio and the maximum drawing load in the cup drawing, Int. J. Mech. Sci. 37 (2) (1997) [6] M.G. El-Sebaie, Mellor, Plastic instability conditions in the deep drawing of a circular blank of sheet metal, Int. J. Mech. Sci. 14 (1972) [7] S. Kobayashi, Metal Forming and Finite Element Method, Oxford University Press, New York, 1989, pp [8] C.H. Lee, H. Huh, Three-dimensional multi-step inverse analysis for the optimum blank design in sheet metal forming processes, J. Mater. Process. Technol. 80 (1998) [9] K. Chung, O. Richmond, Ideal forming II: sheet forming with optimum deformation, Int. J. Mech. Sci. 34 (1997) [10] K. Chung, F. Barlat, J.C. Brem, Blank shape design for a planar anisotropy sheet based on ideal forming design theory and FEM analysis, Int. J. Mech. Sci. 39 (1997) [11] A.M. Zaky, A.B. Nassar, M.G. El-Sebaie, Optimum blank shape of cylindrical cups in deep drawing of anisotropic sheet metals, J. Mater. Process. Technol. 76 (1998) [12] X. Chen, R. Sowerby, Blank development and the prediction of earing in cup drawing, Int. J. Mech. Sci. 8 (5) (1996) [13] T. Kuwabara, S.I. Wen-hua, PC-based blank design system for deep drawing irregularly shaped prismatic shells with arbitrarily shaped flange, J. Mater. Process. Technol. 63 (1997) [14] S. Hyunbo, S. Kichan, K. Kwanghee, Optimum blank shape design by sensitivity analysis, J. Mater. Process. Technol. 104 (2000) [15] S.H. Park, J.W. Yoon, D.Y. Yang, YH. Kim, Optimum blank design in sheet metal forming by the deformation path iteration method, Int. J. Mech. Sci. 41 (1999) [16] F. Barlat, J. Lian, Plastic behavior and strechability of sheet metal, Part I: A yield function for orthotopic sheet under plane stress condition, Int. J. Plasticity 7 (1989)