The effect of strain rate sensitivity on theoretical prediction of limiting draw ratio for cylindrical cup drawing process

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Abstract

The plastic instability of cup drawing is usually measured by the limiting drawing ratio (LDR). A new and practically applicable equation for estimating the LDR in the cup (cylindrical) drawing with a flat nosed punch using process parameters namely yield strength (σy), strain hardening exponent (n), strain rate sensitivity (m), normal anisotropy value (R¯), friction coefficient (μ), radius of die arc (rd) and half die opening (r1), has been derived using an integral technique based on the load maximum principle for localization of plastic flow.

The results shows that there is an interaction between the process parameters and the LDR and it is thereby possible to better understand and control cup drawing behavior for optimum press drawability. It is observed that there is an increase in LDR value as the strain rate sensitivity (m) increases.

Introduction

Cup drawing is one of the most important operations utilized for shaping of metals. The condition for plastic instability of cup drawing, using theoretical prediction without carrying out an actual try-out procedure, is urgently pursued in tool design and manufacturing control. The limiting drawing ratio (LDR) is commonly used to provide a measure of the drawability of sheet metal, being the ratio of R0/r1 under the drawing limit without failure, as shown in Fig. 1.

The correlation of the LDR of a sheet metal with its material properties and process parameters has been activated by industrial necessity for improving drawability.

Eric T. Harpell et al. [1] used an explicit dynamic finite element code LS-DYNA for the numerical prediction of limiting draw ratio. A series of 10 different tooling geometries were modeled for a cylinder cup-drawing process with differences being variation in the die and punch profile radii. A phenomenological yield criteria incorporating rolling-induced crystallographic texture effect are considered and their effect on the predicted strain distribution within drawn cups are assessed through comparison with measured strain. Transverse anisotropy is found to have a large influence on the predicted strains distributions within drawn cups are assessed through comparison with measured strain.

Transverse anisotropy is found to have a large influence on the predicted strains where as the influence of in plane anisotropy is small. Reasonable agreement with measured strain is obtained using a non-quadratic equation. The LDR is predicted based on two methods. (i) Proximity to be formed as characterized by a forming limit ratio (FLR) parameter calculated using principle prediction strains. (ii) Attainment of peak punch force (PPF) corresponding to the maximum blank size that can be drawn into the cavity without necking. The predicted LDR is in good agreement from the experiment for the tooling profile radii greater than 3 mm. However, the PPF method was less sensitive to the variation in the punch profile radius than the FLR based predictions. For the sharp 3 mm die radius tooling, the model over predicts the LDR which suggests the change in the failure mechanism to a bending failure, since the bend radius to thickness ratio approaches the bendable limits. Narayanasamy and Sowerby [2] made an experimental study of the drawing on 304 grade Stainless steel, dual phase steel, stainless steel and aluminium killed drawing quality steel which together covers a wide range of mechanical properties through a conical die using flat bottom ended and hemispherical ended punches. They identified that use of conical die can enhance the LDR. The most commonly observed failure mode in this process is the fracture of the material near the punch profile radius or wrinkling in the flange region when a thin sheet blank is drawn. The investigation shows that the LDR obtained for thin sheet materials can be related to a non-dimensional parameter index involving the punch diameter, the sheet material thickness, the average plastic strain and the average strain or work hardening exponent values. Hariharasudan Palaniswamy [3] made a finite element simulation for magnesium alloys for sheet metal drawing at elevated temperatures the use of magnesium alloys offers a significant potential to improve automotive fuel efficiency. However, the application of formed magnesium alloy components in auto body structures is restricted due to materials low formability at room temperatures and lack of knowledge for processing alloys at elevated temperatures. A non-isothermal finite element simulation has been conducted for forming round cups and rectangular pans from magnesium alloy AZ31B sheet at elevated temperature. Simulations and experiments shows that the increase in LDR with increase in temperature. Maximum LDR was obtained at forming temperature of 200 °C. Jain Guo Hu et al. [4] made finite element analysis of damage evolution and the prediction of limiting draw ratio in textured aluminium sheets. Void nucleation and growth models have been incorporated into an elasto-plastic finite element code to wear with an anisotropic fourth order strain grade potential in order to evaluate the damage evolution during deep drawing of textured aluminium sheets. The fourth order strain grade potential is based on the Tailor Model of crystal plasticity and therefore takes the presence of texture into account. The damage evolution is modeled in terms of void nucleations and growth during deformations. The influence of plastic isotropy on damages discussed together with the roles of void nucleation and growth on damage evolution for cold rolled and cold rolled annealed aluminium sheets. It is observed that growth of voids, which opposed the formation of the nucleation, plays a very important role in damage formation during the development of localized necks. More attention is concentrated to void nucleation and growth in analysis of localized necking determination of LDR and fracture during sheet metal forming. Leek et al. [5] compared the performance of titanium nitrite coated punches and the dies in deep drawing with that of uncoated tools in turns of two parameters namely, (1) maximum punch force and (2) LDR. The effectiveness of titanium nitrite coating on tools with different corner radii is also studied. Deep drawing experiments were conducted on aluminium using all possible combinations from two similar sets of tools (uncoated and coated with titanium nitrite) both the sets consists of punches and dies with different corner radii. It was found that the titanium coated dies reduce the maximum punch force by 7.5–13.4%. The coated dies also improve the LDR by 4.59% with the combination of uncoated punch and coated dies yielding the largest limiting draw ratios. The punch force were also reduced when the punch and die corner radii is increased. The LDR were improved with increase in die corner radius while punches of profile radii six to seven times the blank thickness gives the hire limiting draw ratios. The experimental results are in good agreement with the theoretical predictions. Narayanasamy and Sowerby [6] analyzed the wrinkling behavior of cold rolled sheet metals drawn through tractrix die. Deep drawing behavior of circular banks into cylindrical cups was passing through tractrix die using flat bottom and hemispherical ended punches have been analyzed. The use of Tractrix die can enhance the LDR as compared with that of obtainable unconventional drawing options. Thiruvarudchelvan and Lewis [7] made a new method of blank holding in deep drawing in urethane pad is used between the two parts of a drawing punch. As the punch moves to draw, the blank holder automatically applies the blank holding force. An experimental rig with this type of blank holding was constructed and cups of aluminium, copper, and brass were drawn. Tear were also conducted with constant force blank holding for comparison, results indicate lower punch force, modest increase in limiting draw ratios and less thickness variation when compared with formed cups under constant force blank holding. Jain et al. [8] analyzed the deep drawing characteristics of automotive aluminium sheet materials are investigated as a function of die profile radii by experiments and numerical predictions. A procedure for rapid determination of LDR based on the characteristics limit load of the material at fracture is developed and varied other deep drawing characteristics such as punch load vs. displacement plots, flange-drawing, strain distribution along cup profile, flange wrinkling, wall ironing and fracture characteristics are experimentally assessed. The deep drivability of AA5457-0 as measured by cup depth at fracture and LDR is superior to that of AA611-T4. The differences in the deep drawing between the two material is explained in terms of computation in work hardening between the two material in the flange at the die profile region vs the material in the punch profile material, bendability of two material and fracture characteristics. A decrease in LDR and flange drawing is observed as a function of the die profile radius. This decrease is attributed to the increase work hardening in the die profile region resulting from additional bending; unbending and stretching of the material as it enter the die cavity. Yegorov et al. [9] made a study on molecular mobility and breakdown processes in the high density polyethylene and investigated the reasons of the termination of orientation drawing of high density polyethylene and investigation the reasons of the termination of orientation drawing of high density polyethylene at limiting draw ratios, during the process of drawing when the material approaches the limiting draw ratios. During the process of drawing when the material approaches the limiting elongation the sample becomes turbid with the formation of kink band similar to that observed in drawing of low molecular weight solids. Under oriented loads nearer to the limiting draw ratio, a practically complete suppression of segmental mobility in amorphous region was recorded. Seah and Lee [10] studied the effect of titanium nitride coating on punch and dies in deep drawing of cold rolled mild steel. A series of experiments were conducted for the formation of cups by deep drawing process using blanks of cold rolled mild steel. Two similar sets of tools, one with titanium nitride coated, and other without were compared. In each sets, punches and dies of different edges radii were used in various combinations. The effect of these geometrical parameters as well as the benefits of the titanium nitride coating above the limiting draw ratio, the punch force

  • (i)

    Required and the ratio of the equipmental surface area of the cup formed to the theoretical surface area were observed and analyzed.

  • (ii)

    Thiruvarudchelvan and Wang [11] made investigation into hydraulic pressure augmented deep drawing to study the parameter such as draw ratio, the punch and die radii, the diameter of the stem of the punch and lubrication of the hydraulic augmented deep drawing process for the manufacture of very deep cups from sheet metal. Cylindrical cups from 1 mm thickness aluminium blank at draw ratio up to 3.5 mm are used in this process. Conditions leading to possible modes of trough failure and ways of avoiding these failures are discussed. The method of reducing the blank holding forces applied on flange of cups and steps are taken to reduce the speed of the emerging cups to acceptable values are described.

The investigations mentioned above were primarily concentrated to show the effects of material characteristics such as the normal anisotropy value R¯ and the strain hardening exponent n, on the LDR. In the cup drawing process tools with complicated geometries, multiple operations to transform a blank into the shape of a cup. Lubrication at the tool sheet interface, non-linear hardening behavior of plastic deformation, anisotropy property of the sheet metal, bending affect around the die arc and blank holding force all make it difficult to predict the LDR. To establish the correlation between the LDR and these parameters is not an easy task. In the past, little work had been done to theoretically investigate the relationship between the various process parameters and the LDR.

In view of the above situation, the purpose of the present work is to develop a new and more accurate equation based on the load maximum principle for the localization of the plastic flow, to estimate the LDR.

The new equation is a function of normal anisotropy value R¯, the strain hardening exponent (n), the strain rate sensitivity (m), the friction coefficient μ, the die arc radius rd, the half die opening r1 and the yield strength σy (measure of the blank holding stress), so that it may explore the effects of the process parameters on the LDR of the cup drawing in a theoretical manner. No comparison is made with the experimental results. The present work may allow a better understanding of the cup drawing behavior of sheet metals to determine the process parameters for the optimum press drawability and die design.

Section snippets

Analysis

Fig. 1 shows the cup-drawing operation under consideration that a circular blank of original radius R0 and thickness t0 is deep drawn by the flat-bottomed punch through a die opening of radius r1 with a constant clearance blank holder. Planner isotropy, radially symmetrical properties and rigid plastic strain hardening material are assumed. The radial friction stress exerted in the rim of deformed sheet resulting from the blank holding stress is considered. For simplicity the bending effect

The critical condition of cup-wall

A criterion for plastic instability based on tensile instability as explained by Daw-Kwei Leu [13] in the cup wall will be used to establish the critical drawing force to limit the maximum diametrical reduction of cup drawing.

During cup drawing, the drawing force P in the cup wall at a particular stage of the punch stroke is:P=2πr1tσrwhere

P is the drawing force in the cup wall at a particular stage of punch stroke,t=t0eεzandεz=-(1+2R¯p)0.5(1+R¯p)εewhere Rp is the new normal anisotropy parameter

Drawing in the die arc region

Based on the force equilibrium and the increase in rope tension around a capstan with friction between radii r1 and r2, the drawing force per unit circumferential length at r2 can be approximated asPr22πr2=Pc2πr1r1r2e-(π/2)μSubstituting in Eq. (9a), (9b) into the above equation, the drawing stress σr at r2, correlating with the condition of the critical drawing force, can be obtained as:σr2=Kt0Kr1r2e-(π/2)μ[e-Kεeεen{ln(1+εe)}m]where the stress σr2 denotes the radial drawing stress at r2 in

Pure radial drawing in the flange region

The radial equilibrium equation of an element of the flange for constant thickness isdσrdr=σr-σΘrBased on plane strain condition dεz = 0, the constant thickness of the flange, the relationship of the stresses σr and σΘ can be written asσr-σΘ=2(1+R¯p)1+2R¯p0.5σeSubstituting the above equation into Eq. (12) and integrating the equation from r2 to r0 for the maximum radius of blank R0, the radial drawing stress at the position r2 in the flange can be written as:σr2=σr0+KK11+nr2r0lnRrnln1+K1lnRrmdrr

A new equation for estimating the LDR

The LDR can be determined from the condition that the drawing stress σr2 of the die arc region to cause plastic instability in the cup wall, Eq. (11), equals the radial drawing stress σr2 of the flange region, Eq. (14) due to continuity of stress.

Substituting Eq. (11) into Eq. (14), the following relationship for the LDR can be obtained:Kt0Kr1r2e-(π/2)μ[e-Kεeεen{ln(1+εe)}m]=2.2μσyR0r0+KK11+n[1+n+m(2K1-1)+mn(K1-1)]lnr0r2-[2n(1+mK1)+2m(K1-n)]lnR0+r0R2+r2+r0R0+r0-r2R2+r2In Eq. (16) the

Results and discussion

The values of limiting draw ratio (LDR) were obtained from Eq. (16) developed in the above mathematical analysis, by varying the coefficient of friction μ, the strain hardening exponent n, the strain rate sensitivity m, yield stress σy and the normal anisotropy value-power constant, p, by the computation process and then plots were drawn.

Fig. 2 shows the variation of LDR with respect to the friction coefficient value for various values of strain rate sensitivity m. It is clearly observed that

Conclusion

A new accurate equation for estimation of LDR which is a function of normal anisotropy value R¯p, the strain hardening exponent n, the strain rate sensitivity m, the friction coefficient μ, the yield strength σy, the die arc radius rd and the half die opening r1 is derived. It is demonstrated by the above discussion that the R¯p value, the n value, and the μ value have marked effects on LDR.

The LDR value decreases with increase in friction coefficient but increases for higher m values.

The LDR

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