4th ed., Louis Schuler AG, Goppingen. (Wuertt.), 1966. 8 Chiang, D. C , and Kobayashi, S., "The Effect of Anisotropy and Work-. Hardening Characteristics on the ...
A. S. Korhonen1 Helsinki University of Technology, Dept. of Mining and Metallurgy, Lab. of Metal Working and Heat Treatment, Vuorimiehentie 2 A, 02150 ESPOO 15, Finland Assoc. Mem. ASME
Drawing Force in Deep Drawing of Cylindrical Cup With Flat-Nosed Punch A method for estimating the maximum drawing force in the deep drawing of cylindrical cups with flat-nosed punch is developed. The criterion is obtained using the load maximum principle for localization of the plastic flow. The criterion is then applied in an attempt to develop a simple and practically applicable method for deep drawing force evaluation. Simple charts are presented to obtain the maximum drawing force as a function of the tooling geometry and anisotropy for steel, brass and stainless steel.
Introduction Despite its apparent simplicity, deep drawing of a cylindrical cup is a very complicated process. Attempts to analyze it and evaluate the drawing force have been made by many authors. The early work of Siebel and Pomp [1] and Sachs [2, 3] laid the foundation for the subsequent theoretical treatment. The radial drawing problem was later studied by Hill [4] who considered the limiting cases of plane stress and strain. A comprehensive study of the elementary mechanics of the drawing process was carried out by Chung and Swift [5] who improved the model for bending over the die profile radius. They were, however, unable to extend their solution to the punch nose region. The attempts to develop a simple method for the estimation of the drawing force were continued in Germany by Siebel [6] and later by Panknin. The practical aspects of the German work have been summarized in reference [7]. Numerical methods were further applied by Chiang and Kobayashi [8], Budiansky and Wang [9], and ElSebaie and Mellor [10] to analyze the radial drawing problem of anisotropic materials. The analysis was extended into the punch nose region by Woo [11] who also improved the formulation of the blank-holding pressure boundary condition. The through thickness variation of the stresses was recently also taken into account by Odell [12] who, however, came to the conclusion that the membrane theory is adequate when cups of moderate die radius/material thickness ratios-are analyzed. The practical deep drawing processes may roughly be classified according to whether or not blank-holding is used. In the latter method (usually applied to thick sheets in specially shaped dies), considerable ironing occurs in the upper portion of the cup. The maximum diametrical reduction is, however, usually limited by the exceeding of the maximum allowable drawing force at the beginning of the
Formerly Graduate Student, Dept. of Mechanical Engineering and Materials Science, Duke University, Durham, N.C. Contributed by the Production Engineering Division and presented at the Winter Annual Meeting, Washington, D.C., November 15-20, 1981, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received at ASME
Headquarters, December 1980. Paper No. 81-WA/PROD-7.
punch nose rounding. This is generally the case with flatnosed punches when the punch profile radius is not too great and appreciable stretching over the punch nose does not take place. The friction at the punch nose cannot then assist in carrying the drawing load and the maximum force can be easily determined. Traditionally, the maximum force has been taken to be equal to the tensile strength x i x punch diameter x original thickness or slightly greater (1.15 times the value just mentioned) due to the plane strain condition at the cup wall [7,13]. Although it was realized by Chiang and Kobayashi [8], Budiansky and Wang [9], and El-Sebaie and Mellor [10] that the condition may not be as simple as previously stated and must depend on the work-hardening properties of material, little work has been done to investigate the accuracy of the criterion and to develop a simple and more accurate method for practical purposes. The aim of this paper is to reconsider the problem of evaluating the drawing force and to develop a simple method which could be used in practical press forming. Theory The maximum drawing force at the limiting drawing ratio can be obtained by maximizing the drawing force at the beginning of the punch nose rounding with respect to the effective stress. This method assumes that there is no contact between the sheet and the punch at the point of necking (and fracture). Consequently, the frictional forces cannot assist in carrying the drawing load. The drawing force is obtained from the equation F= (jj • 2-KR • t (1) and the appropriate values of ax and t will be obtained by maximizingF with respect to a, i.e., by writing dF =0 (2) da As has been noted by many authors [6, 13], the effect of punch profile radius is to make the stress state around the punch nose rounding triaxial. This is especially evident when
Journal of Engineering for Industry
FEBRUARY 1982, Vol. 104/29
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the sheet thickness/punch profile radius ratio (t/p) is not vanishingly small. However, as will be shown later, the neglecting of this triaxiality will lead to an overestimation of the drawing force. Let us therefore first consider the plane stress case. Plane Stress State. When the t0/p-mt\o is small, the plane stress assumption can be regarded as valid. It follows from equations (1) and (2) (see Appendix 1 for details) that the maximum drawing force is • R„ • 2-wR • t„
Wl+2/-/
F=
m
2-V3 2m
>
( ^ )
2irRtn
A,B
D d e F K
m n
1.6
1.5 r = 1.5 1.4
(>^™)(l*?") R„
Rmttdt0
(3)
if the material obeys the Hollomon strain-hardening equation a = Kt". Equation (3) is graphically displayed in Fig. 1 where the cross-hatched area denotes the expected scatter band for steel sheets. Some experimental points taken from the literature are also shown. These confirm the expected trend quite well. It appears that the drawing force is very sensitive to the anisotropy which may not completely be described by the single mean /--value. The exponent n may be taken as 0.2 for most steel sheets [14, 15]. The value of R,„ (tensile strength) may also have an influence on the accuracy of the results. As can be seen from Figs. 2(a) and (b), the tensile strength for Al-killed steel sheet increases with increasing testing speed. For stainless steel, the contrary is true. Since, however, the testing is usually for economical reasons carried out at relatively high speeds, it may be thought that the R,„ -values thus obtained rather closely correspond to the situation in the wall of the cup during drawing. Thus, reported tensile strength values could safely be used. It is, however, evident that if the strain-hardening behavior of the sheet metal cannot be described by the Hollomon equation, equation (3) is not valid. Using Voce's equation a = B — (B — A) exp ( — me) and assuming isotropy, the following equation is obtained (see Appendix 1) (1 + m)\
experimental data: • Thorp / 1 6 / , several steals x De Goede & v. Harinelli mi, r = 1.20 • Wilson .Sunrer a H a r t i n /18/, rs1.14 — Backofen's formula /13/ r= 2
(4)
= parameters in Voce's strain-hardening equation a = B (B-A) exp ( - w e ) = initial blank diameter = punch diameter = engineering strain (or natural logarithm base) = drawing force = strength coefficient in Hollomon's strainhardening equation a = Ke" = parameter in Voce's strain-hardening equation = strain-hardening exponent in Hollomon's equation
30 / Vol. 104, FEBRUARY 1982
0
R R/n r
t a
P
e,ei,e 2) e 3
1.3 r=1 1.2 sapproximare scatter band f o r steel sheets which obey the equation 9 s K i n
1.1
1.0 0
0.1
0.2
0.3
_L
_L
0.4
0.5
0.6
n
Fig. 1 Nondimenslonal fracture load corresponding to the limiting drawing ratio in cup-drawing. In plotting the experimental points from references [17] and [18], work-hardening exponent n = 0.2 was assumed.
Since it is known that the strain-hardening of stainless steel and copper metals may be better described by Voce's equation, equation (4) should be applied to them. The results from equations (3) and (4) are compared with the experimental drawing forces obtained from the literature in Table 1. The agreement is very good. In only one case, the value for stainless steel differs considerably from the measured one, but this may be due to some experimental error. The strain-hardening of stainless steels depends strongly on temperature and strain-rate. The coefficients in the force formula (equation (4)) are shown in Table 2 for different drawing speeds and loading conditions. It is somewhat surprising to see that the coefficient in the drawing force formula does not vary very much. The tensile test results at the cross-head speed of 200 mm/min are thought to
subscript referring to original dimension (if necessary) punch radius, 2R = d tensile strength (ISO recommendation) = UTS plastic strain ratio (width to thickness strain ratio in uniaxial tensile testing) thickness angle between punch profile radii as explained in Fig. 3 drawing ratio Did logarithmic strain,
P (7,(7!
=
effective strain and components of principal strains in the plane of the sheet and in the trough thickness direction, respectively angle between punch profile radii as explained in Fig. 3 punch profile radius effective stress and principal stress components, respectively superscript referring to a value corresponding to the limiting drawing ratio ((3*)
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400
Al- kilted s t e e l
Rm(MPa)
numerically c a l c u l a t e d : deformation hearing without cooling (conduction and n a t u r a l convection only) a = 10J/(sm 2 K)
& 350
i
n m n
400
O
Al- killed steel
a ^
numerically calculated •' conduction and free convection a - 10 J / (sm2K)
r
•n ON ^o ^f fN > S
on B
B
1 !
if
o o o oo o
•*t -— m
—i tS d
si
SB
300 WO
as
C-l fN (N
i P
o o o -. 1 1
B ^ B
ii
^H r^
O
-H -H —
Fig. 2 in ro oo Tf O N
m m ro
"-
.
2
o
describe best the strain-hardening of the cup wall and the drawing force for stainless steel in Table 1 was consequently obtained from the formula:
ON
' O OO m •—< >—< e n
rq (N
*sD \o \o *o
—
B^^s-g
«
• o a >- b
„o\ooo •*
coooo
1
& 2 •- -S 3 „ i-. •-• a) -3 > n
CB
w
rj **=
-<
-^
5 ^)
Vl+2r / a \ 2 VTT/V2 + 7- \ K )
F
ffj
• t
(A-ll) Rm • 2itRtn The tensile strength R,„ can be expressed in the case of Voce's and Hollomon's equations, respectively, as B-(B-A)e-mld
(A-l)
Assuming that a % 90 deg (i.e., necking and fracture take place at the beginning of the punch profile radius), which is usually the case for flat nosed punches, one obtains the mean through thickness stress as
< >=
ai R+ +ai p+
(A-12)
^ -2k[ ( i) ( i)]
"
while the definition of the effective stress [4] gives T [(g2 - "if + (^3 ~
