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Elastic-plastic deformation of sheet metal-forming with a specified angle of rotation
Theoretical and Applied Fracture Mechanics 14 (1990) 71-79 Elsevier
71
Elastic-plastic deformation of sheet metal-forming with a specified angle of rotation Y.J. H w n * a n d C.T. H s u Steel and Aluminum Research and Deoelopment Department, China Steel Corporation, Kaohsiun~ Taiwan, Republic of China
This paper considers the elastic-plastic stress analysis of a thin AOO-H steel sheet bent around a ninety degree (90 °) circular die. The positions analyzed after forming are specified as 18 o, 36 ° and 48 °. Constant contours of the effective stress and strain energy density are obtained to identify the localized regions of high distortion and dilatation are identified as the contact between the sheet and die increases with the angle of rotation. As expected, yielding would first initiate near the contact where the sheet is bent while fracture is anticipated to take place on the tension side of the sheet where volume change is the largest. The contact area increased slowly with the angle of rotation up to about 30 o for the present configuration, beyond which it increased much more rapidly. This general feature of metal.forming is also reflected by the stored energy density; it increased almost linearly up to 30 o of rotation and then remained nearly constant. A range of angles prevails within which most of the energy is used to form the bend in the steel and it depends on the combination of sheet material and geometry in addition to the rate of forming.
1. Introduction W h i l e f o r m i n g is a w e l l - k n o w n process for s h a p i n g metals, the p r o c e d u r e , for the m o s t part, involves trial a n d error. O n l y recently has m e t a l f o r m i n g b e e n a n a l y z e d a n a l y t i c a l l y [1-3]. I n o r d e r to establish limits o n the m a t e r i a l a n d g e o m e t r i c p a r a m e t e r s , effective results are n e e d e d n o t o n l y in s t r e s s analysis, b u t also in p r e d i c t i n g failure so t h a t it c o u l d b e a v o i d e d in p r o d u c t i o n . T o this end, inconsistencies a n d a r b i t r a r i n e s s b e t w e e n the stress a n d failure analyses should b e minimized. I n p a r t i c u l a r , the yield a n d fracture criteria s h o u l d n o t b e a d d r e s s e d i n d e p e n d e n t l y , a situation that has p r e v a i l e d t r a d i t i o n a l l y . T h e s i m u l t a n e o u s d e s c r i p t i o n o f yield a n d fracture is r e g a r d e d essential in m e t a l - f o r m i n g . T h e s e two m e c h a n i s m s of m a t e r i a l d a m a g e i n t e r p l a y a n d d e t e r m i n e the final state o f the p r o d u c t . It is therefore the objective o f this w o r k to assess the c h a n g e s in local d i s t o r t i o n a n d d i l a t a t i o n assuming t h a t they c o u l d b e used to forecast y i e l d i n g a n d fracture [4,5]. P a r t i c u l a r e m p h a s e s are thus given to the d e t e r m i n a t i o n o f the effective stress a n d strain e n e r g y d e n s i t y c o n t o u r s in two d i m e n * Also, Graduate Student, Department of Mechanical Enginecring, National Sun Yat-Sen University, Kaohsiung, Talwan, Republic of China. 0167-8442/90/$03.50 © 1990 - Elsevier Science Publishers B.V.
sions. C h a n g e s in local s t r a i n rates are a c c o u n t e d for in a h o m o g e n e o u s m a n n e r a c c o r d i n g to the classical t h e o r y of plasticity. T h a t is, the s a m e constitutive relation is iased for all e l e m e n t s in the sheet. These c o n s i d e r a t i o n s a n d those y e t to b e d e v e l o p e d i offer new findings t h a t w o u l d e n h a n c e c o m p u t e r s i m u l a t i o n of the m e t a l - f o r m i n g process. A s t e p - b y - s t e p p r o c e d u r e is followed to e s t a b l i s h sensitivity o f the w a y s with w h i c h m a t e r i a l p a r a m eters w o u l d i n t e r a c t with those arising f r o m the process of m e t a l - f o r m i n g .
2. Preliminaries on plasticity and finite elements M a t e r i a l e l e m e n t s are s u b j e c t e d to large d e f o r m a t i o n in m e t a l - f o r m i n g , a feature that is i n c o r p o r a t e d in the P E R D A p r o g r a m [7]. T h e JE-flow rule will b e used in the i n c r e m e n t a l t h e o r y of p l a s t i c i t y [8] which assumes that d ~ / ' - - OfFTM + O~//'P = o~j(dcej + d ~ [ P j ) = o i j d c i j ,
(1)
s Inlaomogeneous plastic deformation has been considered [6] and will be applied to metal forming in subsequent work. The constitutive relation can differ from element to element.
Y.J. Hwu, C T. Hsu /Elastic-plasticdeformationof sheet metal-forming
72
in which the superscripts e and p are used to denote quantities associated with elastic and plastic deformation, respectively. The stresses are o o and strains are % .
2.1. Requirements in plasticity The application of external stresses a n d / o r displacements gives rise to positive work, and hence for work hardening, it is required that
do G d~i / > 0.
(2)
The net work done by the application and removal of external loads must be zero or positive, i.e. do/j(d,ij-d,:j)>_0
E
)
Oys
- - -E
a
'
foro>Oy~, for o < oys , (10)
which may be referred to as the uniqueness condition. The development of stress-strain relations requires that at each stage a function f(oij) exists such that further plastic deformation can take place only for
in which a, fl and Oys depend on a particular material and the technique of data-fitting.
or
do, j d , P > _ 0 ,
{o+o[ot
(3)
df(oij)>O
or
A nonlinear stress and strain response is modelled into PERDA [7] with the capability to account for small changes on the stress and strain curve, particularly in regions near the yield point where the linear response terminates and the nonlinear behavior begins. Large differences in the end results can occur if care is not taken in crossing the y i e l d p o i n t . A convenient way of modelling the nonlinear stress and strain curve is to use the uniaxial expression of Ramberg-Osgood:
?fdo~j>0.
(4)
OOij
A specific form of stress and strain could be the Prandtl-Reuss relation in the form
a!
d,P. = ~
d)~,
(5)
2.2. Finite element procedure Application of the finite element method would result in a governing equation solving for the nodal displacements: [ K ( u ) ] ( ~ } = {/~},
(11)
where u is the nodal velocity vector and R is the equivalent nodal loading rate vector. The stiffness matrix is given by [K(u)] = E £[B(u)l~[M(u)l[S(u)l
dA.
where dX =
(12) (6)
oe d ~ p
o,(af/oo.)
The equivalent or effective plastic strain increment is defined as d,p = l a *
Oe tj
d , p,
(7)
where oij are the deviatoric stress components
1
(8)
Oij = Oij - - ~ Okk~i j .
The effective stress, %, takes the expression
1
o¢= " ~ - I ( O 1 - - 02) 2 q- (02--03) 2 -I- (03--01
)2
(9) with o~ ( j = 1, 2, 3) being the principal stresses.
The material matrix [M(u)] relates to the stress and strain, and the components in the matrix [B(u)] are derivatives of the interpolation functions N~, with i being the number of nodes for a given element. The integral in eq. (12) applies to an element, while [K(u)] is obtained by summing for all the elements in a system as implied by the sign Z. An iterative procedure is adopted to solve eq. (11) incrementally. The details can be found in [7] and will not be repeated here. Convergence criteria are based on comparing the mean square of the residual force vector, ] F I 2 = F - F, with the mean squares of the increment load vector, I a R 12 = ai R- A R, and the total load vector, I R 12 = R • R, i.e. I F I 2/1AR I 2 < C1
(increment equilibrium)
(13)
Y.J. Hwt~ C.T. Hsu / Elastic-plasticdeformation of sheet metal-forming
25ram
l .i mmI
~
AOO-H
73
-I STEEL
5mm--~
I . O'xy
Fig. 1. Flat sheet and circular die configuration.
and
I F I 2 / I R I 2 < C: o~ 0
2
+, I-*
0
I
~ 8
I
I 16
STRAIN
t
I 24
t
J ~ L ~ 32 40
x lO'=(m/m)
Fig. 2. True stress and true strain diagram for A O O - H steel.
(overallequilibrium).
(14)
The constants C 1 and C2 can be preassigned. (71 = 0.05 and C 2 = 1 0 - 7 have been used. An additional feature of P E R D A is that points on the metal sheet do not penetrate into the die. The displacements of all points on the metal surface come into contact near the die with a preassigned numerical accuracy. Physical contact is assumed to take place when a limiting distance is reached.
0=0 °
I
Hill/
I
--~'~'~'~" ".,.
N-..'<-,,---.-..
---....
\
,
Fig. 3. Deflection position of sheet metal specimen.
4,8
Y.J. Hwu, C T. Hsu / Elastic-plastic deformation of sheet metal-forming
74
3. Statement of problem
4. Discussion of results
Figure 1 shows a thin metal sheet 25 m m long and 1 m m thick. It rests on a flat portion of a circular die 5-mm long beyond which the surface curves with a constant radius of R 0 ---5 mm. A system of (x, y ) coordinates is used to locate all points in the metal sheet made of A O O - H steel. It has a yield strength of %, = 15 k g / m m 2, a Young's modulus of E = 2500 k g / m m z and a Poisson's ratio of v = 0.3. A graphical display of the true stress and true strain curve is given in Fig. 2. A downward force is applied at x = 25 m m and y -- 0 such that an arc distance equivalent to angle of 6 ° is approximately equal to the length of contact on the circular surface with radius R o = 5 ram. Three deflected positions with 0 = 18 °, 36 ° and 48 o, as shown in Fig. 3, will be analyzed by application of the P E R D A program [7]. The twelve node isoparametric elements are used to discretize the flat sheet metal specimen. This is shown in Fig. 4. Fourteen elements are interconnected by 26 corner nodes, while altogether 116 nodes are used. Node no. 116 corresponds to the end point (25, 0) in Fig. 1. Referring to Fig. 4, the portion x = 5 m m and y = 0, which is in contact with the die at the ten nodal points 4, 6, etc., terminating at 28, remains flat at all times. As the sheet metal is bent around the die, one of the nodal points in element 4 will first come into contact with the curved surface. For O = 18 o, only one nodal point comes into contact. As 0 is increased to 36 o, the contact points are increased to seven. Eleven contact points are found for 0 = 48 o.
Since the concentrations of stress, strain and energy density are localized in elements 3, 4 . . . . . 8, it suffices to isolate them for the presentation of numerical data. Three sets of numerical results, corresponding to 8 = 18 °, 36 ° and 48 °, are obtained and summarized in Table 1. The stresses o~, Oy and Oxy, as shown on the element in Fig. 1, are given. Also obtained are the effective stress, Oe, in eq. (9) and the strain energy density function, dW/dV, for plane strain given by
dW dV
1 [1-v[o2+
1'
-
(15)
where # is the shear modulus of elasticity and g is Poisson's ratio.
4.1. Eighteen degree rotation From the data in Table 1 for 0= 18 ° , the contours 1 to 5 for ox and Oy in the upper half of the sheet are in tension, while the remaining contours, from 6 to 10, in the lower half are in compression, with the larger amplitudes occurring near the surface as in beam bending. The shear stress, Oxy,changes sign near the raid-plane where contours 3 and 4 are located. This is expected. Figure 5 reveals that oe is highest near the contact where contour 10 is situated. Yielding, therefore, is expected to occur first in the local region where the metal comes into contact with the die. The constant contours of the strain energy density function are displayed in Fig. 6. A m a x i m u m of (dW/dV)min occurs near the top surface, near
,I!T
115 114
2
Fig. 4. Finite element grid pattern for sheet metal.
Y.J. Hw~ C.T. Hsu / Elastic-plastic deformation of sheet metal-forming Table 1 Constant stress and energy density contours in units of k g / m m 2 Contour
ox
Oy
oxy
oe
dW/dV
NO.
0 =18 ° 1 2 3 4 5 6 7 8 9 10
- 37.460 -23.315 -21.180 -13.046 -4.911 3.224 11.359 19.498 27.628 35.763
- 6.491 -4.910 -3.329 -1.749 -0.168 1.413 2.994 4.574 6.155 7.736
- 4.223 -2.571 -0.918 0.734 2.386 4.038 5.690 7.343 8.995 10.647
2.010 4.318 7.826 10.738 13.641 16.549 19.457 22.364 25.272 28.180
0.000 0.112 0.223 0.384 0.446 0.556 0.667 0.778 0.888 1.000
0 = 36 ° 1 2 3 4 5 6 7 8 9 10
-49.849 - 38.021 -28.194 -17.366 -6.589 4.289 15.116 25.944 36.771 47.599
-15.481 - 12.396 -9.310 -6.225 -3.140 -0.054 3.031 6.116 9.202 12.287
-13.507 - 10.427 -7.347 -4.267 -1.187 1.894 4.974 8.054 11.134 14.214
2.464 6.170 9.886 18.602 17.318 21.035 24.751 28.467 32.183 35.899
0.004 0.376 0.747 1.119 1.490 1.862 2.283 2.605 2.976 3.348
-63.036 -42.566 - 37.097 -21.627 -11.158 -0.688 9.781 20.251 30.720 41.190
-23.746 -19.196 - 14.647 -10.037 -5.548 -0.999 3.550 8.100 12.649 17.198
-18.434 -14.151 - 9.869 -6.586 -1.303 2.973 7.262 11.546 15.827 20.110
3.057 7.631 12.210 16.789 21.369 25.948 30.527 35.106 39.686 44.265
0.004 0.383 0.762 1.141 1.520 1.839 2.278 2.657 3.036 3.415
0 =
1 2 3 4 5 6 7 8 9 10
75
node 31 in Fig. 4, where the tension is also the largest. This is the location of (dW/dV)m~ and is predicted as the fracture initiation site. The results confirm with the strain energy density criterion assumption [4,5] that yield and fracture initiation do not occur at the same location.
4.2. Thirty-six degree rotation As the bending of the sheet is increased to 0 = 36 °, the stresses and energy densities increase accordingly, as indicated in Table 1. Again, both ox and Oy change sign near the mid-plane of the sheet. The sign change of Oxy occurred at locations closer to the lower portion of the metal sheet where contacts are made. As the contact surface now extends further away from the point (5, 0), the location of maximum distortion also changes accordingly. This is shown in Fig. 7. On the other hand, the location of [(dW/dV)m~]L in Fig. 8 remained near node 31, even though 0 is increased to 36 °
4.3. Forty-eight degrees of rotation
48 °
The qualitative features of the stress and energy distributions are preserved as the angle 0 is further increased to 48 o. This can be seen from the values of ox, Oy and Oxy in Table 1. The results on % and d W / d V for 0 = 48 ° did not change appreciably from those for 0 = 36 °. This can be seen from the contour plots in Figs. 9 and 10 or the data in Table 1.
10
Fig. 5. Constant contours of oe in elements 3 - 8 for 0 = 1 8 °.
76
Y.J. Hwt6 C T. Hsu / Elastic-plastic deformation of sheet metal-forming
~
_
.- MAXIMUM
/
DILATATION
Fig. 6. Constant contours of d W/dV in elements 3-8 for 0 = 18o.
4. 4. Contact length and energy density
The computed data for the increase in contact lengths with the angle of solution 0 are summarized in Fig. 11. The contact length increased slowly with 0 at first and then more quickly as 0 passed beyond approximately 30 o. This feature is also reflected by the variations of ( d W / d V ) m ~ near node 31 against 0 in Fig. 12. The energy density increased with 0 almost linearly up to 30 ° and then the curve flattened out to a constant value of approximately 3.4 k g / m m 2. This implies
i/4
~6
that the energy accumulation process almost ceased for 0 > 30 o. This is the range where most of the energy is used in forming the bend in the sheet. Additional rotation will not alter the energy state appreciably.
5.Concludinremarks g This work identifies the locations of potential yield and fracture initiation sites. The former correspond to the region of maximum effective stress
5
Fig. 7. Constant contours of oe in elements 3-8 for 0 = 36 °.
Y.J. Hwu, C 1", Hsu / Elastic-plastic deformation of sheet metal-forming
77
/--~---- MAX I MUM
Fig. 8. Constant contours of dW/dV in elements 3-8 for 0 = 36 °.
and the latter to the region of minimum strain energy density function, or (dW/dV)mm, where dilatational effect tends to dominate. Changes in the stress and energy states in metal-forming suggest that the effect of nordaomogeneous deformation might be significant. That is to say, strains
are not only concentrated in small regions but they depend on the change in the local strain rates and strain rate history. More precisely, the local behavior of each dement may depend on its previous history, whereas the dements at large may undergo small deformation. A wide spread of de-
'MAXIMUM
DISTORTION AT CON TACT
Fig. 9. Constant contours of oe in elements 3-8 for 0 = 48 °.
78
Y.J. Hwu, C T. Hsu / Elastic-plastic deformation of sheet metal-forming
/
------
MAX IMUM
DILATATION
2
~
2
Fig. 10. Constant contours of dW/dV in elements 3-8 for 0 = 48 °.
formation range and rate could exist that may not be adequately covered by a single constitutive relation. A scheme is needed to describe the behavior of each element by a different constitutive relation, in addition to accounting for changes in local strain rates and strain rate history. This necessitates a modification of the current theory of plasticity which may be summarized as follows: • Provide a scheme where the constitutive relations for each local element can be derived instead of preassigned.
f
E
/
~3.2~
• The derived constitutive relations should include the change in local strain rates and strain rate history. Such capabilities can be achieved by the development of a material data bank. The response of a particular element at a given time would be determined from knowledge of current strain c or % and rate i or ie- Here, % is an equivalent or effective plastic strain that can be related to an g-,
z4i "
I
z~2 f 0
o
12 °
24 °
36 u
48"
60 ~
r
ANGLE e ( d e g . ) ANGLE e (deg.)
Fig. 11. Contact length versus angle of rotation.
Fig. 12. Variations of local maximum of minimum strain energy density function near node 31 with angle of rotation.
Y.J. Hwu, C T. Hsu / Elastic-plastic deformation of sheet metal-forming
effective stress, oe. The details of the m e t h o d can be f o u n d in [6]. Application to metal-forming is being made.
Acknowledgement This work presents part of the results obtained u n d e r project TRC-78-25 supported by the China Steel Corporation. The senior author wishes to express his gratitude to the Vice President of the T e c h n o l o g y Division, Dr. Tsou Jo-chi, and General M a n a g e r of the Steel and A l u m i n u m Research and D e v e l o p m e n t Department, Dr. H u a n g Jong-Yuh, for their support and interest in this project. Special acknowledgement is also due to the Lehigh University Branch Institute of Fracture and Solid Mechanics in the R O C for using the P E R D A c o m p u t e r p r o g r a m to obtain the numerical results.
79
References [1] G.M. Goodwin, "Application of strain analysis to sheet metal forming problems in the press shop", Society of Automotive Engineers, Paper No. 680093 (1968). [2] S.S. Hecker, "Formability of high-strength low-alloy steel sheets", Metals Eng. Q. 13, pp. 42-48 (1973). [3] Y.J. Hwu and C.T. Hsu, "A large strain and elastic-plastic finite element analysis of stretching process", Sixth Nat. Conf. on Mech. Eng., CSME, Tainan, 1323-1333 (1989). [4] G.C. Sih, Introductory Chapters of Mechanics of Fracture, Vols 1 to VII, edited by G.C. Sih, Martinus Nijhoff Pubfishers, The Hague (1972-1983). [5] G.C. Sih, "Fracture mechanics of engineering structural components", Fracture Mechanics Methodology, edited by G.C. Sill and L.O. Faria, Martinus Nijhoff Publishers, The Netherlands, 35-101 (1984). [6] G.C. Sih and D.Y. Tzou, "Plastic deformation and crack growth behavior", Plasticity and Failure Behavior of Solids, Series on Fatigue and Fracture, Vol. III, edited by G.C. Sih, A.J. Ishfinksy and S.T. Mileiko, Kluwer Academic Pubfishers, The Netherlands (1990). [7] "Path evaluation and rate dependent analysis (PERDA) computer program", Institute of Fracture and Solid Mechanics, Lehigh University, USA (Dec. 1989). [8] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London (1950).
71
Elastic-plastic deformation of sheet metal-forming with a specified angle of rotation Y.J. H w n * a n d C.T. H s u Steel and Aluminum Research and Deoelopment Department, China Steel Corporation, Kaohsiun~ Taiwan, Republic of China
This paper considers the elastic-plastic stress analysis of a thin AOO-H steel sheet bent around a ninety degree (90 °) circular die. The positions analyzed after forming are specified as 18 o, 36 ° and 48 °. Constant contours of the effective stress and strain energy density are obtained to identify the localized regions of high distortion and dilatation are identified as the contact between the sheet and die increases with the angle of rotation. As expected, yielding would first initiate near the contact where the sheet is bent while fracture is anticipated to take place on the tension side of the sheet where volume change is the largest. The contact area increased slowly with the angle of rotation up to about 30 o for the present configuration, beyond which it increased much more rapidly. This general feature of metal.forming is also reflected by the stored energy density; it increased almost linearly up to 30 o of rotation and then remained nearly constant. A range of angles prevails within which most of the energy is used to form the bend in the steel and it depends on the combination of sheet material and geometry in addition to the rate of forming.
1. Introduction W h i l e f o r m i n g is a w e l l - k n o w n process for s h a p i n g metals, the p r o c e d u r e , for the m o s t part, involves trial a n d error. O n l y recently has m e t a l f o r m i n g b e e n a n a l y z e d a n a l y t i c a l l y [1-3]. I n o r d e r to establish limits o n the m a t e r i a l a n d g e o m e t r i c p a r a m e t e r s , effective results are n e e d e d n o t o n l y in s t r e s s analysis, b u t also in p r e d i c t i n g failure so t h a t it c o u l d b e a v o i d e d in p r o d u c t i o n . T o this end, inconsistencies a n d a r b i t r a r i n e s s b e t w e e n the stress a n d failure analyses should b e minimized. I n p a r t i c u l a r , the yield a n d fracture criteria s h o u l d n o t b e a d d r e s s e d i n d e p e n d e n t l y , a situation that has p r e v a i l e d t r a d i t i o n a l l y . T h e s i m u l t a n e o u s d e s c r i p t i o n o f yield a n d fracture is r e g a r d e d essential in m e t a l - f o r m i n g . T h e s e two m e c h a n i s m s of m a t e r i a l d a m a g e i n t e r p l a y a n d d e t e r m i n e the final state o f the p r o d u c t . It is therefore the objective o f this w o r k to assess the c h a n g e s in local d i s t o r t i o n a n d d i l a t a t i o n assuming t h a t they c o u l d b e used to forecast y i e l d i n g a n d fracture [4,5]. P a r t i c u l a r e m p h a s e s are thus given to the d e t e r m i n a t i o n o f the effective stress a n d strain e n e r g y d e n s i t y c o n t o u r s in two d i m e n * Also, Graduate Student, Department of Mechanical Enginecring, National Sun Yat-Sen University, Kaohsiung, Talwan, Republic of China. 0167-8442/90/$03.50 © 1990 - Elsevier Science Publishers B.V.
sions. C h a n g e s in local s t r a i n rates are a c c o u n t e d for in a h o m o g e n e o u s m a n n e r a c c o r d i n g to the classical t h e o r y of plasticity. T h a t is, the s a m e constitutive relation is iased for all e l e m e n t s in the sheet. These c o n s i d e r a t i o n s a n d those y e t to b e d e v e l o p e d i offer new findings t h a t w o u l d e n h a n c e c o m p u t e r s i m u l a t i o n of the m e t a l - f o r m i n g process. A s t e p - b y - s t e p p r o c e d u r e is followed to e s t a b l i s h sensitivity o f the w a y s with w h i c h m a t e r i a l p a r a m eters w o u l d i n t e r a c t with those arising f r o m the process of m e t a l - f o r m i n g .
2. Preliminaries on plasticity and finite elements M a t e r i a l e l e m e n t s are s u b j e c t e d to large d e f o r m a t i o n in m e t a l - f o r m i n g , a feature that is i n c o r p o r a t e d in the P E R D A p r o g r a m [7]. T h e JE-flow rule will b e used in the i n c r e m e n t a l t h e o r y of p l a s t i c i t y [8] which assumes that d ~ / ' - - OfFTM + O~//'P = o~j(dcej + d ~ [ P j ) = o i j d c i j ,
(1)
s Inlaomogeneous plastic deformation has been considered [6] and will be applied to metal forming in subsequent work. The constitutive relation can differ from element to element.
Y.J. Hwu, C T. Hsu /Elastic-plasticdeformationof sheet metal-forming
72
in which the superscripts e and p are used to denote quantities associated with elastic and plastic deformation, respectively. The stresses are o o and strains are % .
2.1. Requirements in plasticity The application of external stresses a n d / o r displacements gives rise to positive work, and hence for work hardening, it is required that
do G d~i / > 0.
(2)
The net work done by the application and removal of external loads must be zero or positive, i.e. do/j(d,ij-d,:j)>_0
E
)
Oys
- - -E
a
'
foro>Oy~, for o < oys , (10)
which may be referred to as the uniqueness condition. The development of stress-strain relations requires that at each stage a function f(oij) exists such that further plastic deformation can take place only for
in which a, fl and Oys depend on a particular material and the technique of data-fitting.
or
do, j d , P > _ 0 ,
{o+o[ot
(3)
df(oij)>O
or
A nonlinear stress and strain response is modelled into PERDA [7] with the capability to account for small changes on the stress and strain curve, particularly in regions near the yield point where the linear response terminates and the nonlinear behavior begins. Large differences in the end results can occur if care is not taken in crossing the y i e l d p o i n t . A convenient way of modelling the nonlinear stress and strain curve is to use the uniaxial expression of Ramberg-Osgood:
?fdo~j>0.
(4)
OOij
A specific form of stress and strain could be the Prandtl-Reuss relation in the form
a!
d,P. = ~
d)~,
(5)
2.2. Finite element procedure Application of the finite element method would result in a governing equation solving for the nodal displacements: [ K ( u ) ] ( ~ } = {/~},
(11)
where u is the nodal velocity vector and R is the equivalent nodal loading rate vector. The stiffness matrix is given by [K(u)] = E £[B(u)l~[M(u)l[S(u)l
dA.
where dX =
(12) (6)
oe d ~ p
o,(af/oo.)
The equivalent or effective plastic strain increment is defined as d,p = l a *
Oe tj
d , p,
(7)
where oij are the deviatoric stress components
1
(8)
Oij = Oij - - ~ Okk~i j .
The effective stress, %, takes the expression
1
o¢= " ~ - I ( O 1 - - 02) 2 q- (02--03) 2 -I- (03--01
)2
(9) with o~ ( j = 1, 2, 3) being the principal stresses.
The material matrix [M(u)] relates to the stress and strain, and the components in the matrix [B(u)] are derivatives of the interpolation functions N~, with i being the number of nodes for a given element. The integral in eq. (12) applies to an element, while [K(u)] is obtained by summing for all the elements in a system as implied by the sign Z. An iterative procedure is adopted to solve eq. (11) incrementally. The details can be found in [7] and will not be repeated here. Convergence criteria are based on comparing the mean square of the residual force vector, ] F I 2 = F - F, with the mean squares of the increment load vector, I a R 12 = ai R- A R, and the total load vector, I R 12 = R • R, i.e. I F I 2/1AR I 2 < C1
(increment equilibrium)
(13)
Y.J. Hwt~ C.T. Hsu / Elastic-plasticdeformation of sheet metal-forming
25ram
l .i mmI
~
AOO-H
73
-I STEEL
5mm--~
I . O'xy
Fig. 1. Flat sheet and circular die configuration.
and
I F I 2 / I R I 2 < C: o~ 0
2
+, I-*
0
I
~ 8
I
I 16
STRAIN
t
I 24
t
J ~ L ~ 32 40
x lO'=(m/m)
Fig. 2. True stress and true strain diagram for A O O - H steel.
(overallequilibrium).
(14)
The constants C 1 and C2 can be preassigned. (71 = 0.05 and C 2 = 1 0 - 7 have been used. An additional feature of P E R D A is that points on the metal sheet do not penetrate into the die. The displacements of all points on the metal surface come into contact near the die with a preassigned numerical accuracy. Physical contact is assumed to take place when a limiting distance is reached.
0=0 °
I
Hill/
I
--~'~'~'~" ".,.
N-..'<-,,---.-..
---....
\
,
Fig. 3. Deflection position of sheet metal specimen.
4,8
Y.J. Hwu, C T. Hsu / Elastic-plastic deformation of sheet metal-forming
74
3. Statement of problem
4. Discussion of results
Figure 1 shows a thin metal sheet 25 m m long and 1 m m thick. It rests on a flat portion of a circular die 5-mm long beyond which the surface curves with a constant radius of R 0 ---5 mm. A system of (x, y ) coordinates is used to locate all points in the metal sheet made of A O O - H steel. It has a yield strength of %, = 15 k g / m m 2, a Young's modulus of E = 2500 k g / m m z and a Poisson's ratio of v = 0.3. A graphical display of the true stress and true strain curve is given in Fig. 2. A downward force is applied at x = 25 m m and y -- 0 such that an arc distance equivalent to angle of 6 ° is approximately equal to the length of contact on the circular surface with radius R o = 5 ram. Three deflected positions with 0 = 18 °, 36 ° and 48 o, as shown in Fig. 3, will be analyzed by application of the P E R D A program [7]. The twelve node isoparametric elements are used to discretize the flat sheet metal specimen. This is shown in Fig. 4. Fourteen elements are interconnected by 26 corner nodes, while altogether 116 nodes are used. Node no. 116 corresponds to the end point (25, 0) in Fig. 1. Referring to Fig. 4, the portion x = 5 m m and y = 0, which is in contact with the die at the ten nodal points 4, 6, etc., terminating at 28, remains flat at all times. As the sheet metal is bent around the die, one of the nodal points in element 4 will first come into contact with the curved surface. For O = 18 o, only one nodal point comes into contact. As 0 is increased to 36 o, the contact points are increased to seven. Eleven contact points are found for 0 = 48 o.
Since the concentrations of stress, strain and energy density are localized in elements 3, 4 . . . . . 8, it suffices to isolate them for the presentation of numerical data. Three sets of numerical results, corresponding to 8 = 18 °, 36 ° and 48 °, are obtained and summarized in Table 1. The stresses o~, Oy and Oxy, as shown on the element in Fig. 1, are given. Also obtained are the effective stress, Oe, in eq. (9) and the strain energy density function, dW/dV, for plane strain given by
dW dV
1 [1-v[o2+
1'
-
(15)
where # is the shear modulus of elasticity and g is Poisson's ratio.
4.1. Eighteen degree rotation From the data in Table 1 for 0= 18 ° , the contours 1 to 5 for ox and Oy in the upper half of the sheet are in tension, while the remaining contours, from 6 to 10, in the lower half are in compression, with the larger amplitudes occurring near the surface as in beam bending. The shear stress, Oxy,changes sign near the raid-plane where contours 3 and 4 are located. This is expected. Figure 5 reveals that oe is highest near the contact where contour 10 is situated. Yielding, therefore, is expected to occur first in the local region where the metal comes into contact with the die. The constant contours of the strain energy density function are displayed in Fig. 6. A m a x i m u m of (dW/dV)min occurs near the top surface, near
,I!T
115 114
2
Fig. 4. Finite element grid pattern for sheet metal.
Y.J. Hw~ C.T. Hsu / Elastic-plastic deformation of sheet metal-forming Table 1 Constant stress and energy density contours in units of k g / m m 2 Contour
ox
Oy
oxy
oe
dW/dV
NO.
0 =18 ° 1 2 3 4 5 6 7 8 9 10
- 37.460 -23.315 -21.180 -13.046 -4.911 3.224 11.359 19.498 27.628 35.763
- 6.491 -4.910 -3.329 -1.749 -0.168 1.413 2.994 4.574 6.155 7.736
- 4.223 -2.571 -0.918 0.734 2.386 4.038 5.690 7.343 8.995 10.647
2.010 4.318 7.826 10.738 13.641 16.549 19.457 22.364 25.272 28.180
0.000 0.112 0.223 0.384 0.446 0.556 0.667 0.778 0.888 1.000
0 = 36 ° 1 2 3 4 5 6 7 8 9 10
-49.849 - 38.021 -28.194 -17.366 -6.589 4.289 15.116 25.944 36.771 47.599
-15.481 - 12.396 -9.310 -6.225 -3.140 -0.054 3.031 6.116 9.202 12.287
-13.507 - 10.427 -7.347 -4.267 -1.187 1.894 4.974 8.054 11.134 14.214
2.464 6.170 9.886 18.602 17.318 21.035 24.751 28.467 32.183 35.899
0.004 0.376 0.747 1.119 1.490 1.862 2.283 2.605 2.976 3.348
-63.036 -42.566 - 37.097 -21.627 -11.158 -0.688 9.781 20.251 30.720 41.190
-23.746 -19.196 - 14.647 -10.037 -5.548 -0.999 3.550 8.100 12.649 17.198
-18.434 -14.151 - 9.869 -6.586 -1.303 2.973 7.262 11.546 15.827 20.110
3.057 7.631 12.210 16.789 21.369 25.948 30.527 35.106 39.686 44.265
0.004 0.383 0.762 1.141 1.520 1.839 2.278 2.657 3.036 3.415
0 =
1 2 3 4 5 6 7 8 9 10
75
node 31 in Fig. 4, where the tension is also the largest. This is the location of (dW/dV)m~ and is predicted as the fracture initiation site. The results confirm with the strain energy density criterion assumption [4,5] that yield and fracture initiation do not occur at the same location.
4.2. Thirty-six degree rotation As the bending of the sheet is increased to 0 = 36 °, the stresses and energy densities increase accordingly, as indicated in Table 1. Again, both ox and Oy change sign near the mid-plane of the sheet. The sign change of Oxy occurred at locations closer to the lower portion of the metal sheet where contacts are made. As the contact surface now extends further away from the point (5, 0), the location of maximum distortion also changes accordingly. This is shown in Fig. 7. On the other hand, the location of [(dW/dV)m~]L in Fig. 8 remained near node 31, even though 0 is increased to 36 °
4.3. Forty-eight degrees of rotation
48 °
The qualitative features of the stress and energy distributions are preserved as the angle 0 is further increased to 48 o. This can be seen from the values of ox, Oy and Oxy in Table 1. The results on % and d W / d V for 0 = 48 ° did not change appreciably from those for 0 = 36 °. This can be seen from the contour plots in Figs. 9 and 10 or the data in Table 1.
10
Fig. 5. Constant contours of oe in elements 3 - 8 for 0 = 1 8 °.
76
Y.J. Hwt6 C T. Hsu / Elastic-plastic deformation of sheet metal-forming
~
_
.- MAXIMUM
/
DILATATION
Fig. 6. Constant contours of d W/dV in elements 3-8 for 0 = 18o.
4. 4. Contact length and energy density
The computed data for the increase in contact lengths with the angle of solution 0 are summarized in Fig. 11. The contact length increased slowly with 0 at first and then more quickly as 0 passed beyond approximately 30 o. This feature is also reflected by the variations of ( d W / d V ) m ~ near node 31 against 0 in Fig. 12. The energy density increased with 0 almost linearly up to 30 ° and then the curve flattened out to a constant value of approximately 3.4 k g / m m 2. This implies
i/4
~6
that the energy accumulation process almost ceased for 0 > 30 o. This is the range where most of the energy is used in forming the bend in the sheet. Additional rotation will not alter the energy state appreciably.
5.Concludinremarks g This work identifies the locations of potential yield and fracture initiation sites. The former correspond to the region of maximum effective stress
5
Fig. 7. Constant contours of oe in elements 3-8 for 0 = 36 °.
Y.J. Hwu, C 1", Hsu / Elastic-plastic deformation of sheet metal-forming
77
/--~---- MAX I MUM
Fig. 8. Constant contours of dW/dV in elements 3-8 for 0 = 36 °.
and the latter to the region of minimum strain energy density function, or (dW/dV)mm, where dilatational effect tends to dominate. Changes in the stress and energy states in metal-forming suggest that the effect of nordaomogeneous deformation might be significant. That is to say, strains
are not only concentrated in small regions but they depend on the change in the local strain rates and strain rate history. More precisely, the local behavior of each dement may depend on its previous history, whereas the dements at large may undergo small deformation. A wide spread of de-
'MAXIMUM
DISTORTION AT CON TACT
Fig. 9. Constant contours of oe in elements 3-8 for 0 = 48 °.
78
Y.J. Hwu, C T. Hsu / Elastic-plastic deformation of sheet metal-forming
/
------
MAX IMUM
DILATATION
2
~
2
Fig. 10. Constant contours of dW/dV in elements 3-8 for 0 = 48 °.
formation range and rate could exist that may not be adequately covered by a single constitutive relation. A scheme is needed to describe the behavior of each element by a different constitutive relation, in addition to accounting for changes in local strain rates and strain rate history. This necessitates a modification of the current theory of plasticity which may be summarized as follows: • Provide a scheme where the constitutive relations for each local element can be derived instead of preassigned.
f
E
/
~3.2~
• The derived constitutive relations should include the change in local strain rates and strain rate history. Such capabilities can be achieved by the development of a material data bank. The response of a particular element at a given time would be determined from knowledge of current strain c or % and rate i or ie- Here, % is an equivalent or effective plastic strain that can be related to an g-,
z4i "
I
z~2 f 0
o
12 °
24 °
36 u
48"
60 ~
r
ANGLE e ( d e g . ) ANGLE e (deg.)
Fig. 11. Contact length versus angle of rotation.
Fig. 12. Variations of local maximum of minimum strain energy density function near node 31 with angle of rotation.
Y.J. Hwu, C T. Hsu / Elastic-plastic deformation of sheet metal-forming
effective stress, oe. The details of the m e t h o d can be f o u n d in [6]. Application to metal-forming is being made.
Acknowledgement This work presents part of the results obtained u n d e r project TRC-78-25 supported by the China Steel Corporation. The senior author wishes to express his gratitude to the Vice President of the T e c h n o l o g y Division, Dr. Tsou Jo-chi, and General M a n a g e r of the Steel and A l u m i n u m Research and D e v e l o p m e n t Department, Dr. H u a n g Jong-Yuh, for their support and interest in this project. Special acknowledgement is also due to the Lehigh University Branch Institute of Fracture and Solid Mechanics in the R O C for using the P E R D A c o m p u t e r p r o g r a m to obtain the numerical results.
79
References [1] G.M. Goodwin, "Application of strain analysis to sheet metal forming problems in the press shop", Society of Automotive Engineers, Paper No. 680093 (1968). [2] S.S. Hecker, "Formability of high-strength low-alloy steel sheets", Metals Eng. Q. 13, pp. 42-48 (1973). [3] Y.J. Hwu and C.T. Hsu, "A large strain and elastic-plastic finite element analysis of stretching process", Sixth Nat. Conf. on Mech. Eng., CSME, Tainan, 1323-1333 (1989). [4] G.C. Sih, Introductory Chapters of Mechanics of Fracture, Vols 1 to VII, edited by G.C. Sih, Martinus Nijhoff Pubfishers, The Hague (1972-1983). [5] G.C. Sih, "Fracture mechanics of engineering structural components", Fracture Mechanics Methodology, edited by G.C. Sill and L.O. Faria, Martinus Nijhoff Publishers, The Netherlands, 35-101 (1984). [6] G.C. Sih and D.Y. Tzou, "Plastic deformation and crack growth behavior", Plasticity and Failure Behavior of Solids, Series on Fatigue and Fracture, Vol. III, edited by G.C. Sih, A.J. Ishfinksy and S.T. Mileiko, Kluwer Academic Pubfishers, The Netherlands (1990). [7] "Path evaluation and rate dependent analysis (PERDA) computer program", Institute of Fracture and Solid Mechanics, Lehigh University, USA (Dec. 1989). [8] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London (1950).