effects of shape and size of tensile specimens on the stress-strain relationship of sheet-metal
TRANSCRIPT
Journal o f Mechanical Working Technology, 7 (1982) 23--37 23 Elsevier Scient if ic Publishing Company , Ams te rdam -- Printed in The Nether lands
EFFECTS OF SHAPE AND SIZE OF TENSILE SPECIMENS ON THE STRESS--STRAIN RELATIONSHIP OF SHEET-METAL
T.N. GOH
Department o f Industrial and Systems Engineering, National University o f Singapore (Singapore)
and
H.M. S H A N G
Department o f Mechanical and Production Engineering, National University o f Singapore (Singapore)
(Received Sep tember 7, 1981; accepted in revised form March 30, 1982)
Industrial Summary
In the industrial forming of sheet-metals, knowledge of the mechanical proper t ies of the work-meta l is of major impor tance , as these proper t ies de te rmine the sui tabil i ty of the material for a part icular forming route , and the l imita t ions of the processes employed . Likewise, the effects of the shape and size of the specimens of the uniaxial tensile test - - by far the mos t c o m m o n l y employed test - - used to de termine the mechanica l propert ies need to be unders tood , in order that appropr ia te al lowances can be made for such effects.
In this paper is r epor ted an investigation carried o u t on the stress--strain relat ionship of sheet-metal derived f rom uniaxial tensile tests in which the process parameters , namely, specimen width , thickness, as well as the angle be tween the loading di rec t ion and the roll ing direct ion of the sheet, are varied. By f i t t ing the exper imenta l data to the familiar H o l l o m o n equa t ion , a = Ke n, and using statistical significance tests, it is found tha t values of the co- eff icients K and n are inf luenced by specimen thickness, loading direct ion, and width, in that order o f significance. Tha t the values o f work-hardening coeff ic ients in the Ludwik equat ion , o = ay + Ce rn , are dependen t on these variables is also observed. A mathemat ica l mode l based on the H o l l o m o n equa t ion is a t t empted to fur ther depic t the response of K and n to process parameter changes.
Introduction
The uniaxial tension test is perhaps the commones t testing technique for both ducti l i ty and strength of a material. Testing standards include specifica- tions for specimens having circular and rectangular cross-sections. For test specimens having a rectangular cross-section such as sheet-metal or plates, standard as well as non-standard specimens are permitted in both British and ASTM Standards Specifications [1, 2]. Perusal of these Standards shows that specimens having different widths~may be used: this implies that, for a given material undergoing the uniaxial tension test, the stress--strain diagrams ob- tained by using different specimen widths are expected to be identical.
0 3 7 8 - 3 8 0 4 / 8 2 / 0 0 0 0 - - 0 0 0 0 / $ 0 2 . 7 5 © 1982 Elsevier Scientif ic Publishing Company
24
The work-hardening characteristic of sheet-metal during forming is frequent- ly derived from the uniaxial tension test. In many published papers, the shape and size of the tensile testpiece are not mentioned [3--5]. Hsu et al. [6] observed that the maximum uniform strain, length of neck, maximum strain due to necking, and strain distribution in a fractured testpiece were affected by the shape and size of the testpiece. They also observed the change of failure mode with different specimen width-to-thickness ratios.
The present investigation attempts to determine how, and to what extent, the specimen width does affect the stress--strain relationship in the plastic range and before the onset of tensile instability. Since sheet-metals are rolled to various thicknesses, the effects of thickness and those of the angle between loading direction and rolling direction on the stress--strain curves are also examined.
The experimental data in this investigation are fitted by the familiar empir- ical expression frequently known as the Holloman equation [7] :
o = K e n , (1)
where K and n are constants; o and e denote, respectively, the true stress and true strain.
Experimental technique
The material used in this investigation was rolled copper sheets (H3101, half-hard) in the as-received condition. Specimens of various combinations of 8 different widths (7.5--25.0 mm in steps of approximately 2.5 mm)~ 5 thick- nesses (0.56--1.63 mm in steps of approximately 0.2 mm) and 7 loading directions (0--90 ° in steps of 15 °) were machined and tested by the Tinius Olsen Universal Testing Machine set to a loading range of 0--600 kg. The term "loading direction" refers to the angle between the longitudinal axis of the testpiece and the rolling direction of the copper sheet. The shape and dimen- sions of the testpieces used are shown in Fig. 1. The combinations of shapes and sizes were devised in accordance with a statistical experimental design for a three-way analysis of variance [8] described in a later section.
The gauge length was not taken as a parameter since its variation does not affect the stress--strain relationship before the onset of plastic instability [6]. In this investigation, a Tinius Olsen extensometer having a 50 mm gauge length and a sheet-metal a t tachment were used to record the load--extension curves for all the test specimens.
The true stress--strain curve was derived from the recorded load--extension curve in the following manner. From the condition of incompressibility, the relationship between the current and original cross-sectional areas of a test- piece is given by
W ' T ' = W T ( L / L ' ) , (2)
where W, T, and L denote, respectively, the original dimensions along the
25
r Rolling Direchon 12.5 R
- f ~Lo5O" ~ ~w 5
. . . . 2 6 0
AIr Linear Dimensions In ram.
Loading Direction, 8 Deg
0
15
30
/*5
60
75
90
7 Direct ions
Width, W mm
7.5
10.0
12.7
t5.0
17.0
20.0
23.0
25.0
B Widths Accuracy: 11±0.025}W
Nominal Thickness, T
SW6 mm
24 05588
22 0.7122
20 0.9114
18 1.2192
1 6 1. 6256
5 Thicknesses Accuracy
(1±O08)T
Fig. 1. Shape and dimensions of testpieces.
width , th ickness , and longi tudinal axis o f the spec imen during the test. W', T' and L' d e n o t e the current d imens ions during de format ion . The true stress a corresponding to a load P applied in the longi tudinal d irect ion may thus be expressed b y the equat ion
P a = - - exp (e), (3)
WT
where e is def ined as the true strain of a f ibrous e lement along the longi tudinal axis o f the testpiece , and it may be expressed as
e = In (L'/L)
= In (L + e)/L, (4)
where L d e n o t e s the 50 m m gauge length and e deno te s the e longat ion during testing.
Hence , the true stress and true strain corresponding to any po in t on the l o a d - - e x t e n s i o n curve could be calculated from eqns. (3) and (4) . Subsequent - ly, the coef f ic ients K and n in eqn. (1) were c o m p u t e d by the m e t h o d of least squares after l inearisation o f the equat ion . The port ion o f the l oad- - ex tens ion curve used for c o m p u t i n g these coef f ic ients was be tween t h e yield po in t and the m a x i m u m load - corresponding to instabi l i ty - - where the m a x i m u m strain was observed to be approx imate ly 0 . 1 3 in some spec imens and 0 .25 in others.
t~
TA
BL
E 1
Var
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on
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ess
T
(W/T
) (m
m)
(ram
) L
oad
ing
dir
ecti
on
0 (
deg
)
0 1
5
30
8.2
7
.5
0.9
11
4
K
36
9
n 0
.14
8
10
1
.21
92
K
3
42
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37
14
10
0
.71
22
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3
48
.9
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.16
6
12
.7
0.9
11
4
K
38
2
n 0
.16
0
17
1.2
19
2
K
34
2
n 0
.12
6
16
.4
15
0.9
11
4
K
38
8.7
n
0.1
51
20
1
.21
92
K
3
49
n
0.1
35
45
6
0
75
9
0
33
4.4
3
44
.6
34
4.3
3
44
3
38
.5
33
5.3
0
.12
1
0.1
26
0
.12
6
0.1
30
0
.12
5
0.1
22
33
8
33
8
33
8
32
8
33
2
33
4
0.1
33
0
.11
6
0.1
41
0
.12
5
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24
0
.11
8
37
3.7
3
50
3
48
.7
34
2
34
3
34
8
0.2
11
0
.18
0
0.1
72
0
.16
8
0.1
62
0
.17
0
34
2.7
3
52
.5
35
4.7
3
51
.2
35
4.3
3
49
.5
0.1
21
0
.13
5
0.1
33
0
.13
6
0.1
34
0
.12
7
33
9
33
0
32
2
32
5
33
0
33
4
0.1
23
0
.12
3
0.1
15
0
.11
3
0.1
18
0
.11
7
34
7.9
3
47
.6
35
5.6
3
46
.9
34
3.7
3
58
0
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2
0.1
28
0
.13
6
0.1
31
0
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0.1
36
33
5
33
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32
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33
1
32
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.13
2
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.12
9
0.1
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.11
7
27
Geometrical effects on K and n
The computed values of K and n are found to vary for the various combina- tions of process parameters. From the point of view of the geometry of test specimens these values may be grouped according to the aspect ratio, i.e. ratio of width to thickness. This is shown in Table 1, where K and n for differ- ent aspect ratios and loading directions are shown. The dependence of K and n on loading direction is obvious, and this will be discussed in a later section. It is also observed that for a given aspect ratio, different combinations of specimen width and thickness give rise to different values of K and n. This suggests that a one-to-one relationship between the work-hardening coeffi- cients, K and n, and the aspect ratio (W/T) is non-existent~ and that effects of specimen width and thickness on these coefficients may be examined separate- ly.
Effects of specimen w i d t h o n K and n
Figure 2(a) shows the variation of K with specimen width for a fixed thick- ness of 0.5588 mm and with loading direction as the running parameter. It is readily seen that, for all load directions, the value of K increases gently as the specimen width increases. This trend is elucidated by the solid line which represents the average values of K for each specimen width. Figure 2(b) shows also a gradual increase of K with specimen width for different thicknesses. Again, this response is illustrated by the average values of K for each specimen width.
The value of n is also found to increase gradually with specimen width, as seen from the solid lines indicating the average n values in Figs. 3(a) and 3(b).
Effects of specimen thickness on K and n
As shown in Fig. 4(a), the value of K increases with thickness to a thickness value between 22 SWG (0.7122 mm) and 20 SWG (0.9114 mm). As the thick- ness is further increased, the value of K drops before it increases again. This trend also exists for other specimen widths, as illustrated in Fig. 4(b).
Figures 5(a) and 5(b) show the variation of n with thickness. As thickness is increased f rom 24 SWG (0.5588 mm) to 22 SWG (0.7122 mm), the value of n increases; thereafter, it decreases up to a thickness of 1.2192 mm (18 SWG) before gradually increasing again with thickness.
Effects o f loading direction on K and n
The effect of loading direction on K is shown in Figs. 6(a) and 6(b). It is clearly seen tha t K decreases when the loading direction varies from 0 ° to about 45°; thereafter, up to a loading direction of 90 °, it remains practically constant.
28
400
380
%
z 360
3~0
3 2 0 - -
300
280
3 6 0 - - - -
340
32O
3OO
280 0
x 1 5 0
e 30 °
- - o L~S o •
[ ] 60 °
o 9 0 o _ _ • x
x
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o
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o
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15 20 25
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• 1 6 2 5 6 m m
× 1 2 1 0 2 m m
r O ' ) 1 ' ~ m m
® O ~ l l ? m m
, , 0 5 0 8 S mrn
15 20 2
W, mm
30
(a)
(b)
Fig. 2. Variat ion o f K wi th width, (a) for 24 SWG spec imens; (b) for 0 ° loading direct ion.
The variation of n with loading direction is such that n decreases when the loading direction deviates from 0°; but from 45 ° onwards, the change becomes gradual, as indicated in Figs. 7(a) and 7(b).
Statistical analysis o f effects of process parameters on the stress--strain curve
The experimental results indicate that the coefficients K and n are affected by the process parameters. To formally ascertain the statistical significance of the effects of these parameters beyond those associated with uncontrolled and
29
0.20
0.1B
,5.16 - -
0.I~
0.12
0.10
0
0.22
0.20
0.1B - -
0.16
014
0.12
0.10
I o
• o o
x 15 °
e 3 0 °
• L. 5 o
o 6 0 °
7 5 °
• 9 0 °
×
× I
x
9
o • o •
A o
', I I I 10 15 20 25
W, m m
T
• 1 6 2 5 6 m m
x 1 2 1 9 2 m m
o 0 9 7 1 i , m r n
0 . 7 1 2 2 m m
[ ] 0 5 5 8 8 m m
@
$
F ®
: , _- . I ° I o o
o o o
× ×
10 15 20 25
W, mm
(a)
(b)
Fig. 3. Variation of n with width, (a) for 24 SWG specimens; (b) for 0 ° loading direction.
random fluctuations, a three-way analysis of variance (ANOVA) was perform- ed with the tensile test data. The mathematical model for the three-parameter experiment is
Y i j k l = P . . . + ~ i . . + l~.j. + P . . k + P i j . + P i . k + P . j k + P i j k + e i j k l , (5)
where the indices i, j, k are for widths, thicknesses, and loading directions respectively; index l is for replicates with a given combination of the three process parameters. In this experiment, duplicate tests were carried out for each combination, i.e. l = 1,2. This was considered an adequate "within-cell" sample size as it provides a degree of freedom of 166 for estimation of random errors and also, for a given total number of tensile tests, leaves enough data points to cover sufficiently wide ranges of process parameter values. The terms
3 0
z
I 28o t i I i 04 06 08 10 1.2 14 16 1
T, mm
Z, 0"
38 %
E
z 36
3~
(a)
(b)
32
30
28 O.& 0.6 08 1.0 12 14 16 18
T mm
Fig . 4. Variat ion o f K wi th thickness , (a) for 7 .5 m m wide spec imens; (b) for 45 ° loading direct ion.
on the right-hand side of eqn. (5) stand successively for the mean, the three main effects due to the process parameters, the three two-parameter interactio~ effects, the three-parameter interaction effect, and the residual error.
Results o f statistical tests of hypothesis at 5% level of significance reveal thai the process parameters, namely thickness, loading direction and width, in- variably have statistically significant effects on the values of K and n, with thickness the most significant parameter and width the least significant (see Table 2). This suggests once more that the aspect ratio alone is not satisfactory in explaining K and n changes: if it were, both thickness and width should
0.20
n
n 022 I e ~
• 0 o
0.20 x iso -
o 3 0 °
45 °
o 60 °
018' ~ 7so
0.16 - - I A o 9o °
I / 0.14
0.12
010 04 06 08 1.0 12 1.4 1.6 18
T mm
I ' W
• 75ram
x IOmm 0.18
0.1z. ~ . / • o
012
0.10 0.4 0 6 0.8
31
8
10 1.2 1.4 1.6 1.8 T mm
(a)
(b)
Fig. 5. Variation of n with thickness, (a) for 7.5 mm wide specimens; (b) for 45 ° loading direction.
exhibit comparable statistical significance in influencing K and n values. It can be reasoned that when thickness is the overwhelming parameter, its effect will dominate even though width is adjusted to keep the aspect ratio constant.
The analysis of variance also provides information on possible interaction effects resulting from simultaneous variations of given combinations of process parameters. Among the four interaction effects, the thickness--loading direc- tion interaction has the strongest tendency to exert statistically significant in- fluence on K and n values; in fact, the magnitude of its test statistic (the F-val- ue} is always higher than that of the effect due to specimen width. The overall conclusion from the formal statistical analysis is that thickness and loading direction have both separate and joint effects on the coefficients K and n, while
32
400, a
I 3 80°~ -
3 6 o ' - -
3 4 0 - "
320 - -
300
280 2O
• 75 mm
x 10 mm -1 oo!, 3 8 O [ - -
~. 360
3401
320
300
280
x o
1~-- L I • ! 6 2 5 6 m m
* ; 2 ' g 2 m m
o O q ' l L , m m
* ',' s 1 2 2 m m
0 { 5 c : 8 8 m m
-t ~ °
&O 50 80 100 0 20 &O 60 80
(a) 0, Deg (b) e, Oeg
Fig. 6. Var ia t ion of K with loading direction (a) for 24 SWG spec imens ; (b) for 15 m m wide specimens.
100
018 n
0.16
014
012
0.10 o 20 40 6 o 80 lOO
(a) 8, Deg
° ! 1 0.20* . . . . . .
0 .18 . • O q S a S m T i
° I 0.16 ! ~ l - - - ~
0 1 4 ~ ~ ~ ]
010 _ _ 0 20 40 50 80 100
(b) 0, Deg
Fig. 7. Var i a t ion o f n with loading direction, (a) for 24 SWG spec imens ; (b) for 15 m m wide specimens.
w i d t h has a less p r e d o m i n a n t e f fect ; these e f fec t s have caused variat ions in K and n t o an e x t e n t that c a n n o t be exp la ined in t erms o f exper imenta l errors.
Modi f i ed stress--strain e q u a t i o n
Analys i s o f variance is usefu l to the e x t e n t that it es tabl ishes t h e statist ical s igni f icance o f t h e e f fec t s o f process parameters o n K and n, but the analys is
TABLE 2
F-values extracted from ANOVA tables for three-factor tests (Hollomon equation)
33
Parameters All 7 loading All 7 loading All 7 loading directions directions directions All 8 widths All 8 widths 4 widths (7.5, 3 thicknesses 3 thicknesses 10, 12.7, 15 mm) (20, 22, 24 SWG) (18, 20, 24 SWG) 4 thicknesses
(16, 18, 20, 24 SWG)
loading direction (0) K 18.2" 122.7" 77.8* n 34.9* 33.6* 22.1"
width (W) K 3.5* 8.5* 14.1" n 4.2* 7.7* 7.9*
thickness (T) K 37.7* 228.1" 172.0" n 642.8* 37.1" 36.2*
0 --W interaction K 1.0 1.8" 2.0 n 0.9 1.6 1.1
O--T interaction K 7.0* 25.3* 10.5" n 12.2" 21.0" 5.0*
W--T interaction K 1.1 1.8 1.4 n 1.3 4.1 0.8
o--W--T interaction K 1.5 2.3 1.6 n 2.1" 1.7" 1.3
*Denotes a significant F-value.
necessar i ly a s sumes a l inear addi t ive m o d e l fo r these ef fec ts , as ind ica ted b y eqn. (5), and does n o t b y i tself suggest a n y possible func t i ona l re la t ionsh ips b e t w e e n the process p a r a m e t e r s and K or n. The graphical analysis , descr ibed in t h e earl ier sec t ions , has p rov i ded n o t on ly a visual c o n f i r m a t i o n of the s ignif icance o f t he e f fec t s , b u t also revealed the non- l inear i ty of the unde r ly ing re la t ionships . T h e n e x t s tep in this s tudy , t he re fo re , is to exp lo re the possibili- t y o f bui ld ing a non- l inear m o d e l t o descr ibe the r e sponse o f K and n to p rocess p a r a m e t e r var ia t ions . A m a t h e m a t i c a l m ode l , based on the empir ica l s t ress- - s t ra in e q u a t i o n and i nco rpo ra t i ng the t w o m o s t s ignif icant pa r ame te r s , n a m e l y , th ickness and rol l ing d i rec t ion , could be p o s t u l a t e d fo r K and n. The shapes o f t h e K- a n d n-graphs suggest the fo l lowing m o d e l fo rms :
K = (a3T 3 + a2 T2 + a l T + ao)(O + 1) -~ exp(3'0 ),
n = (c~'3T ~ + a ' :T 2 + a ' lT + ao)(O + 1) -~' exp(7 '0 ), (6)
where T and 0 are s p e c i m e n th ickness in m m and loading d i rec t ion in degrees respec t ive ly , a n d a i , a~, ~, ~ ' , 7, 3", i = 0, 1, 2, 3, are cons t an t s to be e s t i m a t e d f r o m e x p e r i m e n t a l da ta ; n is a d imens ion less q u a n t i t y , and K is expressed in N / m m 2.
3 4
Since Figs. 5(a), (b) and others reveal unusually high values of n for the 22 SWG (0 .7122 mm) specimens, for simplicity in mathematical modelling, ex- perimental data for specimens of this thickness are treated as statistical outliers and are removed. Thus variations in K and n are assumed in this investigation to be linear with respect to thickness. Equations (6) are thus reduced to
K = (a lT + a0)(0 + 1) -~ exp(70) ,
n = ( c J l T + a'o)(O + 1) -~' exp( 7 '0 ). (7)
With the available experimental data, the constants in eqns. (7) were estimated by the method of least squares. This yields a modified equation for copper sheets with the effects of thickness and loading direction taken into consideration:
o = K e n,
where
K = (332T + 21.25)(0 + 1) -o.024 exp(0 .000720)
and (8)
n = (0 .13T + 0.015)(0 + 1) -°'°s2 exp(0 .000970) .
The standard errors for K and n are 18.51 and 0 .0124 respectively, estimated with a degree of freedom 192. The estimation errors are found to be approxi- mately normally distributed with zero mean in both cases, suggesting that the equations constitute a statistically adequate model for describing the experi- mental data.
T A B L E 3
T e s t r e s u l t s f o r p r e d i c t i v e a c c u r a c y o f m o d i f i e d H o l l o m o n e q u a t i o n
T h i c k - W i d t h L o a d i n g K - v a l u e s e r r o r n - v a l u e s e r r o r h e s s ( m m ) d i r e c t i o n (%) (%) ( S W G ) ( d e g r e e s ) o b s e r v e d a e s t i m a t e d b o b s e r v e d a e s t i m a t e d b
16
1 8
7.54 30 365 344 .47 - - 5 . 6 2 5 0 .1 3 5 0 .1336 - -0 .9 6 8 9.97 30 355 3 4 4 . 9 0 2 .845 0 .137 0 . 1 3 3 4 1.89
12 .63 0 370 3 6 6 . 3 4 - -0 . 9 3 9 0 .153 0 . 1 5 5 0 1.31 12 .83 15 376 384 .93 2 .370 0 .139 0 . 1 3 6 5 - -1 .88 14.9 0 372 3 6 6 . 0 4 - -1 .603 0 .1 5 5 0 . 1 5 5 0 0 .00
7 .54 45 316 339 .09 7.306 0 .119 0 . 1 2 7 8 7.39 9 .98 60 330 340 .93 3 .312 0 .120 0 .1279 6 .58
10 45 337 339 .09 0 . 62 0 0 .114 0 .1277 12 .01 12.6 75 333 3 4 3 . 0 0 3 .000 0 .124 0 .1283 3 .47 12 .79 45 352 338 .89 - -3 .724 0 .123 0 .1277 3 .82 14 .95 45 349 338 .89 - - 2 . 9 0 5 0 .126 0 .1277 1 .35 15 60 312 340 .93 9 .272 0 .133 0 . 1 2 7 9 - -0 .509 17 60 324 340 .93 4 . 94 0 0 .119 0 . 1 2 7 9 7 .48 20 30 348 336 .67 - -3 .256 0 .121 0 . 1 2 8 6 - -1 .76 23 30 345 336 .67 - -2 . 41 4 0 .143 0 . 1 2 8 6 - -1 0 .0 7 23 45 335 338 .89 1 .161 0 .1 3 4 0 . 1 2 7 7 - -4 .72
3 .456 c 4 . 0 7 5 c
a D e n o t e s v a l u e s d e d u c e d f r o m t h e u n i a x i a l t e n s i l e t e x t . b D e n o t e s v a l u e s c o m p u t e d f r o m e q n s . (8). C D e n o t e s a v e r a g e s o f a b s o l u t e v a l u e s o f p e r c e n t a g e error .
35
To ascertain the predictive accuracy of the modified equation, another 16 tests with specimens of various process parameters were performed, and the results, shown in Table 3, indicate that the errors incurred in estimating K and n values with eqns. (8) to be in the order of 4%, the maximum errors actually encountered being 9.27% and 12.01% for K and n respectively. No systematic over,estimation or under-estimation is evident.
Tests with the Ludwik equation
The study based on the Hollomon equation was also carried out with the widely used Ludwik equation [9],
0 = O y + C e m , ( 9 )
where Oy is the yield stress, and C and m are the work-hardening coefficients. The stress value at the limit of proportionali ty was taken to be the yield stress (Oy). Results of the analysis, summarised in Table 4, again indicate that thick- ness, loading direction, width and thickness--loading direction interaction have statistically significant effects on C and m, in that order of signifieanee. The patterns of variation of C and m with process parameters have also been observed to be similar to those of K and n, respeetively, and hence will not be further elaborated.
T A B L E 4
F-values e x t r a c t e d f r o m A N O V A tab les for t h ree - f ac to r tes t s ( L u d w i k e q u a t i o n )
Pa rame te r s All 7 loading d i rec t ions All 8 w id ths 3 th icknesses (20, 22, 24 SWG)
loading d i r ec t ion (0) C 17 .1" m 4.0*
wid th (W) C 2.3* m 2.2*
th ickness (T) C 84 .4* m 61 .4"
0--W in t e r ac t i on C 2.0 m 2.3
0 - - T i n t e r a c t i o n C 3.6* m 5.7*
W - - T i n t e r ac t i on C 0.9 m 1.3
o - - W - - T i n t e r ac t i on C 1.1 m 1.7
* D e n o t e s a s igni f icant F-value.
36
Discussions and conclusion
In this investigation, the variation of the true stress--strain relationship of copper sheets in the uniaxial tensile test with thickness, loading direction and width as process parameters has been studied through accepted procedures of formal statistical analysis. The study shows that thickness, loading direction and width have statistically significant effects, i.e. non-random influence, on the coefficients K and n of the Hollomon equation as well as on C and m of the Ludwik equation, and their order of significance follows that sequence. Moreover, if the interactions of these process parameters are considered, the interaction of thickness and loading direction tends to be a more significant factor than specimen width. Of the two coefficients K and n, the value of n is of more practical significance for obvious reasons.
The variations of n with specimen width and thickness may be attributed to the variation of strain paths due to the geometrical effects. In the uniaxial tensile t e s t a specimen having a large aspect ratio (W/T) would result in a strain path corresponding to the plane strain condition with zero strain in the trans- verse (W) direction, whilst the strain path for a small aspect ratio would corre- spond to the plane strain condition with zero strain in the thickness (T) direc- tion. The strain paths corresponding to the specimens used in this investigation would then lie between the two above-mentioned types of strain paths. The dependence of n on strain path [10] therefore explains the non-existence of a one-to-one relationship between n and the aspect ratio.
In addition to the geometrical effects, different loading directions and differ- ent rolling histories of sheet metals of different thicknesses would affect the material properties, and consequently the work-hardening characteristic. Any change in the loading direction implies a change in the angle between the direction of applied force and the grain axes so that both the strain path and the work-hardening characteristic are affected [11, 12]. The thickness effect observed in this investigation may be explained by the increase of limiting strain with n [13] and initial sheet thickness [14]. However, it is not known how the strain path is affected by the rolling history.
As for patterns of changes in magnitude, it is observed that n increases with specimen thickness and width but decreases as the angle 0 between the rolling and loading directions increases. The low statistical significance of the effect of specimen width on n is also corroborated by an earlier investigation [15]. The decrease of n with an increasing 0 shows that the stress necessary to cause the same strain is the highest for 0 = 0 °. This phenomenon has also been observed previously [11], but in some cases maximum n occurs at 0 = 45 ° [12].
By leaving out specimen width so as to simplify the mathematical modelling the variation of K and n may be expressed by eqns. (8) with reasonably good accuracy. However, it has not been the intent of this investigation to postulate definitive or universal equations for describing the stress--strain relationship of sheet-metal under uniaxial tensile loading; the objective is to explore the
37
s i g n i f i c a n c e o f t h e e f f e c t s o f p r o c e s s p a r a m e t e r s a n d t h e f e a s i b i l i t y o f ex - p r e s s i n g K a n d n in t e r m s o f t h e s e p r o c e s s p a r a m e t e r s . F u r t h e r g e n e r a l i s a t i o n s c a n b e e s t a b l i s h e d o n l y a f t e r m o r e e x t e n s i v e s t u d i e s w i t h d i f f e r e n t m a t e r i a l s .
A c k n o w l e d g e m e n t s
T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e t h e e n c o u r a g e m e n t o f P r o f e s s o r T .C. Hsu~ D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y o f A s t o n in B i r m i n g - h a m , U . K . , a n d P r o f e s s o r B. B a u d e l e t , U n i v e r s i t 6 d e M e t z , F r a n c e , d u r i n g t h e c o u r s e o f t h i s s t u d y .
R e f e r e n c e s
1 British Standards Specifications BS 18: Part 1, 1970. 2 ASTM Standards ANSI/ASTM E8-79. 3 J. Chakrabarty, A theory of stretch forming over hemispherical punch heads. Int. J.
Mech. Sci., 12 (1970) 315--325. 4 B. Kaftanoglu, Plastic instability of thin shells deformed by rigid punches and by
hydraulic pressure. ASME Winter Annual Meeting, New York, Paper No. 72-WA/MAT-4, 1970.
5 J. Chakrabarty and J.M. Alexander, Hydrostatic bulging of circular diaphragms. J. Strain Anal., 5 (1970) 155--161.
6 T.C. Hsu, G.S. Litt lejohn and B.M. Marchbank, Elongation in the tension test as a measure of ductil i ty. Proc. ASTM, 65 (1965) 874--898.
7 H.H. Hollomon, Tensile deformation. Trans. AIME, 162 (1945) 268--290. 8 R.E. Walpole and R.H. Myers, Probabili ty and Statistics for Engineers and Scientists
(2nd edn.). Macmillan, New York, 1978, Chapter 11. 9 P. Ludwik, Elemente der Technologischen Mechanik, J. Springer, 1909, p. 32.
10 J.M. Jalinier, J.H. Schmitt, R. Argemi, J.L. Salsmann and B. Baudelet, Different damage behaviours and their influence on forming process. M~m. Sci. Rev. M~tall., 77(3) (1980) 313--325.
11 A.N. Bramley and P.B. Mellor, Some strain-rate and anisotropy effects in the stretch- forming of steel sheet. Int. J. Mach. Tool Des. Res., 5 (1965) 43--55.
12 K.J. Pascoe, Directional effects of prestrain steel. J. Strain Anal., 6 (1971 ) 181--184. 13 Z. Marciniak and K. Kuczynski, Limit strains in the processes of stretch-forming sheet
metal. Int. J. Mech. Sci., 9 (1967) 609--620. 14 J.H. Schmitt, J.M. Jalinier and B. Baudelet, Effect of initial thickness of sheets on the
F.L.D. 's at necking, International Symposium on New Aspects on Sheet Metal Form- ing, Tokyo, Japan, May 14--15, 1981.
15 Private discussion with B. Baudelet, Universit4 de Metz, France, 1981.