- Email: [email protected]
Tensile plastic instability of thin tubes—II
Int. d. Mech. 8el. Pergamon Press
TENSILE
Ltd. 1965. Yol.7, pp. 539-549. Printed in GreatBritain
PLASTIC
INSTABILITY
OF
THIN
TUBES--II
M. J . HILLIER Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario (Received 3 N o v e m b e r
1964)
Summary--The results of calculations on the loads and strains at instability for tubes subjected to various loading paths are presented. A critical comparison is made with simple theory and with experiment. I t is concluded that, as expected, the loads and strains at instability will not in general correspond with those at fracture. Further tests are required in order to choose between alternative theoretical formulations of the problem. 1. I N T R O D U C T I O N THE i n s t a b i l i t y of long t h i n t u b e s s u b j e c t e d to a n i n t e r n a l pressure p a n d a n i n d e p e n d e n t axial l o a d P has b e e n considered in t h e p r e c e d i n g P a r t I of this p a p e r . A m e t h o d of c o m p u t a t i o n was outlined which enables t h e loads a n d t o t a l strains a t i n s t a b i l i t y to be d e t e r m i n e d for (a) sequential loading a n d (b) p r o p o r t i o n a l loading. W h e n loading in sequence the t o t a l axial load Pz m a y be applied alone u p to an assigned c o n s t a n t value, followed b y increasing i n t e r n a l pressure. A l t e r n a t i v e l y , t h e pressure a n d i n d e p e n d e n t axial load P m a y be applied first in such a m a n n e r as to hold P~ zero, a n d t h e t o t a l axial l o a d m a y t h e n be increased s u b s e q u e n t l y , while k e e p i n g t h e pressure c o n s t a n t . N o t e t h a t t h e initial loading of t h e sequence is essentially uniaxial a b o v e or below t h e yield load. I f a b o v e t h e yield l o a d it w o u l d be u n r e a s o n a b l e to e x p e c t a simple t h e o r y to a p p l y directly. I t w o u l d t h e n be n e c e s s a r y to consider t h e d e f o r m a tion as t a k i n g place in t w o stages. T h e t o t a l e q u i v a l e n t plastic strain a t t h e end of t h e first stage would serve to a d j u s t t h e v a l u e of t h e empirical c o n s t a n t B in t h e relation = A(B+g)
~
T h a t is, for the second stage of biaxial stressing, B would be r e p l a c e d b y B + g l , where gl is t h e t o t a l s t r a i n a t t h e end of t h e first stage of sequential loading. I f t h e initial loading is below t h e yield p o i n t value, a simple t h e o r y applies directly, since o n l y one p h a s e of plastic d e f o r m a t i o n would t h e n exist. I n t h e case of p r o p o r t i o n a l loading t h e r a t i o p / P ~ is held c o n s t a n t . T h a t is, the r a t i o of t h e n o m i n a l stresses, b a s e d on original cross-sectional areas, is held constant. 2. S E Q U E N T I A L
LOADING
Table 1 refers to the calculated loads and strains at instability for the case where the pressure is applied first. The value of B" in the second column is the yield point value of the dimensionless stress 80~ acting alone. The values of s0¢ given in the third column are the assigned constant values of circumferential stress, g, e¢, ez are the integrated total 539
M. J . H I L L I E R
540
TABLE 1 B
0.05
0.10
0"10
0'20
n
So4~
~
e¢
0.20 (0.549)
0.1 0.2 0'3 0.4 0.5
0.146 0.152 0.168 0.192 0.194
--0.055 --0.037 --0.016 40'015 40'055
0.144 0.146 0.152 0.158 0.133
0-665 0-700 0.726 0.741 0.728
0.135 0.265 0.400 0.555 0'766
0"30 (0.407)
0.1 0"2 0.3 0.4
0.242 0.254 0.290 0-322
--0-086 -- 0.049 0.000 + 0-079
0.238 0-240 0.250 0.230
0"583 0.619 0"650 0.663
0-144 0.293 0.460 0.705
0.40 (0.275)
0.1 0"2 0.3 0-4
0'338 0.360 0.428 0.096
--0.114 -- 0"050 + 0.048 + 0.096
0.332 0.332 0-341 -- 0.048
0"528 0.569 0.611 -- 0-0025
0.151 0.318 0.541 ( -- 0.005)
0'50 (0.224)
0-1 0.2 0"3
0.436 0.472 0"060
-- 0" 137 -- 0.035 40'060
0.425 0-421 --0"030
0.490 0.540 --0"030
0" 155 0.346 (--0.013)
0.20 (0.631)
0.1 0-2 0"3 0.4 0'5 0"6
0"096 0.100 0.116 0-138 0"150 0"118
--0.037 --0"026 --0"015 40"004 +0"031 40'048
0-095 0"096 0.107 0.117 0"111 0'068
0.698 0.732 0.757 0.770 0"763 0"712
0.133 0"259 0.384 0.524 0"695 0"926
0"30 (0"501)
0"1 0"2 0"3 0"4 0"5
0-192 0-204 0"234 0.272 0"226
--0"071 --0"045 -- 0'012 +0"042 +0'094
0'190 0"194 0"208 0"211 0'130
0"611 0'646 0"674 0"887 0"648
0"142 0"283 0.435 0.634 0"935
0"40 (0"398)
0"1 0'2 0"3 0"4
0"288 0-308 0"362 0"400
--0-101 -- 0"054 + 0"012 + 0"004
0"284 0"289 0'306 -- 0"002
0'552 0'590 0"625 -- 0"004
0"50 (0"316)
0"1 0"2 0"3 0"4
0"386 0"414 0"504 0.156
--0"129 -- 0"054 +0'067 +0.156
0"379 0"381 0"396 --0"078
0"512 0"555 0"602 --0' 001
0"2 (0"457)
0'1 0.2 0-3 0"4 0"5 0"6
0"002 0'002 0"002 0.002 0.002 0"002
--0"001 -- 0"001 -- 0"001 -- 0.001 -- 0-001 -- 0"001
0"002 0"002 0'002 0"002 0"002 0"002
0"769 0.804 0"826 0"836 0"831 0"805
0"130 0"248 0-362 0-478 0-599 0"747
0"3 (0"309)
0"1 0"2 0"3 0.4 0"5 0"6
0"092 0.102 0"126 O"160 0"170 0"106
--0"035 --0"026 --0"014 + 0.008 40"041 40"048
0"091 0"098 0"115 O"134 0"122 0"057
0"673 0.706 0"730 0.742 0"730 0"663
0'139 0"269 0.400 0.546 0.74I 0"996
0"4 (0"209)
0"1 0"2 0-3 0"4 0"5
0"190 0"204 0"244 0"296 0"230
-- 0'070 -- 0"045 -- 0-013 +0"045 + 0"093
0"188 O"195 0"217 0"230 0"134
0"608 0"642 0"670 0-684 0"643
0"143 0-284 0"436 0"641 0-935
0"5 (0.141)
0"1 0"2 0-3 0"4
0"286 0"308 0"370 0"408
-- 0"102 --0"059 + 0"001 +0"110
0"282 0.291 0"319 0"284
0"562 0"599 0-634 0"647
0"145 0-296 0.474 0"770
(l~)
ez
Soz
(lio<)
0"148 0'304 0'492 ( -- 0"0096) 0-151 0"324 0"571 (0"505)
Tensile plastic instability of thin tubes--II
541
TABLE 2 B
n (B '~)
soz
~
e~
0.05
0.2 (0.549)
0.1 0.2 0"3 0.4 0-5 0"6
0.070 0.062 0"062 0"070 0"080 0"052
0-069 0.059 0"056 0"057 0"062 -- 0"026
0"3 (0"407)
0"1 0"2 0"3 0"4 0"5
0'126 0"116 0"120 0"140 0"082
0"4 (0.275)
0"1 0.2 0"3 0"4
0"5 (0-244)
0"10
0"10
0-20
ez
so~
a
--0.026 --0.014 -- 0-005 +0"006 +0"023 + 0"052
0.624 0.650 0"666 0"670 0"659 -- 0.0002
7.16 3.66 2'48 1"877 1'491 -- 0.0039
0.124 0"109 0"102 0"101 -- 0"041
--0-044 --0"020 + 0"003 + 0"032 + 0"082
0"519 0"539 0"551 0"553 -- 0"003
6'65 3"35 2"25 1"69 -- 0"005
0"184 0"170 0"180 0"074
0"I80 0.155 0"140 -- 0.037
--0"059 -- 0"018 +0"024 + 0"074
0"445 0"462 0.474 -- 0"004
6.38 3"15 2'09 -- 0.009
0"1 0"2 0"3 0"4
0"240 0"224 0"052 0"180
0'233 0"195 --0"026 --0"090
--0"070 --0"007 +0"052 +0"180
0"391 0"041 --0"005 --0"002
6"22 3"00 --0"016 --0"005
0"2 (0"631)
0"1 0"2 0"3 0-4 0"5
0"018 0"012 0'012 0"020 0'036
0"018 0.012 0.011 0"017 0"027
--0"007 --0"003 -- 0"001 +0'001 +0" 007
0"676 0"707 0"726 0"730 0"717
7.01 3"62 2"47 1-89 1.51
0"3 (0"501)
0-1 0"2 0'3 0'4 0"5
0"076 0"066 0"070 0"088 0"122
0-075 0"062 0"061 0"068 0"076
--0"027 --0-013 -- 0-002 +0"014 +0"043
0"563 0"587 0"600 0-600 0"583
6"54 3'33 2'26 1"72 1"36
0"4 (0-398)
0-1 0"2 0"3 0"4 0"5
0"132 0"120 0"130 0"002 0-174
0-130 0"111 0"108 --0"002 --0'087
--0"045 --0"018 +0"007 +0"004 +0-174
0"484 0-503 0-513 --0"005 --0"001
6"27 3"14 2"12 0-0116 --0.0019
0"5 (0"316)
0"1 0"2 0"3 0"4
0'190 0"176 O"190 0"096
0.186 0.160 O"147 -- 0'048
--0'060 --0"017 + 0"028 + 0"096
0"425 0"441 0.451 -- 0'003
6"16 3"03 2-02 -- 0.0063
0"2 (0"457)
0"1 0"2 0-3 0"4 0"5 0"6
0*002 0"002 0"002 0"002 0"002 0"002
0"002 0-002 0-002 0"002 0"002 0-001
--0"0008 --0"0006 -- 0"0003 -- 0-0005 + 0"0002 + 0"0005
0'769 0.803 0'825 0"835 0.830 0'805
7"72 4"03 2"76 2" 10 1.67 1.34
0"3 (0.309)
0.1 0-2 0'3 0.4 0.5 0.6
0.002 0"002 0.002 0"002 0.016 0"062
0.002 0.002 0.002 0.002 0.011 0"036
-- 0"0007 -- 0-005 -- 0.0002 + 0" 0001 +0'0039 +0"026
0-661 0.692 0.709 0.710 0.690 0.646
6.63 3.47 2-37 1.78 1.41 1.16
0-4 (0.209)
O' 1 0"2 0-3 0.4 0.5
0'032 0'020 0"030 0-056 0"102
0.032 0.019 0.026 0.043 0"065
-- 0.011 --0'004 -- 0.0006 + 0.009 + 0"036
0.570 0.597 0.609 0.604 0-582
6.07 3.10 2-14 1.65 1.33
0.5 (0-141)
O. 1 0.2 0"3 0"4 0"5
0"090 0.076 0"090 O"128 0"116
0.088 0.071 0"076 0"092 --0"058
0.502 0.524 0'532 0'528 --0"0013
5.99 3-01 2"07 1.59 --0"0024
-- 0.031 --0"012 + 0'003 + 0"031 +0"116
542
M.J.
HllmmR
strains a t the p o i n t of instability, s0= is t h e dimensionless nominal axial stress at instability and ~ t h e corresponding t r u e stress ratio s~,/s,. F o r values of 8o4 r a t h e r larger t h a n those t a b u l a t e d , instability occurs first due to uniaxial stress alone. E l e m e n t a r y t h e o r y predicts t h e following m i n i m u m v a l u e of engineering stress for which this s i t u a t i o n occurs: 80¢(crit)
----- n n
exp [ - ( n - - B ) ]
(l)
Fig. 1 plots this critical n o m i n a l stress for a range of values of n and B. T a b l e 2 presents t h e results of calculations for t h e case where t h e sequence is r e v e r s e d ; stress 8o= being applied first, a n d s0¢ second. The assigned c o n s t a n t values of dimensionless circumferential stress are t a b u l a t e d in c o h m m 3. The critical v a l u e of s0z at which yield first occurs in simple tension is again g i v e n b y t h e r i g h t - h a n d side of e q u a t i o n (1) and
Fig. 1. So long as instability does n o t first occur in simple tension t h e c o m p u t e d z--a c u r v e agrees w i t h t h e e l e m e n t a r y t h e o r y of P a r t I to w i t h i n 1 per cent.
o.s 0.7
\
0,7
~
0.3
o.z
~~0.6 0.5} ~ ~'--~
---'----- ~
0.6
\
0.4
0.8
- -
0.6
0.3
-------- -----~'- "-'-'--- 0.2
J.
0.5
So~ (crit)
0.4 0.20
~
0.10
02 t
0.3
0.2 I i 0 . 2
0.2
06
0.05 03
0.0 02
-"
~
------'~ -------~ -----'---
0.6
0.4
~
I
0.0
0.3
0.4
0.5
0
R
0.05
QIO
0~5
0.20
B
F i e . 1.
F I e . 2. T r u e stress ratio at instability.
3. P R O P O R T I O N A L
LOADING
Figs. 2 t h r o u g h 6 s u m m a r i z e t h e results of calculations w h e n t h e ratio of pressure to t o t a l axial load is held constant. This ratio is expressed here as C = a ~ / a o z = 21rr*o P / P z
T h e critical v a l u e of t h e t r u e stress ratio at instability is p l o t t e d as a f u n c t i o n of B, n a n d C in Fig. 2. Figs. 3(a) t h r o u g h 3(c) plot t h e corresponding v a l u e of dimensionless n o m i n a l axial stress so= = _P,/(2rrr o t e a ). The t o t a l t h i n n i n g strain a t instability is p l o t t e d in Figs. 4(a) t h r o u g h 4(e); a n d Figs. 5 a n d 6 p l o t t h e corresponding ratios of t h e t o t a l strains. T h e c o m p u t e d z - a c u r v e for this loading path, Fig. 7, does n o t correspond so closely to simple t h e o r y as for t h e p r e v i o u s case considered. I n particular, for a greater t h a n u n i t y t h e r e is no correspondence b e t w e e n t h e c o m p u t e d results and those of simple theory. A
543
Tensile plastic instability of thin t u b e s - - I I
similar comparison between Swift's theory z and computed results is also shown: a discrepancy again appears. The method of computation in this latter case will be outlined in the next section. There remains a further discrepancy for values o f ~ less than unity. The possibility of error in numerical integration was checked b y halving the increment in g in the step-by-step computation. The identical discrepancy remained.
0.9
0.9 n=0.2
0.8
C=
So~
0.4
n=0.3
~
0.8
Soz
_...:
~
.
-:I0.6 0.5 0
./ -11.0
o s :222::i
C=~3.4 i.~
~'"
./-J.o
0.5
OD5
0.1o
0.15
0
CL20
0.05
0.10
B (a)
B
0.15
0.20
(b) O.S n=0.5 0.7
0.8
C=
n=0.4
0.6
So=, 0.6
0.7 C = ~ . ~ _ ~ 0.4
Soz 0.6
0.5
0.5
0.4
0.4 o
0.05
0,10
0.15
f
0
005
0.10
B
0.15 B
(e)
(d)
0.8 n=0.6
0.7 ci 0.6 Soe
o.
0.5 L0/ 0.4
O.3 0
/ - f
0.05
~°
0,10
0.15
0.20
B
(e) FzO.
3.
1,0
j°J
0.5
0.20
/
Nominal axial stress at instability (dimensionless).
0.20
544
M.J.
HILLIER
0.4
0.4
n=0.3i n =0.2
!
02
0.5
-e r er 0.2
0.~
......
0.6
~'"
0.1
0.1
0.05
0`10
0.15
0.C6
0.20
I
0`10
0,15
0,20
B
B
(a)
(b) 0.6
""'-..
n= 0.5
0.5
0.5 n =0.4
0.4""-.
0.4
-el" 0.3
-er 0.2
0,2
o.!
0,1
0,05
(210
0`15
"~'~"
0.05
0,20
0,1o
0.15
B
B
(d)
(c) 0.7
""
0.6
n.0.6
~.6
0.5 -e r 0.4
Q3 ~-
'1 0
0.2
~ 0
0`05
0`10
0,15
0.20
B
(e) F I G . 4. T h i c k n e s s
strain at instability.
0.20
545
Tensile plastic instability of thin tubes---II 1.8
0.8 R= 0.6 .......
C
I1==
U.t)
1.6 . . . . . .
0.4
0.2
C: 0.6
0.2
0.4
0.2
0.4
0.4
0.4
0.6
_ _ 0.4
0.8
0.4
1.0
0.2
0.4
1.4
0.2
0.6
0.4
02
0. =-
I.Z
0.4
o.z
er 0
I
e=
0.2 0.4 -0.2 . . . . . . . .
1.0
- e"~
02
0.6
~
0.6
0.8
0.6"
0.2
0.2 ~
~
n
a
n=
0.6 =.. . . . .
r .....
-o4
..... -0.6
~----~
L.....
~
,0.4
~'--'-,--
:.0
~
U.q5
--~.---.----~
0 4 __.~1 ~
-0.8
0
.
6
0.2
- 1.0 0
Q05
(210
(215
G20
0
0.05
OJO
B
0.15
0.20
B
F r o . 5. S t r a i n r a t i o a t i n s t a b i l i t y .
F I e . 6. S t r a i n r a t i o a t i n s t a b i l i t y .
1.4 COMPUTED
RESULTS
-----~P~=o . . . . .
dP=O
0.8
O~
0.5 0
0.5
LO
,C
0.5
V&
F r o . 7. Critical s u b t a n g e n t a t i n s t a b i l i t y .
3. C O M P A R I S O N
WITH
EXPERIMENT
T h e m o s t useful a n d r e l e v a n t e x p e r i m e n t a l r e s u l t s a r e t h o s e d u e t o D a v i s o n t h e b u r s t i n g o f c o p p e r S a n d steel s t u b e s . T h e t e s t s were c a r r i e d o u t u n d e r t h e c o n d i t i o n s o f p r o p o r t i o n a l loaxiing c o n s i d e r e d in t h e p r e s e n t p a p e r . I t m u s t b e e m p h a s i z e d , h o w e v e r , t h a t t h e m e a s u r e m e n t s were m a d e o n t u b e s a t f r a c t u r e a n d n o t a t i n s t a b i l i t y . H e n c e a close correlation between present theory and experiment cannot be expected. In particular, for r e l a t i v e l y low pressures, t h e b e h a v i o u r o f t h e t u b e s w o u l d a p p r o x i m a t e t o s i m p l e t e n s i o n ; t h e localized t h i n n i n g o f t h e t u b e s u b s e q u e n t t o i n s t a b i l i t y a n d p r e c e d i n g f r a c t u r e is n o t negligible. T h u s , for e x a m p l e , i t w o u l d b e e x p e c t e d t h a t t h e t o t a l t h i n n i n g a t fracture would be greater than that predicted by any instability theory.
546
M.J.
1-[ILLIER
T a b l e 3 s u m m a r i z e s t h e e x p e r i m e n t a l r e s u l t s a t f r a c t u r e , t a k e n f r o m t h e references c i t e d a n d a d d i t i o n a l i n f o r m a t i o n s u p p l i e d b y D r . E . A. D a v i s . T h e e x p e r i m e n t a l s t r e s s s t r a i n c u r v e s for t h i n t u b e s i n s i m p l e t e n s i o n g i v e n i n t h e a b o v e references were f i t t e d b y t h e e m p i r i c a l r e l a t i o n 5 = A ( B + $)n, a n d t h e following r e s u l t s o b t a i n e d : Material
A (lb/in 2) 110,000 45,000
Steel Copper
B
n
0.00 0.02
0.23 0.58
TABLE 3. EXPERIMENTAL RESULTS Nominal stress ratio ~o~/(7o~
Inside diameter at fracture (in.)
Tube wall a t fracture (in.)
M e d i u m c a r b o n steel I n i t i a l o u t s i d e d i a m e t e r 1.45 in. I n i t i a l t u b e t h i c k n e s s 0' 10 in. 0-0 0.902 0.064 0.500 1.302 0.060 0.750 1.566 0.0505 0-762 1.567 0.052 0.775 1-510 0.065 0.800 1-480 0.070 0-875 1.444 0.078 1-00 1.489 0.078 1/0-5 1.430 0.087 1/0-0 1.500 0.090
Load P at fracture (lb)
Pressure p at fracture (lb)
T r u e stress ~4, a t fracture (lb/in s)
T r u e stress ~z a t fracture (lb/in 2)
21900 20800 15100 15150 16000 I5500 14400 11800 0 --
0 5100 6560 6740 7290 7450 7980 8280 8650 8580
0 55200 101400 101400 84500 78500 74500 79000 71200 72000
113000 107500 108300 106600 90600 83000 74400 68300 33500 10400
12600 11600 10900 10200 8250 5200 0
0 1235 1860 2500 3590 3660 3760
0 15800 25400 33000 38200 42000 39200
68900 63000 67800 63600 43700 35500 18700
Copper I n i t i a l o u t s i d e d i a m e t e r 1.45 in. I n i t i a l t u b e t h i c k n e s s 0.10 in. 0.0 0-910 0.060 0.250 1-225 0.048 0.375 1.281 0.047 0.500 1.318 0.050 0.750 1.449 0.068 1.00 1.546 0.067 1/0.5 1.570 0.075
T a b l e s 4 a n d 5 give t h e c o r r e s p o n d i n g e x p e r i m e n t a l v a l u e s o f a, so,, - e , e4,/er, %/e t c a l c u l a t e d f r o m t h e d a t a o f T a b l e 3. T h e t h e o r e t i c a l v a l u e s c o r r e s p o n d i n g t o t h e calculat i o n s o f S e c t i o n 3 a r e t a b u l a t e d u n d e r t h e h e a d i n g " T h e o r y ( v ) " . T h e case B = 0.00 w a s c o m p u t e d b y s e t t i n g B = 0.002; a n y r e s u l t i n g e r r o r w a s j u d g e d t o b e small. F o r c o m p a r i s o n a f u r t h e r s e t of c a l c u l a t i o n s c o r r e s p o n d i n g t o t h e c o n s t r a i n t s o f S w i f t ' s t h e o r y , as d i s c u s s e d i n P a r t I o f t h e p r e s e n t p a p e r , is g i v e n i n c o l u m n s h e a d e d " T h e o r y (i)". T h e i n s t a b i l i t y c o n d i t i o n a s s u m e d h e r e is dp = d P = 0. I n t h i s case dPz is n o n - z e r o a t i n s t a b i l i t y , a n d i t m a y b e s h o w n t h a t t h e i n s t a b i l i t y c o n d i t i o n dso, ~<0 m u s t now be replaced by the condition
dso, <~CSo, e x p (2e4,) de4, As e x p e c t e d , t h e r e is o n l y q u a l i t a t i v e a g r e e m e n t and experiment. The experimental thinning strains than the theoretical thinning strains at instability. r o u g h l y t h e s a m e for b o t h t h e o r i e s a n d for e x p e r i m e n t .
i n t r e n d s , as b e t w e e n e i t h e r t h e o r y at fracture are substantially larger T h e g e n e r a l t r e n d o f t h e r e s u l t s is A l t h o u g h t h e v a l u e of t h e e m p i r i c a l
547
Tensile plastic i n s t a b i l i t y o f t h i n t u b e s - - I I TABLE 4
O~ =
ao = s~/S,o
80~
84ff8 ~
Theory
Theory Exp.
Exp.
(i)
(v)
(i)
(v)
0-566 0"677 0"652 0"648 0.643 0.633 0"599 0"540 1.15
0"566 0"677 0"658
0.408 0"463 0-489 0.523 0"470 0.335 0"119
0.408 0"463 0"490 0"523 0-481 0.345
Steel 0-0 0.50 0.75 0.762 0.775 0.80 0"875 1.0
0.0 0.50 0.852 0.868 0.883 0.915 1/0.997 1/0.876 I/0.492
1/o.5 1/o.o Copper 0.0 0.25 0"375 0.50 0.75 1.0
1/o.5
0.0 0-523 0.936 0-952 0-937 0.946 1/0.999
0-0 0.50 0-898 0.933 0.968
1/0.865
1/0.828
1/0.471 1/0-144
0.0 0.172 0-299 0.5
0.0 0-176 0.305 0.50
0-991
1/0.905
1/0.772 1/0.428
1/0.697
0.0 0"232 0"374 0"519 0.874
1/o.841 1/0.477
TABLE
O~0
Theory
;reel 0.0 0.50 0.75 0.762 0.775 0-80 0.875 1.0 1/0.5 1/0.0 ;opper 0-0 0.25 0.375 0-50 0.75 1.0 1/0.5
36
(v)
0.115 0.229 0.193 0.190 0-184 0.177 0.153 0.121 0.007
0.115 0-229 0.259
0-28 0.392 0.471 0.563 0.369 0.219 0.072
0.248 0.239 0.166
0.28 0-369 0.440 0.563 0.470 0.288
0"547
0.660 0"684 0"696 0"713 0-741 0"631 0.380
5
e~/e.
Theory
Theory
Exp. (i)
0-649 0.640
e¢/e.
-- e r
0"47 0.593 0"595 0"603 0.623 0.593 0"589 0-562 0.298 0-0
Exp. (i)
(v)
0.446 0.551 0.683 0.653 0.425 0.358 0.248 0.248 0.140 0.105
1.0 0.00 --0-330 --0-343 0-355 --0.381 --0.449 --0.547 -- 1.00
1.0 0'00 --0.347
0.511 0.733 0.755 0.693 0.386 0.400 0.287
1.0 0.479 0.241 0.00 --0.379 --0.591 -- 1.05
1.0 0.474 0.237 0.00 --0.588 --0.376 --
--0.374 --0.40 --0-567
Exp. (i)
(v)
0.751 --0.018 --0.260 --0.277 --0.362 --0.441 --0.484 --0.597 --0.822 -- 1.553
--2.0 -- 1.00 --0.670 --0-657 --0-644 --0.619 --0.551 --0-453 --0.034
--2.0 -- 1-00 --0.653
0.648 0.079 0-023 --0.017 --0-298 --0-442 --0-686
--2.0 -- 1.48 -- 1-24 -- 1.00 --0.621 --0.409 --0.047
--2-0 -- 1-47 -- 1-24 -- 1-00 --0.412 --0.624
--0.626 --0.600 --0.433
--1.975 --0.982 --0-739 --0-722 --0.638 --0"559 --0"516 --0.403 --0.178 + 0.55 --1-650 -- 1.080 -- 1-022 --0.982 --0.702 --0.558 --0-314
548
M.J.
HILLIER
index n for copper is rather larger than would normally be expected it does appear to fit the experimental yield stress curves fairly well. Note also that, as discussed in the previous section, there are theoretical reasons to believe that neither theory gives results valid for a greater than unity. 4. DISCUSSION In Part I of this paper it was seen t h a t a number of theories of instability arc consistent with a more general formulation. Each theory is in fact an application of the more general theory subject to certain constraints on deformation and/or load rates. The elementary theory does not give the actual stress ratios at instability excepting in the simple case of proportional stressing. A method of calculating the critical stress ratios for general loading paths has been outlined; and this method gives also the total strains and loads at instability. The proposed method of numerical integration by a step-by-step computation is not the only one; in certain cases the method of integration using finite differences might well be more convenient and appropriate. Experiments on fracture do not necessarily yield results completely consistent with theories of instability. To verify the general theory requires further tests to be made using measurements at instability, not at fracture. To distinguish between particular theories within the framework of the more general theory requires similar tests, specifically designed with this purpose in mind. There would appear to be little doubt, however, t h a t the general theory is probably sufficient to allow prediction of the critical instability conditions preceding fracture. Their importance lies in the fact t h a t the instability condition marks the limit of any useful deformation of the structure, whether required for load-carrying purposes or whether it is subject to some forming process. The loads on structures at instability form the true ultimate load system for the structure for the given loading path. The strains at instability mark the true limit of useful ductility of the structure. The question remains: how is it possible to choose between alternative theories satisfying the same general constraints on the deformation but corresponding to a different choice of maximum in load ? In particular, which of the conditions dP~ = 0 or d P = 0 should apply ? Now P~ = P + rrr2p
hence 2 dr dp dP~=dP+~rr p [r+p]
For 1o constant it follows t h a t dP~ < d P ;
dr < O
dP~ > d P ;
dr > O
I t will be suggested that, since P~ attains a maximum first when the tube radius is decreasing, the condition d Pz - - 0 is the correct choice. Similarly, when the radius is increasing the condition d P = 0 is the correct one. Similar
Tensile plastic instability of t h i n t u b e s - - I I
549
arguments apply when the pressure is changing. A rational choice between Swift's theory and the theory proposed in the present paper can therefore be made in any given ease. This tentative conclusion must be tested, however, in the light of suitable experimental studies. REFERENCES 1. H. W. SWIFT, J. Mech. Phys. Solids 1, 1 (1952). 2. E. A. DAvis, J. Appl. Mech. Trans. A S M E , {}5, A-187 (1943). 3. E. A. DAVXS, J. Appl. Mech. 12, A-13 (1945).
TENSILE
Ltd. 1965. Yol.7, pp. 539-549. Printed in GreatBritain
PLASTIC
INSTABILITY
OF
THIN
TUBES--II
M. J . HILLIER Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario (Received 3 N o v e m b e r
1964)
Summary--The results of calculations on the loads and strains at instability for tubes subjected to various loading paths are presented. A critical comparison is made with simple theory and with experiment. I t is concluded that, as expected, the loads and strains at instability will not in general correspond with those at fracture. Further tests are required in order to choose between alternative theoretical formulations of the problem. 1. I N T R O D U C T I O N THE i n s t a b i l i t y of long t h i n t u b e s s u b j e c t e d to a n i n t e r n a l pressure p a n d a n i n d e p e n d e n t axial l o a d P has b e e n considered in t h e p r e c e d i n g P a r t I of this p a p e r . A m e t h o d of c o m p u t a t i o n was outlined which enables t h e loads a n d t o t a l strains a t i n s t a b i l i t y to be d e t e r m i n e d for (a) sequential loading a n d (b) p r o p o r t i o n a l loading. W h e n loading in sequence the t o t a l axial load Pz m a y be applied alone u p to an assigned c o n s t a n t value, followed b y increasing i n t e r n a l pressure. A l t e r n a t i v e l y , t h e pressure a n d i n d e p e n d e n t axial load P m a y be applied first in such a m a n n e r as to hold P~ zero, a n d t h e t o t a l axial l o a d m a y t h e n be increased s u b s e q u e n t l y , while k e e p i n g t h e pressure c o n s t a n t . N o t e t h a t t h e initial loading of t h e sequence is essentially uniaxial a b o v e or below t h e yield load. I f a b o v e t h e yield l o a d it w o u l d be u n r e a s o n a b l e to e x p e c t a simple t h e o r y to a p p l y directly. I t w o u l d t h e n be n e c e s s a r y to consider t h e d e f o r m a tion as t a k i n g place in t w o stages. T h e t o t a l e q u i v a l e n t plastic strain a t t h e end of t h e first stage would serve to a d j u s t t h e v a l u e of t h e empirical c o n s t a n t B in t h e relation = A(B+g)
~
T h a t is, for the second stage of biaxial stressing, B would be r e p l a c e d b y B + g l , where gl is t h e t o t a l s t r a i n a t t h e end of t h e first stage of sequential loading. I f t h e initial loading is below t h e yield p o i n t value, a simple t h e o r y applies directly, since o n l y one p h a s e of plastic d e f o r m a t i o n would t h e n exist. I n t h e case of p r o p o r t i o n a l loading t h e r a t i o p / P ~ is held c o n s t a n t . T h a t is, the r a t i o of t h e n o m i n a l stresses, b a s e d on original cross-sectional areas, is held constant. 2. S E Q U E N T I A L
LOADING
Table 1 refers to the calculated loads and strains at instability for the case where the pressure is applied first. The value of B" in the second column is the yield point value of the dimensionless stress 80~ acting alone. The values of s0¢ given in the third column are the assigned constant values of circumferential stress, g, e¢, ez are the integrated total 539
M. J . H I L L I E R
540
TABLE 1 B
0.05
0.10
0"10
0'20
n
So4~
~
e¢
0.20 (0.549)
0.1 0.2 0'3 0.4 0.5
0.146 0.152 0.168 0.192 0.194
--0.055 --0.037 --0.016 40'015 40'055
0.144 0.146 0.152 0.158 0.133
0-665 0-700 0.726 0.741 0.728
0.135 0.265 0.400 0.555 0'766
0"30 (0.407)
0.1 0"2 0.3 0.4
0.242 0.254 0.290 0-322
--0-086 -- 0.049 0.000 + 0-079
0.238 0-240 0.250 0.230
0"583 0.619 0"650 0.663
0-144 0.293 0.460 0.705
0.40 (0.275)
0.1 0"2 0.3 0-4
0'338 0.360 0.428 0.096
--0.114 -- 0"050 + 0.048 + 0.096
0.332 0.332 0-341 -- 0.048
0"528 0.569 0.611 -- 0-0025
0.151 0.318 0.541 ( -- 0.005)
0'50 (0.224)
0-1 0.2 0"3
0.436 0.472 0"060
-- 0" 137 -- 0.035 40'060
0.425 0-421 --0"030
0.490 0.540 --0"030
0" 155 0.346 (--0.013)
0.20 (0.631)
0.1 0-2 0"3 0.4 0'5 0"6
0"096 0.100 0.116 0-138 0"150 0"118
--0.037 --0"026 --0"015 40"004 +0"031 40'048
0-095 0"096 0.107 0.117 0"111 0'068
0.698 0.732 0.757 0.770 0"763 0"712
0.133 0"259 0.384 0.524 0"695 0"926
0"30 (0"501)
0"1 0"2 0"3 0"4 0"5
0-192 0-204 0"234 0.272 0"226
--0"071 --0"045 -- 0'012 +0"042 +0'094
0'190 0"194 0"208 0"211 0'130
0"611 0'646 0"674 0"887 0"648
0"142 0"283 0.435 0.634 0"935
0"40 (0"398)
0"1 0'2 0"3 0"4
0"288 0-308 0"362 0"400
--0-101 -- 0"054 + 0"012 + 0"004
0"284 0"289 0'306 -- 0"002
0'552 0'590 0"625 -- 0"004
0"50 (0"316)
0"1 0"2 0"3 0"4
0"386 0"414 0"504 0.156
--0"129 -- 0"054 +0'067 +0.156
0"379 0"381 0"396 --0"078
0"512 0"555 0"602 --0' 001
0"2 (0"457)
0'1 0.2 0-3 0"4 0"5 0"6
0"002 0'002 0"002 0.002 0.002 0"002
--0"001 -- 0"001 -- 0"001 -- 0.001 -- 0-001 -- 0"001
0"002 0"002 0'002 0"002 0"002 0"002
0"769 0.804 0"826 0"836 0"831 0"805
0"130 0"248 0-362 0-478 0-599 0"747
0"3 (0"309)
0"1 0"2 0"3 0.4 0"5 0"6
0"092 0.102 0"126 O"160 0"170 0"106
--0"035 --0"026 --0"014 + 0.008 40"041 40"048
0"091 0"098 0"115 O"134 0"122 0"057
0"673 0.706 0"730 0.742 0"730 0"663
0'139 0"269 0.400 0.546 0.74I 0"996
0"4 (0"209)
0"1 0"2 0-3 0"4 0"5
0"190 0"204 0"244 0"296 0"230
-- 0'070 -- 0"045 -- 0-013 +0"045 + 0"093
0"188 O"195 0"217 0"230 0"134
0"608 0"642 0"670 0-684 0"643
0"143 0-284 0"436 0"641 0-935
0"5 (0.141)
0"1 0"2 0-3 0"4
0"286 0"308 0"370 0"408
-- 0"102 --0"059 + 0"001 +0"110
0"282 0.291 0"319 0"284
0"562 0"599 0-634 0"647
0"145 0-296 0.474 0"770
(l~)
ez
Soz
(lio<)
0"148 0'304 0'492 ( -- 0"0096) 0-151 0"324 0"571 (0"505)
Tensile plastic instability of thin tubes--II
541
TABLE 2 B
n (B '~)
soz
~
e~
0.05
0.2 (0.549)
0.1 0.2 0"3 0.4 0-5 0"6
0.070 0.062 0"062 0"070 0"080 0"052
0-069 0.059 0"056 0"057 0"062 -- 0"026
0"3 (0"407)
0"1 0"2 0"3 0"4 0"5
0'126 0"116 0"120 0"140 0"082
0"4 (0.275)
0"1 0.2 0"3 0"4
0"5 (0-244)
0"10
0"10
0-20
ez
so~
a
--0.026 --0.014 -- 0-005 +0"006 +0"023 + 0"052
0.624 0.650 0"666 0"670 0"659 -- 0.0002
7.16 3.66 2'48 1"877 1'491 -- 0.0039
0.124 0"109 0"102 0"101 -- 0"041
--0-044 --0"020 + 0"003 + 0"032 + 0"082
0"519 0"539 0"551 0"553 -- 0"003
6'65 3"35 2"25 1"69 -- 0"005
0"184 0"170 0"180 0"074
0"I80 0.155 0"140 -- 0.037
--0"059 -- 0"018 +0"024 + 0"074
0"445 0"462 0.474 -- 0"004
6.38 3"15 2'09 -- 0.009
0"1 0"2 0"3 0"4
0"240 0"224 0"052 0"180
0'233 0"195 --0"026 --0"090
--0"070 --0"007 +0"052 +0"180
0"391 0"041 --0"005 --0"002
6"22 3"00 --0"016 --0"005
0"2 (0"631)
0"1 0"2 0"3 0-4 0"5
0"018 0"012 0'012 0"020 0'036
0"018 0.012 0.011 0"017 0"027
--0"007 --0"003 -- 0"001 +0'001 +0" 007
0"676 0"707 0"726 0"730 0"717
7.01 3"62 2"47 1-89 1.51
0"3 (0"501)
0-1 0"2 0'3 0'4 0"5
0"076 0"066 0"070 0"088 0"122
0-075 0"062 0"061 0"068 0"076
--0"027 --0-013 -- 0-002 +0"014 +0"043
0"563 0"587 0"600 0-600 0"583
6"54 3'33 2'26 1"72 1"36
0"4 (0-398)
0-1 0"2 0"3 0"4 0"5
0"132 0"120 0"130 0"002 0-174
0-130 0"111 0"108 --0"002 --0'087
--0"045 --0"018 +0"007 +0"004 +0-174
0"484 0-503 0-513 --0"005 --0"001
6"27 3"14 2"12 0-0116 --0.0019
0"5 (0"316)
0"1 0"2 0"3 0"4
0'190 0"176 O"190 0"096
0.186 0.160 O"147 -- 0'048
--0'060 --0"017 + 0"028 + 0"096
0"425 0"441 0.451 -- 0'003
6"16 3"03 2-02 -- 0.0063
0"2 (0"457)
0"1 0"2 0-3 0"4 0"5 0"6
0*002 0"002 0"002 0"002 0"002 0"002
0"002 0-002 0-002 0"002 0"002 0-001
--0"0008 --0"0006 -- 0"0003 -- 0-0005 + 0"0002 + 0"0005
0'769 0.803 0'825 0"835 0.830 0'805
7"72 4"03 2"76 2" 10 1.67 1.34
0"3 (0.309)
0.1 0-2 0'3 0.4 0.5 0.6
0.002 0"002 0.002 0"002 0.016 0"062
0.002 0.002 0.002 0.002 0.011 0"036
-- 0"0007 -- 0-005 -- 0.0002 + 0" 0001 +0'0039 +0"026
0-661 0.692 0.709 0.710 0.690 0.646
6.63 3.47 2-37 1.78 1.41 1.16
0-4 (0.209)
O' 1 0"2 0-3 0.4 0.5
0'032 0'020 0"030 0-056 0"102
0.032 0.019 0.026 0.043 0"065
-- 0.011 --0'004 -- 0.0006 + 0.009 + 0"036
0.570 0.597 0.609 0.604 0-582
6.07 3.10 2-14 1.65 1.33
0.5 (0-141)
O. 1 0.2 0"3 0"4 0"5
0"090 0.076 0"090 O"128 0"116
0.088 0.071 0"076 0"092 --0"058
0.502 0.524 0'532 0'528 --0"0013
5.99 3-01 2"07 1.59 --0"0024
-- 0.031 --0"012 + 0'003 + 0"031 +0"116
542
M.J.
HllmmR
strains a t the p o i n t of instability, s0= is t h e dimensionless nominal axial stress at instability and ~ t h e corresponding t r u e stress ratio s~,/s,. F o r values of 8o4 r a t h e r larger t h a n those t a b u l a t e d , instability occurs first due to uniaxial stress alone. E l e m e n t a r y t h e o r y predicts t h e following m i n i m u m v a l u e of engineering stress for which this s i t u a t i o n occurs: 80¢(crit)
----- n n
exp [ - ( n - - B ) ]
(l)
Fig. 1 plots this critical n o m i n a l stress for a range of values of n and B. T a b l e 2 presents t h e results of calculations for t h e case where t h e sequence is r e v e r s e d ; stress 8o= being applied first, a n d s0¢ second. The assigned c o n s t a n t values of dimensionless circumferential stress are t a b u l a t e d in c o h m m 3. The critical v a l u e of s0z at which yield first occurs in simple tension is again g i v e n b y t h e r i g h t - h a n d side of e q u a t i o n (1) and
Fig. 1. So long as instability does n o t first occur in simple tension t h e c o m p u t e d z--a c u r v e agrees w i t h t h e e l e m e n t a r y t h e o r y of P a r t I to w i t h i n 1 per cent.
o.s 0.7
\
0,7
~
0.3
o.z
~~0.6 0.5} ~ ~'--~
---'----- ~
0.6
\
0.4
0.8
- -
0.6
0.3
-------- -----~'- "-'-'--- 0.2
J.
0.5
So~ (crit)
0.4 0.20
~
0.10
02 t
0.3
0.2 I i 0 . 2
0.2
06
0.05 03
0.0 02
-"
~
------'~ -------~ -----'---
0.6
0.4
~
I
0.0
0.3
0.4
0.5
0
R
0.05
QIO
0~5
0.20
B
F i e . 1.
F I e . 2. T r u e stress ratio at instability.
3. P R O P O R T I O N A L
LOADING
Figs. 2 t h r o u g h 6 s u m m a r i z e t h e results of calculations w h e n t h e ratio of pressure to t o t a l axial load is held constant. This ratio is expressed here as C = a ~ / a o z = 21rr*o P / P z
T h e critical v a l u e of t h e t r u e stress ratio at instability is p l o t t e d as a f u n c t i o n of B, n a n d C in Fig. 2. Figs. 3(a) t h r o u g h 3(c) plot t h e corresponding v a l u e of dimensionless n o m i n a l axial stress so= = _P,/(2rrr o t e a ). The t o t a l t h i n n i n g strain a t instability is p l o t t e d in Figs. 4(a) t h r o u g h 4(e); a n d Figs. 5 a n d 6 p l o t t h e corresponding ratios of t h e t o t a l strains. T h e c o m p u t e d z - a c u r v e for this loading path, Fig. 7, does n o t correspond so closely to simple t h e o r y as for t h e p r e v i o u s case considered. I n particular, for a greater t h a n u n i t y t h e r e is no correspondence b e t w e e n t h e c o m p u t e d results and those of simple theory. A
543
Tensile plastic instability of thin t u b e s - - I I
similar comparison between Swift's theory z and computed results is also shown: a discrepancy again appears. The method of computation in this latter case will be outlined in the next section. There remains a further discrepancy for values o f ~ less than unity. The possibility of error in numerical integration was checked b y halving the increment in g in the step-by-step computation. The identical discrepancy remained.
0.9
0.9 n=0.2
0.8
C=
So~
0.4
n=0.3
~
0.8
Soz
_...:
~
.
-:I0.6 0.5 0
./ -11.0
o s :222::i
C=~3.4 i.~
~'"
./-J.o
0.5
OD5
0.1o
0.15
0
CL20
0.05
0.10
B (a)
B
0.15
0.20
(b) O.S n=0.5 0.7
0.8
C=
n=0.4
0.6
So=, 0.6
0.7 C = ~ . ~ _ ~ 0.4
Soz 0.6
0.5
0.5
0.4
0.4 o
0.05
0,10
0.15
f
0
005
0.10
B
0.15 B
(e)
(d)
0.8 n=0.6
0.7 ci 0.6 Soe
o.
0.5 L0/ 0.4
O.3 0
/ - f
0.05
~°
0,10
0.15
0.20
B
(e) FzO.
3.
1,0
j°J
0.5
0.20
/
Nominal axial stress at instability (dimensionless).
0.20
544
M.J.
HILLIER
0.4
0.4
n=0.3i n =0.2
!
02
0.5
-e r er 0.2
0.~
......
0.6
~'"
0.1
0.1
0.05
0`10
0.15
0.C6
0.20
I
0`10
0,15
0,20
B
B
(a)
(b) 0.6
""'-..
n= 0.5
0.5
0.5 n =0.4
0.4""-.
0.4
-el" 0.3
-er 0.2
0,2
o.!
0,1
0,05
(210
0`15
"~'~"
0.05
0,20
0,1o
0.15
B
B
(d)
(c) 0.7
""
0.6
n.0.6
~.6
0.5 -e r 0.4
Q3 ~-
'1 0
0.2
~ 0
0`05
0`10
0,15
0.20
B
(e) F I G . 4. T h i c k n e s s
strain at instability.
0.20
545
Tensile plastic instability of thin tubes---II 1.8
0.8 R= 0.6 .......
C
I1==
U.t)
1.6 . . . . . .
0.4
0.2
C: 0.6
0.2
0.4
0.2
0.4
0.4
0.4
0.6
_ _ 0.4
0.8
0.4
1.0
0.2
0.4
1.4
0.2
0.6
0.4
02
0. =-
I.Z
0.4
o.z
er 0
I
e=
0.2 0.4 -0.2 . . . . . . . .
1.0
- e"~
02
0.6
~
0.6
0.8
0.6"
0.2
0.2 ~
~
n
a
n=
0.6 =.. . . . .
r .....
-o4
..... -0.6
~----~
L.....
~
,0.4
~'--'-,--
:.0
~
U.q5
--~.---.----~
0 4 __.~1 ~
-0.8
0
.
6
0.2
- 1.0 0
Q05
(210
(215
G20
0
0.05
OJO
B
0.15
0.20
B
F r o . 5. S t r a i n r a t i o a t i n s t a b i l i t y .
F I e . 6. S t r a i n r a t i o a t i n s t a b i l i t y .
1.4 COMPUTED
RESULTS
-----~P~=o . . . . .
dP=O
0.8
O~
0.5 0
0.5
LO
,C
0.5
V&
F r o . 7. Critical s u b t a n g e n t a t i n s t a b i l i t y .
3. C O M P A R I S O N
WITH
EXPERIMENT
T h e m o s t useful a n d r e l e v a n t e x p e r i m e n t a l r e s u l t s a r e t h o s e d u e t o D a v i s o n t h e b u r s t i n g o f c o p p e r S a n d steel s t u b e s . T h e t e s t s were c a r r i e d o u t u n d e r t h e c o n d i t i o n s o f p r o p o r t i o n a l loaxiing c o n s i d e r e d in t h e p r e s e n t p a p e r . I t m u s t b e e m p h a s i z e d , h o w e v e r , t h a t t h e m e a s u r e m e n t s were m a d e o n t u b e s a t f r a c t u r e a n d n o t a t i n s t a b i l i t y . H e n c e a close correlation between present theory and experiment cannot be expected. In particular, for r e l a t i v e l y low pressures, t h e b e h a v i o u r o f t h e t u b e s w o u l d a p p r o x i m a t e t o s i m p l e t e n s i o n ; t h e localized t h i n n i n g o f t h e t u b e s u b s e q u e n t t o i n s t a b i l i t y a n d p r e c e d i n g f r a c t u r e is n o t negligible. T h u s , for e x a m p l e , i t w o u l d b e e x p e c t e d t h a t t h e t o t a l t h i n n i n g a t fracture would be greater than that predicted by any instability theory.
546
M.J.
1-[ILLIER
T a b l e 3 s u m m a r i z e s t h e e x p e r i m e n t a l r e s u l t s a t f r a c t u r e , t a k e n f r o m t h e references c i t e d a n d a d d i t i o n a l i n f o r m a t i o n s u p p l i e d b y D r . E . A. D a v i s . T h e e x p e r i m e n t a l s t r e s s s t r a i n c u r v e s for t h i n t u b e s i n s i m p l e t e n s i o n g i v e n i n t h e a b o v e references were f i t t e d b y t h e e m p i r i c a l r e l a t i o n 5 = A ( B + $)n, a n d t h e following r e s u l t s o b t a i n e d : Material
A (lb/in 2) 110,000 45,000
Steel Copper
B
n
0.00 0.02
0.23 0.58
TABLE 3. EXPERIMENTAL RESULTS Nominal stress ratio ~o~/(7o~
Inside diameter at fracture (in.)
Tube wall a t fracture (in.)
M e d i u m c a r b o n steel I n i t i a l o u t s i d e d i a m e t e r 1.45 in. I n i t i a l t u b e t h i c k n e s s 0' 10 in. 0-0 0.902 0.064 0.500 1.302 0.060 0.750 1.566 0.0505 0-762 1.567 0.052 0.775 1-510 0.065 0.800 1-480 0.070 0-875 1.444 0.078 1-00 1.489 0.078 1/0-5 1.430 0.087 1/0-0 1.500 0.090
Load P at fracture (lb)
Pressure p at fracture (lb)
T r u e stress ~4, a t fracture (lb/in s)
T r u e stress ~z a t fracture (lb/in 2)
21900 20800 15100 15150 16000 I5500 14400 11800 0 --
0 5100 6560 6740 7290 7450 7980 8280 8650 8580
0 55200 101400 101400 84500 78500 74500 79000 71200 72000
113000 107500 108300 106600 90600 83000 74400 68300 33500 10400
12600 11600 10900 10200 8250 5200 0
0 1235 1860 2500 3590 3660 3760
0 15800 25400 33000 38200 42000 39200
68900 63000 67800 63600 43700 35500 18700
Copper I n i t i a l o u t s i d e d i a m e t e r 1.45 in. I n i t i a l t u b e t h i c k n e s s 0.10 in. 0.0 0-910 0.060 0.250 1-225 0.048 0.375 1.281 0.047 0.500 1.318 0.050 0.750 1.449 0.068 1.00 1.546 0.067 1/0.5 1.570 0.075
T a b l e s 4 a n d 5 give t h e c o r r e s p o n d i n g e x p e r i m e n t a l v a l u e s o f a, so,, - e , e4,/er, %/e t c a l c u l a t e d f r o m t h e d a t a o f T a b l e 3. T h e t h e o r e t i c a l v a l u e s c o r r e s p o n d i n g t o t h e calculat i o n s o f S e c t i o n 3 a r e t a b u l a t e d u n d e r t h e h e a d i n g " T h e o r y ( v ) " . T h e case B = 0.00 w a s c o m p u t e d b y s e t t i n g B = 0.002; a n y r e s u l t i n g e r r o r w a s j u d g e d t o b e small. F o r c o m p a r i s o n a f u r t h e r s e t of c a l c u l a t i o n s c o r r e s p o n d i n g t o t h e c o n s t r a i n t s o f S w i f t ' s t h e o r y , as d i s c u s s e d i n P a r t I o f t h e p r e s e n t p a p e r , is g i v e n i n c o l u m n s h e a d e d " T h e o r y (i)". T h e i n s t a b i l i t y c o n d i t i o n a s s u m e d h e r e is dp = d P = 0. I n t h i s case dPz is n o n - z e r o a t i n s t a b i l i t y , a n d i t m a y b e s h o w n t h a t t h e i n s t a b i l i t y c o n d i t i o n dso, ~<0 m u s t now be replaced by the condition
dso, <~CSo, e x p (2e4,) de4, As e x p e c t e d , t h e r e is o n l y q u a l i t a t i v e a g r e e m e n t and experiment. The experimental thinning strains than the theoretical thinning strains at instability. r o u g h l y t h e s a m e for b o t h t h e o r i e s a n d for e x p e r i m e n t .
i n t r e n d s , as b e t w e e n e i t h e r t h e o r y at fracture are substantially larger T h e g e n e r a l t r e n d o f t h e r e s u l t s is A l t h o u g h t h e v a l u e of t h e e m p i r i c a l
547
Tensile plastic i n s t a b i l i t y o f t h i n t u b e s - - I I TABLE 4
O~ =
ao = s~/S,o
80~
84ff8 ~
Theory
Theory Exp.
Exp.
(i)
(v)
(i)
(v)
0-566 0"677 0"652 0"648 0.643 0.633 0"599 0"540 1.15
0"566 0"677 0"658
0.408 0"463 0-489 0.523 0"470 0.335 0"119
0.408 0"463 0"490 0"523 0-481 0.345
Steel 0-0 0.50 0.75 0.762 0.775 0.80 0"875 1.0
0.0 0.50 0.852 0.868 0.883 0.915 1/0.997 1/0.876 I/0.492
1/o.5 1/o.o Copper 0.0 0.25 0"375 0.50 0.75 1.0
1/o.5
0.0 0-523 0.936 0-952 0-937 0.946 1/0.999
0-0 0.50 0-898 0.933 0.968
1/0.865
1/0.828
1/0.471 1/0-144
0.0 0.172 0-299 0.5
0.0 0-176 0.305 0.50
0-991
1/0.905
1/0.772 1/0.428
1/0.697
0.0 0"232 0"374 0"519 0.874
1/o.841 1/0.477
TABLE
O~0
Theory
;reel 0.0 0.50 0.75 0.762 0.775 0-80 0.875 1.0 1/0.5 1/0.0 ;opper 0-0 0.25 0.375 0-50 0.75 1.0 1/0.5
36
(v)
0.115 0.229 0.193 0.190 0-184 0.177 0.153 0.121 0.007
0.115 0-229 0.259
0-28 0.392 0.471 0.563 0.369 0.219 0.072
0.248 0.239 0.166
0.28 0-369 0.440 0.563 0.470 0.288
0"547
0.660 0"684 0"696 0"713 0-741 0"631 0.380
5
e~/e.
Theory
Theory
Exp. (i)
0-649 0.640
e¢/e.
-- e r
0"47 0.593 0"595 0"603 0.623 0.593 0"589 0-562 0.298 0-0
Exp. (i)
(v)
0.446 0.551 0.683 0.653 0.425 0.358 0.248 0.248 0.140 0.105
1.0 0.00 --0-330 --0-343 0-355 --0.381 --0.449 --0.547 -- 1.00
1.0 0'00 --0.347
0.511 0.733 0.755 0.693 0.386 0.400 0.287
1.0 0.479 0.241 0.00 --0.379 --0.591 -- 1.05
1.0 0.474 0.237 0.00 --0.588 --0.376 --
--0.374 --0.40 --0-567
Exp. (i)
(v)
0.751 --0.018 --0.260 --0.277 --0.362 --0.441 --0.484 --0.597 --0.822 -- 1.553
--2.0 -- 1.00 --0.670 --0-657 --0-644 --0.619 --0.551 --0-453 --0.034
--2.0 -- 1-00 --0.653
0.648 0.079 0-023 --0.017 --0-298 --0-442 --0-686
--2.0 -- 1.48 -- 1-24 -- 1.00 --0.621 --0.409 --0.047
--2-0 -- 1-47 -- 1-24 -- 1-00 --0.412 --0.624
--0.626 --0.600 --0.433
--1.975 --0.982 --0-739 --0-722 --0.638 --0"559 --0"516 --0.403 --0.178 + 0.55 --1-650 -- 1.080 -- 1-022 --0.982 --0.702 --0.558 --0-314
548
M.J.
HILLIER
index n for copper is rather larger than would normally be expected it does appear to fit the experimental yield stress curves fairly well. Note also that, as discussed in the previous section, there are theoretical reasons to believe that neither theory gives results valid for a greater than unity. 4. DISCUSSION In Part I of this paper it was seen t h a t a number of theories of instability arc consistent with a more general formulation. Each theory is in fact an application of the more general theory subject to certain constraints on deformation and/or load rates. The elementary theory does not give the actual stress ratios at instability excepting in the simple case of proportional stressing. A method of calculating the critical stress ratios for general loading paths has been outlined; and this method gives also the total strains and loads at instability. The proposed method of numerical integration by a step-by-step computation is not the only one; in certain cases the method of integration using finite differences might well be more convenient and appropriate. Experiments on fracture do not necessarily yield results completely consistent with theories of instability. To verify the general theory requires further tests to be made using measurements at instability, not at fracture. To distinguish between particular theories within the framework of the more general theory requires similar tests, specifically designed with this purpose in mind. There would appear to be little doubt, however, t h a t the general theory is probably sufficient to allow prediction of the critical instability conditions preceding fracture. Their importance lies in the fact t h a t the instability condition marks the limit of any useful deformation of the structure, whether required for load-carrying purposes or whether it is subject to some forming process. The loads on structures at instability form the true ultimate load system for the structure for the given loading path. The strains at instability mark the true limit of useful ductility of the structure. The question remains: how is it possible to choose between alternative theories satisfying the same general constraints on the deformation but corresponding to a different choice of maximum in load ? In particular, which of the conditions dP~ = 0 or d P = 0 should apply ? Now P~ = P + rrr2p
hence 2 dr dp dP~=dP+~rr p [r+p]
For 1o constant it follows t h a t dP~ < d P ;
dr < O
dP~ > d P ;
dr > O
I t will be suggested that, since P~ attains a maximum first when the tube radius is decreasing, the condition d Pz - - 0 is the correct choice. Similarly, when the radius is increasing the condition d P = 0 is the correct one. Similar
Tensile plastic instability of t h i n t u b e s - - I I
549
arguments apply when the pressure is changing. A rational choice between Swift's theory and the theory proposed in the present paper can therefore be made in any given ease. This tentative conclusion must be tested, however, in the light of suitable experimental studies. REFERENCES 1. H. W. SWIFT, J. Mech. Phys. Solids 1, 1 (1952). 2. E. A. DAvis, J. Appl. Mech. Trans. A S M E , {}5, A-187 (1943). 3. E. A. DAVXS, J. Appl. Mech. 12, A-13 (1945).