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Theoretical prediction of tensile instability conditions in rotating disks
Int. J. Mech. Sci. Pergamon Press Ltd. 1964. Vol. 6, pp. 421-43-1. Printed In Great Britain
THEORETICAL PREDICTION OF TENSILE INSTABILITY CONDITIONS IN ROTATING DISKS ]~[. J . PERCY* a n d P. B. ~[ELLOR'~ (.Received 15 June 1964)
Stmrmary--An analysis of conditions at instability in rotating disks of uniform thickness is presented. Two plasticity theories are applied. The first analysis based on the Tresca yield criterion and its associated flow rule is an extension of the work of Prager and Weiss. The results of this analysis are correlated with results based on the Heneky deformation theory of plasticity using a finite difference method of solution. Both theories show that if fracture is preceded by a tensile instability condition then large strains occur at the bore of a hollow disk. NOTATION
A a, b
ao, be er, eo, ez 11 re 8 r, 80, 8z
t to Y 0s s
CO, Ez
O (Yf~ 0"0~,
(3"z
O)
constant in strain-hardening relation current inner and outer radii of disk initial inner and outer radii of disk engineering strains in radial, circumferential and axial directions constant in strain-hardening relation current radius to particle initial radius to particle nominal stress in radial, circumferential and axial direction current thickness of disk of radius r initial uniform thickness of disk radial displacement of particle tensile yield stress radius ratio, bo/ao natural straits in radial, circumferential and axial directions u[ao ro/ao density of material true stress in radial, circumferential and axial directions angular velocity of rotating disk INTRODUCTION
WHE~ the angular velocity of a thin circular disk at room temperature is g r a d u a l l y increased the disk first o f all b e h a v e s elastically then, a t a certain velocity, plastic yielding begins a n d procceds, w i t h w o r k - h a r d e n i n g , until all t h e m a t e r i a l o f the disk h a s r e a c h e d t h e initial yield point. F u r t h e r increase in speed causes u n r e s t r i c t e d plastic flow to t a k e place. T h e s y s t e m is stable as long as a n increase in speed is n e c e s s a r y to p r o d u c e f u r t h e r plastic flow. T h e following analyses are r e s t r i c t e d to conditions of large plastic s t r a i n a n d it will be a s s u m e d t h a t the elastic strains are negligible in c o m p a r i s o n w i t h the plastic strains. I t will also be a s s u m e d t h a t t h e m a t e r i a l is h o m o g e n e o u s a n d isotropie. F o r the a b o v e conditions it is generally agreed t h a t the L d v y - v o n Mises i n c r e m e n t a l s t r e s s - s t r a i n relationships are the m o s t a c c e p t a b l e but, to a v o i d * Department of l~Iechanical Engineering, University of Liverpoo!. Department of Mechanical Engineering, University of Birmingham. 421
422
i%I.J. PERCY and P. ]3.1%~ELLOR
the large amount of computation that these relationships entail, methods of reducing the mathematical work have been sought. Lee %Vu I, ~Ianson z and Zaid 3 simplified the computation by assuming a yon Mises yield criterion and t h e H e n c k y d e f o r m a t i o n t h e o r y o f p l a s t i c i t y i n s t e a d o f t h e L 4 v y - v o n 5Iises flow t h e o r y . L e e W u a n d Z a i d j u s t i f i e d t h i s s i m p l i f i c a t i o n f o r t h e c a s e o f s o l i d disks of uniform thickness by showing a posteriori that the strain ratios remain p r a c t i c a l l y c o n s t a n t a s t h e s p e e d is i n c r e a s e d . W e i s s a n d P r a g e r ~ m a d e a n alternative assumption that the disk material obeys Tresca's yield criterion a n d t h e a s s o c i a t e d flow r u l e , a n d b a s i n g t h e i r s o l u t i o n o n t h e n o m i n a l t e n s i l e stress-strain curve derived results for hollow disks of uniform thickness. This p a r t i c u l a r c h o i c e o f y i e l d c r i t e r i o n a n d flow r u l e s i m p l i f i e s t h e a n a l y s i s s i n c e then one of the strain components reduces to zero. Smith 5 adopted the same assumption in an analysis of rotating plastic disks of variable thickness, and Mellor and Percy G extended the theory of Weiss and Prager to larger strain values. I n t i l e f o l l o w i n g w o r k r e s u l t s b a s e d o n t h e T r e s c a flow r u l e a r e c o m p a r e d , for the case of hollow disks of initial uniform thickness, with results from the H e n c k y t y p e s o l u t i o n . T h e c o n d i t i o n s a t i n s t a b i l i t y , t h a t is a t t h e m a x i m u m speed attained by the disk, are of particular interest. THEORETICAL ANALYSIS BASED ON THE TRESCA YIELD CRITERION AND THE ASSOCIATED FLOW RULE The principal stress components, for a circular disk rotating about the central axis, are the circumferential stress a0, the radial stress ar and the axial stress az. The axial stress, (;~, will be assumed to be zero for the thin disks to be considered here. F o r a hollow disk of initial uniform thickness unloaded a t the boundaries it can be shown 4 t h a t ae and (7r are both tensile and t h a t a0 is every~vhere greater t h a n at. Tresca's yield criterion and flow rule can then be written ao= a>a,>0, az= 0 (1) d e r = O, dee = - - d e , ~ O (2) where a is twice the critical shearing stress and dee, der and de, are respectively the circumferential, radial and axial strain increments. I t is necessary to enquire if the above relations hold when ar = 0, t h a t is a t the unloaded boundaries of the disk where the stress system is one of uniaxial tension. This corresponds to " a corner" of the Tresca hexagonal yield surface. Koiter: suggested t h a t the above equations should hold even when ar = 0, b u t this restriction can be shown to be too strong. (See, for example, BlandS.) :However, to avoid a disconthmity in the solution of this particular problem it is advantageous to a d o p t Koitcr's restriction and assume t h a t the above equations are applicable a t all points on the disk. Further, if it is assumed t h a t the above relations hold for simple tension then the strain-hardening relation can logically be assumed to be based on the simple tensile stress-strain curve. The consequences of following this course will be discussed later. Let u be the radial displacement of a particle on the disk surface. Then, since from the first equation (2) the radial strain increment is zero, this displacement u does n o t depend on the initial radius of the particle r 0 b u t only on the angular velocity. Thus an annulus bounded initially b y radii r o and r o + dro is a t a particular speed (after undergoing a radial displacement, u) bounded b y radii r = r o + u and r + d r = r o + u + d r o from which it follows t h a t d r = dro. Also, suppose t h a t the initial thickness of the disk t o has now a t this particular radius a thickness t. Since the material is assumed to be incompressible to ro dro = tr d r
or
to ro = tr
(3)
Theoretical predictions of tensile instability conditions in r o t a t i n g disks
423
The t r u e stress, a,, is t r a n s m i t t e d across a section t h a t is proportional to tr and since tr = t o re trr is e q u a l to t h e nominal radial stress s t. The t r u e circumferential stress, a0, is t r a n s m i t t e d across a section p r o p o r t i o n a l to t d r and i s therefore related to the nominal circumferential stress, s e, b y t h e relation so(to dro) = uo(t dr)
a n d since dr o = d r so = no(tire)
(4)
This n o m i n a l stress is related to t h e circumferential engineering strain U
e0 = -
(5)
s =1(0
(6)
re
b y t h e s t r a i n - h a r d e n i n g law where s is the n o m i n a l stress and e t h e longitudinal engineering strain in t h e simple tensile test. On the basis of equations (1) a n d (2) s will be i n t e r p r e t e d as t h e c o n v e n t i o n a l hoop stress so a n d e as the engineering hoop strain eo. T h e n so = f ( e o )
(7)
Finally, t h e e q u a t i o n of radial e q u i l i b r i u m in the deformed s t a t e is ~(rtcrr)]Sr = t a o - - t p o J ~ r 2
(8)
where m is the a n g u l a r v e l o c i t y a n d p is the d e n s i t y of the material. W i t h reference to the u n d e r f o r m e d state, e q u a t i o n (8) can be w r i t t e n as 8(re sr)/~ro = ao-- poJ 2 r e ( r e + U )
(9)
= f ( u / r o ) -- p w 2 ro(r o + u )
I f t h e disk is n o t loaded at t h e bore, r o = ao, or a t t h e periphery, r o = be, t h e n s r is zero at these radii a n d the integral o f t h e r i g h t - h a n d side of e q u a t i o n (9) b e t w e e n t h e l i m i t s a o and b o m u s t vanish. P u t t i n g c~ = bo/a o, 71 = u / a o and ~ = ro/ao, this condition yields pay oJ~16 =
f/
f ( T / ~ ) d ~ / [ 2 ( a3 - I) + 3T(a 2 - 1 ) ]
(I0)
This is t h e e q u a t i o n derived b y Weiss a n d PragerL F o r a k n o w n strain-hardening characteristic a n d g i v e n initial dimensions t h e r i g h t - h a n d side of e q u a t i o n (10) m u s t be e v a l u a t e d , a n a l y t i c a l l y or numerically, for various values of T = u/ao" E a c h assumed v a l u e of T gives t h e corresponding v a l u e o f m ~ and a plot of T against ra 2 can be carried o u t to d e t e r m i n e t h e m a x i m u m v a l u e of angular v e l o c i t y and hence t h e p o i n t of instability. E q u a t i o n (10) is based on t h e nominal stress-engineering strain c u r v e d e t e r m i n e d in simple tension a n d t h e e q u a t i o n used in this form gives valid and easily derived results p r o v i d e d the longitudinal strain a t instability in simple tension is greater t h a n t h e m a x i m u m circumferential strain at instability of the r o t a t i n g disk. This is so for t h e p a r t i c u l a r e x a m p l e g i v e n b y Weiss and P r a g e r 4 for an a l u m i n i u m alloy w i t h a = 2. I{owever, it will be shown t h a t in general w h e n a > 3 t h e n t h e m a x i m u m strain a t instability o f t h e disk is g r e a t e r t h a n t h e instability strain in simple tension. T h e r e is no reason w h y the solution should be limited b y t h e a m o u n t of tnfiform strain t h a t can be a t t a i n e d in simple tension a n d in t h e following analysis t h e solution will be derived in t e r m s of a n empirical t r u e stress, n a t u r a l strain relationship. A s t r a i n . h a r d e n i n g law o0 = A ~
(H)
will be a d o p t e d , where A a n d n are c o n s t a n t s for a p a r t i c u l a r disk. This relation does n o t give full r e p r e s e n t a t i o n of a rigid-plastic material, since there is no initial yield point, b u t it is a simple a n d useful expression for discussion of conditions at large plastic strains a n d in p a r t i c u l a r a t instability.
424
M . J . PERCX" a n d P . B . MELLOR
,The r e l a t i o n b e t w e e n t r u e c i r c u m f e r e n t i a l stress a n d n o m i n a l c i r c u m f e r e n t i a l s t r e s s is
ao -- ~o(1+eo) = So(1 +'11~)
(12)
a n d t h e n a t u r a l s t r a i n is r e l a t e d to t h e e n g i n e e r i n g s t r a i n b y eo = In ( l + e o ) = In ( 1 + , / [ ~ )
(13)
Therefore, m a k i n g use o f e q u a t i o n s (11), (12) a n d (13), t h e f u n c t i o n i n e q u a t i o n (10) can be written
f:.f
('T/D d~ = A
f:[In(l+v/D]"d$ (1 + ~ / D
9 N o w c h a n g e t h e v a r i a b l e f r o m ~ t o In (1 + ~ / ~ ) = e0. D e n o t i n g t h e c i r c u m f e r e n t i a l s t r a i n a t t h e b o r e b y Eo, a n d a t t h e p e r i p h e r y b y COba n d r e m e m b e r i n g t h a t co. = In ( 1 + ~ )
eeb = In (1 +:q[=) the expression becomes
f~J('Tff)d$= A af'o% ~ dee % (exp co-- 1) ~ F o r a g i v e n v a l u e o f co~, ,/ -- u[r o is c o n s t a n t a n d t h e r e f o r e e q u a t i o n (1O) c a n b e w r i t t e n
pa~ co~
"qJ'ob(exp-7--- 1) ~ dee
6A
[2(c~3 - l) + 3,7(az - I)]
(14)
To determine the integral of the above equation the term e0/(exp e0- I) will be expressed as a series eo = 1-(exp e0 -- 1)
+
Ble~ B3e~ Bst~ -+-2!
4!
w h e r e to < 2rr a n d t h e B e r n o u i l l i n u m b e r s h a v e t h e v a h m s B z [ , Bs = ~ , B5 U s i n g t h i s series it c a n b e s h o w n t h a t t h e r i g h t - h a n d side o f t h e e q u a t i o n 5
(15)
6!
2
I (expeo ~' e0-- I)1' ~ l--ee+~-~eo---~
_!_
(16)
is correct to better than 0.06 per cent for 0
a'O~
6A
2(a a -- 1) + 3~(a 2 -- 1)
(17)
I n t e g r a t i n g e q u a t i o n (17)
lea-'
eb'+ 5 eg+'
eg+= ]'o,
~- ~qt~--~----1 n 12"n+l 12(-~2)J,0b Pa~176= 6A [2(a 3 -- 1) + 3r)(a 2 -- 1)] =
(18)
I f for a p a r t i c u l a r r a d i u s r a t i o a a v a l u e o f r/ is a s s u m e d , t h e n co, a n d eo~ c a n b e c a l c u l a t e d . H e n c e t h e c o r r e s p o n d i n g v a l u e of w ~ c a n b e d e t e r m i n e d f r o m e q u a t i o n (18). V~rhen t h i s is r e p e a t e d a p l o t o f oJ2 a g a i n s t eo, c a n b e o b t a i n e d . S u c h c u r v e s are s h o w n in Fig. 1 for m a t e r i a l s w i t h n = 0.05 a n d n = 0-1 a n d for a v a l u e of a = 10. ( T h e v a l u e of t h e s t r a i n - h a r d e n i n g i n d e x n for s o m e t u r b i n e disk m a t e r i a l s lies b e t w e e n 0-05 a n d 0.1.) T h e v a l u e of t h e m a x i m u m v e l o c i t y c a n b e a c c u r a t e l y d e t e r m i n e d f r o m c u r v e s of r ~a g a i n s t es,, b u t i n s p e c t i o n of Fig. 1 will s h o w t h a t i t is n o t possible to d e t e r m i n e a c c u r a t e l y
Theoretical prediction of tensile instability conditions in r o t a t i n g disks
425
t h e v a l u e of co,, a t instability. T o d e t e r m i n e t h e instability strain e q u a t i o n (18) is differentiated w i t h respect to ~7 and e q u a t e d to zero,
d(oJ~)
2(=~_1)
.-z_
d - - ~ = O = 2(oca_l)T3~((x.,_l )L
._
.-z
0~--1)
.
n
5 I~nq-1 ~.COa - - ~.n+l~ Ob I " 1 I~n-l-2 ,o0o -
+ 12 -I-
[(~Oa "-'* --F~Oa "-' )
( n + 1)
12
~n+2~l 0b //
~
(eOlb'-2--eOlb'-l) 1 1, s e no.-- ~ no.§ ,
J I
(~. .
. .
§
!]
(19)
,004~1
MAX.. ] ~ - " fl-0"0$
JJ f
MAX. ~
"oo3(
"~
/
fl=~lO
9002
*0020
0
0-05
0.I0
0.I$
0.20
0.25
0-30
FIG. 1. Variation of bore t a n g e n t i a l strain w i t h speed p a r a m e t e r for disk of uniform thickness and radius ratio a = 10.
Several values of co, are assumed a n d the v a l u e of e0, satisfying e q u a t i o n (19) is d e t e r m i n e d graphically. D e r i v e d instability strains for materials w i t h values of n b e t w e e n 0-01 a n d 0-10 a n d for values of c0 b e t w e e n 1 a n d 20 are sho~xt in Fig. 2. F o r a m a t e r i a l w i t h a strain-hardening characteristic a = A e " , t h e axial strain a t instability in t h e tension test is n. Fig. 2 shows t h a t the circumferential hoop strain a t the bore of a disk at instability can be m u c h greater t h a n n. I n fact, for t h e two values o f n considered, t h e m a x i m u m instability strain in the disk is greater t h a n t h e instability strain in simple tension for all values of a > 3.3. 29
M. J . PEI,..cr a n d P . B . ~rIELLOR
426
T h e v a l u e s for a = 1 h a v e b e e n c a l c u l a t e d as a s e p a r a t e case. ~ n c n t h i n r o t a t i n g ring, t h e c i r c u m f e r e n t i a l stress is
c~ = 1, t h a t is a
~0 = p~o~ r ~
(20)
w h e r e r is t h e c u r r e n t m e a n r a d i u s o f t h e ring. A t m a x i m m n v e l o c i t y __dao = 2 __dr = 2 de o (~8
(21)
r
or
da.._.oo= 2so dee F o r a m a t e r i a l w i t h a s t r a i n - h a r d e n i n g c h a r a c t e r i s t i c ao = Ar~ t h i s m e a n s t h a t t h e c i r c u m f e r e n t i a l s t r a i n a t i n s t a b i l i t y is n]2, t h a t is h a l f t h e a m o u n t of s t r a i n a t t a i n e d a t i n s t a b i l i t y in s i m p l e t e n s i o n . .45
n=O-lO
= .35
/"
~
I n-o.oa
.
.f
J
~
z
z 92 o
I ~
ri= o . o 6
~
n-o-os
I
i -
n=o.o4
/
I
S.
10
IS RADIUS
20
2S
RATIO
FIG. 2. V a r i a t i o n of t a n g e n t i a l i n s t a b i l i t y s t r a i n a t b o r e w i t h r a d i u s ratio. 9K n o w i n g t h e i n s t a b i l i t y s t r a i n for a g i v e n v a l u e o f a, t h e m a x i m u m s p e e d is d e t e r m i n e d f r o m e q u a t i o n (18). T h e h o o p s t r e s s d i s t r i b u t i o n across t h e d i s k a t a p a r t i c u l a r s p e e d c a n b e d e t e r m i n e d f r o m e q u a t i o n s (12), (13) a n d (18). T h e r a d i a l s t r e s s d i s t r i b u t i o n c a n b e o b t a i n e d b y n u m e r i c a l i n t e g r a t i o n o f e q u a t i o n (9). THEORETICAL ANALYSIS BASED ON THE ~IENCKY DEFORMATION THEORY OF PLASTICITY T h e finite difference m e t h o d o f s o l u t i o n s e t d o ~ b y M a n s o n 2 h a s b e e n a d o p t e d t o obtain results based on the ttencky theory. This theory assumes a one to one relationship b e t w e e n s t r e s s a n d s t r a i n a n d will b e t h e c o r r e c t s o l u t i o n i f i t c a n b e s h o w n a poster~ori t h a t t h e s t r e s s r a t i o s h a v e r e m a i n e d c o n s t a n t t h r o u g h o u t t h e s t r a i n i n g . B e c a u s e of t h e
Theoretical prediction of tensile instability conditions in rotating disks
427
large a m o u n t of computation that is necessary using this method the theory was programmed for an electronic digital computer. Since the Hencky theory as applied to rotating disks has been discussed by m a n y previous authors only results will be quoted here. :It is only necessary to define equivalent stress and strain for principal directions as =
{(~1 - a~) 2 + (a~ - ~3) ~ + (c'3 - ~1)2) z/2 (22)
~2
e= T
{(ez-~2)
2
+
z
(~.~-~3)
+
(e3-~z)q
"~
For the particular case of a thin rotating disk a, = 0 and equation (22) reduces to = ( o ~ - a~ a o + aD i n /
~: ~ (~.f.Crr CORRELATION
BETWEEN
THE SOLUTIONS
(23)
TRESCA
AND
HENCKY
For the purposes of correlation two specific materials were chosen for analysis. The two materials were characterized by ]laving the same tensile strength of 77.5 ton/in s and tensile stress-strain curves represented by a = I07.8e ~176 and a = 94.5~ ~176 respectively. The initial outer diameter of the disks was 4.8 in. BursL speed and associated stress and strain distributions were determined by both theories for hollow disks with ratios of outside radius to inside radius of 5, 10, 15 and 20. Similar calculations were cai'ried out, using the IIencky theory, for solid disks of both materials. I
"
rl m 0 . 1 0
.~0
~
_ .~s
~
7
,,
~.05
0
I
5
I0
RA m u s
IS RA'rIo
20
25
~r
Fla. 3. Correlation of theoretical values of instability strain.
Fig. 3 shows the values of the tangential bore strains at instability, as calculated b y both theories, for differeng values of radius ratio a. There is a m a x i m u m difference between the two theories of approximately 10 per cent in the strain. Also, the trend of
428
~I. J. PERCY and P. B. ~[ELLOIt
the curves is very different at high values of radius ratio; the curves for the H e n c k y solution appear to be asymptotic to a lower value of strain than do the curves for the Tresca solution. However, both theories predict that large tangential bore strains m u s t bo attained before failure can occur by art instability condition. I t will be seen later (Fig. 11) t h a t the tangential strain at instability at the centre of a solid disk is mrtch less t h a n t h a t for the case of a hollow disk with a value of a > 5. T h a t is, drilling a hole at the centre of a disk gives rise to a high strain concentration at the bore. liO
fJJ *o
:2,
.
|os
f
"
{
0
~
~
f
J
f
f
/
J
J
I
/,,
HENCKY "I'RESCA
" /;/
. . . .
NOM. AV. TAN. STRESS = U. 1". S.
DIAMETER I . - - 0 "
/
:L
4
S
I0
4.SINS.
= 1 0 7 . 8 ~-IO ~" ~
IS RADIUS
OF DISK
9 4 . 5 ~-OS
I
20
RATIO
Fro. 4. Correlation of theoretical values of burst speeds.
Tile instability strains are independent of the absolute diameter of tile disk b u t of course the angular velocity at instability does depend on the size of the disk and on the strength of the material. Fig. 4 shows the variation in burst speed with radius ratio for disks made of the two chosen materials and having art initial outside diameter of 4.8 in. The burst speed calculated on the basis t h a t at burst the nominal average tangential stress in the disk corresponds to the tensile strength of the material is also included on this diagram. This latter method of predicting the burst speed has been used extensively as a criterion of the performance of rotating disks. ~obinson 9 and Winne and V~rtmdt1~ tabulated experimental results on disk burst tests and concluded t h a t the burst speed is usually less t h a n b u t occasionally greater t h a n the speed predicted by the above method. Recent experimental work b y ~Valdren and W a r d " on v a c u u m melt material confirmed t h a t the burst speed cart be higher t h a n the prediction based on tensile strength. Taking these points into consideration it would appear t h a t the l=[encky solution gives the more realistic value of burst speed, ttowever, it must be noted t h a t there is only about 6 per cent
Theoretical prediction of tensile instability conditions in r o t a t i n g disks
429
difference in tile p r e d i c t e d speed b e t w e e n the two theories. I f fracture of the disk is preceded b y a n instability condition t h e n the b u r s t speed is affected b y the dimensional changes of t h e disk. F o r two materials w i t h the s a m e tensile s t r e n g t h b u t different rates o f w o r k - h a r d e n i n g t h e b u r s t speed will be higher for t h e m a t e r i a l w i t h t h e lower rate o f work-hardening. This p o i n t is illustrated in Fig. 4. The bttrst speed also varies w i t h t h e a b s o l u t e initial size of t h e disk a n d it can be shown t h a t for a g i v e n v a l u e of radius ratio d o u b l i n g the outside d i a m e t e r of tim disk reduces the b u r s t speed b y half.
4-_ TANGs
~
'~
TANGENTIAL
8C HENCKY TRESCA
%
eJ 9 Z I-0
HENCKY TRESCA
o
0.2 (>4 o~ o-a ,o PROPORTIC~ATC P.A01AL ms'rA~c~.~
Fza. 5. Stress distribution in disk of u n i f o r m thickness, a = 5.
MATERIAl.
-- -
0"= 94,5E"Os
~ /
o
0.2 o.4 0.6 0.8 I.o mOF~C~T~NATE RAOIAL 0~STANCC--~
Fzc. 6. Stress distribution in disk of tmiform thickness, c~ = 15.
Tile stress distribution a t instability across two disks of the s a m e m a t e r i a l b u t h a v i n g radius ratios a = 5 and a = 15 are shown in Figs. 5 and 6 respectively. The shape of t h e t a n g e n t i a l stress distribution d e r i v e d f r o m t h e Tresca t h e o r y is n o t affected v e r y m u c h b y t h e radius ratio. I n d e e d for a n o n - w o r k - h a r d e n i n g m a t e r i a l t h e t a n g e n t i a l stress w o u l d always be distributed t m i f o r m l y across the disk. The shape o f t l m t a n g e n t i a l stress distribution derived using the t t e n c k y t h e o r y changes g r e a t l y as the v a l u e of radius ratio increases. T h e s m o o t h c u r v e of Fig. 5 (o~ = 5) is replaced b y a c u r v e w i t h a n additional inflexion, Fig. 6, and a m a x i m u m t h a t m o v e s closer to tile bore w i t h increasing values of radius ratio. T h e f o r m of t h e radial stress distributions are similar for t h e two theories b u t it will be noticed, f r o m Figs. 5 a n d 6, t h a t t h e m a x i m u m radial stress increases g r e a t l y as increases f r o m a v a l u e of 5 to one of 15. The level of t a n g e n t i a l stress on the o t h e r h a n d ehangcs v e r y little.
430
M. J . PERCY and P. B. ~IELLOrt
The effect of rate of work-hardening on the shape of the tangential stress distribution, as derived by the t t e n c k y theory, is sho~T, qualitatively in Fig. 7 for a disk with radius ratio a = 5. As the rate of work-hardening increases, measured by increasing values of the index n, the m a x i m u m in the tangential stress curves moves towards the bore. For a value n ~0.3 the m a x i m u m tangential stress is at the bore.
r ~ TAN
\ 200
i.os
-o
e'-- ~ rAN.
200
\
-o_
o, Z IZ uJ n. IOO
0"5 S
IOC
ni-
/ 0
I NO ~h~RKHARDENING rl=O NAI~I & DONNE.LL
\
O
/ I
LOW Ih~RKHA.q'I~I~ING n. 0"0 $ MELLOR I PERCY
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'\ I
HiGH VfORKHAROE.NING 1"1-O.3 LEE WU
F i e . 7. Dependence of stress distribution on work-hardening, a = 5. Figs. 8 and 9 give the strain distributions at instability corresponding to the stress distribution in Figs. 5 and 6. I t should be noted that the scale of the strain axis is different in each figure. I t will be remembered t h a t according to the Tresca theory the radial strain component is zero at every point in the disk and t b a t therefore from the assumption of incompressibility the thickness strain is numerically equal to but of opposite sign to the tangential strain. The t t e n c k y solution is more realistic in that at the bore and the periphery the radial and thickness strains are equal and each numerically equal b u t of opposite sign to half the tangential strain. The two theories give fairly close agreement over t h a t part of the disk where the H e n e k y theory predicts a small value for tile radial strain component. Near to the bore there is a very rapid increase in both radial and tangential strains. This strain gradient is especially severe for disks with largo values of radius ratio. The stress and strain distributions at instability in a solid disk, as given by the }Iencky theory, are shown in Figs. 10 and 11 respectively. The predicted burst speeds are 109,370rev/mial for the material a = 9t'5e ~176and 108,710rev/min for the material cr = 107-8e ~176 I t would appear from Fig. 4 t h a t the burst speed c u r v e s for hollow disks of the same materials are asymptotic to the respective speed values for the solid disks. I t should also be noted t h a t the values of tangential stress at the bore of the hollow disks, Figs. 5 and 6, and at the centre of the solid disk, Fig. 10, are almost identical, although of course the radial stress component has now changed from z e r o to a value equal to the tangential stress. Another point to notice is t h a t the instability strain at the centre of the solid disk, Fig. 11, is much less than t h a t at the bore of a hollow disk (see Fig. 3) and the tangential strain is more uniformly distributed across the disk. I f it can be demonstrated t h a t the principaI stress ratios at every point in the disk, as derived by the t I e n c k y theory, are constant throughout the straining then the t t e n c k y and L d v y - v o n Mises equations become identical and more reliance can be placed on the
T h e o r e t i c a l p r e d i c t i o n of tensile i n s t a b i l i t y c o n d i t i o n s i n r o t a t i n g disks
431
*08
*07
-O~
,05
TANGENTIAL
_~ * 0 2
Z
.OI
\
/
-.01
RADIAL
/
/ --.03
l
--.O4 0
TANG~ NTIAL
-
-
HENCI(Y TRESCA
...~-'--'~ ---......~....~ RADIAL
--
1 MATERIAL r "~ 0.2 O'4 0.6 0.8 I-O PROPORTIONATE RADIAL DISTANCE r-
b
FIG. 8. S t r a i n d i s t r i b u t i o n in disk o f u n i f o r m t h i c k n e s s , c~ = 5.
H ENCKY TRE 5C/k
I O
.2 .4 .6 PROPORTIONATE RADIAL
.8 I-O DISTANCE ..~
b
Fzo. 9. S t r a i n d i s t r i b u t i o n i n disk o f u n i f o r m t h i c k n e s s , a = 15.
432
BI. J . PERCY a n d P . B . ~IELLOR
results. The variation of stress ratio across the disk as strain proceeds is shown in Fig. 12 for a solid disk and for a hoUow disk with a = 10. I t can be sccn t h a t there is some small change in the stress ratios as the deformation proceeds, b u t it is thought t h a t for practical purposes the effect on the final results will be negligible. .15
ENTI L
I.
r
2.-'--- Ir'=lO7.B t "I0
I
"z'
g
~
~
f TANGEN'T IAL
N4 o
20"
2.
r
IO7"8
DtAMF_.TEROFDISK
4.8~NS
I DIAMETER OF DISK 4.8 INS
1
O .2 0.4 0-6 Oe I-O PROPORTIONATE RADIAL DISTANCE r'
0"2 0"4 0"6 0"8 I-O PROPORTIONATE RADIAL DISTANCE.r_
Fro. 10. Stress distribution in solid disk.
Fro. l l . Strain distribution in solid disk.
O
b
b
CONCLUSIONS The Hencky deformation theory has often been criticized as being basically incorrect and indced no one nowadays would attempt to give this theory general application. The tIcncky equations have of course been applied to problems where the L6vy-von Miscs equations have proved intractable and in
most cases no attempt has been made to justify their use. S o m e a u t h o r s h a v e t a k e n t h e s t a n d t h a t i t is b e t t e r t o u s e t h e T r c s c a y i e l d
criterion and the associated flow rule where this simplifies the mathematics. While of course a flow rule theory is basically sound the Trcsca flow rule introduces its own peculiar complexities. In the first place experiment shows t h a t t h e T r e s c a y i e l d c r i t e r i o n is n o t a s a p p r o p r i a t e
as the yon Mises yield
criterion. Secondly, when the problem includes a singular point on the yield locus further complications arise. The flow rule is not uniquely determined at such a singularity., In this problem this complication has been avoided by the simple device of applying Koiter's restriction. O n e way of looking at this
Theoretical prediction of tensile instability conditions in r o t a t i n g disks RAD~$1N~
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---~
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0"8 1.00
_
I
-- I,SO
I
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1.75
I
O'4
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~--
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I 0.2
240
0"~
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EQUIVALENT STRAIN AT CENTRE O.
o.~
!
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i i ~--o78
0",
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t I I ~---o.~,
, ~
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i
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(
2.13
i
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I
-- 0.27 1 )'24, 2.4
0.2
STRAIN AT BORE
b.
/7Ic. 12. Variation of stress ratio w i t h bore or centre strain. Material: a = 107e ~ (a) Solid disk. (b) t t o l l o w disk.
433
434
M . J . P E R C Y and P. B. I%IELLOI%
problem is to imagine a v e r y small b u t positive radial stress a t the bore a n d a t the periphery. P e r h a p s a greater error is i n t r o d u c e d b y assuming t h a t the hoop stress and strain can be identified with the longitudinal stress and strain in the simple tensile test. I t is obvious t h a t the strain relations in a simple tensile test do n o t conform to e q u a t i o n (2). T a k i n g into a c c o u n t the above r e m a r k s it is perhaps surprising t h a t there is a fair degree of correlation between the two thcorics. T h e main discrepancies appear, as would be expected, a t the bore and the p e r i p h e r y where the Tresca t h e o r y holds the radial strain to zcro. Strictly, the H e n c k y t h e o r y is n o t valid if the stress ratios v a r y t h r o u g h o u t the deformation. H o w e v e r , the variations in the cases investigated are fairly small a n d it is proposed to test the practical usefulness of the t h e o r y b y detailed correlation with e x p e r i m e n t . Acknowledgement--This work was supported by the Ministry of Aviation.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
REFERENCES !%I.H. LEE %Vu, N.A.C.A. Tech. Note 2217 (1950). S. S. MA~SOh-, Proc. 1st U.S. 1Vat. Gongr. Appl. Mech. p. 569 (1951). M. ZAXD,J . Aero. Sci. 20, 369 (1953). H. J. %VEIss and %V. P~AeER, J . Aero. Sci. 21, 196 (1954). M. M. S.~nTH, Office of Naval Research, Tech. Report No. 35 (1958). P. B. i%IELLORand M. J. PERCY, A.R.C. Current Paper No. 692 (1963). W. T. KOITEn, Quart. J . Appl. Math. 11,350 (1953). D. R. BriAnD, J . Mech. Phys. Solids 6, 71 (1957). E. L. ROBIIqSOh', Trans. Amer. Soc. l]iech. Engrs 66, 373 (1944). D. It. %VIhn~'Eand B. BI. %VUNDT,Trans. Amer. Soc. Mech. Engrs 80, 1643 (1958). N. E. %VALD~E~"and D. E. %VARD,A.R.C. Current Paper No. 660 (1963).
THEORETICAL PREDICTION OF TENSILE INSTABILITY CONDITIONS IN ROTATING DISKS ]~[. J . PERCY* a n d P. B. ~[ELLOR'~ (.Received 15 June 1964)
Stmrmary--An analysis of conditions at instability in rotating disks of uniform thickness is presented. Two plasticity theories are applied. The first analysis based on the Tresca yield criterion and its associated flow rule is an extension of the work of Prager and Weiss. The results of this analysis are correlated with results based on the Heneky deformation theory of plasticity using a finite difference method of solution. Both theories show that if fracture is preceded by a tensile instability condition then large strains occur at the bore of a hollow disk. NOTATION
A a, b
ao, be er, eo, ez 11 re 8 r, 80, 8z
t to Y 0s s
CO, Ez
O (Yf~ 0"0~,
(3"z
O)
constant in strain-hardening relation current inner and outer radii of disk initial inner and outer radii of disk engineering strains in radial, circumferential and axial directions constant in strain-hardening relation current radius to particle initial radius to particle nominal stress in radial, circumferential and axial direction current thickness of disk of radius r initial uniform thickness of disk radial displacement of particle tensile yield stress radius ratio, bo/ao natural straits in radial, circumferential and axial directions u[ao ro/ao density of material true stress in radial, circumferential and axial directions angular velocity of rotating disk INTRODUCTION
WHE~ the angular velocity of a thin circular disk at room temperature is g r a d u a l l y increased the disk first o f all b e h a v e s elastically then, a t a certain velocity, plastic yielding begins a n d procceds, w i t h w o r k - h a r d e n i n g , until all t h e m a t e r i a l o f the disk h a s r e a c h e d t h e initial yield point. F u r t h e r increase in speed causes u n r e s t r i c t e d plastic flow to t a k e place. T h e s y s t e m is stable as long as a n increase in speed is n e c e s s a r y to p r o d u c e f u r t h e r plastic flow. T h e following analyses are r e s t r i c t e d to conditions of large plastic s t r a i n a n d it will be a s s u m e d t h a t the elastic strains are negligible in c o m p a r i s o n w i t h the plastic strains. I t will also be a s s u m e d t h a t t h e m a t e r i a l is h o m o g e n e o u s a n d isotropie. F o r the a b o v e conditions it is generally agreed t h a t the L d v y - v o n Mises i n c r e m e n t a l s t r e s s - s t r a i n relationships are the m o s t a c c e p t a b l e but, to a v o i d * Department of l~Iechanical Engineering, University of Liverpoo!. Department of Mechanical Engineering, University of Birmingham. 421
422
i%I.J. PERCY and P. ]3.1%~ELLOR
the large amount of computation that these relationships entail, methods of reducing the mathematical work have been sought. Lee %Vu I, ~Ianson z and Zaid 3 simplified the computation by assuming a yon Mises yield criterion and t h e H e n c k y d e f o r m a t i o n t h e o r y o f p l a s t i c i t y i n s t e a d o f t h e L 4 v y - v o n 5Iises flow t h e o r y . L e e W u a n d Z a i d j u s t i f i e d t h i s s i m p l i f i c a t i o n f o r t h e c a s e o f s o l i d disks of uniform thickness by showing a posteriori that the strain ratios remain p r a c t i c a l l y c o n s t a n t a s t h e s p e e d is i n c r e a s e d . W e i s s a n d P r a g e r ~ m a d e a n alternative assumption that the disk material obeys Tresca's yield criterion a n d t h e a s s o c i a t e d flow r u l e , a n d b a s i n g t h e i r s o l u t i o n o n t h e n o m i n a l t e n s i l e stress-strain curve derived results for hollow disks of uniform thickness. This p a r t i c u l a r c h o i c e o f y i e l d c r i t e r i o n a n d flow r u l e s i m p l i f i e s t h e a n a l y s i s s i n c e then one of the strain components reduces to zero. Smith 5 adopted the same assumption in an analysis of rotating plastic disks of variable thickness, and Mellor and Percy G extended the theory of Weiss and Prager to larger strain values. I n t i l e f o l l o w i n g w o r k r e s u l t s b a s e d o n t h e T r e s c a flow r u l e a r e c o m p a r e d , for the case of hollow disks of initial uniform thickness, with results from the H e n c k y t y p e s o l u t i o n . T h e c o n d i t i o n s a t i n s t a b i l i t y , t h a t is a t t h e m a x i m u m speed attained by the disk, are of particular interest. THEORETICAL ANALYSIS BASED ON THE TRESCA YIELD CRITERION AND THE ASSOCIATED FLOW RULE The principal stress components, for a circular disk rotating about the central axis, are the circumferential stress a0, the radial stress ar and the axial stress az. The axial stress, (;~, will be assumed to be zero for the thin disks to be considered here. F o r a hollow disk of initial uniform thickness unloaded a t the boundaries it can be shown 4 t h a t ae and (7r are both tensile and t h a t a0 is every~vhere greater t h a n at. Tresca's yield criterion and flow rule can then be written ao= a>a,>0, az= 0 (1) d e r = O, dee = - - d e , ~ O (2) where a is twice the critical shearing stress and dee, der and de, are respectively the circumferential, radial and axial strain increments. I t is necessary to enquire if the above relations hold when ar = 0, t h a t is a t the unloaded boundaries of the disk where the stress system is one of uniaxial tension. This corresponds to " a corner" of the Tresca hexagonal yield surface. Koiter: suggested t h a t the above equations should hold even when ar = 0, b u t this restriction can be shown to be too strong. (See, for example, BlandS.) :However, to avoid a disconthmity in the solution of this particular problem it is advantageous to a d o p t Koitcr's restriction and assume t h a t the above equations are applicable a t all points on the disk. Further, if it is assumed t h a t the above relations hold for simple tension then the strain-hardening relation can logically be assumed to be based on the simple tensile stress-strain curve. The consequences of following this course will be discussed later. Let u be the radial displacement of a particle on the disk surface. Then, since from the first equation (2) the radial strain increment is zero, this displacement u does n o t depend on the initial radius of the particle r 0 b u t only on the angular velocity. Thus an annulus bounded initially b y radii r o and r o + dro is a t a particular speed (after undergoing a radial displacement, u) bounded b y radii r = r o + u and r + d r = r o + u + d r o from which it follows t h a t d r = dro. Also, suppose t h a t the initial thickness of the disk t o has now a t this particular radius a thickness t. Since the material is assumed to be incompressible to ro dro = tr d r
or
to ro = tr
(3)
Theoretical predictions of tensile instability conditions in r o t a t i n g disks
423
The t r u e stress, a,, is t r a n s m i t t e d across a section t h a t is proportional to tr and since tr = t o re trr is e q u a l to t h e nominal radial stress s t. The t r u e circumferential stress, a0, is t r a n s m i t t e d across a section p r o p o r t i o n a l to t d r and i s therefore related to the nominal circumferential stress, s e, b y t h e relation so(to dro) = uo(t dr)
a n d since dr o = d r so = no(tire)
(4)
This n o m i n a l stress is related to t h e circumferential engineering strain U
e0 = -
(5)
s =1(0
(6)
re
b y t h e s t r a i n - h a r d e n i n g law where s is the n o m i n a l stress and e t h e longitudinal engineering strain in t h e simple tensile test. On the basis of equations (1) a n d (2) s will be i n t e r p r e t e d as t h e c o n v e n t i o n a l hoop stress so a n d e as the engineering hoop strain eo. T h e n so = f ( e o )
(7)
Finally, t h e e q u a t i o n of radial e q u i l i b r i u m in the deformed s t a t e is ~(rtcrr)]Sr = t a o - - t p o J ~ r 2
(8)
where m is the a n g u l a r v e l o c i t y a n d p is the d e n s i t y of the material. W i t h reference to the u n d e r f o r m e d state, e q u a t i o n (8) can be w r i t t e n as 8(re sr)/~ro = ao-- poJ 2 r e ( r e + U )
(9)
= f ( u / r o ) -- p w 2 ro(r o + u )
I f t h e disk is n o t loaded at t h e bore, r o = ao, or a t t h e periphery, r o = be, t h e n s r is zero at these radii a n d the integral o f t h e r i g h t - h a n d side of e q u a t i o n (9) b e t w e e n t h e l i m i t s a o and b o m u s t vanish. P u t t i n g c~ = bo/a o, 71 = u / a o and ~ = ro/ao, this condition yields pay oJ~16 =
f/
f ( T / ~ ) d ~ / [ 2 ( a3 - I) + 3T(a 2 - 1 ) ]
(I0)
This is t h e e q u a t i o n derived b y Weiss a n d PragerL F o r a k n o w n strain-hardening characteristic a n d g i v e n initial dimensions t h e r i g h t - h a n d side of e q u a t i o n (10) m u s t be e v a l u a t e d , a n a l y t i c a l l y or numerically, for various values of T = u/ao" E a c h assumed v a l u e of T gives t h e corresponding v a l u e o f m ~ and a plot of T against ra 2 can be carried o u t to d e t e r m i n e t h e m a x i m u m v a l u e of angular v e l o c i t y and hence t h e p o i n t of instability. E q u a t i o n (10) is based on t h e nominal stress-engineering strain c u r v e d e t e r m i n e d in simple tension a n d t h e e q u a t i o n used in this form gives valid and easily derived results p r o v i d e d the longitudinal strain a t instability in simple tension is greater t h a n t h e m a x i m u m circumferential strain at instability of the r o t a t i n g disk. This is so for t h e p a r t i c u l a r e x a m p l e g i v e n b y Weiss and P r a g e r 4 for an a l u m i n i u m alloy w i t h a = 2. I{owever, it will be shown t h a t in general w h e n a > 3 t h e n t h e m a x i m u m strain a t instability o f t h e disk is g r e a t e r t h a n t h e instability strain in simple tension. T h e r e is no reason w h y the solution should be limited b y t h e a m o u n t of tnfiform strain t h a t can be a t t a i n e d in simple tension a n d in t h e following analysis t h e solution will be derived in t e r m s of a n empirical t r u e stress, n a t u r a l strain relationship. A s t r a i n . h a r d e n i n g law o0 = A ~
(H)
will be a d o p t e d , where A a n d n are c o n s t a n t s for a p a r t i c u l a r disk. This relation does n o t give full r e p r e s e n t a t i o n of a rigid-plastic material, since there is no initial yield point, b u t it is a simple a n d useful expression for discussion of conditions at large plastic strains a n d in p a r t i c u l a r a t instability.
424
M . J . PERCX" a n d P . B . MELLOR
,The r e l a t i o n b e t w e e n t r u e c i r c u m f e r e n t i a l stress a n d n o m i n a l c i r c u m f e r e n t i a l s t r e s s is
ao -- ~o(1+eo) = So(1 +'11~)
(12)
a n d t h e n a t u r a l s t r a i n is r e l a t e d to t h e e n g i n e e r i n g s t r a i n b y eo = In ( l + e o ) = In ( 1 + , / [ ~ )
(13)
Therefore, m a k i n g use o f e q u a t i o n s (11), (12) a n d (13), t h e f u n c t i o n i n e q u a t i o n (10) can be written
f:.f
('T/D d~ = A
f:[In(l+v/D]"d$ (1 + ~ / D
9 N o w c h a n g e t h e v a r i a b l e f r o m ~ t o In (1 + ~ / ~ ) = e0. D e n o t i n g t h e c i r c u m f e r e n t i a l s t r a i n a t t h e b o r e b y Eo, a n d a t t h e p e r i p h e r y b y COba n d r e m e m b e r i n g t h a t co. = In ( 1 + ~ )
eeb = In (1 +:q[=) the expression becomes
f~J('Tff)d$= A af'o% ~ dee % (exp co-- 1) ~ F o r a g i v e n v a l u e o f co~, ,/ -- u[r o is c o n s t a n t a n d t h e r e f o r e e q u a t i o n (1O) c a n b e w r i t t e n
pa~ co~
"qJ'ob(exp-7--- 1) ~ dee
6A
[2(c~3 - l) + 3,7(az - I)]
(14)
To determine the integral of the above equation the term e0/(exp e0- I) will be expressed as a series eo = 1-(exp e0 -- 1)
+
Ble~ B3e~ Bst~ -+-2!
4!
w h e r e to < 2rr a n d t h e B e r n o u i l l i n u m b e r s h a v e t h e v a h m s B z [ , Bs = ~ , B5 U s i n g t h i s series it c a n b e s h o w n t h a t t h e r i g h t - h a n d side o f t h e e q u a t i o n 5
(15)
6!
2
I (expeo ~' e0-- I)1' ~ l--ee+~-~eo---~
_!_
(16)
is correct to better than 0.06 per cent for 0
a'O~
6A
2(a a -- 1) + 3~(a 2 -- 1)
(17)
I n t e g r a t i n g e q u a t i o n (17)
lea-'
eb'+ 5 eg+'
eg+= ]'o,
~- ~qt~--~----1 n 12"n+l 12(-~2)J,0b Pa~176= 6A [2(a 3 -- 1) + 3r)(a 2 -- 1)] =
(18)
I f for a p a r t i c u l a r r a d i u s r a t i o a a v a l u e o f r/ is a s s u m e d , t h e n co, a n d eo~ c a n b e c a l c u l a t e d . H e n c e t h e c o r r e s p o n d i n g v a l u e of w ~ c a n b e d e t e r m i n e d f r o m e q u a t i o n (18). V~rhen t h i s is r e p e a t e d a p l o t o f oJ2 a g a i n s t eo, c a n b e o b t a i n e d . S u c h c u r v e s are s h o w n in Fig. 1 for m a t e r i a l s w i t h n = 0.05 a n d n = 0-1 a n d for a v a l u e of a = 10. ( T h e v a l u e of t h e s t r a i n - h a r d e n i n g i n d e x n for s o m e t u r b i n e disk m a t e r i a l s lies b e t w e e n 0-05 a n d 0.1.) T h e v a l u e of t h e m a x i m u m v e l o c i t y c a n b e a c c u r a t e l y d e t e r m i n e d f r o m c u r v e s of r ~a g a i n s t es,, b u t i n s p e c t i o n of Fig. 1 will s h o w t h a t i t is n o t possible to d e t e r m i n e a c c u r a t e l y
Theoretical prediction of tensile instability conditions in r o t a t i n g disks
425
t h e v a l u e of co,, a t instability. T o d e t e r m i n e t h e instability strain e q u a t i o n (18) is differentiated w i t h respect to ~7 and e q u a t e d to zero,
d(oJ~)
2(=~_1)
.-z_
d - - ~ = O = 2(oca_l)T3~((x.,_l )L
._
.-z
0~--1)
.
n
5 I~nq-1 ~.COa - - ~.n+l~ Ob I " 1 I~n-l-2 ,o0o -
+ 12 -I-
[(~Oa "-'* --F~Oa "-' )
( n + 1)
12
~n+2~l 0b //
~
(eOlb'-2--eOlb'-l) 1 1, s e no.-- ~ no.§ ,
J I
(~. .
. .
§
!]
(19)
,004~1
MAX.. ] ~ - " fl-0"0$
JJ f
MAX. ~
"oo3(
"~
/
fl=~lO
9002
*0020
0
0-05
0.I0
0.I$
0.20
0.25
0-30
FIG. 1. Variation of bore t a n g e n t i a l strain w i t h speed p a r a m e t e r for disk of uniform thickness and radius ratio a = 10.
Several values of co, are assumed a n d the v a l u e of e0, satisfying e q u a t i o n (19) is d e t e r m i n e d graphically. D e r i v e d instability strains for materials w i t h values of n b e t w e e n 0-01 a n d 0-10 a n d for values of c0 b e t w e e n 1 a n d 20 are sho~xt in Fig. 2. F o r a m a t e r i a l w i t h a strain-hardening characteristic a = A e " , t h e axial strain a t instability in t h e tension test is n. Fig. 2 shows t h a t the circumferential hoop strain a t the bore of a disk at instability can be m u c h greater t h a n n. I n fact, for t h e two values o f n considered, t h e m a x i m u m instability strain in the disk is greater t h a n t h e instability strain in simple tension for all values of a > 3.3. 29
M. J . PEI,..cr a n d P . B . ~rIELLOR
426
T h e v a l u e s for a = 1 h a v e b e e n c a l c u l a t e d as a s e p a r a t e case. ~ n c n t h i n r o t a t i n g ring, t h e c i r c u m f e r e n t i a l stress is
c~ = 1, t h a t is a
~0 = p~o~ r ~
(20)
w h e r e r is t h e c u r r e n t m e a n r a d i u s o f t h e ring. A t m a x i m m n v e l o c i t y __dao = 2 __dr = 2 de o (~8
(21)
r
or
da.._.oo= 2so dee F o r a m a t e r i a l w i t h a s t r a i n - h a r d e n i n g c h a r a c t e r i s t i c ao = Ar~ t h i s m e a n s t h a t t h e c i r c u m f e r e n t i a l s t r a i n a t i n s t a b i l i t y is n]2, t h a t is h a l f t h e a m o u n t of s t r a i n a t t a i n e d a t i n s t a b i l i t y in s i m p l e t e n s i o n . .45
n=O-lO
= .35
/"
~
I n-o.oa
.
.f
J
~
z
z 92 o
I ~
ri= o . o 6
~
n-o-os
I
i -
n=o.o4
/
I
S.
10
IS RADIUS
20
2S
RATIO
FIG. 2. V a r i a t i o n of t a n g e n t i a l i n s t a b i l i t y s t r a i n a t b o r e w i t h r a d i u s ratio. 9K n o w i n g t h e i n s t a b i l i t y s t r a i n for a g i v e n v a l u e o f a, t h e m a x i m u m s p e e d is d e t e r m i n e d f r o m e q u a t i o n (18). T h e h o o p s t r e s s d i s t r i b u t i o n across t h e d i s k a t a p a r t i c u l a r s p e e d c a n b e d e t e r m i n e d f r o m e q u a t i o n s (12), (13) a n d (18). T h e r a d i a l s t r e s s d i s t r i b u t i o n c a n b e o b t a i n e d b y n u m e r i c a l i n t e g r a t i o n o f e q u a t i o n (9). THEORETICAL ANALYSIS BASED ON THE ~IENCKY DEFORMATION THEORY OF PLASTICITY T h e finite difference m e t h o d o f s o l u t i o n s e t d o ~ b y M a n s o n 2 h a s b e e n a d o p t e d t o obtain results based on the ttencky theory. This theory assumes a one to one relationship b e t w e e n s t r e s s a n d s t r a i n a n d will b e t h e c o r r e c t s o l u t i o n i f i t c a n b e s h o w n a poster~ori t h a t t h e s t r e s s r a t i o s h a v e r e m a i n e d c o n s t a n t t h r o u g h o u t t h e s t r a i n i n g . B e c a u s e of t h e
Theoretical prediction of tensile instability conditions in rotating disks
427
large a m o u n t of computation that is necessary using this method the theory was programmed for an electronic digital computer. Since the Hencky theory as applied to rotating disks has been discussed by m a n y previous authors only results will be quoted here. :It is only necessary to define equivalent stress and strain for principal directions as =
{(~1 - a~) 2 + (a~ - ~3) ~ + (c'3 - ~1)2) z/2 (22)
~2
e= T
{(ez-~2)
2
+
z
(~.~-~3)
+
(e3-~z)q
"~
For the particular case of a thin rotating disk a, = 0 and equation (22) reduces to = ( o ~ - a~ a o + aD i n /
~: ~ (~.f.Crr CORRELATION
BETWEEN
THE SOLUTIONS
(23)
TRESCA
AND
HENCKY
For the purposes of correlation two specific materials were chosen for analysis. The two materials were characterized by ]laving the same tensile strength of 77.5 ton/in s and tensile stress-strain curves represented by a = I07.8e ~176 and a = 94.5~ ~176 respectively. The initial outer diameter of the disks was 4.8 in. BursL speed and associated stress and strain distributions were determined by both theories for hollow disks with ratios of outside radius to inside radius of 5, 10, 15 and 20. Similar calculations were cai'ried out, using the IIencky theory, for solid disks of both materials. I
"
rl m 0 . 1 0
.~0
~
_ .~s
~
7
,,
~.05
0
I
5
I0
RA m u s
IS RA'rIo
20
25
~r
Fla. 3. Correlation of theoretical values of instability strain.
Fig. 3 shows the values of the tangential bore strains at instability, as calculated b y both theories, for differeng values of radius ratio a. There is a m a x i m u m difference between the two theories of approximately 10 per cent in the strain. Also, the trend of
428
~I. J. PERCY and P. B. ~[ELLOIt
the curves is very different at high values of radius ratio; the curves for the H e n c k y solution appear to be asymptotic to a lower value of strain than do the curves for the Tresca solution. However, both theories predict that large tangential bore strains m u s t bo attained before failure can occur by art instability condition. I t will be seen later (Fig. 11) t h a t the tangential strain at instability at the centre of a solid disk is mrtch less t h a n t h a t for the case of a hollow disk with a value of a > 5. T h a t is, drilling a hole at the centre of a disk gives rise to a high strain concentration at the bore. liO
fJJ *o
:2,
.
|os
f
"
{
0
~
~
f
J
f
f
/
J
J
I
/,,
HENCKY "I'RESCA
" /;/
. . . .
NOM. AV. TAN. STRESS = U. 1". S.
DIAMETER I . - - 0 "
/
:L
4
S
I0
4.SINS.
= 1 0 7 . 8 ~-IO ~" ~
IS RADIUS
OF DISK
9 4 . 5 ~-OS
I
20
RATIO
Fro. 4. Correlation of theoretical values of burst speeds.
Tile instability strains are independent of the absolute diameter of tile disk b u t of course the angular velocity at instability does depend on the size of the disk and on the strength of the material. Fig. 4 shows the variation in burst speed with radius ratio for disks made of the two chosen materials and having art initial outside diameter of 4.8 in. The burst speed calculated on the basis t h a t at burst the nominal average tangential stress in the disk corresponds to the tensile strength of the material is also included on this diagram. This latter method of predicting the burst speed has been used extensively as a criterion of the performance of rotating disks. ~obinson 9 and Winne and V~rtmdt1~ tabulated experimental results on disk burst tests and concluded t h a t the burst speed is usually less t h a n b u t occasionally greater t h a n the speed predicted by the above method. Recent experimental work b y ~Valdren and W a r d " on v a c u u m melt material confirmed t h a t the burst speed cart be higher t h a n the prediction based on tensile strength. Taking these points into consideration it would appear t h a t the l=[encky solution gives the more realistic value of burst speed, ttowever, it must be noted t h a t there is only about 6 per cent
Theoretical prediction of tensile instability conditions in r o t a t i n g disks
429
difference in tile p r e d i c t e d speed b e t w e e n the two theories. I f fracture of the disk is preceded b y a n instability condition t h e n the b u r s t speed is affected b y the dimensional changes of t h e disk. F o r two materials w i t h the s a m e tensile s t r e n g t h b u t different rates o f w o r k - h a r d e n i n g t h e b u r s t speed will be higher for t h e m a t e r i a l w i t h t h e lower rate o f work-hardening. This p o i n t is illustrated in Fig. 4. The bttrst speed also varies w i t h t h e a b s o l u t e initial size of t h e disk a n d it can be shown t h a t for a g i v e n v a l u e of radius ratio d o u b l i n g the outside d i a m e t e r of tim disk reduces the b u r s t speed b y half.
4-_ TANGs
~
'~
TANGENTIAL
8C HENCKY TRESCA
%
eJ 9 Z I-0
HENCKY TRESCA
o
0.2 (>4 o~ o-a ,o PROPORTIC~ATC P.A01AL ms'rA~c~.~
Fza. 5. Stress distribution in disk of u n i f o r m thickness, a = 5.
MATERIAl.
-- -
0"= 94,5E"Os
~ /
o
0.2 o.4 0.6 0.8 I.o mOF~C~T~NATE RAOIAL 0~STANCC--~
Fzc. 6. Stress distribution in disk of tmiform thickness, c~ = 15.
Tile stress distribution a t instability across two disks of the s a m e m a t e r i a l b u t h a v i n g radius ratios a = 5 and a = 15 are shown in Figs. 5 and 6 respectively. The shape of t h e t a n g e n t i a l stress distribution d e r i v e d f r o m t h e Tresca t h e o r y is n o t affected v e r y m u c h b y t h e radius ratio. I n d e e d for a n o n - w o r k - h a r d e n i n g m a t e r i a l t h e t a n g e n t i a l stress w o u l d always be distributed t m i f o r m l y across the disk. The shape o f t l m t a n g e n t i a l stress distribution derived using the t t e n c k y t h e o r y changes g r e a t l y as the v a l u e of radius ratio increases. T h e s m o o t h c u r v e of Fig. 5 (o~ = 5) is replaced b y a c u r v e w i t h a n additional inflexion, Fig. 6, and a m a x i m u m t h a t m o v e s closer to tile bore w i t h increasing values of radius ratio. T h e f o r m of t h e radial stress distributions are similar for t h e two theories b u t it will be noticed, f r o m Figs. 5 a n d 6, t h a t t h e m a x i m u m radial stress increases g r e a t l y as increases f r o m a v a l u e of 5 to one of 15. The level of t a n g e n t i a l stress on the o t h e r h a n d ehangcs v e r y little.
430
M. J . PERCY and P. B. ~IELLOrt
The effect of rate of work-hardening on the shape of the tangential stress distribution, as derived by the t t e n c k y theory, is sho~T, qualitatively in Fig. 7 for a disk with radius ratio a = 5. As the rate of work-hardening increases, measured by increasing values of the index n, the m a x i m u m in the tangential stress curves moves towards the bore. For a value n ~0.3 the m a x i m u m tangential stress is at the bore.
r ~ TAN
\ 200
i.os
-o
e'-- ~ rAN.
200
\
-o_
o, Z IZ uJ n. IOO
0"5 S
IOC
ni-
/ 0
I NO ~h~RKHARDENING rl=O NAI~I & DONNE.LL
\
O
/ I
LOW Ih~RKHA.q'I~I~ING n. 0"0 $ MELLOR I PERCY
O
'\ I
HiGH VfORKHAROE.NING 1"1-O.3 LEE WU
F i e . 7. Dependence of stress distribution on work-hardening, a = 5. Figs. 8 and 9 give the strain distributions at instability corresponding to the stress distribution in Figs. 5 and 6. I t should be noted that the scale of the strain axis is different in each figure. I t will be remembered t h a t according to the Tresca theory the radial strain component is zero at every point in the disk and t b a t therefore from the assumption of incompressibility the thickness strain is numerically equal to but of opposite sign to the tangential strain. The t t e n c k y solution is more realistic in that at the bore and the periphery the radial and thickness strains are equal and each numerically equal b u t of opposite sign to half the tangential strain. The two theories give fairly close agreement over t h a t part of the disk where the H e n e k y theory predicts a small value for tile radial strain component. Near to the bore there is a very rapid increase in both radial and tangential strains. This strain gradient is especially severe for disks with largo values of radius ratio. The stress and strain distributions at instability in a solid disk, as given by the }Iencky theory, are shown in Figs. 10 and 11 respectively. The predicted burst speeds are 109,370rev/mial for the material a = 9t'5e ~176and 108,710rev/min for the material cr = 107-8e ~176 I t would appear from Fig. 4 t h a t the burst speed c u r v e s for hollow disks of the same materials are asymptotic to the respective speed values for the solid disks. I t should also be noted t h a t the values of tangential stress at the bore of the hollow disks, Figs. 5 and 6, and at the centre of the solid disk, Fig. 10, are almost identical, although of course the radial stress component has now changed from z e r o to a value equal to the tangential stress. Another point to notice is t h a t the instability strain at the centre of the solid disk, Fig. 11, is much less than t h a t at the bore of a hollow disk (see Fig. 3) and the tangential strain is more uniformly distributed across the disk. I f it can be demonstrated t h a t the principaI stress ratios at every point in the disk, as derived by the t I e n c k y theory, are constant throughout the straining then the t t e n c k y and L d v y - v o n Mises equations become identical and more reliance can be placed on the
T h e o r e t i c a l p r e d i c t i o n of tensile i n s t a b i l i t y c o n d i t i o n s i n r o t a t i n g disks
431
*08
*07
-O~
,05
TANGENTIAL
_~ * 0 2
Z
.OI
\
/
-.01
RADIAL
/
/ --.03
l
--.O4 0
TANG~ NTIAL
-
-
HENCI(Y TRESCA
...~-'--'~ ---......~....~ RADIAL
--
1 MATERIAL r "~ 0.2 O'4 0.6 0.8 I-O PROPORTIONATE RADIAL DISTANCE r-
b
FIG. 8. S t r a i n d i s t r i b u t i o n in disk o f u n i f o r m t h i c k n e s s , c~ = 5.
H ENCKY TRE 5C/k
I O
.2 .4 .6 PROPORTIONATE RADIAL
.8 I-O DISTANCE ..~
b
Fzo. 9. S t r a i n d i s t r i b u t i o n i n disk o f u n i f o r m t h i c k n e s s , a = 15.
432
BI. J . PERCY a n d P . B . ~IELLOR
results. The variation of stress ratio across the disk as strain proceeds is shown in Fig. 12 for a solid disk and for a hoUow disk with a = 10. I t can be sccn t h a t there is some small change in the stress ratios as the deformation proceeds, b u t it is thought t h a t for practical purposes the effect on the final results will be negligible. .15
ENTI L
I.
r
2.-'--- Ir'=lO7.B t "I0
I
"z'
g
~
~
f TANGEN'T IAL
N4 o
20"
2.
r
IO7"8
DtAMF_.TEROFDISK
4.8~NS
I DIAMETER OF DISK 4.8 INS
1
O .2 0.4 0-6 Oe I-O PROPORTIONATE RADIAL DISTANCE r'
0"2 0"4 0"6 0"8 I-O PROPORTIONATE RADIAL DISTANCE.r_
Fro. 10. Stress distribution in solid disk.
Fro. l l . Strain distribution in solid disk.
O
b
b
CONCLUSIONS The Hencky deformation theory has often been criticized as being basically incorrect and indced no one nowadays would attempt to give this theory general application. The tIcncky equations have of course been applied to problems where the L6vy-von Miscs equations have proved intractable and in
most cases no attempt has been made to justify their use. S o m e a u t h o r s h a v e t a k e n t h e s t a n d t h a t i t is b e t t e r t o u s e t h e T r c s c a y i e l d
criterion and the associated flow rule where this simplifies the mathematics. While of course a flow rule theory is basically sound the Trcsca flow rule introduces its own peculiar complexities. In the first place experiment shows t h a t t h e T r e s c a y i e l d c r i t e r i o n is n o t a s a p p r o p r i a t e
as the yon Mises yield
criterion. Secondly, when the problem includes a singular point on the yield locus further complications arise. The flow rule is not uniquely determined at such a singularity., In this problem this complication has been avoided by the simple device of applying Koiter's restriction. O n e way of looking at this
Theoretical prediction of tensile instability conditions in r o t a t i n g disks RAD~$1N~
1
---~
o.rQ
0"8 1.00
_
I
-- I,SO
I
0"5 --
1.75
I
O'4
0"~
~--
2.00
I 0.2
240
0"~
0"1
o
I O'1
EQUIVALENT STRAIN AT CENTRE O.
o.~
!
I
i i ~--o78
0",
1.32
t I I ~---o.~,
, ~
I I r I
0.3
0-~
i
o., I 0
0.1
EQUIVALENT
(
2.13
i
I
I
-- 0.27 1 )'24, 2.4
0.2
STRAIN AT BORE
b.
/7Ic. 12. Variation of stress ratio w i t h bore or centre strain. Material: a = 107e ~ (a) Solid disk. (b) t t o l l o w disk.
433
434
M . J . P E R C Y and P. B. I%IELLOI%
problem is to imagine a v e r y small b u t positive radial stress a t the bore a n d a t the periphery. P e r h a p s a greater error is i n t r o d u c e d b y assuming t h a t the hoop stress and strain can be identified with the longitudinal stress and strain in the simple tensile test. I t is obvious t h a t the strain relations in a simple tensile test do n o t conform to e q u a t i o n (2). T a k i n g into a c c o u n t the above r e m a r k s it is perhaps surprising t h a t there is a fair degree of correlation between the two thcorics. T h e main discrepancies appear, as would be expected, a t the bore and the p e r i p h e r y where the Tresca t h e o r y holds the radial strain to zcro. Strictly, the H e n c k y t h e o r y is n o t valid if the stress ratios v a r y t h r o u g h o u t the deformation. H o w e v e r , the variations in the cases investigated are fairly small a n d it is proposed to test the practical usefulness of the t h e o r y b y detailed correlation with e x p e r i m e n t . Acknowledgement--This work was supported by the Ministry of Aviation.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
REFERENCES !%I.H. LEE %Vu, N.A.C.A. Tech. Note 2217 (1950). S. S. MA~SOh-, Proc. 1st U.S. 1Vat. Gongr. Appl. Mech. p. 569 (1951). M. ZAXD,J . Aero. Sci. 20, 369 (1953). H. J. %VEIss and %V. P~AeER, J . Aero. Sci. 21, 196 (1954). M. M. S.~nTH, Office of Naval Research, Tech. Report No. 35 (1958). P. B. i%IELLORand M. J. PERCY, A.R.C. Current Paper No. 692 (1963). W. T. KOITEn, Quart. J . Appl. Math. 11,350 (1953). D. R. BriAnD, J . Mech. Phys. Solids 6, 71 (1957). E. L. ROBIIqSOh', Trans. Amer. Soc. l]iech. Engrs 66, 373 (1944). D. It. %VIhn~'Eand B. BI. %VUNDT,Trans. Amer. Soc. Mech. Engrs 80, 1643 (1958). N. E. %VALD~E~"and D. E. %VARD,A.R.C. Current Paper No. 660 (1963).