Plastic and visco-plastic instability of a thin tube under internal pressure, torsion and axial tension

Int. J. Me~h. S¢i. Pergamon Press. 1968. Vol. 10, pp. 519-529. Printed in Great Britain

PLASTIC AND VISCO-PLASTIC I N S T A B I L I T Y OF A THIN T U B E U N D E R I N T E R N A L P R E S S U R E , TORSION AND A X I A L TENSION BERTIL

STOR/~KERS

The Royal Institute of Technology, Stockholm, Sweden

(Received 21 August 1967, and in revised form 18 January 1968) S u m m a r y - - A n analysis is presented of the stress state at instability of a thin tube which is subject to internal pressure, torsion and axial tension, all of which are independent of each other. The analysis is based on a work principle postulated b y Drucker and is carried out for a strain-hardening material. I t is shown that stability m a y be lost without either o n e of the loads attaining a m a x i m u m and that depending on the strain-hardening characteristics of the material, a twist of a tube subject to additional loads may increase or decrease the magnitude of axial stress and hoop stress at instability. I t is shown for some cases t h a t a twist accelerates the onset of instability in respect to other external loads. Finally, it is shown how superposed viscous flow influences the stability of a tube. NOTATION A ~j e~ e, F~ h l m M p P P~ r R s~ a~ ae a0 T T T~ u~ V z

area Kronecker's delta strain tensor effective strain body forces thickness of tube length of tube strain-hardening index torsional moment hydrostatic pressure axial force discrete forces radius of tube force ratio stress deviator stress tensor effective stress constant in stress-strain relation (equation (21)) shear stress twisting force surface tractions displacements volume subtangent to strain-hardening curve INTRODUCTION

THE FINAL f r a c t u r e o f a s t r u c t u r e , w h i c h u n d e r a l o a d i n g s y s t e m u n d e r g o e s large plastic straining, might be preceded by a period in which the mode of deformation changes from homogeneous to heterogeneous. 519

520

BERTrL STOR~.KERS

A well-known example of the change from homogeneous to heterogeneous deformation is the formation of a neck on the specimen in an ordinary uni-axial tensile test. The onset of plastic instability in a system depends not on the intrinsic strength of the material but on the geometry of the structure, the loading program and the stress-strain characteristics of the material. Failure criteria in the case of large plastic strains should therefore not be based on simple stress conditions; instead an analysis must be performed in order to determine when plastic instability of the actual structure will occur under the current circumstances.

•

EFFECTIVE STRAIN

Ee

FIG. 1. S u b t a n g e n t z t o t h e s t r e s s - s t r a i n curve.

In metal forming the occurrence of plastic instability often sets the limit for how far the process m a y be continued. The need of a method to calculate the stress state at instability has inspired many authors to present a variety of instability criteria. The problem is very simple when a structure is subjected to only one load variable. Instability will occur then simply when this load attains a maximum. I f several independent load variables are applied to the structure, the problem needs a more rigorous treatment. Swift 1 approached the problem of a thin tube under internal pressure and independent axial load b y calculating the rate of strain-hardening needed for a spontaneous change of the deformation state to occur under constant loads. A similar approach was made independently b y Marciniak ~. Some investigators in the field, e.g. Lankford and Saibel 3, Mellor 4 and especially 14illier~-~, have treated the problem from a different starting-point. Their method is based on the assumption that at instability some intuitively chosen load variables will reach maximal values. Consequently, different

Plastic and visco-plastic instability of a thin tube

521

results are arrived at regarding the stress state a t instability, depending u p o n which load variables are considered. The c o m m o n w a y to present t h e result of an instability calculation is t o give the critical m a g n i t u d e of the s u b t a n g e n t z t o the stress-strain curve o f the material in question. The s u b t a n g e n t is defined in Fig. 1. Hillier 5 has derived a general expression for the s u b t a n g e n t in t e r m s of c u r r e n t stress ratios a n d the rates o f load change. To determine w h e n this s u b t a n g e n t becomes critical is, however, risky as the a r g u m e n t involves assumptions a b o u t the rate of the change of i n d e p e n d e n t loads at instability. The use of this m e t h o d seems limited to the case when only one load variable is varied independently. I n the following a general solution of the title problem is given based on D r u c k e r ' s s-x° postulate for stability in the small. I t is shown t h a t for the case of no twist the result o b t a i n e d b y Swift, a l t h o u g h criticized, is quite consistent with D r u c k e r ' s definition of stability. ANALYSIS According to Drucker, a system consisting of a structure subject to arbitrary loading is fully stable in the small if small loading perturbations cause only small changes in configuration. This statement has been expressed in the form of a work principle. I f a body of time-independent deformable material is subject to arbitrary discrete forces Pi, surface tractions Ti and body forces F~ and small perturbation loads ~Pi, ~T~ and 8Fi are added, the system is fully stable in the small if all possible perturbation systems do positive work on the displacements But they produce. This implies that

~P, 3u,+ ( ~T,~u, dA + ( 3F, Su, dV>O jA JV

(1)

if ~Pi points outwards from the yield surface in load space as in Fig. 2.

~

~u i 8P i (~ii)

FIG. 2. Y i e l d surface in load space.

In general it is necessary to find that perturbation system for which the energy dissipation is smallest. If condition (1) is satisfied for this system it is satisfied for any other system. If the body is in an inhomogeneous state of stress it might be laborious to find the critical disturbance, but in the present case of a tube in a homogeneous stress state it is sufficient to consider a disturbance acting on the loaded areas.

522

B E R T I L STOR~;rERS

Consider a t u b e a c c o r d i n g t o Fig. 3 w i t h closed e n d s of r a d i u s r, l e n g t h l a n d t h i c k n e s s h s u b j e c t t o a n i n t e r n a l p r e s s u r e 19, a n a x i a l l o a d P a n d a t w i s t M a r o u n d t h e t u b e axis w h i c h is p r o d u c e d b y a force couple 2 T r o w h e r e r 0 is k e p t c o n s t a n t . I t is a s s u m e d t h a t h < r a n d t h a t t h e r a t i o of l e n g t h t o r a d i u s is o f s u c h a m a g n i t u d e t h a t end-effects m a y b e neglected. F o r s i m p l i c i t y elastic s t r a i n s a r e neglected. T

\ \

2r~

/ / J

1 FIG. 3. T u b e u n d e r a p p l i e d e x t e r n a l loads. The~true.'stressesVandWdeviator stresses in t h e t u b e are in c y l i n d e r c o - o r d i n a t e s : O'r ~ 0 pr pr

~=

P

~+~-~

~-~ = ~-~

s¢ =

(2)

Tr o =

pr 2h

pr s t = 2h

P 67rrh P 6~rrh

(3)

P Sz -- 3zr~h Tr o

S t r a i n i n c r e m e n t s are defined as dh de r _- ~ dr de~ = r de,

d/

= y

w h e r e ~ is t h e t o r s i o n a n g l e of t h e t u b e .

(4)

Plastic and vlsco-plastie instability of a thin tube

523

When the tube is in a state r, l, h and is twisted to the angle ~ under the loads p, P and T a n external agency 8p, 8P and 8T is introduced. I f the perturbation is infinitesimal, full stability in the small prevails according to condition (1) ff

dP dl + dp dl Trr2+ dp dr 2~rl + dT 2ro d¢ > O

(5)

Introduction of yon Mises flow rule yields de~__ d = 3 s~ dee 2 a,

(6)

a~ = ~s~j s,j

(7)

~. = a.-

(8)

}ak~ 8 .

de~ = ~ deij det~

(9)

The differentials dr and d e may now be expressed as functions of dl by making use of the flow rule of equation (6), the definition of strain increments in (4) and stress deviators in (3). This procedure leads to

dr = r [3~rr2 p _ 1] dl 2l \

P

(10)

]

and

de = ~6Tr° r ~ dl

(11)

Introduction of equations (10) and (11) into condition (5) yields [dP + dp rrr2 + d p 2zrrl 2ll r ( 3 ~ c r ' P - 11 + dT 2ro ~6Tro T ] dl > 0

(12)

I f the perturbing agency is infinitesimal, stability is lost when the expression within the brackets in condition (12) equals zero, i.e. P ro 2T dP+3(rrr')2ffdp+12(r ) ~dT = 0

(13)

I t can be seen from equation (13) that ff P , p and T are varied independently, instability m a y well occur even ff one of the loads has passed through a maximum. I t has been stated b y Mellor a t h a t this fact does not seem physically acceptable, b u t there is no immediate reason why it should not be. Unfortunately, reports of tests with thin tubes under internal pressure and axial load are rather rare in the literature. Davis n, 1~ has reported results from m a n y tests in the field, b u t in his experiments the ratio between pressure and axial load was always kept constant during the loading. This means that both loads always attained a m a x i m u m at instability and the results cannot be used to give a general experimental verification of the instability condition (13). I t is of interest to express the quantities in the instability condition in terms of stresses. Introduction of equation (3) into equation (7) and differentiation yields

~a. da,

Pr2(dp+pdr

dh~

P

[dp-pdr

pdh~

2Tr~ ( d T _ 2 T d r

T (14)

I f the incompressibility condition dr dl dh -7-+T+-E = 0

(15)

and the instability condition (13) are introduced, equation (14) reduces to

~a, dae = ~ h l ~dr+ [1 ~h )/Pr~ ~+-6-~ ~)1/P\~l T _ 2~_/~)_Tro \ 2 /dr~ r _T)dl\ at instability.

(16)

524

BERTIL STORAKERS

B y m a k i n g use of t h e flow rule and the stress definitions, e q u a t i o n (16) m a y be w r i t t e n in t e r m s of stresses as

The stress ratios are defined as a¢ = a(~z

(18)

r¢~ = flaz

{19)

where a and fl m a y v a r y i n d e p e n d e n t l y during t h e loading. I t is now possible to calculate t h e critical m a g n i t u d e Zcr of t h e s u b t a n g e n t a, de.

Zcr -

da,

4( 1 - a + a 2 + 3fl~) t

- 4aa+3a 2-6a+4+

1Stir(i-a)

(20)

I n the case of fi = 0 this result is identical w i t h t h a t of Swift and Mareiniak a l t h o u g h their works are n o t based on an energy principle. I t should be p o i n t e d out here t h a t the significant result of the preceding analysis is the expression for t h e critical m a g n i t u d e of the s u b t a n g e n t in e q u a t i o n (20). E q u a t i o n (13) is of no general i m p o r t a n c e for the solution of the problem. The disturbing agency d P , dp a n d d T i n t r o d u c e d here could simply be interpreted as the result of a change of the applied loads. Full stability according to D r u c k e r prevails, however, only if there is no possibility of e x t r a c t i n g energy f r o m the s t r u c t u r e b y the action of a n y k i n d of disturbing agency. Consequently t h e effect on the stress state of e v e r y possible p e r t u r b a t i o n should be investigated. I t is possible to show, however, t h a t in this case w h e n t h e t u b e is in a homogeneous state of stress, a n y p e r t u r b a t i o n causing a homogeneous increase in stress will result in the same expression for the critical r a t e of strain-hardening. The chosen agency d P , dp and d T in this case serves only as an example.

EFFECT

OF

A

TWIST

Plastic instability occurs when the rate of strain-hardening of the material is not large enough to c o m p e n s a t e for the increase of stress due to the geometrical changes in a struct u r e which occur u n d e r e x t e r n a l loads. The derived expression for t h e critical s u b t a n g e n t according to e q u a t i o n (20) takes a n e g a t i v e v a l u e if a > 1 a n d f12 > (4a8 + 3a, _ 6a + 4)/(a - 1). I f these conditions are fulfilled, geometrical changes during d e f o r m a t i o n will f a v o u r a decrease in stress a n d c o n s e q u e n t l y instability will n e v e r occur. This m e a n s t h a t t h e r i g h t - h a n d side of e q u a t i o n (17 ) is n e g a t i v e a n d t h e instability condition (13) will n e v e r be reached. Thus for a t u b e of an o r d i n a r y stable material, for which t h e s u b t a n g e n t to t h e s t r e s s - s t r a i n c u r v e is a m o n o t o n i c a l l y increasing f u n c t i o n of t h e effective stress, equation (20) gives t h e critical m a g n i t u d e of t h e s u b t a n g e n t if t h e d e n o m i n a t o r is positive. Otherwise t h e t u b e should always be geometrically stable. H o w e v e r , only external agencies p e r t u r b i n g the m e m b r a n e state h a v e been considered. W h e n large compressive stresses appear, in this case due to torsion, buckling m a y occur. This state of affairs needs a special investigation for e v e r y particular shell g e o m e t r y considering also the elastic properties of the m a t e r i a l and is not t r e a t e d in the following. W i t h t h e aid of e q u a t i o n (20) it can be p r o v e d t h a t t h e presence of a twist always increases t h e m a g n i t u d e of t h e critical s u b t a n g e n t and t h u s t h e effective stress at instability. The effect of the twist on the capability of the t u b e to carry o t h e r loads requires, however, a special investigation. Assume for simplicity t h a t the stress-strain c u r v e of the m a t e r i a l can be represented b y a parabola a, = a0e~ (21) and t h a t the r a t e of strain-hardening is a unique function of the effective stress only, which implies t h a t ~l/m dcrs m -o de~ a~x-m~/m (22)

Plastic a n d visco-plastic instability of a t h i n t u b e

525

I f a~ is chosen as t h e reference stress t h e critical v a l u e of a, m a y be solved f r o m e q u a t i o n s (7), (18), (19), (20) a n d (22) to give 4(1 - a + ~ + 3 8 ~ ) (Sm-1)/2m

a~/'~ =

ma°~+'~ 40~ 3 + 3 ~ ~ - 60~ + 4 + 1 8 8 3 ( 1 - -

(23) ~)

>-

_~ 0.8 co

Z

.~ 0.7 U3 if3

~,

0.6

~ 0.5 Z 0 z 0.4 ~J ]E

"17.

0

0.2

0.4

0.6

STRESS RATIO /~ =

0-8

1.0

q-t.

FIG. 4. Dimensionless axial stress at instability as f u n c t i o n of t h e ratio of shear stress to axial stress. Fig. 4 shows for t h e case of a --- 0 t h e dimensionless axial stress a,/ao a t instability as a f u n c t i o n of 8 for some different values of t h e strain-hardening index. A twist of the t u b e always decreases the critical axial stress for 0 < m < 1. F r o m a design p o i n t of view, however, it is m o r e interesting to investigate t h e influence of a twist n o t on t h e critical stress state b u t on t h e critical m a g n i t u d e s of o t h e r e x t e r n a l loads. The d e t e r m i n a t i o n of t h e critical e x t e r n a l load requires knowledge of t h e strain state a t instability. I n general w h e n calculating t h e strain s t a t e a n i n c r e m e n t a l a p p r o a c h m u s t be m a d e as t h e strain state a t a certain stress state depends on t h e loading p a t h . Such calculations h a v e been p e r f o r m e d b y Hillier 7 for some loading p a t h s in t h e case of axial tension a n d i n t e r n a l pressure. To illustrate t h e effect of a twist in one case t h e ratio R b e t w e e n t h e axial force a t instability Per (8 ~ 0) w i t h a twist a n d Per (8 = 0) w i t h o u t a t w i s t has b e e n calculated w i t h t h e aid of t h e flow rule and e q u a t i o n (23) for t h e simple case of p r o p o r t i o n a l stressing and no internal pressure. I t is easily shown t h a t

Per(8#o) R = Per (8 = 0) =

(I+38~) `3'~-1,'~ (1 +4.58~)~

( 3,-83~ " exp \ 2 + 9 8 2 ]

(24)

I n Fig. 5 t h e r a t i o R is shown as a f u n c t i o n of 8 -- ~r~/a, for t h e s a m e values of t h e s t r a i n - h a r d e n i n g i n d e x as in Fig. 3. Fig. 5 shows t h a t u n d e r t h e assumed circumstances a twist always decreases t h e c a p a b i l i t y of c a r r y i n g an axial load. I n t h e preceding analysis t h e t u b e was subject to a twisting m o m e n t 2Tr o where r 0 was k e p t c o n s t a n t during t h e d e f o r m a t i o n program. I f t h e twisting forces act at t h e current

526

BERTIL STORIKERS

p e r i p h e r y o f t h e t u b e , t h e m o m e n t b e c o m e s 2 T r a n d t h e r e s u l t i n g s h e a r stress T/~rrh. T h e a r m o f l e v e r 2r n o w varies d u r i n g d e f o r m a t i o n . T h e s t a b i l i t y c o n d i t i o n (5) t h e n c h a n g e s t o d P d l 4- d p dl rrr ~ 4- d p d r 2rrrl 4- d T 2r d ¢ > 0

(25)

l'0

0.9 >I-J

~, 0-8 =7

-~ 0-7

~'0"6 -

-

It

o_

"~ 0 . 5 - fl,."

U,J ~_) n-" 0 IJ_

0

2" 0

0"2

0"4

0"6

0"8

1"0

STRESS IRATIO ~ : q"r',#~. .--

FIQ. 5. R a t i o b e t w e e n a x i a l force a t i n s t a b i l i t y w i t h a n d w i t h o u t a t w i s t as f u n c t i o n of t h e r a t i o of s h e a r stress t o a x i a l stress. A n a n a l y s i s of t h e critical m a g n i t u d e of t h e s u b t a n g e n t for t h i s case r e s u l t s i n z =

4(a ~- a 4- 1 4- 3fl2) t 4a a + 3a 2 - 6a + 4 + 6fl2(2 - ~)

(26)

I t c a n b e s h o w n for t h i s ease, too, t h a t w h e n i n t e r n a l p r e s s u r e is zero, t h e critical a x i a l s t r e s s a n d force d e c r e a s e w i t h i n c r e a s i n g s h e a r stress. T h e v a r i a t i o n of critical a x i a l force w i t h s h e a r stress is v e r y s i m i l a r t o t h a t o f t h e p r e c e d i n g ease. Still a n o t h e r w a y o f i n t r o d u c i n g t h e t w i s t i n g m o m e n t is b y a p p l y i n g d i r e c t l y c o n t r o l l e d s h e a r stresses in t h e t u b e . T h e critical s u b t a n g e n t is t h e n z =

4(a ~- a + 1 + 3fl2) | 4~ 3 + 3c~2 6~ + 4

(27)

- -

I t is e a s y t o s h o w t h a t i n t h i s case t h e critical a x i a l a n d h o o p stresses will increase w i t h f~ for m > ] . T h e critical a x i a l force will d e c r e a s e w i t h ft. T h i s w a y of i n t r o d u c i n g t h e t w i s t ing m o m e n t is, h o w e v e r , s o m e w h a t tmrealistic.

THE

EFFECT

OF

SUPERPOSED

VISCOUS

FLOW

In the preceding analysis the flow rule has been used to determine the response of the system to the perturbing agency. If plastic flow is coupled with viscous flow, the flow rule might take the form deij = g((~,, e~) ~ dae + h(a,, e,, t) s o dt 0"¢

(28)

Plastic and visco-plastic instability of a thin tube

527

F o r the determination of stability, however, the perturbing agency should be applied in a quasi-static manner, i.e. slowly enough to neglect inertia effects but rapidly enough to neglect viscous response. This means t h a t in equation (28) the t e r m representing viscous flow should be neglected when determining the response to the perturbation. Thus the critical magnitude of the subtangent according to equation (20}, (26) or (27) is not affected by superposed viscous flow. The stre~s state at instability, however, will be different from t h a t in the pure plastic case, unless the tube is subject to proportional stressing and the rate of strain-hardening is a unique function of the effective stress. The time to creep failure for a tube which undergoes creep coupled with plastic deformation can be calculated as the time needed for the critical magnitude of the subtangent to be reached, provided t h a t brittle fracture does not intervene. R i m r o t t etal. 13 have presented a theory capable of predicting creep failure time for a pressure vessel with closed ends. Their basic assumption, originating from Hoff 1~, is t h a t actual fracture time and the time to reach infinite theoretical strain are essentially the same. Their theory does not take into account the time-independent plastic properties of the material. I t is clear, however, t h a t if the time-independent plastic strain is included in addition to the creep strain as is done in equation (28), infinite strain will never be reached, as the strain rate will already have become infinite at a finite strain. This will happen when the subtangent to the stress-strain curve becomes critical. When this strain state is arrived at the new mode of deformation is not unique and t h a t a rapid failure will follow is plausible. I t has been pointed out by Carlson 15 t h a t the difference in the value of the critical time computed by a method based on creep effects only and a method considering both plastic and creep effects m a y be significant as shown for an alumininm alloy in uniaxial tension. CONCLUSIONS The application of Drucker's postulate to the problem of plastic instability of a structure requires t h a t the effect of every possible perturbation is examined. The critical perturbation is t h a t one which does the smallest positive work on the displacements it produces. Consequently if under applied external constraints a certain stress state has been reached in a body, the critical rate of strain-hardening is by no means affected b y the way in which this stress state was reached. The magnitude of the subtangent at instability, as calculated in equation (20), is independent of the stress history and consequently of any initial straining. There is no reason to make separate analyses for proportional stressing, proportional loading and so on as hes been done by some authors. For the s t r e s s s t a t e at instability the argument above holds if the rate of strain. hardening is a unique function of the effective stress. I f this is not the case, different strain histories might for physical reasons produce different rates of strain-hardening at the same effective stress. This strain-dependence will, of course, affect the stress state at the onset of instability. Numerous tests of the limiting load-carrying capacity of thin nickel tubes under tensile force, torsional m o m e n t and internal pressure have been performed b y Mikhailov and Yagn 1~. The results arc presented very briefly and no stress-strain curves of the material are given which makes it impossible to draw any quantitative conclusions from the performed experiments. Mikhailov and Yagn state, however, t h a t " t h e character of the loading (proportional or complex}, which terminated in nearly the same values of ft, did not noticeably affect the values of the limiting stresses". The q u a n t i t y ft is defined as 2a2_1 £r 1

where al, a S and a a are the principal stresses with a x> a~ and a a = 0. This means t h a t if due to applied external loads a certain stress state was prevailing at instability the loading path had no influence on the critical effective stress in conformity with what has been maintained above. I t is important, however, when applying Drucker's postulate, to take into account the effect of all applied external loads and the way they are introduced. The importance of

528

BERTIL STOI~KERS

this has been illustrated by introducing a torsional m o m e n t in different ways which leads to different results in respect of the subtangent at instability. Another example is an open tube under hydrostatic pressure which will not become instable at the same pressure as a capped end tube with an axial force balancing the axial stress to zero value. This becomes obvious from the stability condition (5). Equation (13) shows t h a t if the tube is subject to loads which are varied independently of each other, instability m a y well occur even after one or two loads have passed through a maximum. F or the treated cases a calculation of the effect of a change in the applied external loads gives the same resulting critical rate of strain-hardening as any possible disturbing agency would do. In general, however, these two concepts should be kept apart. For example, a uniaxial creep test m a y be performed under decreasing axial load P but with increasing stress due to the change of the cross section of the specimen. Even though P is decreasing during the test, when investigating the stability of the specimen, the effect of a positive disturbance d P should be investigated. I t has been shown t h a t a twist will always increase the effective stress at instability but will decrease the critical external force in the case of axial tension with proportional stressing. I f a tube undergoes creep coupled with plastic flow, complete knowledge of the flow rule makes it possible to compute the time needed to reach the critical subtangent according to equation (20) or equivalently to the time to instability. I t is likely, however, that in the ease of low initial stress, when instability occurs at very large strains, fracture due to brittle cracking might intervene. The occurrence of brittle fracture m ay be predicted by one of the various theories of cumulative damage, e.g. the one given by Kachanov 17 in which it is assumed t h a t the rate of damage is governed by the m a x i m u m tensile stress and the current damage. Reports on tubular creep rupture tests have been presented by, among others, Rowe et al. 18. I n their experiments "tubes at high stresses, which ruptured in something like 10 hr, produced an 'open-door' fracture resulting from violent action of rupture, ''is while tubes at lower stresses and times beyond 100 hr fractured due to the generation of one, or occasionally several, small seam-like cracks, which did not cause a gross deformation of the tube as a whole. The reported rupture behaviour indicates t h a t rupture m ay be caused by two different mechanisms; at high initial stress the creep straining of the tube might induce instability, but at low initial stress fracture occurs in a brittle way. A combination of the theory of viscoplastic instability outlined above and a theory of cumulative damage provides a basis to predict creep failure of a tube under an arbitrary loading program in the entire range of initial stress from zero to a stress causing in~nediate instability. F o r a certain loading program the theory which gives the most conservative result in respect of failure time should be chosen. I t would be of interest to perform a theoretical calculation of the time after which viscoplastic instability occurs in a pressurized tube for comparison with experimentally found rupture times. Rowe et al. is report yield strength and ultimate tensile strength of their test material but not the complete stress-strain characteristics, which excludes this possibility for experimental verification. For the case of uniaxial tension such calculations have been performed by Storhkers ~°. Drucker's postulate for stability in the small is very convenient to use when dealing with simple structures like the one considered here and it has been shown that the presence of a viscous flow component leads to no additional fundamental difficulties in determining when stability is lost in a system.

REFERENCES 1. H. W. SWIFT, J . Mech. P h y s . Solids 1, 1 (1952). 2. Z. M)mCrNIAK, Rozpr. inz. 110, 529 (1958). 3. W. T. LA:NKFORD and E. S~BEL, A I M M E M e t a l s Technology Tech. Pbn. No. 2238 (1947). 4. P. B. M_ET.LO~,J . mech. E n g n g Sci. 4, 251 (1962).

Plastic and visco-plastic instability of a thin tube

529

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