Plastic instability and growth of grooves and patches in plates or tubes

Int. J. mech. Sci. Pergamon Press. 1972. ¥ol. 14, pp. 325-337. Printed in Great Britain

PLASTIC I N S T A B I L I T Y AND G R O W T H OF GROOVES A N D P A T C H E S IN P L A T E S OR T U B E S E. DU~COMBE Westinghouse Electric Corporation, Bettis Atomic Power Laboratory, P.O. Box 79, West ~ , Pennsylvania 15122, U.S.A.

(Received 19 August 1971, and in revised form 21 January 1972) Summary--Plastic instability is defined as the growth of a locally thinned region or neck in a material upon the application of stresses. I n the case of sheets and tubes the thinned regions can be groove-like (concentrated) or patch-like (diffuse). Conditions for instability in these two modes are compared over a range of stress states and material properties by examining the roots of linearized dynamic equations. The material properties are strain hardening, strain rate sensitivity and anisotropy. The stress state is defined by the ratio of normal stresses in two orthogonal directions (biaxiality ratio). Regions of stress biaxiality ratios are determined for which one or the other deformation mode dominates. I t is often more realistic to define stability in terms of an acceptably small rate of thinning rather t h a n zero rate. The effects of anisotropy appear to be small in the ranges considered. NOTATION a constant differential D i equal to ~ag/~a~ Dij equal to a s ~2 a~/Oa~~aj Flo, F2a parameters defined by equations (14) and (15) GO subscript denoting absolute critical value for grooves G1 subscript denoting effective critical value for grooves h subscript indicating thickness direction l subscript indicating longitudinal direction rn strain rate sensitivity [equation (1)] n exponent equal to e, y 0 subscript denoting initial value P anisotropy [equation (3)] P 0 subscript denoting critical value for patches R anisotropy parameter [equation (3)] s the operator d/dr t subscript indicating width direction t time a biaxiality ratio a~/at fl parameter defined by equation (16) strain hardening parameter [equation (1)] $( ) spacial difference in values of a variable A( ) increment, over a period of time At eg(~g) generalized strain {rate) defined b y equations (4) and (5) ~ value of normal strain in direction i a~ normal stress in direction i au generalized stress [equation (3)] A superscript q u a n t i t y indicates a n exponent. A d( )

INTRODUCTION

TH~ co~c~.PT of failure caused by plastic instability in a bar undergoing a tensile test dates back to proposals of Consid~re~ in 1885. A qualitative 325

326

E. DUNCOMBE

description of the process starts with the supposition t h a t a very small neck has formed. Instability occurs when the strain hardening in the neck is insufficient to offset the increase in stress, and the material in the neck thins down faster than the material outside it. The extension of this concept to sheets and tubes under biaxial stresses requires consideration of the twodimensional character of the localized thinning and the corresponding boundary conditions at the interface with the neighboring unthirmed material. Two approximations to the actual mode of failure, described as groove-type and patch-type, respectively, are illustrated in Fig. 1. Stresses are assumed to be

f (7"t ~

I

(a)

~

I ""~"O't

O"t

% % (b)

~h :5 ~

)

* BE.

?

FIG. 1. Specification of geometry for instability of patches and grooves. (a) Groove with a thickness inhomogeneity Sea; (b) patch with a thickness inhomogeneity 8eh.

uniform inside the patch or groove as determined by compatability of forces or displacements at the boundary as described later. Azrin and Backofen ~ have reported the results of punch and die tests which show that instability can begin by diffuse straining in a patch although continued straining causes a groove-like defect to develop. The avoidance of whichever type of instability t h a t occurs first is therefore an objective to the designer. Other modes of failure, such as crack propagation, are not considered here. Stability under biaxial stress conditions, under the implied assumption of zero sensitivity of strain rate to stress, has been studied by Hillier3 for the diffuse (patch-type) mode of instability of tubes. Studies have also been made by Marciniak and Kuczynski, 4 and, later, by Azrin and Backofen, ~ for groovetype instability, their approaches being the solution of non-linear equations unique to specific cases. A comprehensive analysis of stability conditions for the isotropic case in a bar undergoing a tensile test has been undertaken by

Plastic instability and growth of grooves and patches in plates or tubes

327

Hart s who considered not only the effects of strain hardening but also the influence of the sensitivity of strain rate to the stress level. Groove-type instability is analyzed here using linearization techniques. The effects of strain hardening and strain rate sensitivity and material anisotropy are included. These stability conditions are compared with those for patch-type instability by extending previously published results a to include anisotropy effects. MATERIAL

PROPERTY

RELATIONSHIPS

I t is assumed here that elastic strains are always much smaller t h a n plastic strains and can therefore be neglected. Let the material outside the locally thinned region be at a generalized stress ag, and a generalized strain and strain rate e~ and ~g. These terms will be defined more fully later. !~'ow if a functional relation between ag, ea aazd ~ exists, then for sufficiently small changes

(1) where 8 represents a small change in going from the locally thinned region to the unperturbed or "nominal" region surrounding it. The coefficients rn and ~ are defined as the strain rate sensitivity and the strain hardening parameter respectively. A physical interpretation of y is seen from Fig. 2 which is a schematic plot of the relation between ag ~g and eg. The value of i/y is represented by the subtangent of the

j

,

_ ~ . ~/, .

i

Eg = constant

X/)/

N

i //

/ L__ /

.V

/

/

/

/

/

,, ooe( ).l I

o

Generalil~dstrain, ~g

FIG. 2. Schematic representation of mechanical properties.

operating point Q for the stress-strain curve passing through Q at the operating strain rate ~e. Also the strain rate sensitivity m can be represented as shown. E q u a t i o n (1) can be conceived as a local version of a general flow law

where A, m and n are constants, n being equal to so~' in (1). However, for the purposes of stability analysis it is sufficient to work with (1), where m and y are not universal constants b u t merely characterize the local functional relationship between as, ~a and 8g. 23

328

E. DUNCOMBE

L e t ah, at a n d a~ b e t h e s t r e s s e s in t h e t h i c k n e s s , t r a n s v e r s e a n d l o n g i t u d i n a l d i r e c t i o n s a n d a s s u m e t h a t s h e a r stresses are a b s e n t or c a n b e neglected. F o r t h e case of a n i s o t r o p i c m a t e r i a l s t h e g e n e r a l i z e d s t r e s s ag is defined in a f a s h i o n s i m i l a r to t h a t of Hill ~ b y t h e quadratic relationship

ag = [R(a~-a,)'+ RP(a,-a,)~+Rp+p

P(a~-a~)~]"

(3)

w h e r e R a n d P are a ~ i s o t r o p y p a r a m e t e r s , b e i n g u n i t y in t h e isotropie ease. T h e d e n o m i n a t o r (RP+P) in (3) is h e r e a n o r m a l i z a t i o n f a c t o r w h i c h is u s e d to m a k e ag e q u a l to ~ for a u n i a x i a l t e s t in t h e l d i r e c t i o n . T h e g e n e r a l i z e d s t r a i n eg is n o w defined b y t h e p l a s t i c work relation,

together with eg = ~ o dr. !

(5)

B y u s i n g t h e h o m o g e n e o u s p r o p e r t y of (3) a n d t h e o r t h o g o n a l i t y h y p o t h e s i s a s s o c i a t e d with Drucker: 7 ~o o ba~

~ag

~6)

i t c a n b e s h o w n + t h a t (3)-(6) lead t o t h e r e l a t i o n

~,

=

(?ag/~a~)~ -D, ~,

(7)

w h e r e i s t a n d s for d i r e c t i o n s h, t or l. T h e p a r a m e t e r s R a n d P c a n n o w b e defined i n t e r m s of s t r a i n s in a u n i a x i a l t e s t R = det/den

(aa = a t =

0),

(S)

P=

(am = ch = 0).

(9)

deJde~

T h e s e e x p r e s s i o n s earl be d e r i v e d f r o m (3) a n d (7). I f t h e axes of a n i s o t r o p i c s y m m e t r y do n o t lie in d i r e c t i o n s h, t a n d l of Fig. 1, t r a n s f o r m e d v a l u e s of R a n d P a p p r o p r i a t e t o t h e n e w d i r e c t i o n s are n e e d e d . S u c h v a l u e s m a y b e defined in t e r m s of (8) a n d (9). T h i s is o n l y a n a p p r o x i m a t e r e p r e s e n t a t i o n since t h e f o r m of % in (3) is n o t a s i m p l e q u a d r a t i c in s u c h cases, s S u c h a p p r o x i m a t i o n s are justifiable for s m a l l a m o u n t s of a n i s o t r o p y . F o r cases of s h e e t s a n d t u b e s c o n s i d e r e d h e r e t h e stress an in t h e t h i c k n e s s d i r e c t i o n is a s s u m e d to b e negligible. I n d i f f e r e n t i a l f o r m e q u a t i o n (7) c a n b e w r i t t e n di~ = Oi dg~ + d~ d D e GROOVE

STABILITY

(I0)

RELATIONSHIPS

Analysis T h e g e o m e t r y of a g r o o v e is s h o w n in Fig. l(a). T h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n is t h a t t h e p l a s t i c s t r a i n r a t e i n t h e d i r e c t i o n of t h e g r o o v e is t h e s a m e inside t h e g r o o v e as it is outside. T h i s c o n d i t i o n , i n c o n j u n c t i o n w i t h t h a t of c o n s t a n t v o l u m e , leads t o /i~ = -- ~ h -- ~e~ = 0.

( I 1)

A s e c o n d b o u n d a r y c o n d i t i o n a p p l i e s in t h e d i r e c t i o n t r a n s v e r s e t o t h e groove. T h e s t r e s s i n t h i s d i r e c t i o n is i n c r e a s e d i n p r o p o r t i o n to t h e a m o u n t o f t h i n n i n g , i.e.

dgh/at =

-8~.

(12)

Plastic instability and growth of grooves a n d patches in plates or t u b e s

329

N o w if (11) is used w i t h t h e p r o p e r t y relation (1) and w i t h s t r e s s - s t r a i n relationships {3) a n d (7), a d y n a m i c e q u a t i o n in ~ results, describing t h e r a t e of increase or decrease in 8es, and stability conditions o b t a i n e d b y e x a m i n a t i o n of its roots. This e q u a t i o n is d e r i v e d in t h e A p p e n d i x , being as follows, d~

. d

(13)

(~+F,~) 3-~ (8~) = (F~o- 7 ) e0 ~ (8~s), where Flo a n d .F~o are defined as

FIG

-----

F~a =

(D~ D. + D~ D~, - 2D, D~ D,z)~/D,,,

(14 i

D~/D..

(15)

H e r e D~ and D~j are defined as ~as/~a i a n d ( ~ as/~ai ~a~) as. A l s o fl is t h e ratio a,/a s a n d can be expressed in t e r m s of R, P , and t h e b i a x i a l i t y ratio a = az/a~ as t~ = aJos = [ ( R P + P ) t / ( R + R P ( a -

1)'+Pa~)~]

a(~).

(16)

I n this linearized a p p r o a c h all q u a n t i t i e s o t h e r t h a n 8e s are assumed c o n s t a n t or slowly v a r y i n g . T h e variable 8es is chosen for convenience r a t h e r t h a n t h e thickness strain difference 8e~. These are r e l a t e d t h r o u g h (10), on replacing t h e differential (d) b y t h e difference (8). Since t h e e q u a t i o n s are linearized t h e characteristic roots do n o t d e p e n d on t h e v a r i a b l e chosen. T h e r e l e v a n t d e r i v a t i v e s D~ a n d D ~ are Dt = ~a,/~at = R + R P ( 1 - a ) RP + P

as,

(17)

ers

D~ = ~as/~a ~ = - R P ( 1 - c~) + Po~ a~, .RP + P ug

(18)

Du = ~-a2 as = 1 - D ~ ,

(19)

~2a s RP + R D. = ~ as = R p + p ~ a D. = D~ = ~tas

(20)

D2,, •

-- R P = RP+P

DiDt.

(21)

T h e formal solution of (13) is given, in t e r m s of a n initial v a l u e (8~s) 0 b y

8'ss = exp [(Flo_7) ~

(8~s)o

r

= exp L(Flo-7)

At] ' _]

( 31

This solution is of course only valid for t i m e i n t e r v a l s At during which changes in Flo , F~o, 7, ~ a n d es are sui~ciently small. Resu/~ To d e t e r m i n e h o w r a p i d l y t h e groove increases in size it is necessary to e x a m i n e t h e root ~ = (Flo-7)~g/(rn-F.F~o ) of e q u a t i o n (13). Values of this r o o t are shown for t h e illustrative cases R = P = 1, m = 0-01; R = P = 1, m = 0.1; R = P = 1, rn = 1; a n d R = 2, P = 3, m = 0.1 in Figs. 3-6. Increases in t h e strain r a t e sensitivity p a r a m e t e r ~n

330

E . DUNCOMBE I000

!GO I

tO!-

'

~-

~

Effective ~o..,oo,,,,, /

='-oo,

i

~,bsolute

/ i

~"X

h,

[ I\\

~_o,

\. I

~~

~=,,

i~

i' ~I0

-05

0

05

05

O

-C5

-iO

- a=o- /o-,

Ilez= o'tl~. ~ E3iexialfty ratio

FIG. 3. Stability field for longitudinal grooves, R

=

P

=

l, m

=

0.01

I000

i00

-

~0

0 i

Effective stability;

o

Y,

oo,L_ - ~ J ~ - - ~

~

-o.,r'oo'"J \\\

\

,j//\\\

-hO

-0 5

0

C'5

0

v

/

06

r/a = o-t lot

i

0 ~:

-O 5

-~ 0

~,Io-~

Bia×ialitymT~o

FIQ. 4. S t a b i l i t y f i e l d for l o n g i t u d i n a l g r o o v e s , R = P = I, m = 0.1.

Plastic instability and growth of grooves and patches in plates or tubes

I000 t tCX) I0 Y'=O'l

• i

l~Effectiv e

°'i-"°~'i"

f~l

ooi

T-I T. I

\

._oo,

f

/

Y-3

_

_,'o -tO

-05

0

IA~.G,

05

IG ~

I0

O~

BiQxiali ty ratio

~

0

-05

-IO

a'G,.Iat

F i a . 5. S t a b i l i t y field for l o n g i t u d h ~ l grooves, R = _P = 1, m = 1,0.

I 0OO

I00 10

I

7's 0.01

,-D,-7.o,::

0 ~- stability

i o goJot, -001

o,

•-

~r~

A kt

i saioility~il, l | ~ II

~ -o

\

),~_o-e~

-IOC -I

0

-05

0

05

" ~0

05

I/Q: o'tI G,:...,m--_.

~ Biaxiality ratio

FIG.

6.

Stability field for longitudinal

grooves,

0

-05

-I.0

~ • o'<~/o"t R =

2, P

=

3 , ~n =

0. I.

331

332

E. D U N C O ~ E

or in t h e strain h a r d e n i n g p a r a m e t e r ~ are seen to cause increased s t a b i l i t y in the sense t h a t these increases lead to algebraically smaller roots ~. The condition for absolute s t a b i l i t y is t h a t $ ~ does n o t increase w i t h time, and this can be derived directly from (13) w i t h o u t i n t e g r a t i n g o v e r time. This defines a critical v a l u e of ~, for grooves, Y~0, given. f r o m (13) a n d (14), b y ~ o = -Fza = (D[ D,, + D~ D~z -- 2D, D, D,,) fl/Dm

(24)

T h u s 7a0 is t h e v a l u e of F on t h e line )t/~g = 0 in Figs. 3-6. I t will be o b s e r v e d t h a t w h e n I / a -- 0, i.e. at = 0 the condition is always theoretically u n s t a b l e (~ = 0). F u r t h e r u n d e r s t a n d i n g of this s i t u a t i o n can be o b t a i n e d b y assuming t h a t some finite v a l u e of $~g/(8~g)0 in (23) is permissible, a reasonable a n d eonvenien~ value being the N a p e r i a n base e = 2"718. Also a practical limit on t h e n o m i n a l strain A~g mtmt exist. Assume this limit is 1-0. E q u a t i o n (23) can t h e n be solved for the value of ~,, say Fa~, r e q u i r e d to reach "effective s t a b i l i t y " , i.e. y ~ = F~a - (F~a+ m)

(25)

where Fin a n d F~a are g i v e n b y (14) and (15). This is y on the line h / ~ = 1 in Figs. 3-6. Figs. 3-6 show t h a t t h e v a l u e YGz for effective s t a b i l i t y is often quite different from Ya0 for absolute stability. I n fact w h e n 1/a = 0, i.e. ~t = 0, h = 0 only expresses the obvious c o n d i t i o n t h a t S~g is neither increasing nor decreasing.

PATCH

STABILITY

RELATIONSHIPS

Analysis

T h e g e o m e t r y of a p a t c h of m a t e r i a l in a b i a x i a l l y loaded plate or p o r t i o n of a t u b e is specified in Fig. l(b). The local i n h o m o g e n e i t y in thickness is defined b y t h e t e r m Seh. The a s s u m p t i o n is m a d e t h a t the loading on the p a t c h is not affected by changes in p a t c h thickness. This is physically realizable in the case of pressurized tubes where a t h i n n e d p a t c h becomes a slight bulge. This a s s u m p t i o n leads essentially to the condition t h a t the m e m b r a n e stresses in t h e t h i n n e d p a t c h are increased in p r o p o r t i o n to t h e r e d u c t i o n in thickness, i.e. this limiting condition can be expressed as ~e~ = - Sat~at -- - Sal/(~.

(26)

The d y n a m i c e q u a t i o n g o v e r n i n g t h e g r o w t h of an in_homogeneity Sea can now be derived w i t h t h e aid of the p r o p e r t y relation (1), stress a n d strain relationships (3) a n d (7) and geometrical relationship (26). The condition for stability u n d e r such conditions has been d e r i v e d b y Hillier a (his case (ii)) for isotropic properties. This can be e x t e n d e d to the m o r e general anisotropic case (R ~ P ¢ 1), asstuning t h a t Dz and D< r e m a i n constant. I n this case the critical value of y for s t a b i l i t y is g i v e n b y ~po = ( D ~ a + D~)~,

(27)

where, as before, a is t h e b i a x i a l i t y ratio a~/,7~, fl is the ratio atla~ a n d D~ is the d e r i v a t i v e Oag/~ai. A l t h o u g h Hillier's analysis did n o t cover non-zero values of m it can be sho~-a t h a t yr~ is not d e p e n d e n t on m u n d e r t h e a s s u m p t i o n s for this m o d e of instability. This is n o t at v a r i a n c e w i t h H a r t ' s analysis a of tensile tests on a bar, which did show a dependence on m, because H a r t used a c t u a l thickness i n h o m o g e n e i t y r a t h e r t h a n strain i n h o m o g e n e i t y as the variable. I f strain i n h o m o g e n e i t y is the v a r i a b l e his condition is also n o t d e p e n d e n t on m. Of course, this is n o t necessarily t r u e for sm~icienty rapidly v a r y i n g loading, b u t only for q u a s i - s t a t i o n a r y loads considered here. Results Values of the critical s u b t a n g e n t 1/yv0 from (19) are g i v e n in Table 1. These values are only m o d e r a t e l y d e p e n d e n t on the values of b o t h R and P . T h e y are m i n i m i z e d (worst case) for as ~ 0-25at and at ~ 0.25a~.

333

P l a s t i c i n s t a b i l i t y a n d g r o w t h o f grooves a n d p a t c h e s in p l a t e s or t u b e s TABLE 1. PATCH STABILITY--VALUE OF CRITICAL SUBTANGENT (ypo) -1 Aniso tropy factors

a

=

ai/at

a = 1/a = 1

R

P

-0.9

-0.7

-0-5

0

0.5

0'8

1.0

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

13.0 4.6 15-0 3.8 8.0 16.0 3.4 6"5 11.0 16.0

4.0 12.0 2.7 4.5 6-3 8.1 2.5 3.7 4.7 5.7 2-4 3.4 4-2 4.9

2.2 3.2 4.0 4.8 1.9 2.4 2.8 3.1 1"8 2.2 2.5 2'8 1-7 2'2 2.4 2.6

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1-0 1.0 1.0

1.2 1.2 1.3 1.3 1.0 0.97 0.95 0.93 0.94 0"88 0.84 0.81 0-92 0-84 0.78 0.75

1.8 2-0 2.1 2.0 1.5 1.9 2.1 2.3 1.4 1.6 1.8 1.9 1.3 1.5 1"6 1.7

2.0 1-8 1.6 1.5 2.1 2.4 2.4 1.3 2.0 2-6 2.8 2"9 1.9 2.5 3-0 3.2

COMPARISON

BETWEEN STABILITY

l / a = at/~l 0.8 1.8 1.3 1.1 1.0 2"3 1"9 1.5 1"3 2"5 2.2 1.8 1.5 2.6 2"5 2.0 1.7

GROOVE COR~DITIONS

0-5

0

-0.5

--0.8

-0"91

1.2 0.87 0.77 0.72 1"4 0.97 0"83 0.77 1.5 1.0 0.84 0.76 1"3 1-0 0"83 0.75

1"0 0.87 0.82 0.79 1"2 1.0 0"94 0.91 1.2 1-1 1.0 0.97 1.3 1-1 1.0 1-0

2.2 1.6 1.4 1.4 3.7 2.4 2.1 2.0 5.0 3.0 2.5 2.4 6-1 3.4 2.9 2.6

6-3 3.0 2.5 2.2 7"1 4-8 4.1 14.0 7.5 6.0 28.0 10-0 7.8

15.0 4-1 3.1 2-8 16.0 7.9 6-1 17.0 11.0 46.0 18.0

AND

PATCH

A comparison between the stability of grooves and patches can be m a d e b y examination o f t h e d a t a f r o m Figs. 3-6 a n d T a b l e 1, w h i c h are b a s e d on (24), (25) a n d (27). T h e isotropic case (R = P -- 1) is s h o w n in Fig. 7. T h e critical s u b t a n g e n t s , 1/Tao, 1/yG1, for

°'t 5

f-

2

If

/ / /

Porches A-O, all values of m Grooves Xl~¢= = I, m • 0.0t /~/%-I,m -0'25

/

-%

IO'

b

2

~ L

it,

~ x

i // #

\\ \\

/

/

# I

10% 5

U~

/

2 j

// / I

- .

-0-5

/%Graoves

X-O,all values

{

!

i

!

t

0

0"5

1.0

05

0

l/a-=,/~

"--" ~=t/=,= Biax~lity ratio

afro

I -0-5

-1.0

a

FiG. 7. C o m p a r i s o n b e t w e e n s t a b i l i t y o f g r o o v e s a n d p a t c h e s , R = P = 1, g r o o v e in d i r e c t i o n l.

334

E. DUNCOMBE

g r o o v e s ( b r o k e n lines) a n d liYp0 for p a t c h e s (full lines) are s h o w n as f u n c t i o n s of t h e b i a x i a l i t y r a t i o a. w i t h v a l u e s of m as a p a r a m e t e r . T h e lowest c u r v e in t h e f o r m e r case r e p r e s e n t s " a b s o l u t e " s t a b i l i t y , i.e. ~ = (8~g)0. T h e t w o a d d i t i o n a l b r o k e n lines i n Fig. 7 r e p r e s e n t t h e s o l u t i o n of (25) for m = 0.01 a n d m = 0.25. I t w o u l d b e e x p e c t e d t h a t s t a b i l i t y c o n d i t i o n s a r e t h e s a m e for g r o o v e s as for p ~ t c h e s u n d e r t h o s e c o n d i t i o n s w h e r e t h e b o u n d a r y e q u a t i o n s (11) a n d (12) are e q u i v a l e n t to (26). T h e s e o c c u r w h e r e a is e i t h e r zero or Rt(R + 1), t h e so-called p l a n e s t r a i n c o n d i t i o n . A f u r t h e r c o n c l u s i o n f r o m Fig. 7 is t h a t p a t c h s t a b i l i t y is a m o r e l i m i t i n g c r i t e r i o n (occurs a t a lower s u b t a n g e n t 1/~ in Fig. 2) t h a n g r o o v e s t a b i l i t y e x c e p t in a s m a l l region of b i a x i a l i t y r a t i o ~. T h e least s t a b l e c o n d i t i o n for g r o o v e p r o p a g a t i o n o c c u r s for t h e p l a n e s t r a i n case. F o r u n i a x i a l t e s t s o n a s h e e t (isotropic case) t h i s occurs w h e n t h e g r o o v e is a t a n a n g l e o f a b o u t 55 ° to t h e d i r e c t i o n of a p p l i e d stress, b e i n g in a g r e e m e n t w i t h e x p e r i m e n t a l o b s e r v a t i o n s a n d p l a s t i c flow p r e d i c t i o n s . 9 S t a b i l i t y c o m p a r i s o n s s i m i l a r to t h o s e o f Fig. 7 are g i v e n in Figs. 8 - 1 0 for a n i s o t r o p i c m a t e r i a l . Figs. 8 a n d 9 s h o w t h e s a m e c o m p a r i s o n for a v l a u e of R e q u a l to 2 a n d P e q u a l to 4. I n t h e first case t h e g r o o v e is l o n g i t u d i n a l as s h o w n in Fig, HaL w h e r e a s in

,o'~; L-

.... Grooves

5L

~"

!

"~

lOt--

"b

k/l~Q :1, m=O 0 1 ~

1

"~ co

,'~ , \ '1 \ ~ \.

Patches 5 k / k = O , oll I-~( valuesofm

X

s,

/

2 ~

i

/

' I / / /l

/ /

I /

l

]

\V.

/ % G r o c , ves k:O, all volues of m

I iCS't

-10-05

,

/

0

05

~0

O~

~

O

' -05

~lO×l(~tlty re1riO Fro. 8. C o m p a r i s o n b e t w e e n s t a b i l i t y of grooves a n d p a t c h e s , R = 2. P = 4, g r o o v e in d i r e c t i o n I.

t h e s e c o n d case t h e g r o o v e is p e r p e n d i m d a r t o t h e d i r e c t i o n s h o ~ . A g a i n it is e v i d e n t t h a t g r o o v e s t a b i l i t y is l e a s t for p l a n e s t r a i n c o n d i t i o n s , i.e. w h e n aUat is R/(R+ 1) ( l o n g i t u d i n a l groove) or P/(P+I) ( t r a n s v e r s e groove). C h a n g e s in t h e s t a b i l i t y loci a s s o c i a t e d w i t h a n i s o t r o p y c h a n g e s a r e p a r t i c u l a r l y e v i d e n t i n t h e case of grooves. F i n a l l y t h e c o m p a r i s o n of p a t c h a n d g r o o v e s t a b i l i t y for t h e case R = 4, P = 4 is s h o ~ m in Fig. 10. H i g h v a l u e s of R a n d P m e a n t h a t texttLre is t h e s a m e in b o t h l o n g i t u d i n a l a n d t r a n s v e r s e direction, w i t h c o n s i d e r a b l e t e x t u r e h a r d e n i n g in tile t h i c k n e s s d i r e c t i o n [ e q u a t i o n s (8) a n d (9)]. T h e n a r r o w e d s t r e s s b i a x i a l i t y r a n g e o v e r w h i c h i n s t a b i l i t y in t h i s m o d e is d o m i n a t i n g c a n p o s s i b l y b e identified w i t h t h i s fact.

Plastic instability and g r o w t h o f grooves and patches in plates or tubes

¢ -'~

5 /

II

/ /Patches

X = 0 , ollgroovesVOiues of m

/

/ ~'X/~ =i, m : 0.25

/J

7

I0'

,(//

/.X,~o.,, m . o a

\

//

t,

i

2 N

E n

I0 ° .5

/

/

•~

_,m O0

!I I

I

i o

-~o-o.s

Grooves X - O , all values of m

i o.s

I ~.o

I os

I o

i -os-a.o

Bioxiality ratio

F r o . 9. Comparison between stability of grooves and patches, _R = 2, P

=

4, g r o o v e

L~ d i r e c t i o n

t.

i0 z

Patches X = 0 , all volues ofm

c II

?

Grooves • k / % - I , m =0.01

2

,I/, /

/X/,g- I ,m- 0"25

c~ c~ s II

2

l

I

II II

I

~

/

~

/ ii

7

p,,

,o

E 0.

5

:-: .0

2

I// ! I

u~

IOZlo_oi5

~ ~'I k = 0 , all values of

I I 0 05

Z I I 0 05

I 0 0-5

I-0

Biaxiality ratio FIC.

I0.

Comparison between stability of grooves and patches, R = P = 4, groove in direction L

335

336

E. D u~'colwB E CONCLUSIONS

Methods of analyzing the plastic stability and propagation of grooves as a function of material strain hardening, strain rate sensitivity and anisotropy have been used to compare stability characteristics of grooves with those of patches. The analysis is strictly valid only for the case where the principal stress axes are coincident with the axes of anisotropy and where the groove direction coincides with one of the principal stress axes. The conclusions are as follows. (1) Over most biaxiality conditions patch-type instability occurs rather than groove-type instability, when it may be expected that diffuse (patch-type) thinning will precede concentrated (groove-type) thinning. However in regions where the strain rate in the direction of the groove is small the groove-type instability is expected to occur first. (2) Within the assumptions of the analysis and with the choice of thickness strain as the variable, the strain rate sensitivity influences propagation rates after instability has occurred rather than the instability conditions themselves. (3) In the case of groove propagation, absolute stability (zero roots) is sometimes found to be an unrealistic criterion. A criterion can be based on the degree of local thinning reached after a given amount of strain is reached in the nominal section. A dependence on strain rate sensitivity exists for such criteria. (4) Over the range of parameters explored, anisotropy slightly changes the regions of stress ratio in which the onset of groove-type instability precedes patch-type instability. Acknowledgements--This work was performed as part of the L W B R development program under the U.S. Atomic Energy Contract AT-11-1-GEN.14 {Division of Naval Reactors) and permission to publish the results is gratefully acknowledged. REFERENCES A. CONSIDERS., Ann. ponts et chaussdes 9 (Sd,r. 6), 574 (1885). M. Azl~I~ and W. A. BACKOFEN, Metallurgical Trans. 1, 2857 (1970). M. J. H I ~ E R , Int. J. mech. Sci. 7, 531 (1965). Z. M~RCI~L~K a n d K. KUCZ¥~SKI, Int. J. mech. Sci. 9, 609 (1967}. E. W. HART, Acta 3letaUurgica 15, 351 (1967). R. HILL, The Mathematical Theory of Plasticity, pp. 317-320. Clarendon Press, Oxford (1950). 7. D. C. DRUCKER, J. appl. Mech. 26, 101 (1959). 8. D. C. BOGUE, The Yield Stress and Plastic Strain Theory for Anisotropic Materials, Oak Ridge National Laboratory Report ORNL.TM-1869 (July 1967). 9. R. HILL, The Mathematical Theory of Plasticity, p. 324. Clarendon Press, Oxford {1950). 1. 2. 3. 4. 5. 6.

APPENDIX

Stability relations for a groove The geometry is defined by Fig. l(a). Equations for strains in a specific direction, using (7), are ~ t = Dt 8'~e+ ~g Du 3a~ + ~ D a 8o'~, (A1)

Plastic instability and growth of grooves and patches in plates or tubes

337

where derivatives D i and D~j are defined b y (17)-(21). The first boundary condition (12), together with (10) becomes The second b o u n d a r y condition (11 ) becomes 8(Ds ~g) = D1 ~ g + sa ~Dl = 0.

(A4)

By expanding $Dt and SDs in terms of derivatives D u, D,l and Dl,, equations (A3) and (A4) become D, 8~g + ~g Du Sort+ ~g D,t ~crl --- SaJa,, (A5) Ds 8~g Dn Strt + ~g Du 8al = 0.

(A6)

Equations (A5) and (A6) can be solved for the variables ~et and 6a,, yielding 6at = [3(Dl Dn - Dt Du) ao 8~,

(AT)

- On s

$as = [3(Dl Dn - Dt Du) - Ds s/~g a, $~g, Dn s

(A8)

where • emd s axe defined as crt/ag and the time operator d/dt. The identity D~ = Du Du

(A9)

has been employed. This m a y be verified b y direct substitution. Now the p r o p e r t y relation (1) becomes, b y expanding Sag,

L (Dr Sat + D~ 8al) = m~.,~g+ y~Sg. (A10) ag ~o The relations (AT) and (AS) for 8a~ and 8at m a y now be substituted in (A10), yielding [(m+ Fxa) s - ( F l a - 7) ~g] $~g = 0,

(All)

F~a = D~/D~l

(h12)

where and

D,2 D n - 2Dr Dt Dn + Ds2 Dn. fl

Fla =

Dn

"

(A13)