Exact Solutions for Large Elastoplastic

8/2/2019 Exact Solutions for Large Elastoplastic

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PergamonInternational Journal of Plasticity, Vol. 11, No. 1, pp. 99-118, 1995Copyright 1995 Elsevier Science LtdPrinted in the USA. All rights reserved0749-6419/95 $9.50 + .00

0749-6419(94)00040-9

E X A C T S O L U T I O N S F O R L A R G E E L A S T O P L A S T I CD E F O RM A T I O N S O F A T H I CK - W A L L E D T U BE

U N D E R I N T E R N A L P R E S S U R E

ROMAN BONN and PETER HAUPTUniversity of Kassel

(Communicated by Romesh Batra, Virginia Polytechnic and State University)

Ab st ra ct -A rate-independent plasticity theory based on the concept of dual variables and dualderivatives is utilized to describe finite elastic-plastic deformations including kinematic and iso-tropic hardening effects. Application of this theory to the problem of the thick-walled tube underinternal pressure leads to a system of partial differential equations of hyperbolic type. The exis-tence and uniqueness of the solution of the boundary value problem is guaranteed, as well asthe convergence of its numerical approximation. The exact solution of this problem is calcu-lated by means of an extrapolation technique. This integration method turns out to be appli-cable for rather general hardening models of rate-independent plasticity. On the basis of thecomputed solutions the influence of the hardening parameters is investigated. As finite defor-mations are of special interest , this investigation is carried out not only for the partially yieldedtube but also for the completely plastified tube. Furthermore, the onset of secondary plastic flowduring unloading as well as residual stress distributions are studied.

I. INTRODUCTIONBecause of its geometrical and physical nonlinearity, finite elastoplasticity admits onlya few exact solutions. Among these are special homogeneous deformations leading toa system of ordinary differential equations, which can be explicitly solved in particu-lar cases. An example is the problem of simple shear, which has been extensively stud-ied in the literature to investigate the properties of objective time derivatives (see e.g.HAUPr & TSAKMAKIS [1986] and the literature cited there).Inhomogeneous deformation processes are governed by nonlinear partial differentialequatiLons, and it cannot be expected that an explicit solution will be available. How-ever, there are particular classes of nonlinear partial different ial equations, where notonly existence and uniqueness of solutions of boundary value problems have been rig-orously proved, but also the possibility to calculate these solutions principally with anydesired numerical accuracy has been demonstrated. If a special problem of continuummechanics leads to a boundary value problem of this kind, we call the accurate numer-ical solution of this problem an exact solution.In this article, the problem of a thick-walled cylindrical tube under internal pressureloading is investigated in the context of large elastic-plastic deformations and generalhardening properties o f the material. Elastoplastic deformations of thick-walled tubeshave been analysed in several papers, which can be divided into two classes. The firstis concerned with small deformations: assuming nonhardening material, plane strain,and a yield criterion o f Tresca the paper of HILL, LEE, and TOPPER [1947] is one of the

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1 0 0 R . B O N N a n d P . H A U P T

first taking compressibility into account. A review of papers investigating the partiallyyielded thick-walled tube under internal pressure can be found for example in Snm-CmCrItr [1972]. The second class of papers investigates finite straining of pressurized tubes.MACGREGOR, COFFIN,and FISHER [1947] derive a solution in integral form on the as-sumptions of a von Mises yield condition, plane strain, total incompressibility, and iso-tropic hardening. Incorporating the yield criterion of Tresca, linear isotropic hardeningtogether with elastic compressibility and plane strains, FISCHER [1977] is led to a systemof ordinary differential equations, which he has solved for cyclic loading processes.In more recent works of DURBAN [1979, 1988] and DURBAN and KUBI [1992] solutionsin the form of quadratures have been developed for the yield criteria of yon Mises andof Tresca considering general isotropic hardening rules based on deformation theory.The exact formulat ion of the boundary value problem for plane strain, developed insections III.2 to III.4, leads to a system of quasilinear partial differential equations. Thissystem is of hyperbolic type, and its characteristic directions are known a priori. Theexistence and uniqueness of the solution was established in HARTMAN and WINTNER[1952]. Moreover, SMITFI [1970] and HACKBUSCI-I[1977a] were able to prove that for spe-cial classes of finite difference schemes the numerical solution has an asymptotic expan-sion in terms of powers of the stepsize. These results can be applied to realize a far moreaccurate calculation of the solution by means of an extrapolation technique. We call thisextrapolated solution an e x a c t s o l u t i o n of the problem.While the geometry of the problem is quite special, the constitutive modelling maybe very general, however, within the theory of rate-independent plasticity. It turns outthat the presented procedure applies to constitutive models incorporating general kine-matic and isotropic hardening properties. The applicability of the e x a c t s o l u t i o n s devel-oped in this article is twofold: On the one hand they may be utilized to discuss thephysical meaning of advanced constitutive equations of elastoplasticity. On the other handthey may serve as testing examples to evaluate the efficiency of finite element schemes.

I I. C O N S T I T U T I V E A S S U M P T I O N SII. 1. I n t e r m e d i a t e c o n f i g u r a t i o n

The formulation of constitutive assumptions applies the concept of an intermediateconfiguration, which implies the multiplicative decomposition of the total deformationgradient 17 into an elastic part Fe and a plastic part 17u:

F = F e F o (1)(see HAUPT [1985] and the literature cited there). In its physical interpretation the plas-tic deformation "gradient" Fp is the result of a local unloading process. In view of thepolar decomposition

F p = R p U p (2)and the nonuniqueness of the decomposition of F (F = ~'eFp = Fe( ~r ( ~Fp for all orthog-onal tensors ~)) it can be argued that only the plastic stretch tensor or, equivalently, theplastic Green strain

E p T~(FpFp-1) (3)


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T h i c k - w a l l e d t u b e 1 01

c a n b e r e l a te d t o t h e h i s t o r y o f t h e t o t a l s t ra i nE = ( F r F - 1 )

b y a c o n s t i tu t i v e fu n c t i o n a l( 4 )

E p ( t ) = (R [ E ( r ) ] . (5 )r < tT h i s f a n c t i o n a l is a s s u m e d t o b e r a t e - i n d e p e n d e n t ; i t w il l b e d e f i n e d i m p l i c i t ly i n th eu s u a l w a y , i .e . b y c o m b i n a t i o n o f a n e l a st ic i ty r e l a ti o n w i t h a y i e l d a n d a l o a d i n g c o n -d i t io n , a f l o w r u l e , a n d h a r d e n i n g m o d e l s . I n t h e s e q u e l , t h e s e i n g r e d i e n ts a r e p r o v i d e di n t e r m s o f v a ri a b l e s r e la t e d t o t h e i n t e r m e d i a t e c o n f i g u r a t i o n . T h e p a r t i c u l a r c h o i c e o fs t r a in a n d s tr e s s te n s o r s a n d t h e i r t im e d e r iv a t iv e s f o l l o w s th e c o n c e p t o f d u a l v a r i a b le sa n d d u a l d e r i v a t iv e s , d e v e l o p e d i n H A U P T a n d T SA KM A KIS[1989].

W h i l e th e t o t a l G r e e n s t r a in t e n s o r E is a c ti n g o n t h e r e f er e n c e c o n f i g u r a t i o n , t h ee q u i v a l e n t t o t a l s t r a i n t e n s o r

~ 1 7 T - 1 l ~ l T - 1= P - - - - P ~___ 1 ( ~ T ~ e - - - p i ~ r - I p Fl ) ( 6 )o p e r a t e s o n t h e i n t e r m e d i a t e c o n f i g u r a t i o n . F r o m t h e d e f i n i t i o n o f 1~ w e i n f e r th e a d d i -t i v e d e c o m p o s i t i o n

l~ = l~e + A p (7)i n t o a n e l a s t i c G r e e n s t r a i n ,

E e 1 ^ T ^---- ~ ( F e F e - 1 ) ( 8 )

a n d a p l a s t i c A l m a n s i s t r a i nA p 1 - 1 r - I !~-~= p - - p , . (9)

T h e c o r r e s p o n d i n g t o t a l s t r a i n r a t e i s g i v e n b yAE " = = p r- ll ~ r -' = = p = ! ~ + L ~ I ~ + l ~ L p , ( 1 0 )

w h e r e [ , p = ~ 'p F p ~ is t h e p la s t ic v e l o c i t y " g r a d i e n t . " T h e t o t a l s t r a in r a t e d e c o m p o s e sa c c o r d i n g t o

E = E e J r - A p , ( 1 1 )w i t h

a n d= ^ ^ T ^ A p L p . ~ . < L p - .[ -) p A . = ~ p + L o A p + [ J r ) .


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102 R. BONNand P. HAUPT

I n t h e c o n t e x t o f t h e s t r a i n E ~ t h e d u a l s t re s s is d e f i n e d b y( 12 )

w h e r e S = ( d e t F ) T is t h e w e i g h t e d C a u c h y s tr es s t e n s o r , T t h e C a u c h y s t re s s t e n s o r , a n dT = ( d e t F ) F - l T F T -1

t h e 2 n d P i o l a - K i r c h h o f f s tr e ss t e n s o r . I n th i s c o n t e x t , t h e c o r r e s p o n d i n g s t re s s r a t eis g i v e n b y

( 13 )T h e s t r a i n !~ a n d t h e s t r es s S a r e d u a l i n t h e s e n s e t h a t t h e f o l l o w i n g q u a n t i t i e s a r e i n v a r i -a n t w i t h re s p e c t t o a c h a n g e o f t h e c o n f i g u r a t i o n : S c a l a r p r o d u c t o f s tr e ss a n d s t ra i nt e n s o r s , s t re s s p o w e r , c o m p l e m e n t a r y s t re s s p o w e r , i n c r e m e n t a l s t re s s p o w e r ( s e e H A U P Ta TSAKMAKIS [19 89]) .

I1 .2 . C o n s t i t u t i v e r e l a t i o n sN o w , t h e c o n s t i t u t i v e a s s u m p t i o n s a r e la i d d o w n i n t e r m s o f d u a l v a r i a b le s a n d

d e r i v a t i v e s :

O ( l~ e ) ( P ( t r i ~ e ) l ) ( 1 4 )l a s t i c i t y r e l a t i o n S = g (l ~ e ) = t5 - ~ - 2/~ ! ~ + 1 - 2-------~/~ a n d u a r e t h e s h e a r m o d u l u s a n d P o i s s o n ' s r a t i o , r e sp e c t iv e l y , a n d is t h e s t r a i n e n e r g yf u n c t i o n .

Y i e ld c o n d i t io n f ( S , X , s ) = 0(~ D _ x D ) . ( ~ D _ x D ) _ 2 k 2 ( s ) = 0 (1 5)

i s t h e b a c k s t r e s s t e n s o r a n d s i s t h e p l a s t i c a r c l e n g t h ; t h e s u p e r s c r i p t ( ) D d e n o t e s t h ed e v i a t o r i c p a r t o f t h e t e n s o r i a i q u a n t i t y ( ) a n d k ( s ) t h e u n i a x i a l y i e l d s tr e ss .

F l o w r u l e l ) p = A --~Of _ 3 ~ ( ~ D _ x D ) ( 1 6 )O S 2 k ( s )P l a s t i c a r c l e n g t h ~ = ( 2 D p . ~ ) p ) l /2 ( 17 )

T h e b a c k s t r e s s t e n s o r , m o d e l l i n g k i n e m a t i c h a r d e n i n g b e h a v i o u r , c a n b e e x p r e s s e da s a l i n e a r a n d i s o t r o p i c f u n c t i o n a l o f t h e p l a s t i c s t r a i n h i s t o r y ( H A u P X , K R O ZE ~ r, &T S A K ~ K I S [ 1 98 7 ]) . F o r s p e c ia l c h o i c e s o f t h e k e r n e l , t h e b a c k s t r e s s t e n s o r c a n b e s p l itu p i n t o a s u m ( C r t ~ o c r m [1 97 7]) :

= ~ X / w i t h ~ , = c i f ) p - b , ~ ) ( .i = l


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Thick-walled tube 103

I n t h e f o l l o w i n g , w e r e s tr i c t o u r s e l v e s t o m = 1 , i .e . w e a s s u m e a k i n e m a t i c h a r d e n i n gr u l e d u e t o A R US TR O ~rG a n d F m ~ DE R IC K [1 96 6 ] w i t h t h e m a t e r i a l p a r a m e t e r s c a n d b "

VK i n e m a t i c h a r d e n in g X = c f ) p - b ~ X (18)l s o t r o p i c h a r d e n i n g k = ~ / k ( s ) ( k o . - k ( s ) ) ~ . (19)

T h e i s o tr o p i c h a r d e n i n g m o d e l d e s c r ib e s a m o n o t o n i c i n c r e a se o f t h e u n i a x ia l y ie l ds t r e ss k ( s ) a n d i n c lu d e s a s a t u r a t i o n p r o p e r t y . T h e m a x i m u m v a l u e o f k is k ~ a n d "r isa m a t e r i a l c o n s t a n t .

T o o b t a i n a n e x p l ic i t f o r m f o r t h e c o n s t i t u t i v e e q u a t i o n o f t h e p l a s ti c a r c l e n g t h ( 1 7)t h e c o n s i s te n c y c o n d i t i o n ( f = 0 ) h a s t o b e e v a l u a t e d b y m e a n s o f t h e c o n s t i t u t iv e r e la -t i o n s ( 1 4 - 1 9 ) :

2 2 # ( S D - i D) !~= ~ k ( s ) 1V (20)T h e d e n o m i n a t o r ~ r o f e q n (2 0) is g iv e n i n th i s c a s e b y

2 [ 2 d k ( s ) ] + 8 v t r ( ~ )]Q = - ~ k 2 (s ) 2 # + c + ~ - ~ s 3 k 2 (s ) 2( 1 + v~-----)_ 2 ( ~ D _ ~ D ) . [ (~ D _ ~ D ) ( ~ D _ ~ D ) ] _ 2 k ( s ) b ( g O _ ~ o ) . i

v t r ( S ) l ) 4 k 2 ( s ) t r ( ~ _ ~ ) .4 ( ~ D _ ~ D ) 2 . ~ 1 + V -- (21)

T h i s c o n s t i t u t iv e m o d e l h a s t h e f o l l o w i n g t w o p r o p e r t i e s ( se e G R E E N & N A G H D I [1 97 1],SrDOROFF [1973]):

I t f u l f il l s t h e p r i n c i p l e o f o b j e c t i v i t y , I t o b e y s f u l l i n v a r i a n c e w i t h r e s p e c t t o a r b i t r a r y r o t a t i o n s o f t h e i n t e r m e d i a t e

c o n f i g u r a t i o n .T h i s i m p l i e s t h a t t h e t w o f u n c t i o n s f a n d g ( o r ~b) i n e q n s (1 4) a n d ( 15 ) a r e i s o t r o p i c te n -s o r f u n c t i o n s .

I II . T H I C K - W A L L E D T U B E U N D E R I N T E R N A L P R E S S U R EI n t h e f o l lo w i n g w e f o r m u l a t e t h e b o u n d a r y v a l u e p r o b l e m f o r a t h i ck - w a l l e d t u b e

u n d e r i n t e r n a l p r e ss u r e c o n s i d e r i n g t h e c o n d i t i o n o f p l a n e s t ra i n .I I I . 1 . K i n e m a t i c s

A c c o r d i n g t o t h e c y l in d r i c a l s y m m e t r y o f t h e c o n s i d e r e d p r o b l e m , c y l in d r i c a l c o -o r d i n a t e s a r e i n t r o d u c e d . T h e n o r m a l i z e d c y l i n d r ic a l b a s e v e c t o rs G I , G 2 , a n d G 3( r a d ia l , c i r c u m f e r e n t i a l , a n d a x ia l d i re c t io n ) o f t h e r e f e re n c e c o n f i g u r a t i o n f o r m a n


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104 R. Boron and P. HAUPT

o r t h o - n o r m a l s y s te m . I t is e v i d e n t th a t i n t h is c a se t h e c o m p o n e n t s A * o f a t e n s o r Ai n t h e r e p r e s e n t a t i o n

A = A ~ G k G ma r e ph y s i c a l c o m p o n e n t s . R e f e r r in g t o t h e m a t e r i a l r e p r e s e n t a t i o n , a l l m a t e r i a l d e r i v a -t i v e s o f t e n s o r s r e d u c e t o p a r t i a l t i m e d e r i v a t i v e s o f t h e p h y s i c a l c o m p o n e n t s .

T h e d e f o r m a t i o n p r o c e s s i s g i v e n t h r o u g h a d i s p l a c e m e n t f i e l d u r e p r e s e n t i n g a r a d i a le x t e n s i o n o f t h e t u b e :

u = u ( R , t ) G 1 . (22)H e r e , t is t i m e a n d R t h e d i s t a n c e o f a m a t e r i a l p o i n t f r o m t h e r o t a t i o n a l a x is in t h e r e f -e r e n c e c o n f i g u r a t i o n . F r o m (2 2) w e d e ri v e t h e m a t r ix r e p r e s e n t a t i o n o f t h e d e f o r m a t i o ng r a d i e n t , l 0 i l~ /3 , (23 )

0w h e r e t h e f o l l o w i n g d e f i n i t i o n s h a v e b e e n u s e d :

o (1 = ~ , (3 = 1 + = ~ . (24)T h e v a r i a b l e r r e p r e s e n ts i n th i s c o n t e x t t h e d i s t a n c e o f a m a t e r i a l p o i n t f r o m t h e r o t a -t i o n a l a xi s i n th e c u r r e n t c o n f i g u r a t i o n . D u e t o t h e s y m m e t r y o f t h e p r o b l e m w e f i n dt h e m a t r i x r e p r e s e n t a t i o n o f t h e p l a s t i c a n d e l a s t i c d e f o r m a t i o n " g r a d i e n t s " t o b e

Otp 0 0o ~p o

10 0t e n

a__ o ooLpo - - o

o o e ,p~p

(25)

I t i s w e l l k n o w n (KOITER [ 19 5 3] ) t h a t t h e a p p l i c a t i o n o f a T r e s c a y i e l d c o n d i t i o n i m p l i e sa l so a p l a n e p l a st ic s t ra i n f i e ld . I n c o n t r a r y t o t h is , a v o n M i s e s y i e ld c o n d i t i o n i n c o m -b i n a t i o n w i t h t h e a s s o c i a t e d f l o w r u l e le a d s t o a n o n v a n i s h i n g p l a st ic s t r a i n r a t e c o m -p o n e n t i n d i re c t i o n o f t h e s y m m e t r y a x is , i. e .

~3

T h i s is s a t i s fi e d b y e q n s ( 2 5 ) i f B p . 1 /o ~p .


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Thick-wall ed tub e 105

111.2. E q u a t i o n s o f eq u i li b r iu m a n d c o m p a t i b il it yIn the following we consider only quasistatic processes and neglect body forces. Then,

the condition of equilibrium reduces in terms of physical components of the Cauchystress tensor T to! 0 T ~ + 1.2_. ( T I - T 2 ) = 0 . ( 2 6 )Or r

Because all constitutive relations are formulated with respect to the intermediate con-figuration, eqn (26) should be transformed into a relation between the physical compo-nents of S:2

- - + 2 + = 0 . ( 2 7 )OR a OR otp OR RThe components or,n of the deformation gradient must satisfy the compatibility condition

an ~ - n- - - ( 2 8 )a R R

Inserting the relation of elasticity (14) into (27), one is led to the equilibrium condition

. r i p ~R ~ n p -= 2/z ~ aR

[ 1 - v a S I ] & ~ 2 ~ n o n+ ~ 2 # i - - 2 ~ (~--zp + ~ j O R + 1 - 2 . n z O R

2lz UO tpn - (1 - p) - 2+ ~ OlpJOR'

(29)

which will be used in the sequel together with eqn (28).

III.3. C o n s t i t u t i v e e q u a t i o n sIncorporating the kinematic assumptions o f the deformation process (eqns (22-25))into the constitutive equations (eqns (15-19)) leads to evolution equations for the plas-tic art: length, the plastic strain components, and the hardening components, which aresummarized below:

= i k ( s ) - E ( ~ '~ - ~ ' ~ ) ~ + ~ ( - ~ ' ~ ) ~ (30)

~ p _ 3 1 ( ~ n _ X n ) I s (31)~ p 2 k ( s )


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106 R. BONN an d P. HAUPT

f3p 3 13 p 2 k ( s )- - ( ~ - ~ ' o ) ~ (32)

]~'I = k - ~ (;~O _ ~.-O)l (C + 2)~'I ) -- b-~'l (33)

[c = 3 ' k ( s ) ( k o . - k ( s ) ) ~ .F u r t h e r m o r e , w e

1~ D _ f ( O ) 2 ( C + 2.1~-2) _ b)~-2 S(c + 2 .,X '] ) - b ) ( ] ]$

f i n d t h e d e n o m i n a t o r ~ r i n e q n ( 3 0 ) t o h a v e t h e f o r m

(34)

(35)

(36)

-3 - - - ~ s J + 2 Iz - - + I z a 2 3 2 + . f ( l [ ( ~ n _ , e~O) l ]23 o~

2 2 2 ~ [ ( ~ 2 ) ~ 1 2+2 #-g-~ + I z a p 3 ~ , +3p2 2 __ __+ 4 ( /~ p3 ~ + )~ -3 ) [ (,~O ~ - D ) I (~O ~- O)Z] . (37)

I n v ie w o f a m a t h e m a t i c a l d i s c u s s i o n o f t h e c o n s t i t u t i v e e q u a t i o n s ( e q n s ( 3 0 -3 6 ) ) e a c ho f t h e s e e q u a t i o n s c a n b e e x p r e s s e d i n t h e s y m b o l i c n o t a t i o n

9 Oym~,, Akin(Y) ~ = 0 , (k = 3 ,4 . . . . . 9 ) , (38)m = l

w h e r e y i s t h e v e c t o r o f t h e d e p e n d e n t v a r i a b l e s :y = ( o , , t L o , , , , ~ p , S , k , ~ l , Y c ~ , , k ~ ) (39)

T h i s m a t h e m a t i c a l s t r u c t u r e o f e q n s (3 8 ) i s d u e t o th e r a t e - i n d e p e n d e n c e o f t h e c o n -s t i tu t i v e m o d e l ; i t is n o t a f f e c t e d b y th e s p e c i al c h o i c e o f h a r d e n i n g m o d e l s T h u s ,e v e n m o r e s o p h i s t i c a t e d m o d e l s o f f i n i t e e l a s t o p l a s t i c i t y , w h i c h a r e f o r m u l a t e d a s r a t e -i n d e p e n d e n t e v o l u t i o n e q u a t i o n s , c a n b e e x p re s s e d w i t h i n th e s t r u c t u r e o f e q n s (3 8 ) a n dt h e r e f o r e s o l v e d w i t h t h e i n t e g r a t i o n t e c h n i q u e p r o p o s e d i n s e c t i o n I V . 2 . F u r t h e r m o r e ,a s a c o n s e q u e n c e o f t h e r a te - i n d e p e n d e n c e , t h e ti m e t a s i n d e p e n d e n t v a r i a b le c a n b er e p l a c e d i n e q n s ( 3 0 - 3 6 ) b y a n y m o n o t o n i c f u n c t i o n z , ; ~ ( t ) > 0 . I n t he seque l , t h ree d i f -f e r e n t f u n c t i o n s z w i l l b e u s e d a s s u b s t i t u t e s f o r t i m e t , n a m e l y : t h e i n t e r n a l p r e s s u r ep ( t ) , t h e r a d i u s C ( t ) o f t h e c y l i n d ri c a l s u r f a c e , w h i c h s e p a r a t e s e l a s t ic f r o m p l a s t icd o m a i n s , a n d t h e p l a s t i c a r c l e n g t h s ( t ) .

T o g e t h e r w i t h t h e e q u a t i o n o f e q u i l i b r i u m ( 2 9) a n d t h e c o m p a t i b i l i t y r e l a t i o n ( 2 8 ),h a v i n g t h e s t r u c t u r e


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Thick-wall ed tub e 107

9 COym~a Akin(Y) = Bk(y ,R) (k = 1,2), (40)m=l O Reqns (38) form a system of partial differential equations determining all unknownvariables. This system has to be completed by appropriate boundary conditions.III.4. B o u n d a r y v a lu e p r o b l e m

For the pressurized thick-walled tube, boundary conditions are prescribed on the innersurface of the tube Ri as well as on the outer surface R o . These are formulated withrespect to the Cauchy stresses,

T 1 ( R i , t ) = - p ( t ) and T l ( R o , t ) = O, (41)where p ( t ) is the applied internal pressure and TI the radial stress component.

In 1Lhe case of purely elastic loading and unloading processes, eqns (28) and (29),together with (41), determine the complete boundary value problem. This two-pointboundary value problem can be solved by standard methods of integration (see for exam-ple PRESSet al . [1988]).If elastic-plastic deformations take place, two different situations may occur, namelythe partially yielded tube and the completely plastified tube. For the partially yieldedtube (see HILI. [1950]) two regions o f different material behaviour exist: An inner ringof elastic-plastic material behaviour is separated by a cylindrical surface with radiusC ( t ) (with respect to the reference configuration) from a purely elastic outer ring.Boundary conditions for the elastic domain are given at the outer radius R o and atthe radius C ( t ) of the separating surface; these are the vanishing outer pressure and theyield condition, respectively:

S l ( R o , t ) = 0 and f ( S ( C ( t ) ) , X ( C ( t ) ) , s ( C ( t ) ) ) = O. (42)This boundary value problem o f the purely elastic domain will be solved as ment ionedabove. For the inner elastic-plastic region, the complete set of partial differential equa-tions (eqns (28-36)) has to be solved. As will be shown in section IV. l, this system hasa unique solution, so that the boundary value problem can be replaced by an equiva-lent initial value problem. The initial data are prescribed on the separating surface. Forthe first loading process, the plastic variables have the following values:

X ( C ) = O, s( C) = O, k ( C ) = ko, otp( C ) = 13p(C ) = I. (43)On the other hand, the components of the total deformation gradient (o~,/~) follow fromthe solution within the outer elastic ring calculated at C. The radius C ( t ) of the sepa-rating: surface cannot explicitly be given as a function of time t or internal pressure p.Therefore, to avoid an iterative calculation of C(t ) , an inverse technique (see e.g. SHm-CHI CI-rO [1972]) is applied. The idea thereby is to make use of the rate-independence inthe sense that in the system of partial differential equations (eqns (30-36)) time t is re-placed by the radius C ( t ) . Due to this change of the integration variable, the domain ofsolution changes to a square (see Fig. 1), where the upper triangle represents the elastic-plastic region. Thus, the separating surface is now a straight line and known a priori.


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p ( t ) I I I I _ I ~- e l a s t i c - /- p l a s t i c / . . . .

L I i J ' iJ " " C i t )J - - e l a s t i c - -/

Ca o

p ( t ) ~ C ( t )

R i _

I I I l _ /e l a s t i c - # #I- - - ~ - P l ~ a s t t c ~

/ I f- - e l a s t i c - -/ l l l l/ I I I II R i R o l - = " R I R i R o l m R

Fig. 1. Cha nge of dom ain of solution after change of integration variable.

I f t h e c y l i n d e r i s c o m p l e t e l y p l a s ti f ie d , e l a s ti c - p la s t ic d e f o r m a t i o n s t a k e p l a c e t h r o u g h -o u t t h e t u b e . I n t h is c a s e t h e c o m p l e t e s e t o f c o n s t i tu t i v e e q u a t i o n s h a s t o b e s o l v e d i nt h e w h o l e r eg i o n . H o w e v e r , u p t o n o w n o f o r m u l a t i o n o f a n e q u i v a l e n t in i ti al b o u n d -a r y v a lu e p r o b l e m h a s b e e n p r o p o s e d i n t h e l i te r a t u re . S o m e a u t h o r s r e d u c e t h e b o u n d -a r y v a lu e p r o b l e m t o a s y st e m o f t w o o r d i n a r y d i f f e re n t i a l e q u a t i o n s o n t h e b a si s o f am o r e s p e ci a l c h o i c e o f c o n s t i t u t iv e m o d e l s , l ik e D U I~ A N [1 98 8] ( d e f o r m a t i o n t h e o r y , v o nM i s e s y i e l d c o n d i t i o n , a n d g e n e r a l i s o t r o p i c h a r d e n i n g ) a n d F IS CH E R [1 97 7] ( T r e s c a y i e ldc o n d i t i o n a n d l i n e a r i s o t r o p i c h a r d e n i n g ) .

I n t h e f o l lo w i n g w e p r o p o s e a p r o c e d u r e t o d e f i n e a n e q u i v a l e n t in i ti a l b o u n d a r y v a l u ep r o b l e m f o r c o n s t i t u t i v e e q u a t i o n s i n t h e f o r m o f e q n s ( 3 8) . F i g . 2 i l lu s t ra t e s t h e d o m a i no f s o l u t i o n a n d t h e c u r v e s a t w h i c h d a t a h a v e t o b e p r e s c ri b e d . O n t h e o r d i n a t e a x is ,t h e r a d i u s C o f t h e s e p a r a t i n g s u r f a c e i s r e p l a c e d b y t h e p l a s t ic a r c l e n g t h So = s (Ro , t ),w h i c h s t a r t s w i t h So = 0 a n d d e v e l o p s m o n o t o n i c a l l y . O n t h e c u r v e CA b o u n d a r y v a l-u e s a t th e o u t e r r a d i u s Ro h a v e t o b e g i v e n , a n d o n t h e c u r v e C B a t t h e v a l u e So = 0 o ft h e p l a s t i c a r c l e n g t h i n i t i a l d a t a m u s t b e p r e s c r i b e d a c r o s s t h e w a l l - t h i c k n e s s . T h e l a t -t e r d a t a f o l lo w f r o m t h e s o l u t i o n o f t h e p a r t ia l l y y i e ld e d t u b e a t t h a t i n s ta n t w h e r e y ie l d-i n g h a p p e n s t o c o m m e n c e a t t h e o u t e r ra d i u s. T o o b t a i n a s et o f a p p r o p r i a t e b o u n d a r yv a l u e s , w e a g a i n c h a n g e t h e i n d e p e n d e n t v a r ia b l e : W e n o w r e p l a c e t h e t i m e t b y t h e p l as -t i c a r c l e n g t h So = S(Ro,t) .

A s a c o n s e q u e n c e , a l l t i m e d e r i v a t i v e s i n e q n s ( 3 0 - 3 6 ) a r e r e p l a c e d b y d e r i v a t i v e s w i t hr e s p e c t t o So. R e w r i ti n g e q n s ( 3 0 - 3 6 ) i n th i s w a y , w e o b t a i n a s e t o f o r d i n a r y d i f f e r e n -t ia l e q u a t i o n s a t t h e f i x e d ra d i u s Ro, v a l i d f o r So > 0 :

s I I I Il a s t i c -p l a s t i c0 I I I\ C B

~ , C A

I R i R o I RFig. 2. Initial conditions for com pletelyplastified thick-walled tube.


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Thick-walled ube 109

d ~p 3dso 2 k(s o)

_ _ ( ~ , _ / : t , ) ]

d/~p = 3 {3p ( ,~ D _ ~ D )~dso 2 k(so )d k- - = 7 k ( s o ) ( k ~ - k (S o )) (44)dso

a 2 1 [ 3 1dso 2 k(s o) ( ~ D _ X o ) [ ( c + 2 P~ 'l) - b ~ ' l ]

dso k ( so)_ _ _ ( , ~ o _ / ( o ) ~ ( c + 2 / ' 2 ) - b . ~ " 2 ]

d / . / ~ = [ ( ~ o _ / . o ) l + ( ~ o _ / - o ) ~dso L ] k ( so ) ( c + 2 ~ ) - b / ' ~ ] .

B y c o m b i n a t i o n o f t h e b o u n d a r y c o n d i ti o n ,S~(Ro, t ) = 0 wi th eqn (30) a fu r ther d i f -f e r en t i a l eq u a t i o nd a f i/ 2 # ( s D - - ) ( D ) 2 I 1 - v ot 2 ]d so = M ~ ,, o,~ 4 ~ ( S D - / ~ ' ) ~

2 /z (S D - ) ( ) i [ ~ 2 } (45)

can b e d ed u ced , w h e r e N i s g i v en b y ( 3 7 ) an d M i s d e f i n ed a sM = 2 k ( s o ) 2 # [ ( ~ D 2 o ) I

3- 2 ] . ( 46 )

In t eg ra t i ng eqn s (44) an d (45 ) a long CA (see F ig . 2 ) l eads t o t he c om ple t e se t o f bo un d-a r y v a l u es f o r R = R o .I V . N U M E R I C A L I N T E G R A T I O N A N D R E S U L T S

IV. 1. E x i s te n c e a n d u n iq u en e s s o f t h e s o lu ti o nA s , ;h o w n in s ec t i o n II I . 3 , t h e c o n s i d e r ed p r o b l em o f a t h ick - w a l l ed t u b e u n d e r i n t e r-na l p ressu re l eads t o a sys t em o f f i r s t -o rder par t i a l d i f fe ren t i a l equa t ion s (eqns (28-36)) .

T h i s s y s t em i s a s p ec i a l c a s e o f t h e m o r e g en e r a l s y s t emN OyraA k m ( Y , R , t ) ~ = B k ( y , R , t )

m = l( k = 1 . . . . n )

m a y m~ _ ~ A k m ( Y , R , t ) = B k ( y , R , t ) ( k = n + 1 . . . . N ) ,m=l OR( 4 7 )


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w h e r e t h e c o e f f i c i e n t f u n c t i o n s A k in a s w e l l a s t h e r i g h t h a n d s i d e s B k a r e c o n t i n u o u sf u n c t i o n s o f t h e s o l u t i o n v e c t o r y = ( y~ . . . . . Y N ) a n d o f t h e i n d e p e n d e n t v a r ia b l e s Ra n d t . I f fo r a ll y , R , a n d t

d e t A k in =/: 0 (48)h o l d s , a n d i f a ll Z k m a n d B k p o s s e s s c o n t i n u o u s f i r s t - a n d s e c o n d - o r d e r d e r i v a t i v e sw i t h r e s p e c t t o a l l a r g u m e n t s , t h e s y s t e m ( 47 ) is o f h y p e r b o l i c t y p e . F u r t h e r m o r e , i tc an b e p r o v e n ( H ~ W IN T r~ R [1 95 2], SM m~ [197 0]) t ha t a u n i q u e s o l u t i o n e x i s t sf o r s y s t e m ( 4 7 ) .

I V . 2 . E x t r a p o l a t i o n t o t h e l i m i tI n o r d e r t o o b t a i n a n u m e r i c a l a p p r o x i m a t i o n o f t h e e x a c t s o l u t io n o f th e s y s t e m (4 7 ),

w h i c h i s a s c lo s e a s p o s s i b l e , w e a p p l y a s p e c i a l i n t e g r a t i o n t e c h n i q u e , w h i c h g o e s b a c kt o R IC HA RD SO N [1 91 1 ]. T h e m a t h e m a t i c a l c o r r e c t n e s s o f t h is m e t h o d h a s b e e n e s t a b l i s h e db y S m r a [ 19 7 0] a n d H A C rB U S C H [ 1 9 7 7 a , 1 9 7 7b ] f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s a n db y G P ,A o o [ 1 96 5] a n d H A m E R a n d L U m C H [1 98 4] f o r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s .

T h e R i c h a r d s o n - e x t r a p o l a t i o n o r e x t r a p o l a t i o n t o t h e l i m i t is b a s e d o n c e r t a i n f in i te -d i f f e r e n c e i n t e g r a t i o n s c h e m e s . A c c o r d i n g t o S M IT H [ 19 7 0] a n d H A C KB U SC H [ 1 9 7 7 a ], i tis n e c e s s a r y t h a t t h e d i s c r e ti s a t io n e r r o r c a n b e e x p r e s s e d a s a p o w e r s e r ie s o f t h e d i s-c r e t i s a t i o n s te p s i ze h . T h i s is e q u i v a l e n t to a n a s y m p t o t i c e x p a n s i o n o f t h e s o l u t i o n Ydiso f t h e d i s c re t i s e d sy s t e m , v a l i d f o r a ll p o i n t s i n t h e s p a c e o f i n d e p e n d e n t v a r i a b l e s R , t :

Y d i s ( R , t ; h ) = Y ex ac t ( R , t ) + e p ( R , t ) h p + e q ( R , t ) h q + . . . + E ( R , t ; h ) hz + l. (49)H e r e , q , p . . . . . z a r e p o s i t i v e n u m b e r s w i t h q < p < . . . < z , Y exact i s t h e e x a c t s o lu -t i o n o f s y s te m ( 4 7) a n d e q . . . . . e z d e p e n d o n l y o n t h e i n d e p e n d e n t v a r ia b l e s . T h e e r r o rt e r m E i s b o u n d e d .

F o r s i m p l i c it y , w e r e s tr i c t o u r c o n s i d e r a t i o n t o o n e s p e c ia l f i n i t e - d i ff e r e n c e s c h e m e ,n a m e l y t h e m e t h o d o f c h a r a c t e r is t ic s , w h i c h i s c o n v e r g e n t o f f ir st o r d e r . I n t h is c a s e,w e o b t a i n t h e f o l l o w i n g s y s t e m o f d i f fe r e n c e e q u a t i o n s :

N y m ( R , t ) - y m ( R , t - h )~ _ a A k m ( R , t - h ) = b k ( R , t - h ) k = l , . . . , nm = l hN y m ( R , t ) - y m ( R + h , t )A k m ( R + h , t ) = - b k ( R + h , t ) k = n + l . . . . . N .m=l h

(50)

F o r c o n v e n i e n c e , t i m e t h a s b e e n s c a l e d , s o t h a t t h e s t e p s i z e 4 t i n t i m e i s e q u a l t o t h es t ep s iz e A R o f t h e s p a t ia l c o o r d i n a t e :A t = A R = h .

I t h a s b e e n p r o v e n b y S m 'rt- i [1 97 0 ] t h a t t h e s o l u t i o n o f s y s t e m ( 50 ) p o s s e s se s t h e r e q u i r e da s y m p t o t i c e x p a n s i o n ( 4 9 ) . F o r a l l p o i n t s ( R , t ) w e h a v eY d i s ( R , t ; h ) = Y e xa c t( R ,t ) + e l ( R , t ) h l + e 2 ( R , t ) h 2 + . . . + E ( R , t ; h ) h z+ l. ( 5 1 )


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Thick-wal led tube 111

I n o r d e r t o i m p r o v e t h e o r d e r o f c o n v e r g e n c e , i . e . t o c a l c u l a t e a n e x t r a p o l a t e d s o l u t i o nY ~xp w i t h a l e a d i n g e r r o r t e r m h i g h e r t h a n h ~ , s o l u t i o n s o f t h e d i s c r e t i s e d s y s t e m ( 50 )a r e c a l c u l a t e d w i t h d i f f e r e n t s t e p s i z e s h k . T h e s e s t e p s i z e s h k a r e g i v e n a s d e c r e a s i n gp a r t s , of a b a s i c s t e p s i z e H ,

h k = H / k ,w h e r e k a r e p o s i t iv e n u m b e r s , f o r e x a m p l e k E {1 , 2 ,3 , 4, 5 . . . . ] ( se e D E U I ~ [ 19 83 ]) .T h e w a lu e s o f t h e s o l u t i o n s ( 51 ) c o r r e s p o n d i n g t o s t ep s iz e s h k a r e d e n o t e d a s

T k ] ) = Ydis( R , t ; h k ) . (52)

T h e s e v a l u e s T k ] ) c a n b e u s e d a s s t a r t in g v a l u e s o f a p o l y n o m i a l e x t r a p o l a t i o n i n s e r t e di n t o t i l e A i t k e n - N e v i l l e a l g o r i t h m ( se e H A n ~ R , N ~ R S E r r , ~ W AN N ER [1 98 7] ). T h e n , t h ed i a g o n a l t e r m T~ k) o f t h e A i t k e n - N e v i l l e t a b l e i s a k - t h o r d e r a p p r o x i m a t i o n o f t h ee x a c t s o l u t i o n Y exact i n t e r m s o f t h e b a s i c s t e p s iz e H :

yexp(R, t ) = T~ k) = Yex~ct(R , ) + gk (R , t ) H k + G ( R , t ; H ) H k + l . (53)E n l a r g i n g k r e d u c e s t h e d i s c r e t i s a t i o n e r r o r a n d i n t h e l i m i t o f k a p p r o a c h i n g i n f i n i t y( 5 3 ) c o n v e r g e s t o t h e e x a c t s o l u t i o n . I n t h i s s e n s e , t h e s o l u t i o n ( 5 3 ) i s c a l l e d t h e e x a c ts o l u t i o n o f s y s t e m (4 7). O f c o u r s e , t h e a t t a i n a b l e a c c u r a c y o f ( 53 ) is re s t r i c te d b e c a u s er o u n d - o f f e rr o r s o c c u r a n d t h e m a c h i n e p r e c i si o n is li m i te d .T o i l lu s t r a te t h e e f f i c ie n c y o f t h e p r o p o s e d n u m e r i c a l m e t h o d , F i g . 3 s h o w s t h e i n te r -

o .

J}u }O .Er -

3 . 0 0

2 . 8 0

2 . 6 0

2 . 4 0

2 . 2 0

- - - - h = 1 . e -2. ~ w - h = 1 . e - 2 , e x t r a p o l a t e d. . . . h : 5 e - 3. . . . . . . . h = 2 . 5 e - 3 i~ h : 2 . e - 4 i

2 . 0 00 . 0 0 0 5 0 . 0 01 0 .0 0 1 5 0 . 0 0 2 0 . 0 0 2 5 0 . 0 0 3

R a d i a l d i s p l a c e m e n t U ( R o ) / R Fig. 3. Interna l pressure versus displacement at the outer radius.

. : .

f/ . - ~ . . . . . . . . . . . . . . . . . . . . . . : . . . .. f " ...........

f k o / 2 ~ . = k / 2 p . = 4 . 2 4 e - 3~ , 5000 / 21~ , v O .3c / 2 r t = 0 . 0 , b = O . O

i I I i I0 . 0 0 3 5


14/20

1 1 2 R . B O N N a n d P . H A U P T

n a l p r e s su r e a s a f u n c t i o n o f t h e d i s p l a c e m e n t a t t h e o u t e r r a d i u s u p t o t h e v a l u e w h e r et h e t u b e b e c o m e s c o m p l e t e l y p l a s ti f ie d . T h e n u m e r i c a l s o l u t i o n s h a v e b e e n c a l c u l a t e df o r d i f f e r e n t d e c r e a s i n g s t ep s i ze s h . E x t r a p o l a t i o n h a s b e e n a p p l i e d t o t h e r e s u l t s o f t h ec a l c u l a t i o n s w i t h h = 1 - 1 0 - 2 , 5 . 1 0 - 3 , 2 . 5 - 1 0 - 3 . T h e e x t r a p o l a t e d s o l u t i o n ( th e T 3 3)t e r m o f t h e c o r r e s p o n d i n g A i t k e n - N e v i l l e ta b l e ) c o in c i d e s w i t h t h e s o l u t i o n d u e t o t h es te p s iz e h = 2 . 1 0 - 4 . A s t h e n u m b e r o f p o i n t s P f o r m i n g t h e f i n i t e - d i f fe r e n c e m e s hi n c re a s e s w i t h d e c r e a s i n g s te p s iz e ( P - h - 2 / 2 ) t h e n u m b e r o f n u m e r i c a l o p e r a t i o n sn e e d e d t o c a l c u l a te th e e x t r a p o l a t e d s o l u t i o n i s m o r e t h a n 1 00 t i m e s le ss t h a n t h e n u m -b e r o f o p e r a t i o n s t o o b t a i n t h e s o l u t i o n f o r h = 2 . 1 0 - 4 .

I n c o m p a r i s o n t o s t a n d a r d f ir st - a n d s e c o n d - o r d e r f in i t e - d i f f e r e n c e s c h e m e s , u s e o ft h e e x t r a p o l a t i o n t e c h n i q u e l e a d s t o s i g n i f ic a n t s av i n gs i n c o m p u t i n g t i m e . F u r t h e r -m o r e , i t is c l e a r t h a t f e w e r n u m e r i c a l o p e r a t i o n s i m p l y a r e d u c t i o n o f t h e i n f l u e n c e o fr o u n d - o f f e r r o r s. A l l th e s e e ff e c ts c a n b e i m p r o v e d f u r t h e r , i f i n s t e a d o f th e m e t h o do f c h a r a c t e r i s t ic s a s e c o n d - o r d e r f i n i t e - d i f f e r e n c e s c h e m e is u t i l iz e d a s a b a s i s f o r t h ee x t r a p o l a t i o n ( s e e S M I TH [ 1 97 0 ] a n d H A C K B U SC H [ 1 97 7 a 1 ).

I V . 3 . L o a d i n g a n d u n l o a d i n g p r o c e s s e sT h e f o l l o w i n g d i s c u ss io n s r e f e r t o t h e e x a c t s o l u t io n s o f e q n s ( 2 8 - 3 6 ) f o r t h e t h i c k -

w a l l ed t u b e s u b j e c t e d t o i n t e r n a l p re s su r e . A l t h o u g h t h e p r o p o s e d i n t e g r a t i o n t e c h n i q u ee n a b l e s u s t o c o m p a r e a v a r i e t y o f m o d e l s o f r a t e - i n d e p e n d e n t p l a s ti c i ty , w e r e st r ic t o u r -s el ve s t h r o u g h o u t t h is a r t ic l e t o t h e m a t e r i a l b e h a v i o u r a c c o r d i n g t o e q n s ( 1 8 ). T h e r e i ni n c l u d e d a r e t h e s p e c i a l ca s e s o f l i n e a r k i n e m a t i c h a r d e n i n g a s w e l l a s p e r f e c t l y p l a s t icm a t e r i a l b e h a v i o u r . O u r c a l c u l a t i o n s w i ll o u t li n e s e v e r al e f fe c t s w h i c h h a v e b e e n f ir s tn o t i c e d f o r p e r f e c t l y p l a st ic m a t e r i a ls . A m o n g t h o s e a r e t h e f o l lo w i n g :

( i ) T h e r e e x i s t s a c r i t i c a l d i s p l a c e m e n t u ( R o ) o f t h e o u t e r s u r fa c e b e y o n d w h i c h ad e c r e a s e o f t h e i n t e r n a l p r e s s u r e c a n b e o b s e r v e d ( se e P R A G ER & H O D G E [ 1 95 1 ], C RO SS -L A N D & B O N E S [ 19 5 8 ] , F ISCHER [ 19 77] ).

( i i ) F o r r a t h e r t h i c k - w a l l e d t u b e s ( R o / R i > 2 ) th e c i r c u m f e r e n t i a l st re s s m a y c h a n g ef r o m t e n s i o n t o c o m p r e s s i o n ( s e e M A C G R E C ,OR, C of f iN , & F ISrm R [ 19 4 8 ] , Bo ur n [ 19 9 2 ] )e v e n f o r p a r t ia l l y y i e ld e d t u b e s .

( ii i) D u r i n g u n l o a d i n g s e c o n d a r y p l a s t ic f l o w m a y o c c u r ( se e P R A TE R & H O D ~ E [ 1 95 1 ],BONN [ 19 9 2 ] ).

F u r t h e r m o r e , w e d i sc u ss t h e i n fl u e n c e o f th e h a r d e n i n g p a r a m e t e r s o n l o a d - d i s p l a c e -m e n t r e l a t i o n s , t h e s t re s s d i s t r i b u t i o n s , a n d t h e d i s t r i b u t i o n o f r e s i d u a l s tr e s se s a f t e ru n l o a d i n g . I n F i gs . 4 a n d 5 t h e c a l c u l a t e d s t r e s s- s t r a i n c u r v e s a re p l o t t e d f o r a t h i c k -w a l l e d t u b e ( R o / R i = 5 ) . A t t h e i n n e r r a d i u s s t r a i n s o f a b o u t 2 5 7o c o r r e s p o n d t o t h ev a l u e o f t h e n o n d i m e n s i o n a l i z e d d i s p l a c em e n t Uo = u ( R o ) / R o = 0 . 0 1 a t t h e o u t e rr a d i u s . T h e v e r t i c a l l in e a t t h e d i s p l a c e m e n t

ko 2(1 _--_v)_ ~1/2flo = l + 2 # ~/l ~ u + v2 ] - - 1 ( 5 4 )

( se e B o ~ r~ [ 1 99 2 ]) i n d i c a t e s t h a t t h e t u b e i s j u s t c o m p l e t e l y p l a s t i fi e d . I n c a s e o f i d e a lp l a s t i c i t y t h e l i m i t p r e s s u r e is re a c h e d a t t h i s s t a g e o f l o a d i n g . I f h a r d e n i n g w i t h s a t u -r a t i o n p r o p e r t y i s i n c o r p o r a t e d , t h i s l i m i t p r e s s u r e s t i l l e x i s t s ; h o w e v e r , i t h a s s i g n i f i -c a n t l y h i g h e r v a l u e s a n d i s r e a c h e d a t m u c h h i g h e r s t r a i n s .


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7 . 0

6 . 0

-- . . 5 .0t"t.

~ 4 . 0t , t . ~t , t , bE ~-E~ 3.0ECp, . . . ,_ci 2 .0

1 . 0

0 . 0

k o /2 1 ~ = 2 . 0 e - 3- R o / R i = 5

v = 0 . 3- 2 ~ 7 = 5 0 0 0

b = 1 0

I b = 0 . 0 , k / 2 ~ = 1 0 e - 3. ~ . . . . . . . = 6 o~ 2 . , o

~ ~ ' ] " . . . . . . . . . . . . . . . . . . . . . . . . . : : ~ : o . . . . . . . . . . . . . . o , ~ , = o o- - - - c / 2 p . = 0 . 1

- I i c / 2 ~ - 0 .2uo 0.002 0.004 0.006 0.008 0.01 0.012

R a d i a l d i s p l a c e m e n t U ( R o ) / R Fig. 4. Internal pressure versus displacement of external radius.

2 . 5

2 . 0k0/2~ = 2 .0e -3 . .- -R o / R i = 5 / " " " b = 10

t r - v = 0 . 3 f2 1 ~ = 5 0 0 01 . 5 - - - / - . . . . , . .v "~ , . 7 " . b = 0,0, k /21~= 10e -3

. ~ . . . . . . . . . . C / 2 p = 0 . 0~ 1-- 1 . 0 L ' " ' . . . . C / 2 p = 0 . 1

= 0 5 L - ~ - - - ~ - - ~ 2 , - , , . , . ~ 2 ~ = 0 2' ~ b = 2 0 - - .- .. ., ,, . " ~ - : . , . . .

o o . . . . : . , ~ . 2 . ~ ~ . - - - . . . . . . . . . . . . . . ~ . = o o" "- 1 . 0 I b = 6o

I- 1 . 5 J I I I I0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 , 0 0 8 0 . 0 1 0 . 0 1 2

R a d i a l d i s p l a c e m e n t U ( R o ) / R Fig. 5. Circumferential stress at inner radius.


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114 R. BONN and P. HAUPT

M a t e r i a l p a r a m e t e r s o f k i n e m a t i c h a r d e n i n g i n f l u e n c e t h e p re s s u r e in th e f o l l o w -i n g w a y s :

( i) A n i n c r e a s e o f c (l i n e a r k i n e m a t i c h a r d e n i n g p a r a m e t e r ) l e a d s t o a n i n c r e a s e o f th es l o p e o f t h e p r e s s u r e c u r v e a t t h e b e g i n n i n g o f p l a s t i c f l o w .( i i ) T h e c o n s e q u e n c e o f a n i n c r e a s e i n t h e v a l u e o f t h e s a t u r a t i o n p a r a m e t e r b i s ad e c r e a s e i n t h e l i m i t p r e s s u r e . F o r b - -- 0 n o l i m i t p r e s s u r e e x i s t s .

T h i s p ri n c ip a l b e h a v i o u r is w e ll k n o w n f r o m o n e d i m e n s i o n a l h o m o g e n e o u s d e f o r -m a t i o n s . O n t h e o t h e r s i de i s o t ro p i c h a r d e n i n g c a u s e s a f u r t h e r i n c r e a s e o f t h e l i m i t p r e s-s u re . A c o m b i n a t i o n o f b o t h h a r d e n i n g m o d e l s l e ad s t o a r e p r e s e n t a t i o n o f o n e a n d t h es a m e p r e s s u r e c u r v e o n t h e b a s is o f d i f f e r e n t , i .e . n o t u n i q u e l y d e f i n e d , s et s o f m a t e -r ia l p a r a m e t e r s . O f c o u r s e t h e s e se ts l ea d t o d i s t r i b u t i o n s o f t h e t w o o t h e r s t re ss c o m -p o n e n t s , w h i c h s h o w s i g n i f i c a n t d i f f e r e n c e s . F i g . 5 s h o w s e . g . t h e c i r c u m f e r e n t i a l st re s sa t th e i n n e r r a d i u s . N o t o n l y f o r th e p a r t i a l l y y i e l d e d t u b e b u t a l so f o r t h e c o m p l e t e l yp l a s ti f ie d t u b e a v e r y st r o n g i n f l u e n c e o f t h e m a t e r i a l p a r a m e t e r s is o b s e r v e d . I t t u r n so u t t h a t i n g e n e r a l th e a x i al a n d c i r c u m f e r e n t i a l st re s se s a r e m o r e s e n si ti v e t o c h a n g e so f t h e h a r d e n i n g p a r a m e t e r s t h a n t h e r a d i a l s tr es s i s.

T h i s c a n a l s o b e i l l u s t ra t e d i n F i g . 6 , w h i c h s h o w s t h e c i r c u m f e r e n t i a l s tr e ss d i s tr i b u -t i o n a c r o s s t h e w a l l t h i c k n e s s f o r a t u b e w h i c h i s j u s t c o m p l e t e l y p la s t i f ie d . A t t h e i n n e rr a d i u s , w h e r e l ar g e pl a st ic d e f o r m a t i o n s o c c u r , t h e i n f lu e n c e o n t h e m a t e r i a l b e h a v i o u rd u e t o t h e m a t e r i a l p a r a m e t e r s i s m o s t o b v i o u s . I n t h e c a s e o f i d e a l p la s t ic i t y a n d f o ra h i g h v a l u e o f t h e s a t u r a t i o n p a r a m e t e r b t h e c i rc u m f e r e n t i a l st re s s c h a n g e s f r o m t e n -s i o n a t t h e o u t e r r a d i u s t o c o m p r e s s i o n a t t h e i n n e r r a d i u s .

B e s id e s th e i n v e s t i g a t io n o f t h e l im i t p r e ss u r e , t w o o t h e r q u e s t i o n s c o n c e r n i n g t h e

3 .02 . 5

" ~ . 2 .0tO

1 . 5

. ~ 1 . 0e

' , , - 0 .5E' - Iot 3 o .o

- 0 . 5

- 1 . 0

" ' : 3 . . . . . . . . . . . . . . . . . . . .. . ~ c / 2 p = 0 . 2 :

- . - b = 0 . 0. . . . . . . b = 1 0

b = 6 0 .... !

0 . 2

I " ~ v = 0.3. . . . . . . . . . . . . k o /2 p = k / 2 p = 2 . 0 e - 3 ......c / 2 p = 0 . 0

t = i I J ~ i0 , 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

R a d i u s R / R 0Fig. 6. Circumferentia l stress distribution.


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unloading process of a pressurized tube have been discussed in the literature, namelywhether secondary plastic flow occurs and how the residual stresses look like. Assum-ing ideal plasticity P~OER and HODOE [1951] pointed out tha t secondary plastic flowstarts only for tubes with a ratio Ro/Rt > 2.22. From Fig. 7 it can be seen that for lin-ear isotropic hardening only tubes with ratios Ro/Ri > 2.5 show an onset of secondaryplastic: flow. For kinematic hardening the opposite tendency can be seen. For compar-ison with the results obtained by FIscrmR [1977] we have chosen the same linear isotro-pic hardening rule for the calculations presented in Fig. 7:

k (s ) =/Co + o~s. (55)

Fig. 7 illustrates the pressure p for which during unloading the yield condi tion is ful-filled :for the first time at the radius R. One result is that secondary plastic flow startsat the inner radius. The radius R/Ro at which the pressure is equal to zero representsthe position of the separating surface, i.e. an elastic-plastic domain has spread out dur-ing unloading up to the radius R/Ro. Another interpretation of this effect is that fortubes with ratio RJRo larger than the ratio of the radius R/Ro no secondary plasticflow happens. For ideal plasticity, the critical ratio Ri/Ro is 0.45, which is exactly thevalue given by PRAGER and HODG~ [1951] in the context o f small deformations.The evolution of an elastic-plastic domain during unloading can also be inferred froma study of the residual stress distributions. In this case, a jump in the slope of the cir-cumferential stress at the radius of the separating surface occurs (see Fig. 8). Further-

~ , : : >

( / }t/ }Q .

1 . 0

0 . 5

0 . 0

- 0 . 5

\. . . . . . . r .J 2 p . = 0 . 0 , b = 0 . 0 , ( z / 2 p . = 5 . 0 e - 2~ - c / 2 p . = 0 . 2 , b = 0 . 0 , 0 ./2 1 ~ = 5 . 0 e - 2. . . . c / 2 p . = 0 . 2 , b = 6 0 , c t /2 p . = 5 . 0 e - 2~ c / 2 p . = 0 . 0 , b = 0 . 0 , o J 2 1 ~ = 0 . 0

\ ~ - - - c / 2 1 ~ = 0 . 2 , b = 6 0 , c z / 2 p . = 0 . 0\ . . . .

. x ~ x x , k o /2 P . = 4 . 2 4 e - 3. . . . . ." ' - - ~-'-......~

k ( s ) = k o + 1 / 2 ~ t s

I I I I0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

R a d i u s R / R o

Fig. 7. Pressure for which yielding starts at the radius R during unloading.


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116 R. BONN and P. HAUPT

o 1.. Q" Om 0 . 5N -O" ~ 0 -t- "(DEV, -0 .5 -e-"OCO' }I Dm -109

t~ -1 .5r r -2 0 .2

- R a d i a l s t r e s s- C i rc u m f e r e n t ia l s t re s s . . . . _ , . . . , . . . . . . , , .

. . . . . . . A x i a l s t r e s s ~ , : . . . . . . . i i i i : i . . i i i i ' " ' i i i ..........\ ::: . . . . . . . .

i . . ~ . . . - . .. .. .. .. - -" / k o /2 p , = 4 . 2 4 e - 3

k o o /2 1 ~ = 6 . 0 e - 37 = 5 0 0 0 / 2 r t

Sma x(R o ) = 2 .5 e - 3I I I I I0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

R a d i u s R / R oFig. 8. Re sidual stress distribution.

m o r e , t h e r e s i d u a l s t r e s s e s m a y b e l a r g e r t h a n t h e i n i t i a l y i e l d s t r e s s k 0 . T h e r a d i a l s t r e s st u r n s o u t t o b e n e g a t i v e ( c o m p r e s s io n ) w i t h i ts m a x i m u m v a l u e n e a r t h e s e p a r a t in g s u r-f a c e . B o t h a x ia l a n d c i r c u m f e r e n t i a l s tr es se s m a y c h a n g e f r o m c o m p r e s s i o n a t t h e i n n e rr a d i u s t o t e n s i o n a t t h e o u t e r r a d i u s. F o r a t u b e , w h i c h h a s b e e n l o a d e d u p t o t h e p o i n tw h e r e it i s j u s t c o m p l e t e l y p l a s t i fi e d , t h e r e s i d u a l c i r c u m f e r e n t i a l s tr e s s d i s t r i b u t i o n i ss h o w n i n F i g . 9. T h e j u m p i n th e s l o p e in d i c a t e s h o w f a r t h e z o n e o f s e c o n d a r y p la s ti cf l o w h a s s p r e a d o u t a c r o s s t h e w a l l t h i c k n e s s . A s m e n t i o n e d a b o v e , k i n e m a t i c h a r d e n -i n g le a d s t o l a r g e r p la s t ic d e f o r m a t i o n s c o m p a r e d t o i d e a l p l a st i c it y . F o r i s o t r o p i c h a r d -e n i n g t h e o p p o s i t e s t a t e m e n t i s t r u e .

V. CONCLUSIONST h e b e h a v i o u r o f a th i c k - w a l l e d c y l i n d r ic a l t u b e h a s b e e n i n v e s t i g a te d u n d e r t h e

a s s u m p t i o n o f a r a t h e r g e n e r a l c o n s t it u t iv e m o d e l o f r a t e - i n d e p e n d e n t f i n i te e l as t o p la s -t ic i ty . T h e b o u n d a r y v a l u e p r o b l e m l e a d s t o a s y s t e m o f h y p e r b o l i c p a r ti a l d i f f e r e n t i a le q u a t i o n s . T h e e x i st e n c e a n d u n i q u e n e s s o f t h e s o l u t i o n o f t h e se e q u a t i o n s i s k n o w nf r o m t h e l i te r a tu r e . U s i n g a c o n v e r g e n t f i n it e - d if fe r e n c e m e t h o d i n c o m b i n a t i o n w i t ha n e x t r a p o l a t i o n t e c h n i q u e , a n a p p r o x i m a t i o n o f t h e e x a c t so l u t io n i s c a l c u la t e d . T h ei n v e s t ig a t i o n s s h o w a s t r o n g d e p e n d e n c e o f t h e s tr e ss d i s t ri b u t i o n a n d t h e l o a d - d i s p l a c e -m e n t b e h a v i o u r o n t h e m a t e r ia l p a r a m e t e r s a n d i n p a r t ic u l a r o n t h e k i n e m a t i c h a r d e n -i n g m o d e l . T h e p r e s e n t e d m e t h o d m a y b e a p p l i e d t o e x p l o r e p h y s i c a l i m p l i c a ti o n s o f s p e-c i a l c o n s t i t u t i v e t h e o r i e s a n d t o v a l i d a t e a f i n it e e l e m e n t c o d e .


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T h i c k - w a l l e d t u b e 1 17

.= = qt'xL

F -

t/ }

E

E

ocD

0 . 5

_

- 0 . 5 -

- 1 -

- 1 . 5

- 2 0 . 2

R / R i = 5v = o . 3 . . . . .

k ^ /2 p . = 2 . 0 e - 3 , ~ , ~ . r . . . . ~ i " I

/ " .... / , . . : . : " . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ / ~ - ~ 2 ~ = 0 . 2 , b -- -6 0 , ko = k/ / - . . . . . . c / 2 p . = 0 . 1 , b = 6 0 , k o = k

c / 2 ~ = o , b = 0 . k = 2 k 0i t i J i i t0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9

RadiusR/R OF i g . 9 . C i r c u m f e r e n t i a l re s i d u a l s t re s s e s .

Acknowledgement -This w o r k w a s s u p p o r t e d b y t h e D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t ( D F G ) a s a p r o j e c tw i t h i n t h e r e s e a rc h g r o u p " I n g e n i e u r w i s s e n s c h a f t li c h e u n d M a t h e m a t i s c h e A n a l y s e B r u c h m e c h a n i s c h e r u n dI n e l a s t i s c h e r P r o b l e m e . "

REFERENCES1 9 1 1 R IC h aR D S O N , L . F ., " T h e A p p r o x i m a t e A r i t h m e t i c a l S o l u t i o n b y F i n it e D i f fe r e n c es o f P h y s i c a l P r o b -l e m s I n c l u d in g D i f f e r e n t ia l E q u a t i o n s , W i t h A p p l i c a t i o n t o t h e S t re s s es in a M a s o n a r y D a m , " P h i l .

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I ns t i t u t f i i r M e c h a n ikUn iv e r s i ty o f Ka ss elM 6nc he be rg s t r . 7D-34109 Kasse l , Germany(Received in f in al revised fo rm 24 A p r i l 1994)

Exact Solutions for Large Elastoplastic

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