ON THE INTRODUCTION OF RETURN MAPPING SCHEMES IN ELASTO-PLASTIC FINITE ELEMENT SIMULATIONS FOR ...

7/28/2019 ON THE INTRODUCTION OF RETURN MAPPING SCHEMES IN ELASTO-PLASTIC FINITE ELEMENT SIMULATIONS FOR IS

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O N TH E IN TR OD UC TI ON O F R ET UR N MA PP IN G S CH EM ES I NE L A ST O - P L AS T IC F I N IT E E L EM E NT S I MU L AT I ON S F O R

ISOTROPIC AND KINEMATIC HARDENING

INTRODUCTION

T h e i n t r od u c ti o n o f l i n ea r -e l a st i c f i n i te e l e me n t t e c h n i qu e s

c an be re garde d now aday s as a st anda rd s ubje ct in

m e c ha n ic a l, c iv i l a n d a e r os p a ce e n g in e e ri n g d e g re e

progra ms. However, th e con s ideration of non lin earities

s u c h a s e l a st o - p la s t ic m a t er i a l b e ha v i ou r i n f i n i t e e l e me n tt ec hn iq ue s i s s ti ll no t so e st ab li sh ed i n t he a ca de mi c

c u r ri c u lu m s. T h e r e a so n m i g ht b e t h a t t h e d e ma n d s o n t h e

c o n st i t u ti v e l a w s a r e m u c h h i g he r s i n ce t h e e l a st i c l a w m u st

b e e x te nd e d t o t h e p l as t ic p a rt w hi c h u su a ll y i n vo l ve s t h e

c ons ide rat io n of a y ie ld co ndi tio n, a f lo w rul e and a

h a rden ing ru le. On the oth er h an d, a n on lin ear s ys tem of

e qu at io ns i s o bt ai ne d w hi ch r e qu ir es t he a pp li ca ti on o f

i t er at i ve s ol u ti o ns s c he m es s u ch a s t h e N e wt o n- Ra ph s on

m e t h o d . F u r t h e r m o r e , s t r e s s p a t h s i n g e n e r a l i s e d

c oo rd in at e s ys te ms ( e. g. a p ri nc ip al s tr es s s ys te m) a red i f f ic u l t t o r e p re s e nt . T hu s, t h e c o ns i d er a t io n o f m a t e r ia l

n o n li n e ar i t i es i n t h e f r a me w o rk o f t h e f i n i te e l e me n t m e t h od

r eq ui re s s om e i np ut f r om c o nt in uu m m ec ha ni cs a nd

h i g h e r e n g i n e e r i n g m a t h e m a t i c s . F u r t h e r m o r e , t h e

m a t he m at i c a l n o t at i o n i n t h e t h r ee d i m en s i on a l c a s e

migh t be difficu lt on th e firs t glan ce. Neverth eles s, th ere are

e x ce l le nt t e xt b oo k s o n t he m a rk et w h ic h i nt r od uc e t h is

By

t op ic a nd c ov er t he g en er al c as e. P ro ba bl y t he f ir st

t ex tb oo k w hi ch co ve rs t he t op ic o f fi ni te e le me nt s a nd

plas ticity was written by Owen an d Hin ton in 1980 [1]. Later

o n , s e v er a l s p e c ia l i se d t e x t bo o k s w e r e w r i t te n f o r e x a m p l e

b y C r is f ie l d i n 1 9 91 a nd 1 9 97 [ 2 ,3 ] a n d S i mo a nd H u gh es i n

1 99 8 [ 4] . Th e m or e r ec en t te xt bo ok s o n th at t op ic w er ewritten by Belyts ch ko et al. in 2000 [5], Du n n e an d Petrinic in

20 05 [6 ] and de S ouza e t a l. in 20 08 [7 ]. Many mo re

t ex tb oo ks w hi ch a re n ot c it ed h er e c ov er t he t op ic o f

p la st ic it y a nd f in it e e le me nt s i n s pe ci al is ed c ha pt er s.

F ur t he r mo r e, t o d e te r mi ne t h e L em a it r e d am a ge m od e l

for du ctile material, th e au th ors pres en ted s imilar algorith m

in [8]

I t s ho ul d b e n ot ed t ha t t he a bo ve m en ti on ed t ex tb oo ks

partially refer to th e on e-dimen s ion al cas e. However, th e

i n f or m a ti o n i s o f t e n s ca t t e re d o v e r t h e e n t ir e t e x t bo o k o r t h ea ut h or s m ov e q ui c kl y t o t h e g e ne r al c as e. A c o ns e qu en t

in trodu ction of th e gen eral concept of plas ticity in the

f ra me wo rk o f t he f in it e e le me nt m e th od b as ed o n on e-

d i m en s i on a l st r e ss a n d s t r ai n s t at e s m a y he l p t o o v e rc o m e

t hi s p ro bl em a nd f ac il it at e t he u nd er st an di ng o f th e

gen eral th ree-dimens ion al cas e. For th is pu rpos e, a on e-

d im e ns i on a l m o de l c as e i s c o ns i de r ed w h ic h d oe s n ot

* School of Engineering, U niversity of A berdeen, U K.

** D eportment of A pplied M echanics, U niversity of Technology of M alay sia, M alay sia.

ABSTRACT

Elasto -plastic constitut ive equation s are integra ted in the framewor k of nonlin ear finite element techniqu es based on

s o - c a l l e d p r e d i ct o r c o r re c t o r s c h e m e s . A n I n t r o d u ct i o n o f m a t e r i a l n o n l i n e a r i t i e s ( e l a s t o - p l a s t i c b e h a v i o u r ) a p p l i e d i n

t h e f r am e wo r k o f f i ni t e e l em e nt a n al y si s . T h e r es t ri c ti o n t o t h e o n e- d im e ns i on a l c a se f a ci l it i es s i gn i fi c an t ly t h e

mathema tical notation while the steps remain the same as in the general three dimensi onal cases. In addition , the

e n t i re s o l u ti o n pr o ce d u re c a n be e a s i ly o b se r v e d in t h e c l a s si c a l s t r es s - st r a i n d i a g ra m . Th e c o n ce p t of t h e p r e di c t or

c o r re c t or s c h e me i s p r es e n t ed f o r t h e c a s e o f i s o t ro p i c, k i n e m at i c a n d c o mb i n e d h a rd e n i n g. N u m e r ic a l e x a m pl e s

i l l u s t r a t e t h e i n f l u e n c e o f d i f f e r e n t b o u n d a r y c o n d i t i o n s f or d i f f e r e n t h a r d e n i n g l a w s .

Keywords: Finite Element Method, Plastici ty, Numerica l S imulat ion, Predictor-Correcto r Procedure, Return-Ma pping

S c h e m e s , I s o t r o p i c H a r d e n i n g , K i n e m a t i c H a r d e n i n g .

ANDREA S C H SN E R * *

R E S E A R C H P A P E R S

M OO SA E SM AE IL I *

l

i-manager s Journal o Mechanical Vol . No. 1 2013l

n E ngi neeri ng, 3 N ovember 2012 - J anuary 17


2/13

s ho w a ny c on tr ac ti on p er pe nd ic ul ar t o t he l oa di ng

d i re c ti o n. T hi s i s a s i mp l if i ca t io n to r e al u ni a xi a l l o ad i ng

cases where a specimen w ould s how a deformation

p er pe nd ic ul ar t o t he l oa di ng d ir ec ti on. H ow ev er, t hi s

s i m pl i f i ca t i o n ma k es t h e e n t ir e m a t he m a ti c a l d es c r ip t i ono f t he p r ob l em m uc h si mp l er w hi l e t h e b as i c s t ep s d o n ot

c h a ng e . T o f u r t he r s i m pl i f y a n d m a ke t h e e n t ir e d e r iv a t i on s

m o r e t r a ns p a re n t, m a t ri c e s a r e i n t he f o l l ow i n g a n al y t i ca l l y

in verted an d fu n ction al derivatives are an alytically

c a lc ul a te d. T hi s i s d if f er e nt t o r e al f in it e e l em e nt c od e s

wh ere s u ch in vers ion s an d derivates are n u merically

determined.

T he f ol lo wi ng s ec ti on s s um ma ri se s b ri ef ly t he c on ti nu um

mech an ica l des cription of elas to-plas tic beh aviou r.

C h a pt e r 2 i n t ro d u ce s t h e i n t eg r a t io n o f t h e e l a st o - pl a s ti cc o ns t it u t iv e e qu a ti o n b as e d o n s o -c a ll e d p r ed i ct o r

corrector s ch emes. It is importan t to h igh ligh t h ere th at th e

c o n ce p t i s i n t r od u c ed a s i n t h e g e n er a l t h r ee - d im e n si o n al

c a s e b u t on t h e l e v el o f o ne - d im e ns i o na l st r e s s a n d s t r ai n

s t a t es . T hi s f o r ma l i sm i s o f c o u rs e n o t r eq u i re d i f s o me o n e i s

o n l y i n t er e s t ed i n th e s o l ut i o n o f t he p r o bl e m. F or a g i v en

s t r ai n s t a t e, t h e e n g in e e ri n g s t r e ss - s tr a i n d i ag r a m a l l o ws t o

conclude to the acting stress state and vice versa.

H o we v er, t he m ai n id e a o f t he a p pr o ac h is t o p r ov i de t h e

b a s is f o r a n e a s y u nd e r st a n di n g o f t h e m o r e c o m pl i c at e dt h r e e - d i m e n s i o n a l c a s e . N u m e r i c a l e x a m p l e s a r e

p re se nt ed i n c ha pt er 3 . T he se e xa mp le s c an b e

p r og r am m ed a n d t h e r e su lt s c a n b e t r ac e d i n s t r es s -s t ra i n

coordin a te s ys tem.

1 . C on ti nu um M e ch an ic s o f E la s to - pl as ti c M at er ia l

Behaviour

U n d er t h e c l a ss i c al a s su m pt i o ns o f s m al l s t ra i n s a n d l i n ea r

relation s hip b etween th e s econ d-order s train tens or ean di jt h e s t re s s t e ns o r s, t h e e l as t ic s t ra i n- s tr e ss r e la t io n i s g i ve ni j

by the gen eralis ed Hooke s law

(1)

Wh ere E is You n gs modu lu s, C t h e f o u rt h - o rd e r el a s ti ci j k l

c o mp l ia nc e t e ns or a nd ni s Po i ss on s r at i o. T he s ym bo l di jrepres en ts th e Kron ecker ten s or (di s e qu al to 1 i f i = j, a nd 0i jif i j) . I n a p ur e u ni ax ia l s tr es s a nd s tr ai n s ta te, th is

r el at io ns hi p i s s im pl if ie d t o t he o ne -d im en si on al H oo ke s

l a w i n t h e f o rm

- 1e= E s. (2)

T he t hr ee e ss en ti al s i ng re di en ts o f p la st ic a na ly si s a re t he

yield criterion , th e flow ru le an d th e h arden ing ru le Armen

[ 9 ]. Th e y ie l d c r it e ri o n r e la t es t h e s t at e o f s t re s s t o t h e o n se t

o f y i el di n g. Th e f l ow r ul e r e la t es t h e s t at e o f s t re s s st o t hei jc o r re s po n d in g i n c re m e nt s o f p l a st i c s t r ai n w h e n a n

i nc re me nt o f p l as ti c f lo w o cc ur s. T h e h ar de ni ng r ul e

d es cr ib es h ow t he y ie ld c ri te ri on i s m od if ie d b y st ra in in g

b ey o nd i n it i al y ie l d. T he y i el d c r it e ri o n f o r an i so t ro pi c

m at er ia l c an ge ne ra ll y b e e xp re ss ed i n t he c as e o f

combin ed h arden ing as

(3)

Wh ere ki s t h e i s o t ro p i c h a r de n i ng p a r am e t er a n d aarei jt he k in em at ic h ar de ni ng p ar am et er s ( so -c al le d b ac k-

s t r es s t e n so r ) , r e s pe c t i ve l y. T h e e l a st i c r e l at i o n ( 1 ) p r e va i l s

wh ile F < 0. Wh en th e s tres s es are s u ch th at F = 0, yieldin g

b eg in s o r i s a lr ea dy in pr og re ss. Th e c as e o f F > 0 i s

ph ys ically n ot pos s ible for rate in depen den t materials. For a

on e-dimen s ion al s tate, E q. (3) reads as

(4)

The f low rul e is st at ed in t er ms o f a f unc ti on Q, which is

desc ri bed in uni ts o f st res s, and i s c all ed t he pl ast ic

potential. With d la s ca l ar c a ll e d p l as t ic m ul t ip l ie r, t h e

p l as t i c s t r ai n i n c r e m en t s a r e g i v en b y

(5)

T he f lo w r ul e i s s ai d t o b e a ss oc ia te d i f Q= F, oth erwis e it is

n o na s s oc i a te d . H ar d e ni n g c an b e m o de l l e d a s i s o tr o p ic

(i.e. in itial yield s u rface expan ds u n iformly with ou t dis tortion

a nd t r an s la t io n a s p la s ti c f l ow o cc u rs ) o r a s k i ne ma t ic ( i .e .

i ni ti al yi el d s ur f ac e t ra ns la te s a s a r ig id b od y i n t he s tr es s

s p ac e, m a i nt a i ni n g it s s i z e, s h ap e a n d o r i en t a ti o n ), o r i n

c o m bi n at i o n. T h e c o n tr i b ut i o n of i s o t ro p i c h a r de n i ng i s

c o n si d e re d b y t he e v ol u t i on o f t he e x p er i m en t a l l y

d e t e rm i n ed f l o w c u r v e ( y i el d s t r es s ) , i . e.k = k( k), (6)

wh ile th e con tribu tion of kin ematic h arden ing can be

i n t r od u c ed b y a c o r re c t i on o f t h e s t r es s t e n so r a s :

(7)

T h e i n t er n a l v a r ia b l e w h i ch d e s cr i b es i s o tr o p ic h a r de n i ng

c an b e t he e ff ec ti ve p la st ic s tr ai n, , ( so -c al le d s tr ai n

h a r de n i ng ; o n e -d i m en s i on a l c a s e: ) o r t h e s p e ci f i c


18 i - ma n ag e r s J o ur n al o M e ch a ni c al ll

n E ngi neeri ng, V ol . 3

No . 1 N o ve m be r 2 0 12 - J an u ar y 2 0 13

11

1,i j i j i j kk ij kl kl C

E

n ne s ds s

n-+ = - = +

( ), , 0i j ijF ska=

( ), , 0F ska=

p ld dijij

Qel

s

=

p le f f

pl pleff e=

( )i j i j i jass-


3/13

R E S E A R C H P A P E R S pl

p l a s t i c w o r k , w , ( s o - c a l l e d w o r k h a r d e n i n g ; o n e -

d i m en s i on a l c a s e: ) . I n t h e c a s e o f a u n i ax i a l

s t a t e, t h e y i e ld c r i t er i o n ( 4 ) c a n b e w r i t te n a s :

F = |s- a| - k(k) = 0. (8)

B a se d o n t h is f o rm ul a ti o n, t h e a s so c ia t ed f l ow r u le c a n be

d e r iv e d a s f o r t h e o n e -d i m en s i on a l c a s e a s :

(9)

Wh ere s gn is th e s ign u m fu n ction (s gn (+s) = +1 ; s gn (-s) =- 1 ; s g n( 0 ) = 0) . T o c o mp l et e t h e s e t o f e q ua t io ns , i t i s s t il l

r eq ui re d t o s ta te t he e vo lu ti on eq ua ti on s f or t he i nt er na l

varia bles kan d a. Assuming strain hardening, theevolu tion equ ation for th e is otropic h arden ing parameter kc a n b e e x p re s s ed u n de r c o n si d e ra t i o n o f t h e a s s oc i a t ed

f l o w r u l e ( 6 ) f o r t h e c a s e o f c o m bi n e d h a r d e n in g a s :

(10)

I t sh ou ld b e n ot ed h er e t ha t i n t he c as e o f p ur e i so tr op ic

hardening, parameter ai s e qu al t o z er o i n E q. (1 0) . A c o mm o n ap pr o ac h t o m od e l t he k i ne m at i c h a rd e ni ng

parameter ai s b as ed o n Prager s k i ne m at i c r u le [ 4 ] w h er eth e evolu tion equ a tion is given as:

(11)

wh ere H is th e kin ema tic h arden ing modu lu s.

2. Fin ite Elemen t Procedures

2.1 The principal f inite elem ent equation f or rod elem ents

a n d i t s s o lu t i on p r oc ed u re i n t h e e l as t ic - pl a st i c r a n g e

I n an e l as t i c- p l as t i c a n al y s is , t h e c o n st i t u ti v e r e l at i o ns h i p

d e p en d s o n t he d e f or m a ti o n hi s t or y a n d a n in c r em e n ta l

a pp ro ac h i s r eq ui re d w he re t he a pp li ed l oa d i s a dd ed i n

i n cr e me n ts s t ep b y st e p. T h er e fo r e, t h e p r in ci p al f i ni t e

elemen t equ ation for a on e-dimens ion al rod elemen t with

l i n ea r sh a p e f u n ct i o n s ( t w o n o de s ) [ 1 0 , 1 1 ] c a n b e w r i t te n i n

t h e c a se o f e l a s t ic - p la s t ic a n al y s is a s :

(12)Wh ere Du a n d DF in d icate th e in cremen tal dis placemen tsi i

a nd l oa ds, r es pe ct iv el y. T he m od ul us i s i n t he e la st ic

r a n ge o f t he s i m ul a t i on e qu a l t o Yo u n g s m o du l u s E. In th ee l p l

p l a st i c r a n ge , t h e s o - ca l l ed e l a st i c - pl a s ti c m o d ul u s E (e)

m u s t b e u s e d i n s t e a d. T h i s m o d ul u s i s o b t ai n e d a s t h e s l o pe

i n t he p l as t i c p a rt o f t h e s t r e ss - s tr a i n d i a gr a m ( a n d s h o ul dp ln ot b e c on fu se d w it h t he s o- ca ll ed p la st ic m od ul us E

wh ich is defin ed as th e s lope in a diagram yield s tres s overp l p l p l is otropic h arden ing variable: E = E (e)) . Si n c e t h e s t i f f ne s s

em a t ri x i s n o w a f u n ct i o n of t h e n o da l d i sp l ac e m en t, i . e. K =

e eK (u ), i t e ra t iv e m et h od s a r e u su al l y em pl o ye d t o s o lv e t h e

s ys tem of equ ations given in (12).

L et u s c o ns i de r n ow i n t h e f o ll o wi ng t h e c a se o f a s i ng l e r o d

e le me nt wh ic h i s f ix ed a t n od e 1 (u = 0 ) and at no de 21

subj ect ed t o a pr es cri bed di sp lace ment b oun dar y

con dition u . For th is cas e, E q. (12) is s implified to

(13)

Th u s, all n odal dis placemen ts (u an d u ) a re i mm e di a te l y 1 2k no wn f or ea ch s te p ( in cr em en t) a nd a r ea l solution

procedure f or Eq. (12) is not required. In a post-

compu tation al s tep (pos t-proces s in g), th e con s tan t s train

i n t h e r o d e l e me n t c a n be c a l cu l a te d b a s ed o n [ 1 0 , 1 1]

(14)

Wh ich can be fu rth er s implified u n der con s ideration of th e

b o un d ar y c o n d it i o n at n o d e 1 (Du =1 ) . T h e t o t al va l u es o f1dis placemen t an d s train can be obtain ed by su mmin g u p

t h e i n c re m e nt a l v al u e s g i v en i n E q. ( 1 3 ) a n d ( 1 4 ). B a se d o n

th e given s train in cremen t De, t h e r e m ai n i ng f i e l d v a r ia b l esp l

(s, e, k, a) ca n be upd at ed base d o n the alg ori thms

i n t r od u c ed i n s e c t io n 2 . 2 a n d 2 . 3 .

L et us h av e i n t h e f o ll o wi n g a l o ok o n t h e c as e t h at a s i ng l eforce F i s a c t i ng a t n o de 2 . T h us , E q. ( 1 2 ) i s s i m pl i f i ed t o :2

(15)

S i nc e t h e m o du l us d ep e nd s n o w o n t h e d e fo r ma t io n, a n

i t e r at i v e s o l ut i o n p r o c e du r e i s r e q ui r e d. A p p ly i n g a N e w t on -

R a ph s o n s ch e m e [ 1 1 ], E q . ( 1 5 ) c a n b e w r i t te n i n t h e f o r m o f

a res idu u m as :

(16)

A Taylor s eries expan s ion of th e las t equ ation an d

n eglectin g h igh er order expres s ions res u lt in the followin g

s tatemen t:

(17)

Wh ere

(18)

t ha n d t h e s o -c al l ed t a ng e nt st i ff n es s i n t h e i i t er a ti o n s t ep i s

given by:

(19)

l



pl pldw se=

( ) pld d d sgnF

l lsas

= = -

( ) pld d d s gn dke lsal== -=

( )pld d d sgnH Hae lsa== -

1 1

2 2

1 1

1 1

u F AE

u F L

-

= -

%

2 uu =

( )2 11

u uL

e= -

2 2 AE

u FL

=%

()2 2 2 2 2 0AE A

r u F E u u F L L

= -= -=%

%

()()()()()

()12 2 22

i

i i rr u r u u

ud+

= + +

()() ()12 2 2 i iu u ud += -

()()

T

2

i

i r K

u

=

E%


4/13


Th u s , E q. (16) ca n b e written as :

(20)

wh ich ca n be fin ally written as :

(21)

I n t he e la st ic r an ge, t he m od ul us i s e qu al t o Yo un g s

m o d ul u s E . I n t h e c a s e o f l i n e ar h a r d en i n g, t h e m o d ul u s ise l p li n th e f u ll y pl as t ic r an g e e q ua l to E = c o n s t. T h e e l a st i c

p l a st i c t r a ns i t io n r a n ge , i . e. th e r a ng e w h e n t h e l a s t p u r e

e l a st i c s t e p o c c ur r e d ( s t ep n) a nd t h e s i m u la t io n s e n t e rs t h e

p l a st i c r a n ge , t h e f o l l ow i n g s ch e me ( i t er a t i on i n d ex j ) can

b e u s ed t o i t e r at i v el y c al c u la t e t h e n e x t s t e p n + 1:

forj = 0 , (22)

forj > 0. (23)

wh ere for j > 0 th e d efin ition of a s ecan t modu lu s is u s ed, cf.

F ig ur e 1 a ). Th is l in ea ri sa ti on in th e f or m o f t he s ec an t

mod u lu s implies th a t th e n on lin ear term in E q. (19), i.e.

(24)

c an b e d is ca rd ed f ro m t he t an ge nt st if fn es s a nd t hi s

s pec if ic met hod is som et imes c all ed t he g ener al is ed

Newton -Raph s on meth od [1].

T h e s a me d e f in i t i on of a s e c an t m o d ul u s c a n b e a p pl i e d i f

t he s tr es s- st ra in c ur ve i s n on li ne ar i n t he p la st ic r an geel pl e lpl

(E =E (e)) , cf. F ig u re 1 b ). I n t h is c as e, t he s t ep n i s t h e l a st

k n o wn s t a t e i n t h e p l a st i c r a n ge .

2.2 Integration of Rate Equation

C o m p a r e d t o a p u r e l i n e a r - e l a s t i c f i n i t e e l e m e n t

s im ul at io n, t he c om pu ta ti on o f t he p la st ic m at er ia l

b eh av io ur ca n n o m or e b e p er fo rm ed i n a s in gl e s te p

s i nc e t h er e i s n o m o re a u ni q ue r e la t io ns hi p b e tw e en s t r e ss

a n d s t r ai n [ 1 ]. T h e l o a d m u st b e i n c re m e nt a l l y a p p li e d a n d

a n on li ne ar s ys te m o f e qu at io ns h as t o b e s ol ve d i n e ac h

in cremen t (e.g. b a s ed on a Newton -Raph s on algorith m).

Bas ed on the stres s state at the end of the per vious

i n c re m e nt ( n ) a n d t h e g i v en s tr a i n i n c re m e nt De, th e f i le dnvariables (e.g. Stres s s) m us t b e c al cu la te d f or ea chn +1increment (n +1) at each in tegration poin t (Gau s s poin t), cf.

F i g ur e 2 .

To t hi s end, i t is req uir ed t o nume ri cal ly i nt eg rat e t he

d i f f er e n ti a l f o r m o f th e e x p l ic i t c o n st i t u ti v e b e h av i o ur.

E x p li c i t i n t e gr a t i on s c he m e s, s uc h a s t h e f o r w a rd - E ul e r

p r o c e d u r e , a r e h o w e v e r i n a c c u r a t e a n d u n d e r

c ir cu ms ta nc es i ns ta bl e s in ce a g lo ba l e rr or ca n b e

a c c um u l at e d, C r i s f i e ld [ 2 ] . I n s te a d o f e x p l i ci t i n t e g r at i o n

s c he m e s, s o - ca l l ed p r e di c t o r- c o rr e c to r a l g or i t hm s ( c f.

Figu re 3) wh ich con s is ts of an in itial explicit elas tic-predictor

s tep, in volvin g a deviation away from th e yield s u rface, an d

a p l a st i c -c o r re c t o r st e p ( i m pl i c i t co r r ec t i on ) w h i ch r e t ur n st h e s t r es s t o t h e u p d at e d y i e ld s u r fa c e.

L e t u s i l l u st r a t e t h e g e n er a l c o nc e p t f o r t he c a se o f i s o t r op i c

h arden in g firs t. Un der as s u mption of pu re lin ear-elas tic

m at er ia l b eh av io ur, t he e la st ic p re di ct or i s u se d t o

c a l cu l a te t h e t r i al s t a te ( s o -c a l le d t r i al s t r e ss ) :

F i gu r e 1 . D e fi n it i o n o f t h e a v er a ge E : a) Elastic- PlasticTr a ns i t io n R e gi o n; b ) N o nl i n ea r P l as t i c R e gi m e

F i gu r e 2 . S c he m at i c R e pr e se n ta t io n o f t h e I n te g ra t i on A l go r it h mf o r P l as t i c M a te r i al B e h av i o ur i n t h e F i ni t e E l em e nt M e t ho d [ 4 ,1 4 ]

F i gu r e 3 . S c he m at i c R e pr e se n ta t io n o f t h e I n te g ra t i on A l go r it h mi n t h e S t re s s- s tr a i n D i ag r am [ 4 ]




()() ()()() ()2 T 2 2 i i i iAEu K r u F uL

d =- =-%

() ()12

i i Lu F

AE

+=%

E E=%()

()1

1

j

n nj

n n

Ess

ee+

+

-=

-%

()() ()

T 22 2

nonl i near

i

i ir AE A E K u

u L u L

==+

% %

E%


5/13


(25)

T he c or re sp on di ng h ar de ni ng s ta te c or re sp on ds t o t he

s t a t e o f t h e p e r v i ou s i n c re m e nt . Th u s, i t i s a s s um e d t h a t th e

l o ad s t ep i s p ur e e l as t ic , i . e. wi t ho ut pl a st i c d e fo r ma t io n

an d th u s , with out h a rden in g:

(26)

D e pe nd e nt o n t he l o ca t io n of t h e t r ia l st a te i n t he s t re s s

s pa ce, t he y ie ld c on di ti on a ll ow s t o d is ti ng ui sh t wo

elemen ta l s tates :

T he s tr es s i s i n t he e la st ic r an ge o r o n t he y ie ld s ur fa ce

( v a li d s t r es s s t a t e, c f. F i g ur e 3 ) :

(27)

I n th is c as e, t he t ri al s ta te c an b e t ak en a s t he n ew s tr es s

a n d h a r de n i ng s t a t e s i n c e i t co r r es p on d s t o t h e r e a l s t a t e:

(28)

(29)

F i n al l y, o n e c a n m o v e t o t h e n e x t l o a d i n c r e m en t.

T he s tr es s s ta te i s o ut si de t he y ie ld s ur f ac e ( in va li d s tr es s

s tate, cf. Figu re 3)

(30)

I n t h is c a se , t he s e co n d p ar t o f t h e a l go r it h m ca l cu la t es

f rom t he inv ali d s tat e a val id st at e on t he y ie ld sur fac e

(F (s, k) = 0 ) . T he r e q ui r e d s t r es s d i f f er e n cen+1 n +1(31)

i s c a l le d t h e p l a st i c c o r re c t or.

Altern a tive exp res s ions for th e predictor corrector meth od

a r e r e t u rn m ap p in g o r c a t c hi n g -u p. I n th e f o l l ow i n g, t h e

aut hors are going to have a closer look on t he retu rn

m a p pi n g p r o c ed u re . D e t a i le d d e s cr i p t io n s o f t h e g e n er a l

c a se c a n b e f o un d i n Si m o a nd H u gh es [ 4 ]; B e ly t sc hk o, L i u

a nd Mo ra n [ 5] ; C ri sf ie ld [ 2, 3] ; d e S ou za N et o, Peri an d

Owen [7].

T h e d i ff e re n ce b et w ee n t h e i ni t ia l an d t a rg e t s t at e ( s tr e ss

increment)

(32)

r e s ul t s a c c or d i ng t o H o o ke s l a w f r om th e elas tic p ar t o f t he

s train increment wh ich can be obt ai ne d as t he di ff ere nt

between th e tota l s train in cremen t an d th e plas tic part as :

(33)

T he t o ta l s t ra i n i nc r em e nt c a n b e e x pr es s ed w i th t h e t r ia l

s t r es s s t a t e a c c o r d in g t o F i g ur e 3 a s :

(34)

I n t ro d u ci n g t h e l a s t e q ua t i o n a nd t h e f l o w r u l e a c co r d in g t o

E q. ( 9) i nt o E q. ( 33 ), t he f in al s ta te c an be o bt ai ne d i n

d e pe n d en c e o f t h e t r i al s t r es s s t a te a s :

(35)

D e p en d i ng o n t h e p o i nt f o r t h e e v a lu a t io n o f t h e f u n ct i o n

sgn() , d i ff e re n t m e th o ds c an be o b ta i ne d t o c al c ul a te

t he i ni ti al v al ue f or t h e p la st ic c or re ct or a nd t o i te ra ti ve ly

d e t er m i ne t h e f i n a l s t a te . T o o b t ai n t h e i n i t ia l v a l ue f o r t h e

plas tic corrector, s gn () c a n be e i th e r e va l ua t ed i n t h e t r ia l

st re ss st at e (B ac kwar d-E ule r) o r o n t he yi el d sur face

( Fo r wa rd -E ul er ; f or t he t ra ns it io n f ro m t he e la st ic t o t he

p la s ti c r a ng e, t h is i s t h e i ni t ia l yi e ld s t re s s) . E va l ua t in g t he

function sgn() d u r in g t he i t e r at i o n i n t he f i n al s t at e e n s ur e s

t h a t t he n o r ma l r u le i s f u l f il e d i n t h e f i n al s t a t e. F o r t h i s i m p l i c it

B a c kw a r d- E u le r a l g or i t h m ( a l so k n o wn a C l o se s t- Po i n t-

P ro je ct io n ( CP P) ), S im o a nd H ug he s [ 4] , i t is r eq ui re d t o

c al c ul a te h i gh er o r d er d e r iv a ti v es i n t h e g e ne r al t h r ee -

d im e ns i on al ca s e. Fo r t h e s o -c a ll e d c u tt i ng pl a ne

a lg or it hm , Si mo a nd O ri tz [ 12 ], t he f un ct io n sg n() is

e v al u at e d f o r t h e s t re ss s t at e o f t he i t h i t er at i on s te p. T he

fu n ction s gn () i s f o r t h e m i d po i n t ru l e i n b o th s t a te s , i. e. t h e

f i n a l s t a te a n d o n t h e y i e ld s u r f a ce , e v a lu a t ed a n d e q u al l y

weigh ted. Th e evalu ation of th e s gn () o nl y o n t he y ie ld

s u r f a ce y i e ld s t h e s e m i- i m pl i c it B a c k w a rd - E ul e r a l g o ri t h m,

M o ra n, O rt i z a nd S hi h [1 3 ], w hi c h r e qu ir e s o nl y t he

d e t er m i na t i on o f f i r s t o r d e r d er i v at i v e s. T h e d i f f er e n t

T a bl e 1 . S u m ma r y o f P re d i c t or - C o rr e c t or A l g o r it h m s [ 1 5 ]

l



el

t r i a l1

pr edi ct or

E

n

n n n

s

ss+=+

t r i a l1n nkk+=

( )t ri al t ri al1 1, 0n nF sk+ +

tria l1 1n nss+ +=

tria l1 1n nkk+ +=

( )t ri al t ri al1 1, 0n nF sk+ +>

pl tria l1 1 n nsss+ +=-

1 n n nsss+=-

( )el pl n n n nE Ese ee= = -

( )1 t ri al1 1 n n n n nEeee ss-+ +=-= -

()trial1 1 1 sgnn n n Essl s+ + +=-

State fo r e valuatio n o f sg n(s)

Initial value fo r co rre c to r

Trial state o n t h e f l ow

Curve

During ite ratio n

Final state (Full.-im pl. B ac k ward-Eule r alg .

Clo se st-Po int-Pro je c tio n)

On the flo w cur ve (Se mi.-i mpl.

F i n a l s t a t e a n d

o n f l o w c u r v e ( m i d po i n t -r u l e )

S t re s s s t at e o f i th ite ratio n ste p

acc. Eq. (22)

B ac k ward-Eule r alg .)

(Cutting -Plane alg o rithm )

1ns+

() ()( )t r i a l1 1 11

sg n sgn2

n n n n E Esl s s+ + +- +

trial1 1 sgn( )n n nE sl s+ +-

()trial1 1 1 sgnn n nEsl s+ + +-

()()trial1 1 sgn in n Esl s+ +-

()t r i a l1 1 sgnn n nEsl s+ +-

tri al tri al1 1 1 s gn( )n n n E sl s+ + +-


6/13


a p p ro a c he s a r e s u m ma r i ze d i n T ab l e 1 . C o n si d e ri n g t h e

d e pe n de nc y o f t h e y i el d c o nd it i on o n t h e h ar de n in g

varia ble, requ ires th e application of a fu rth er equ ation to

des crib e th e h a rd en in g. Th e followin g in cremen tal

r el at io ns hi ps f or t he d et er mi na ti on of t he h ar de ni ngvaria ble ca n be ob ta in ed from th e evolu tion equ ation (10):

(36)

F in al ly, it sh ou ld b e n ot ed t ha t t hr ee o f t he i nt eg ra ti on

p r o ce d ur e s g i v e n i n T ab l e 1 c a n b e s u mm a r is e d b a s ed o n

th e followin g eq u ation :

(37)

Wh ere pa rameter ht a k es t h e n va l u es o f 1 , 0 o r 0 . 5.

2.3 Num erical schem es

2 . 3 . 1 I s o t r o p ic H a r de n i n g

T h i s r e t ur n m a p pi n g a l g o ri t h m c a lc u l at e s t h e c l o se s t p o i n t

p ro je ct io n o f t he t ri al st at e o nt o t he y ie ld s ur f ac e i n t he

c o m pl e m en t a r y e n e rg y. It is n o t a s o n e c o u ld a s s um e t h e

c a l cu l a ti o n o f t he g e o me t r i ca l c l o se s t p oi n t. T h e b a s is o f

t h is p ro c ed ur e i s t h e a ss um pt i on t h a t t h e p l as t ic w o rk t a ke s

i t s m ax i mu m f or a g i ve n st r ai n st a te . To g et h er w i th t h e

e l em en t al r e qu e st th at th e f i na l s t re s s s t at e m u st be

l o c at e d o n t h e y i e ld s u r f a ce , t h e r e t ur n m a p p in g a l g o r it h m

c an b e u nd er st oo d a s t he s ol ut io n o f a o pt im is at io n

p ro bl em ( ma xi mu m of p la st ic w or k) u nd er a c on ve x c o n st r a i nt ( f i n al s t r e ss s t a te m u st b e o n t h e y i e ld s u r f a ce ) [ 4 ] .

T h e p ro c ed u re i s i m pl i ci t f o r t h e c a lc ul a ti o n o f t h e f u nc t io n

sgn() s i nc e t h e e v al u at i on i s p er f o rm e d f o r t h e f i na l s t at e

n + 1. Clos es t Point Projection (CPP) or fu lly implicit

B a c kw a r d E u l er a l g o r i t hm a re c o m mo n e x p r es s i on s f o r t h i s

s tr es s u pd at e p ro ce du re. In th e f in al st at e (n + 1), th e

equations

(38)

Are fu lfilled. However, a res idu u m r r e m ai n s f o r e a c h o f

t h e se e q u at i o ns o u t si d e t h e f i n al s t a t e :

(39)

T hu s, t he f in al st re ss a nd h ar de ni ng s ta te i s t he r oo t of a

vector fu n ction m wh ich is compos ed of th e res idu al

fu n ction s. Fu rth ermore, it is u s efu l to collect all variables in a

s i n gl e v a r ia b l e v e c t o r v:

(40)

Th e root is fou n d by a Newton iteration (iteration in dex: i) [ 4 ] :

(41)

Wh ere

(42)

c a n b e u s ed a s a n i n i ti a l v a l ue . T h e J a c ob i a n m a t ri x o f

the residual functions is obtained f rom the partial

derivatives of Eqs . (39) as :

(43)

I n a d d it i o n t o t h e f u l f il l m en t o f t h e p l a st i c it y e q u a ti o n s ( 3 8 )1

t o ( 3 8 )3 i n e ac h i nt e g r at i o n po i n t, t h e g l o ba l e qu i l ib r i um

m us t be f u lf i ll e d. To b e a bl e t o a pp l y t h e N e wt o n i t er at i on

f o r t hi s t a s k, i t i s r e q ui r e d e v e n f o r s m al l s t ra i n s i n t h e g e n er a l

t h re e -d im e ns i on a l c a se t o d e ri v e t h e a pp r op r ia t e e l as t o-

p l as t i c t a n ge n t m od u l us , P. W r i gg e r s [ 1 4 ]. T h e c o n si s t en t

e la st o- pl as ti c m od ul us r es ul ts f or t he o ne -d im en si on al

c a se a s :(44)

T he i nv er si on o f t he J ac ob ia n m at ri x w hi ch h as t o b e

e v al u a te d i n t h e c o n ve r g ed s t a t e o f t h e a b o ve m e n ti o n ed

Newton iteration gives :

(45)

F o r t he s p ec i a l c as e o f l i n ea r i s o t ro p i c h a r de n i ng , w e h a v e

= E = cons t. wher e E i s t he k ine mat ic harde ni ng

modulus.

As an altern ative s olu tion approach, th e con s titu tive

r el at io ns c an b e c om bi ne d t o o bt ai n a s in gl e n on li ne ar

e q u at i o n t o s o l ve t h i s p r o bl e m. T o t h i s e n d, le t u s i n t ro d uc e

t h e e qu at i on f o r t h e s t re s s u pd at e ( 3 8) 1 a nd t h e u pd a te o f

t he i nt er na l va ri ab le a cc or di ng t o E q. ( 38 )2 i n th e y ie ld

c o n di t i o n ( 3 8 ) 3 t o o b t ai n :

(46)

= pl p l




1 1n n nkkl+ +=+

[]() ()( )trial1 1 1 1 1 sgn sgnn n n n nEssl hshs+ + + +=- - +

()trial1 1 1 1 sgnn n n nEssl s+ + + +=-1 1n n nkkl+ +=+

()1 1 1 0n n nF ksk+ + +=- =

( ) ()1 1 tria l1, , sgn 0nr E Essklssls- -+=- + (), 0nrkklkkl=-++V

(), ( ) 0 F r ksksk=-

()( )

()

()()

3 3 ,

F

r

r

r

s

k

s

k

l

= =

R R

v

vm v v

v

() () () ()1

1 ( )i i i i-

+ =-

m

v v v m v

v

()

()

()

()

0t r i a l

10 0

0 0

n

n

s s

k k

l

+

= =

v

m

v

( )()

()

1 0 s gn

, , 0 1 1

sgn 0

F F F

r r r

E r r r

k r r r

s s s

a a a

skl s

sklskl

sk

skl

-

= = - -

m

v

elpl 11

1 1

n nn

n n

Es s

e e+

++ +

==

m

v

() ()

()

()

1 11 1 2 1 31

2 1 22 231

1 13 1 32 33 1

1

d ds g n sgn

d ds g n 1

ddd s gn

d

n

n

n

k k

m m m E

m m m E k

E m m m k E E

s sk k

s

ksk

-

-

+- -+

+

- - +

= =

m

v

d

d

k

k

( ) ()( )trial trial1 1 1 1 1 1 1 , sgn 0n n n n n n n nF D E kl sl s kl+ ++ + + + +=- -+=


7/13


I t sh ou ld b e n ot ed h er e t ha t i n t he d er iv at io n o f t he l as t

e q u at i o n th e f a c t wa s u s e d t h a t th e u p d at e d a n d t r i al s t r es s

a r e c o -l i ne a r: s g n(s) = s gn . E qu ation (45) can ben+1expres s ed for arb itra ry Dli n t h e f o r m o f a r e s id u um r as : F

(47)

A Taylor s eries exp res s ion of firs t order, i.e.

(48)

With gives fin ally th e n u merical s ch eme

t o s o l ve t h i s e q ua t i o n:

(49)

I n t h e l a s t eq u at i o n, t h e f o l l ow i n g t wo r e l at i o ns h i ps c a n be

applied:

(50)

an d

(51)

wh ere th e followin g s implification can be applied:

(52)

2 . 3 . 2 K i n e m a t i c H a r de n i n g

I n t he c a se o f k i ne m a ti c h a r de n i ng , th e s y st e m of r e si d u a

e q u at i o ns r e a ds a s :

(53)

wh ich ca n b e arra n ged in vector in a s imilar way to E q. (40)

as

(54)

Wh ere th e in itia l va lu e of v can be taken as :

(55)

To a pp ly t he N ew to n s ch em e g iv en in Eq . ( 41 ), t he

J ac ob ia n m at ri x o f t he r es id ua l f un ct io ns c an b e

calcu lated as

(56)

a n d t h e i n v er s e i s f i n al l y g i v en i n s y m b ol i c n o t at i o n b y :

(57)

2.3.3 Combined Hardening

T he c as e o f c om bi ne d h ar de ni ng , i. e. a s im ul ta ne ou s

c on si de ra ti on o f i so tr op ic a nd k in em at ic h ar de ni ng ,

r e q ui r e s t h e c o m bi n at i o n o f t he t w o s e t o f eq u at i o ns g i v en

i n ( 3 9) a nd ( 5 3) . T he r e su lt i ng fo ur re s id ua e qu at i on s a re

given by:

(58)

T he s am e v ec to r n ot at io n a s i n E q. (4 0) o r ( 54 ) c an be

i n t ro d u ce d w h i ch g i v e s t h e f o l l ow i n g 4 - c o mp o n en t v e c t o r

fu n ction

(59)

with in itial valu es of:

(60)

T he J ac ob ia n ma tr ix i s n ow g iv en b y th e f ol lo wi ng 4 X 4

matrix

(61)

wh ich can be s till s ymbolically in verted to obtain :

(63)

3. Numerical Examples

L e t u s c o ns i d er i n t h e f o l l ow i n g a n i d e al i s ed t e n si l e t e s t t o

i l lu st r at e t h e d e ri v ed n um e ri c al s c he me s. T he s ch e ma t ic

r e p re s e nt a t i on o f t h e p r o bl e m i s s h o wn i n F i g ur e 4 w h e re

l



() ()( )tri al t ri al

1 1 sgn 0 F n n nr E klsls kl+ +=- -+

()() ()()()

()1 0i

i i FF F

rr rl l dl

l+ = + += V

()() ()1 i idll l+=-

() ()()

()()1

1 .

i

i i iFF

rrl l l

l

-

+ =+ -

()() () () ()( )t ri al t ri al1 1 sgn i i iF n n nr E klsl s kl+ +=- -+

() ()( ) ()( )tria l t rial tr ia l1 1 1d dsgn sgn sgnd di i

iFn n n

r k E E sl s s

l l+ + + = - - -

() ()( ) ()( )t rial tr ial tri al1 1 1sgn sgn sgnin n nE E Esl s s+ + +- - =-

( ) ( )1 1 t rial1, , sgn 0nr E Esslasslsa- -

+=- + -

( ) (), , sgn 0nr Haalsaalsa=-++ -

(), 0 F r ksksa=--

()( )()()()

3 3,

F

r

r

r

s

a

s

a

l

= =

R R

v

vm v v

v

()

()

()

()

0 t r i al1

0 0

0 0

n

n

s s

a a

l

+

= =

v

m

v

( )

, ,

F F F

r r r

r r r

r r r

s s s

a a a

sal

salsal

sal

= =

m

v

()()

( ) ()

1 0 sgn

0 1 s g n

s g n s g n 0

E

H

sa

sa

sa sa

- -

= - - ---

1 11 1 2 1 3

2 1 22 231

3 1 32 33 1

1

n

n

m m m

m m m E H

m m m

-

+

+

=

=+

m

v

()()

( ) ( )1

sgn

1 s g n

s gn s gn 1n

H E E E

H E H

E

sa

sasa sa

+

- -

- - - - -

( ) ()1 1 tria l1,,, sgn 0nr E Esskalss lsa- -+=- + -

(), 0nrkklkkl=-++

( ) ( ),, sgn 0nr Hasalaalsa=-++ -

( ), , ( ) 0 F r kskasak=--

()( )

()()()

()

4 4 ,

F

r

r

r

r

s

k

a

s

k

a

l

= =

R R

v

v

m v v

v

v

()

()

()

()

()

0trial

10

0

0

0 0

n

n

n

s sk k

aa

l

+ = =

v

( )

, , ,

F F F F

r r r r

r r r r

r r r r

r r r r

s s s s

k k k k

a a a a

skal

skalskal

skal

skal

=

m

v

()

()

() ()

1 0 0 s gn

0 1 0 10 0 1 H s gn

ds gn s gn 0

d

E

k

sa

sa

sa sak

- -

- = - -

----

11 1 2 1 3 1 41

2 1 22 23 2 4

3 1 32 33 3 41

4 1 42 43 4 4 1

1

d

d

n

n

m m m m

m m m m

k m m m m E H

m m m m k

-

+

+

= = ++

m

v

() ( )

() ( ) ()

() ( )

() ()1

d dsgn sgn

d d

sgn sgn 1

d dsgn sgn

d d

dsgn sgn 1

d n

k k H E E E E

E E H

k k HE E H

k E

sa sak k

sa sa

sa sak k

sa sak +

+ - - - -

- -+ - -

- --+ - -

- - -


8/13


t he g eo me tr ic d im en si on s o f t he u nl oa de d a nd l oa de d

s p ecimen are s h own . In addition , the common bou n dary

c o n d i t i o n s , i . e . f o r c e b o u n d a r y c o n d i t i o n ( F) an d

dis p la cemen t b ou n d ary con dition (u ) are in dicated.

I t i s i m po r ta nt t o r e mi n d t h at a n i d ea li s ed t e ns i le s a mp l e i s

as sumed where the specimen deforms only in its

lon gitu din al direction (i.e. th e x-direction as s h own in Figure

4 ) a n d d o e s n o t s h ow a n y c on t r ac t i on p e r pe n d ic u l ar t o t h e

loa din g direction. Su ch a problem can be s olved bas ed on

t h e r o d e l e me n t w h i c h i s d e s cr i b ed i n s e c t i on 2 . 1 .

Let u s firs t con s ider th e cas e of linear is otropic h arden ing (cf.

s ection 2.3.1) u n d er d ifferen t types of bou ndary con ditions .

T h e i m pl e m en t e d m a t er i a l p a r am e t er s a r e s u mm a r is e d i n

T ab l e 2 .

Th e effect of differen t bou n dary con dition s , cf. Figu re 5, on

t h e s t re s s s t ra i n di a gr a m is i l lu s tr at e d i n F ig u re 6 . F ig u re 6 a )

s ho ws t he c as e o f a s ym me tr ic d is pl ac em en t b ou nd ar y

c on di ti on ( cf. F ig ur e 5 a ) w it h | u | = | u | = 0 .0 08 m. A smax mi n

c a n b e s e e n, t he a b so l u te v a l ue o f th e s t r es s i s i n c re a s in g i n

t h e p l as t ic r a ng e a nd a f te r a n e n ti r e c yc l e, t he s t re ss e s f o r

e x a mp l e i n t h e f i r s t q u ad r a nt a re h i g he r t h a n i n th e i n i t ia l

l o a di n g. T h e s y mm e t ri c b o u nd a r y c o n di t i o ns c a us e s t h a t t h e s t re ss s t ra in cy c li ng is b ou nd e d b y t h e s am e a b so l ut e

s t r ai n v a lu e o n t h e p o s it i v e a n d n e g at i v e s t r ai n a x i s. F i g ur e 6

b ) s ho ws t he r es ul ts o f a s li gh tl y d if f er en t d is pl ac em en t

b ou nd a r y c on d it i on (c f. Fi g ur e 5 a ) w i th u = 0. 0 08 m an dmax

u = -0 .0 03 1 m. Th e n eg at iv e d is pl ac em en t v al ue w it hmin

|u | > |u | was chosen in such a way that the stressmax min

F i g u re 4 . S c h em a t i c R e p r es e n t a ti o n o f a U n l o ad e d a n d Lo a d e d

I d e a l T e ns i l e S p e c i me n

Yo ung' s m o dulus in Mpa E = 70000

P oi s s on ' s r a ti o - n= 0.3

L i n e ar i s o t r o p i c h a r d en i n g m o d u lu s i n M p a

F l ow c u r ve i n M p a

L i n e ar k i n e m a t i c h a r d en i n g m o d u lu s i n M p a

2

k () = 3 5 0 + 12 9 00-125000

H = 7000

Epl = 7000

Ta b le 2 . M a te r i al P a r am e te r s U s ed f o r t h e S i m ul a t io n [ 7 ]

F i g u r e 5 . D ef i n i t i on o f t h e D i f f e r en t B o un d a r y C on d i t i o ns :a) Displacement umin = -umax; b) Displacement| umin | < umax; c) Force | Fmin | > Fmax




(a)


9/13

c y c le p a ss e s t h r ou g h t h e o r i gi n o f t h e s t r e ss - s tr a i n d i a gr a m,

i . e. t h e p l a st i c s t r ai n i s z e r o i n t h i s p a r t of t h e d i a gr a m. F i g ur e

6 c ) c on s id e rs t h e c a se o f a f o rc e b o un da r y c o nd i ti o n ( cf.

Figu re 5 c) with F = 4450 N and F = -6190 N wh ich wasmax mi n

c ho se n in s uc h a w ay t ha t th e s am e m ax im um a ndm in im um s t ra in i s o bt ai ne d a s i n ca se a ). S i nc e t he l oa d

i n c re m e nt f o r e a c h s t e p i s c o n st a n t (DF = const.), fewer

d a t a p o i nt s ( e a ch p o in t i s r e p re s e nt e d b y a s m a ll c i r cl e ) a r e

c a lc ul a te d i n t he p la s ti c l o ad i ng r an ge . Fi g ur e 6 d ) s ho w s

t he c as e o f a f or ce b ou nd ar y co nd it io n w it h F = 44 50 Nmax

a n d F = -5 4 00 N w h ic h w a s c h o se n i n s uc h a w a y t h at t hemin

s t re s s c y cl e p as se s t h ro ug h t h e o r ig i n o f t h e s t re s s- s tr ai n

diagram, i.e. n o p las tic s train in th is part of th e diagram.

L et u s c o ns i de r i n t he f o ll o wi n g t he c a se o f i de a l pl a st i ci t y

wh ich can be des cribed bas ed on th e th eory of is otropic

h arden ing with a con s tan t yield s tres s of k = 350 M Pa, cf. E q.

(6). Impos in g th e s ame dis placemen t bou n dary con dition s

a s i n c as e a ) , a s t r es s - st r a i n d i a gr a m a s s h o wn i n F i g ur e 7 a )

i s o bt ai ne d w he re t he c yc le i s b ou nd ed o n t he s tr ai n s id eb y t he s am e a bs ol ut e v al ue a nd o n t he s tr es s s id e b y t he

valu es of k. It s h ou ld be n oted h ere th at th e cas e of ideal

p l as t i c it y w i t h a f o r ce b o u nd a r y c o n di t i on f a i l s i n t h e p l a st i c

r an ge s in ce t he re i s n o m or e u ni qu e r el at io ns hi p b et we en

t h e g i ve n l o ad in g a nd t h e s t ra i n. F ig u re 7 b ) i l lu st r at e s t h e

cas e of n on lin ear is otropic h arden ing with a dis placemen t

b o un d ar y c o n di t i o n a s i n Fi g u re 6 a ) . Co m pa r e d t o F i g ur e 6

a ) , a s i m il a r s t re s s -s t r ai n b e ha v i ou r i s o b t ai n e d e x c e pt t h e

f a ct t h at t h e s t re s s i n t h e h ar d en i ng r a ng e i s n o nl i ne a rl y

F i gu r e 6 . S t re s s- S tr a i n D i ag r am s f o r t h e C a se o f L i ne a r I s ot r op i c H a rd e ni n g


l




10/13

i n c re a s in g . F i g ur e 7 c ) s h o ws t h e c a s e o f n o nl i n ea r i so t r op i c

h a rd e ni ng w i t h t he s a me f o rc e b ou nd ar y c o n di t io n a s i n

F i g ur e 6 c ) . D u e t o t h e d e f i ni t i o n o f th e n o nl i n ea r ha r de n i ng

c u r v e, t h e m a x im u m st r e s s i n t he f i r st q u ad r a nt i s r e a ch e d

a t a l o w e r s t r ai n co m p ar e d t o t h e c a s e s h o w n in F ig u r e 6 c ) .

T h e c a s e o f l i n ea r k i n em a t ic h a rd e n in g ( c f. S e c t io n 2 . 3. 2 ) i s

s how n i n Fi gure 7 d) f or a dis plac eme nt bound ar y

c on di ti on a s i n F ig ur e 6 a ). S in ce t he l in ea r k in em at ic

h ar de ni ng m o du lu s H w a s e qu al t o t he l in ea r i so tr op ic

h ar de ni ng m od ul us E pl o f Fi gu re 6 a ), t he s tr es s- st ra in

valu es for th e firs t cycle in th e firs t qu adrant are iden tical for

th e case of Figure 6 a) and Figure 7 d) and only in the

c o m pr e s si v e p l as t i c r e g im e a d e v ia t i on b e t w e e n t h e s e t w o

c a s es s t a rt s t o d e v el o p.

T he c as e o f c om bi ne d h ar de ni ng ( c f. s e ct io n 3 .3 .3 ) i s

s h ow n i n F i g ur e 8 w h e re t h e s a m e d i s pl a c em e n t b o u nd a r y

c on di ti on as i n Fi gu re 7 a ) w as a ss ig ne d t o b ot h c as es.

F ig u re 8 a ) i l lu st r at e s t h e c a se l i ne a r c o mb i ne d h ar d en i ng

wh ile Figu re 8 b) illu s trates th e cas e of n on lin ear combin edh a rd e n in g . Un d e r th e c h o se n p a ra m e te r s, q u i te s i m il a r

s tres s s train diagrams are obtain ed.

Th e evolu tion of th e in tern al h arden in g variables , i.e. kan d

a, i s i l lu st r at e d i n Fi g ur e 9 f o r th e c as e s o f l in e ar h ar d en i ng .

F ig u re 9 a ) i s g i ve n f o r i s ot r op i c h ar d en i ng in th e c a se o f a

f or ce b ou nd ar y c o nd it io n ( ca se a s F ig ur e 6 c ) a nd s ho ws

t h e e v o lu t i on o f a n d f o r c o m pa r i so n t h e e v o lu t i on o f t h e

p la st ic s tr ai n. A s c an b e s ee n, t he i nt er na l is ot ro pi cp l

h arden in g variable (dk= |de|) i s a v a ri a bl e w h ic h c a n o nl y

k

F i gu r e 7 . S t re s s S t ra i n D i ag r am s f o r t h e C a s es o f a ) I d ea l P l a st i ci t y ; b ) c ) N o nl i n ea r I s ot r op i c H a rd e ni n g; d ) L i ne a r K i ne m at i c H a rd e ni n g






11/13

increase while t he plas tic strain c an d ecrease and

in creas e du rin g a cycle.

T h is i s t h e r e as on w hy i s t a ke n a s t h e i n te r na l v ar i ab le a ndp l

n ot e. T h e p l a st i c s t r ai n f o r e x a mp l e c a n t a k e d u ri n g c y c li cl o a di n g se v e ra l t i me s a v a l ue o f z e ro w h i le kn ev er f or ge ts

i t s h i st o r y. F i gu re 9 b ) s ho w s t h e c as e f o r t h e d i sp l ac em e nt

b o u nd a r y c o nd i t io n w h ic h c o rr e s po n d s t o F i g ur e 6 a ) . T he

e v o lu t i o n of t h e k i n em a t ic h a r de n i ng v a r i ab l e ( da=Hde)i s p re se nt ed i n F ig ur e 9 c ) f or t h e c as e o f l in ea r k in em at ic

h a rden ing u n der a dis placemen t bou n dary con dition . Th is

r es ul t c or re sp on ds t o F ig ur e 7 d ). As c an be s ee n, th e

k i n em a t i c h a rd e n in g v ar i a bl e c a n ch a ng e i t s s i g n as i n t h e

c a s e o f t h e p l a st i c s t r ai n .

As des cribed in s ection 2.1, th e tran s ition between th e

k

pl

e l as t ic a nd p l as t ic r a ng e ( c f. F i gu r e 1 ) i n t he c a se o f a f o rc e

b ou nd a r y co n di t io n ne e ds s o me c yc l in g i n or d er t o

d e t er m i ne t h e f i n al v a lu e o f t he s e c an t m od u l us a s g i v en i n

E q . ( 23 ) . T h is e v a lu a t io n o f t h e m o d ul u s i s s h o wn i n F i g ur e 1 0

f or t he c as e o f l in ea r i so tr op ic h ar de ni ng a nd a f or ce

b o u n d a r y c o n d i t i o n ( t h e c o r r e s p o n d i n g s t r e s s - s t r a i n

d ia g ra m i s s ho w n i n F i gu r e 6 c ). F ig u re 1 0 a ) i l lu st r at e s t h e

F i gu r e 8 . S t re s s S t ra i n D i ag r am s f o r t h e C a s es o f a ) L i ne a rC o m b i ne d H a r d e n in g ; b ) N o n l i n e ar C o m b i n ed H a r d e n i n g

F i gu r e 9 . E v o lu t i on o f P l a s ti c S t ra i n a n d I n te r na l V a ri a bl e f o ra ) L i n e ar I s ot r op i c H a rd e ni n g w i t h f o r ce B o un d ar y

C o n d i t io n ; b ) L i n e ar I s o t r op i c H a r d e ni n g w i t h D i s p l ac e m e n tB o u n d ar y Co n d i t i on ; c ) l i n e a r K i n e m a ti c H a r d e ni n g

With Displacement Boundary Condition


l




12/13

s itu ation in th e firs t qu adran t of th e s tres s s train diagram, i.e.

t h e t r a ns i t i on b et w e en t h e e l a st i c a n d t h e p l a st i c p a r t i n t he

t e ns i le r e gi m e. F i gu r e 1 0 b ) i l lu st r at e s a s i mi l ar s i tu at i on b ut

n ow in th e fou rth q ua d ran t of th e s tres s s train diagram, i.e. in

t he t en si le r eg im e. As c an be s ee n f or bo th ca se s,

c o n ve r g en c e i s a c h ie v e d a f t e r f e w c y c le s.

Con clusion

T h e p r i nc i p al i n g r ed i e nt s o f c l a ss i c al p l a st i c i ty t h e o r y a s

a p p li e d i n f i n i te e l e me n t c o d es w e r e i n t ro d u ce d b a se d o n

o n e -d i m en s i on a l s t r es s a n d s t r ai n s t a t es . T h i s i s a s i m pl i f y in g

a n d i d ea l is e d c a se b ut al l t h e c o mp o ne nt s o f t h e p l as t ic

c o n st i t ut i v e d e s cr i p ti o n c a n b e i n t ro d u ce d . I n t h e g e n er a l

t h re e d i me n si o na l c a se t h e s a me e qu at i on s a r e r e qu i re d( i. e. a y i el d c o nd i ti o n, a f l ow r ul e a nd a h ar de n in g r u le ) a nd

on ly th e math ema tical des cription becomes more

c o m p l i c a t e d . H o w e v e r , t h e c o n c e p t u a l a p p r o a c h

r e m ai n s t h e s a me .

T h e i n te g ra t io n sc he m e f o r t h e c o ns t it u ti v e e q ua t io ns w a s

b a s ed o n t h e p r e di c t o r co r r ec t o r co n c ep t w h ic h i s u s ed i n

t h e t h r ee d i m en s i on a l c as e. I f o n e i s i n t er e s te d i n t h e s i m pl e

s ol ut io n of t he p ro bl em , t he re i s n o n ee d f or a p re di ct or

c o r re c t or c o n ce p t f o r o ne - d im e n si o n al s t a t es s i n ce t h e

st ress st rain c ur ve g ives nor mal ly t he uni que rel at ion

bet wee n st res s and st rain and o ne can det ermi ne

i m me di a te l y t h e s t re ss i f t h e s t ra i n i s g i ve n o r v i ce v e rs a.H ow ev er, t h e b as ic i de a w as t o i nt ro du ce t he i nt eg ra ti on

p ro ce du re a s i n th e g en er al c as e b ut e xp la in ed o n th e

i d e al i s ed c a s e o f a o n e d i m en s i on a l s t r es s a n d s t r ai n s t a t e.

T h i s a p p ro a c h r e v ea l s t h e a d va n t ag e t h a t a l l t h e s t e ps c a n

b e e a s il y u n d er s t oo d a n d t r a ce d i n t h e c l a ss i c al s t r e s s s t r ai n

diagram an d th e s olu tion procedu re can be eas ily realis ed

b y si m pl e c o de s o r by t he a pp l ic a ti o n o f c om m er c ia l

compu ter algebra s ys tems.

T he p ri nc ip al fe at ur es a nd p ro ce du re o f t he p re se nt ed

s c he m e s d o n o t c h a ng e i n t h e g e n er a l t h r ee d i m en s i on a lcas e. Again , on ly th e math ematical n otation mu s t be

adopted. Fu rth ermore, an y addition of a fu rth er in tern al

variable (for example a s calar damage variable) res u lts

on ly in th e addition of a fu rth er equ ation wh ich in creas es

t h e s i z e o f t h e f i n al s y s t em s o f e q ua t i on s b y o n e.

It is believed th at th is on e-dimens ion al approach facilitates

t h e u n de r s ta n di n g o f t h e t h r ee d i me n s io n al ca s e a n d t h a t

t h e i m p le m e nt a t i on o f t he p r e se n t ed s c h em e s s h ou l d b e a

g o o d p r e pa r a to r y wo r k f o r t h e h a nd l i ng of el a s to - p la s t ic

problems in fin ite elemen t codes.

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[1]. O wen, D. R. J., & Hin ton , E. (1980). F in it e e le me nt s i n

p l as t ic i ty : t h eo r y a nd p ra c ti c e. S wansea, P i ne r i dg e P r es s

Limited.

[2]. Crisfield, M. A . (1991). Non -lin ear fin ite elemen t an alys is

o f s o li d s a nd s t ru ct u re s ( Vo l. 1) . Chichester, John Wiley &

Sons.

[3]. Crisfield, M. A . (1997). Non -lin ear fin ite elemen t an alys is

o f so l id s a n d s t ru c tu r es ( Vo l. 2) . Chichester, John Wiley &

Sons.

[4]. Simo, J. C., & Hughes, T. J. R. (1998). Computational

Inelasticity. New York, Springer-Verlag.

[5]. Belytschko, T. W. K ., L iu, & Moran , B. (2000). Nonlinear

f i n i t e e l e m en t s f o r c o n ti n u a a n d s t r u ct u r es . Chichester,

Joh n Wiley & Son s .

[ 6 ] . D un n e , F. , & P e t ri n i c, N . ( 2 0 0 5 ). Introduction to

F i gu r e 1 0 . Ev o lu t i on o f t h e A v er a ge M o du l u s E : a ) T e ns i l e R e g i on ;b ) C o m p re s s i v e R e g i o n






13/13

computational plasticity. Oxford, Oxford Un ivers ity Pres s.

[7]. de Souza Neto, E. A ., Peri, D., & O wen , D. R. J. (2008).

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Applications. Ch ich es ter Joh n, Wiley & Son s .

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J., 42(5), 44445.

[ 9 ] . A r m e n , H . ( 1 9 7 9 ) . A s s u m p t i o n s , m o d e l s , a n d

comp u ta tiona l meth ods for plas ticity, Comput. S truct.,

10(1), 161-174.

[10]. Reddy, J. N. (2006). An introduction to the finite

element method . Sin gapore, M cGraw Hill.

[11]. Cook, R. D., Malkus, D. S., Plesha, M. E., & Witt, R. J.

(2002). C o n c ep t s an d a p pl i c a ti o n s of f i n i te e l e me n t

analysis. Ch ich es ter, Joh n Wiley & Son s.

[12]. Simo, J. C., & O rtiz, M. (1985). A u n ified approach to

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[13]. Moran , B., O rtiz M., & Shih, C. F. (1990). Formulation of

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(1990), 483514.

[14]. Wriggers, P. (2001). Nichtlineare Finite-Element-

Methoden. Berlin, Springer-Verlag.

[15]. Merkel, M., chsn er, A . (2010). Eindimensionale Finite

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ABOUT THE AUTHORS

M oosa Esmaeili is a Ph.D student w ithin the School of Engineering at the U niversity of A berdeen, U K . He obtained his M aster D egree in M echanical

Engineering at the Technical U niversity of M alay sia, M alay sia (2011).

A ndreas Ochsner is Professor in the D epartment of A pplied M echanics at the Technical U niversity of M alay sia, M alay sia. Having obtained a M aster

D egree in A eronautical Engineering at the U niversity of Stuttgart (1997), Germany, he spent the time from 1997-2003 at the U niversity of Erlangen-

N uremberg as a research and teaching assistant to obtain his Ph.D in Engineering Sciences. From 2003-2006, he w orked as A ssistant Professor in

the D epartment of M echanical Engineering and Head of the Cellular M etals Group affiliated w ith the U niversity of A veiro, Portugal.


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ON THE INTRODUCTION OF RETURN MAPPING SCHEMES IN ELASTO-PLASTIC FINITE ELEMENT SIMULATIONS FOR ...

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