Numerical Modelling in Large Strain Plasticity With Application to Tube Collapse Analysis

8/12/2019 Numerical Modelling in Large Strain Plasticity With Application to Tube Collapse Analysis

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NUMERICAL MODELLING IN LARGE STRAIN PLASTICITYWITH APPLICATION TO TUBE COLLAPSE ANALYSIS

By

JOSEsM. GOICOLEA RUIGOMEZIng. Caminos

A Thesis submitted for the degree ofDoctor of Philosophy

In the Faculty of Engin eering of theUniversity of London

King'sCollege London

October 1985


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ABSTRACT

N u m e r i ca l m e th o d s are p r o p o s e d f o r t h e a n a l y s i s o f 2 o r 3 -d i m e n s io n a l l a r g e s t r a i n p l a s t i c i t y p r o b le m s . A F i n i t e D i f f e r e n c ep r o g ra m , w i t h 2 - d i m e n s i o n a l c o n t i n u u m e l e m e n t s and e x p l i c i t t i m ein te g r a t i o n , has been deve loped and ap p l i ed to model the a xis ym m etr icc r u m p l i n g o f c i r c u l a r t u b e s .

New t yp e s o f m i xe d e l e m e n ts (T r i a n g le s - Q u a d r i l a te ra l s f o r 2 -D ,T e t r a h e d r a - B r i c k s f o r 3 -D ) a r e p r o p o s e d f o r t h e s p a t i a ld i s c re t i z a t i o n . These e le m e n ts m odel a cc u ra te l y i n co m p re ss ib l e p l a s t i cf l o w , w i t h o u t u nw an te d " z e r o - e n e r g y " d e f o r m a t i o n modes o r t a n g l i n go v er o f t h e m esh . E l a s t i c - p l a s t i c , r a t e d ep en de nt l aw s are m o d e l l e dw i th a " ra d ia l r e t u r n " a l g o r i t h m . T he t ra n s m is s io n o f he at g e n e rate dby p l a s t i c w o rk an d m a t e r i a l d ep en de n ce on t e m p e r a t u r e are a l s oi n c l u d e d , e n a b l i n g a f u l l y co u p le d the rm o -m e ch a ni ca l a n a l y s i s .

A 2 -D a nd a x i s ym m e t r i c co m p u te r p ro g ra m has b ee n d e ve lo p e d ,i m p l e m e n t i n g t h e n u m e r i c a l t e c h n i q u e s d e s c r i b e d . C o m p u t a t i o n a le f f i c i e n c y was e s s e n t i a l , as l a r g e s c a l e , c o s t l y a p p l i c a t i o n s w erein te n d e d . An im po r tan t pa r t o f the p rog ram was the con tac t a lg o r i th m ,e n a b l i n g t h e m o d e l l i n g o f i n te ra c t i o n b e tw e e n su r fa ce s .

The a x i s y m m e t r i c c r u m p l i n g o f t u b e s u n d e r a x i a l c o m p r e s s i o n( " c o n c e r t i n a " mode) has been a n a l y z e d N u m e r i c a l l y . Q u a s i - s t a t i cexper imen ts on A lumin ium tubes were mo de l led , us ing ve lo c i t y s ca l in g .Mery l a r g e s t r a i n s are d e v e l o p e d i n t h e c r u m p l i n g p r o c e s s ; w i t h t h ehe lp o f tens ion te s ts , m a te r ia l laws v a l i d fo r such s t r a in ranges weredevelop ed. Good agreement was ob tain ed between nu m eric al p re d ic t i o n sand expe r imen ta l re su l ts . M ode l l ing cho ices such as mesh re f ine m en t ,e lement type and v e lo c i t y sc a l in g were st u d i ed , and found to have ani m p o r t a n t i n f l u e n c e on t h e n u m e r i c a l p r e d i c t i o n s . F i n a l l y , a l a r g esc a le im p a c t a n a l y s i s o f a s te e l t u b e a t 17 6 m /s w as p e r fo rm e d . Ther e s u l t s co mpa re d w e l l w i t h e x p e r im e n t , i n d i c a t i n g d i f f e r e n c e s w i t h t h ebehav iou r o f low ve loc i ty c rump l ing mechan isms.

To c o n c l u d e , F i n i t e D i f f e r e n c e p r oc e d u re s w i t h e x p i i c i t t i m e -


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m a rch in g t e ch n iq u e s a re p ro po se d fo r l a rg e s t r a i n p l a s t i c i t y p ro b le m s ,a t l o w o r m e diu m i m p a c t v e l o c i t i e s . A f a i r l y r o b u s t c od e has b ee ndeve loped and ap p l ie d su cc es sf u l ly t o a range o f la rg e s t r a in and tubecrump l ing p rob lems.


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ACKNOWLEDGEMENTS

I w i sh t o e xpress my s i n ce re g r a t i t u d e to th e f o l l o w in g :

D r . G .L. E n g l an d who s u p e r v i s e d t h i s w o r k , f o r h i s c o n s t a n t s u p p o r tand gu idance throu gho ut the per iod o f re se ar ch ;

D r. J . M a r t i f r o m P r i n c i p i a , f o r h i s i n v a l u a b l e e x p e r t a d v i c e ont h e o r e t i c a l and n u m e r i c a l m a t t e r s , and h i s f r i e n d l y s u p p o r t andsu g g e s t i o n s ;

O r. . A l a r c o n , f o r h i s h e lp and m o t i v a t i o n i n t h e i n i t i a l s t a g e s ofwork ;

The s t a f f and t e c h n i c i a n s o f t h e C i v i l E n g i n e e r i n g D e p a r t m e n t a tK in g ' s C o l l e g e , f o r t h e i r h e lp and co o p e ra t i o n ;

My w i f e , Te reca M ar i n , fo r he r he lp in the p repa ra t io n o f f i g u re s , andher unend ing pat ienc e and encouragement du r in g the research pe r i od .

I wou ld a lso l i k e to thank the fo l l o w in g bod ies fo r sp onsor ing myr e s e a r c h a t t h e v a r i o u s s t a g e s : The B r i t i s h C o u n c i l , The S p a n i s hGovernment (M in is t r y o f Ed uc at io n) , P r i n c ip ia Mechan ica L t d . (UK), andthe Commit tee o f V icec han ce l lo rs and P r in c i pa ls (UK) .


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TABLE OF CONTENTSPa

TITLE

ABSTRACT

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

BASIC NOTATION 10

CHAPTER 1 - INTRODUCTION 13

1.1 Ob jec t ive s 141 .2 N o n - l i n e a r m o d e l l i n g 161.3 La yo ut 17

CHAPTER 2 - CONTINUUM MECHANICS DESCRIPTIONS 18

2.1 Introduction 192.2 Kinematics 20

2.2.1 Configurations 212.2.2 Deformation tensors 212.2.3 Deformation and spin rates 232.2.4 Strains 232 .2 .5 T ransfo rm a t ions 24

2.3 Stre ss 252 .3 .1 Cauchy 252 . 3 . 2 P i o l a - K i r c h h o f f 25

2.4 Balance laws 262 . 4 .1 Balan ce of momentum 26


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Pa2.4.2 Bal anc e of mas s 262.4.3 Bala nce of energ y 27

2.5 Consti tutive relations 282.5.1 Rate equa tio ns 292.5.2 Elas tici ty 30

2.5.2.1 Hyperelast ic mater ials 312.5.2.2Hypoelastic materials 31

2.5.3 Plast icity 322.5.3.1 Von Mis es model 342.5.3.2Other plasticity models 35

CHAPTER 3 - NON-LINEAR NUMERICAL MODELS FOR SOLID MECHA NICS 37

3.1 Intro ducti on 383.2 Finite Differ ence method s 393.3 Finit e Elemen t meth ods 413.4 Mesh des cri pti ons 44

3.4.1 Lagr angi an 443.4.2 Eule rian 453.4.3 Arbitrar y Lagran gian-E uleria n 45

3.5 Large displa cement formul ation s 463.5.1 Total Lagr angi an 463.5.2 Cauchy stre ss - veloc ity strain 473.5.3 Updated Lagrang ian 49

3.6 Time inte grat ion 493.6.1 Central Diffe rence (explicit) 503.6.2 Trapezoi dal rule (imp lici t) 523.6.3 Operator split method s 54

3.7 Practical consid erati ons for discrete mehes 553.7.1 "Locki ng-up" for incomp ressibl e flow 553.7.2 "Hourgl assing " 58

3.8 Conclus ions 59

CHAPTER 4 - EXPL ICIT FINITE DIFFERENC E NUMERI CAL MODE L 614.1 Introd uction 63


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Pa4.1.1 General methodology 63

4.2 Spatial semidiscretization 664.2.1 Constant Strain Triangles andTetrahedra (CSTelements)674.2.2 Mixed Discretization (MTQ, MTBelements) 724.2.3 Prevention ofnegative volumes (MTQC, MTBC el ement s) 754.2.4 Mass lumping procedure 78

4.3Momentum balance 784.4Central difference time integration 794.5 Constitutive models 79

4.5.1 Hypoelasticity 804.5.2 Plasticity; radial return algorithm 804.5.3 Hardening anduniaxial stress-st rain laws 844.5.4 Objective stress rates 87

4.6 Damping 884.7 Stability oftime integration 904.8Modelling ofcontacts 91

4.8.1 Contact interface laws 924.8.2 Contact detection algorithm 94

4.9Heat conduc tion 964.10 Energy computations 1004.11 Implementat ion into Fortran program 102

CHAPTER 5 -BENCHMARKS AND VALIDATION EXAMPLES 106

5 .1 In t r od uc t io n 1075.2 Wave pr op ag at ion 107

5. 2 .1 E la s t i c waves in bars 1075 .2 .2 E la s t i c -p l a s t i c waves 1105.2 .3 E la s t i c waves in cone 112

5 .3 V ib ra t i o n o f a ca n t i l e ve r 1145 .4 S t a t i c e l a s t i c -p l a s t i c p ro ble m s 117

5 .4 .1 Punch t e s t 1175 .4 .2 E la s t i c -p l a s t i c sph ere u nd er i n te rn a l p re ssu re 12 1

5.5 Heat co nd uc tion 1235. 5 . 1 Coupled thermomechanica l an a ly sis 1235 .5 .2 Tempera tu re re d is t r ib u t i o n in a s lab 124

5 .6 La rge s t ra in s and ro ta t io ns 124


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Pa5.7 Impact o f c y l i n d e r 1265.8 Co nclus ions 128

CHAPTER 6 - TENSION TESTS ON ALUMINUM BARS 129

6.1 Introduction 1306.1.1 Constitut ive idealization 1326.1.2 Tension tests - a review 133

6.2 Theoretical interpretation of tension tests 1366.2.1 Strain distribution at minimum neck section 1376.2.2 Stress distribution 138

6.3 Tension tests 1406.3.1 Specimens and material 1406.3.2 Programme 1416.3.3 Procedure 1426.3.4 Results 1426.3.5 Microhardn ess tests 144

6.4 Material hardening law 1476.5 Numerical calculation s for tension tests 151

6.5.1 Model 1516.5.2 Analysis 1536.5.3 Results 154

6.6 Conclusions 156

CHAPTER 7 - CONCERTINA TUBE COLLAPSE AN ALYSIS 1647 . 1 I n t r o d u c t i o n 166

7 .1 .1 Scope 1667.2 Overview o f energy d is s ip a t i n g devices 167

7 . 2 . 1 D e f i n i t i o n and c r i t e r i a 1677.2 .2 Types o f energy d is s ip a t i n g devices 1687. 2. 3 Tubes as energy abs orb ers 169

7 . 2 . 3 . 1 La te r a l compression 1697 .2 .3 . 2 Ax isymmet r i c ax ia l c rum p l ing 1707 .2 .3 . 3 D iamon d- fo ld ax ia l c rum p l ing 1717 .2 .3 . 4 Tube inv e rs ion 171


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Page7 .3 Q ua s i -s ta t i c co nc e r t ina tube co l lap se mechanisms 172

7.3.1 Related experimental work 1727.3.1.1 Experimental programm e andmethod 1737.3.1.2Typical experiment al results 1747.3.1.3Microhardness tests 177

7.3.1.3.1 Equipment andprocedure 1777.3.1.3.2Derivation ofmaterial strength , Y 178

7.3.2 Numerical model 1827.3.2.1Discretization andmaterial 1837.3.2.2Velocity scaling 1847.3.2.3Interpretation ofoutput 185

7.3.3 Results form numerical calculations andexperiment 1877.3.3.1 Tube geometry A: ID=19.05mm, t=1.64mm, L=50.8mm 1887.3.3.2 Tube geometry B: ID=19.05mm, t=1.17mm, L=50.8mm 1937.3.3.3 Tube geometry C:0D=38.10mm, t=1 .65m m, L=50.8mm 1977.3.3.4 Tube geometry D:0D=25.40mm, t= 0.95m m, L=25.4mm 203

7.3.4 Parametric studies innumerical analys es 2087.3.4.1 Influence offriction 2087.3.4.2 Influence ofvelocity scaling 2137.3.4.3 Influence ofmesh refinement 2167.3.4.4 Influence ofelement type 223

7.3.5 Discussion 2287.4Medium velocity (176m/s) tube impact anal ysis 233

7.4.1 Description ofproblem 2337.4.2 Numerical idealization 2357.4.3 Numerical results 2377.4.4 Discussion 243

7.5Conclusions 249

CHAPTER 8 -CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH 251

8.1Conclusions 2528.2Suggestions forfurther research 253

8.2.1 Theoretical andnumerical develop ments 2548.2.2 Additional applications 255

REFERENCES 256


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BASIC NOTATION

A Area ; Mass damping c o e f f ic ie n t (eqn. 4 .56)B S t i f f n e s s d am pin g c o e f f i c i e n t ( eq n. 4 . 56 ) ; s t r a i n - r a t e

parameter (eqn. 4 .50)B (B. j - ) Le f t Cauchy-Green defo rm at ion tens or (eqn . 2 .12)B.:j Gra d ient op era tor fo r F in i t e E lements (eqn . 3 .6)c S t ress wave v e lo c i t yC (C j - jk i ) Co n s t i tu t i v e ten so r fo r Jaumann ra te o f Cauchy st re ss (eqn.

3.23)C ( C J J ) R igh t Cauchy-Green ten so r (eqn . 2 .12)C Damping m at r ix (eq n. 3.30 )(^ ( C j- jk i ) C o n s t i t u t i v e t e n s o r f o r T r u e s d e l l r a t e o f C auc hy s t r e s s

( s e c t . 3 . 5 . 2 )CST Constant s t r a in e lementsd (d. j- j ) Rate of de form atio n ten so r (eqn. 2.15 )d (d.j ) Pe ne tra t ion in co nta ct (eqn. 4 .68 )D (DJJKL) C o n s t i t u t i v e t e n so r ( t o t a l L a g ra n g ian ) (e q n s . 2 .4 7 , 3 .21 )D DiameterE Young 's modulus o f E la s t i c i t yE (E j j ) Green 's s t r a in tens o r (eqn . 2 .17 )F Fo rce ; Y ie ld fu nc t io n (eqn . 2 .54 )F (F j ) De fo rma t ion g ra d ien t tenso r (eqn . 2 .7 )FD F i n i t e D i f f e re n c e (m eth od )FE F i n i t e E l e m e n t ( m e t h o d )G E l a s t i c s h e a r m o d u lu s ( e q n . 2 . 4 5 )g ( g -j -j ) M e t r i c t e n s o r ( e q n . 2 . 2 )h , h a , h Y , h ' H e i g h t ; P l a s t i c h a r d e n i n g m o d u li ( e q n s . 4 . 4 4 , 4 . 4 5 )h ( h i ) Hea t f l o w ra t e (eq n . 2 . 3 2 )

I d e n t i t y t e n s o rInner d iameterJacobian of motion (eqn. 2.10)S t i f f n e s s , s t i f f n e s s m a t r i x ( e qn s . 3 . 1 0 , 3 .1 1 )LengthVe loc i ty g rad ien ts (eqn . 2 .14 )Mass, mass m atr ice s (eq n. 3.8 )M ix ed D i s c r e t i z a t i o n ( s e c t . 4 . 2 . 2 )

MTB(C) Mixed Te tra he dra -B r ic k (Co rrec ted ) e lements

IIDJK ,KL1 O im , M , mMD

j >i,M


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MTQ(C)N,nNOCPqr,Rs,sstTuuU

n

i

R( R i j )s( S I J )( s i d )

( U - j )( U I J )

M ixed T r i a n g le s - Q u a d r i l a te ra l (C o r re c te d ) ele m e ntsNormal vectorsT ime in s ta n t co r respond ing to n tShape fu nc t io ns fo r FE (eqn . 3 .5)Outer DiameterI n t e r n a l f o r c e sBody heat supplyRad ius ; Ex te rn a l fo r ce (eqn . 3 .8 )Ro ta t ion tens o r (eqn . 2 .11 )Su r face ; D is tance a long a cu rve2nd P i o l a -K i r c h h o f f s t re s s t e n so r (e q n . 2.2 5 )Cauchy de v i a t o r i c s t ress es (eqn . 2 .59 )T ime; Th icknessTemperatureIn te rna l ene rgyDisp lacementsR igh t s t re tch tenso r (eqn . 2 .11 )

V VolumeV (Vi-j) Left stretc h ten sor (eqn. 2.11)v,v (v.j ) Velocity (eqn. 2.6)w (w-j ^) Spin tensor (eqn. 2.15)W Work, Energyx (xn-) ,

(x,y,z) Spatial coord inat es (eqn. 2.5)X Particle (sect.2.2.1)X (Xj) Lagrangian coordinate s (eqn. 2.4)X Vector product (eqn. 2.30)V Y ie ld s t re s sa T h e rm a l e x p a n s i o n c o e f f i c i e n t ; Back s t r e s s ( eq n . 2 . 6 1 );

M ix ed D i s c r e t i z a t i o n c o r r e c t i o n c o e f f i c i e n t ( e qn . 4 . 19 )(3 P r o p o r t i o n o f c r i t i c a l d a m p i n g ; R a d ia l r e tu r n c o e f f i c i e n t

(eqn. 4 .32)T P l a s t i c f l o w a r b i t r a r y m u l t i p l i e r ( e q n. 2 . 55 )A Increment5-j -; Kro nec ker d e lt aeP E f f e c t i v e p l a s t i c s t r a i n ( e q n . 2 .6 0 b)e(e-j- j) Small s t r a in ten so r (eqn . 2.43)X Lame 's E las t i c cons tan t\i C o e f f i c i e n t o f Coulom b f r i c t i o n (e q n . 7 .11 )


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vP9a (c r i : j)dx0)

P o isso n ' s ra t i oMass d e ns i t y ; Radius o f c ur va tu reAngu la r coo rd ina teCauchy stress tensor (eqn. 2.24)P a r t i a l d e r i v a t i v ePu l l -b ac k , push - fo rward o f tenso rs (eqns. 2 .2 1 , 2 .22^Mesh co ord ina tes (s ec t . 3 .4)Angu lar f requency


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CHAPTER 1

INTRODUCTION

1.1 OBJECTIVES

1.2 NON-LINEAR MODELLING

1.3 LAYOUT


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1.1 OBJECTIVES

T h i s w ork c o n s i s t s o f a t h e o r e t i c a l p a r t ( m a t h e m a t i c a l andn u m e r i c a l m o d e l s , c h a p t e r s 2 - 4 ) , and a p r a c t i c a l p a r t ( a p p l i c a t i o n sand n u m e r i c a l s i m u l a t i o n of l a r g e s t r a i n t u b e c o l l a p s e a n a l y s i s ,chap te rs 5 -7 ) .

The m o t i va t io n fo r the th eo re t i ca l pa r t o f the work l ie s in thea u th o r ' s i n te re s t i n n o n - l i n e a r so l i d m e ch an ics m o d e l l i n g , u n d e rs to o dbroad ly as encompassing the fo l lowing phenomena:

- l a rg e s t r a i n s and l a rg e d i sp lace m e n ts (ge o m e t r ic n o n l i n e a r i t i e s ) ;

- p l a s t i c and v i s c o p l a s t i c b e ha v io u r ( m a t e r i a l n o n l i n e a r i t i e s ) ;

- con ta c ts and impact (n on l in ea r boundary co n d i t io n s ) ;

- thermomechanica l co u p l in g .

On the p r a c t i c a l s i de , the sou rce o f mo t iv a t i on was the researchp r o g r a m on t u b e c o l l a p s e m e c ha n is m s b e i n g c a r r i e d o u t a t t h e C i v i lE n g i n e e r i n g d e p a r t m e n t o f K i n g 's C o l l e g e , U n i v e r s i t y o f Lo nd on(A n d re w s , E n g la n d a nd G h a n i , 1 9 8 3 ) . S uch m e ch a n ism s a re e f f i c i e n tenergy d iss ipat ing systems (Johnson and Re i d , 1978), for use in impacts i t u a t i o n s . A d d i t i o n a l l y , t u b e s a r e f r e q u e n t s t r u c t u r a l c om po ne nts f o raerospace ve hi cl es and oth er equipment or components which may s u ff e ra c c i d e n t a l c o l l i s i o n s .

The o b j e c t i v e o f t h i s w o r k was t h e d e v e l o p m e n t o f n u m e r i c a lm eth od s o f s i m u l a t i o n f o r n o n l i n e a r a n a l y s i s , c a p a b le o f m o d e l l i n gt u be c o l l a p s e m e ch an is m s . M ore s p e c i f i c a l l y , t h e a t t e n t i o n wasr e s t r i c t e d t o c o l l a p s e t h r o u g h a x i s y m m e t r i c s e q u e n t i a l f o l d i n g(Conc ert ina mode). Numer ica l p r ed ic t i on s fo r tube co l la ps e shou ld beob ta ined and compared to exper im en ta l re s u l t s , a va i la b l e f rom p rev iousw o rk on a l u m in iu m tu b e s by G h an i (1 9 8 2 ) . T h i s o b je c t i v e p o se d som eim po rta nt ch a l l en ge s, such as the deve lopment o f a num er ica l model fo rl a r g e s t r a i n s and l a r g e d i s p l a c e m e n t s , w i t h e l a s t i c - p l a s t i c b e h a v io u r ,c a p a b le o f m o d e l l i n g a r b i t r a r y t u b e - t u b e and t u b e - p l a t e n c o n t a c t s( ch a p te r s 2 , 3 , 4 ) . R e l i a b le d a ta w o u ld h ave to be o b ta in e d fo r t h e


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TIME = 0.00 ms

TIME = 1.37 ms

TIME = 0.77 ms

TIME = 2.09 ms

3=TIME = 2.93 ms TIME = 3.62 ms

igur 1, TYPICAL RESULTS FOR AXISYMMETRIC TUBE COLLAPSE ANALYSIS0D = 38..1mm.t.= 1. 22mm .L =8 8. 9 mm - TUB E9, GEO METR Y F (SEE TABLE 7.4)


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1.3 LAYOUTIn chap ter 2 a number of e ss en t ia l s o l i d mechanics concepts are

i n t r o d u c e d and d i s c u s s e d b r i e f l y . N o n - l i n e a r n u m e r i c a l m o de ls andt e c h n i q u e s t o i m p l e m e n t t h o s e c o n c e p t s i n t o n u m e r i c a l c od es a r er e v i e w e d i n c h a p t e r 3. The E x p l i c i t F i n i t e D i f f e r e n c e m od el andcomputer code deve loped here are descr ibed in chapter 4 , wh i le chapter5 co nta ins some v a l i d a t i o n examples wh ich te s t the main aspects o f th ef o r m u l a t i o n . C h a pt er 6 c o n c e rn s t h e d e r i v a t i o n o f a m a t e r i a lc o n s t i t u t i v e l aw f o r A l u m in iu m a l l o y t h ro u g h t e n s i l e t e s t s , w i t h somea p p l i c a t i o n s t o t h e n u m e r i c a l s i m u l a t i o n o f t h e t e n s i l e t e s t sth e m se l ve s . C ha p te r 7 co n ta in s a p p l i c a t i o n s t o tu b e co l l a p se a n a l y s i s ,co m p a ri ng t h e re s u l t s w i t h e xp e r im e n ta l d a ta f o r q u a s i - s t a t i c co l l a p seo f A l u m i n i u m t u b e s . The c o n s t i t u t i v e l aw f r o m c h a p t e r 6 i s u s ed f o rthe numer ica l p re d i c t io ns . An ana lys is fo r a med ium v e lo c i t y (176 m/s)t u b e i m p a c t i s a l s o d e s c r i b e d . F i n a l l y , c o n c l u s i o n s a nd som esuggest ions fo r fu r ther work are g iven in chapter 8 .


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CHAPTER 2

CONTINUUM MECHANICS DESCRIPTIONS

2.1 INTRODUCTION

2.2 KINEMATICS

2.2.1 Configurations2.2.2 Deformation tensors2.2.3 Deformation and spin rates2.2.4 Strains2.2.5 Transformations

2.3 STRESS

2.3.1 Cauchy

2.3.2 Piola-Kirchhoff

2.4 BALANCE LAWS

2.4.1 Bal anc e of m a s s2.4.2 Balance of momentum

2.4.3 Balance of energy

2.5 CONSTITUTIVE RELATIONS

2.5.1 Rate equations2.5.2 Elasticity2.5.2.1 Hyperelastic materi als2.5.2.2Hypoelastic mat erials

2.5.3 Plasticity2.5.3.1 Vo n Mises model2.5.3.2Other plasticity models


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2.1 INTRODUCTION

The advances made in d i g i t a l com put ing w i t h in th e la s t decadesh av e o pe ne d up new f i e l d s f o r e n g i n e e r s an d s c i e n t i s t s . P r o b le m spre v io us ly rega rded as un so lva b le , on ly app roached th rough exper imen tsand s im p l i f i e d e m p i r i c a l f o rm u la e , ca n now be a n a lyze d n u m e r i ca l l y i ng r e a t d e t a i l . I n t h e f i e l d o f c o n t i n u u m m ec h a ni cs t h i s h as g r e a t l yi n c rea se d th e i n te re s t i n d e ta i l e d m a th e m a t i ca l d e sc r i p t i o n s , a me na blet o be u se d i n n u m e r i c a l m o de ls w i t h d i s c r e t i z a t i o n t e c h n i q u e s ( e . g.F in i t e Elemen t o r F in i t e D i f fe ren ce m e thods).

H a vi ng s a i d t h i s , t h e r e s t i l l e x i s t s a c e r t a i n d eg re e o fc o n f u s i o n i n t h e s p e c i a l i s t l i t e r a t u r e . On t h e p a r t o f t h ema them at ic ians , r igo ro us mechan ica l de sc r ip t io ns a re o f te n p resen tedi n w ay s d i f f i c u l t t o be g r a s p e d by e n g i n e e r s an d i m p l e m e n t e d i nnumer ica l p rodu ct ion codes. As a re s u l t , many eng inee rs s t i l l c l i n g ont o o u t d a t e d an d m uch l e s s p o w e r f u l n o t a t i o n s . On t h e o t h e r h a n d ,t h e o r e t i c a l p r e s e n t a t i o n s a r e n o t u n i q u e , c a u s i n g some d e gr ee ofco n fu s io n t o re se a rch e rs f i r s t a p p ro ach in g se r i o u s l y t h e se to p i c s .

An e f f o r t h as b ee n m ade i n t h i s ch a p te r t o p re s e n t a b r i e fove rv iew o f c e r ta in con t inuum mechanics concep ts , ind is pen sab le in ar igo rou s t r ea tm en t , w i tho u t unnecessa ry ma them at ica l fu ss . The purposeo f t h i s e x p o s i t i o n i s :

- t o i n t r o d u c e t h e n o m e n c l a t u r e and d e f i n i t i o n s o f c o n c e p t s u se d i nl a t e r c h a p t e r s ;

- t o d i s c u s s t h e s i g n i f i c a n c e o f and i n t e r p r e t s ome c o n c e p t s w i t h av iew to numer ica l m ode l l ing (bas is o f th is wo rk) ;

- t o e n s u re c e r t a i n c o m p l e t e n e s s f o r t h e i d e a s p r e s e n t e d i n t h i st h e s i s .

I t m u st be s t r e s s e d , h o w e v e r , t h a t t h i s e x p o s i t i o n does n o tpre ten d to be co mp le te . On ly the concepts wh ich are re le va nt fo r ther e s t o f t h i s t h e s i s w i l l be d w e l t u p o n . I n p a r t i c u l a r , e m p h a si s i sl a i d on s o l i d m e c h a n ic s and e l a s t i c - p i a s t i c b e h a v i o u r . A n um be r ofr e s u l t s w i l l be p r e s e n t e d w i t h o u t p r o o f . F or a m ore c o m p l e t e


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d iscu ss ion o f these to p ic s , the in te re s t ed reader i s re fe r r ed to Fung(1965), Malvern (1969), and B i l l i n g t o n and Tate (1981) fo r the genera lco n c e p ts , a nd to M a rsd en an d H ug he s (1 9 8 3 ) f o r a m o re d e t a i l e dand u p - to -d a te m a th em a t ica l d e s c r i p t i o n .

In the fo l l o w in g p re se n t a t ion , the amb ien t space is assumed to bean E u c l i d e a n p o i n t s p ac e ( i . e . i n t e r i o r p r o d u c t d e f i n e d ) , and w h er enecessa ry th is w i l l be p a r t i c u la r i z e d to R3. The co o rd ina t e bases maybe c u r v i l i n e a r a nd a r b i t r a r y , a l t h o u g h when e q u a t i o n s a r e g i v en i ncom ponent f o r m , o f t e n o r t h o no r m a l b as es ( n o t n e c e s s a r i l y c a r t e s i a n )a re assumed fo r s im p l i c i t y . The usua l conven t ions f o r tens o r no ta t iona r e e m p l o y e d : r e p e a t e d i n d i c e s i n d i c a t e s u m m a t io n o v e r t h e i r r an g eu n le s s e x p l i c i t l y s t a t e d , and commas i n d i c a t e c o v a r i a n t d e r i v a t i v e s .Vectors and tens ors are repre sente d by bo ld face ch ar ac te rs . Superposedd o t s i n d i c a t e m a t e r i a l t i m e d e r i v a t i v e s .

G i ve n tw o te n s o r s A a nd B th e p ro d u c t AB i s u n d e rs t o o d to beco n t ra c t i n g t h e n e a r i n d i ce s w i t h o p p o s i t e va r i a n ce :

( A B )i j = A

i kB k

j ( 2 .1 )

I f t h e i n d i c e s h ave th e sam e v a r i a n c e , e .g . b o th are c o n t r a v a r i a n t ,the metr ic t ens or g is necessary to lower one:

(A B ) 1 J - A l k g k l B 1 j = A l k 8 k j (2 .2 )

When th e t e n s o r co m p o n e n ts a re r e fe r re d t o o r th o n o rm a l b a se s th ev e r t i c a l p o s i t i o n o f t h e i n d i c e s i s i r r e l e v a n t , as t h e m e t r i c t e n s o ri s u n i t y .

A co lo n i n d i c a te s d o u bl y co n t ra c te d p ro d u c t :

A:B = A ^ B ^ ( 2 .3 )

2.2 KINEMATICSA bo dy ( o r c o n t i n u u m ) i s a s e t wh os e e l e m e n t s , c a l l e d m a t e r i a l

p a r t i c l e s , h av e a o n e - t o - o n e c o r r e s p o n d e n c e w i t h a r e g i o n V o f t h e


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Euclidean point space. Thefollowing kinematics concepts are intendedto provide adescription of themotion ofdeformable bo dies .

2.2.1 CONFIGURATIONS

Each particle X of thebody B may beidentified by itspositionXintheoriginal confi gurat ion, V 0,which istaken asreference:

X= k( X) (2.4)

X (components X i ) a re ca l le d Ma te r ia l o r lag r ang ian coo rd ina tes o f thep a r t i c le . The mo t ion o f the body a t a la te r t im e is g iven by the t im e -d e p e n d e nt p o s it io n s x o f t h e p a r t i c l e s i n t h e c u r r e n t c o n f i g u r a t i o n ,V:

x = x ( X , t ) ( 2 . 5 )

x (components xi) are t h e s p a t i a l or E u le r i a n co o rd in a te s . H e re a f te rupper c a se i n d i c e s s h a l l r e f e r t o L a g r a n g i a n c o o r d i n a t e s , and l o w e rcase to Eu le r ia n . The v e lo c i t ie s a re de f in ed as

:2 .6 '

w here t h e d ot s i g n i f i e s a m a t e r i a l t i m e d e r i v a t i v e , i . e . f o l l o w i n g t h ep a r t i c l e X .

2.2.2 DEFORMATION TENSORS

Central to deformation measuremen ts is thedeform ation gradienttensor:

F =d x / d X (2.7)

with components F n j = x 1 j .

The te ns o r F i s used as th e base f o r a number o f s t r a i n andd e f o r m a t i o n m e a s u r e s . An e l e m e n t o f a c u r v e dX i s t r a n s f o r m e d b y


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dx = FdX. Theinverse of F gives thespatial gradients of thematerialcoordi na tes:

F" 1= dX/dx(2.9)

( ( F " 1 ) 1 = X s i )

F c o n s t i t u t e s a t w o - p o i n t t e n s o r . A no th er i n t e r p r e t a t i o n t h a t r e l a t e sF to t r a n s fo rm a t i o n s b etwe en co n f i g u ra t i o n s i s g i ven i n se c t i o n 2 .2 .5 .

The Jaco bian of the m otion isJ = de t ( F ) (2 .10 )

The polar decomposition of Fgives

F = RU = VR (2.11)

w h e re R i s a n o r th o g o n a l ( r o t a t i o n ) t e n s o r , RRT = I . U a nd V a rep o s i t i v e d e f i n i t e , and a re c a l l e d t h e r i g h t a nd l e f t s t r e t c h t e n s or sr e s p e c t i v e l y . E q u a t i o n s (2 .1 1 ) r e p r e s e n t t w o w ay s t o v i s u a l i z e t h ed e f o r m a t i o n : f i r s t s t r e t c h i n g (U) and t he n r o t a t i n g ( R ), o r f i r s tro ta t i n g (R) and then s t re tc h i ng (V )

Other deformation measures are the Cauchy-Green tensors:

C = F 'F ( r ig h t Cauchy-Green)(2 .12 )

B = FF1 ( l e f t Cauchy-Green)

The le n g th o f an e lem en t o f cu rv e i s g iv en by ds2 = dxdx in th ecu r re n t co n f i g u r a t i o n , and dS2 = dXdX i n th e o r i g i n a l co n f i g u r a t i o n .The s i gn i f i ca nc e o f C and B is g iven by the re la t i on s

ds2 = dXCdX( 2 . 1 3 )

dS2 = dxB-ldx


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2.2.3 DEFORMATION ANDSPIN R ATES

The spatial velocity gradient tensor isdefinedas:

1 =dv/ dx (2.14)

which can be decomposed in to symm etric and skew-sym metric p a r t s :

d = ( l + l T ) / 2(2 .15 )w = ( l - l T ) / 2

These a re ca l led the ra te o f de fo rma t ion (o r ve loc i ty s t ra in ) and sp inra te tenso rs re sp ec t i v e l y . The ra t e o f change o f leng th o f an e lementof curve is g iven by

ds = (dxd dx) /d s (2 .1 6)

2.2.4 STRAINS

A m e a su re o f t h e t o t a l s t r a i n i s g i v e n by t h e Gre en s t r a i ntens o r , de f in ed as

E = (C - I ) / 2 (2 .17 )

w here I i s t h e I d e n t i t y t e n s o r . I t i s t r i v i a l t o see th a t

ds2-dS2 = 2dXEdX (2 .1 8 )

and th a t the ra te o f E is g iven by

E = FTdF ( 2 . 1 9 )


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2 . 2 . 5 TRANSFORMATIONS

F or t h e t e n s o r s d e f i n e d a b o v e , som e o f t h e i n d i c e s r e f e r t o t h eo r i g i n a l c o n f i g u r a t i o n (up pe r ca se ) , w h i l e o th e rs a re re l a te d t o t h ecu r ren t con f ig u r a t io n ( lower case ) . Here some t ra ns fo rm a t ion laws a reg i v e n t o f i n d t h e c o r r e s p o n d i n g t e n s o r i n t h e a l t e r n a t i v ec o n f i g u r a t i o n .

A l th o u g h th e t ra n s fo rm e d te n so rs w i l l be co n s id e re d as d i f f e re n tt e n s o r i a l e n t i t i e s , one way t o v i s u a l i z e t he t r a n s f o r m a t i o n i s as am e re ch a ng e o f b a se . Im a g in e a b ase ( O . e ^ f i x e d i n sp a ce t h r o u g h o u tt h e m o t i o n , a nd a n o t h e r b a s e ( 0 ' ( t ) , e ' i U ) ) w h i c h d e f o r m s a n dt r a n s l a t e s w i t h t he b od y. In t h i s c o n v e c te d c u r v i l i n e a r b a s e, t h ecoord ina tes o f a m a te r ia l po in t rema in cons tan t th roughou t the mo t io n ,and equa l to the m at er ia l c oo rd in at es , X I . The s p at ia l components o f Fp r o v i d e t h e m a t r i x f o r t h e c h an ge o f c o o r d i n a t e s b e t w e en t h e t w ob a s e s . G i ve n a 2nd o r d e r c o n t r a v a r i a n t t e n s o r a by i t s c o n v e c t e dm at er ia l components, a_U, the sp at ia l components are

a i j = O x V d x M O x J / a x ^ a U ( 2 . 2 0 )Hence F prov ides a means fo r t r an sf or m in g between sp at ia l and m at er ia lcoordinates, a"iJ and a_U are the components in d i f f e r e n t bases o f thesame t e n s o r , a . I f now one assumes com pon ents a_U to a pp ly to th esp a t ia l ba s is , a new tens o r i s o b ta ined :

A = a_ IJ e jB e j (2 .21a )

w he re s s i g n i f i e s a t e n s o r i a l p r o d u c t . A i s c a l l e d t h e p u l l - b a c k of a ,and may be obtained as

A = F _ 1 a F " T = (d(a) (2 .21 b)

wh i le the push-forward i s d e f i n e d by t h e i n ve rse r e l a t i o n :a = 0 t * (A ) = FAFT (2 .2 2)

These re la t io ns may be t r i v i a l l y gen era l i ze d to tenso rs o f any rank.


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E l e m e n t s o f a r e a a nd v o l u m e i n r e f e r e n c e a nd c u r r e n tco n f i g u ra t i o n s a re t r a n s fo rm e d by t h e f o l l o w in g t ra n s p o r t f o rm u la e :

nda = JF"TNdA (2 .23a)dv - JdV (2. 23 b)

T he se r e l a t i o n s may be u s ed t o e x p r e s s i n t e g r a l b a l a n c e l a w s ( s e c t .2 .4 ) i n e i t h e r c o n f i g u r a t i o n . E qn . ( 2. 23 a ) c o n d i t i o n s t h e f o r m o f t h eP io la t ra ns fo r m a t i on s fo r the s t r es s tenso r (eqn . 2 .25)

2.3 STRESS

2.3.1 CAUCHY

The co n c e p t o f s t re ss re s t s up on th e C au ch y p o s tu la te t h a t t h eac t io n o f the res t o f the ma te r ia l upon any vo lume e lement o f i t is o ft h e same f o r m as d i s t r i b u t e d s u r f a c e f o r c e s . A t r a c t i o n v e c t o r t ( n )may be d e f i n e d a t e a c h p o i n t , a s t h e f o r c e e x e r t e d p e r u n i ti n f i n i t e s i m a l a r e a , f o r each o r i e n t a t i o n n.

A p p l y i n g e q u i l i b r i u m c o n s i d e r a t i o n s , i t may be d ed uc ed t h a t as t re ss t e n so r a m ust e x i s t , such th a t f o r e ve ry o r i e n t a t i o n n

t ( n ) = n a (2 .2 4 )

a i s ca l le d the Cauchy o r t ru e s t res s tens o r , and i t i s re la te dt o t he c u r r e n t c o n f i g u r a t i o n .

2 .3 .2 PIOLA-KIRCHHOFF

I f b o th t h e f o rc e an d th e area com ponen ts o f the co nc ep t o fs t r e s s are t r a n s f o r m e d back i n t o th e o r i g i n a l c o n f i g u r a t i o n , a news t re ss t e n so r i s o b ta in e d :

S = JF -1 (7 F _ t = J 0 t * ( c r ) (2 .25 )


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T h is re l a t i o n i s ca l l e d t h e b ackw a rd P io l a t r a n s fo r m a t i o n . I t d e f i n e st h e 2nd P i o l a - K i r c h h o f f s t r e s s t e n s o r S, w h i c h i s a s t r e s s m e as ur er e f e r r e d t o t he o r i g i n a l c o n f i g u r a t i o n .

2 . 4 BALANCE LAWS

Balance laws (mass, momentum, angular momentum, and energy) maybe s t a t e d a l t e r n a t i v e l y i n i n t e g r a l f o r m o r as f i e l d e q u a t i o n s .In teg ra l fo rms p rov ide "weaker " exp ress ions fo r the same p r inc ip les .Th is w i l l be commented fu r t h e r i n sec t io n 4 .3 .

2 . 4 . 1 BALANCE O F MASS

C o n s e r v a t i o n o f m ass i m p l i e s t h a t t h e m ass o f t h e m a t e r i a lo c u p y i n g a c e r t a i n r e g i o n V o f t h e bo dy r e m a i n s c o n s t a n t t h r o u g h o u tthe mo t ion :

(d/d t)fp dV = 0 (2.26)where p is themass density.

As a f i e l d e qu a t i o n, b a l a nc e of m a s s is e x p r e s s e d b y t h econtinuity equation:

p+p d i v (v ) = 0 (2.27)

2 . 4 . 2 BALANCE O F MOMENTUM

For a reg ion V of the body w it h boundary S, the i n te g r a l form ofthe equ atio n of l in e a r momentum balance is

(d /d t ) f pvdV = I p fdV + f n a d S (2 .2 8 )V V S

w h e r e f i s t h e bo dy f o r c e p e r u n i t m a s s . The c o r r e s p o n d i n g


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d i f f e re n t i a l e xp re ss io n i s C a uchy 's e q u at i o n o f m o t i o n ,

p v = d i v ( a) + p f (2 .29 )

The i n t e g r a l s i n e q n. ( 2 .2 8 ) i n v o l v e v e c t o r s , a nd as p o i n t e d o u tby Marsden and Hughes (1978), may not p ro vi de a c o v a ri a n t sta tem en t ofthe momentum ba lance p r in c ip le in a genera l m an i fo ld . However, f o r theEuc l idean space to wh ich th is e xp os i t io n r e fe rs , the ob jec t io n i s no tr e l e v a n t . The i n t e g r a l e x p r e s s i o n i s b e t t e r s u i t e d f o r f i n i t ed i f fe re nc e numer ica l models (se c t ion 4 .3 ) , fo r wh ich weak v a r i a t i o na lg loba l expres sions (as employed in F in i t e E lements) are no t ob ta ined .

T a k i n g m om en ts i n e q n . ( 2 . 28 ) w i t h r e s p e c t t o t h e o r i g i n , t h ebalance of angular momentum is expressed by

(d /d t ) / p ( xX v )d V = f p (x X f )d V + / xX (n c7 )d S (2 .3 0 )

w h e r e x X vd e n o t e s v e c t o r p r o d u c t of x and v.T h e c o r r e s p o n d i n g f i e l de q u a t i o n s t a t e s s i m p l y t h e s y m m e t r y of O":

G J ( 2 . 3 1 )S y m m e t r y of Sm a ybed e d u c e d f r o m e q n s . ( 2 . 2 5 )and( 2 . 3 1 ) .

2 . 4 . 3 B A L A N C E OFE N E R G YIna c o n t i n u u m , t h e f i r s t law oft h e r m o d y n a m i c s may bee x p r e s s e d

as( d / d t ) J p u d V = /( p q + a : d ) d V + f h n d S ( 2 . 3 2 ;

Vw h e r eu ist h e i n t e r n a l e n e r g y p e r u n i t m a s s

q ist h e r a t eofb o d y h e a t s u p p l y p e r u n i t m a s shist h e h e a t f l u x v e c t o r ; f o r ano r i e n t e d i n f i n i t e s i m a l a r e a

t h e h e a t f l o w r a t e isg i v e nby H =h n d ST h e c o r r e s p o n d i n g f i e l d e q u a t i o nis


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p u = aid + p q + d i v ( h ) ( 2 . 3 3 )

The term aid r e p r e s e n t s t h e s t r e s s w o rk p e r u n i t v o l u m e a ndt i m e , a and d are sa id to be co n jug ate st r es s and s t r a in measures. Ana l t e r n a t i v e r e p r e s e n t a t io n o f t h e e n e rg y b a l a n c e p r i n c i p l e i n v o l v e sthe use of S and E, a ls o c on ju ga te :

p 0 u = S:E + p 0 q + & - d i v ( h ) ( 2 . 3 4 )

2.5 CONSTITUTIVE RELATIONSThe b a la n ce l a w s p ro v i d e a se t o f e q u a t i o n s w h i ch a re n o t

s u f f i c ie n t to de te rm ine the behav iou r of a m a te r ia l body. Some fu r t h e re q u a t i o n s a r e n e c e s s a r y , s t a t i n g t h e r e l a t i o n be tw ee n k i n e m a t i c andd yn a m ic va r i a b le s ( co n s t i t u t i ve e q u a t i o n s ) .

C on st i tu t i v e equa t ions a re based on judgemen t , a - p r i o r i know ledgeof how the m at er ia l behaves. However, ce r t a in genera l p r i n c ip le s mustbe s a t i s f i e d i n t h e i r f o rm u la t i o n . For o ur p u rp ose , t h e mos t im p o r ta n tp r i n c i p l e i s t h a t o f o b j e c t i v i t y , wh ic h s t a t e s t h a t c o n s t i t u t i v eequa t ions must be in v a r i a n t under changes o f re fe renc e f ra m e, in ord erto re p re se n t t h e m a te r i a l b e h a v io u r o b j e c t i v e l y .

For a homogeneous m a te ri a l i t may be seen ( B i l l in g t o n and Ta te ,1 9 8 1 ), t h a t an o b j e c t i v e r e l a t i o n b e tw e e n C au ch y s t r e s s anddeformat ion takes the form:

a = R p ( C t ( s ) , T t ( s ) ) RT (2 .35 )

where R is the ro ta t i o n te nso r (eqn. 2 .11) and T the tem pe rat ure . Then o t a t i o n C t ( s ) s i g n i f i e s t h e h i s t o r y o f C ( e qn . 2 .1 2) f r o m - o < s < t .Note th a t in genera l the com ple te h i s to ry o f the de for m at io n C (or o fE e q u i v a l e n t l y , e q n . ( 2 . 1 7 ) ) a r e r e q u i r e d , w h i l e f o r R o n l y t h ein s ta n ta n e o u s cu r re n t va lu e i s u se d , f o r r o t a t i n g t h e s t re sse s .

The second P io la -K i rc hh o f f s t re ss S is o b j ec t i v e as such (be ingr e l a t e d t o a f i x e d r e f e r e n c e c o n f i g u r a t i o n ) . I n t e r m s o f i t e qn .


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(2.35) may be rephrased asS - B ( C t ( s ) , T t ( s ) ) ( 2 .3 6 )

The advan tage o f us in g the 2nd P io la -K i rc hh o f f s t re ss tenso r f o r to ta lf o rm u la t i o n s i s e v id e n t (see a l so se c t . 3 . 5 . 1 ) .

Some t y p e s o f E l a s t i c a nd P l a s t i c r a t e e q u a t i o n s a r e d i s c u s s e db e lo w . F or s im p l i c i t y , a t t e n t i o n i s ce n t re d on th e i so th e rm a l ca se .

2 .5 .1 RATE EQUATIONS

Materia ls without memory or with smooth memory may be describedw i th ra te e q u a t i o n s , e .g .

a= g ( d , a , F ) ( 2 .3 7 )

w he re a i s a s t re s s ra te w h i ch i s o b je c t i v e f o r r i g i d body ro ta t i o n s .The c h o i c e o f o b j e c t i v e r a t e i s n o t u n i q u e . A v a r i e t y o f o p t i o n s a r eavai lable, the two most widely used being the Jaumann rate

a = a + aw + wTa (2 .3 8)

and the T rue sde l l ra te ,

G =


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S = g.(E,E,S) (2 .4 0)

S i s t he m a te r i a l t im e ra te o f a t e n so r on t h e cu r re n t co n f i g u ra t i o n(2nd P i o l a - K i r c h h o f f ) , w h ich i s a l re a d y o b je c t i ve .

2.5.2 ELASTICITY

E l a s t i c m a t e r i a l s a re t h os e f o r w hi ch a n a t u r a l , s t r e s s - f r e es ta te ex is ts , to wh ich the body re tu rns upon removal o f a l l ex t e rn a lfo rc es . The s t re ss depends on the de fo rma t ion f rom th is na t u ra l s ta te :

S = f ( C , t ) ( 2 . 41 )

A pe r fe ct memory o f the natu ra l s ta te , w i th no memory o f in te rm ed ia tes t a t e s , i s e x h i b i t e d .

F or l i n e a r e l a s t i c i t y and s m a l l s t r a i n s th e r e l a t i o n i s asfo l 1 o w s :

o = c:e (2 .42 )

(in component form er1J= c ^ - j e *' )

c istermed theelasticity tensor,and e is thesmall strain te nso r:

e i j= (

ui,j

+ uj , i >

/ 2 (

2'43>

where u a re d isp lacem en ts . For i s o t r o p i c m a te r i a ls , and p rov ided aand e are both sym me tric, c must tak e the form

c i j k l " X i j 5 k l + 2 G 5 i k S j l ( 2 .4 4 )w h e re X a nd G a r e c a l l e d La m e's c o n s t a n t s . T h i s g i v e s r i s e t o t h ec la ss ic gene ra l i ze d Hooke 's la w :

ffij = ^ k k 5 i j + 2 G e i j ( 2 - 4 5 )


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2 . 5 . 2 . 1 HYPERELASTIC MATERIALS

The concept o f H y p e re la s t i c i t y was in t ro du ce d by Green and g iveni t s p re se n t nam e by T ru e s d e l l ( e .g . T ru e s d e l l a n d T o u p in , 19 6 0 ) . I tp o s tu l a te s t h e e x i s te n ce o f a s t ra i n -e n e r g y f u n c t i o n f r o m w h ich t h estresses may be derived as

S = ft, ( d W /d E ) (2 .4 6 )

A ss um in g t he n ec es sa ry d i f f e r e n t i a b i 1 i t y , t h e e l a s t i c i t y t e n s o r i sdef ined as

D = P0 (d^/d^) (2 .47)and arate equationmay bewrittenas

S = D:E (2.48)

(i n component form S = D \ \ E )

F or a c o n s t a n t v a l u e o f D, a l i n e a r h y p e r e l a s t i c t o t a l e q u a t i o ni s o b ta in e d :

S = D:E (2 .49 )

2.5.2.2 HYPOELASTIC MATERIALS

The te rm Hypo e las t i c , a lso in t rod uce d by T rues de l l (T rues de l l andT o u p i n , 1 9 6 0 ) , c h a r a c t e r i z e s a m a t e r i a l f o r w h i c h t h e b e h a v i o u r i sd e f i n e d i n t h e c u r r e n t c o n f i g u r a t i o n by an i n c r e m e n t a l l y l i n e a rre l a t i o n sh ip o f t h e f o rm :

0= c:d (2 .5 0)( in component form & 1 J = c1Jk-]d )

An o b je c t i v e s t re ss ra te m u s t be u se d fo r e q n . (2 .5 0) ( see s e c t i o n


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2 . 5 . 1 ) .

Hyp oe las t i c behav iou r i s ve ry conven ien t fo r de sc r ip t io ns basedon t h e c u r r e n t c o n f i g u r a t i o n . M a t e r i a l d a t a b as ed on t r u e s t r e s s -n a tu ra l s t ra i n re l a t i o n sh ip s (see se c t i o n 6.1 ) g i ve r i s e n a tu ra l l y t oh y p o e l a s t i c i n t e r p r e t a t i o n s .

Fo r i so t rop ic ma te r ia ls eqn . (2 .50 ) takes the fo rm

a = \ t r ( d ) I + 2Gd ( 2 .5 1 )

( in orthonormal components o^- = A d ^ 8 + 2Gd^ )

2 .5 .3 PLASTICITYFor most s o l i d s , behaviour may be assumed e la s t i c on ly w i t h in a

c e r t a i n s t r e s s r an ge . B ey on d t h e e l a s t i c r a ng e y i e l d o c c u r s ,d e fo rm a t i o n s b e in g ch a ra c te r i z e d by p erm a ne nt cha ng es o ccas io n e d bys l i p o r d i s l o c a t i o n s a t t h e a to m ic l e v e l ( P l a s t i c f l o w ) .

A f t e r y i e l d , E l a s t i c a nd P l a s t i c d e f o r m a t i o n s a r e a ss um ed t ohappen c o n c u r r e n t l y ( E l a s t i c - P l a s t i c m a t e r i a l s ) . M ore r e s t r i c t i v ei d e a l i z a t i o n s a re p r o v i d e d by r i g i d - p l a s t i c m od els ( o n l y p l a s t i cde fo rm a t ions ) . An ad d i t i v e decom pos i t ion of the ra te of de fo rm a t ion i sassumed here:

d = d e + dP (2. 52 )w h e re s u p e r i n d i c e s e and p i n d i c a t e e l a s t i c and p l a s t i c c o m p on e nt sr e s p e c t i v e l y . A d d i t i v e d e c o m p o s i t i o n of s t r a i n s i n t h i s f a sh i on wasp r o po s e d by H i l l ( 1 9 50 ) . Lee ( 19 6 9) has pr o po s e d a m u l t i p l i c a t i v edecom pos i t ion o f de fo rma t ion g rad ie n ts in s t ea d , F = Fepp , w h i le Greena nd N a gh di ( 1 9 65 ) h av e a d v o c a t e d an a d d i t i v e d e c o m p o s i t i o n o f t o t a ls t r a i n , E = Ee + EP.

C l a s s i c a l p l a s t i c i t y i s f o r m u l a t e d in t e rm s o f t h e c u r r e n tc o n f i g u r a t i o n ( H i l l , 1 9 5 0 ). H ence t h e p o p u l a r i t y o f an a d d i t i v edecompos i t ion o f the ra tes o f de fo rma t ion , eqn . (2 .52 ) , coup led w i th


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h y p o e l a s t i c b e h a v io u r , f o r E l a s t i c - P l a s t i c m a t e r i a l d e s c r i p t i o n s ( e .g .Wi 1k ins (1964), H i b b i t , Marca l and Rice (1970)). In th is case


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2 . 5 . 3 . 1 VON MISES MODEL

P a r t i c u la r l y u s e fu ] and s im p le m od els are d e r i ve d f ro m th e y i e l dc r i t e r i o n o f Von Mises (191 3). Th is may be w r i t t e n as

F = (3 /2 )s :s - Y2 = 0 (2 .58 )

wheres are thedeviatoric Cauchy stresses,

s= a -(l/3)tr( a)I (2.59a)( i n or thonormal components, s ^ = a^- - ( 1 /3 ) o - ^ 5 . . ( 2 .5 9 b )

Y i s t h e y i e l d s t r e n g t h o f t h e m a t e r i a l , w h i c h c o i n c i d e s w i t h t h ey i e l d s t r e s s i n u n i a x i a l t e n s i o n (s ee s e c t i o n 6 . 1 ) . The Von M i s e sy i e l d c o n d i t i o n i s i n de p e n de n t o f v o l u m e t r i c s t r e s s e s , w h i c h a r eassumed to behave e la s t i c a l l y .

An is o tr o p ic harden ing model is obta ined by making Y a fu nc t io no f t he e f f e c t i v e p l a s t i c s t r a i n , e P :

Y = Y(e p ) (2 .60a)w i t h eP = J 6 e p = J A 2/3) dp: dP d t (2 .60 b)

A more general ha rden ing mode l , in co rp o r a t in g Bausch inge r e f f e c t ,may be ob ta ined by comb in ing i s o t r o p i c ha rden ing w i th the k inem at ich a r d e n i n g p r op o s e d by P r a g e r ( 1 9 56 ) a nd Z i e g l e r ( 1 9 5 9 ) , g i v i n g t h ey i e l d c o n d i t i o n

F = ( 3 / 2 ) ( s - a ) : ( s - a ) - Y2 ( 2.6 1)o: i s c a l l e d t h e b a c k - s t r e s s and r e p r e s e n t s a k i n e m a t i c h a r d e n i n gp a ra m e te r ( t r a n s la t i o n o f t h e Von M ises c i r c l e ) . The a s so c ia t i ve f l o wru1e i s

dP = 7 ( s - a ) ( 2. 62 )


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a n d theh a r d e n i n g l a w s ,

a= (2/3)hadP ( 2 . 6 3 a )Y = h yeP ( 2 . 6 3 b )

I m p o s i n g t h e c o n s i s t e n c y c o n d i t i o n (F = 0 ) d u r i n g l o a d i n g , andc o m b i n i n g e qn s . ( 2 . 5 1 ), (2 . 5 3 ) , ( 2 . 6 1 ) - ( 2 . 6 3 ) , t h e s t r e s s - s t r a i nre la t ion i s found to be

a =c : L d - ( s -a ) ( 3 / 2 ) d : ( s - a ) / Y 2 ( i + h ' / 3 G ) J ( 2. 64 )whe re h ' = h y + n a ( p l a s t i c m o d u l u s ). P u r e l y i s o t r o p i c h a rd e n i n g i sob ta ined w i th h a =0 , and pu re ly k ine ma t ic w i th h Y=o.

In a u n i a x i a l t e s t , l a w ( 2.6 4) w i l l p r o v i d e an E l a s t o p l a s t i chardening modulus of

h = 1/L1/E + 1 /h ' J (2 .6 5 )

w h e r eE -G ( 3 X + 2 G ) / ( X + G ) ( Y o u n g ' s m o d u l u sofe l a s t i c i t y ) .

2.5.3.2 O T H E R P L A S T I C I T Y M O D E L S

P l a s t i c i t y i n s o i l s i s g e n e r a l l y c o n s i d e ra b l y more c o m p l ic a t e dthan the above Von Mises model. Pressure dependent y i e l d , a n i s o t r o p y ,d i l a t a t i o n and n o n - a s s o c i a t i v i t y , h y s t e r e t i c c y c l i c be ha vio ur , porep r e s s u r e , a re i m p o r t a n t f e a t u r e s f o r s o i l p l a s t i c i t y . An e x c e l l e n trev iew of th is to p ic has been given by M ar t i and Cun dal l (1980).

The m a t h e m a t ic a l t h e o r y o f p l a s t i c i t y i s a f i e l d s t i l l u nd erdevelopment. Very ref ined phenomenological models have been proposed(e.g. Mroz (19 67 ) , P re vo st (19 78 ) ) . These mod e ls are based onm u l t i p l e - s u r f a c e i d e a l i z a t i o n s . They p ro v id e e la b o ra t e s t r e s s - s t r a i nla ws r e q u i r i n g c o n s i d e r a b l e c o m p u t a t i o n a l c o s t f o r n u m e r i c a lm o d e l l i n g , t h u s i n p r a c t i c e t h e y a r e h a r d l y u s e d . T h i s f a c t has bee na ckn o w le d g e d by O r t i z a n d P op ov (1 9 8 3 ) , who p ro p o se s im p le r , o n e -su r fa ce m odels f o r m etal p l a s t i c i t y .


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F i n a l l y , a p r i n c i p l e t o be h e l d p r e s e n t w hen c h o o s i n g ap l a s t i c i t y m o d e l , i s t h a t i t c an o n l y be as r e l i a b l e a nd ass o p h i s t i c a t e d as t h e e x p e r i m e n t a l i n f o r m a t i o n on w h i c h t h edeterminat ion o f the model parameters is based. For example , there isl i t t l e p o i n t i n u s i n g a n y t h i n g o t h e r th a n a Von M i se s i s o t r o p i ch a r d e n i n g m odel i n a m e t a l , i f a l l t h e i n f o r m a t i o n a v a i l a b l e i s au n ia x ia l s t r e ss -s t r a i n l a w . On th e o th e r h a nd , t h e added co m p l i c a t i o no f som e m o d e l s may n o t be n e c e ss a ry i f t h e l o a d i n g i s m a i n l ymono ton ic .


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CHAPTER 3

NONLINEAR NUMERICAL MODELS FO R SOLID MECHANICS

3.1 INTRODUCTION

3.2 FINITE DIFFERENCE METHODS

3.3 FINITE ELEMENT METHODS

3.4 MESH DESCRIPTIONS

3.4.1 Lagrangian3.4.2 Eulerian

3.4.3 Arbitrary Lagran gian-E ulerian

3.5 LARGE DISPLACEMENT FORMULATIONS

3.5.1 Total Lagrangian3.5.2 Cauchy stress -velocity strain3.5.3 Updated Lagrangian

3.6 TIME INTEGRATION

3.6.1 Central difference (explicit)3.6.2 Trapezoid al rule (i mpl ici t)3.6.3 Operator split methods

3.7 PRACTICAL CONSIDERATIONS FORDISCRETE MESHES

3.7.1 "Locking up" forincompressible flow3 . 7 . 2 " H o u r g l a s s i n g "

3.8 CONCLUSIONS


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3.1 INTRODUCTION

The G o v e r n i n g e q u a t i o n s i n s o l i d m e c h a n i c s a r e t h e e q u a t i o n s o fm o t i o n , w h i c h ca n be w r i t t e n i n c om p o n e n t f o r m a s

a i j 5 j + p ( f i - i i i ) = 0 (3 .1 )

where cr^ j - j s t h e C au ch y s t r e s s t e n s o r , p t he m ass d e n s i t y , f - j a re bodyf o r c e s pe r u n i t mass ( t y p i c a l l y g r a v i t y ) , a nd u i d i s p l a c e m e n t s . T he see q u a t i o n s o r i g i n a t e f r o m t h e b a l a n c e o f m om en tu m p r i n c i p l e ( s e c t i o n2 . 4 .2 ) . T h i s p r i n c i p l e may be s t a t e d a l t e r n a t i v e l y i n i n t e g r a l f o r m( e q n . 2 . 2 8 ) .

The p a r t i a l d i f f e r e n t i a l e q n s . o f m o t i o n ( 3 .1 ) d ep en d u po n 3s p ac e a nd 1 t i m e v a r i a b l e s . T he n u m e r i c a l m o d e l s d e s c r i b e d h e r ep e r f o r m in d e p en d e n t s e m i d i s c r e t i z a t i o n s i n sp ac e and t i m e . F i r s t e q n s .( 3 . 1 ) a r e d i s c r e t i z e d i n s p a c e , y i e l d i n g a s y s t e m o f o r d i n a r yd i f f e r e n t i a l e q ns . i n t i m e . T he s e a r e t h e n i n t e g r a t e d w i t h a t i m e -s t e p p i n g p r o c e d u r e .

The d i s c r e t i z a t i o n o f t h e c o n t i n u u m may be a c h i e v e d e i t h e r w i t hF i n i t e E l e m e n t (F E) o r F i n i t e D i f f e r e n c e ( FD ) m e t h o d s . B o t h m e th o d sh av e h ad s e p a r a t e h i s t o r i c a l d e v e l o p m e n t s , a l t h o u g h som e d e g r ee o fc o n v e r g e n c e h a s b ee n r e a c h e d l a t e l y i n t h e l i t e r a t u r e ( e . g .B e l y t s c h k o , 1 9 8 3 ). The t h e o r e t i c a l p r i n c i p l e s f o r bo t h m e th od s a r ed i f f e r e n t : l o c a l t r u n c a t i o n e r r o r s f o r FD, g l o b a l e r r o r n orm s f o r FE.How ever, FE me thods are a l s o ba s ed on i n d e p e n d e n t s h ap e f u n c t i o n s f o re a ch e l e m e n t . As a r e s u l t , FE a nd FD f o r m u l a t i o n s o f t e n p r o d u c ee q u i v a l e n t a l g o r i t h m s ( e . g . K u na r a nd M i n o w a , 1 9 8 1 ) .

O t h e r n u m e r i c a l m e th o d s n ee d o n l y a d i s c r e t i z a t i o n i n t h eb o u n d a r y : t h e B o u n d a r y E l e m e n t M e t h o d s ( B EM ) . T h e s e w e r e f i r s tp r o p o s e d f o r s o l i d m e c h a n i c s by R i z z o ( 1 9 6 7 ) a n d C r u s e ( 1 9 6 9 ) .C o n s i d e r a b l e a d v a n t a g e c a n be g a i n e d by t h e r e d u c t i o n a n ds i m p l i f i c a t i o n of t h e d i s c r e t i z a t i o n . F or n o n l i n e a r p r o b l e m s , ho w e ve r,BEM l o s e m uch o f t h e i r a p p e a l . V o lu m e i n t e g r a l s a p p e a r w h i c h r e q u i r ean a d d i t i o n a l d i s c r e t i z a t i o n o f t h e c o n t i n u u m ( e .g . G a r c i a , 1 98 1) . F ort h i s reason BEM w i l l no t be rev i ew ed he re .


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T im e i n t e g r a t i o n may be p e r f o r m e d e i t h e r by m od al a n a l y s i sm eth ods o r by d i r e c t i n t e g r a t i o n ( t i m e - m a r c h i n g ) . M odal a n a l y s i sr e q u i r e s t r a n s f o r m a t i o n s i n t o t h e f r e q u e n c y do m a in w h i c h a r e o n l yv a l i d in a l i n e a r re g im e, f o r w h ich reason they must be ru le d out f o rnon l inea r mode ls . As to t ime-m arc h ing p rocedures two ma in a l te rn a t i v e se x i s t , e x p l i c i t or i m p l i c i t m e th o d s . B o th ha ve a d v a nt a ge s andd isadvan tages , wh ich w i l l be rev iewed b r i e f l y in th is chap te r . Recen ta l t e rn a t i v e p roce d ure s b ased on o p e ra to r s p l i t t i n g m e th ods w i l l a l sob e co n s id e re d .

3 .2 FIN IT E DIFFEREN CE METHODS

F i n i t e D i f f e re n c e m e th o d s ha ve b ee n u se d f o r a l o n g t im e byeng ineers w i t h in re la xa t io n p rocedures (e .g. So u th we l l , 1940). F in i t eD i f f e r e n c e o p e r a t o r s p r o v i d e l o c a l a p p r o x i m a t i o n s f o r a s y s t em o fc o u p l e d d i f f e r e n t i a l e q u a t i o n s . Due t o t h i s f a c t , a o n e - s t e p g l o b a ls o lu t i o n is not pos sib le and recourse must be made to r e l a x at io n andi t e r a t i v e t e c h n iq u e s . A d d i t i o n a l l y , FD m e th o ds ha ve been a s s o c i a t e dn o r m a l l y w i t h r e g u l a r z o n in g ( at l e a s t t o p o l o g i c a l l y r e g u l a r ) . Forthes e two re aso ns, FD methods were ec l ip s ed by the F i n i t e Element boomin the 1960 's fo r s t r u c tu ra l and s o l i d mechanics app l i c a t io n s . FD hasa l w a y s be en p o p u l a r , h o w e v e r , i n o t h e r a re a s s u ch as E u l e r i a n f l u i dmechan ics (e .g . N ic ho ls , H i r t , and H otc hk iss , 1980).

F or a r e g u l a r m esh w i t h " I " a nd " J " l i n e s a l o n g t h e t w oc o o r d i n a te d i r e c t i o n s , s ta n d ar d f i n i t e d i f f e r e n c e a p p r o x im a t i o n s f o rthe gra d ien t o f a ve cto r u are g iven by :

u I + l / 2 , J + l / 2 a J _ ( u I + l , J + l / 2 _ u I , J + l / 2 )u | : i / 2 , J + l / 2 = 1 _ ( u I + l / 2 , J + l _ u j + l / 2 , J )

Ax.E qn s. ( 3 . 2 ) r e q u i r e t h e m esh t o b e t o p o l o g i c a l l yr e g u l a r .

The use o f con tou r i n t eg ra l fo rmu las (W i lk in s , 1964) a l low s thea p p l i c a t i o n o f FD a p p r o x im a t i o n s t o t o p o l o g i c a l l y a nd g e o m e t r i c a l l yir r e g u l a r meshes. The bas ic idea i s to employ Gauss' theorem in orde r

(3 .2 )

a n d g e o m e t r i c a l l y


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t o e x p r e s s t h e g r a d i e n t o f a f i e l d i n a c e l l i n t e r m s o f a c o n t o u rin te g r a l . Co nside r ing ce l l VE enclosed by con tour SE, and the gra d ie nto f the d isp lacemen t vec to r u ,

J ui5JdV = [ u^jdS (3.3)If the gradient is assumed constant in thecell,then1f

u i,j= - I u^ jd S (3.4)V E J S E

The con tou r in te g r a l may be eva lua ted assuming a l i ne a r v a r i a t i o n o f ualong the edge of the c e l l .

Con tou r i n t e g r a l s may be used f o r any 2 -0 po ly gon o r 3 -Dp o l y h e d r o n , t o i n t e r p o l a t e a v a l u e f o r t h e g r a d i e n t a t t h e c e n t r e oth e c e l l , k n o w i n g t h e v a l u e s a t t h e c o r n e r n od e s . F or t h e p a r t i c u l a rc as es o f t r i a n g l e s and t e t r a h e d r a , an a l t e r n a t i v e t e c h n i q u e i sa v a i l a b le (e.g. M a r t i , 1 98 1) , i n w h ich t h e g ra d ie n ts a re i n te rp o la te dd i r e c t l y by i n v e r t i n g t h e s p a t i a l f i n i t e d i f f e r e n c e e q u a t i on s . T h istechn ique has been fo l lo w ed in the present work, and w i l l be de ta i le di n se c t i o n 4 . 2 . 1 .

F or e x p l i c i t t im e -m a rch in g m o de ls t h e se m i -d i s c re te e q u a ti o n s o fmot ion become unco up led. Th is means th a t on ly lo ca l app rox ima t ions tot h e 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 (3 .1 ) a r e p e r f o r m e d w i t h i n eacht i m e - s t e p , no i t e r a t i o n s b e i n g n ee de d f o r a FD o p e r a t o r . Such a f a c twas ex p l o i te d in the deve lopment o f the f i r s t FD "Hydrocodes" a t theU.S. n a t i o n a l l a b o r a t o r i e s i n t h e 1 9 5 0' s . T he se w e r e o r i e n t e d m a i n l yt ow a rd s s e n s i t i v e n u c l ea r and d efe nc e a p p l i c a t i o n s . L i t t l e p u b l i c i t ywas giv en u n t i l the 1960's (W il k in s (1964 ), Maenchen and Sack (19 64),Noh (1 9 6 4 ) ) . A t t h i s t i m e F i n i t e E le m e n t M e th o d s h ad j u s t b ee ni n t r o d u c e d f o r s o l i d m e c h a n ic s ( C l o u g h , 1 9 6 0 ) , a nd t e c h n i q u e s w e r ebe ing deve loped fo r l i n e a r a na ly si s. Not much a t t e n t i o n was g iven toFD f o r s o l i d m e c h a n ic s by t h e e n g i n e e r i n g c o m m u n i t y , as in d e e d FEmethods seemed much more po w er fu l and indee d advantageous fo r l i n e a rsys tems, be ing ab le to p rov ide a one -s tep g loba l s o lu t i o n .

I n t e r e s t i n t h e n o n l i n e a r and w a v e - p r o p a g a t i o n r e g im e s f o rs p e c i a l i z e d e n g i n e e r i n g a p p l i c a t i o n s i n t h e l a t e 1 9 60 's and 1 97 0's


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c re a te d a res u rg en ce o f the H ydrocodes ( Be r th o l f and Benz ley (1968) ,W ilk in s (1975)), and a c e r t a in degree of convergence between FD and FEl i t e r a t u r e ( B e l y t s c h k o ( 1 9 7 8 ) , K r i e g and Key ( 1 9 7 6 ) , G ou dre au andH a l l q u i s t ( 19 82 )) . E x p l i c i t f i n i t e - d i f f e r e n c e m ethods w ere p o p u l a ri z e dto w i d e r se c t o r s o f t h e e n g i n e e r i n g co m m u n i t y , a nd new co d es w e recr e a te d such as P ISCES (Hancoc k, 19 76) , th e rock me chan ic s codes o fCundal l and M ar t i (1979), and PR3D fo r s o l i d mechanics impa ct by M ar t i(1981).

3 . 3 FI N IT E ELEMENT METHODS

The f i r s t a p p l i ca t i o n o f F i n i t e E lem e nt t e ch n iq u e s fo r co n t i n u aw as by C lo u gh (1 9 6 U ), a l t h o u g h th e t h e o r e t i c a l b a se s f o r t h e m e th o dhad a l read y been set by Courant (1943) and ap p l ic a t io ns to s t r u c tu ra lan a ly sis had been proposed e a r l i e r (A rgy r is and Ke lse y, 1954).

F i n i t e E le men t d i s c r e t i z a t i o n s r e l y on t w o e s s e n t i a l i n g r e d i e n t s :a v a r i a t i o n a l o r w eak fo rm o f t h e e q n s . o f m o t i o n (3 .1 ) , a nd ac o n s t r u c t i o n o f a p p r o x i m a t e s o l u t i o n s b as ed on g e n e r a l i z e d n od alco ord ina tes and independent e lement shape fu n c t i on s .

The d o m a in V i s s u b d i v i d e d i n t o e l e m e n t s VE, i n t e r c o n n e c t e d bynodes. An approx ima te s o l u t io n i s co nst ruc ted w i t h in an e lemen t E as ap roduct o f shape fun c t i on s N^ x) and the noda l d isp lacem en ts u f ( t ) :

u ( x , t ) = u ^ ( t ) N j ( x ) ( 3 . 5 )

where I i s summed over the nodes of th e elem en t. The shape f u n c ti o n sare c h o s e n so t h a t u i s c o n t i n u o u s o v e r t h e e l e m e n t b o u n d a r i e s ,a l t h o u g h i t s g r a d i e n t need n o t be c o n t i n u o u s (CO c o n t i n u i t y ) . Thes hap e f u n c t i o n s N i s o d e f i n e d are i n d e p e n d e n t o f t i m e ; e q n . ( 3 .5 )c o n s t i t u t e s i n f a c t a l o c a l s e p a r a t i o n o f v a r i a b l e s(semi d is c re t i za t i on ) .

The d i sc re te fo rm of the gr ad ien t o per a tor may be w r i t t e n asu i ' j " B j l u i l (3 .6 )


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whereBJi

dxiL e t th e s o l i d c o n t i n u u m be V w i t h b o u n d a ry S , c o n s i s t i n g o f Sy

and S j ' w h e r e

u = u * on Su

a n = T* o n s TA weak fo r m o f th e eqns. o f m o t ion (3 .1 ) may be ob ta in ed by us in ge i t h e r G a le rk i n w e ig h ed re s id u a l s o r t h e v i r t u a l w o rk p r i n c i p l e , b o tho f wh ich y ie ld the same r e s u l t :

f V i , j C T i j d V + r p v i u i d V = J p V i f i d V + [ v- jT ^ dS ( 3 . 7 )V V V ST

w he re v i s t h e t e s t f u n c t i o n (o r va r i a t i o n ) and u t h e t r i a l f u n c t i o n .

E qn s. ( 3. 7) r e q u i r e o n l y C c o n t i n u i t y f o r b o t h t r i a l and t e s tf u n c t i o n s , as o pp os ed t o ( 3 . 1 ) , f o r w h i c h C l c o n t i n u i t y i s n ee de d. I fthe app rox ima t ions de f ined in (3 .5 ) are used for u and v, and because( 3.7 ) m us t h o l d f o r a r b i t r a r y v , th e g l o b a l d i s c r e t e e q u a t i o n s arededuced:

Mu + P(u) = R (3 .8 )

w h e re t h e g l o b a l c o e f f i c i e n t m a t r i c e s M , P , R a r e a s s e m b le d f r o min d i v i du a l e lement ma t r i ces th a t take the fo rm :

ME= / pN jN j& j jdV (mass m at r ix ;VE

PE= / B j j ^ j d V ( i n t e r n a l f o r c e s ) ( 3 . 9 ;J VERE= / f-j NjdV +f N IT id S (e x te rn a l f o r c e s )

v Js$


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3 .4 MESH DE SCR IPTIO N S

L e t a p a r t i c l e X o f b od y B be d e f i n e d b y i t s p o s i t i o n a t t = 0( r e f e r e n c e c o n f i g u r a t i o n ) , X . A t t i m e t ( c u r r e n t c o n f i g u r a t i o n ) t h ep o s i t i o n o f t h e p a r t i c l e w i l l b e

x = x ( X , t ) ( 3 . 1 4 )

x arecalled thespatial coord inate s, and X the material coordina tesof X.Eqns. (3.14) describe themotion of 8.

Fort h ed i s c r e t i z a t i o n of Bt h r e e t y p e s ofm e s h e s m a y b eused ,d e p e n d i n g o n t h em o t i o n o f t h eno d e s o f t h em e s h .T h ep o s i t i o n of apoint of themes h, initially coincident with particle X,will begivenby X = X (X,t ).

3.4.1 LAGRANGIAN

Inal a g r a n g i a n d e s c r i p t i o n t h em e s h f o l l o w s t h em o t i o n o f t h ebody,

X (X,t )=x(X,t) (3.15)

A g i v e n node r e m a i n s c o i n c i d e n t w i t h t h e same m a t e r i a l p a r t i c l ethrough out the m ot io n . Each e lement w i l l co nta in the same domain o fm a t e r ia l t hr ou gh ou t t he d e f o r m a t i o n , t h u s e n f o r c i n g i m p l i c i t l y t h ec o n t i n u i t y e q u a t i o n .

M o t i o n o f t h e b o un d ar y does n o t p r e s e n t d i f f i c u l t i e s , as i ta l w a y s c o i n c i d e s w i t h t h e mesh b o u n d a r y. F or a s c a l a r f i e l d g ( X , t ) ,t h e m a t e r i a l t i m e d e r i v a t i v e ( i . e . f o l l o w i n g t h e p a r t i c l e ) c o i n c i d e sw i t h t h e p a r t i a l t i m e d e r i v a t i v e :

dgg = - (3 .1 6 )dtThe on ly d isadvantage o f th is de sc r i p t io n comes f rom the fac t tha t them esh c an becom e e x c e s s i v e l y d i s t o r t e d f o r c e r t a i n p r o b le m s ( e . g .f l u i d s , h i g h v e l o c i t y i m p a c t ) . I n some c a s e s , " r e z o n i n g " t e c h n i q u e s


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W i t h an ALE d e s c r i p t i o n p r o p e r t i e s s t i l l need t o be f l u x e d t h r o u g hc e l l s , and some sm earing may occ ur as a r e s u l t .

A c r u c i a l a s p ec t i n ALE d e s c r i p t i o n s i s th e d e f i n i t i o n o f t h ea r b i t r a r y m o t i o n o f t h e m esh X f o r i n t e r n a l p o i n t s . G e n e r a l l y acomplex rezo n ing a lg or i t h m is necessary fo r op t i m iz in g the new meshp o s i t i o n s a t ea ch s t e p . S uc h a g e n e r a l r e z o n i n g a l g o r i t h m has be enp ro p o se d fo r 2 -D b y G iu l i a n i ( 1 9 8 2 ) . S c h re u rs (1 9 83 ) h as p ro p o se d amesh o p t i m i z i n g a l g o r i t h m b as ed on t h e d e f o r m a t i o n o f a f i c t i t i o u sm a t e r i a l f r o m an " i d e a l " m e sh . In f a c t ALE t e c h n i q u e s w o u l d beequ iva len t to Lag rang ian descr ip t ions in wh ich rezon ing i s pe r fo rmeda t e ve ry s te p . Some a p p l i c a t i o n s (e .g . m e ta l f o rm in g ) may n o t ne eds uc h f r e q u e n t r e z o n i n g , an d L a g r a n g i a n t e c h n i q u e s w i t h r e z o n i n g a tw i de r i n t e r v a l s c o u l d be p r e f e r r a b l e .

3 . 5 LARGE DISPLACEM EN T FORMULATION S

S e v e r a l f o r m u l a t i o n s are p o s s i b l e d e p e n d i n g on w h i c hc o n f i g u r a t i o n s t h e s t r e s s a nd d e f o r m a t i o n t e n s o r s are r e f e r r e d t o .Three a l te rnat ives wide ly used in so l id mechan ics are presented be low.

3.5.1 TOTAL LAGRANGIAN

The 2nd P i o l a - K i r c h h o f f s t r e s s t e n s o r S and t h e G ree n s t r a i nt e n s o r E, b o th o f w h i c h r e l a t e t o 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 , a reu s ed t o d e s c r i b e t h e m a t e r i a l b e h a v i o u r . H i b b i t , M a r c a l and R i c e(1970) p ro po se d t h i s d e s c r i p t i o n i n t h e f i r s t p u b l is h e d l a r g e - s t r a i n ,la rge -d isp lace me n t non l inea r fo rm u l a t io n fo r general pu rpose FE codes.

A c o n s t i t u t i v e r e l a t i o n i s g iv e n by

S = S(E) (3 .20 )


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w he re C i s t h e c o n s t i t u t i v e t e n s o r . For an e l a s t i c - p l a s t i c m a t e r i a '( h y p o e l a s t i c w i t h a s s o c ia t e d p l a s t i c i t y ) C t ak e s t h e fo rm

C i j k l = ^ i j 5 k i + 2G (5 i k 5 J 1 - ^ i j n k i ) (3 .24 )where rj >0 f o r p l a s t i c l o a d in g , =0 o th e rw i se

n is the u n i t norma l to the y ie ld su r face

The J au m an n d e r i v a t i v e u s e d i n e q n . ( 3 . 2 3 ) p r o v i d e s t h ec o n s t i t u t i v e p a r t of t h e s t re ss r a te . To o b ta in t h e t o ta l s t re ss ra tethe ro ta t i o n a l components must be added:

a = o- + w n + w- a 1J 1J W1PCTPJ WJP pi ( 3 . 2 5 )

I f t h e m a t e r i a l b e h a v i o u r i s a n i s o t r o p i c , C m u st be u p d a t ed w i t h as i m i l a r o b j e c t i v e r a t e :

c i j k l _ c i j k l + w i p c p j k l + w j p c i p k l + w k p c i j p l + w l p c i j k p (3 .2 6 )

An a l t e rn a t i v e f o rm u la t i o n re s u l t s f r o m th e use o f t h e T ru e sd e l lra te in eqn . (3 .23 ) :

( J = C : d :3.27

(3 .23) and (3 .27) are e q u i v a l e n t i f on e se ts

C i j k l = C i j k l + CTij5kl-^ik5jl+oril5jk+(7jk5il + CT il5ik)/2 (3- 2 8)The T r u e s d e l l s t r e s s r a t e i s t h e f o r w a r d P i o l a t r a n s f o r m a t i o n

( e q n s . 2 . 22 , 2 .2 5) o f t h e r a t e of t h e 2nd P i o l a - K i r c h h o f f s t r e s st e n s o r :

O = 0 t * ( J - 1 S ) [ 3 . 2 9 )

Pins ky, O rt iz and P is te r (1983) have suggested th at the Tru es de l lr a t e f o r m u l a t i o n i s t h e n a t u r a l one t o u s e ( i n t h e c u r r e n tc o n f i g u r a t i o n ) f o r h y p e r e l a s t i c i t y . I n t h i s ca se t he c o n s t i t u t i v ete n so r i s o b ta in e d d i r e c t l y , f r o m th e t o ta l L a g ra n g ia n te n so r D, as


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= 0 t * ( j - l D ) .

C l a s s i c a l p l a s t i c i t y i s d e s c r i b e d on t he c u r r e n t c o n f i g u r a t i o n( H i l l , 1950). Jaumann Cauchy st re ss fo rm u la t i o n s have been w id el y ands u c c e s s f u l l y u se d f o r e l a s t i c - p l a s t i c b e h a v io u r ( W i l k i n s ( 1 9 64 ) ,M a e n c h e n a n d S ac k ( 1 9 6 4 ) , K r i e g a n d K ey ( 1 9 7 6 ) ) . W i t h s u c hf o r m u l a t i o n s h y p e r e l a s t i c b e h a v i o u r ( r e l a t e d t o t he o r i g i n a lco n f ig u r a t io n ) may a lso be des cr ib ed , a lb e i t in a less conven ien t way,as t h e c o n s t i t u t i v e r e l a t i o n s need t o be p us he d f o r w a r d i n t o t h ec u r r e n t c o n f i g u r a t i o n .

F i n a l l y , one p r o b le m w i t h t h i s f o r m u l a t i o n i s t h a t d i s n oti n t e g r a b l e ( i . e . i t i s n o t t h e r a t e o f any v a l i d s t r a i n t e n s o r ) .Ad d i t io na l s t r a in com pu ta t ions must be done i f a to ta l s t r a in measureis requ i re d .

3.5.3 UPDATED LAGRANGIAN

I n t h i s f o r m u l a t i o n t h e mo de l i s d e s c r i b e d on a r e f e r e n c ec o n f ig u r a t i o n , wh ich i s upda ted a t each incremen t to co in c id e w i th thec u r r e n t c o n f i g u r a t i o n . From t h i s u p d at ed r e f e re n c e , t he in c re m e n ta lc o n f i g u r a t i o n i s d e s c r ib e d w i t h a t o t a l L a g ra n gi an f o r m u l a t i o n . T h ismethod was f i r s t p ro po se d by Yaghmai and Popov (197 1) , and has beenw i d e l y u se d s i n c e f o r i n c r e m e n t a l n o n l i n e a r a n a l y s i s : O s i a s andSwedlow ( 1974 ), B athe et a l . (197 5), Nagtegaal and de Jong (1981).

F or t h i s d e s c r i p t i o n , F = I ( i d e n t i t y ) an d J = 1 . H en ce , e q n .( 3 .2 9 ) i m p l i e s S = a. I t i s a l s o easy to see f ro m eqn . (2 .19 ) th a t E =d . I n f a c t t h i s f o r m u l a t i o n r e v e r t s t o t h e T r u e s d e l l C auchy s t r e s sr a t e f o r m u l a t i o n , e q n . ( 3 . 2 7 ) . T h i s m eans t h a t t h e t e n s o r t o be u s edf o r t h e t a n g e n t i a l s t i f f n e s s i s .

3 .6 T I M E I N T E G R A T I O N

Using e i t h e r F in i t e D i f fe ren ce o r F in i t e E lement Me thods fo r thes p a t i a l s e m i d i s c r e t i z a t i o n , t he p a r t i a l d i f f e r e n t i a l e qns. o f m o tio n( 3.1 ) may be t r a n s f o r m e d i n t o a s y st em of o r d i n a r y d i f f e r e n t i a l


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U s i n g e q n . ( 3.3 1 b) an d l e t t i n g On = ( u n - l / 2 + u n+l / 2 ) / 2 , e q n . ( 3 .3 2 ) mayb e s o l v e d , y i e l d i n g :

u " + l / 2 = ( M M t - C / 2 ) -1 [ M M t - C / 2 ) u n - 1 / 2 + R n - P ( u n ) ] (3 .33)T he new d i s p l a c e m e n t s u n+ 1 a r e t h e n f o u n d f r o m e q n . ( 3 . 3 1 a ) . I f t h esys t em has no dam p i ng (C = 0 ) and t h e m ass m a t r i x M i s d i a g o n a l , eqns .( 3 . 3 3 ) b e co m e u n c o u p l e d :

n + l / 2 = - n - l / 2 + M - l [ R n _ p ( u % t ( 3 . 3 4 )T h es e e q n s . c a n t h e n be s o l v e d i n d e p e n d e n t l y f o r e ac h d e g r e e o ff reedom I :

n + l / 2 = G n - l / 2 + A ( R n _ p n ) / l t ) i ( 3 > 3 5 )

T he eq ua t i on s a l so becom e unc oup l ed i f t he dam p i ng i s assum ed t obe o f t he Ray l e i g h t y p e , as shown i n s e c t i o n 4 . 6 .

T h i s u n c o u p l i n g o f t h e e q u a t i o n s o f m o t i o n i s t h e m a j o r a d v a n t a g eo f e x p l i c i t i n t e g r a t i o n p r o c e d u r e s . No m ass o r s t i f f n e s s m a t r i c e s n ee dbe i n v e r t e d o r e ve n a s s e m b l e d , as a l l t h e i n c r e m e n t a l c a l c u l a t i o n s f o re a ch d e g r e e o f fr e e d o m c an be d on e i n d e p e n d e n t l y a t t h e l o c a l l e v e l .T h i s n ot o n l y a l l o w s f o r a s i m p l e r a r c h i t e c t u r e i n c o m p u t e r c o d e s , b u ti t e n a b le s t h e t r e a t m e n t o f n o n - l i n e a r i t i e s (b e i t o f C o n s t i t u t i v e ,G e o m e t r i c o r B o u n d ar y t y p e ) w i t h v i r t u a l l y no a d de d c o s t f r o m t h el i n e a r c a s e . The n u m be r o f o p e r a t i o n s p e r t i m e - s t e p i s m uch s m a l l e rt h a n f o r i m p l i c i t m e th o d s ( s e c t i o n 3 . 6 . 2 ) , a nd s t o r a g e r e q u i r e m e n t sg ro w o n l y l i n e a r l y w i t h t h e s i z e o f t h e p r o b l e m .

The m a in d i s a d v a n t a g e o f t h e c e n t r a l d i f f e r e n c e an d o t h e re x p l i c i t m ethods i s t h a t c o m p u t a t io n s a re o n l y c o n d i t i o n a l l y s t a b l ed e p e n d i n g on t h e t i m e - s t e p s i z e . The t i m e - s t e p m u s t be s m a l l e r t h a n ac e r t a i n c r i t i c a l v a l u e f o r n u m e r i c a l e r r o r s n o t t o g ro w u n b o un d e d .T h i s c o n s t i t u t e s a m a j o r o b s t a c l e f o r c e r t a i n p r o b l e m s w he re ane x c e s s i v e nu mb er o f t i m e - s t e p s m akes t h e a n a l y s i s t o o c o s t l y .

The s t a b i l i t y o f t h e c e n t r a l d i f f e r e n c e m e th o d i s c o n s i d e r e d i ns e c t i o n 4 .7 . The t i m e - s t e p i s l i m i t e d by t h e C o u r a n t c r i t e r i o n , i . e .


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the t ime i t takes the st re ss waves to t ra v e l across one e lem ent . Th isl i m i t a t i o n i s c o n s i s t e n t w i t h t h e l o c a l , u nc ou p le d i n t e g r a t i o n o f t hee q u a t i o n s o f m o t i o n . I f t h e t i m e - s t e p was l a r g e r t h a n t h e C o u r a n tc r i t i c a l va lu e , s t re ss waves wou ld t r a ve l across an e lemen t w i th in onet i m e - s t e p , a f f e c t i n g th e s u r r o u n d i n g e l e m e n t s . The in c r e m e n t a lbehaviour o f th a t e lem ent wou ld no long er be independent f rom the re stof the model.

C e n t r a l d i f f e r e n c e s ch em es h av e be en w i d e l y u s ed i n n o n l i n e a rn u m e r i c a l c o d e s , f r o m t h e e a r l y FD h y d r o c o d e s o f W i 1 k i n s ( 1 9 6 4 ) a ndMaenchen and Sack (1964), to th e FE codes of H a ll q u is t (1982a, 1 982c),Key (1974 ) , and B e l y ts c hk o and Ts ay( 198 2) . The ac cu rac y ands t a b i l i t y o f ce n t r a l d i f fe re nc e me thods has been s tu d ied and d iscussedby v a r i o u s a u t h o r s ( e . g . B e l y t s c h k o , H olm e s a nd M u l l e n ( 1 9 7 5 ) ,Be lytsc hko(19 78), Kr ie g and Key(1973)) . The c en tr a l d i f f e r e nc e methodis cons ide red as the most conven ien t w i th in the ex p l i c i t c la ss .

3.6.2.TRAPEZOIDAL RULE (IMPLICIT)

The s o - c a l l e d t r a p e z o i d a l r u l e i s an e xa m p le o f i m p l i c i ti n t e g ra t i o n m e th od s. I n f a c t i t co n s t i t u te s a p a r t i c u la r ca se o f t h eNewmark fa m i l y , p roba b ly the most popu lar o f the i m p l i c i t schemes. Ac o n s t a n t a v e ra g e a c c e l e r a t i o n i s a ss um ed f o r ea ch i n c r e m e n t At. Thed i f f e re n ce e q u a t i o n s a re :

u n + h = ( i i n + u n + 1 ) /2 (0 s< h < 1)u n + 1 = u n+ ( i i n + i i n + 1 )At/Z (3 .36 )u n + l = u n + a n 4 t + ( .jjn + yn+1 ) A t 2 / 4

For ob ta in ing un+ l the equa t ions o f mo t ion a re en fo rced fo r t imet+At. In an undamped ca se ,

M iin + 1+ P (u n + 1 ) = R n + 1 (3 .37 )Eqn. (3 .3 7) i s an i m p l i c i t r e l a t i o n f o r u n + l . S u b s t i t u t i n g t h ed i f f e r e n c e e x p r e s s i o n s ( 3. 36 ) i n (3 .3 7 ) t h e f o l l o w i n g sy s t e m i so b ta in e d :


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L ( 4 M t 2 ) M + K n + 1 j u n + 1 = Rn + 1 + M [ 4 M t 2 ) u n+( 4 M t ) u n + i i n J ( 3 . 3 8 )

w h er e K n+ 1 i s t h e s e c a n t s t i f f n e s s m a t r i x ( p n + l = K n + l u n + 1 ) .

F o r a l i n e a r sys t em (Kn = co n s t a n t ) e qns . (3 . 38 ) may be so l ve d byi n v e r t i n g t h e m o d i f i e d s t i f f n e s s m a t r i x K *, d e f i n e d as

K* = ( 4 / 4 t ^ M + K n + 1 (3 .39)

F o r n o n l i n e a r s y s te m s t h e e q u a t i o n s a r e g e n e r a l l y s o l v e d byd i r e c t e l i m i n a t i o n t e c h n i q u e s , e .g . N e w to n -R a p hs o n t y p e m e th o ds ( w h i c hr e q u i r e t r i a n g u l a r i z a t i o n o f K *) .

T h e t r a p e z o i d a l r u l e i s u n c o n d i t i o n a l l y s t a b l e ( s e e e .g .B e l y t s c h k o an d S c h o e b e r l e , 1 9 7 5 ), t h e t i m e - s t e p b e i n g l i m i t e d o n l y bya c c u r a cy c o n s i d e r a t i o n s . T h i s i s t h e m a in a d v a n t a g e o f i m p l i c i ts c h em e s , w h i c h ma ke s t h e m m o re a p p r o p r i a t e f o r p r o b l e m s i n w h i c h l a r g et i m e - s t e p s c a n be u s e d . I f th e t i m e - s t e p s a r e l i m i t e d t o s m a l l v a l u e sf o r re as on s o t h e r t h an s t a b i l i t y ( e.g . s te e p n o n l i n e a r i t i e s , a c c u r a c y ,s t r e s s - w a v e s ) i m p l i c i t m eth od s l o s e t h e i r a d v a n t a g e .

A n o t h e r l i m i t a t i o n i s t h e l a r g e s t o r a g e r e q u i r e d f o r t h e m a t r i xc o e f f i c i e n t

Numerical Modelling in Large Strain Plasticity With Application to Tube Collapse Analysis

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