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UBC Theses and Dissertations

A finite element and experimental study of plastic compression for metal forming Houlston, Robin 1984

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A F I N I T E E L E M E N T AND E X P E R I M E N T A L S T U D Y OF P L A S T I C C O M P R E S S I O N FOR M E T A L F O R M I N G b y R O B I N H O U L S T O N B . S c , P h y s i c s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 6 8 M . S c . , A p p l i e d M e c h a n i c s , U n i v e r s i t y o f M a n c h e s t e r , 1 9 7 5 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F DOCTOR OF P H I L O S O P Y i n T H E F A C U L T Y OF G R A D U A T E S T U D I E S ( D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A 1 9 8 4 © R o b i n H o u l s t o n , 1 9 8 4 In presenting this thesis in par t ia l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shal l make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of this thesis for f inancial gain shall not be allowed without my written permission. Department O f Mechanical Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date April 9, 1984. DE-6 (3/81) ABSTRACT Me ta l - fo rm ing i n v o l v e s the de fo rmat ion o f meta ls f o r the purposes o f manufac tu r ing a p r o d u c t . An unde rs t and ing o f t he e f f e c t o f p r o c e s s v a r i a b l e s on m e t a l f o r m i n g o p e r a t i o n s i s o f fundamental importance t o the e n g i n e e r . However, l i t t l e i n f o r m a t i o n i s a v a i l a b l e r e g a r d i n g the e f f e c t o f s t r a i n h a r d e n i n g and s t r a i n r a t e s e n s i t i v e m a t e r i a l p r o p e r t i e s , dynamic l o a d i n g , s u r f a c e f r i c t i o n and specimen dimensions on the way components deform d u r i n g the f o r g i n g p r o c e s s . In t h i s work the p l a n e s t r a i n c o m p r e s s i o n o f an i n i t i a l l y r e c t a n g u l a r specimen between f l a t , p a r a l l e l and r i g i d p l a t e n s i s s e l e c t e d f o r i n v e s t i g a t i o n as b e i n g r e p r e s e n t a t i v e o f a b a s i c d i e f o r g i n g o p e r a t i o n . T h i s c o n f i g u r a t i o n a l l ows the complete de fo rmat ion h i s t o r y o f q u a s i - s t a t i c a l l y o r d y n a m i c a l l y deformed specimens to be r e c o r d e d p h o t o g r a p h i c a l l y and t he e f f e c t o f p r o c e s s v a r i a b l e s i d e n t i f i e d . A f i n i t e element model i s deve loped f o r t h i s c a s e . The code accounts f o r l a r g e s t r a i n s , n o n l i n e a r m a t e r i a l p r o p e r t i e s , i n e r t i a e f f e c t s and s u r f a c e f r i c t i o n on a l l b o u n d a r i e s . The r e s u l t s o f dynamic compress ion t e s t s on p l a s t i c i n e and q u a s i - s t a t i c compress ion t e s t s on aluminum are compared t o the f i n i t e element code p r e d i c t i o n s . They g i v e good c o r r e s p o n d e n c e o v e r a l a r g e r ange o f s t r a i n h i s t o r y . F u r t h e r s t u d i e s conducted w i th the f i n i t e element model i d e n t i f y many o f the fundamental c h a r a c t e r i s t i c s o f the f o r g i n g o p e r a t i o n . I t i s shown tha t r i g i d - p e r f e c t l y p l a s t i c m a t e r i a l tends t o d e f o r m a l o n g l i n e s o f i n t e n s e s h e a r and can be approx imated by c e r t a i n upper bound s o l u t i o n s . The normal i n t e r f a c e s t r e s s d i s t r i b u t i o n f o r t h i s c a s e i s v e r y d i f f e r e n t f rom the c l a s s i c a l f r i c t i o n h i l l s t h a t may sometimes be assumed. A more normal type o f f r i c t i o n h i l l s t r e s s d i s t r i b u t i o n i s o b t a i n e d w i th s t r a i n ha rden ing and s t r a i n r a t e s e n s i t i v e m a t e r i a l s . The l i n e s o f i n t ense shear become wider and g i ve more homogeneous de fo rmat ion f o r t h i s c a s e . With dynamic l o a d i n g inhomogeneous de fo rma t ion occurs as the energy o f impact i s r a p i d l y d i f f u s e d throughout the spec imen. I t i s p o s s i b l e t h a t an i n v e r s e f r i c t i o n h i l l deve lops on the lower p l a t e n and a f r i c t i o n h i l l on the top p l a t e n . i v ACKNOWLEDGEMENTS The author wishes t o thank P r o f e s s o r s G. W. V i c k e r s and D. L. Anderson f o r j o i n t l y s u p e r v i s i n g the work. The author i s a l s o t h a n k f u l f o r the i n t e r e s t and c o o p e r a t i o n shown by P r o f e s s o r s J . P. Duncan, M. D. Olson and H. Ramsey. The author a l s o extends h i s g r a t i t u d e to a l l the f a c u l t y and s t a f f o f the c i v i l and mechanical e n g i n e e r i n g departments, without whose c o o p e r a t i o n the work would not have been p o s s i b l e . In p a r t i c u l a r , he would l i k e t o thank John R i c h a r d s , E l e c t r o n i c s T e c h n i c i a n . Mr. R i c h a r d s designed and b u i l t the e l e c t r o n i c c i r c u i t s r e q u i r e d f o r h i g h speed photography i n the experimental work. A c e r t a i n amount of work i n t h i s t h e s i s was done at the Defence Research Establishment S u f f i e l d (DRES). The author acknowledges the co o p e r a t i o n o f Mr. N. B a n n i s t e r , Miss J . Smith, and Mr. H. L i g h t f o o t o f the DRES computing c e n t r e . The author was a l s o encouraged by the i n t e r e s t shown i n the work by the members of the M i l i t a r y E n g i n e e r i n g S e c t i o n at DRES . The author g r a t e f u l l y acknowledges an H. R. MacMillan Family F e l l o w s h i p (award 0326) awarded t o him by the U n i v e r s i t y o f B r i t i s h Columbia. V TABLE OF CONTENTS A b s t r a c t i i Acknowledgements i v L i s t Of F i g u r e s i x CHAPTER 1: INTRODUCTION AND LITERATURE SEARCH 1 1.1. METAL FORMING 1 1.2. SOLUTION TECHNIQUES 2 1 .2 .1 . I n t r o d u c t i o n 2 1 .2 .2 . L i m i t A n a l y s i s 2 1 .2 .3 . S l i p L i ne F i e l d Technique 2 1 .2 .4 . V i s i o p l a s t i c i t y 3 1 .2 .5 . The F i n i t e D i f f e r e n c e Technique . . . 3 1.3. APPLICATION OF THE FINITE ELEMENT METHOD TO THE SOLUTION OF METAL FORMING PROBLEMS 3 1 .3 .1 . I n t r o d u c t i o n 3 1 .3 .2 . Genera l f i n i t e element f o r m u l a t i o n fo r problems of l a rge s t r a i n 4 1 .3 .3 . F i n i t e element a p p l i c a t i o n s to f low problems 5 1 .3 .4 . A p p l i c a t i o n of the f i n i t e element method to p l a s t i c compress ion 6 1 . 3 . 5 . Dynamic Compress ion 8 1.4. PURPOSE AND SCOPE OF THE THESIS 8 CHAPTER 2: APPROXIMATE MODELS 11 2 . 1 . HOMOGENEOUS PLANE DEFORMATION 11 2 . 1 . 1 . I n t r o d u c t i o n 12 v i 2.1.2. The V e l o c i t y and S t r e s s F i e l d s 12 2 .2 . HOMOGENEOUS PLANE DEFORMATION WITH FRICTION .- 13 2 . 3 . VELOCITY DISCONTINUITY PATTERNS 16 2 . 3 . 1 . A v e l o c i t y d i s c o n t i n u i t y p a t t e r n fo r H/D>2 17 2 . 3 . 2 . A v e l o c i t y d i s c o n t i n u i t y p a t t e r n fo r 1<H/D<2 18 2 . 3 . 3 . A v e l o c i t y d i s c o n t i n u i t y p a t t e r n fo r H/D<1 18 CHAPTER 3: THE FINITE ELEMENT FORMULATION 27 3 .1 . INTRODUCTION 2 7 3.2 . FORMULATION OF THE FINITE ELEMENT EQUATIONS 2 7 3 .3 . DISCUSSION OF ELEMENT SELECTION 3 3 3.4. SPECIFICATION OF GEOMETRY 35 3 .5 . MATERIAL CONSTITUTIVE BEHAVIOR 36 3.6 . ELEMENT ASSEMBLY 3 8 3 .7 . BOUNDARY CONDITIONS 3 9 3 . 7 . 1 . S p e c i f i e d V e l o c i t y And Mean Normal S t r e s s 40 3 . 7 . 2 . Su r f ace T r a c t i o n s 41 3 . 7 . 3 . P l a t e n F r i c t i o n 42 3.7.4. G l a s s F r i c t i o n 47 3.8 . NONLINEAR GEOMETRY 52 CHAPTER 4: THE EXPERIMENTAL STUDY 5 3 4 . 1 . INTRODUCTION 53 4 . 2 . EQUIPMENT 53 4 . 3 . TEST PROCEDURE 57 4 . 3 . 1 . I n t r o d u c t i o n 57 4 . 3 . 2 . P r e p a r a t i o n o f specimens 58 v i i 4.3.3. Photographing the specimens and developing the fi l m 58 4.3.4. Viewing the f i l m and d i g i t i s i n g the results 60 4.4. CALIBRATION OF PLASTICINE 61 4.5. EXPERIMENTAL RESULTS. 6 3 4.5 . 1 . Introduction. 63 4.5.2. Experimental tests 63 CHAPTER 5: FINITE ELEMENT MODELLING OF PLANE STRAIN COMPRESSION 85 5 . 1 DETERMINATION OF THE FINITE ELEMENT PARAMETERS. 86 5 . 1 . 1 . Homogeneous compression 86 5 .1.2. Number of elements 86 5 . 1.3. Determination of allowable s t r a i n increment 86 5 .1.4. Constitutive Relations 87 5.2 EXPERIMENTAL COMPARISONS 9 8 5.2 . 1 . Comparison of the f i n i t e element model with experimental tests on p l a s t i c i n e 88 5.2.2. Comparison of the f i n i t e element model with experimental test on aluminum 91 5.3. RIGID PERFECTLY PLASTIC RESULTS 92 5.3 . 1 . Case 1 . H0/D0 = 4 93 5.3.2. Case 2. He/Do = 0.839 97 5.3.3. Load predictions for the r i g i d - p e r f e c t l y p l a s t i c case 98 5.4. RIGID PLASTIC,STRAIN HARDENING AND STRAIN RATE SENSITIVE RESULTS ..101 v i i i 5 . 4 . 1 . S t r a i n harden ing r e s u l t s f o r H/D>1 102 5.5 . THE DYNAMIC COMPRESSION OF ALUMINUM 104 5 .6 . STRESS RESULTS 106 5 . 6 . 1 . S t r e s s r e s u l t s f o r r i g i d - p e r f e c t l y p l a s t i c q u a s i - s t a t i c compress ion 106 5 . 6 . 2 . S t r e s s r e s u l t s f o r dynamic compress ion . . . . 1 0 7 CHAPTER 6: CONCLUSIONS. 152 REFERENCES • 15a» ix LIST OF FIGURES 2 . 1 . 1 . 1 . I l l u s t r a t i o n Of The Homogeneous P lane Deformat ion Of A B lock Of M a t e r i a l Between Two F r i c t i o n l e s s And R i g i d P l a t ens 19 2 . 2 . 1 . Homogeneous P lane Deformat ion With F r i c t i o n 20 2 . 2 . 2 . I l l u s t r a t i o n Of A F r i c t i o n H i l l Normal S t r e s s D i s t r i b u t i o n In Non D imens iona l Form 2 1 2 . 3 . 1 . A V e l o c i t y D i s c o n t i n u i t y P a t t e rn For The Case H/D=1 22 2 . 3 . 1 . 1 . A V e l o c i t y D i s c o n t i n u i t y Pa t t e rn For The Case H/D=2. 2 3 2 . 3 . 1 . 2 . One P o s s i b l e V e l o c i t y D i s c o n t i n u i t y P a t t e rn For P e r f e c t l y P l a s t i c P lane S t r a i n Compression Between R i g i d And P a r a l l e l P l a t e n s . H/D>2 Is Assumed In The Diagram 24 2 . 3 . 2 . 1 . A P o s s i b l e V e l o c i t y D i s c o n t i n u i t y P a t t e rn For P e r f e c t l y P l a s t i c P lane S t r a i n Compression Between R i g i d And P a r e l l e l P l a t e n s . 1<H/D<2 Is Assumed In The Diagram 25 2 . 3 . 3 . 1 . A P o s s i b l e V e l o c i t y D i s c o n t i n u i t y P a t t e rn For P e r f e c t l y P l a s t i c P lane S t r a i n Compression Between R i g i d And P a r a l l e l P l a t e n s . H/D<1 Is Assumed In The Diagram 26 3.3.1 I l l u s t r a t i o n Of The Mapping To s 1 , s 2 C o o r d i n a t e s Of A 4 Node Q u a d r i l a t e r a l Element 34 3.4.1 I l l u s t r a t i o n Of Nodal Po in t Numbering For X Elements ^5 3.5.1 I l l u s t r a t i o n Of The C o n s t i t u t i v e Behav ior Assumed 37 3 .7.3.1 I l l u s t r a t i o n Of The Compression In P lane S t r a i n Of A Block" Of M a t e r i a l Between Two P l a t e n s . . . . 4 3 3 .7 .3 .2 I l l u s t r a t i o n Of The Top R ight Hand Quadrant Of A Specimen With The S lave F r i c t i o n Nodes And The Master F r i c t i o n Node Shown 46 3 .7.4.1 I l l u s t r a t i o n Of M a t e r i a l Deforming In P lane S t r a i n Between G l a s s P l a t e s 48 4 .2 .1 ( a ) Photograph Of The Impact Press And High Speed Camera 66 4 . 2 . 1 . (b) The Impact P ress Used For The Deformat ion T e s t s 67 4 . 2 . 2 . Log i c Of The E l e c t r o n i c C i r c u i t For Camera S y n c h r o n i z a t i o n 68 4 . 2 . 3 . The Specimen Ho lder On The Impact P r e s s . A P l a t en V e l o c i t y Aga ins t D isp lacement Curve Is A l s o Shown 69 4 . 2 . 4 . Photographs Of Impact P ress With Specimen In P o s i t i o n 70 4 . 2 . 5 . Graphs Of F i l m V e l o c i t y Aga ins t T ime, And Frames Per Second Aga ins t Feet Of F i l m Through The H igh Speed Camera 71 4 . 2 . 6 . a) M i l l i v o l t Output Aga in s t V e l o c i t y From The V e l o c i t y T r ansduce r . b) V o l t Output Aga in s t D isp lacement From The D isp lacement Transducer 72 4 . 2 . 7 . P l o t Of Output Vo l t age From The Load C e l l x i Aga ins t A p p l i e d Load 73 4.4.1 I n i t i a l And F i n a l Deformed Shapes Of Two Ax isymmetr i c P l a s t i c i n e Specimens From T e s t s For De te rmin ing M a t e r i a l P r o p e r t i e s . . . . . ' 7 4 4 . 4 . 2 . Dynamic Compression Of A C y l i n d r i c a l P l a s t i c i n e Specimen 75 4 . 4 . 3 . P l a t en D isp lacement Aga in s t Time Curve 76 4 . 4 . 4 . T y p i c a l Load Trace For Impact Onto A C y l i n d r i c a l P l a s t i c i n e Specimen At 4m/s 76 4 . 4 . 5 . C a l i b r a t i o n Curve Of 6 Aga ins t ~D For P l a s t i c i n e 77 4 . 5 . 2 . 1 . Photographs Of A P l a s t i c i n e Specimen At V a r i o u s Times A f t e r Impact With Graphs Of Load And Specimen He ight Aga ins t Time (v 0=4m/s Dur ing Def ormat ion) 78 4 . 5 . 2 . 2 . Expe r imen ta l Path L i n e s For A T y p i c a l P l a s t i c i n e Specimen Be ing Compressed In P lane S t r a i n 79 4 . 5 . 2 . 3 . P lane S t r a i n Compress ion Of P l a s t i c i n e -Boundary Behav ior At Impact S u r f a c e . No L u b r i c a t i o n . V 0 = 4m/s Dur ing Deformat ion 80 4 . 5 . 2 . 4 . A Superimposed T r a c i n g From Two Photographs In F i g . 3 . 5 . 2 . 3 81 4 . 5 . 2 . 5 . P lane S t r a i n Compress ion Of P l a s t i c i n e Boundary Behav ior At Impact S u r f a c e . L u b r i c a t i o n : L i g h t O i l . V 0 = 4m/s Dur ing Deformat ion 82 4 . 5 . 2 . 6 . A Superimposed T r a c i n g From Two Photographs In F i g . 3 . 5 . 2 . 5 83 x i i 4 . 5 . 2 . 7 . Photographs Of The Deformat ion P r o f i l e s Of A P l a s t i c i n e Specimen Be ing Deformed In P lane S t r a i n . 84 5.1.1.1 V e l o c i t y F i e l d For Homogeneous Compress ion Between F r i c t i o n l e s s P l a t e s 1 0 8 5.1.2.1 Comparison For Aluminum Between 15 Element And 50 Element P r e d i c t i o n s Of Deformed Shapes For H/H o=0.582 ( s o l i d L i ne ) And H/H o=0.559 (do t ted L ine ) 1 0 9 5 . 1 . 3 . 1 . Gauss Po in t Boundary Shear S t r e s s e s For V a r i o u s S i z e s Of Incrementa l S t r a i n Steps To Ach ieve A F r a c t i o n a l He ight Reduct ion Of 0.831 1 1 0 5 . 1 . 4 . 1 . C a l i b r a t i o n Curves Of <=> Aga ins t e For Aluminum I l l 5 .2 .1 .1 , Expe r imen ta l Deformed P r o f i l e s For The P lane S t r a i n Compression Of P l a s t i c i n e 1 1 2 5 .2 .1 .2 Exper imenta l Deformed P r o f i l e s For The P lane S t r a i n Compression Of P l a s t i c i n e . 1 1 3 5 . 2 . 1 . 3 ( a ) . T h e o r e t i c a l Deformed P r o f i l e s From T=0 To 6.32ms For The Plane S t r a i n Dynamic Compress ion Of P l a s t i c i n e At 4.4m/s With ?^=0 1 1 4 5 . 2 . 1 . 3 ( b ) . T h e o r e t i c a l Deformed P r o f i l e s From T=8.l2ms To 10.40ms For The P lane S t r a i n Dynamic Compress ion Of P l a s t i c i n e At 4. 4m/s With "7^  =0 1 1 5 5 . 2 . 1 . 4 ( a ) . T h e o r e t i c a l Deformed P r o f i l e s From T=0 To 6.30ms For The Plane S t r a i n Dynamic Compress ion Of P l a s t i c i n e At 4.4m/s With 7^=0.1 1 1 6 5 . 2 . 1 . 4 ( b ) . T h e o r e t i c a l Deformed P r o f i l e s From x i i i T=8.1ms To 10.39ms For The P lane S t r a i n Dynamic Compression Of P l a s t i c i n e At 4.4m/s With ^=0.1 . ...117 5 . 2 . 1 . 5 ( a ) . T h e o r e t i c a l Deformed P r o f i l e s From T=0 To 6.41ms For The P lane S t r a i n Dynamic Compression Of P l a s t i c i n e At 4.4m/s With ^ = 0 . 2 3 5 118 5 . 2 . 1 . 5 ( b ) . T h e o r e t i c a l Deformed P r o f i l e s From T=8.1ms To 10.36ms For The P lane S t r a i n Dynamic Compression Of P l a s t i c i n e At 4.4m/s With 7^=0.235 . 119 5 .2 .1 .6 ( a ) Expe r imen ta l R e s u l t s For The P lane S t r a i n Compression Of P l a s t i c i n e 120 5 .2 .1 .6 (b ) T h e o r e t i c a l Re su l t s For The Dynamic P lane S t r a i n Compression Of P l a s t i c i n e For Three Va lues Of G l a ss F r i c t i o n C o e f f i c i e n t , / ^ , Of 0 .235, 0 . 1 , And 0 120 5 .2 .1 .7 ( a ) Exper imenta l And C a l c u l a t e d Loads For P l a s t i c i n e 121 5 .2 .1 .7 (b ) C a l c u l a t e d Loads For P l a s t i c i n e 121 5 .2 .1 .8 ( a ) Power D i s t r i b u t i o n For The P lane S t r a i n Compression Of P l a s t i c i n e 122 5 .2 .1 .8 (b ) Power D i s t r i b u t i o n For The P lane S t r a i n Compression Of P l a s t i c i n e 122 5.2.2.1 Comparison For Aluminum Of Expe r imen ta l And F i n i t e Element Deformed P r o f i l e s With V e l o c i t y V e c t o r s 123 5 .2 .2 .2 ( a ) An Expe r imen ta l Flow P l o t For The P lane S t r a i n Compression Of Aluminum 124 5 .2 .2 .2 (b ) T h e o r e t i c a l Flow P l o t Cor respond ing To The x i v Exper imenta l Flow P l o t Given In F i g . 5 .2 .2 .2 (a ) 125 5 . 3 . 1 . 1 . Deformed Shapes With V e l o c i t y Vec to r P l o t s At V a r i o u s Stages Of Compression From H 0 /D o =4 For Case 1 1 2 6 5 . 3 . 1 . 2 ( a ) . Graphs Of -V y Aga in s t Y A long X G r i d L i n e s For Case 1 127 5 . 3 . 1 . 2 ( b ) . Futher Graphs Of - V y Aga ins t Y A long X G r i d L i n e s For Case 1 128 5 . 3 . 1 . 3 ( a ) . Graphs Of - V y / V 0 Aga ins t Y A long X G r i d L i n e s For The Upper Bound Model For Case 1 At H/D=2.023 . 129 5 . 3 . 1 . 3 ( b ) . Graphs Of - V y / V 0 Aga ins t Y A long X G r i d L i nes For The Upper Bound Model For Case 1 At H/D=1 .117 1 3 0 5 . 3 . 1 . 4 . F i n i t e Element D i s p l a c e d Shape P l o t s For The S t a r t Of P lane S t r a i n Compress ion Of Rec tangu la r B locks Of Aluminum 1 3 1 5 . 3 . 1 . 5 . Graphs Of - V y Aga ins t Y A long X G r i d L i n e s For V a r i o u s H 0 / D o R a t i o s (undeformed B locks ) And One H/D Ra t i o For A B lock With A Deformed Boundary. 132 5 . 3 . 1 . 6 . -V y Aga ins t Y For Two S l i g h t l y S t r a i n e d B locks Of H 0 / D 6 = 3 And 3.5 1 3 3 5 . 3 . 1 . 7 . Flow L i n e s For Case 1 1 3 4 5 . 3 . 2 . 1 . Deformed Shapes For The Q u a s i - s t a t i c P lane S t r a i n Compression Of A R i g i d - p e r f e c t l y P l a s t i c M a t e r i a l Between R i g i d And P a r a l l e l P l a t ens 1 3 5 5 . 3 . 2 . 2 . Graphs Of -v y Aga ins t Y A long X G r i d L i n e s XV For Case 2 1 3 5 5 . 3 . 2 . 3 . A Comparison Of V y Aga ins t X From The F i n i t e Element C a l c u l a t i o n s With The S imple Mode l . P r e d i c t i o n s For The Case H/D=0.25 137 5 . 3 . 2 . 4 . A Comparison Of Vy Aga ins t X From The F i n i t e Element C a l c u l a t i o n s With The S imple Model P r e d i c t i o n s For The Case H/D=0.125 13"8 5 . 3 . 3 . 1 . A Comparison Of Normal P l a t en T r a c t i o n From The F i n i t e Element C a l c u l a t i o n s For A R i g i d - p e r f e c t l y P l a s t i c M a t e r i a l With The P r e d i c t i o n s Of Approximate C l o sed Form S o l u t i o n s . ...139 5 . 4 . 1 . 1 . Deformed Shapes With V e l o c i t y Vec to r P l o t s At V a r i o u s Stages Of Compression From H /D =4 For P l a s t i c i n e 140 5 .4 .1 .2 Graphs Of -v Aga ins t Y Along X G r i d L i n e s For P l a s t i c i n e 141 5 . 4 . 2 . 1 . Deformed Shapes With V e l o c i t y Vec to r P l o t s At V a r i o u s Stages Of Compression From H /D =4 For Aluminum 142 5 .4 .2 .2 Graphs Of -v Aga ins t Y A long X G r i d L i n e s For Aluminum 143 5 . 4 . 2 . 3 . A Futher Graph Of -v Aga ins t Y A long X G r i d L i n e s For Aluminum 144 5 . 5 . 1 , D i s p l a c e d Shape P l o t s For The Dynamic Compress ion Of Aluminum At I00m/s 145 5 . 5 . 2 . D i s p l a c e d Shape P l o t s Cont inued For The Dynamic Compression Of Aluminum At I00m/s 146 5 . 6 . 1 . 1 . Graphs Of (element Averages) Aga in s t X xv i For H o r i z o n t a l Rows Of E lements On Uns t r a i ned B locks Of Aluminum 147 5 .6 .1 .2 6 2 . (element Average) Aga ins t X For Three Va lues Of H o / D o For U n s t r a i n e d B locks Of Aluminum. ..148 5 . 6 . 1 . 3 . Normal S t r e s s Aga ins t X A long The Top And Bottom Boundar ies AB For Seve ra l H/D R a t i o s Ach ieved By S t r a i n i n g From H /D =0.84 149 5 . 6 . 2 . 1 . - 62_(element Average) Aga ins t X For The Dynamic Compression In Plane S t r a i n Of Aluminum At I00m/s 150 5 . 6 . 2 . 2 . Graphs Of - C ^ A g a i n s t X For Dynamic Compress ion Of Aluminum At I00m/s ( c o n t . ) . A Graph For Un load ing Is A l s o Shown 151 x v i i NOMENCLATURE OF SELECTED SYMBOLS USED Symbol D e s c r i p t i o n D.. E f f e c t i v e d e f o r m a t i o n r a t e 1D S.. D e v i a t o r i c S t r e s s e s 13 V Y i e l d s t r e s s V G r a d i e n t o p e r a t o r P Mean n o r m a l s t r e s s V P l a t e n v e l o c i t y o D E f f e c t i v e d e f o r m a t i o n r a t e a" E f f e c t i v e s t r e s s a^_. Cauchy s t r e s s e s p Mass d e n s i t y v. V e l o c i t y v. A c c e l e r a t i o n i T. S u r f a c e t r a c t i o n I XVIII DEDICATED T O MY PARENTS 1 CHAPTER 1 INTRODUCTION AND LITERATURE SEARCH.  1.1. METAL FORMING. Meta l- fo rming i n v o l v e s the de fo rmat ion of meta ls fo r the purpose of manufac tu r ing a p r o d u c t . A v i t z u r ( 1 , 1 9 6 8 ) and Johnson and M e l l o r (2,1973) o u t l i n e some o f the meta l f o rm-ing p roces ses c u r r e n t l y used i n i n d u s t r y . An unders tand ing of the e f f e c t of p rocess v a r i a b l e s on meta l- fo rming o p e r a t i o n s must be c o n s i d e r e d of fundamental importance to the eng ineer in the des ign and development of manufac tu r ing p r o c e s s e s . For example, one may ask the f o l l o w i n g q u e s t i o n s . What loads are r e q u i r e d to deform the metal in a p a r t i c u l a r forming p rocess ? How do metal p r o p e r t i e s such as s t r a i n ra te s e n s i t i v i t y and s t r a i n harden ing a f f e c t the s t r e s s and de fo rmat ion p a t t e r n s ? How do l u b r i c a t i o n c o n d i t i o n s on the boundar ies of the meta l in con tac t wi th the forming s u r f a c e s a f f e c t the s t r e s s d i s t r i b u t i o n s in the meta l? What g u i d e l i n e s can one e s t a b l i s h fo r e f f e c t i v e des ign in meta l- forming p rocesses ? Very l i t t l e i n f o rma t i on i s a v a i l a b l e in the l i t e r a t u r e r ega rd ing the d e t a i l e d e f f e c t of i n e r t i a , s t r a i n ra te s e n s i t i v i t y , and s t r a i n harden ing on the way components deform du r i ng the f o r g i n g p rocess and the r e s u l t i n g s t r e s s e s and loads i n v o l v e d . T h i s i s due p a r t i a l l y to the d i f f i c u l t i e s i n v o l v e d in measur ing s t r a i n or s t r e s s w i t h i n m a t e r i a l that i s undergoing very l a r g e , p o s s i b l y r a p i d , d i s t o r t i o n s . A d d i t i o n a l l y , d e t e r m i n a t i o n of the f r i c t i o n a l boundary c o n d i t i o n s that e x i s t i s ext remely d i f f i c u l t . 2 1.2. SOLUTION TECHNIQUES. 1 .2 .1 . I n t r o d u c t i o n . A f i r s t s t ep in the a n a l y t i c a l t reatment of a meta l- fo rming problem i s to propose a mathemat ica l mode l . The model w i l l u s u a l l y i n vo l v e l a r g e d e f o r m a t i o n s , n o n l i n e a r m a t e r i a l s , and unknown f r i c t i o n boundary c o n d i t i o n s . In gene ra l there are no c l o s e d form s o l u t i o n s . T h e r e f o r e , to so l ve p r a c t i c a l p r o b l e m s , s p e c i a l p rocedures are employed to f i n d approximate numer i ca l s o l u t i o n s . F i v e p r a c t i c a l methods fo r f i n d i n g approximate s o l u t i o n s are l i m i t a n a l y s i s , the s l i p l i n e f i e l d t e c h n i q u e , the v i s i o p l a s t i c i t y t e c h n i q u e , the f i n i t e d i f f e r e n c e t e c h n i q u e , and the f i n i t e element t e chn ique . The f i r s t four are d i s c u s s e d in s e c t i o n s 1.2.2 to 1.2.5 i n c l u s i v e . In s e c t i o n 1.3 the f i n i t e element method i s d i s c u s s e d . 1 .2 .2 . L i m i t A n a l y s i s The Upper Bound and the Lower Bound Theorems of p l a s t i c i t y a l low one to p l a ce bounds on forming loads in metal forming p r o c e s s e s . These theorems are d i s c u s s e d and proved by H i l l ( 3 , 1 9 5 0 ) . 1 .2 .3 . S l i p L i ne F i e l d Techn ique . The s l i p l i n e f i e l d method i s a p p l i c a b l e to p lane s t r a i n problems and was i n t r oduced in 1923 by Hencky (4 ) . B a s i c a l l y the method i n v o l v e s c o n s t r u c t i n g a mesh of o r thogona l l i n e s a long which the shear s t r e s s i s a maximum. It i s assumed that s l i p can occur on ly a long these l i n e s . From the l i m i t theorems an upper bound on a p p l i e d loads w i l l be o b t a i n e d . Johnson, Sowerby, and Ven te r (5 ,1982 ) 3 presen t numerous a p p l i c a t i o n s of the s l i p l i n e f i e l d t e c h n i q u e . 1 .2 .4 . V i s i o p l a s t i c i t y . Thomsen (6,1963) appears to have f i r s t i n t r o d u c e d the v i s i o p l a s t i c i t y t e c h n i q u e . More r e c e n t l y Shabaik (7,1972) used the techn ique to study v a r i o u s types of meta l forming p r o c e s s e s . C .M .Lee , G.W V i c k e r s , and S.N Dwivedi (8,1983) a l s o fu the r extended the method to i n c l u d e dynamic p rob lems . The v i s i o p l a s t i c i t y method uses the equa t ions of motion to determine the s t r e s s f i e l d from an e x p e r i m e n t a l l y measured and n u m e r i c a l l y smoothed v e l o c i t y f i e l d . 1 .2 .5 . The F i n i t e D i f f e r e n c e Techn ique . The f i n i t e d i f f e r e n c e techn ique i s a we l l known compute r-or i en ted method of n u m e r i c a l l y s o l v i n g 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 . Shaba ik(9,1975) g i ves an example of the a p p l i c a t i o n of the method to p lane s t r a i n e x t r u s i o n . One of the d i sadvan tages of the f i n i t e d i f f e r e n c e techn ique i s tha t an e q u a l l y d i v i d e d mesh i s r e q u i r e d to ach ieve optimum a c c u r a c y . 1.3. APPLICATION OF THE FINITE ELEMENT METHOD TO THE  SOLUTION OF METAL FORMING PROBLEMS. 1 .3 .1 . I n t r o d u c t i o n . The f i n i t e element techn ique i s an a p p r o p r i a t e numer i ca l method fo r s o l v i n g meta l- fo rming p rob lems . I r r e g u l a r meshes can be used and very complex boundary c o n d i t i o n s can be s p e c i f i e d . The method i s based on the c o n s t r u c t i o n of a v e l o c i t y f i e l d over a sma l l r eg ion (element) of m a t e r i a l in terms of a number of g e n e r a l i s e d 4 c o o r d i n a t e s . A l l the g e n e r a l i s e d c o o r d i n a t e s from the e lements are then determined to produce a s o l u t i o n . The r e s u l t i n g problem reduces to a set of s imul taneous equa t ions w i th the g e n e r a l i s e d c o o r d i n a t e s as unknowns. Z i e n k i e w i c z ( 1 0 , 1 9 7 7 ) shows t h a t , sub jec t to q u i t e gene ra l r equ i r emen t s ; as the number of e lements spanning a r eg ion i s i n c r e a s e d to make more g e n e r a l i s e d c o o r d i n a t e s a v a i l a b l e , the f i n i t e element s o l u t i o n w i l l approach the exact s o l u t i o n to the mathemat ica l mode l . In the f o l l o w i n g s e c t i o n s the cu r r en t l i t e r a t u r e on the a p p l i c a t i o n of the f i n i t e element techn ique to metal forming problems i s rev iewed . S e c t i on 1.3.2 c i t e s some of the l i t e r a t u r e concerned wi th problems i n v o l v i n g l a r g e s t r a i n f i n i t e element a n a l y s e s . S e c t i on 1.3.3 c i t e s some of the l i t e r a t u r e concerned wi th flow prob lems . S e c t i on 1.3.4 c i t e s some of the l i t e r a t u r e concerned wi th p lane s t r a i n and ax isymmetr i c compress ion . S e c t i on 1.3.5 c o n s i d e r s dynamic compress i on . 1 .3 .2 . Genera l F i n i t e Element Fo rmu la t i on For Problems  Of Large S t r a i n . In a l a r g e s t r a i n i nc rementa l approach i t i s usua l to c o n s i d e r a c o o r d i n a t e system embedded in the m a t e r i a l . H i b b i t t ( 1 1 , 1 9 7 0 ) gave one of the f i r s t t rea tments in the con tex t of f i n i t e e l ements . K i tagawa, S eguch i , and Tomita (12,1972) gave a s i m i l a r t reatment us ing the t e chn iques of d i f f e r e n t i a l geometry. Re f . 10 a l s o c o n s i d e r s g e o m e t r i c a l l y n o n l i n e a r p rob lems . These r e f e r e n c e s a l l i n d i c a t e that a d d i t i o n a l terms a r i s e r e l a t i n g deformed geometry to i n i t i a l geometry. Gotoh and I s h i s e (13, 1978) fo rmu la ted 5 and a p p l i e d the l a r g e s t r a i n case to a deep drawing p rob lem. It was found that the l a r g e s t r a i n terms made a s i g n i f i c a n t d i f f e r e n c e in t h e i r a p p l i c a t i o n . 1 .3 .3 . F i n i t e Element A p p l i c a t i o n s To Flow Prob lems. Z i e n k i e w i c z , J a i n , and Onate(14,1978) gave a review of f i n i t e element numer i ca l s o l u t i o n methods fo r s tudy ing the f low of s o l i d s . They f i r s t rev iewed the gene ra l f o r m u l a t i o n fo r f low problems and i d e n t i f i e d two methods of m e r i t : the V/P f o rmu l a t i on w i th Lagrang ian c o n s t r a i n t and the pena l t y f u n c t i o n approach . Seve ra l examples were g i ven of e x t r u s i o n , r o l l i n g , and cup f o r m i n g . Z i enk i ew i cz and Godbole (15,1974) used a stream f u n c t i o n r e p r e s e n t a t i o n to so l v e problems i n v o l v i n g l a rge d i s t o r t i o n s . A p p l i c a t i o n s such as e x t r u s i o n and i n d e n t a t i o n were t r e a t e d as those of non-Newtonian f low. The method was a p p l i e d to both steady s t a t e and t r a n s i e n t p rob lems . Z i enk i ew i cz and Godbole (16,1975) a l s o i n t roduced a pena l t y f u n c t i o n approach f o r s o l v i n g l a rge de fo rmat ion prob lems . They o u t l i n e d how a s t anda rd e l a s t i c i t y program wi th a va lue of P o i s s o n ' s r a t i o approach ing 1/2 cou ld be used to so l ve q u a s i - s t a t i c f low prob lems . Examples of c a l c u l a t i o n w i th an i s o p a r a m e t r i c element wi th 2x2 Gauss i n t e g r a t i o n were g i ven fo r punch i n d e n t a t i o n , p lane s t r a i n compress ion of a b l o c k , and e x t r u s i o n . 6 1.3.4. Application Of The F i n i t e Element Method To  Pl a s t i c Compression. Lee and Kobayashi(17,1973) applied the f i n i t e element method to the problem of the r i g i d - p l a s t i c axisymmetric compression of a cylinder. F r i c t i o n on the top contact boundary was treated by applying a spec i f i e d shear stress V both as 7"=0.2Yo and as ^  = 0.3Y0r/R, where Y 0 i s the constant y i e l d stress, r i s the r a d i a l coordinate and R is radius of the top boundary. They studied cylinders of height to diameter r a t i o (H 0/D 0) of 0.25, 0.5, and 1.0. They concluded that for Ho/Do=0.25 and 0.5 there was a f r i c t i o n h i l l on the top boundary as predicted by elementary theory. For the case of H0/D0=1 they found an inverse f r i c t i o n h i l l . This reference also assessed the effect of work hardening. Price and Alexander(18,1979) presented a paper on isothermal forging. Deformed geometries were presented for axisymmetric cylinders with height to diameter ratios of 1/4, 2/3, 1, 3/2, and 4. The calculations were carried out with the penalty function method. The predicted geometries were found to conform well with experiments for a variety of specimen configurations. Hartley, Sturgess, and Rowe(19,1979) gave a technique of handling surface f r i c t i o n . They introduced a thin layer of f r i c t i o n elements on the top surface of an axisymmetric specimen to be deformed by a r i g i d platen. The top nodes were kept fixed to the r i g i d platen while the lower nodes were free to move ho r i z o n t a l l y . The s t i f f n e s s of the f r i c t i o n elements were then modified by a viscous factor. 7 They i d e n t i f i e d the problems tha t they encountered wi th the a p p l i c a t i o n of the approach and d e s c r i b e d t h e i r s o l u t i o n . Good agreement was ob ta ined w i th the r i n g t e s t g iven by Hawkyard and Johnson (20 ,1967) . A f u l l e l a s t i c - p l a s t i c f o r m u l a t i o n was p r e s e n t e d . H a r t l e y , S t u r g e s s , and Rowe(21,1980) a l s o p resen ted a f o l l o w i n g paper in which they gave r e s u l t s f o r the e l a s t i c - p l a s t i c compress ion (up to 40%) of a c y l i n d e r w i th H 0 / D 0 = 1 . They c o n f i r m e d , as d i d Lee and Kobayash i (17 ,1973 ) that an i n ve r se f r i c t i o n h i l l occured on the top boundary fo r t h i s case fo r c e r t a i n v a lues of the f r i c t i o n c o e f f i c i e n t . The low va lues of f r i c t i o n tended to g i ve an i n ve r se f r i c t i o n h i l l . H igh va lues of f r i c t i o n tended to make the boundary su r f a ce t r a c t i o n more un i f o rm . In the case of h igh f r i c t i o n a s i n g l e l a r g e r i g i d zone occu r r ed below the p l a t e n wi th h igh p ressu re areas near the cen t r e and at the o u t s i d e edge. Rooyen and Backofen (22 ,1960) gave many exper imenta l r e s u l t s f o r f r i c t i o n on t h i n d i s c s . In agreement wi th the c a l c u l a t i o n s of Lee and Kobayash i (17,1973) f r i c t i o n h i l l s were observed in a l l c a s e s . However, t h e i r t e s t s were on l y „ f o r sma l l H 0 / D 0 . R e f s . 17 and 21 ag ree , however, that fo r H 0/D 0=1 inve r se f r i c t i o n h i l l s can r e s u l t on the top boundary . B 1 .3 .5 . Dynamic Compression Johnson(23,1972) has g i ven an ex t ens i v e t reatment of dynamic impact fo r e n g i n e e r i n g a p p l i c a t i o n s . Of p a r t i c u l a r i n t e r e s t to the p resen t work i s Hawkyard's energy method and s i m i l a r approaches (see Chapter 5 of r e f . 23 ) . Hutch ings and 0 ' B r i en (24 ,1981 ) p resen ted expe r imen ta l r e s u l t s f o r impact of meta l p r o j e c t i l e s a g a i n s t r i g i d t a r g e t s at low v e l o c i t i e s ( l e s s than 100ms" 1 ) . A compar ison was made to the t h e o r i e s of T a y l o r , Hawkyard, and H u t c h i n g s . G .R . Johnson(25 , 1976,26, 1977) c o n s i d e r e d e l a s t i c - p l a s t i c impact at h i gh v e l o c i t y . A code was deve loped fo r three d imens iona l t e t r a h e d r a l e l ements . The type of problems c o n s i d e r e d were: impact of a n i c k e l c y l i n d e r onto an aluminum p l a t e at 1500ms" 1 , impact of a n i c k e l sphere onto an aluminum p l a t e at 1500ms" 1 , normal impact of a n i c k e l t r unca t ed cone onto an aluminum v a r i a b l e t h i c k n e s s p l a t e at 1500ms" 1 , and o b l i q u e impact of an aluminum rod onto a r i g i d s u r f a c e at 1000ms" 1 . A gene ra l review of impact dynamics was p resen ted by Zukas(27,1980) fo r m a t e r i a l s sub j e c t ed to i n tense impu l s i ve l o a d i n g . 1.4. PURPOSE AND SCOPE OF THE THESIS. The o b j e c t i v e of t h i s work i s to ga in some fundamental i n s i g h t i n t o the gene ra l e f f e c t tha t m a t e r i a l p r o p e r t i e s , boundary f r i c t i o n , and i n e r t i a have on the way f o r g i n g specimens deform. Of i n t e r e s t a l s o i s the s t r e s s d i s t r i b u t i o n s , s t r a i n d i s t r i b u t i o n s , and loads i n v o l v e d . With t h i s in mind a s imple type of d i e f o r g i n g o p e r a t i o n 9 was s e l e c t e d ; namely, the p lane s t r a i n compress ion of an i n i t i a l l y r e c t a n g u l a r specimen between f l a t , p a r a l l e l , and r i g i d p l a t e n s . The p rocess i s d e p i c t e d in F i g . 4 . 2 . 4 . The reason fo r the s e l e c t i o n of t h i s case i s that the complete de fo rmat ion h i s t o r y can be recorded p h o t o g r a p h i c a l l y by obse r v i ng the d i s t o r t i o n of the mesh on the specimen through the g l a s s p l a t e s . The e f f e c t of the p rocess v a r i a b l e s can thus be r e a d i l y r e c o r d e d . P l a s t i c i n e was s e l e c t e d as the t e s t m a t e r i a l . T h i s has been used e x t e n s i v e l y f o r meta l forming s t u d i e s and i s known to g i ve a reasonab le assessment of the de fo rmat ions that occur in metal f o rm ing . I t s main advantage in the present s tudy i s that the loads i n v o l v e d for specimens of a s i z e s u i t a b l e fo r o b s e r v a t i o n are not e x c e s s i v e l y l a r g e . It i s , however, a s t r a i n r a te s e n s i t i v e m a t e r i a l which has to be accounted fo r in any numer i ca l m o d e l l i n g . With the exper imenta l r e s u l t s as a b a s i s f o r compar i son , a f i n i t e element model was deve loped . To do the c a l c u l a t i o n s a f i n i t e element code was p repared to model p lane s t r a i n p l a s t i c compress ion . The code which w i l l be r e f e r r e d to as FELEM-RH has su r f a ce f r i c t i o n r o u t i n e s fo r both s t a t i c and dynamic c o n d i t i o n s and p l o t t i n g r o u t i n e s fo r graph p l o t s and p i c t o r i a l d i s p l a y s . The code was in tended to be compact in na ture a l though g e n e r a l l y a p p l i c a b l e to a range of bulk meta l - forming prob lems . In the work the r e s u l t s of dynamic compress ion t e s t s on p l a s t i c i n e and the q u a s i - s t a t i c compress ion of aluminum are compared w i th code p r e d i c t i o n s . These i n i t i a l compar isons are then fo l l owed by a s ys t emat i c study of the e f f e c t of s t r a i n r a t e s e n s i t i v i t y , s t r a i n h a r d e n i n g , and i n e r t i a on specimens o f v a r i o u s dimensions-The r e s u l t s o f t h e s t u d y i n d i c a t e t h a t a l l t h e v a r i a b l e s mentioned have s i g n i f i c a n t e f f e c t on the r e s u l t i n g de fo rma t i on and s t r e s s p a t t e r n s . The a p p l i c a b i l i t y o f approximate upper bound models t o many o f the cases are d i s c u s s e d . The chap te r con ten ts are summarized below. Chapter 2 p r e s e n t s some s i m p l i f i e d s o l u t i o n s f o r p l a n e s t r a i n compress ion . Chapter 3 p r e sen t s a rev iew o f the f o r m u l a t i o n o f the f i n i t e element e q u a t i o n s . Chapter 4 p resen t s some expe r imen ta l work f o r the p lane s t r a i n de fo rmat ion o f p l a s t i c i n e f o r compar ison w i th numer i ca l s o l u t i o n s . Chapter 5 p r e sen t s the r e s u l t s o f c a l c u l a t i o n s done w i th the code . Both s t a t i c and dynamic cases were cons ide r ed and compar ison w i th expe r imen ta l r e s u l t s made. In a d d i t i o n , compar ison o f q u a s i - s t a t i c s o l u t i o n s found f o r a r i g i d - p l a s t i c m a t e r i a l were compared w i th s l i p l i n e f i e l d s o l u t i o n s and v e l o c i t y d i s c o n t i n u i t y p a t t e r n s . Chapter 6 p r e sen t s an o v e r a l l summary and c o n c l u s i o n o f the s tudy . 11 CHAPTER 2  APPROXIMATE MODELS. Approximate c l o s e d form s o l u t i o n s are used w ide ly in meta l forming p rob lems . A summary of these approximate s o l u t i o n s i s g i ven by B i shop (28 ,1958 ) . In t h i s r e f e r ence s o l u t i o n s f o r p lane s t r a i n compress ion are summarised. In t h i s work the main i n t e r e s t in these approximate s o l u t i o n s ' i s f o r compar ison purposes w i th the f i n i t e element code p r e d i c t i o n s and fo r de te rm in ing t h e i r a p p l i c a b i l i t y over a v a r i e t y of c o n d i t i o n s . The f o l l o w i n g s e c t i o n s of t h i s chap te r p resent approximate s o l u t i o n s fo r the p lane s t r a i n compress ion of a r i g i d - p e r f e c t l y p l a s t i c m a t e r i a l between f l a t and p a r a l l e l p l a t e n s . In s e c t i o n 2.1 the s o l u t i o n fo r homogeneous p lane s t r a i n de format ion i s g i v e n . Whi le t h i s s o l u t i o n i s s imple and e x a c t , i t i s on l y of l i m i t e d use in p r a c t i c e . It w i l l on ly occur in the absence of p l a t e n f r i c t i o n . In s e c t i o n 2.2 i t i s assumed, however, tha t an homogeneous v e l o c i t y f i e l d s t i l l e x i s t s even wi th f r i c t i o n . The s o l u t i o n can then be e a s i l y found but i t i s approx imata te s i n ce an assumed d i sp lacement f i e l d i s u sed . In s e c t i o n 2.3 v e l o c i t y d i s c o n t i n u i t y p a t t e r n s are p resen ted and used to f i n d upper bound s o l u t i o n s . 2 . 1 . HOMOGENEOUS PLANE DEFORMATION.  2.1.1 I n t r o d u c t i o n . For the case where p l a t e n f r i c t i o n and g l a s s - p l a t e f r i c -t i o n i s zero homogeneous de fo rmat ion r e s u l t s and the f r e e boundar i es o f the specimen remain s t r a i g h t and p a r a l l e l . F i g . 2.1.1.1 d e p i c t s the p r o c e s s . The s o l u t i o n f o r t h i s case i s w e l l known — the v e l o c i t y f i e l d i s l i n e a r and the s t r e s s e s are c o n s t a n t . 2.1.2 The V e l o c i t y and S t r e s s F i e l d s . The equa t i on o f c o n t i n u i t y i s VCiC s o (2.1.2.1) w h e r e y-^ £ ^  s a r e t h e v e l o c i t y components i n p l a n e s t r a i n r e f e r r e d to the f i x e d r e c t a n g u l a r c a r t e s i a n axes x^ , x^ as shown i n F i g . 2.1.1.1. The r e p e a t e d i n d e x i m p l i e s summation and the comma denotes p a r t i a l d i f f e r e n t i a t i o n w i th r e s p e c t to For the homogeneous de fo rmat ion o f a r e c t a n g u l a r b l o ck the v e l o c i t y components s a t i s f y i n g equa t ion 2.1.2.1 are V> - (2.1.2.2) ^ = V* • (2.1.2.3) L where V q i s v e l o c i t y o f the p l a t e n s r e l a t i v e to the cen t r e o f the specimen and 2L i s the c u r r e n t specimen h e i g h t . T h e C a u c h y s t r e s s e s a. . r e f e r r e d t o t h e f i x e d r e c t a n g u l a r coo rd i na t e a x i s x^ i s V c ° ° / (2.1.2.4) [ ° where C^, i s the u n i f o r m p l a t e n p r e s s u r e . For a p e r f e c t l y p l a s t i c m a t e r i a l w i l l be equa l to the y i e l d s t r e s s , Y, o f the m a t e r i a l . In gene ra l the y i e l d s t r e s s Y w i l l be a f u n c t i o n o f t h e e f f e c t i v e s t r a i n e and t h e e f f e c t i v e d e f o r m a t i o n r a te D. The d e f i n i t i o n o f the e f f e c t i v e s t r a i n 13 i s g i v e n a s ( 2 . 1 . 2 . 5 ) w h e r e JL = f £> dt r £ = J (%) bcj Oij ' ( 2 . 1 . 2 . 6 ) 2>y =/''*)( ViJ + V-j.i) ( 2 . 1 - 2 . 7 ) I n t h e p r e s e n t c a s e t h e e f f e c t i v e s t r a i n i s g i v e n b y e = In (L/S-°)\ ( 2 . 1 . 2 . 8 ) w h e r e 2 L q i s t h e i n i t i a l s p e c i m e n h e i g h t . A g e n e r a l f o r m ( 3 ) f o r Y w h i c h c a n b e a d a p t e d t o t h e s p e c i f i c c a s e s i s »<D i f "D<B~O Y = ( l + c 1 e ) T [ Y 0 + c l ( D - D 6 } C l ] i f D>D, ( 2 . 1 . 2 .9) w h e r e *D"0=( 1+c , " e ) T Y 0 A c (2.1.2.9) ^ i s a l a r g e n u m b e r c h o s e n t o a p p r o x i m a t e r i g i d - p l a s t i c b e h a v i o r . Y 0 i s t h e s t a t i c a n n e a l e d y i e l d s t r e s s . c 1 f c 2 , c 3 , a n d a r e c o n s t a n t s . 2.2 HOMOGENEOUS P L A N E D E F O R M A T I O N W I T H F R I C T I O N H o m o g e n e o u s d e f o r m a t i o n a s d i s c u s s e d i n S e c t i o n 2.1 c a n n o t o c c u r i f t h e r e i s f r i c t i o n o n t h e b o u n d a r i e s . A n a p p r o x i m a t e r e s u l t g i v e n b y H i l l ( 3 ) c a n b e o b t a i n e d , h o w e v e r , b y a s s u m i n g h o m o g e n e o u s d e f o r m a t i o n a s f a r a s t h e v e l o c i t y f i e l d i s c o n c e r n e d b u t i n c l u d i n g a s h e a r s t r e s s o n t h e b o u n d a r i e s f o r t h e s t r e s s c a l c u l a t i o n . T h i s m o d e l i s i l l u s t r a t e d i n F i g . 2 . 2 . 1 . 14 A f o r c e b a l a n c e on t h e d i f f e r e n t i a l e l e m e n t w i t h = C X , tj ) g i ves (aGxx/dxJ-TVL (2 .2 .1 ) where x i s the boundary shear s t r e s s , and the h o r i z o n t a l s t r e s s w h i c h i s i n d e p e n d e n t o f s p e c i m e n h e i g h t y . I f Coulomb f r i c t i o n i s assumed then /"=-?6 y y (2 .2 .2 ) w h e r e Q y y i s t h e y c o m p o n e n t o f s t r e s s a n d J i s t h e c o e f f i c i e n t o f f r i c t i o n . The nega t i ve s i g n i s i n t r o d u c e d s i n c e Qyy i s c o n s i d e r e d p o s i t i v e when t e n s i l e . The y i e l d c o n d i t i o n f o r a r a te — i n s e n s i t i v e n o n - s t r a i n ha rden ing i s o t r o p i c m a t e r i a l i s 6~=Y0 (2 .2 .3 ) where ~G « / ( 3 / 2 ) ( S £ x + S y y ) ( 2 .2 .4 ) The Levy-Von Mises f low r u l e can be used to show t h a t ( 2 .2 .5 ) where GZ2\ t * i e s t r e s s i n the d i r e c t i o n normal t o the p lane o f d e f o r m a t i o n . T h i s r e q u i r e s tha t P= (1 /2 ) ( G x x + Gyy) ( 2 .2 .6 ) The r e s u l t 2 .2 .6 can be used w i th equa t ions 2 . 2 . 4 , 2 .2 .3 and t h e f a c t t h a t 6" =S„ +P and € y v = S y v + P t o w r i t e the y i e l d 15 c o n d i t i o n i n the form lG X x~Gyyl=2K (2 .2 .7 ) where K=Y0/J~3. K i s the y i e l d shear s t r e s s . Equa t ion 2 .2 .7 i m p l i e s tha t ( d £ * x / d x ) = (d Gyy/dx) s i n ce K i s c o n s t a n t . Equa t ion 2.2.1 can thus be w r i t t e n , wi th 9^ rep laced from equa t ion 2 .2 .2 dGyy/dx = - ? £ y y / L (2 .2 .8 ) T h i s can now be i n t e g r a t e d as (dGyy/Gyy )=-(?/L) X = W U dx (2 .2 . 9 ) W The i n t e g r a t i o n can be c a r r i e d out us ing the f a c t that Gyy at x=W (on the boundary) i s -2K. S ince K i s r e q u i r e d to be p o s i t i v e the negat i ve s ign i n d i c a t e s a compress ive s t r e s s . The i n t e g r a t i o n i s c a r r i e d out as l n ( £ y y ) | * = ( ? / D (W - x ) Thus ln[Gyy/(-2K)]=(7/D[W-x] or £ y y = - 2 K [ e x p { ( ? / L ) ( W - x ) } ] (2 .2 .10 ) Equa t ion 2 .2 .10 i n d i c a t e s a f r i c t i o n h i l l type of d i s t r i b u t i o n ; tha t i s , the magnitude of 6 y y i n c r ea ses towards the c en t r e of the spec imen. The r e s u l t can be w r i t t e n in non-d imens iona l form as [-6 y y / (2K)3=exp [ % e/L] (2 .2 .11 ) where Ji i s the d i s t a n c e inwards from the r i g h t boundary of the spec imen. A s imple non d imens iona l d i ag ramat i c r e p r e s e n t a t i o n of a f r i c t i o n h i l l d i s t r i b u t i o n i s g iven in F i g . 2 . 2 . 2 . There i s an e x p o n e n t i a l i n c rease in Gyy towards 16 the cen t r e of the spec imen. From the d e r i v a t i o n a q u a l i t i t a t i v e o b s e r v a t i o n can be made. Dur ing compress ion 6xx must always i n c r ea se towards the c e n t r e . I f the average Sxx over a v e r t i c a l p lane in the m a t e r i a l such as AD in F i g . 2.2.1 i s used , then even fo r inhomogeneous de fo rmat ion the average Gxx. must i n c r e a s e a c c o r d i n g to equat ion 2 . 2 . 1 1 . 2 .3 . VELOCITY DISCONTINUITY PATTERNS. Ref . 2 d e s c r i b e s the method by which v e l o c i t y d i s c o n t i n u i t y pa t t e rn s and a s s o c i a t e d hodographs can be c o n s t r u c t e d . For conven ience a b r i e f d e s c r i p t i o n of the method i s p resen ted he re . In the model c o n s i d e r e d in s e c t i o n 2.2 Gxxand Gyy were p r i n c i p a l s t r e s s e s . Under these c o n d i t i o n s , the l i n e s of maximum shear s t r e s s occur at 45° to the x a x i s i n F i g . 2 . 2 . 1 . A v e l o c i t y d i s c o n t i n u i t y pa t t e rn i s c o n s t r u c t e d by assuming tha t the b lock of m a t e r i a l be ing compressed i s formed of r i g i d b l o cks of m a t e r i a l d e f i n e d by boundar ies which are the 45° l i n e s of maximum shear s t r e s s . The b l o cks are c o n s i d e r e d to s l i d e over each other wi th a shear s t r e s s K a c t i n g between the b l o c k s . The power d i s s i p a t i o n at these s l i d i n g i n t e r f a c e s thus r e p r e s e n t s the i n t e r n a l power which can be equated to the power of the e x t e r n a l l oads to o b t a i n an es t imate of the forming l oad r e q u i r e d . Thus f o r a b lock w i th H/D=1, a v e l o c i t y d i s c o n t i n u i t y p a t t e r n and hodograph can be c o n s t r u c t e d as shown in F i g . 2 . 3 . 1 . The r i g i d b l o c k s ABC and EBF in F i g . 2.3.1 move v e r t i c a l l y w i th un i t v e l o c i t y . The b locks FBC and AEB move h o r i z o n t a l l y . S l i p thus o c c u r s , f o r example, between b l o c k s 17 ABC and CBF a long BC. The hodograph i s c o n s t r u c t e d by f i r s t drawing an arrow to r ep resen t the v e r t i c a l v e l o c i t y of b lock ABC. An arrow V^ 3 i s then drawn to represen t the h o r i z o n t a l v e l o c i t y of b lock CBF. V 2 2 i =^2~on the hodograph i s the r e l a t i v e v e l o c i t y between the two b l o c k s ABC and CBF. I t i s next assumed that the t r a c t i o n due to the f r i c t i o n between the b locks i s K, the y i e l d s t r e s s of the m a t e r i a l in shea r . For a b lock of unit t h i c k n e s s the t o t a l power d i s s i p a t i o n i n BC i s thus ^2K t imes the l eng th BC which i s {T. 2K i s thus the t o t a l power d i s s i p a t i o n in BC, and 8K i s the t o t a l power d i s s i p a t i o n in a l l the v e l o c i t y d i s c o n t i n u i t i e s . Let 6yy be the normal t r a c t i o n s on AC and EF . The e x t e r n a l power a s s o c i a t e d wi th compress ing the b lock i s 4Cyy. S ince t h i s must equa l the i n t e r n a l power d i s s i p a t i o n 8K, thenGyy=2K i s the es t ima ted t r a c t i o n on AC and EF 2 . 3 . 1 . A V e l o c i t y D i s c o n t i n u i t y Pa t t e rn For H/D^2 For i n t e g r a l v a lues of H/D£2, the b lock in F i g . 2.3.1 can be used to b u i l d up a v e l o c i t y d i s c o n t i n u i t y p a t t e r n . For H/D=2, f o r example, the p a t t e r n shown in F i g . 2 .3 .1 .1 c o u l d be used . I t i s c l e a r tha t EF has ze ro v e l o c i t y here and hence the power d i s s i p a t i o n in AEFC i s on l y one h a l f of tha t in F i g . 2 . 3 . 1 . S ince b lock EHIF has equa l power d i s s i p a t i o n to b lock AEFC however, the t o t a l power d i s s i p a t i o n in the b lock AHIC in F i g . 2 .3 .1 .1 i s the same as i t was in b lock AEFC in F i g . 2 . 3 . 1 . Gvy=2K i s thus s t i l l the compress ive su r f a ce t r a c t i o n on the p l a t e n f a c e s . For non i n t e g r a l v a lues of H/D£2, the s i t u a t i o n 18 becomes s l i g h t l y more complex. F i g . 2 . 3 . 1 . 2 shows one p o s s i b l e v e l o c i t y d i s c o n t i n u i t y p a t t e r n f o r H/D=3.35. B a s i c a l l y t h i s i s c o n s t r u c t e d by f i r s t drawing the pa t t e rn fo r H/D=4 and then r educ ing the he igh t by a l l o w i n g the l i n e s to o v e r l a p in a way tha t i s obv ious from F i g . 2 . 3 . 1 . 2 . The hodograph shows tha t the r e l a t i v e v e l o c i t i e s between a l l the b l o cks i s /2?4. The t o t a l power d i s s i p a t i o n a long BE and BG, fo r example, i s ijlk/l/'4)K or K. The t o t a l power d i s s i p a t i o n in AOIE i s thus 2K, and fo r the whole b lock JKLE i s 8K. S ince the same i n t e r n a l power as found p r e v i o u s l y h o l d s , the same forming load i s o b t a i n e d . 2 . 3 . 2 . A V e l o c i t y D i s c o n t i n u i t y Pa t t e rn For 1^H/D^2 F i g . 2 .3 .2.1 shows a proposed v e l o c i t y d i s c o n t i n u i t y p a t t e r n and hodograph f o r t h i s c a se . Us ing the same procedure as p r e v i o u s l y , £ y y = 2 K i s aga in ob t a i ned for the compress ive s t r e s s . 2 . 3 . 3 . A V e l o c i t y D i s c o n t i n u i t y Pa t t e rn For H/DS1 A v e l o c i t y d i s c o n t i n u i t y p a t t e r n and hodograph fo r H/D<1 i s shown in F i g . 2 . 3 . 3 . 1 . In t h i s p a r t i c u l a r case H/D=0.516 was chosen fo r i l l u s t r a t i o n . Aga in Gyy=2K i s ob t a i ned fo r the compress ive s t r e s s . 19 • F I G . Z . \ . \ . \ . ILUVJ fTT KTI 0 N 0 F THE. H O K O G F . M E Q U S Pi_ftMEl E F F O R M A T I f l N O f A R L O C k O F J ^ n T E R T P\V_ R F T W F F N T W O E"R T C T T ON'.L.E S <; A N D R T f n i n Pl A - T F ^ «=, 20 I - M e - M o W I T H F R I C T I O N A S S U M E D T L C L . A C T O N T H E P L A T E N S . 21 I L L y S T R W I O N O F ^ F U C T 1 0 N H T U U N o K t A A U S T R E S S U I S T M B U T I O N 22 F I G . 2..3>. 1 A V E L O C I T Y D I 5 C O N T l N U I T Y P A T T E R M F O R T H E . C A 5 EL V\/D = 1 23 F I G . 2 . . 3 > . 1 . 1 . A V E L O C I T Y D I S C O N T l K i U X T y P A T T E R N F O R T H EL C A S E L H / D = 2L . 24 V 0=V,3 = \ F I G . 2 . 3 . 1 . 2 . . O N E P O S S I B L E . V E L O C I T Y D I S C O N T I N U I T Y P A T T E R N F O R P E R F E C T L Y ? L A S T I C P L A N E S T R A I N C O M P R E S S I O N B E T W E E N R I G I D A N D P A R A L L E L P L A T E us . H / D > 2 _ I S A S S U M E D I N T H E DIAGRAM . 25 i Vo =V.2-= \ F I G . 2. 3. a. V -/K POSSIBLE. V E L O C I T y D I S C ONTIN UIT)' P A T T E R N FOR P E R F E C T L Y P L A S T I C P L A N S T R A I N C O M P R E S S I O N BETWEEN! R I G I D AND P A R A L L E L P L A T E N S - \ ^ H /t) ^  2. I S A S S U M E D IN THE DIAGRAM . V l o d o o j r a \ D L F I G - 2 . 3 . 3 , I . A P O S S I B L E : V E u o c i r y D I S C O N T I N U I T Y P A T T E R N F O R P E R F E C T L Y P L A S T I C P L A N E S T R A I N C O M P R E S S I O N B E T W E E N R I G I D AHb P A R A L L E L P L A T E N S . H/D < | 1 s A S S U M E D I M T H E T^I^G RAM .• 27 CHAPTER 3  THE FINITE ELEMENT FORMULATION. 3 .1 . INTRODUCTION The F i n i t e Element Method (FEM) i s now we l l e s t a b l i s h e d as a power fu l numer i ca l techn ique fo r s o l v i n g problems i n s o l i d mechanics wi th complex boundary c o n d i t i o n s . One purpose of t h i s work i s to deve lop a s p e c i a l i s e d e f f i c i e n t code capab le of be ing e a s i l y adapted to m o d e l l i n g the complex problems that occur in metal forming a p p l i c a t i o n s . These a p p l i c a t i o n s i n vo l ve very l a rge p l a s t i c de fo rmat ions and very complex f r i c t i o n boundary c o n d i t i o n s . These f a c t o r s must be c o n s i d e r e d in c o n j u c t i o n wi th s t r a i n harden ing and s t r a i n ra te s e n s i t i v i t y of the m a t e r i a l . In a d d i t i o n , dynamic e f f e c t s must be taken i n t o account f o r r a p i d meta l forming a p p l i c a t i o n s . The f o rmu l a t i on of the code i s based on e x i s t i n g methods of a n a l y s i s . Tomita and Sowerby(29) have a p p l i e d the f o rmu l a t i on to metal forming prob lems . A b r i e f o u t l i n e of the f i n i t e element theory i s g iven in t h i s chapte r as the r a t i o n a l e f o r the s e l e c t i o n of the approach . S p e c i a l i s e d procedures fo rmula ted by the author w i l l be d i s c u s s e d in f u l l but s tandard ones w i l l on l y be d i s c u s s e d b r i e f l y . 3 .2 . FORMULATION OF THE FINITE ELEMENT EQUATIONS. A rev iew of continuum mechanics i s g i ven by Mase(30) . However, f o r completeness and fu tu re r e f e r e n c e r e l e van t equa t ions of continuum mechanics that are to be so l ved are g i v en h e r e . I t i s r e c a l l e d t h a t the r e c t a n g u l a r c a r t e s i a n c o o r d i n a t e s , x . , and Cauchy s t r e s s e s , (7*. . , a re u s ed . 28 The equation of motion governing mechanical behaviour i s - f 0"; = o (3.2.D and the continuity equation for an incompressible material i s v - ^ i =0 (3.2.2) It i s r e c a l l e d that the components of deformation rate are given by D.l3 = ( l / 2 ) ( V i ^ +v. <. ) (3.2.3 and the e f f e c t i v e deformation rate i s given by D= tf(2/3)D; JDj J (3.2.4) The e f f e c t i v e p l a s t i c s t r a i n i s "e^Ddt (3.2.5) and the y i e l d function to be sp e c i f i e d for each material i s given by Y=Y(e,D) (3.2.6) The d e f i n i t i o n of a generalized v i s c o s i t y i s given by y = (2Y)/(3D) (3.2.7) and the Levy-Von Mises flow rule by S U S F D"U (3.2.8) I n t h e a b o v e e q u a t i o n s C ; j i s t h e C a u c h y S t r e s s T e n s o r , p t h e m a s s d e n s i t y , v . t h e v e l o c i t y , a n d S . . t h e d e v i a t o r i c C a u c h y s t r e s s t e n s o r . T h e f i n i t e e l e m e n t m e t h o d t r a n s f o r m s t h e e q u a t i o n s 3.2.1 a n d 3.2.2 t o a s y s t e m o f a l g e b r a i c e q u a t i o n s . T h e e q u i v a l e n t p r i n c i p l e s o f v i r t u a l p o w e r g i v e n b e l o w a r e u s e d f o r t h i s p u r p o s e . T h e e q u a t i o n t h u s d e r i v e d i s \i>5?Cr;Sff-. JV- k ; i i v = 0 ( 3 - 2 - 9 ) JPdV=0 (3.2.10) V w h e r e t ^ i s s u r f a c e t r a c t i o n p e r u n i t a r e a A , v^ d e n o t e s t h e d e r i v a t i v e o f v w i t h r e s p e c t t o t i m e , V i s v o l u m e . 6 v . d e n o t e s a r b i t r a r y v a r i a t i o n s o f t h e x v e l o c i t y c o m p o n e n t s w h i c h a r e s u b j e c t t o t h e c o n s t r a i n t g i v e n i n 3.2.10. T h e a r b i t r a r y v a r i a t i o n i n p r e s s u r e , 6 P , a n d d e f o r m a t i o n r a t e c o m p o n e n t s 6 D ^ _. c a n b e e x p r e s s e d i n t e r m s o f 5 v . u s i n g e q u a t i o n 3.2.3. T h e c o m p o n e n t s o f s u r f a c e t r a c t i o n t ^ a r e o b t a i n e d f r o m t h e C a u c h y s t r e s s c o m p o n e n t s Q~y' b y £ I - tyy Hj w h e r e n.. a r e t h e c o m p o n e n t s o f t h e o u t w a r d u n i t n o r m a l t o t h e s u r f a c e . v ^ i s n o w e x p a n d e d w i t h r e s p e c t t o a s e t o f d i s c r e t e n o d a l v a l u e s v > p, r e f e r r e d t o a. s e t o f b a s i s f u n c t i o n s v Tf<*i>v- ( 3 . 2 . 1 1 ) w h e r e s u m m a t i o n o v e r t h e n o d e s Q i s i m p l i e d . 1 " f ( j / x i ) i s t h e b a s i s f u n c t i o n o f t h e g l o b a l c o o r d i n a t e s x ^ f o r g l o b a l n o d e T h e s u m m a t i o n r u l e o n r e p e a t e d i n d i c e s w i l l b e a s s u m e d t h r o u g h o u t t h i s c h a p t e r a n d f u t u r e c h a p t e r s u n l e s s o t h e r w i s e s t a t e d . Q and s a t i s f i e s t h e c o n d i t i o n i ^ ( x ^ ) = l when x^ are the co o r d i n a t e s f o r node Q. At a l l other nodes the value of t h i s f u n c t i o n i s zero. The f u n c t i o n v a r i e s c o n t i n u o u s l y between nodes and i s chosen t o g i v e only l o c a l support t o the v e l o c i t y f i e l d . S i m i l a r l y , the h y d r o s t a t i c p r essure i s an independent v a r i a b l e i n a r i g i d - p l a s t i c f o r m u l a t i o n . L e t a d i f f e r e n t s e t o f nodes L be d e f i n e d f o r p r e s s u r e , and l e t P be the pre s s u r e at the L'th p r e s s u r e node. These pressures can be expanded with r e s p e c t t o a s e t of b a s i s f u n c t i o n s , ^ as f o l l o w s P=Vl_Pl_ (3 .2 .12 ) I f the expansions 3.2.11 and 3.2.12 are s u b s t i t u t e d i n t o e q u a t i o n s 3.2.9 and 3.2.10 r e s p e c t i v e l y , and c o n s i d e r a t i o n i s given of the f a c t t h a t the v a r i a t i o n s Cv^ and £p are a r b i t r a r y , the r e q u i r e d f i n i t e element equations r e s u l t . These equations are l i s t e d below. In these equations QR and S_T rep r e s e n t s i n g l e i n d i c e s denoting, r e s p e c t i v e l y , the g l o b a l v a r i a b l e number of the R'th degree o f freedom a t g l o b a l node Q and the T' t h degree of freedom at g l o b a l node S. L denotes g l o b a l v a r i a b l e number of the s i n g l e degree of freedom p r e s s u r e , at pressure nodes. The summation r u l e on repeated i n d i c e s i n a product term a l s o h o l d s . ( M ^ § z ) ( V s ; r ) + ( C i £ § z ) ( V ^ : ) + ( G ^ J ( P u ) = ( F ^ ) (3 .2 .13 ) < G S l J ( V a t ) - 0 (3 .2 .14 ) 31 [(f )(Ts)(Y'9)]dV (3.2.15) U V ( C « E _ ^ ) = \ [ ( ^ ) ( B " 9 a ) ( B i 3 s T n d V (3.2.16) V t ( f 9 ' R ) ( y j ^ d v (3.2.17) Z T ) (3.2.18) ( V J = (T9)(Y^) (3.2.19) ( D x f f ) B ( B T 3 S T ) ( V £ L ) (3.2.20) Here V S T , V S T , and F S T represent the a c c e l e r a t i o n , v e l o c i t y , and e x t e r n a l f o r c e components i n the T'th degree of freedom at node S. P L i s the h y d r o s t a t i c pressure at the L'th pressure node. ( M ^ ) , ( C ^ ^ ) , and ( G ^ J are the mass, s t i f f n e s s , and c o n s t r a i n t m a t r i c e s , r e s p e c t i v e l y , which w i l l subsequently be denoted [M], [C], and [G]. S i m i l a r l y , ( V S T ) , ( V S T ) , and ( P u ) are the a c c e l e r a t i o n , v e l o c i t y , and pressure m a t r i c e s , r e s p e c t i v e l y , which w i l l subsequently be denoted by {V}, {V}, and {P}. £ R T i s the Kronecker D e l t a (£ R X=1 i f R=T and £ ^ = 0 i f R*T) . A comma denotes d i f f e r e n t i a t i o n with respect to the g l o b a l c o o r d i n a t e s x x . In order to solve equations 3.2.13 and 3.2.14 the a c c e l e r a t i o n must be r e l a t e d to the v e l o c i t y . I n order f o r t h i s to be done a f i n i t e d i f f e r e n c e i n t e g r a t i o n formulae must be adopted . Le t {v} =[(1-0){V} + &{V}. ]A.t+{v} (3 .2 .21 ) where {V}, and {Vj^ are the g l o b a l v e l o c i t y and a c c e l e r a t i o n v e c t o r s at t ime t+At and the v e c t o r s wi thout a s u b s c r i p t " 1 " are those at t ime t . 0 i s a cons tan t tha t can be chosen f o r optimum a c c u r a c y . If equa t i on 3.2.21 i s s u b s t i t u t e d i n t o equa t ion 3.2.13 and then the equa t ions 3.2.13 and 3.2.14 are w r i t t e n in mat r ix form the r e s u l t can be expressed as " # J [ G ] [ G ] T [ 0 ] {v}- "(FT _{0}_ (3 .2 .22 ) where {F}={F}+b c[M](V}+b,[M]{V} (3 .2 .23 ) and [C]=[C]+b e [M] (3 .3 .24) b e =1/[(0) (At ) ] (3 .2 .25) b, = d-0)/0 (3 .2 .26 ) For t h i s case i s symmetric r c ] [ G ] [ G ] T [ 0 ] (3 .2 .27 ) The system 3.2.22 can now be w r i t t e n [A]{Y}={B} (3 .2 .28 ) where {Y}T = [ {V}T , {0}T ] . 33 3.3. DISCUSSION OF ELEMENT SELECTION. r/v, . . . The global matrix [A] in equation 3.2.28 i s b u i l t conveniently by joining the nodal points by l i n e s so as to form elements of the material. The required matrices can then be formed for each element in turn and assembled into the global matrix. The standard four node isoparametric q u a d r i l a t e r a l element was chosen for the application in th i s work (ref. 10). This element was selected since i t i s the simplest f i n i t e element that can deform under plane s t r a i n conditions with constant volume. The basis functions to be l i s t e d below are b i - l i n e a r for the v e l o c i t y f i e l d and unity - for the pressures (or the mean normal stresses). The pressures w i l l thus be constant for each element and hence discontinuous between elements. The element matrices for the four node qua d r i l a t e r a l are formed conveniently by mapping a distorted qu a d r i l a t e r a l to a unit square as depicted otv t h e ^OWOVJiftc 34 1 * v F IG. -3.3.1 I l l u s t r a t i o n of the mapping to s 1 , s 2  c o o r d i n a t e s of a 4 node q u a d r i l a t e r a l element The b a s i s f u n c t i o n s are w r i t t e n in the s, and s L c o o r d i n a t e system as ?!l = ( l - s | - s 1 + s l s l ) / 4 (3 .3 .1 ) 0z.= ( 1 + s v - s^-s , s t ) / 4 (3 .3 .2 ) ^ 3 = ( 1 + S , + S 1 + S , S l ) / 4 (3 .3 .3 ) ' ^ = ( 1 - S , + S t - S , s t ) / 4 ( 3 .3 .4 ) The mapping f u n c t i o n from the l o c a l s x c o o r d i n a t e s to the g l o b a l x x c o o r d i n a t e s i s d e f i n e d as x 1 = (s, , S ; l ) X ( J ) I (3 .3 .5 ) where i s the I' th c o o r d i n a t e of node Q and x x i s the 1 1 t h c o o r d i n a t e of any o ther p o i n t in the m a t e r i a l . A l l element i n t e g r a t i o n s can then be c a r r i e d out over s x c o o r d i n a t e s i n s t e a d of x x on us ing the J acob ian of the t r a n s f o r m a t i o n 3 . 3 . 5 . The i n t e g r a t i o n s are c a r r i e d out u s i n g Gauss ian i n t e g r a t i o n which i n v o l v e s a sum of va lues over s e l e c t e d p o i n t s in the e lement . 35 3 .4 . SPECIFICATION OF GEOMETRY. The f i g u r e below i l l u s t r a t e s how the noda l po in t numbering and c o o r d i n a t e data i s used to s p e c i f y the m a t e r i a l geometry. F IG. 3.4.1 I l l u s t r a t i o n of noda l po in t numbering  fo r e l ements . The c o n n e c t i v i t y data c o n s i s t s of an a r ray C e c j which g i ves the g l o b a l node number of the q ' t h l o c a l node in element e. For element 6 f o r example, C (6 ,1 )=11 , c (6 ,2 )=6 , C (6 ,3 )=5 , C (6 ,4 )=12, and C(6 ,5 )=22. The p r e s s u r e node, f o r the l i n e a r q u a d r i l a t e r a l element ( l o c a l element node number 5) i s always assumed to be at the c en t r e of the e lement . The c o o r d i n a t e data can be s p e c i f i e d in the a r r ay as X 9 1 which g i v e s the I ' t h c o o r d i n a t e at the g l o b a l node Q. Thus X 9 Z w i th Q=Ceq would g i ve the I ' t h c o o r d i n a t e of the q ' t h l o c a l node (Q ' t h g l o b a l node) i n element e. The c o o r d i n a t e and c o n n e c t i v i t y data i s thus s u f f i c i e n t to d e f i n e the r eg ion over which the s o l u t i o n i s r e q u i r e d . 36 3 .5 . MATERIAL CONSTITUTIVE BEHAVIOR. Equat ion 3.2.8 r e l a t e s d e v i a t o r i c s t r e s s to de fo rmat ion r a t e wi th a c o e f f i c i e n t of v i s c o s i t y . A l l q u a n t i t i e s are d e f i n e d except the f u n c t i o n a l r e l a t i o n of the y i e l d s t r e s s i n d i c a t e d i n equa t ion 3 . 2 . 6 . The f u n c t i o n a l r e l a t i o n to be assumed i s s p e c i f i e d below. The f low r u l e and other d e f i n i t i o n s are r e- s t a t ed fo r comp le teness . s i j * = y D ; j ( 3 . 5 . D y*=(2Y)/(3D) (3 .5 .2 ) Y = ( 1 + C l e ) r [ y 0 + c t ( D - D 0 ) C 3 ] i f D>D 0 (3 .5 .3 ) Y=<D i f D<D 0 ( 3 .5 .4 ) where D 0 = ( 1 + c , e ) T y a / o < _ ( 3 .5 .5 ) Here c , , Y » c^, c 5 are cons tan t s fo r the m a t e r i a l . <*c i s an i n i t i a l cons tan t s lope on the Y a g a i n s t "TT curve to prevent Y / D from becoming i n f i n i t e as D—•O. G r a p h i c a l l y the assumed c o n s t i t u t i v e behaviour i s as i l l u s t a t e d \n F I ^ . 2>.S*.t o n -tW«_ ioNovvf\<j Ipa^e. 3 7 D F IG . 3.5.1 I l l u s t r a t i o n of the c o n s t i t u t i v e behav io r assumed. The s t a t i c y i e l d s t r e s s fo r m a t e r i a l that has an e f f e c t i v e s t r a i n e i s Y 0 ( 1 + c , e ) ' . If c ,*0 s t r a i n harden ing w i l l thus o c c u r . The y i e l d s t r e s s may a l s o be s t r a i n ra te s e n s i t i v e . In t h i s case i t i s assumed that the annea led y i e l d s t r e s s i n c r e a s e s by c t ("S"-D"0 ) C a . The suggested combined e f f e c t s of s t r a i n r a t e s e n s i t i v i t y and s t r a i n harden ing was not used in any of the c a l c u l a t i o n s . Only the separa te cases of s t r a i n harden ing wi th no s t r a i n ra te s e n s i t i v i t y ( c ^ O ) or s t r a i n r a te s e n s i t i v i t y w i th no s t r a i n harden ing (c,=0 )were assumed. I t was e s t a b l i s h e d tha t equa t ion 3.5.3 was s u f f i c i e n t to account fo r the v a r i a t i o n of the y i e l d s t r e s s of aluminum wi th s t r a i n harden ing and a l s o the y i e l d s t r e s s of p l a s t i c i n e wi th s t r a i n r a te s e n s i t i v i t y . These m a t e r i a l s were used in t h i s work to compare the computer code p r e d i c t i o n s wi th expe r imen ta l r e s u l t s . 38 3.6 . ELEMENT ASSEMBLY The p rocess of element assembly can be i n d i c a t e d as e e e. f o l l o w s . I f A g r ^ = B q r r ep re sen t s the a l g e b r a i c equa t ions f o r a s i n g l e element on l y and A ( ? r s t Y s ^ . = B ( j ) r tha t fo r the t o t a l s t r u c t u r e , then A i * " s t = ^ A ^ s i (3 .6 .1 ) e. and B ^ = ^ B ^ . (3 .6 .2 ) e. where Q=C e ^ and S = C e s - 1 On r e f e r r i n g to equa t ion 3.2.22 i t can be seen that [A] c o n t a i n s e lements of [C] and [G ] . In the computer code i t i s important to be ab le to s e l e c t i v e l y assemble [ C ] or [G] , For example, on i t e r a t i n g w i th respec t to m a t e r i a l v i s c o s i t y , on ly [C] changes from one i t e r a t i o n to the nex t . [G] and [G]^ are independent of v i s c o s i t y and on ly change when the m a t e r i a l i s i n c r e m e n t a l l y s t r a i n e d . Thus on ly [ C ] needs to be updated on an i t e r a t i o n . I f i n e r t i a i s not be ing c o n s i d e r e d then the update f o r v i s c o s i t y change i s e a s i l y done s i n ce [M]=0 and [C]=[C] from equa t ion 3 .2 .24 . 1 Here A^-vt , which i s an element of [A ] , can be c o n s i d e r e d an i n f l u e n c e c o e f f i c i e n t between the O r ' t h and S t ' t h g l o b a l s o l u t i o n v a r i a b l e s . Qr r e f e r s to the r ' t h s o l u t i o n v a r i a b l e at g l o b a l node Q and jS_t to the t ' t h s o l u t i o n v a r i a b l e at g l o b a l node S. S i m i l a r l y , at the element l e v e l , A ^ . ^ , which i s an element of [A] 6-, can be c o n s i d e r e d an i n f l u e n c e c o e f f i c i e n t between the q r ' t h and s t ' t h element s o l u t i o n v a r i a b l e s . qr r e f e r s to the r ' t h s o l u t i o n v a r i a b l e at element node q and st to the t ' t h s o l u t i o n v a r i a b l e at element node S. In the p resen t f o rmu l a t i on there are 2 v e l o c i t y components at v e l o c i t y nodes and one p r e s su re at p r e s su re nodes . 39 If i n e r t i a i s be ing c o n s i d e r e d , however, i t i s u s e f u l to be ab l e to take advantage of the f a c t that [M] i s independent of v i s c o s i t y and does not have to be r e c a l c u l a t e d on i t e r a t i o n s . The u sua l procedure f o l l owed in the code fo r assembly i s as f o l l o w s . On the f i r s t i t e r a t i o n of an increment (which occu rs on the f i r s t run or j u s t a f t e r the m a t e r i a l has been i n c r e m e n t a l l y s t r a i n e d ) assemble , f o r each e lement , [ C ] e , [G] 6", [M] 5", and [F ]^ . Then form [ C ] ^ from equa t ion 3.2.24 and {F} from equa t ion 3 . 2 . 2 3 . Then form the t o t a l element ma t r i c e s [A] and {B} and assemble i n t o A / Ay the g l o b a l ma t r i c e s to g i ve [A] and [B ] . Whi le assembl ing the element ma t r i c e s a l s o form [M] as a separa te g l o b a l m a t r i x . On the remain ing i t e r a t i o n s a f t e r the f i r s t on an increment assemble on ly [C] i n t o [A ] . A f t e r t h i s assembly add b Q [M] i n t o the [C] submatr ix of [A] to form [A ] , The procedure o u t l i n e d above avo ids a r e f o r m u l a t i o n of IV [M] and {B} which do not change when v i s c o s i t y on ly i s updated a f t e r an i t e r a t i o n on an inc rement . They on l y change when the m a t e r i a l i s i n c r e m e n t a l l y s t r a i n e d . 3 .7 . BOUNDARY CONDITIONS. Much of the m a t e r i a l in t h i s s e c t i o n has been s p e c i a l l y f o rmu la ted fo r t h i s work. Reasonably f u l l d e t a i l s w i l l thus be g i v e n . 40 3 . 7 . 1 . S p e c i f i e d V e l o c i t y and Mean Normal S t r e s s As a r e s u l t of the d i s c r e t i s a t i o n p r o c e s s , the cont inuum problem governed by the equa t ions of motion and c o n t i n u i t y has been r ep l a ced by a d i s c r e t e problem governed by a system of a l g e b r a i c equa t i ons of the form. [A]{Y}={B} ( 3 . 7 . 1 . 1 ) Before i n s e r t i o n of any boundary c o n d i t i o n s the matr ix [A] w i l l be s i n g u l a r . S u f f i c i e n t boundary c o n d i t i o n s must be s u p p l i e d to d e f i n e the p rob lem; in p a r t i c u l a r , a l l r i g i d body mot ions must be removed. Seve ra l d i f f e r e n t types of boundary c o n d i t i o n s w i l l be o u t l i n e d in subsequent s e c t i o n s . Let {Y} T be w r i t t e n [ {Y}^ , {Y}"£ ] such tha t {Y}"J"contains unknown v e l o c i t i e s and p r e s su re s and (Y ) ^ c o n t a i n s s p e c i f i e d v e l o c i t i e s and p r e s s u r e s . The sub-system [ A ] U { Y } ) ={B} t - f A ^ j Y } ^ ( 3 . 7 . 1 . 2 ) r ep r e sen t s the system of equa t ions that i s to be s o l v e d fo r non s p e c i f i e d v a r i a b l e s . The s o l u t i o n {Y}, w i l l a u t o m a t i c a l l y r e f l e c t the requi rement of the s p e c i f i e d nodal v a r i a b l e s . The s o l u t i o n {Y} ( found sub jec t to the s p e c i f i e d {Y}^ w i l l not in gene ra l s a t i s f y the rema in ing pa r t of the system 3 . 7 . 1 . 1 namely [ A L ^ f Y } , = { B } 2 _ - [ A y Y}^ ( 3 . 7 . 1 . 3 ) The r e s i d u a l s fo r each equa t i on in 3 . 7 . 1 . 3 can be i n t e r p r e t e d as the r e a c t i o n (a f o r c e fo r v e l o c i t y or f low ra te fo r p r e s su re ) tha t i s r e q u i r e d to g i ve the s p e c i f i e d v a r i a b l e to which the equa t ion r e f e r s . From a computa t i ona l s t andpo in t i t may be conven ien t to so l ve fo r these r e a c t i o n s d i r e c t l y . T h i s can be.done by 41 w r i t i n g the system 3.7.1.1 in the form [0] { Y } (B},-[A] ( J Y } ^ {B}j,-{A} l t{Y} t ( 3 . 7 . 1 . 4 ) [A] 21 " [ I ] { R } Thus on l y a t r a n s f o r m a t i o n on the mat r ix [A] and load v e c to r {B} i s r e q u i r e d in order tha t the s o l u t i o n of the t rans fo rmed system y i e l d r e a c t i o n s in p l a ce of s p e c i f i e d v a r i a b l e s . I f r e q u i r e d , the symmetry of [A] can be p r e se r ved by s e t t i n g [& ) z x to zero and r e p l a c i n g { R } by {X} say . A f t e r s o l u t i o n when { Y } . and {X} have been c a l c u l a t e d , {R} can be found from In" p r a c t i c e i t may be more conven ien t to adopt the approach d i s c u s s e d above f o r s o l v i n g f o r the r e a c t i o n s d i r e c t l y r a the r than s o l v i n g the sub-system 3 .7 .1 .2 s e p a r a t e l y . T h i s s i t u a t i o n occurs s i n ce on ly a permuted form of [A] i s s t o r e d i n s i d e the computer and a l s o advantage must be taken of the banded s t r u c t u r e of the ma t r i x . 3 . 7 . 2 . Sur face t r a c t i o n s . To account fo r s u r f a c e t r a c t i o n ^ e q u i v a l e n t nodal f o r c e s must be formed by an i n t e g r a t i o n over the s i d e s of the e l ements . The r e s u l t s are then added i n t o the load v e c t o r {F} in equa t ion 3 . 2 . 2 2 . Should ad jacen t e lements have su r f a ce t r a c t i o n or the same element have su r f a ce t r a c t i o n on more than one s i d e , then t h i s a d d i t i o n p rocess w i l l f i n a l l y r e s u l t in the complete l oad v e c t o r s fo r s u r f a c e t r a c t i o n be ing formed. { R } = [ A ] , . { Y } +{X} ( 3 . 7 . 1 . 5 ) 4 2 3 . 7 . 3 . P l a t en F r i c t i o n In the system of equa t ions 3 .7 .1 .2 1 a subscr ipt 1 v a r i a b l e s and 2 known v a r i a b l e s . [A]„ {Y}v ={B}, ~ [ A ] ^ {Y}^ (3 .7 .3 .1 ) c o u l d be s o l v e d as a separa te system and i s i n t e r p r e t e d as the s o l u t i o n f o r {Y} sub jec t to the a p p l i e d f o r c e v e c to r s {B} and a d d i t i o n a l l y - [ A] l 2_ {Y} ^. The l a t t e r f o r c e vec to r can be i n t e r p r e t e d as the e x t e r n a l f o r c e s tha t must be a p p l i e d to make the v a r i a b l e {Y} c o n s i s t e n t with the a p p l i e d c o n s t r a i n t s . The r e s i d u a l s {R} = [A]T, {Y}( + [A]2J,{Y}2_-{B}2_ ( 3 . 7 . 3 . 2 ) are the g e n e r a l i s e d f o r c e s that are r e q u i r e d to ach ieve the c o n s t r a i n t s . Coulomb f r i c t i o n e s s e n t i a l l y re la tes the normal s t ress between two surfaces, F , to the shear s t r e s s , Y , that r e s u l t s due to s l i d i n g contact by a c o e f f i c i e n t of f r i c t i o n according to the fo l lowing r e l a t i o n s h i p ^ = 7 F N ( 3 . 7 . 3 . 3 ) If now motion of a boundary i s s p e c i f i e d in the normal d i r e c t i o n then a normal r e a c t i o n r e s u l t s . The c o e f f i c i e n t of f r i c t i o n t imes the r e a c t i o n w i l l then be a l oad vec to r tha t shou ld c o n s t r a i n the s l i d i n g motion of the s u r f a c e . Let node W be one of the nodes where f r i c t i o n i s to be s u p p l i e d . Le t 2 be the degree of freedom in which v e l o c i t y i s to be s p e c i f i e d and hence the d i r e c t i o n of the normal r e a c t i o n . The F i gu re below summarises the n o t a t i o n fo r the a p p l i c a t i o n be ing c o n s i d e r e d . 43 F IG. 3.7.3.1 I l l u s t r a t i o n of the compress ion in  p lane s t r a i n of a b lock of m a t e r i a l  between two p l a t e n s . S p e c i f i c a t i o n of v e w i l l g i ve a normal f o r ce on the top face BC of the spec imen. The equa t ion of motion of node W in the second degree of freedom occurs in the system 3 .7 .3 .2 wh i l e the equa t ion that governs motion in the f i r s t degree of freedom at W w i l l occur in the system 3 . 7 . 3 . 1 . Now R i s a v e r t i c a l r e a c t i o n f o r c e that r e s u l t s when the v e l o c i t y v 0 i s s p e c i f i e d . The c o e f f i c i e n t of f r i c t i o n , denoted by ^ p , t imes t h i s r e a c t i o n must be added i n t o the l oad v e c to r of the equa t ion tha t governs motion of node W in the h o r i z o n t a l d i r e c t i o n . T h i s c o n s t r a i n t on s l i d i n g i s conven i en t l y " a p p l i e d by p u t t i n g i n t o the d i a g o n a l of [0] i n equa t i on 3 .7 .1 .4 i n the row fo r the h o r i z o n t a l degree of freedom at node W. Dur ing the course of a s o l u t i o n , a f t e r every i t e r a t i o n on every inc rement , the s t a t u s of a l l p l a t e n f r i c t i o n nodes i s examined. On the f i r s t i t e r a t i o n of the f i r s t increment 44 i n the code a l l su r f a ce nodes under the p l a t e n s are made to s t i c k to the su r f a ce by imposing v e l o c i t y c o n s t r a i n t s . From the h o r i z o n t a l r e a c t i o n s that r e s u l t the d i r e c t i o n that nodes want to move in can be de te rm ined . Depending upon the magnitude of the v e r t i c a l r e a c t i o n a node i s e i t h e r l e f t s tuck to the su r f a ce or a l l owed to s l i d e . S l i d i n g i s a l lowed i f the es t imate of the f r i c t i o n a l f o r c e tha t would ho ld i f the node were r e l e a s e d i s l e s s in magnitude than the s t i c k i n g r e a c t i o n f o r c e . I t i s then assumed that the node c o u l d s l i d e wi th the f r i c t i o n f o r ce a c t i n g in the d i r e c t i o n oppos i t e to that of motion a long the s u r f a c e . On i t e r a t i o n s a f t e r the f i r s t , i f a node i s found to be s l i d i n g in the same d i r e c t i o n as the f r i c t i o n f o r c e , i t i s c o n s t r a i n e d to a s t i c k c o n d i t i o n . The code i n c l u d e s r o u t i n e s that a l low a comp le te l y a r b i t r a r y boundary movement. Nodes can s t i c k to the p l a t e n s , s l i d e a long the p l a t e n s , or move away from the p l a t e n s . Nodes a l r eady f r ee are a u t o m a t i c a l l y r e s t r a i n e d i f they aga in reach the p l a t e n s . U n t i l now boundary f r i c t i o n has been d i s c u s s e d w i th the assumpt ion that the c o e f f i c i e n t s of f r i c t i o n at the p l a t e n f r i c t i o n nodes are known. I f t h i s i s not the case i t i s then of i n t e r e s t to be ab l e to p r e d i c t the c o e f f i c i e n t s of f r i c t i o n based upon i n f o r m a t i o n about boundary v e l o c i t i e s in the s l i d i n g d i r e c t i o n . What we have c a l l e d the Master-S lave Node Method i s one approach and i s d i s c u s s e d . Be fore do ing t h i s i t shou ld perhaps be s t a t e d tha t the p r e s c r i p t i o n of boundary v e l o c i t i e s a l l a long a su r f a ce where f r i c t i o n a c t s would not in gene ra l be 45 s a t i s f a c t o r y . T h i s procedure e f f e c t i v e l y t o t a l l y c o n s t r a i n s the boundary and f o r c e s i t to move in a way tha t may not be c o n s i s t e n t . « For example, w i th the assumed c o n s t i t u t i v e behaviour ' i t i s much b e t t e r to min imise the boundary c o n s t r a i n t s and p r e d i c t the observed m o t i o n . A second approach might be to so l ve the problem s e v e r a l t imes wi th d i f f e r e n t c o e f f i c i e n t s of f r i c t i o n and then to p r e d i c t the best one by i n t e r p o l a t i o n . T h i s techn ique would , however, r e q u i r e s o l v i n g the problem s e v e r a l t imes . The Master-S lave Node Method overcomes both these d i f f i c u l t i e s . The method f i r s t r e q u i r e s the cho i c e of a boundary node where f r i c t i o n a c t s . The success of the method i s dependent upon the s l i d i n g v e l o c i t y of the node be ing s e n s i t i v e to f r i c t i o n c o n d i t i o n s a long the whole p l a t en boundary . In the present a p p l i c a t i o n t h i s node occurs at the top r i g h t hand co rne r of the specimen and i s i l l u s t r a t e d in the f o l l o w i n g f i g u r e on tY\e. n e x t ^page . . 46 1 r \ N o d e s F IG . 3 . 7 .3 .2 I l l u s t r a t i o n of the top r i g h t hand  quadrant of a specimen wi th the s l a ve  f r i c t i o n nodes and the master f r i c t i o n node shown. The b a s i c idea of the method i s to impose the measured x 1 component of v e l o c i t y at the master node and then to a d j u s t the c o e f f i c i e n t s of f r i c t i o n at the f r i c t i o n nodes u n t i l the r a t i o of the t a n g e n t i a l to the normal r e a c t i o n s at the master node becomes equa l to the c o e f f i c i e n t of f r i c t i o n at the s l a ve nodes . The c a l c u l a t i o n w i l l now be b r i e f l y o u t l i n e d . Le t F be the sum of the t a n g e n t i a l f r i c t i o n f o r c e s of the s l i d i n g s l a ve nodes a long a p l a t e n and l e t R be the t a n g e n t i a l r e a c t i o n of the master node. A c o e f f i c i e n t of f r i c t i o n ,J, i s c a l c u l a t e d from = (F+R)/(R 5+R M ) where R s i s the t o t a l v e r t i c a l (or normal) r e a c t i o n of the s l a ve nodes and R M i s the v e r t i c a l r e a c t i o n of the master node. 47 C o r r e c t i o n s to 7 are made on the b a s i s of the r a t i o fo r s l i d i n g nodes of the t o t a l t a n g e n t i a l f o r ce to the t o t a l normal f o r c e . It was found n u m e r i c a l l y that |R/RM| approached 7 on i t e r a t i o n s . 3 . 7 . 4 . G l a s s F r i c t i o n . Du r ing the s o l u t i o n of the system of equa t ions 3.2.22 the noda l p r e s s u r e s are ob ta ined as b a s i c s o l u t i o n v a r i a b l e s . T h i s p r e s su re i s the mean normal s t r e s s and , f o r p lane s t r a i n c o n d i t i o n s , i s the p res su re d i s t r i b u t i o n tha t the g l a s s p l a t e s en fo r ce on the m a t e r i a l . The assumpt ion of Coulomb f r i c t i o n a l l ows a shear s t r e s s to be c a l c u l a t e d from t h i s normal p r e s s u r e . An i n t e g r a t i o n over the face of each element in con tac t w i th the g l a s s p l a t e s •then a l l ows a set of c o n s i s t e n t noda l f o r c e s to be c a l c u l a t e d from the p r i n c i p l e of v i r t u a l power. These nodal f o r c e s w i l l be r e f e r r e d to the v e l o c i t y nodes and w i l l be r e l a t e d d i r e c t l y by ma t r i c e s to be d e r i v e d to p r e s su r e s at p r e s su re nodes . T h i s l a t t e r f a c t w i l l a l low the load terms to be added to the equa t ions on the l e f t s i de r a the r than the r i g h t by a t r a n s f o r m a t i o n on the g l o b a l mat r ix that i n v o l v e s s u b t r a c t i n g terms o f f the c o n s t r a i n t mat r ix [G] in equa t ion 3 . 2 . 2 2 . As t h i s approach i s e n t i r e l y unique to t h i s work f u l l d e t a i l s w i l l now be p r e s e n t e d . F i g . 3 .7.4.1 d e p i c t s m a t e r i a l de forming in p lane s t r a i n between g l a s s p l a t e s . 48 F IG. 3.7.4.1 I l l u s t r a t i o n of m a t e r i a l deforming in  p lane s t r a i n between g l a s s p l a t e s . The sma l l a rea AA i s con t a i ned w i t h i n a f i n i t e element e and i s s l i d i n g between the g l a s s p l a t e s in a d i r e c t i o n g i ven by the u n i t v e c t o r u ^ . The g l a s s f r i c t i o n , which a c t s on both s i d e s of the e lement , must be in the d i r e c t i o n - u ^ . On t a k i n g i n t o account that the p r e s s u r e , P, exe r t ed by the g l a s s p l a t e s i s nega t i ve when compress ive one can deduce tha t the net g l a s s f r i c t i o n f o r c e LF^ a c t i n g on the m a t e r i a l ABCD wi th a rea AA in con tac t w i th the f ron t and back p l a t e s i s A F i = - 2 ^ P u i A A (3.7.4.1) where i s the c o e f f i c i e n t of f r i c t i o n f o r the m a t e r i a l be ing compressed fo r s l i d i n g a long the g l a s s p l a t e s . I f $ v i i s an a r b i t r a r y v i r t u a l v a r i a t i o n of v e l o c i t y f o r the d i f f e r e n t i a l element the v i r t u a l power i n vo l v ed w i l l be 2 (^Pu-j^A) £ V ; L • If t h i s d i f f e r e n t i a l power i s i n t e g r a t e d over the f i n i t e element the v i r t u a l power, , 49 a s s o c i a t e d wi th the element i s ob ta ined as *W e=2 ( ? g P u i § v i ) d A C ( 3 . 7 . 4 . 2 ) where the i n t e g r a t i o n i s taken over one face of the f i n i t e element in con tac t w i th the g l a s s . The s e r i e s expans ions v_^  ='fq)b^)± and T U P L . r o r v e l o c i t y and p ressu re as g i ven in equa t i ons 3.2.11 and 3.2.12 can be s u b s t i t u t e d i n t o 3 .7 .4 .2 to g i ve <jWe=[2 ( ^ u r T 9 V y J d A e ] P u ^ V 9 R ( 3 . 7 . 4 . 3 ) where u. i s l v e l o c i t y v— the u n i t v e c t o r i n the d i r e c t i o n of the p a r t i c l e The g l a s s f r i c t i o n a c t i n g over the element e i s thus e q u i v a l a n t to a g e n e r a l i s e d f o r c e v e c to r F ^ such that $We=F(pr^ , where fcv^r i s a v i r t u a l noda l f o r c e v e l o c i t y v e c t o r . Thus F e =[2 ( ^ u r r 9 y j d A e ] P L A ^ ( 3 . 7 . 4 . 4 ) {F}£ = [X] e{P}^ where F^ r i s the r ' t h component of g e n e r a l i s e d f o r c e at node Q. T h i s can be w r i t t e n in mat r ix n o t a t i o n as ( 3 . 7 . 4 . 5 ) On assembl ing the f o r c e v e c to r in 3 . 7 . 4 . 5 i n t o the equa t ion 3.2.22 one o b t a i n s the system •{vv " [C] [G]-[X] [ G ] T [0] _ ru {F} {0} (3 .2 .22 ) Thus a t r a n s f o r m a t i o n on the mat r ix [A] in equa t ion 3.2 .28 i s r e q u i r e d to account fo r g l a s s f r i c t i o n once the 50 e q u i v a l e n t element l oad v e c t o r s have been c a l c u l a t e d . A d d i t i o n a l c o n s t r a i n t s are r e q u i r e d i f both, s t i c k i n g and s l i d i n g f r i c t i o n are to be accounted f o r . Let v^denote noda l v e l o c i t y , F^nodal f o r ce due to s l i d i n g f r i c t i o n , and R. noda l r e a c t i o n c a l c u l a t e d fo r a s t i c k c o n d i t i o n . The a f o l l o w i n g c r i t e r i a were found to be s a t i s f a c t o r y . A) . I f the node i s s l i d i n g c a l c u l a t e v.F.. If v . F > 0 , app ly a v e l o c i t y r e s t r a i n t of ze ro to s imu la t e a s t i c k c o n d i t i o n . Otherwise con t i nue to a l low the node to s l i d e . In the case of nodes in con t a c t wi th the p l a t e n s t h i s c r i t e r i o n i s not used. In t h i s case the g l a s s f r i c t i o n f o r c e i s added onto the p l a t e n f r i c t i o n fo r ce and then the d e c i s i o n on r ecap tu re to a s t i c k c o n d i t i o n made by the p l a t e n f r i c t i o n r o u t i n e s as o u t l i n e d in s e c t i o n 3.7.3. If an e x t e r n a l boundary c o n d i t i o n e x i s t s at the nodes , such as the downward v e l o c i t y of the top p l a t e n , then the boundary c o n d i t i o n i s ma in ta ined r e g a r d l e s s of the g l a s s f r i c t i o n s i n ce i t i s then i m p l i c i t l y assumed that the r e q u i r e d e x t e r n a l f o r c e to ma in ta in the v e l o c i t y c o n s t r a i n t i s always be ing a p p l i e d . B) . In the case of a node that i s s tuck to the g l a s s p l a t e s the ang le 9 between R. and F. i s c a l c u l a t e d from c o s ' 1 [R L F i / ( \ RJUEJ)) ] . I f 0 i s between 90° and 270° i t i s assumed that the node would move in the d i r e c t i o n of the v e l o c i t y i f r e l e a s e d . In t h i s case i t i s thus c o n s t r a i n e d to ze ro v e l o c i t y . If the angle 0 i s l e s s than 9 0 ° , i t i s a l lowed to s l i d e p r o v i d e d a l s o that \ F .J<) R J C O S ( 8 ) . T h i s 51 l a t t e r c h e c k i s t o e s t i m a t e whether the r e a c t i o n f o r c e i n t h e d i r e c t i o n of F ^ i s l e s s t h a n th e m agnitude of F^. A g a i n p l a t e n f r i c t i o n nodes a r e d e a l t w i t h by p l a t e n f r i c t i o n r o u t i n e s w i t h th e g l a s s f r i c t i o n f o r c e t a k e n i n t o a c c o u n t . An a d d i t i o n a l c o n s i d e r a t i o n t h a t a r i s e s i n t h e t r e a t m e n t of g l a s s f r i c t i o n i s t h e c a s e where an e l e m e n t has a l l of i t s v e l o c i t y nodes c o n s t r a i n e d t o a s t i c k c o n d i t i o n . In t h i s c a s e th e e l e m e n t p r e s s u r e ( w h i c h i s c o n s t a n t f o r t h e f o u r node q u a d r i l a t e r a l ) becomes i n d e t e r m i n a t e . T h i s s i t u a t i o n o c c u r s beacause t h e c o n t i n u i t y e q u a t i o n i s s a t i s f i e d a u t o m a t i c a l l y and t h e i n c o m p r e s s i b i l i t y c o n s t r a i n t c a n n o t be i n v o k e d by d e t e r m i n i n g th e e lement p r e s s u r e . C o m p u t a t i o n a l l y , a p p l i c a t i o n of t h e method d e s c r i b e d i n s e c t i o n 3.7.1 f o r t r a n s f o r m i n g t h e m a t r i x [A] t o c o n s t r a i n b o u n d a r y c o n d i t i o n v e l o c i t i e s f o r t h e e l e m e n t would l e a d t o a l i n e of z e r o s i n t h e m a t r i x making i t s i n g u l a r . In t h i s c a s e t h e e l e m e n t p r e s s u r e p r e v i o u s l y c a l c u l a t e d f o r t h e e l e m e n t i s a p p l i e d as a b o u n d a r y c o n d i t i o n . The p r o c e d u r e o u t l i n e d i n s e c t i o n 3.7.1 t h e n l e a d s t o a f l o w r a t e b e i n g d e t e r m i n e d as a r e a c t i o n . F o r a s t i c k c o n d i t i o n , t h e f l o w r a t e would be d e t e r m i n e d as z e r o t h u s s a t i s f y i n g c o n t i n u i t y . The a d v a n t a g e of t h i s p r o c e d u r e i s t h a t t h e s i n g u l a r i t y i s e a s i l y removed i n t h e c a l c u l a t i o n . 52 3.8 . NONLINEAR GEOMETRY. A p i e cew i se cons tan t i n c rementa l approach was adopted fo r t a k i n g i n t o account the geometry change of the f i n i t e element meshes due to s t r a i n . The v e l o c i t y f i e l d was c a l c u l a t e d keeping the f i n i t e element mesh f i x e d and then the v e l o c i t i e s were m u l t i p l i e d by a t ime increment to g i ve a new d i s t o r t e d mesh. The v e l o c i t i e s would then be r e - c a l c u l a t e d and the p rocess r epea t ed . The t ime increment was a u t o m a t i c a l l y c o n t r o l l e d by the program. The user s p e c i f i e s the maximum s t r a i n he w i l l t o l e r a t e . The program w i l l then a u t o m a t i c a l l y choose the t ime increment to keep the s t r a i n at the s p e c i f i e d t o l e r a n c e . The p i ecew ise cons tan t geometry approach n e g l e c t s c e r t a i n terms that would occur i n a piecewise l i n e a r approach . Dur ing mesh d i s t o r t i o n , f o r example, the p r i n c i p a l s t r e s s v e c t o r s r o t a t e and change in magni tude. Incrementa l s t r e s s e s thus r e s u l t t h a t , in the p i ecew ise cons tan t approach , were not taken i n t o account in the v i r t u a l power e q u a t i o n . S ince these i nc rementa l s t r e s s e s do c o n t r i b u t e to the power the s i z e of the s t r a i n increment must be sma l l enough to make these inc rementa l s t r e s s e s n e g l i g i b l e . 53 CHAPTER 4  THE EXPERIMENTAL STUDY. 4 . 1 . INTRODUCTION. In the expe r imen ta l work a study was undertaken of the r a p i d p lane s t r a i n compress ion of a r e c t a n g u l a r b lock of p l a s t i c i n e between f l a t , p a r a l l e l , and r i g i d p l a t e n s . Specimens of v a r i o u s he igh t to d iameter r a t i o s (H/D) were used and d i f f e r e n t compress ion speeds s e l e c t e d . Specimen deformed p r o f i l e s were examined from h igh speed photographs taken at wide range, and d e t a i l e d boundary motion was examined from h igh speed photographs taken at c l o s e range. The exper imenta l r e s u l t s ob ta ined are used as a b a s i s of compar ison wi th the r e s u l t s of c a l c u l a t i o n s ob ta ined w i th the f i n i t e element code d i s c u s s e d in chapter 3. The d e t a i l s of the expe r imenta l work are o u t l i n e d in the f o l l o w i n g s e c t i o n s . In S e c t i on 4.2 the equipment used i s d i s c u s s e d . In S e c t i on 4.3 an o u t l i n e of the t e s t procedure i s d i s c u s s e d . In S e c t i on 4.4 the procedure fo l l owed to c a l i b r a t e e f f e c t i v e s t r e s s to e f f e c t i v e s t r a i n r a te fo r p l a s t i c i n e i s d i s c u s s e d . F i n a l l y , in S e c t i o n 4 . 5 , the expe r imen ta l r e s u l t s are p resen ted and gene ra l o b s e r v a t i o n s made r ega rd ing the de fo rmat ion p a t t e r n s . 4 . 2 . EQUIPMENT. A photograph of the Dynamic Impact P ress (DIP) i s g i ven in F i g . 4 . 2 . 1 . ( a ) , and a schemat ic drawing of the DIP i s shown in f i g . 4 . 2 . 1 ( b ) . T h i s p r e s s was des igned and b u i l t by the author's a d v i s o r P r o f e s s o r G .W .V i cke r s . It c o n s i s t s of a l a rge motor d r i v e n wheel a t t a ched to a Whitworth qu i ck r e tu rn mechanism. The ma in p u r p o s e of the 54 d r i v e wheel was to d r i v e a cam a long the cam g u i d e s . At each end of the cam gu ides were p l a ced o p t i c a l sensors , OS 1 and 0S2, which enab led a sequence of e l e c t r o n i c a l l y c o n t r o l l e d events to occur d u r i n g a t e s t . P r i o r to i n i t i a t i o n of the t e s t c y c l e the cam jus t f r e e l y moved up and down the cam guides as the d r i v e wheel r o t a t e d . When the t e s t sequence was i n i t i a t e d , 0S1 was a c t i v a t e d . When 0S1 de t e c t ed the cam the s o l e n o i d was e n e r g i s e d . When the cam next reached the top of the cam g u i d e s , the s o l e n o i d mechanism engaged and the c y c l o i d a l cam was p u l l e d down through the cam f o l l o w e r s . T h i s in tu rn caused the upper p l a t e n to move down with cons tan t v e l o c i t y onto the spec imen. When the cam, which was now a t t a ched to the c y c l o i d a l cam, aga in reached the top of the cam gu ides the c y c l o i d a l cam was r e l e a s e d . In order to photograph specimens under dynamic c o n d i t i o n s , h igh speed photography at about 2000 frames per second (FPS) was r e q u i r e d . The h igh speed camera used was a Hycam r o t a t i n g p r i sm type which used J6mm f i l m and was capab le of o p e r a t i n g at 10,000 FPS. The camera was des igned to p u l l the f i l m through a f i l m gate at h igh speed wi th the image r e f l e c t e d onto the f i l m by a r o t a t i n g p r i s m . Of importance to t h i s work was the c a p a b i l i t y of s y n c h r o n i z i n g the camera s t a r t pu l se to the compress ion of p l a s t i c i n e by the p l a t e n . F i g . 4 .2 .2 shows a schemat ic d iagram of the l o g i c of the e l e c t r o n i c c i r c u i t des igned fo r 55 t h i s p u r p o s e . 1 When the system was enab led by p r e s s i n g a but ton (P/B) a p rese t camera s t a r t de lay was i n i t i a t e d when the cam reached 0S2 in F i g . 4 . 2 . 1 ( b ) . The time taken fo r the cam to move between 0S1 and 0S2 was the time base fo r t h i s d e l a y . When the cam reached 0S1 the s o l e n o i d was a c t i v a t e d to enable the cam to engage the c y c l o i d a l cam. The sequence then p r e v i o u s l y d e s c r i b e d p roceeded . The camera s t a r t e d a f t e r the p rese t time de l ay and stopped when the end of f i l m was d e t e c t e d . A f t e r the impact , the s o l e n o i d was r e l e a sed when a mechan ica l mic ro sw i t ch (MS) was reached by the cam. T h i s a c t i o n a l lowed the cam to d isengage when i t next reached the s o l e n o i d . The p r e s e t time de lay fo r the camera was set manual ly us ing the S y n c h r o n i z a t i o n C o n t r o l shown in F i g . 4 . 2 . 2 . The i n d i c a t o r f l a s h e d when the two p u l s e s from 0S1 and 0S2 i n d i c a t e d tha t the camera advance s e t t i n g was c o r r e c t . F i g . 4 .2 .3 shows a schemat ic drawing of the specimen h o l d e r . T h i s arrangement was des igned by the author w i th the o b j e c t i v e of enab l i ng a wide f i e l d of view fo r the camera l ens wi th impact at maximum speed onto the spec imen. Adjacent to the specimen ho lde r i s shown a graph of upper p l a t e n v e l o c i t y aga in s t d i s p l a c e m e n t . The curve shows tha t over most of the d i sp lacement path the v e l o c i t y was c o n s t a n t . F i g . 4 . 2 . 4 . i l l u s t r a t e s the expe r imen ta l arrangement tha t was used fo r compress ing specimens in p lane s t r a i n . 1 Cour tesy of John R i c h a r d s , E l e c t r o n i c s T e c h n i c i a n , U n i v e r s i t y of B.C. 5 6 The f r o n t view shows a p l a s t i c i n e specimen in p o s i t i o n . The specimens were t y p i c a l l y 3 inches h i g h , 1.5 inches wide, and 1.375 inches t h i c k . Squares o f 1/8 i n . were a l s o drawn on the s u r f a c e s to enable the movement of p o i n t s to be f o l l o w e d . S c a l e s were mounted on the g l a s s p l a t e s in f r o n t of the specimens and a l s o on the upper p l a t e n . These s c a l e s enab led v e l o c i t i e s and d i s t a n c e s to be determined on the pho tographs . The s i de view in F i g . 4 .2 .4 shows the specimen between the g l a s s p l a t e s . The p l a t e n shown deformed the specimen by p a s s i n g between the g l a s s p l a t e s . A l s o shown in F i g . 4 .2 .4 are v e l o c i t y and d i sp lacement t r a n s d u c e r s . The d i sp lacement t r ansduce r i s e s s e n t i a l l y a l i n e a r v a r i a b l e d i f f e r e n t i a l t r ans fo rmer wi th a 6V DC power supp ly to a s o l i d s t a t e o s c i l l a t o r . The v e l o c i t y t r ansduce r opera ted by magnetic i n d u c t i o n and t h i s r e q u i r e d no power s u p p l y . The outer c a s i n g i s magnet ised and the s e l f - i n d u c e d v o l t a g e was p r o p o r t i o n a l to the v e l o c i t y of the i n s i d e c o i l through the magnetic f i e l d . A DMS 510 D i g i t a l Memoryscope wi th a s i n g l e channel c o u l d be connected to the d i sp l acement t r a n s d u c e r , v e l o c i t y t r a n d u c e r , or load c e l l f o r d i r e c t r e c o r d i n g and subsequent p l o t t i n g . F i g . 4 .2 .4 a l s o i l l u s t r a t e s a t y p i c a l l i g h t i n g arrangement used w i th the camera in p o s i t i o n . A h igh i n t e n s i t y l i g h t was r e q u i r e d fo r the f i l m speeds used . The camera had a t e l e-pho to l ens a t t a ched which enab led very c l o s e examinat ion of the specimens du r i ng d e f o r m a t i o n . F i g . 4 .2 .5 shows some c h a r a c t e r i s t i c s of the h igh 57 speed camera ob ta ined from an examinat ion of t i m i n g marks on the f i l m . 100 f t of f i l m was ab l e to t r a v e l through the camera in about 1.5s. S ince on ly 26ms was t y p i c a l l y r e q u i r e d f o r the t r a v e r s a l of the upper p l a t e n from the h ighes t to the lowest p o s i t i o n , there was adequate time fo r the specimen de fo rmat ion to be cap tu red on on l y a few fee t of f i l m . Dur ing specimen compress ion the f i l m speed would u s u a l l y be about 2300 FPS. In F i g . 4 .2 .6 i s shown c a l i b r a t i o n curves ob ta ined fo r d i sp lacement and v e l o c i t y t r a n s d u c e r s . It was conf i rmed that the manufacturer 's s p e c i f i c a t i o n s on l i n e a r i t y were met over the r eg ion t e s t e d . F i g . 4 .2 .7 shows a c a l i b r a t i o n curve fo r the load c e l l . The l o a d i n g in t h i s case was c a r r i e d out on a T i n i u s O l sen Mechan ica l Load T e s t i n g Machine capab le of l o a d i n g a c c u r a t e l y to 60,000 pounds. From the s e r i e s of d e t a i l e d c a l i b r a t i o n curves c o n s t r u c t e d the load c e l l was found to be a c c u r a t e l y l i n e a r to the maximum s p e c i f i e d l o a d i n g va lue of 25000 pounds. 4 . 3 . TEST PROCEDURE. 4 . 3 . 1 . I n t r o d u c t i o n . The t e s t procedure i n v o l v e d a ) . P r e p a r a t i o n of spec imens, b ) . Photographing the specimens be ing deformed and d e v e l o p i n g the f i l m , and c ) . V iewing the f i l m and d i g i t i s i n g the r e s u l t s . 58 4 . 3 . 2 , P r e p a r a t i o n Of Specimens. P l a s t i c i n e specimens were p repared by f i r s t c u t t i n g them to shape wi th a t h i n s t r and of w i r e . The g r i d l i n e s were marked on the p l a s t i c i n e w i th a S t a e d t l e r Lumocolor pen . As the ink f lows from the t i p of t h i s pen by v e n t u r i a c t i o n min imal p r e s su re was r e q u i r e d between the t i p and the s u r f a c e . The ink was absorbed i n t o the p l a s t i c i n e . It was found tha t the g r i d p a t t e r n s were q u i t e r e s i s t a n t to shea r ing s t r e s s e s from the g l a s s p l a t e s p r o v i d e d no l u b r i c a t i o n was a p p l i e d to the p l a s t i c i n e spec imen. L u b r i c a t i o n w i th l i g h t o i l tended to absorb a t h i n l a y e r of p l a s t i c i n e w i th the r e s u l t tha t the g r i d p a t t e r n s were des t royed w i th on l y a sma l l amount of movement. 4 . 3 . 3 . Photographing The Specimens And Deve lop ing The  F i l m . In c a r r y i n g out the t e s t s a sys temat i c procedure was e s s e n t i a l s i n ce a c o n s i d e r a b l e amount of p r e p a r a t o r y work c o u l d e a s i l y be l o s t . The f o l l o w i n g s teps were s y s t e m a t i c a l l y f o l l owed du r i ng t e s t i n g . F a i r l y s p e c i f i c d e t a i l s are g i ven fo r comp le teness . 1. The impact p ress d r i v e motor and e l e c t r o n i c c o n t r o l system were connected to the main power s u p p l y . 2. The B r idge A m p l i f i e r Meter (BAM) was b a l a n c e d . The BAM was a Wheatstone B r idge a t t a ched to the s t r a i n gages in the l o ad c e l l . A f i n a l ba lance was c a r r i e d out when the BAM had reached a s teady s t a t e thermal c o n d i t i o n . 3. The p l a s t i c i n e specimen was next p l a ced in the impact p ress specimen ho lde r w i th the g l a s s p l a t e s in 59 p o s i t i o n . If r e q u i r e d , the lower p l a t e n was l u b r i c a t e d be fo re p u t t i n g the specimen in p o s i t i o n . 4. With the f l o o d l i g h t s on fo r as shor t a time as p o s s i b l e , the camera was focused wi th a de tachab le f o c u s i n g p r i sm and the ape r tu re set a f t e r measur ing the l i g h t i n t e n s i t y wi th a 1 i g h t m e t e r . The f o c u s i n g pr i sm was then i n s e r t e d d i r e c t l y in the f i l m gate p o s i t i o n and the image that would reach the f i l m obse r ved . Any p o s i t i o n i n g adjustments were then made. 5. The camera was next loaded wi th the f i l m . 100ft r o l l s of Eastman 4-X Negat ive F i l m 7224 were used . T h i s i s 16mm f i l m prepared s p e c i a l l y fo r h igh speed cameras. 6. The camera supply vo l t age was set ( u s u a l l y to 75V) , and the f i l m t im ing l i g h t genera tor was turned on u sua l l y at 1000 pu l s e s per second (PPS). The t im ing marks p l a ced on the f i l m by the pu l se genera tor a l lowed the f i l m speed to be de te rmined . 7. The BAM ba lance was next r e-checked , and the d i g i t a l s to rage scope made ready fo r a c c e p t i n g the pu l se from the load c e l l d u r i n g impact . The scope was ad ju s t ed to s e l f t r i g g e r at an e a r l y p o i n t on the load pu l se and a sweep time of 10ms per scope d i v i s i o n was used . 8. The impact p ress d r i v e motor was next s t a r t e d to g i ve a specimen impact speed of u s u a l l y 4m/s. The s y n c h r o n i z a t i o n t imer fo r the camera advance was then s e t . 9. The camera supply v a r i a c was next turned on . T h i s was not done p r e v i o u s l y to a vo id hav ing the camera s t a r t e a r l y because of an ext raneous pu l se du r i ng the 60 s t a r t i n g p rocedu re . 10. F i n a l l y the f l o o d l i g h t s were turned on and the scope rechecked to ensure tha t no extraneous pu l se had caused i t to t r i g g e r e a r l y . The automat ic photograph ing sequence p r e v i o u s l y d e s c r i b e d was then e n a b l e d . 11. On comple t ion of the t e s t , a hard copy of the load t r a c e cap tu red by the s to rage scope was ob ta ined by a t t a c h i n g an X-Y r e co rde r (model WX4400, Watanabe Instruments C o r p ) . A photograph of the scope sc reen would a l s o u s u a l l y be taken as an a d d i t i o n a l r e c o r d . 12. F i n a l l y , the f i l m was removed from the camera and deve loped in a deve lop ing tank (Nikor F i l m P r o c e s s i n g Machine) us ing Kodak D-76 deve loper (6 minutes at 7 0 ° F ) , a G l a c i a l A c e t i c A c i d s top bath (1 m inu te ) , and Kodak Rapid F i x e r (10 m i n u t e s ) . Th i s p r o c e s s i n g was f o l l owed by 30 minutes of washing in runn ing water . 4 . 3 . 4 . V iewing The F i l m And D i g i t i s i n g The R e s u l t s . A f t e r the f i l m was deve loped i t was examined us ing a 16mm Athena Model 224 p r o j e c t o r . T h i s p r o j e c t o r c o u l d d i s p l a y s i n g l e frames and be p u l s e d at an a r b i t r a r y ra te in e i t h e r d i r e c t i o n . The images were p r o j e c t e d d i r e c t l y onto drawing paper at a reasonab le s i z e and the g r i d p a t t e r n s ske tched d i r e c t l y over the images. From v a r i o u s v e r t i c a l and h o r i z o n t a l s c a l e s a t t a ched to the specimen ho lde r on the impact p r e s s , the s c a l e f a c t o r for each drawing c o u l d be found . I t was e s t a b l i s h e d tha t there was no s i g n i f i c a n t o p t i c a l d i s t o r t i o n in the p r o j e c t e d f rames. I t was a l s o 61 e s t a b l i s h e d that the heat from the p r o j e c t o r lamp when l e f t on a s t a t i o n a r y frame fo r longer than would be r e q u i r e d in ske t ch ing produced no d e t e c t a b l e movement of the image. A f t e r s k e t c h i n g the specimen g r i d p a t t e r n s fo r a s e r i e s of frames ( u s u a l l y co r r e spond ing to equa l t ime i n t e r v a l s ) and e s t a b l i s h i n g the s c a l e fo r each frame, the c o o r d i n a t e s of the p o i n t s on the su r f a ce d e f i n e d by the i n t e r s e c t i o n of g r i d l i n e s were found w i th the use of a d i g i t i s i n g machine suppor ted by the UBC Computing C e n t r e . The c o o r d i n a t e s of the p o i n t s f o r a s e r i e s of g r i d p a t t e r n s cou ld then be used to p l o t , f o r example, a s e r i e s of pa th l i n e s or a v e l o c i t y f i e l d . 4 . 4 . CALIBRATION OF PLASTICINE. Aku, S l a t e r , and Johnson(31) found that p l a s t i c i n e i s very s t r a i n ra te s e n s i t i v e but e s s e n t i a l l y not s t r a i n ha rden ing . Whi le r e f . 31 gave s e v e r a l c a l i b r a t i o n curves fo r p l a s t i c i n e , i t was dec ided to do , as pa r t of t h i s work, a c a l i b r a t i o n of e f f e c t i v e s t r e s s to e f f e c t i v e de fo rmat ion ra te fo r the p l a s t i c i n e to be used in the expe r imen ta l t e s t s . Ax isymmetr ic c y l i n d e r s of p l a s t i c i n e were used fo r c a l i b r a t i o n pu rposes . The main o b j e c t i v e was to deform c y l i n d e r s of v a r i o u s d iameters but of the same he igh t so as to o b t a i n a range of loads fo r the same de fo rmat ion r a t e s . F i g . 4.4.1 shows two examples of the p l a s t i c i n e c y l i n d e r s used wi th the f i n a l r e s u l t s of the de fo rmat ion shown fo r i n t e r e s t . F i n s were moulded on the s i de s of one of the specimens to enable the boundary to be c l e a r l y d e f i n e d on the h igh speed pho tographs . R a d i a l l i n e s drawn 62 on the tops of the c y l i n d e r s was an i n d i c a t i o n that the de fo rmat ion was reasonab ly ax isymmetr i c over the whole compress ion . The s e r i e s of photographs shown in F i g . 4 .4 .2 are examples of the compress ion of the p l a s t i c i n e c y l i n d e r s . The f i l m t im ing marks and the s c a l e s a t t a ched to the l oad c e l l enab led p l a t e n v e l o c i t i e s to be de te rm ined . The photographs in f i g . 4 .4 .2 show that the boundary remained q u i t e s t r a i g h t du r i ng the d e f o r m a t i o n . The h e i g h t s shown on the photographs were determined from the s c a l e f a c t o r as were a l s o the d i ame te r s . A check on volume constancy was c a r r i e d out by c a l c u l a t i n g the d iameters of the specimens assuming un i fo rm compress ion . The r e s u l t s are shown in b r a cke t s above the measured d iameters in F i g . 4 . 4 . 2 . Comparison of the p r e d i c t e d and measured d iameters show reasonab le agreement. A t y p i c a l d i sp lacement a g a i n s t time curve fo r the upper p l a t e n i s shown in F i g . 4 . 4 . 3 . A s e r i e s of marks a l ong the curve show the p o s i t i o n s that were used f o r c a l i b r a t i o n . The ze ro p o i n t on the time a x i s was chosen as the p o s i t i o n of maximum v e l o c i t y which occured at approx imate l y the c en t r e p o s i t i o n between the l i m i t s of d i s p l a c e m e n t . The v e l o c i t y at any p o i n t i s the i ns tan taneous s lope of the curve which v a r i e s s l i g h t l y over the r eg i on of d e f o r m a t i o n . The p i ecew ise l i n e a r approx imat ionsA-B , B-C, C-D, and D-E were used to es t ima te the v e l o c i t i e s fo r the v a r i o u s pho tographs . F i g . 4 .4 .4 shows the load t r a ce with the p o s i t i o n s of the frames s e l e c t e d f o r c a l i b r a t i o n pu rposes . With the 63 v e l o c i t y of the upper p l a t e n known at any p o i n t in the de fo rmat ion together wi th the specimen he igh t and l o a d , i t was p o s s i b l e to c a l c u l a t e both the e f f e c t i v e s t r e s s and e f f e c t i v e de fo rmat ion r a t e . The c a l i b r a t i o n data ob ta ined i s p l o t t e d in F i g . 4 . 4 . 5 . 4 . 5 . EXPERIMENTAL RESULTS. 4 . 5 . 1 . I n t r o d u c t i o n . The exper imenta l work was done on p l a s t i c i n e to observe the de fo rmat ion p r o f i l e s and measure the p l a t e n loads throughout the s t r a i n i n g h i s t o r y . The r e s u l t s ob ta ined are compared to f i n i t e element p r e d i c t i o n s in chapte r 5. 4 . 5 . 2 . Exper imenta l T e s t s . For a l l t e s t s , the procedure o u t l i n e d in Se c t i on 4.3 was f o l l o w e d . The t e s t s to be d i s c u s s e d are l i s t e d below. Each t e s t w i l l then be c o n s i d e r e d in d e t a i l . Tes t 1. P lane s t r a i n compress ion at a p l a t e n speed of 4.4m/s of an u n l u b r i c a t e d specimen wi th the s tandard d imens ions of 3 i n . h i g h , 1 .5 in . wide, and 1 .375 in . t h i c k between g l a s s p l a t e s wi th the f u l l specimen in the f i e l d of view of the camera. Tes t 2. An i d e n t i c a l t e s t to 1 but with the camera at c l o s e range to the top p l a t e n . T h i s t e s t was c a r r i e d out to observe the d e t a i l s of the motion a long the p l a t e n boundary . Tes t 3. T h i s t e s t was a repeat of t e s t 2 but w i th househo ld o i l as a l u b r i c a n t fo r the p l a t e n s . Tes t 4. T h i s t e s t was i d e n t i c a l to t e s t 1 but wi th a 64 p l a t e n speed of 2m/s. F i g . 4 .5 .2 .1 summarises the r e s u l t s of t e s t 1. The top of the specimen deforms the most i n i t i a l l y but movement at the base of the specimen i n c r e a s e s as the de fo rmat ion p roceeds . The r e s u l t s were d i g i t i s e d from en l a rged sketches taken from the p r o j e c t o r image as d i s c u s s e d in Se c t i on 4 . 3 . 4 . From the d i g i t i s e d r e s u l t s , a computer p l o t of the path l i n e s was o b t a i n e d . T h i s ' p lo t i s g i ven in F i g . 4 . 5 . 2 . 2 . The l i n e s j o i n the i ns t an taneous p o s i t i o n s of s e l e c t e d nodes in the m a t e r i a l and r e v e a l tha t the top su r f a ce remained stuck to the upper p l a t e n fo r the f i r s t three photographs used (frames 3, 5, and 6 in F i g . 4 . 5 . 2 . 1 ) . The remain ing two photographs (frames 7 and 8) r e v ea l s tha t s l i p d i d occur at the top r i g h t hand corner of the spec imen. T h i s s l i p i s shown a l s o by the behav iour of po in t C in F i g . 4 . 5 . 2 . 2 . Except fo r t h i s s l i g h t edge s l i p , however, a n o n - s l i p c o n d i t i o n e x i s t s . On F i g . 4 .5 .2 .1 i s a l s o shown a l oad t r a c e ob ta ined fo r compar ison wi th f i n i t e element p r e d i c t i o n s in chapter 5. Test 2 was in tended to examine the motion of the p l a s t i c i n e a long the u n l u b r i c a t e d top p l a t e n boundary . F i g . 4 . 5 . 2 . 3 shows the r e s u l t s . At t=-0.4ms the p l a t e n i s j u s t above the s u r f a c e . The remain ing s e r i e s of photographs shows tha t no motion takes p l a c e a long the impact boundary except fo r the element at the top r i g h t co rne r as was observed in t e s t 1. F i g . 4 . 5 . 2 . 4 i s a super imposed drawing of two specimen g r i d p o s i t i o n s . These were ob ta ined by p r o j e c t i n g the images onto t r a c i n g paper as d e s c r i b e d in 65 S e c t i o n 4 .3 .4 and then super impos ing each of the d raw ings . F i g 4 . 5 . 2 . 4 shows the r e s u l t and con f i rms a c c u r a t e l y the o b s e r v a t i o n s made e a r l i e r r ega rd ing a f u l l s t i c k c o n d i t i o n except f o r s l i p at the top r i g h t hand co rne r of the spec imen. For t e s t 3 the p l a t e n s u r f a c e was l u b r i c a t e d wi th l i g h t o i l to g i ve a s l i p boundary c o n d i t i o n . F i g . 4 . 5 . 2 . 5 shows the photographs ob ta ined and F i g . 4 . 5 . 2 . 6 shows the super imposed t r a c i n g ob ta ined in the same way as F i g . 4 . 5 . 2 . 4 . A compar ison of F i g . 4 . 5 . 2 . 4 and 4 . 5 . 2 . 6 con f i rms that the de fo rmat ion i s more homogeneous in the s l i p c a s e . In t e s t 4 the specimen was u n l u b r i c a t e d and the p l a t e n v e l o c i t y was 2m/s i n s t e a d of 4.4m/s. The deformed p r o f i l e s are shown in F i g . 4 . 5 . 2 . 7 . At the s t a r t of the compress ion the deformed p r o f i l e s were s i m i l a r to those ob ta ined at 4.4m/s. A f t e r s i g n i f i c a n t s t r a i n i n g had taken p l a c e , however, some d i f f e r e n c e s were appa ren t . These d i f f e r e n c e s w i l l be d i s c u s s e d in chapte r 5. lo U F I G . *r . 2. . \ . . PHOTOGRAPH OF THE. I M P A C T P R E S S . AND H I G H S P E E D C A M E R A . C A M G U I D E S F i 5 , 4r,z. \ .(bV T H E IMPACT P R E S S U S E D F o R THE. DEFORMATION T E S T S . 68 C A M FM&Af-.F . pj OS2. . START STOP ARM V t L o c i T Y S Y N C H R O N I S A T I O N I N D X C A T O H C A M E R A A D V A N C E . C O N X R O V . . T r i T SYNCH RONlZATtOM CONTROL. . CAMERA . E-N^BUg. MS n S OuENOXO CONTROY. . F i e . *c. . L O G I C OF THE ELECTRONIC C I R C U I T F o K CAMERA SYNCHRONIZATION . 6 9 VELOCITY (-m/s) . T H E S P E C I M E N H O L D E R OH T H E I M P A C T P R t S S . <L A f L M E N V E L O C I T Y AG to N S T & P L A C E M E N T cu^Vt I S M i O SHOWN . S C A L E . : OAZS" DIV F L O O D L I S H T SPECIMEN A S V I E W E D FOR HX6H S P E E D P H O T O G R A P H S P U N C H G L A S S P L / W E S T R A N S D U C E R S : DISPLACEMENT L O C I T Y SPECIMEN VIEW FROM F I G . . P H O T O G R A P H S O F I M P A C T P R E S S W I T H S P E C I M E N I M P O S I T10N . H I G H S P E E D C A M E R A 71 7 2 8 0 -1 0 -M I L L I V O L T O U T P U T . \ O -8 -"7-G 5" S-Z , M I L L I V O L T O O T P O T A G A I N S T V E L O C I T Y Fr\oV\ T H E V E L O C X T Y T R A N S D U C E R . 1 I ' i ' "' > 1 f ( * U 7 B 1 V E L O C I T Y c w i inn/ s ) . » o V O L T O U T P U T V O L T O U T P U T AG M U S T D I S P L A C E M E N T F R O M T H E D I S P L A C E M E N T TP.ANS D U C E R . ~i 1 1 ' T~ x 4 0 SO t»0 "70 60 °iO 'OO D I S P L A C E M E N T ( f f i l t l ) F I G . ^ r . Z . U . 73 L O A D ( FOUNDS") . F I G . <T. Z . " 7 . ?LOT OF OOTPUT V O L T A G E - F R o M THE. L O A D CELLL A G A I N S T APf L.1 E D LOAD . INITIAL- . F RONT . VP =i".'iln/s AT IMPACT F l < a . 4-. + . I . I N I T I A L A N D FINAL- D E F O R M E D SHAPES OF T W O AX I S Y M M E T R I C P L A S T I C I N E S P E C I M E N S F R O M T E S T S F o R D E T E R M I N I N G M A T E R I A L P R O P E R T I E S . ( ^1 . 0 ^ -VI . 3") 7 . 3 1 ms. *3 • 8 7 ML& * F16 .4-.*. 2L. DYNAMIC C O M P R E S S I O N OF A C 7 u l N T i R t c A L PuKSTXCINE. SPE.CXN\e.N . 76 5 0 -40-3 0 -20 10-RKtt Tn.t.fLAC.S.MENTt'Yft'flO. S P E C I M E N C A L I B R A T I O N POINTS . —, , 1 1 1 i 1 1 1 • T I M E C w s " ) . F I G . 4-."K 3 • PxAM D I S P L A C E M E N T A G A I N S T T I M E C U R V E . . * L O A D ( P O U N D S ! . 1 a i V t i» h fe H '»«> » f i l s" TIME-Ctr.S). FI6.4-.-fr.^. TYPICAL. LOAvD TRACE. FOR IMPACT O N T O A C K l N O M t A L P L A S T I C I N E S P E C I M E N A T -V . 77 "78 SCALE - 2.0 TO<m . PUNCH V E L o c l T Y C V r ) = T . I T U » / s . .'4 -i T ::J . ; IE = l - 0 . 4 - 5 - f t W S . if"*1* Q. Q ms. PUKCH HUT III I | I' I |'| T t 1 I I | I I I I | I I I I 1 I I 1 I | I I I I | I I I I | I g- v o IS" 2o 25" 10 "iS 4-0 F I G . *".ST.l. » • PHOTOGRAPHS O F ^ ^ S T l c l N t SPECintN AT VARIOUS T I M E S A F T E R I M P A C T WITH GRAPHS OF LOAD AND SPECIMEN HEIGHT, ASAIHST T I M E - Vp APPROXIMATELY •i-m/s DURING DEFORMATION. 2 0 -»—t—i—i i i_ < rmT7 79 0.0 T 1 O.OfiJ 0.16J 0.242 0.323 F I G • T. 5. 2.. 2. . E X P E R I M E N T A L . P A T H L I N E S F O R A T V P I C A L P L A S T I C I N E S P E C I M E N E E l N G . C O M P R E S S E D I N P L A N E S T R A I N l."7.3 I n n s . — \ o mm.. PLANE STRAIN COMPRESSION OF PLASTICINE — B O U N D A R Y B E H A V I Q D R A T I M P A C T -S U R F A C E . - MO L U B R I C A T I O N -V f f t i ^ « i | 5 DURING DEFORMATION SPECIMEN . 81 9 2. £ « • • • • • i i l l l l l 83 r' I T I 1 I J 4--4 1 .4 .4 O H O H J t J o V5 x: u O uJ [X-«* o < V UJ o * f o 9 II »v . — I , vi UJ fc s: H pi — o — . o Ln 4- (\1 o Lo LU t! ; a: M o u i O r H 2 M V/) x: c-a. o o i -P H O T O G R A P H S O F T H E D E F O R M A T I O N P R O F I L E S O F A P L A S T I C I N E S P E C I M E N B E I N G D E F O R M E D I N P L A N E S T R A I N . T H E I N I T I A L P L A T E N V E L O C I T Y W A S E m / s . 85 CHAPTER 5 FINITE ELEMENT MODELLING OF PLANE STRAIN COMPRESSION. In t h i s chapter the r e s u l t s of numer i ca l mode l l i ng of p lane s t r a i n compress ion are p r e s e n t e d . For the purposes of code v e r i f i c a t i o n , the computer code p r e d i c t i o n s were f i r s t compared wi th expe r imen ta l r e s u l t s . Once i t had been e s t a b l i s h e d that the model was s a t i s f a c t o r y , a sys temat i c s e r i e s of mode l l i ng t e s t s were done. The main purpose of the t e s t s was to i n v e s t i g a t e the e f f e c t s of m a t e r i a l p r o p e r t i e s , i n e r t i a , f r i c t i o n boundary c o n d i t i o n s , and specimen d imens ions on the s t r e s s and de fo rmat ion p a t t e r n s in the m a t e r i a l be ing compressed in p lane s t r a i n . The r e s u l t s are then used to make fundamental c o n c l u s i o n s about the e f f e c t of the v a r i a b l e s in gene ra l d i e - f o r g i n g o p e r a t i o n s . The f o l l o w i n g s e c t i o n s are p resen ted in t h i s c h a p t e r . S e c t i on 5.1 p r esen t s the r e s u l t s of some i n i t i a l t e s t s wi th the code fo r the purpose of de te rm in ing s u i t a b l e f i n i t e element paramete rs . S e c t i on 5.2 p re sen t s the r e s u l t s of expe r imenta l t e s t s on p l a s t i c i n e and compares them to the f i n i t e element r e s u l t s . In a d d i t i o n , a compar ison i s made w i th exper imenta l t e s t s on aluminum. S e c t i on 5.3 c o n s i d e r s some r i g i d - p e r f e c t l y p l a s t i c r e s u l t s and Se c t i on 5.4 some s t r a i n r a te s e n s i t i v e and s t r a i n harden ing r e s u l t s . S e c t i o n 5.5 c o n s i d e r s the dynamic compress ion of aluminum at l00m/s. F i n a l l y , s e c t i o n 5.6 c o n s i d e r s the s t r e s s d i s t r i b u t i o n s fo r both the r i g i d - p e r f e c t l y p l a s t i c q u a s i - s t a t i c cases and the dynamic compress ion of aluminum. 86 5.1 DETERMINATION OF THE FINITE ELEMENT PARAMETERS. 5 . 1 . 1 . Homogeneous Compress ion . The program was t e s t e d fo r the case of homogeneous p lane s t r a i n de fo rmat ion as d i s c u s s e d in chapte r 2. In t h i s case the s o l u t i o n i s l i n e a r and the f i n i t e e lements are ab le to r ep resen t the s o l u t i o n e x a c t l y . F i g . 5.1.1.1 shows a v e l o c i t y v e c t o r p l o t fo r t h i s c a s e . 5 . 1 . 2 . Number Of E lements . The e f f e c t of element s i z e was c o n s i d e r e d by t e s t c a l c u l a t i o n s in a l l a p p l i c a t i o n s in t h i s work. F i g . 5.1.2.1 compares the de fo rmat ion of a 15 and 50 element i d e a l i z a t i o n of an aluminum b lock at 44% v e r t i c a l s t r a i n . The 15 element case l a cks d e t a i l but both approx imat ions are q u i t e c o n s i s t e n t . The comparison shows that s i g n i f i c a n t d e t a i l would be l o s t i f the 15 element case were used i n t ead of the 50. For t h i s case 50 elements was taken as an a c cep t ab l e amount. However i t was found tha t 144 e lements were r e q u i r e d to g i ve a s a t i s f a c t o r y compar ison w i th our expe r imenta l t e s t s on p l a s t i c i n e . 5 . 1 . 3 . De te rm ina t i on Of A l l owab l e S t r a i n Increment. Dur ing i n i t i a l program. t e s t i n g a p r e l i m i n a r y assessment was made of the maximimum a l l owab l e s t r a i n inc rement . A 15 element i d e a l i z a t i o n of a b lock of aluminum was mode l l ed wi th a l i n e a r d i s t r i b u t i o n of h o r i z o n t a l v e l o c i t y and a cons tan t v e r t i c a l v e l o c i t y on the top boundary . The b lock was deformed to 83% of i t s i n i t i a l he igh t in 1, 2, 4, 8, and 12 inc rementa l s t e p s . In each case the shear s t r e s s on the top boundary was p l o t t e d a f t e r 87 t h e r e q u i r e d number o f i n c r e m e n t a l s t e p s had been t a k e n t o r e a c h t h e f i n a l h e i g h t . T h i s s h e a r s t r e s s i s p l o t t e d i n F i g . 5.1.3.1. I t was f o u n d t h a t t h e s h e a r s t r e s s d i d not change s i g n i f i c a n t l y between 8 and 12 i n c r e m e n t s . Thus a 2% i n c r e m e n t a l s t r a i n s t e p gave r e a s o n a b l e a c c u r a c y . 5.1.4. C o n s t i t u t i v e R e l a t i o n s . The assumed r e l a t i o n of y i e l d s t r e s s t o s t r a i n h a r d e n i n g and s t r a i n r a t e s e n s i t i v i t y s e l e c t e d f o r t h e f i n i t e e l e m e n t p r o g r a m i s g i v e n i n e q u a t i o n 3.5.3. In t h i s work t h e two r e a l m a t e r i a l s t h a t a r e c o n s i d e r e d a r e aluminum and p l a s t i c i n e . In t h e c o n s t i t u t i v e r e l a t i o n s u s e d aluminum i s c o n s i d e r e d s t r a i n h a r d e n i n g but not s t r a i n r a t e s e n s i t i v e . P l a s t i c i n e i s c o n s i d e r e d t o be s t r a i n r a t e s e n s i t i v e but not s t r a i n h a r d e n i n g . The c o n s t i t u t i v e r e l a t i o n f o r t h e s t r a i n h a r d e n i n g c a s e i s f o u n d from e q u a t i o n 3.5.3 by s e t t i n g c ^ O t o i n d i c a t e no s t r a i n r a t e s e n s i t i v i t y . T h i s t h e n g i v e s Y=( 1+c, e ) r Y 0 (5.1.4.1) where "e i s g i v e n from e q u a t i o n 3.2.5. F i g . 5.1.4.1 shows c u r v e s of "5" a g a i n s t "e f o r aluminum o b t a i n e d from v a r i o u s s o u r c e s . The one u s e d i s i n d i c a t e d and c o r r e s p o n d s t o t h e c o n s t a n t s c,=16.436,T=0.25 w i t h Y 0=106.18MPa, t h e s t a t i c y i e l d s t r e s s f o r a n n e a l e d aluminum. B a s e d upon e x p e r i m e n t a l c a l i b r a t i o n t e s t s f o r p l a s t i c i n e i n c h a p t e r 3 t h e r e l a t i o n g i v e n b e l o w was assumed. G"=465dD (5.1.4.2) H e r e - 6 i s i n Pa and D i s i n s " 1 . T h i s r e l a t i o n was u s e d b y t h e f i n i t e e l e m e n t code f o r t h e m o d e l l i n g of p l a s t i c i n e and 88 i s p l o t t e d in F i g . 4 . 4 . 5 . 5.2 EXPERIMENTAL COMPARISONS. The purpose of the compar ison of the f i n i t e element code p r e d i c t i o n s w i th exper iment was to v e r i f y that the numer i ca l model proposed was s a t i s f a c t o r y . Once a s a t i s f a c t o r y compar ison wi th exper iment had been o b t a i n e d , mode l l i ng r e s u l t s wi thout expe r imen ta l comparison c o u l d be accepted w i th c o n f i d e n c e . 5.2.1. Comparison Of The F i n i t e Element Model With  Exper imenta l Tes t s On P l a s t i c i n e . The exper imenta l t e s t i n g techn ique fo r p l a s t i c i n e was o u t l i n e d in Chapter 4. F i g s . 5.2.1.1 and 5.2.1.2 show the p r o f i l e of e x p e r i m e n t a l l y deformed p l a s t i c i n e specimens fo r a compress ion speed of 2m/s and 4.4,m/s r e s p e c t i v e l y . The exper imenta l r e s u l t s fo r these two speeds show somewhat d i f f e r e n t p a t t e r n s . At 4.4m/s there i s l e s s b u l g i n g at the top than fo r 2m/s. F i n i t e element c a l c u l a t i o n s were done fo r the case of compress ion at a speed of 4.4m/s w i th a c o e f f i c i e n t of f r i c t i o n fo r g l a s s , ? c j , of 0, 0.1, and 0 .235. The c o e f f i c i e n t of f r i c t i o n fo r the p l a t e n s was taken as 0.249 fo r a l l cases based upon t a b u l a t e d v a l u e s . The f i n i t e element model w i l l g i ve s i m i l a r r e s u l t s at 2m/s and 4.4m/s as the dynamic e f f e c t i s shown not to be a f a c t o r at t h i s compress ion speed and the f r i c t i o n law i s v e l o c i t y independent . F i g s . 5.2.1 .3(a) and (b ) , F i g s . 5.2.1 .4(a) and (b ) , and F i g s 5.2.1 .5(a) and (b) show the r e s u l t s f o r each of the cases r e s p e c t i v e l y . F i g . 5.2.1 .6 shows a compar ison of the f i n a l deformed shapes fo r p l a s t i c i n e at 2m/s and 89 4.4m/s with the f i n i t e element results for 4.4m/s. Based upon these figures the following conclusions can be drawn. A) . The f i n i t e element results with ^ = 0 . 2 3 5 predict a deformation pattern that i s a good approximation to the experimental result for a compression speed of 2m/s. B) . The f i n i t e element results with 7^  =0 .1 are a good approximation to the experimental results for a compression speed of 4.4m/s. C) . The f i n a l deformed shapes, which involve very large p l a s t i c deformation, agree remarkably well with the experimental re s u l t s . D) . Glass f r i c t i o n i s an important parameter to include in the analysis i f the correct deformation p r o f i l e s are to be obtained. This conclusion was also reached on the basis of the t o t a l load measured during the experimental tests. E) . A vel o c i t y independent f r i c t i o n law may not be appropriate for the glass f r i c t i o n on the p l a s t i c i n e spec imens. F) . Since, with zero glass f r i c t i o n , the f i n i t e element results predicted a symmetric pattern about EF in F i g . 5.2 . 1 . 3(a), the ef f e c t s of i n e r t i a on deformed shape are ne g l i g i b l e at a platen speed of 4.4m/s. Figs. 5.2.1 .7(a) and (b) summarise theoret i c a l and 90 exper imenta l l oad compar i sons . The exper imenta l and t h e o r e t i c a l l oads agree w e l l . In t h i s case the t o t a l load on the bottom p l a t en i s the sum of the g l a s s f r i c t i o n and p l a s t i c i n e compress ion loads s i n ce t h i s i s the t o t a l l oad that the l oad c e l l would measure. F i g . 5 .2 .1 .7 (b ) shows a compar ison of load on the top and bottom p l a t e n s . In t h i s case the load on the bottom p l a t e n does not i n c l ude the load t r a n s m i t t e d to the g l a s s p l a t e s by g l a s s f r i c t i o n . I n i t i a l l y i n e r t i a makes the loads lower on the bottom p l a t e n than on the t o p . However, they converge toge ther a f t e r about 10 ms from impact . A l s o p l o t t e d on t h i s f i g u r e i s the load curve based upon the s imple model p resen ted in s e c t i o n 2.1 wi th g l a s s f r i c t i o n but not i n e r t i a i n c l u d e d . A f t e r about 80 ms i n e r t i a e f f e c t s s t a r t to become important a g a i n . The curves p r e d i c t tha t the loads w i l l aga in become s i g n i f i c a n t l y d i f f e r e n t on the top and bottom p l a t e n s . The o s c i l l a t i o n s that occur in the cu rves a f t e r about 70ms are due to f o l d i n g of m a t e r i a l onto the p l a t e n s . In r e a l i t y t h i s f o l d i n g p rocess i s a con t i nuous one but in the f i n i t e element model the boundar ies of the element be ing used must remain s t r a i g h t . F o l d i n g thus occurs in d i s c r e t e pa tches which approximate a con t inuous movement of m a t e r i a l onto the bounda r i e s . F i g s . 5 .2 .1 .8 (a ) and 5 .2 .1 .8 (b ) show power d i s t r i b u t i o n s aga in s t t ime . The l a t t e r graph i s p l o t t e d as a l og s c a l e but o therw ise the r e s u l t s p l o t t e d are i d e n t i c a l . F i g . 5 .2 .1 .8 (a ) shows that the important power l o s s e s are p l a t e n l o a d , i n t e r n a l de fo rmat ion power and 91 g l a s s f r i c t i o n . K i n e t i c energy , p l a t e n f r i c t i o n , and su r f a ce t r a c t i o n due to a tmospher ic p r e s su re have n e g l i g i b l y sma l l power a s s o c i a t e d w i th them. F i g . 5 .2 .1 .8 (b ) shows more d e t a i l r ega rd ing smal l power l o s s e s . The r a te of change of k i n e t i c energy of the m a t e r i a l i s i n i t i a l l y h i g h , drops to a minimum at about 10 ms and then con t i nues to i n c r e a s e . T h i s i s c o n s i s t e n t with the c o n c l u s i o n s reached e a r l i e r r ega rd ing the e f f e c t s of i n e r t i a . P l a t en f r i c t i o n a l s o f o l l o w s approx imate l y the same t r e n d . I n i t i a l l y the m a t e r i a l s l i d e s a long the top p l a t e n r a p i d l y a f t e r impact . It then reaches a minimum and s t a r t s to i n c r ea se aga in due to more r a p i d movement a long the top p l a t e n and f o l d i n g of m a t e r i a l onto the top and bottom p l a t e n s as the specimen he igh t r educes . 5 . 2 . 2 . Comparison Of The F i n i t e Element Model With  Exper imenta l Tests On Aluminum. The data p u b l i s h e d by Shaba ik (7 ) fo r q u a s i - s t a t i c compress ion of aluminum was used fo r the exper imenta l t e s t s to be compared wi th p r e d i c t i o n s of the f i n i t e element mode l . The r e s u l t s f o r deformed shapes are shown in F i g . 5 . 2 . 2 . 1 . S t r a i n harden ing was assumed with the c o n s t i t u t i v e r e l a t i o n p resen ted in equa t ion 5 . 1 . 4 . 1 . The mas te r-s l ave node method fo r p l a t e n f r i c t i o n was assumed in t h i s case w i th the master node at the top r i g h t of the specimen as i n d i c a t e d in F i g . 5 . 2 . 2 . 1 . The h o r i z o n t a l v e l o c i t y at the master node was taken from the expe r imen ta l v a l u e s . For t h i s purpose a f a c t o r Z2 was defined as the r a t i o of the h o r i z o n t a l v e l o c i t y at the master node to the v e l o c i t y tha t would occur fo r the f ree 92 boundary i f the compress ion were homogeneous. That i s , i f V, i s the h o r i z o n t a l v e l o c i t y of the master node, and V H the v e l o c i t y of the f r ee boundary of z specimen of the same he ight and volume but deforming homogeneously, then V ,=pV H . T h i s a l l owed a s u i t a b l e smoothing of V, between expe r imen ta l p o i n t s . A good agreement between the f i n i t e element model and the e x p e r i m e n t a l l y deformed specimen was o b t a i n e d . A fur ther compar ison wi th exper iment i s ob ta ined by examining the f low l i n e diagrams fo r each case as shown in F i g s . 5 .2 .2 .2 (a ) and (b ) . 5 .3 . RIGID PERFECTLY PLASTIC RESULTS. In the p r e v i ous s e c t i o n the model was found to g i ve a good agreement to -experimental r e s u l t s . I t i s now p o s s i b l e to proceed to p r e d i c t r e s u l t s for which exper imenta l compar ison i s not a v a i l a b l e . By d e f i n i t i o n , a r i g i d - p e r f e c t l y p l a s t i c m a t e r i a l i s one wi th a cons tan t y i e l d s t r e s s . As such , the r e s u l t s w i l l be independent of the r a te of s t r a i n . C o n s i d e r a t i o n w i l l now be g iven to the compress ion in p lane s t r a i n of a b lock of r i g i d - p e r f e c t l y p l a s t i c m a t e r i a l . The r e s u l t s of the m o d e l l i n g fo r t h i s casewere compared to upper bound s o l u t i o n s g i ven in s e c t i o n 2 .3 . S e c t i o n 5.3.1 c o n s i d e r s case 1: The compress ion of a t a l l b lock of he igh t to width r a t i o (H 0 /D a ) of 4. S e c t i o n 5.3.2 c o n s i d e r s case 2: the compress ion of a shor t b lock wi th H o / D o=0 . 8 3 9 . T h i s s e c t i o n a l s o c o n s i d e r s r e c t a n g u l a r b l o cks w i th H o /D o =0.25 and 0.125 r e s p e c t i v e l y . For a l l cases the m a t e r i a l i s assumed to have a 93 cons tan t y i e l d s t r e s s of Y Q=106.18MPa, the va lue fo r annea led aluminum. S t r a i n harden ing and s t r a i n ra te s e n s i t i v i t y are ignored to i s o l a t e the behaviour f o r & =Y C . A cons tan t c o e f f i c i e n t of f r i c t i o n of 0.174 fo r the p l a t e n s was assumed. T h i s was was the average va lue ob ta ined wi th the mas te r-s l ave node method in s e c t i o n 5 . 2 . 2 . 5 . 3 . 1 . Case 1. H 0 /D o =4. F i g . 5.3.1.1 shows the i n i t i a l mesh used fo r t h i s case and the deformed shapes at v a r i o u s stages of compress i on . From an examinat ion of these deformed shapes i t was conc luded that the upper bound s o l u t i o n d i s c u s s e d in s e c t i o n 2.3 was a p p r o p r i a t e fo r d e s c r i b i n g the p a t t e r n of d e f o r m a t i o n . T h i s upper bound v e l o c i t y d i s c o n t i n u i t y p a t t e r n i s super imposed on the deformed shapes fo r compar i son . F i g s . 5 . 3 . 1 . 2 ( a ) and (b) p resent v e l o c i t y d i s t r i b u t i o n s f o r the f i n i t e element s o l u t i o n . For comparison w i th these curves F i g s 5 .3 .1 .3 (a ) and (b) were c o n s t r u c t e d from the v e l o c i t y d i s c o n t i n u i t y p a t t e r n s and hodographs in F i g s . 2 . 3 . 1 . 2 and 2 . 3 . 2 . 1 . Before p roceed ing to d e s c r i b e how these cu rves were c o n s t r u c t e d some d e f i n i t i o n s w i l l be made. As shown in F i g . 5.3.1.1 an X-Y coo rd i na t e system i s d e f i n e d w i th o r i g i n at o, the c en t r e of mass of the spec imen. The Y a x i s i s v e r t i c a l and the X a x i s h o r i z o n t a l . Each l i n e such as 3-3 i s r e f e r r e d to as a y g r i d l i n e and the r e s u l t s are p l o t t e d at nodes a long these l i n e s . 3-3 r e f e r s to the 3 ' r d y g r i d l i n e as i t i s the t h i r d from the l e f t . OA i s the f i r s t y g r i d l i n e and IE the 9 ' t h . 94 S i m i l a r l y , 01 i s the f i r s t X g r i d l i n e and AE the 9 ' t h . The hodograph shown in F i g . 2 .3 .1 .2 i n d i c a t e s that the no rma l i sed v e r t i c a l v e l o c i t y V y / V 0 i s un i t y in r eg ion ABE, 0.75 i n BHFE, 0.5 in BCH and FGH,0.25 in ICHG, and 0 in IOC. With these v e l o c i t i e s a v a i l a b l e F i g . 5 .3 .1 .3 ( a ) was c o n s t r u c t e d by p l o t t i n g the v e r t i c a l v e l o c i t y a long the 9 ' t h y g r i d l i n e wi th symbol " o " , a long the 5 ' th wi th symbol " O " , and a long the f i r s t w i th " D " . The i n t e r s e c t i o n s of x and y g r i d l i n e s d e f i n e the node p o i n t s . For each of these node p o i n t s the va lue of v e r t i c a l v e l o c i t y i s dependent upon the r eg ion i n t o which i t f a l l s . Thus fo r the 5 ' th y g r i d l i n e in F i g . 5.3.1.1 wi th H/D=2.023, V y / V 0 i s ze ro fo r the f i r s t two nodal p o i n t s in the r eg ion IOC. The v e l o c i t y then jumps to 0.25 fo r the t h i r d node which f a l l s in r eg ion ICHG. The v e l o c i t y then i n c r e a s e s to 0.5 fo r the 4 ' t h node and s tays at that va lue fo r the 5 ' th and 6 ' t h nodes . The v e l o c i t y then i n c r ea ses aga in to 0.75 fo r the 7 ' t h node which f a l l s in r eg ion EBHF. The 8 ' t h and 9 ' t h nodes f a l l in r eg i on BAE and hence have a no rma l i s ed v e r t i c a l v e l o c i t y of u n i t y . These p o i n t s are p l o t t e d in F i g . 5 . 3 . 1 . 3 ( a ) . F i g . 5 .3 .1 .3 (b ) fo r H/D=1.117 was p l o t t e d in a s i m i l a r way. The compar ison of the s imple model r e s u l t s in F i g s . 5 .3 .1 .3 ( a ) and (b) w i th the v e l o c i t i e s in F i g s . 5 . 3 . 1 . 2 ( a ) and (b) f o r the a p p r o p r i a t e case shows some very s i m i l a r t r e n d s . The f i n i t e element r e s u l t s show v e l o c i t i e s fo r a l l g r i d l i n e s but those p l o t t e d from the s imple model are s u f f i c i e n t to show a s i m i l a r , d i s t r i b u t i o n of v e l o c i t y s p a t i a l l y f o r each H/D and a s i m i l a r t r end wi th 95 deformation. I t i s concluded that the simple model gives a s a t i s f a c t o r y comparison with the f i n i t e element r e s u l t s . The H/D r a t i o s f o r comparison i n the previous d i s c u s s i o n were s e l e c t e d because the v e l o c i t y d i s c o n t i n u i t y p a t t e r n s were reasonably w e l l developed f o r those v a l u e s . For each of these cases p r i o r deformation had taken p l a c e . In the present f i n i t e element model the only cause f o r any v a r i a t i o n i n the r e s u l t s f o r the v e l o c i t y f i e l d , p r ovided the m a t e r i a l i s not s t r a i n hardening, i s the shape of the f r e e boundary. It was thus decided to i n v e s t i g a t e the e f f e c t of the boundary shape on the r e s u l t s . F i v e cases were c o n s i d e r e d . U n d i s t o r t e d blocks with H/D r a t i o s of 1.5, 2, 3, and 3.5 were modelled under the same c o n d i t i o n s as p r e v i o u s l y . A 5'th block of H/D=2.023 was c o n s i d e r e d i n which the mesh was e q u a l l y spaced as f o r the others but the boundary was d i s t o r t e d to the shape found e x p e r i m e n t a l l y . T h i s l a t t e r case proved a u s e f u l check on any mesh e f f e c t s which might change the numerical s o l u t i o n . D i s t o r t e d mesh e f f e c t s does not i n t h i s case amount to a change in the p h y s i c a l model. The r e s u l t s are shown in Figs.5.3.1.4 and 5.3.1.5. The d i s c o n t i n u i t y p a t t e r n s are drawn d i r e c t l y onto the d i s p l a c e d shape p l o t s as p r e v i o u s l y . The r e s u l t s without the d i s t o r t e d boundary are q u i t e d i s t i n c t from the r e s u l t s with a d i s t o r t e d boundary. They i n d i c a t e no tendency to the formation of a c o n c a v i t y i n the boundary as i n the previous c a s e . At H/D=2 the r e s u l t with the undeformed boundary i s q u i t e d i s t i n c t from the one with a deformed boundary at 96 H/D=2.023. The case of the r e g u l a r mesh with a deformed boundary i s a l s o shown in F i g . 5.3.1.5 and i s in good agreement w i th the r e s u l t in F i g . 5.3 .1.2 (a) . In c o n c l u s i o n , the m o d e l l i n g has shown tha t the s i m p l i f i e d model assumed on ly ho lds when the boundary i s ab l e to d i s t o r t s u i t a b l y . F i g . 5.3.1.6 shows that the c r i t i c a l H/D va lue for c o n c a v i t y fo rmat ion fo r the boundary i s between H/D=3 and H/D=3.5. The r e s u l t s suggest that i f the specimen i s too shor t fo rmat ion of a concave boundary cannot o c c u r . F i g . 5.3 .1.7 i s a path l i n e p l o t d e p i c t i n g the flow h i s t o r y fo r the case shown in F i g . 5.3 .1 .1 . T h i s p l o t shows q u i t e c l e a r l y the change in the mode of de fo rmat ion at H/D=2.023 suggested by the s imple model . For example, the path l i n e ABC cor responds to the path of a p o i n t on the boundary of the spec imen. A change of cu rva tu re in the path f o l l owed by t h i s m a t e r i a l po in t occurs at the s p a t i a l po in t B when H/D=2.023. T h i s change of cu r va tu re i s c o n s i s t e n t w i th the s imple models shown in F i g . 2.3.1.2 and F i g . 2.3.2 .1. When H/D>2 the v e l o c i t y V y in r eg ion ICHG (see F i g . 2.3.1.2) has a f i n i t e y component of v e l o c i t y . For H/D<2, however, V y i n F i g . 2.3.2.1 has no Y component of v e l o c i t y . I t i s thus the development of the r eg ion FHG in F i g . 2.3.2.1 that causes the Y components of v e l o c i t y to decrease and the X components to i n c r e a s e . T h i s r e s u l t s in the observed change of cu r v a tu r e of the path l i n e s at H/D=2.023. 97 5 .3 .2 . Case 2. H o / D o = 0 . 8 3 9 . F i g . 5.3.2.1 shows the deformed meshes ob ta ined fo r the compress ion of a b lock of r i g i d p l a s t i c m a t e r i a l of i n i t i a l he igh t to width r a t i o of 0 .839 . The s imple model v e l o c i t y d i s c o n t i n u i t i e s from F i g . 2 .3.3.1 are a l s o shown. The v e l o c i t y p a t t e r n s are p l o t t e d in F i g . 5 .3 .2 .2 and are in good agreement wi th the p r e d i c t i o n s of the s imple model . S ince f o r t h i s case the r i g h t boundary of the specimen remains s t r a i g h t , the s t r a i n h i s t o r y of the specimen w i l l not i n f l u e n c e the v e l o c i t y f i e l d s . Thus fo r the purposes here i t w i l l be s u f f i c i e n t to assume that fo r H/D<1 the v e l o c i t y f i e l d fo r an u n d i s t o r t e d b lock can be used fo r one tha t has been s t r a i n e d p rov ided the same volume of m a t e r i a l i s be ing c o n s i d e r e d in each case and the specimen he igh t s are the same. In order to a vo id unnecessary computer c a l c u l a t i o n , u n s t r a i n e d b locks were next c o n s i d e r e d wi th the purpose of comparing the v e l o c i t y p a t t e r n s in each wi th the s imple mode l . F i g . 5 . 3 .2 .3 and F i g . 5 .3 .2 .4 show compar isons of the s imple model d e p i c t e d in F i g . 2 .3.3.1 wi th the f i n i t e element r e s u l t s f o r the case of u n s t r a i n e d b l o c k s of the a p p r o p r i a t e H/D r a t i o . F i g . 5 . 3 .2 .3 shows the r e s u l t s fo r the case H/D=0.25. The top pa r t of t h i s f i g u r e shows both the f i n i t e element mesh used f o r the f i n i t e element c a l c u l a t i o n s and the v e l o c i t y d i s c o n t i n u i t i e s of the s imple model . A l s o shown wi th the s imple model i s a hodograph and the r e s u l t i n g v e l o c i t y v e c t o r s w i th i n each of the r i g i d r e g i o n s . S ince the v e l o c i t y f i e l d must be symmetric about GH in F i g . 98 5 .3 .2 .3 these v e l o c i t y v e c t o r s are p l o t t e d on the lower h a l f of the specimen fo r c l a r i t y . In r eg ion edH the s imple model p r e d i c t s zero y component of v e l o c i t y wh i l e in r eg ion cde a f i n i t e component of v e l o c i t y equa l to the p l a t e n v e l o c i t y i s p r e d i c t e d . The f i n i t e element c a l c u l a t i o n s i n d i c a t e a l o c a l maximum in the V~Y in r eg ion cde fo r the i n t e rmed i a t e y g r i d l i n e s CD and EF . S i m i l a r l y a l o c a l minimum in the f i n i t e element s o l u t i o n on these g r i d l i n e s i s i n d i c a t e d for r eg ion c b d . I t can be conc luded from the compar ison that the s imple model f o r t h i s case does g i ve a s a t i s f a c t o r y compar ison wi th the f i n i t e element r e s u l t s . F i g . 5 .3 .2 .4 shows an even more remarkable agreement of the s imple model w i th the f i n i t e element s o l u t i o n . A long the CD y g r i d l i n e the l o c a l maxima of V from the f i n i t e element r e s u l t s are c l e a r l y d e f i n e d . On the EF y g r i d l i n e the maxima become l e s s we l l r e s o l v e d . T h i s change i s a l s o p r e d i c t e d by the s imple model because the GH y g r i d l i n e i s made up on ly of the s i d e s of the r i g i d r eg ions such as cbd wherein V y = 0 . As in the p r e v i ous case the f i n i t e element s o l u t i o n does not p r e d i c t the r i g i d reg ion aGb. T h i s r i g i d r eg i on i s the crude r e p r e s e n t a t i o n in the s imple model fo r the a c t u a l boundary c o n d i t i o n of V =0. 5 . 3 . 3 . Load P r e d i c t i o n s For The R i g i d - p e r f e c t l y  P l a s t i c Case . F i g . 5.3.3.1 shows a compar ison of the f i n i t e element r e s u l t s f o r the r i g i d - p e r f e c t l y p l a s t i c case wi th the s imp le models p resen ted in chapter 2. The method of c o n s t r u c t i o n * o f t h i s graph w i l l f i r s t be d e s c r i b e d . 99 There are two d i s t i n c t regimes to c o n s i d e r ; namely, H/D>1 and H/D<1. For H/D>1 f i n i t e element c a l c u l a t i o n s were done both fo r a s t r a i n i n g h i s t o r y from H/D=4 to H/D=1.117 and f o r u n s t r a i n e d b l o cks of H/D= 1.5, 2, 3, and 3.5. The on l y s imple s o l u t i o n s c o n s i d e r e d fo r t h i s case were the homogeneous s o l u t i o n and the v e l o c i t y d i s c o n t i n u i t y p a t t e r n . Both these s imple s o l u t i o n s gave Gy/2K=1, where Gy i s the normal p l a t e n t r a c t i o n . The f i n i t e element r e s u l t s fo r t h i s case are p l o t t e d in F i g . 5.3.3 .1 . The f i n i t e element r e s u l t s for the undeformed b l o cks are in e x c e l l e n t aggreement w i th the s imple model r e s u l t s . A va lue of £y/2K=1 with n e g l i g i b l e d e v i a t i o n was o b t a i n e d . The p l a t e n normal t r a c t i o n s fo r the case of the b l o cks be ing s t r a i n e d , however, showed s i g n i f i c a n t d e v i a t i o n . At H/D=4 c o n c a v i t y fo rmat ion fo r the boundary o c c u r r e d . For lower va lues of H/D, the r e s u l t s from the b lock be ing s t r a i n e d d i f f e r from those ob ta ined from an u n s t r a i n e d b lock of the same e q u i v a l e n t H/D. It i s c l e a r that the normal p l a t e n t r a c t i o n i n c r e a s e s w i th s t r a i n f o r the case where a concave boundary forms. At H/D=1.433 the p l a t e n t r a c t i o n i s a maximum wi th the va lue at H/D=1.117 somewhat l e s s . F i g . 5.3.1.1 shows tha t between H/D=1.433 and H/D=1.117 the element at the top r i g h t hand co rne r of the specimen (at p o i n t E in F i g . 5.3 .1 .1) has had one of i t s s i d e s f o l d onto the p l a t e n . The reason fo r the i n c r ease of p l a t e n t r a c t i o n can now be e x p l a i n e d . If a v e r t i c a l l i n e i s drawn down from E in F i g . 5.3.1.1 and the excess m a t e r i a l removed from the r i g h t 100 of t h i s l i n e the p l a t e n normal t r a c t i o n would be 2K. With the excess m a t e r i a l i n c l u d e d , however, the amount of m a t e r i a l in con tac t wi th the p l a t e n s remains the same but a d d i t i o n a l p l a t e n t r a c t i o n i s r e q u i r e d because of the ex t r a power i n v o l v e d in d i s t o r t i n g the a d d i t i o n a l m a t e r i a l . It i s a l s o of i n t e r e s t to compare a r e s u l t quoted in r e f . 28, o r i g i n a l l y due to H i l l , f o r the case of compress ion w i th overhanging p l a t e n s . T h i s s o l u t i o n i s p l o t t e d in F i g . 5.3.3.1 fo r compar i son . In the case of overhanging p l a t e n s D r e f e r s to the width of the p l a t ens r a the r than the specimen w id th . The compar ison shows that the i n c r ease of t r a c t i o n i s we l l below that which would occur fo r a ve ry wide specimen where the p l a t e n s would e s s e n t i a l l y be i n d e n t e r s . In t h i s respec t the compar ison i s c o n s i s t e n t . D i f f e r e n t c o n s i d e r a t i o n s app ly for the case H/D<1 in F i g . 5 . 3 . 3 . 1 . In t h i s case the e f f e c t s of p l a t e n f r i c t i o n w i l l become more important than in the case of H/D>1 because of the l a r g e r amount of m a t e r i a l in con t a c t wi th the p l a t e n s and the i n c r e a s e d v e l o c i t y of s l i d i n g that occurs wi th wide spec imens. The s imple model in F i g . 2 .3 .3.1 can be expected to app ly on ly fo r low va lues of f r i c t i o n s i n c e the r i g i d b l o cks must be ab l e to s l i d e a long the p l a t e n s . Thus6 y =2K w i l l not be a good es t imate fo r s i g n i f i c a n t v a l ues of the c o e f f i c i e n t of f r i c t i o n . P l o t t e d in F i g . 5.3.3.1 are the p o i n t s co r r e spond ing to an u n d i s t o r t e d r i g h t boundary co r r e spond ing to the cases shown in F i g s . 5 . 3 . 2 . 3 and 5 . 3 . 2 . 4 . A l s o p l o t t e d in F i g . 5.3.3.1 are the p o i n t s co r r e spond ing the d i s t o r t e d boundary 101 case which i n v o l v e d s t r a i n i n g from H/D=0.839. T h i s l a t t e r case i s the one shown in F i g . 5 . 3 . 2 . 1 . A l l the f i n i t e element r e s u l t s i n v o l v e d s i g n i f i c a n t va lues of the f r i c t i o n c o e f f i c i e n t w i th the r e s u l t that the p l a t e n t r a c t i o n o f ^ = 2 K was s i g n i f i c a n t l y exceeded . Most of the f i n i t e element r e s u l t s are in good agreement wi th the s l i p l i n e f i e l d s o l u t i o n g iven by H i l l and p l o t t e d in F i g . 5 • 3 • 3 • 1 * It can be conc luded that the f i n i t e element r e s u l t s fo r the r i g i d - p e r f e c t l y p l a s t i c case gave r e s u l t s that are c o n s i s t e n t w i th approximate c l o s e d form s o l u t i o n s . 5.4. RIGID PLASTIC STRAIN HARDENING AND STRAIN RATE  SENSITIVE RESULTS. For t h i s case equat ion 5 .1 .4 .2 for p l a s t i c i n e was used to d e s c r i b e the s t r a i n ra te s e n s i t i v i t y . T h i s r e l a t i o n i m p l i e s tha t there i s no s t r a i n harden ing and the q u a s i - s t a t i c y i e l d s t r e s s i s z e r o . F i g s . 5.4.1.1 and 5 .4 .1 .2 show the de fo rmat ion p a t t e r n s and v e l o c i t y f i e l d s c a l c u l a t e d . These r e s u l t s p a r a l l e l those p resen ted in F i g s . 5.3.1.1 and 5 . 3 . 1 . 2 ( a ) . For c o n s i s t e n c y wi th the r e s u l t s in F i g . 5 .3.1.1 the p l a t e n c o e f f i c i e n t of f r i c t i o n was taken as 0.174 and the g l a s s f r i c t i o n was taken as z e r o . The r e s u l t s in F i g . 5.4.1.1 show that no c o n c a v i t y deve lops in t h i s case and the v e l o c i t y cu rves in F i g . 5 .4 .1 .2 show very l i t t l e d i s p e r s i o n . The v e l o c i t i e s are e s s e n t i a l l y those of the homogeneous s o l u t i o n . C o n c e p t u a l l y the s t r a i n r a te s e n s i t i v i t y of the m a t e r i a l p revents a c o n c e n t r a t i o n of de fo rmat ion from 102 o c c u r r i n g . The v e l o c i t y d i s c o n t i n u i t i e s in F i g . 5.3.1.1 can be regarded as concen t r a t ed l i n e s of d e f o r m a t i o n . The s t r a i n r a t e s e n s i t i v i t y of the m a t e r i a l would make these l i n e s have a l a rge e f f e c t i v e s t r e s s . Homogeneous de fo rmat ion thus occurs r a the r than concen t r a t ed bands of d e f o r m a t i o n . C a l c u l a t i o n s were done f o r other H/D r a t i o s wi th r e s u l t s tha t were c o n s i s t e n t wi th t h i s i n t e r p r e t a t i o n . 5 . 4 . 1 . S t r a i n Hardening R e s u l t s For H/D>1. S t r a i n harden ing can be expected to have the same e f f e c t as s t r a i n r a te s e n s i t i v i t y . However, the e f f e c t can be expected to be de layed somewhat because s t r a i n i s r e q u i r e d to i nc rease the y i e l d s t r e s s of the m a t e r i a l . I n i t i a l l y the s o l u t i o n w i l l be the same as fo r the cons tan t y i e l d s t r e s s case but as s t r a i n i n g takes p l a c e the de fo rmat ion bands in the m a t e r i a l w i l l harden and fo r ce m a t e r i a l away from the bands to d i s t o r t . Thus s t r a i n harden ing shou ld l ead to homogeneous de format ion as does s t r a i n r a te s e n s i t i v i t y . F i g . 5 .4 .2.1 to 5 .4 .2 .3 show the r e s u l t s f o r the s t r a i n ha rden ing case and can be compared wi th 5.3.1.1 to 5 .3.1 .2 (b ) fo r the cons tan t y i e l d s t r e s s case and to 5.4.1.1 and 5 .4.1 .2 fo r the s t r a i n r a te s e n s i t i v e c a s e . The r e s u l t s show t h a t , in the s t r a i n ha rden ing c a s e , a concave boundary does form as p r e v i o u s l y . However, s t r a i n ha rden ing p reven ts i t from forming to the same dep th . F i g . 5 .4 .2 .2 fo r the v e l o c i t y curves when compared wi th those in F i g . 5 .3.1 .2 (a ) shows the e f f e c t of s t r a i n harden ing very c l e a r l y . Even at H/D=3.327 the d i s p e r s i o n in 103 the curves fo r Y<12mm i s reduced in F i g . 5 .4 .2 .2 when compared wi th tha t in F i g . 5 . 3 . 1 . 2 ( a ) . For lower va lues of H/D the e f f e c t i s more pronounced. At H/D= 2.023, f o r example, the curves fo r the cons tant y i e l d s t r e s s case show that the y components of v e l o c i t y are e s s e n t i a l l y an t i symmet r i c about a h o r i z o n t a l l i n e drawn in the specimen at y=l0mm i f t h i s l i n e i s c o n s i d e r e d to have a zero y component of v e l o c i t y . For the s t r a i n harden ing case in F i g . 5 . 4 . 2 . 2 , however, the y components of v e l o c i t y i n d i c a t e tha t the v e l o c i t i e s are c l o s e r to the homogeneous va lues fo r y<l0mm than they are fo r y>l0mm. Comparison of F i g . 5 .4 .2 .3 fo r H/D=1.433 wi th F i g . 5 .3 .1 .2 (b ) i n d i c a t e s that in the s t r a i n harden ing case the i n v e r s i o n in the curves about a common po in t no longer o c c u r s . In the cons tant y i e l d s t r e s s case t h i s i n v e r s i o n occu rs when H/D=1.218. The absence of the i n v e r s i o n in the Vy v e l o c i t y curves at h igher va lues of H/D in the s t r a i n harden ing case than in the cons tan t y i e l d s t r e s s case i n d i c a t e s that the development of the concave boundary p e r s i s t s in the l a t t e r case fo r h ighe r s t r a i n l e v e l s in compress ion . It i s thus conc luded that the expected r e s u l t s fo r the s t r a i n harden ing case h o l d ; namely, that s t r a i n i n g causes the concen t r a t ed bands of de fo rmat ion tha t occur in the cons tan t y i e l d s t r e s s case to ha rden . T h i s , in tu rn f o r c e s the spread of de fo rmat ion and a more un i fo rm energy d e n s i t y to occur s p a t i a l l y in the m a t e r i a l than in the cons tant y i e l d s t r e s s c a s e . 104 5.5. THE DYNAMIC COMPRESSION OF ALUMINUM. The purpose of t h i s s e c t i o n i s to examine the e f f e c t of i n e r t i a on the de fo rmat ion pa t t e rn s in aluminum. The c a l c u l a t i o n s were done us ing equa t ion 5.1.4.1 fo r the v a r i a t i o n of y i e l d s t r e s s wi th s t r a i n h a r d e n i n g . A cons tan t c o e f f i c i e n t of f r i c t i o n of 0.174 was assumed on the p l a t e n b o u n d a r i e s . T h i s was the average va lue ob ta ined from the c a l c u l a t i o n s in s e c t i o n 5.2.2 in which the master s l ave node method was used . The top boundary of the specimen was i n i t i a l l y g iven an a c c e l e r a t i o n of l 0 2 O m / s 2 to a speed of I00m/s to approximate an i n f i n i t e a c c e l e r a t i o n . The speed of I00m/s was ma in ta ined u n t i l t = 37.69r 's . The top p l a t e n v e l o c i t y was then set to zero wi th the r e s u l t tha t the top boundary of the specimen moved away from the p l a t e n . D i s p l a c e d shape p l o t s are p resen ted in F i g s . 5.5.1 and 5 . 5 . 2 . The top of the specimen i n i t i a l l y was d i s p l a c e d the most wi th the lower pa r t of the specimen not be ing a f f e c t e d by the impact . The d i s p l a c e d shape p l o t in F i g . 5.5.2 fo r t = 2 3 . 9 l y s i n d i c a t e s t h i s t ime tha t the bottom par t of the specimen i s moving approx imate l y as much as the t o p . One e f f e c t of i n e r t i a i s thus to make the m a t e r i a l move nonsymmet r i ca l l y about the h o r i z o n t a l c en t r e l i n e of the specimen shown in EF in F i g . 5 . 5 . 1 . In the absence of i n e r t i a the p a t t e r n would be symmetric about t h i s l i n e . At t = 2 3 . 9 l f s the d i sp lacement p a t t e r n has become approx imate l y symmetric about EF but the cu r va tu r e of the y g r i d l i n e s i s d i s t i n c t from that observed in F i g . 5.3.2.1 fo r example. S t r a i n harden ing would account to a c e r t a i n 1 0 5 extent fo r the suppression of any de fo rmat ion bands in the m a t e r i a l but not to the extent observed in F i g s . 5.5.1 and 5 . 5 . 2 . Thus even when the p a t t e r n has become symmetric about EF i n e r t i a e f f e c t s are s t i l l very dominant . At t=37.69^s in F i g . 5 . 5 . 2 , f o r example, the change of cu r v a tu r e of the y g r i d l i n e s at GF i s c o n s i s t e n t wi th the fo rmat ion of de fo rmat ion bands as a compar ison w i th F i g . 5.3.2.1 i n d i c a t e s . At the top of the spec imen, however, the cu r va tu re of the y g r i d l i n e s i s s i m i l a r to that shown in the deformed meshes fo r p l a s t i c i n e p resen ted in s e c t i o n 5 .2.1. P l a s t i c i n e has been shown to be s t r a i n ra te s e n s i t i v e which l eads to homogeneous d e f o r m a t i o n . These o b s e r v a t i o n s show that i n i t i a l l y there i s a t r a n s i e n t e f f e c t as a l l p a r t s of the specimen are unequa l l y a c c e l e r a t e d from a s t a t i o n a r y c o n d i t i o n . When t h i s t r a n s i e n t e f f e c t i s complete i n e r t i a i s s t i l l dominant in caus ing homogeneous d e f o r m a t i o n . In the present case be ing c o n s i d e r e d i n e r t i a e f f e c t s are dominant over e f f e c t s of m a t e r i a l p r o p e r t i e s and the de fo rmat ion p r o f i l e s can be expected to be e s s e n t i a l l y independent of the m a t e r i a l c o n s t i t u t i v e behav io r assumed. The d i s p l a c e d shape p l o t fo r t = 4 5 . 1 l y s in F i g . 5.5.2 i s f o r the un load ing c a s e . The f a c t tha t s i g n i f i c a n t movement occu rs when the top p l a t e n that was i n i t i a l l y deforming the specimen was removed i n the c a l c u l a t i o n i s i n d i c a t i v e of the f a c t that i n e r t i a e f f e c t s are dominant . The top of the specimen tends to move in more at the cen t r e than at the edge. E s s e n t i a l l y an un load ing wave i s moving i n t o the specimen from the top r i g h t c o r n e r . 106 5.6. STRESS RESULTS. 5 . 6 . 1 . S t r e s s R e s u l t s For R i g i d - p e r f e c t l y P l a s t i c  Q u a s i - s t a t i c Compress ion . In t h i s s e c t i o n the normal s t r e s s on the top p l a t e n w i l l be examined on b l o cks of r i g i d - p e r f e c t l y p l a s t i c m a t e r i a l fo r a wide range of H/D. The s t r e s s e s fo r the cases shown in F i g s . 5 .3 .1 .4 , 5 .3 .2 .3 and 5 .3 .2 .4 w i l l be examined. F i g s . 5.6.1.1 and 5 .6 .1 .2 show the normal p l a t e n s t r e s s e s fo r v a r i o u s H/D fo r u n s t r a i n e d b locks of r i g i d - p e r f e c t l y p l a s t i c m a t e r i a l . The r e s u l t s show that for H/D>2 the normal s t r e s s e s on the p l a t e n can be q u i t e d i f f e r e n t than the same s t r e s s components w i t h i n the m a t e r i a l . A f r i c t i o n h i l l does not ho ld in the case of H/D>2. A l s o shown in F i g . 5.6.1.1 are the s t r e s s e s fo r a specimen wi th H/D=2.023 wi th p r e - s t r a i n to show that the c o n c l u s i o n i s not a f f e c t e d by the boundary shape. 5 . 6 . 1 . 2 , however, shows tha t f r i c t i o n h i l l s do ho ld fo r f l a t specimens of sma l l H/D. In t h i s case the boundary s t r e s s e s do not d e v i a t e s i g n i f i c a n t l y from the va lues in the m a t e r i a l away from the p l a t e n s . F i g . 5 .6 .1 .3 summarises the p l a t e n normal s t r e s s e s fo r v a r i o u s H/D r a t i o s and v a r i o u s m a t e r i a l c h a r a c t e r i s t i c s . F i g . 5 . 6 . 1 . 3 ( a ) shows that f o r a r i g i d - p e r f e c l y p l a s t i c m a t e r i a l a f r i c t i o n h i l l on l y occurs when H/D>0.45. For higher H/D va lues an i n ve r se f r i c t i o n h i l l h o l d s . F i g . 5 .6 .1 .3 (b ) shows tha t for the s t r a i n harden ing case a f r i c t i o n h i l l e x i s t s at h ighe r v a l ues of H/D than 1 0 7 f o r the cons tan t y i e l d s t r e s s c a s e . S i m i l a r l y , F i g . 5 .6 .1 .3 ( c ) shows that fo r the s t r a i n ra te s e n s i t i v e case a f r i c t i o n h i l l ho lds at a l l the H/D va lues g i v e n . These o b s e r v a t i o n s are c o n s i s t e n t wi th the c o n c l u s i o n s reached e a r l i e r ; namely, that both s t r a i n harden ing and s t r a i n r a t e s e n s i t i v i t y l ead to homogeneous d e f o r m a t i o n . As the e f f e c t i v e de fo rmat ion ra te becomes un i fo rm throughout the m a t e r i a l a f r i c t i o n h i l l w i l l tend to form on the p l a t e n s . T h i s f o l l o w s from the s imple s o l u t i o n in s e c t i o n 2.2 which assumes that the s t r e s s over a v e r t i c a l p lane in the m a t e r i a l i s un i f o rm . 5 . 6 . 2 . S t r e s s Resu l t s For Dynamic Compress ion . F i g s . 5.6.2.1 and 5 .6 .2 .2 show the s t r e s s r e s u l t s - f o r dynamic compress ion of aluminum c o n s i d e r e d in s e c t i o n 5 .5 . The r e s u l t s i n d i c a t e that a f r i c t i o n h i l l ho lds on the top boundary in a l l cases but that an inve rse f r i c t i o n h i l l ho lds on the lower boundary i n i t i a l l y . At t=23.1 s, however, a f r i c t i o n h i l l a l s o s t a r t s to form on the lower p l a t e n . Inverse f r i c t i o n h i l l on the lower boundary i s a dynamic e f f e c t . 108 \'\ V>\ \ \ \\\\\\\\\\\\\\\\\\\\\K\\^.\\V\\NS{ V E L o c i T y F I E L D FcR HoMoGEWEooS C O M P R E S S I ON B E T W E E N FRICTION! LESS P L A T E S . \| ELOCITy VECTORS COMPARISON Fof\ ALUMINIUM B E T W E E N 15" E L E M E N T A N b 5" O E L E M E N T P R E D I C T I O N S O F t) E Fo R h t O SHAPES FoR. H/Ho ~ o. S% 2_ C S o L X b L I N E ) AND H/Ho = o. ( D O T T E D LINE) . 110 \ 2. 3 .4- i , -7 e s vO u 12- l i 14- » 5 ~ ^ - i — • — i 1 1 1 1 r -T r F I & . 5". (.3.1 . G A U S S P O I N T B O U N D A R Y S H E A R S T R E S S E S F O R V A R I O U S . S I Z E S O F I N C R E M E N T A L S T R A I N S T E P S "T A C H I E V E , A F R A C T I O N A L H E I G H T R E D U C T I O N O F O . 6 ^ 1 . I l l Zo -I 0 -6"= ^>0+c.e 1n 7 U S £ D • SHA3AIK ( Rf-f ." 7 "* HARTLEY U e f S.. I 1 ) ToV\t,sosi (kef.10 Wa.\f h'&.ri . a(t.+"S"'> yD= i o n e iSFa . F I G . r . i . - r - . i . _ r f t L l t f . M l O U COP.VE& o F e AGMKST- e f o * ALVJWxNUM o., o . i o . i U b.Fb.L b.7 o.6 b.1 'i.o \.\ 112 t = n.o.i -b- \4-.\Q.r^.f =n.V^4-F I G . S. Z. \ • \ . E X P E R 1 M E N T A \ _ D E F O R M E D P R O F I L E S F O R T H E P L A N E . S T R A I N C O M P R E S S I O N O F P L A S T I C I N E . 1 1 3 c j i i 1 i — 1 i ; 1 \ ' D t = i-Bims, E=O.\H- t - 3..-72.ms, E=e.2.-v^ 1 -1-B - t = L , . - ? , B m s . F = 0 . ^ 7 r r FIC . S - . 2 . 1 . 2 . . E X P E RI M E NT A.L DEFORCED PROFILES FOR, -r - u r p L A N r . STRAIN COMPRESSION OF ' 1 1 4 1 • tt • c F L 1 T T H ^ 2 F I S . 5". Z..\ . 3 fa). T H E O R E T I C A L D E F O R M E D P R O F I L E S FoR t = O To 1 . - 2 . F O R T H E P L A N E , S T R A I N D Y N A M I C c o n f s a s i i o N O F P L A S T I C I N E . A T +.+ n/5 W I T H 7g = O • 115 116 T . + m/s ''ii. if.-r \ \ \ A 1 / / A.Tm. 's zzzzznzzzm I LT t> Cl ' Vo.. F I G . 5". E . I . cal . T H E 0~R E T I C M - D E F O R M E D P R O F I L E S F R O M "t = o T O t . J i O m s F O R THE. P L A N E S T R A I N D Y N A M I C C O M P R E S S I O N OF PLASTIC1ME. A T T-..4-JivLS W I T H ^3= O . I . 7p=o.a'n t= l O . O B m s , t-~ O .TOT t = \ 0 . - 3 T m s , £, = 0.15"U F I G . S " . E . I . T C O . T H E O R E T I C A L D E F O R M E D P R O F I L E S F R O M "t = ?.Us T O 10 .31m- . F O R T H E P L A N E . S T R A I N D Y N A M I C C O M P R E S S I O N O F P L A S T I C I N E A T +.4- M / S W I T H = o. I . 118 t|4.+m/s 0.122. llTo •X FIG. 5". E . \ . r ( ^  . T H E O R E T I C A L R E F O R M E D P R O F I L E S F R O M t = O T O I <Vml F O R T H E P L A N E S T R A T H D Y N A M I C C O M P R E S S I O N O F P L A S T I C I N E , A T A . A m/s W I T H <^3 ~ -119 fXPFCTMFMTAi;- I Co} . E X P E R I M E N T A L . R E S U L T S F O R , T H E L P L A N E S T R . M N C O M P R E S S I O N 0 F P L A S T I C I N E . (W) T H E O R E T I C A L R E S U L T S F O R T H E . D V H A M t C P L A N E . S T R A I N C O M P R E S S I O N O F P L A S T I C I N E . F o R T H R E E . VALVJE.5 O F G L A S S F R I C T I O N C O E F F I C I E N T , ^ , O F 0 - 2 3 S " , o. \ , AND O . A P L A T E N F R I C T I O N C O E F F I C I E N T , ? P , o F W A S U S E D F O R A L L CASES. THEORETICAL I N I T I A L S H A P E . . OEFoRMED SHAPES A T \ 91 "1 : r t J-I * t (A\umii\orv p l a n s ' ) V i e w f Tom A plates) to o ?« = o F I G , . 5 - . 2 . 1 . 1-121 T«EORETICAL E X P t R I M t W T A L U o A D ( H U W I I C »y LOAD CELL.) 14 21 28 3 5 TIME ( S X 1 0 ) 42 49 56 6 3 F I G . S. Z. \. E X P E R I M E N T A L AND C A L C U L A T E D LOADS FOR P L A S T I C I N E , 122 ro 0 10 20 30 40 50 6 0 70 80 90 100 TIME ( S X 1 0 ) F I G . s . i . \ - 8 C a . i . POWER D I S T R I B U T I O N FOR PLANE S T R A I N COMPRESSION OF P L A S T I C I N E . 0 10 2 0 30 40 50 6 0 70 80 90 100 TIME ( S X I O - ^ ) F I G . 5". L. \. fcCW). POWER D I S T R I B U T I O N FOR P L A N E S T R A I N COMPRESSION OF P L A S T I C I N E . .124 125 R I O C O f O 180 0 7S00 I W O I W O O ' O ( ,-0()U (S31U3WJ X 1 2 6 H / p = 2 . 0 Z 3 H / o = \ . 4 - 3 3 U / D = I . U 7 F I G . FT. - t . \ . DEFORMED SHAPES ' W I T H V E L O C I T y V E C T O R P L O T S A T V A R I O U S S T A G E S OF C O W P t t E S S I O N FROM Ho/t>o= *T F O R C A S E \ . A P R O P O S E L D U P P E R B O U N D S O L V J T X O N I S A L S O G I V E N . T Vo v* y j i—i—J— i • • jf 3 6 9 12 15 18 21 24 27 £ = O.O 2,2. 1 tne. H/D = 3.82.U 2 4 6 B 10 12 14 16 18 20 22 y ( T H X \ 0 - 3 ) e=o.zi7 F I G GRAPHS OF -VY /SGAIUST y CASE. \ . • ALONG X GRID 3 6 9 12 15 IB 21 24 y Cm.x\o-i) £.= 0. o<=\2_ H/O - 3. 3Z7 2 0 6 8 10 12 14 16 IB 20 y cm./s A \ 0"\) fc = p . 34-\ VWD - a . 02-3  L I N E S F O R 128 129 0 I Z 3 - T - S " G > - 7 D y F I G . 5-..3_t.3.Ca') . G R A P H S O F " V y / N J o A G A I N S T Y A L O N G , X G R I D L I N E S F O R T H E U P P E R B O U N D M O u E L F O R C A S E . t A T H / D = £ . 0 2 3 . 130 \ 2. i -T- 5" t "7 8 y FTP,. 5". ->>.\. ^CIP^ . G R A P H S O F V y / V o A G A T N S T Y A L O N G X G R I D L XM E S F O R T H E . U P P E R E>0VJND MODEL F o R C A S E I A T H / D = 1. M ~7 . 131 132 E E OE LI » Z IZ 81 S I Z t E 9 E O EC OE U *l ( Z 81 S I Z I 6 9 t O E E OE t Z « IZ 81 S I Z I 6 o H <£ > o u-H in* 5° In H t - O II o O ( 0\ Y s/tt) EE OE I.Z » Z I Z 81 S I Z I 6 9 E O z 2 lfl o X J u ul i «" o H u- h ° £ >/> a: o 0- o ^ _ o Z o ul o u-o -J co 133 ° 2 4 6 8 10 12 14 16 18 20 22 y c u W s x t o - ' O 3 6 9 12 35 18 21 24 YCWsxvo-"*) F I G . J . l . l . L . - VY A S I M N S T y Fop T W O S L I G H T L Y S T R A I N E D B L O C K ' S O F W o / D o = 2> AVJ D "3. 5" 135 G R A P H S OF - V y A G A I N S T y A L O N G x G R I D L I M E S FoR. C A S E a . 137 Fltvite E\emeTvTS. \ P\a.tev\s>. 1 TOP BOUNDARY_~ ( Vy - O ^  . C E N T R E ( Vy =cO F I G . 5 - . 3 . Z L . 3 . -A COV.PAPISOU O F V y A G A I N S T X FP.OVI T H E F I N I T E E L E M E N T C A L C U L A T I O N S W I T H T H E S I M P L E M O D E L P R E D I C T I O N S FOP. T H E C A S E 138 f f r l ^ W t V\3.\T oi s p e c i m e n QTV\^ ~~) I F l i u t e E \ e*i \ e ? i s E2 tr L c e g Strode WoJel L . L . L . U . 'TOP BOUNDARY. CMy = 0 . V* I N T E R H E D I A T E X P O S I T I O N . C E N T R E ( VY = O F I G . S\ 3 . 2.-T-A C O M P A R I S O N O F V Y A G A I N S T X F P C V. THE.. F I N I T E E L E M E K T C A L C U L A T I O N S W I T H T H E S X K I P L E M O D E L P R E f c X C T I O N S FOR T H E C A S E H/D = O.IZ5 " 139 cvJ \ O UNSTRAINED fcLOCH •+• CONTINUOUS STRAINING FRott t L r V T E N I N D E W T A T I O N T CRe^.s:) < S E C . 2 . 3 . \ . A N D 2. 3 . a • i P/2.K= \ FOR , FVJL\_ STICK ON VL^TCMSV " I O NWRKIV1ED BLOCY* -t" CONTINUOUS STRAINING FRon H/CLW SLIP LINE: F I E L D . ( R e f . XS) C SEC- 2.3-^-N O F R I C T I O N } F X G A C O M P A R I S O N O F NORWAL PLATEN T R A C T I O N F R O W T W E F I N I T E E L E M E N T C A L C U L A T I O N S F O R Av R I G I D — P E R F E C T L Y P L A S X C M A T E R I A L . W I T H T H E P R E D I C T I O N S O F A P P R O X I M A T E C L O S E D F O R M S O L U T I O N S . 140 11 a it n 2H < y ,Tn v6 TTTTTTTT H U TT £.= O. 0 ^  Z-H/D = •.V.1. £ = 0.2-17 M/p = 2. F T fr. 5YT. V . l D E F O R M E D S H A P E S W I T H V E L O C I T Y V E C T O R F L O A T S A T V A P.T 0 0 S S T A G E S O F C O M P R E S S I O N ) r- R O M H j /D 6 = T F O R P L - f V S T I C i ^ E . . 141 EE CE LZ »Z IZ SI SI ZI 6 9 E " ^_0\X S/UJ UJ H M LZ « IZ BI SI Zt 6 9 e ° 142 . £ . = 0 . 0 2 . 2 . L = 0 . 0 = 1 2 . £ . - 0 . 2 . \ 7 H / D = 3 . B Z U H / t J = 3 . T > Z 7 H / D = Z . 5 ^ 3 H / D = Z . 0 2 . 3 H / D = I . FI£.5".-T.2..\ D E F O R C E D S H A P E S W I T H V t ^ o c i T y V E C T O R P L O ~ r S A T V A R I O U S S T A G E S O F C O M P R E S S I O N F R t m Ho/Do = T " F O R ALUMIUUtt. 143 F I G . S~. ^- .2. . ^ . A FVJTHER GrUPH OF ~ V v A.GA1NST y ALONG X G R I D L I M E S FOR A L U M I N V J M . 145 1 1— r / , >• i 1 j-4—, 1 9 g n . n H - , 4 s o . i 7 l , t a U t V 5 . E = o. oas-.-rV^o.^SZ , T - L?.-^ F I 6 . S~. g . \ . r , l s f L A C E D SHAPE. ? L O T S F o R T H E D Y K A M I C COW?RESSXOM OF A L U M I N U M A T \ O O t v \ / S > . 146 t o o w / s 1 F R E E (Mr T\f>1 S k d i v c l ) F T ^ g . g . 2. . •DISPLACED SHAPE PLOTS C O N T I N U E D FOP, T H E DYNAMIC C O M P R E S S I O N OF ALUMINUM AT l o o ro/s. P T 7 ~ i — i — i — i 1 — i — i i r 2. 3 4- r U i 8 1 10 x (mm") HVun.S_J.S- _ ? P - . . Q . '-"7-*.--, \ • I — I — I — I 1 — I — I « <~ O . S 1.2. t.<\ Z . U t . 7 5".<V _ . l ( , . 6 " ' • • s ' T r — i — i 1—i 1—i 1— T r 0 . _ Lb' ZA- 3 . 5 - t . Z . 5". I U.O 7 . 8 8 . 7 1 . U H.J Do = z. _>__.•» 'j.^y..'^ o.b i i r — r — r T 1 r •T 2.3 3.2. +.1 5\0 5'.1 _.8 7.7 8.L 1.5 X ( mm) GR(\?V\S O F G _ ( E L E M E N T M E R A S E S ) A G A I N S T X F O R H O R I Z O N T A L R O W S , O F E L E M E N T S ON \J S T P» A I N E D B L O C K S O F ^ U M I N U M - A P L A T E N F K i c T l d M C O E F F I C I E N T O F 0 . r 7 4 - W A S U S E D I N E . A C H C A S E . . 148 OZZ 0l>"2 C S l Ol"Z <3L\ o t | 7y f t I S i« Ui TP. ^ o p s * . * z x Q Iii Y Ul * ri s * > < U) ui o u. z H <£ < uJ • s> <C h 2 Ul UJ UJ uJ 7 H H u J 0 X Li. o ill J Z H J (A 2 3»> o X ti II .J (3 f ' L. _ 3= ™OPT 150 z H o «0 rf o u 0 > P (4 H ui I- • o r H ul o rf H ul < u-o rf J UJ Ul I 151 152 CHAPTER 6  CONCLUSIONS The f i n i t e element model which was deve loped fo r l a rge p l a s t i c p lane s t r a i n de fo rmat ion proved to be an i n v a l u a b l e t o o l f o r i n v e s t i g a t i n g the fundamental c h a r a c t e r i s t i c s o f d i e - f o r g i n g o p e r a t i o n s . I t d e a l t s u c c e s s f u l l y w i th s t r a i n h a r d e n i n g , s t r a i n r a t e s e n s i t i v i t y , i n e r t i a , i n t e r f a c e and end p l a t e f r i c t i o n over a wide range o f specimen s i z e s . I t i s a f l e x i b l e and compact code which hand les the ve ry l a r g e p l a s t i c de fo rmat ion i n an e f f i c i e n t way. The f i n i t e element code reproduced e x a c t l y the r e s u l t s f o r f r i c t i o n l e s s and hence homogeneous de fo rmat ion i n which p lane s e c t i o n s remain plane- Comparisons o f the b i l l e t shape were a l s o made between the f i n i t e element model p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s f o r q u a s i - s t a t i c compress ion of aluminum and the dynamic compress ion o f p l a s t i c i n e . There proved to be good compar isons over a l a r g e range o f d e f o r m a t i o n , as shown f o r some s e l e c t cases i n F i g s . 6 . 1 , 6.2 and 6 .3 . F u r t h e r s t u d i e s conducted w i th the f i n i t e element model i d e n t i f i e d many o f the fundamental c h a r a c t e r i s t i c s o f the p l ane s t r a i n f o r g i n g o p e r a t i o n s . The c o n c l u s i o n s o f the e f f e c t o f specimen h e i g h t (H) t o width (D) r a t i o , m a t e r i a l p r o p e r t i e s , f r i c t i o n , and i n e r t i a are g i v en below. 1. With r i g i d p e r f e c t l y - p l a s t i c m a t e r i a l and H/D>1 de fo rma t i on tends t o concen t r a t e a long l i n e s o f i n t ense s h e a r . The mode o f de fo rma t ion i s approximated by the upper 153 bound s o l u t i o n g i ven i n s e c t i o n 2 . 3 . 1 . M a t e r i a l tends to s t i c k on the p l a t e n (even wi th low c o e f f i c i e n t s o f f r i c t i o n ) and to genera te normal i n t e r f a c e shear s t r e s s d i s t r i b u t i o n s c l e a r l y d i f f e r e n t from c l a s s i c a l f r i c t i o n h i l l . Indeed the normal s t r e s s d i s t r i b u t i o n s were i n v e r s e f r i c t i o n h i l l s w i t h the maximum shear s t r e s s o c c u r r i n g not at the c en t r e o f the spec imen, but a t the ou te r edges . A symmetr ic b u c k l i n g o r boundary c o n c a v i t y o c cu r r ed w i th H/D>3. Boundary shape at the onset o f de fo rmat ion s i g n i f i c a n t l y a f f e c t s the mode o f d e f o r m a t i o n . 2. With r i g i d p e r f e c t l y - p l a s t i c m a t e r i a l and H/D<1 de fo rma t i on tends to concen t r a t e a long l i n e s o f i n t ense shea r . The mode of de fo rmat ion i s approximated by the upper bound s o l u t i o n g i ven i n s e c t i o n 2-3.3 . M a t e r i a l tends to s l i p on the work p l a t e n and genera te f r i c t i o n h i l l type o f normal i n t e r f a c e s t r e s s d i s t r i b u t i o n s . 3. W i t h s t r a i n h a r d e n i n g m a t e r i a l t h e l i n e s o f c o n c e n t r a t e d d e f o r m a t i o n become w i d e r and g i v e more homogeneous d e f o r m a t i o n . W i th mode ra t e f r i c t i o n t he homogeneous s o l u t i o n g i v e n i n s e c t i o n 2.2 i s more a p p r o p r i a t e . A f r i c t i o n h i l l type o f normal i n t e r f a c e s t r e s s d i s t r i b u t i o n i s o b t a i n e d . There i s a l s o a tendency f o r a concave p r o f i l e t o deve lop f o r t a l l spec imens, but the development i s somewhat l e s s pronounced than i t i s f o r r i g i d p e r f e c t l y - p l a s t i c m a t e r i a l . 154 4. With s t r a i n r a t e s e n s i t i v e m a t e r i a l s the l i n e s o f c o n c e n t r a t e d d e f o r m a t i o n become w i d e r and g i v e more homogeneous d e f o r m a t i o n . The mode o f de fo rmat ion i s s i m i l a r t o the s t r a i n ha rden ing case except t h a t the e f f e c t o c cu r s immed ia te l y the m a t e r i a l i s moving and does not r e q u i r e the m a t e r i a l t o be s i g n i f i c a n t l y s t r a i n e d as i t does i n the s t r a i n ha rden ing case . 5. With dynamic l o a d i n g the i n e r t i a e f f e c t s are i n i t i a l l y d o m i n a n t and cause l o c a l inhomogeneous d e f o r m a t i o n . However, as the energy o f impact i s d i f f u s e d through the spec imen, the de fo rmat ion becomes homogeneous. An i n v e r s e f r i c t i o n h i l l can deve lop on the s t a t i o n a r y p l a t e n wh i l e at the same t ime a t rue f r i c t i o n h i l l deve lops on the moving p l a t e n . 6. P l a t en and g l a s s p l a t e f r i c t i o n are both impor tan t i n d e t e r m i n i n g d e f o r m a t i o n c h a r a c t e r i s t i c s . G l a s s p l a t e f r i c t i o n tends to genera te a r e s t r a i n i n g f o r c e so t ha t m a t e r i a l tends t o r o t a t e around the lower p l a t e n . T h i s r e s u l t s i n the de fo rmat ion becoming l e s s homogeneous. The i n f l u e n c e o f p l a t e n f r i c t i o n depends upon the H/D r a t i o c o n s i d e r e d . Fo r H/D>1 s t i c k i n g o c c u r r e d f o r q u i t e low va lues o f the c o e f f i c i e n t o f f r i c t i o n . For H/D<1 s l i p p i n g occured f o r moderate v a lues o f the c o e f f i c i e n t o f f r i c t i o n . 155 7. Inverse f r i c t i o n h i l l s can develop at large H/D r a t i o s and low v a l u e s of s t r a i n h a r d e n i n g and s t r a i n r a t e s e n s i t i v i t y . 8. The energy of deformation i s used p r i m a r i l y i n p l a s t i c work wi thin the specimen and i n overcoming glass pla te f r i c t i o n . Platen f r i c t i o n and surface t r a c t i o n due to atmospheric pressure require very l i t t l e energy. JESG D E F O R M E D M E S H S T R A I N H A R D E N I N G P L A T E N S 7777 "1 (T S T R A I N H A R D E N I N G E L E M E N T M O D E L E X P E R I M E N T QUASI-STATIC COMPRESSION OF ALUMINUM P L A T E N S <T 1 G L A S S P L A T E S B A | F I N N I T E E L E M E N T M O D E L T7T E X P E R I M E N T ] - s i DYNAMIC COMPRESSION OF PLASTICINE F T C , 1 . 2 -FORHOTE EUEftflEWV E^lPERfl^EMT iXFilHiOR/3EMTAL m& ™i©(8SET0CA!L RESULTS \F01R THE pim\E Bin&m COGMIPIRKESSDOM ©F FLASTOODWI F T C-i . Lo. 3 REFERENCES 1. B . A v i t z u r . Meta l Forming P rocesses and A n a l y s i s . Mcgraw H i l l , 1968. 2. W.Johnson and P . B . M e l l o r . E n g i n e e r i n g P l a s t i c i t y . Van Nost rand Re inho ld Company. 1973. 3. R . H i l l The Mathemat i ca l Theory of P l a s t i c i t y . C la rendon P r e s s . O x f o r d . 1950. 4. H.Hencky. Uber d i e e i n i g e s t a t i s c h bestimmte F a l l e des  G l e i c h g e w i c h t s in P l a s t i s c h e n Ko rpe rn . Z.angew. Math. Mech. 3, 241. 1923. 5. W.Johnson, R.Sowerby, R .D .Ven te r . P l a n e - S t r a i n  S l i p - L i n e F i e l d s For Meta l-Deformat ion P r o c e s s e s . A  Source Book and B i b l i o g r a p h y . Pergamon P r e s s , 1982. 6. E .G.Thomsen. V i s i o p l a s t i c i t y . CIRP Con fe rence . September, 1963. 7. A .H . Shaba i k , E .G.Thomsen. Computer A ided  V i s i o p l a s t i c i t y S o l u t i o n of Some Deformat ion Prob lems. I n t . Symp. On Foundat ion of P l a s t i c i t y . E d i t e d by A.Sawczyk, Noo rdho f f , Leyden. 1972. PP. 177-199. 8. C.Moore, G .W .V i cke r s , S .N .Dw i ved i . • V i s i o p l a s t i c i t y  S o l u t i o n of Deformat ion P rob lems; Data Hand l ing  P rocedures . I n t e r n a t i o n a l J o u r n a l fo r Numer ica l Methods in E n g i n e e r i n g . V o l . 1 9 , 257-269 (1983) . 9. A .H . Shaba i k . F i n i t e D i f f e r e n c e Method fo r the Complete  A n a l y s i s of P lane S t r a i n E x t r u s i o n . P roceed ings of the T h i r d North American Meta lwork ing Con fe rence . V o l . 3. 1975. PP. 127-142. 10. O . C . Z i e n k i e w i c z . The F i n i t e Element Method. 3 ' r d E d i t i o n . M c g r a w - H i l l . 1977. 11. H . D . H i b b i t t , P . V . M a r c a l , and J . R . R i c e . A F i n i t e Element  Fo rmu la t i on f o r Problems of Large S t r a i n and Large D i sp l a cemen t . I n t . J . S o l i d s and S t r u c t u r e s . V o l . 6 . PP. 1069-1086. 1970. Pergamon P r e s s . 12. H.K i tagawa, Y . S e g u c h i , and Y . T o m i t a . An Incrementa l  Theory of Large S t r a i n and Large D isp lacement Problems  and i t s F i n i t e Element F o r m u l a t i o n . I ngen i eu r-A r ch i v 41. 1972. PP. 213-224. S p r i n g e r - V e r l a g . 13. M.Gotoh and F . I s h i s e . A F i n i t e Element A n a l y s i s of  R i g i d - P l a s t i c Deformat ion of the F lange in a  Deep-Drawing P rocess Based on a Fourth-Degree Y i e l d  F u n c t i o n . I n t . J . Mech. S c i . V o l . 20. PP. 423-435. Pergamon Press L t d . , 1978. REFERENCES 1 Uo 14. O . C . Z i e n k i e w i c z , P . C . J a i n , and E. Onate . Flow of S o l i d s  Du r ing Forming and E x t r u s i o n : Some Aspec ts of Numer ica l  S o l u t i o n s . I n t . J . S o l i d s and S t r u c t u r e s . V o l . 14. PP. 15-38. 1978. Pergamon P r e s s . 15. O . C . Z i e n k i e w i c z and P .N .Godbo le . Flow of P l a s t i c and  V i s c o - P l a s t i c S o l i d s w i th S p e c i a l Reference to  E x t r u s i o n and Forming P r o c e s s e s . I n t . J . Numer ica l Methods in E n g i n e e r i n g . V o l . 8. PP. 3-16. 1974. 16. O . C . Z i e n k i e w i c z and P .N .Godbo le . A Pena l t y Func t i on  Approach to Problems of P l a s t i c Flow of Meta l s wi th  Large Sur face De fo rma t i ons . J ou rna l of S t r a i n A n a l y s i s . V o l . 10. No. 3. 1975. 17. C .H .Lee and S .Kobayash i . New S o l u t i o n s to R i g i d - P l a s t i c  Deformat ion Problems Us ing a Ma t r ix Method. J ou rna l of E n g i n e e r i n g fo r I n d u s t r y . Aug. 1973. PP. 865. 18. J . W . H . P r i c e and J . M . A l e x a n d e r . Specimen Geometr ies  P r e d i c t e d by Computer Model of High Deformat ion  F o r g i n g . In t . J . Mech. S c i . Vo l 21. PP. 417-430. Pergamon Press L t d . 1979. 19. P . H a r t l e y . C . E . N . S t u r g e s s , and G.W.Rowe. F r i c t i o n in  F i n i t e-E l emen t A n a l y s i s of Me ta l fo rming P r o c e s s e s . I n t . J . Mech. S c i . Vo l 21 . PP. 301-311. Pergamon Press L t d . , 1979. 20. J .B .Hawkyard and W.Johnson. 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V o l . 20. p p . 361-371. 1978. 30. G .E .Mase , Continuum Mechan i cs . Schaum's O u t l i n e S e r i e s . Mcgraw-Hi l l Book Company. 1970. 31. S . Y .Aku . R . A . C . S l a t e r , and W.Johnson. The Use of  P l a s t i c i n e to S imulate the Dynamic Compression of  P r i s m a t i c B locks of Hot M e t a l . I n t . J . Mech. S c i . Pergamon Press L t d . 1967. V o l . 9. PP. 495-525. 32. G .W .V i cke r s , A . P l umt ree , R.Sowerby, and J . L . D u n c a n . S imu l a t i on of the Heading P r o c e s s . T r a n s . ASME. J ou rna l of E n g i n e e r i n g M a t e r i a l s and Techno logy . 1974. 33. T . A l t o n , S.Oh and H. G e g e l , Meta l Fo rming , American S o c . of M e t a l s , 1983. 34. Open Die F o r g i n g Manua l , F o r g i n g Ind. A s s o c . , 1982. REFERENCES 

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