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"FABRICATION AND MECHANICAL PROPERTIES OF GRAPHITE FIBER REINFORCED ALUMINUM ALLOYS  by  KIYOYUKI ESASHI B.E.(1967) and M.E.(1969), TOHOKU UNIVERSITY, JAPAN  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in the Department of Metallurgy  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1976  \u000CIn  presenting  this  thesis  an advanced degree at the I  Library shall  f u r t h e r agree  for  scholarly  by h i s of  this  written  make  it  freely available  that permission  Columbia,  I agree  r e f e r e n c e and this  It  for financial  i s understood that gain s h a l l  permission.  University of B r i t i s h  Columbia  not  copying or  for  that  study. thesis  purposes may be granted by the Head of my Department  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Date  for  the requirements  f o r e x t e n s i v e copying o f  Depa rtment The  fulfilment of  the U n i v e r s i t y of B r i t i s h  representatives. thesis  in p a r t i a l  or  publication  be allowed without my  \u000Cii  ABSTRACT  A new method to f a b r i c a t e continuous graphite f i b e r r e i n f o r c e d aluminum a l l o y composites has been developed and the t e n s i l e properties of the composites have been i n v e s t i g a t e d .  Composites with 601, 201 and 7178 a l l o y  matrix containing up to 19 volume per cent of Thornel 50 graphite f i b e r were studied. These composites showed lower t e n s i l e strength values than the expected values from the \" r u l e of mixture\".  A t h e o r e t i c a l model i s discussed i n order  to understand the t e n s i l e properties of these composites.  In t h i s mechanism,  graphite f i b e r s are thought to be broken continuously one a f t e r another at maximum loading point of ultimate t e n s i l e strength during the t e n s i l e t e s t . A f u r t h e r attempt has been made to improve the t e n s i l e strength of these composites, based on the above t h e o r e t i c a l work.  \u000Ciii  TABLE OF CONTENTS Page ABSTRACT  ii  LIST OF FIGURES  v  LIST OF TABLES  ix  ACKNOWLEDGEMENTS  *  Chapter I  II  INTRODUCTION  1  1-1. General Background  1  1-2. Previous Work on Fabrication Techniques  2  1-3. Previous Work on Strength of Graphite Fiber/Aluminum and/Alloy Composites  12  I- 4. Purpose of Present Work  14  EXPERIMENTAL PROCEDURE  18  II\u2014 1 - Preparation of Composite Specimens  III  11-2. Tensile Testing  27  II-3. Microscopic Observations  28  II- 4. Micro Probe Analysis  29  EXPERIMENTAL RESULTS  29  111 \u2014 1 - Fiber Volume Fraction  IV  18  29  III- 2. Micro Probe Analysis of the Specimens  29  111-3. Tensile Stress S t r a i n Curves  29  I I I - 4 . Ultimate Tensile Strength  36  111-5. Fracture Elongation  40  III- 6 . Microscope Observations of Tested Specimens  41  DISCUSSION  50  IV- 1. Rule of Mixture  50  IV-l-i.  Rule of Mixture f o r Continuous Fiber Reinforced Materials  I V - 1 - i i . Strength of Discontinuous Fiber Reinforced Materials  50 53  \u000Civ  Page,  IV-2. Propagative Fiber F a i l u r e Model f o r Graphite Fiber Reinforced Aluminum A l l o y Composites IV-2-i.  Ultimate Tensile Strength of Distributed Specimens  Homogeneously  IV-2-ii.  Energy C r i t e r i a of Propagative Fiber F a i l u r e f o r Homogeneously D i s t r i b u t e d Composites  57 57 59  I V - 2 - i i i . Improvement of Tensile Strength in Bundle Structure Composites  76  IV-2-iv.  Estimation of V a r i a b l e s , b and T t s  80  IV-2-v.  Evaluation of Ultimate T e n s i l e Strength of Composites by Propagative Fiber F a i l u r e Model  83  u  IV-3. C h a r a c t e r i s t i c s of Powder S l i p Interpenetration Method  89  V  SUMMARY AND CONCLUSIONS  91  VI  SUGGESTION FOR FUTURE WORK  92  REFERENCES  93  \u000CV  LIST OF FIGURES  Page Figure 1  The V a r i a t i o n of Al^Cg i n 5 wt % Graphite Fiber-Aluminum Composite a f t e r Annealing f o r 24 hr at Various Temperatures (from Ref. 29)  2  9  The V a r i a t i o n of Al^C^ in 5 wt % Graphite Fiber-Aluminum Composite a f t e r Annealing f o r Various Times at 600\u00B0C (from Ref. 29)  3  Strength of Al Coated Graphite Fibers a f t e r Heat Treatment f o r 1 Day at Various Temperatures (from Ref. 40)  4  9  10  Tensile Strength of the Al-Graphite Fiber Composites Fabricated by the Matrix F o i l Method ( P l o t t e d from numerical data i n Ref. 30)  5  T e n s i l e Strength of Graphite Fiber Aluminum Composites Fabr i c a t e d by the Chemical Vapour Deposition Method (from Ref. 2)  6  10  13  Tensile Strength of Graphite Fiber Aluminum A l l o y Composites Fabricated by the I n f i l t r a t i o n Method ( P l o t t e d from numerical data in Ref. 24)  7  13  L i q u i d Phase Hot Pressing Die Configuration to Fabricate Specimens from the Al I n f i l t r a t e d Graphite Fiber Composite Wire (from Ref. 25)  16  8  Flow Sheet of the Specimen Fabrication Process  20  9  Scanning Electron Micrographs of the Powders Used, x 1000, (a) Aluminum, (b) Magnesium, (c) Copper, (d) S i l i c o n , (e) Zinc  21  \u000Cvi  Page 10  Schematic Diagram of the Interpenetration Device  23  11  Separating Boat  24  12  Hot Pressing Die Set  24  13  Tensile Specimen Geometries  24  14  Cross Section of a Uniform Composite, 201T6, #29, 12.1% V , x 114  30  f  15  Cross Section of a Bundle Structure Composite, 7178T6, #39, 10.0% V , x 62  30  f  16  Relation Between the Number of P l i e s f o r the Interpenetration Process and the Fiber Volume Fraction in Composites  31  17  Stress S t r a i n Curves of 601T4 A l l o y and the Composites  33  18  Stress S t r a i n Curves of 201T6 A l l o y and the Composite  34  19  Stress S t r a i n Curve of 7178T6 A l l o y  35  20  Tensile Strength of 601T4 Composites, and T h e o r e t i c a l Curves  37  21  Tensile Strength of 201T6 Composites, and Theoretical Curves  38  22  Tensile Strength of 7178T6, Uniform and Bundle Structure Composites, and Theoretical Curves  39  23  Fracture Surface of 601T4 Composite, #17, 14.5% V , x 1000  42  24  Fracture Surface of 201T6 Composite, #29, 12.1% V , x 1000  43  25  Fiber F a i l u r e Zone near Specimen Shoulder Polished Surface  f  f  of a 201T6 Composite Specimen, x 32 26  Scanning Electron Micrograph of Fiber F a i l u r e Zone Surface Showing Broken F i b e r s , x 200  27  43  43  Scanning Electron Micrograph of Zone Surface Showing S l i p L i n e s , x 1000  43  \u000Cvii  Page 28  Micrograph Indicating Propagation of the F i b e r F a i l u r e p r i o r to F a i l u r e of the Specimen, Longitudinal S e c t i o n , x 32  29  Longitudinal Section around the Zone Showing Matrix Grains, and Broken T i l t e d F i b e r s , NaOH Solution E t c h , x 180  30  44  44  Longitudinal Sections of Fractured 601T4 Specimens, Showing the Difference i n the D i s t r i b u t i o n of the Fracture Points in (a) Low V  f  (8.5%), #9, and (b) High V  (14.5%),  f  #17, Composites 31  45  Longitudinal Sections of Fractured 201T6 Specimens, Showing the Difference i n the D i s t r i b u t i o n of the Fracture Points in (a) Low V  f  (6.4%), #24, and (b) High V  f  (17%),  #29, Composites  46  32  Relation between B , E  33  Schematic Diagram of Stress S t r a i n Curves of the Composite,  f  c  and V  48  f  the F i b e r , and the M a t r i x , Obtained According to \"Rule of Mixture\" 34  51  T e n s i l e Strength of Copper Reinforced with 5 mm Continuous B r i t t l e Tungsten Wires (from Ref. 41)  35  Showing Notation Used i n the K e l l y and Tyson's Theory f o r the Discontinuous Fiber Composites (from Ref. 41)  36  54  Expected V a r i a t i o n of Stress along a Fiber w i t h i n a P l a s t i c Metal Matrix (from Ref. 41)  37  51  54  Stress D i s t r i b u t i o n i n the Discontinuous Tungsten F i b e r Obtained by Moire Technique.  Applied Stress on the Composite  i s Low (a) and High (b), (from Ref. 44, 47)  54  \u000Cviii  Page 38  Schematic Diagram of Stress S t r a i n Curve of the Matrix and the Fiber  60  39  Half E l l i p t i c Fiber F a i l u r e Zone in a Composite  60  40  Stress D i s t r i b u t i o n Change in a Fiber which i s Located in the Fiber F a i l u r e Zone  61  41  Fiber F a i l u r e Zones in a Specimen  63  42  Schematic Curves f o r Model Development  63  43  Schematic Diagram of Stress S t r a i n Curve at F a i l u r e Point  63  44  E l a s t i c Loading Curves f o r Crack Lengths a and a + da  71  45  Schematic Diagram of E l a s t i c Energy Released AABC and A A D E , when the Crack and the Fiber F a i l u r e Zone Traverse the Cross Section  46  73  Geometry of the Fiber F a i l u r e Zone i n the Bundle Structure Composite  77  \u000Cix  LIST OF TABLES  Table  Page  1  F a b r i c a t i o n Techniques of Metal Composites  2  Typical Properties of High Modulus Graphite Fibers Compared with other Reinforcing M a t e r i a l s  3  3  11  Tensile Properties of Various Aluminum-Alloy-Thornel 75 Composites (from Ref. 25)  15  4  The Nominal Composition of Matrix A l l o y s  19  5  Properties of Thornel 50 Graphite Fiber  19  6  Numerical Values of C o n t r o l l i n g Factors i n Interpenet r a t i o n Process  26  7  Hot Press and Heat Treatment  26  8  T e n s i l e Test Data of Specimens  32  9  Allowable L i m i t s of the Matrix Composition  36  10  Data f o r Fiber Fracture Zone C h a r a c t e r i s t i c s  49  11  Observed T  82  12  C a l c u l a t i o n of Strength of 601T4 Composites  84  13  C a l c u l a t i o n of Strength of 201T6 Composites  85  14  C a l c u l a t i o n of Strength of 7178T6 Composites  86  15  C a l c u l a t i o n of Strength of 7178T6 Bundle Structure Composites  87  u t s  Values and C a l c u l a t i o n Results of K'  \u000CX  ACKNOWLEDGEMENTS  The author g r a t e f u l l y acknowledges the helpful discussions with his research d i r e c t o r , Professor E. Teghtsoonian, and with Dr. J.S. Nadeau. He wishes to thank the members of the f a c u l t y and f e l l o w graduate students of the Department of Metallurgy f o r t h e i r continued support and i n t e r e s t in t h i s work. Financial assistance was received i n the form of an a s s i s t a n t s h i p under National Research Council of Canada grant number A-2452, and i s g r a t e f u l l y acknowledged.  \u000C1  I INTRODUCTION  1-1.  General  Background  Over the past 15 y e a r s , much research has been c a r r i e d out in attempts to r e a l i z e in p r a c t i c e the greater potential of high performance f i b e r r e i n forced composites.  The f i b e r reinforcement has been considered f o r the  strengthening of weak p l a s t i c m a t e r i a l s , such as r e s i n and some metals.  The  incorporation of strong f i b e r s into d u c t i l e metal matrices has been shown to bring remarkable increases in the strengths of these metals by some theore t i c a l and experimental work of the e a r l y period in the metal matrix composite history. An important simple expression for the composite t e n s i l e strength and tensile  modulus, so c a l l e d \" r u l e of m i x t u r e \" , was derived i n such work  (1)  and i t has been quite often used to discuss the t e n s i l e properties of various kinds of f i b e r composites. and t e n s i l e  In t h i s \" r u l e of m i x t u r e \" , the t e n s i l e strength  modulus of a composite are expressed as the combination or sum-  mation of contributed amounts from f i b e r s and the m a t r i x , to these p r o p e r t i e s . These contributions from each component are taken to be proportional to t h e i r volume f r a c t i o n in the composite.  This r u l e was derived assuming the o v e r a l l  fracture of f i b e r s at the same time.  The d e t a i l of t h i s expression i s d i s -  cussed in a l a t e r s e c t i o n , IV-1. Metal matrix composites are distinguished from the e x t e n s i v e l y developed r e s i n matrix composites by v i r t u e of t h e i r m e t a l l i c p r o p e r t i e s .  The main  advantages of metal matrix as compared with resin matrix are summarized as follows: a) The strength of metals i s greater than resins b) Metals have higher t e n s i l e  modulus than resins  \u000C2  c) M e t a l s possess e l e c t r i c a l of metals i s higher than  c o n d u c t i v i t y and t h e thermal c o n d u c t i v i t y  resins  d) M e t a l s possess g r e a t e r h i g h t e m p e r a t u r e s t r e n g t h .  3 Commonly used r e s i n s possess t e n s i l e s t r e n g t h v a l u e s o f 7 - 1 5 and t e n s i l e gr/cc)  modulus o f 0 . 4 \u2014 0 . 7 x 10  p.s.i.  x 10  p.s.i.  The d e n s i t y o f r e s i n  i s v e r y low compared w i t h m e t a l s , so t h a t p o i n t s a) and b) a r e n o t  d e f i n i t e advantages o f m e t a l s when the composites are used f o r w e i g h t applications. use.  critical  R e s i n m a t r i x composites a r e a v a i l a b l e o n l y f o r room t e m p e r a t u r e  T h e r e f o r e , a d e f i n i t e advantage o f metal m a t r i x c o m p o s i t e s f o r s t r u c t u r a  material  1-2.  (1.25  i s t h e i r high temperature c a p a c i t y .  P r e v i o u s Work on F a b r i c a t i o n Techniques V a r i o u s k i n d s o f f i b e r composite f a b r i c a t i o n t e c h n i q u e s have been d e v e l -  oped so f a r .  These f a b r i c a t i o n methods can be c l a s s i f i e d as shown i n T a b l e 1.  Most o f t h e s e methods, e x c e p t the u n i d i r e c t i o n a l s o l i d i f i c a t i o n method o f e u t e c t i c a l l o y s , a r e thought t o be c o m b i n a t i o n s o f two p r o c e s s e s , the f i b e r a l i g n m e n t process and t h e c o n s o l i d a t i o n p r o c e s s . o f t h e s e f a b r i c a t i o n methods, hot p r e s s i n g sing (L.P.H.P.) damage.  In the c o n s o l i d a t i o n p r o c e s s  (H.P.) and l i q u i d phase hot p r e s -  t e c h n i q u e s are commonly adopted i n o r d e r t o p r e v e n t f i b e r  Most o f t h e s e methods are not used f o r commercial c o m p o s i t e p r o d u c t i o n  because o f the c o s t o r because o f c e r t a i n problems i n each method as mentioned i n the f o l l o w i n g pages; however, they are a p p l i e d t o f a b r i c a t e composite specimens shown as examples i n t h i s t a b l e , s u c c e s s f u l l y o n l y f o r e x p e r i m e n t a l purposes.  The a p p l i c a t i o n o f the plasma s p r a y i n g method f o r S i C c o a t e d boron  f i l a m e n t aluminum m a t r i x composites i s a r e p r e s e n t a t i v e example o f commercial  \u000CTable 1.  Fabrication Techniques of Metal Composites  Applied Examples Method of Fiber Alignment  Matrix'  Method of Consolidation  Fiber  Reference  Deposition Chemical Vapor Deposition  Al Ni.NiCr  Graphite Fib. A l ^ S i C Whisk.  H.P., L.P.H.P. H.P.  2 3,4  Electro CoDeposition  Ni  A l 0 S i C Whisk.  C P . , H.P., L.P.H.P.  Electroplating  Al Al Ni  Graphite Fib. Boron F i l . A l 0 S i C Whisk.  H.P. H.P. H.P.  6 7 3,7,8  Electroless Plating  Ni Co  Graphite Fib. Graphite Fib.  H.P. H.P.  9,10 10  Plasma Spraying  Al (Alloy) Boron Fil.CSiC Cooted)  H.P.  11,5  Ti  Boron F i l .  H.P.  5  Al Cb Zn Ni Co-Cr Ni3Al Ni  Al Ni Cb2C ZnisTi W (Cr,Co)7C Ni Ta NbC  2  2  3  3  3,5  Metal Matrix Unidirectional Soli d i f i c a t i o n of Eutectic Alloy  Infiltration  3  Solidification 3  3  Al(Alloy) Graphite Fib. Ag AI2O3 Whisk. Al(Alloy) AI2O3 Whisk. Ni(Alloy) Al 03 Whisk. Cu W Wire Al Boron F i l . 2  Solidification of Matrix  12,13,14 15,16 17,18 19,16 20,16 21,16 22,16 23,24,25 26,1,3 27,3 7,3 28,1 31  \u000CMethod of Fiber Alignment  Applied Examples Matrix  Fiber  Method of Consolidation  Reference  Solid State Matrix Extrusion of Powder Matrix  Al Al Hastelloy  Alternate Pile up of Metal Foil &amp; Fibers  Al Al  Short Graphite Fib. S i . ^ Whisk.  Diffusion Bonding  W Wire  29 3 7  Ti-6A1-4V  Graphite Fib. Boron Fib. Boron Fi1.  L.P.H.P. H.P. H.P.  30 7,32 7  Ti-6A1-4V  Be wire  Mechanical Deformation and Diffusion bonding  33  Spinning, Extrusion Drawing of Mixture of Metal Powder, Whisker and Carrier Solution  Ag,Fe,Ni Al Alloy Cu.Mg  S i N Whisk. Sic whisk. Sic whisk.  Burn Off Organic Component and H.P. or L.P.H.P.  34,3 35,3 36,3  Filtering Slurry and Settling out of Nicoated Whiskers and Matrix Powder in Magnetic Field  Ni.Cr Al Alloy  SiC Whisk. A1 0 Whisk.  Clad Wire (matrix block with holes for wires) Slurry or Slip of Powder Matrix  3  4  2  3  37,3  \u000C5  productions  (11).  The chemical vapor deposition method seems to be the most expensive process among other methods in t h i s t a b l e .  This process u s u a l l y involves the  use of halide gas of the matrix m e t a l , so that there i s a l i m i t a t i o n on the v a r i e t y of a p p l i c a b l e matrix metal f o r t h i s process.  The number of f i b e r s in  a bundle which i s produced in t h i s process i s also l i m i t e d .  In order to obtain  a uniform coating f i l m on f i b e r surfaces, good penetration of the gas i n t o the bundles i s  necessary.  The e l e c t r o c o - d e p o s i t i o n , e l e c t r o p l a t i n g and e l e c t r o l e s s p l a t i n g methods often form small pores i n the metal matrix when rather f i n e f i b e r s l i k e whiskers ofgraphite f i b e r s are used.  Solution i s often trapped in these  pores, so that i t i s rather d i f f i c u l t to e l i m i n a t e these pores by the subsequent consolidation process. The plasma spraying method can be adopted only f o r large diameter continuous f i b e r s l i k e boron f i b e r s (11).  Melted metal powder i s sprayed c o n t i n -  uously on the f i b e r s aligned on a t h i n tape of the same metal. The u n i d i r e c t i o n a l s o l i d i f i c a t i o n method of e u t e c t i c a l l o y s has been e x t e n s i v e l y studied because of the high p o t e n t i a l to produce high temperature resistance metals f o r gas turbines e t c . (39). such as ( C r ^ o ^ C g  Not only the f i b r o u s e u t e c t i c s ,  r e i n f o r c e d (Co, Cr) e u t e c t i c a l l o y , Ni^Ta r e i n f o r c e d Ni^Al  a l l o y and NbC r e i n f o r c e d Ni a l l o y , but also the l a m e l l a r e u t e c t i c s , such as NigAl ( X J - N i ^ N b ^ ) e u t e c t i c a l l o y were reported to possess higher strength and more creep resistance than the s o - c a l l e d \"super a l l o y s \" (16).  These types of  composites have been expected to be s u i t a b l e f o r high temperature a p p l i c a t i o n s because of the thermodynamic s t a b i l i t y in the e u t e c t i c systems and the small e f f e c t of grain boundaries due to t h e i r large columnar s t r u c t u r e s .  This  process has advantages of easy f a b r i c a t i o n ; however, the a l l o y s are l i m i t e d to  \u000C6  the e u t e c t i c a l l o y s which can form s u i t a b l e second phase and the volume f r a c t i o n of reinforcements i s l i m i t e d consequently.  The i n f i l t r a t i o n method  i s used f o r composites using small diameter f i b e r s .  The matrix metal must  have good wetting property with the f i b e r in t h i s method.  In order to prevent  the degradation of f i b e r s by chemical a t t a c k , the proper control of i n f i l t r a t i o n condition i s necessary.  Graphite f i b e r aluminum a l l o y composites  were s u c c e s s f u l l y f a b r i c a t e d by t h i s method (23)(24)(25). The powder matrix extrusion process tends to damage f i b e r s . foil  The matrix  process i s s u i t a b l e f o r rather large diameter f i b e r s which can be e a s i l y  handled and a l i g n e d .  I t seems to be d i f f i c u l t to increase the f i b e r volume  f r a c t i o n of composites and control the f i b e r spacing uniformly by t h i s method, e s p e c i a l l y i n the case of f i n e f i b e r s l i k e graphite f i b e r s .  Clad wire process  can be used only f o r ordinary metal wire of high d u c t i l i t y . F i n a l l y , two methods based on metal powders i n an organic s o l u t i o n were developed to a l i g n f i n e whiskers in a matrix with l i t t l e damage to them.  In  the f i r s t method, the green composites are f a b r i c a t e d into a strand shape by some mechanical deformation, such as extrusion.  The organic components of  the s l u r r y are burned o f f p r i o r to hot pressing. In the second method, green composites are f a b r i c a t e d i n t o the shape of a mat by s e t t l i n g out and f i l t e r i n g the s l u r r y .  P r i o r to the s e t t l i n g process,  whiskers are coated with magnetic metal i n order to permit high alignment of these whiskers by magnetic force during the process.  This process seems to  have some d i f f i c u l t i e s to obtain uniform d i s t r i b u t i o n of f i b e r s through the t o t a l thickness of the mat because of a large d i f f e r e n c e in the s e t t l i n g speed of these two m a t e r i a l s . Boron f i b e r s have already been s u c c e s s f u l l y incorporated i n t o metals such  \u000C7  as aluminum, magnesium and t i t a n i u m . The a p p l i c a t i o n s o f these composites are l i m i t e d t o s p e c i a l f i e l d s because o f t h e i r high c o s t . The main p o t e n t i a l advantage f o r g r a p h i t e f i b e r composites i s the much lower f i b e r c o s t . As a matter o f f a c t , l a r g e amounts o f g r a p h i t e f i b e r s have been used i n r e s i n matrix composites, such as g o l f c l u b s h a f t s and turbine blades which must have high s t i f f n e s s (Young's modulus) - weight r a t i o .  The f u t u r e progress o f g r a p h i t e  f i b e r metal matrix composites g r e a t l y depends on the development o f r e l i a b l e and low c o s t f a b r i c a t i o n techniques. It i s very d i f f i c u l t to f a b r i c a t e metal matrix composites with g r a p h i t e f i b e r s because o f the small f i b e r diameter (6-9y) (100-125y).  compared with boron f i b e r s  The alignment and c o n s o l i d a t i o n processes f o r such f i n e f i b e r  composites have to be c a r e f u l l y designed i n order t o prevent any mechanical f i b e r damage. The chemical a t t a c k a t the f i b e r - m a t r i x i n t e r f a c e may a l s o give severe damage to the f i n e f i b e r s , i f the f a b r i c a t i o n process i n v o l v e s high temperature o p e r a t i o n s . As a matter o f f a c t , n i c k e l , c o b a l t and s t e e l d i s s o l v e g r a p h i t e a t high temperature and degrade the f i b e r s .  Copper i s expec-  ted to be a good matrix because o f the low carbon s o l u b i l i t y ; however, not many s t u d i e s have been done with copper due to i t s high d e n s i t y r e l a t i v e t o aluminum, and i t s l i m i t e d range o f high temperature use comapred with n i c k e l and c o b a l t . In recent y e a r s , aluminum o r aluminum a l l o y s have been thought to be the most promising matrix f o r g r a p h i t e f i b e r s because o f the high s t r e n g t h and s t i f f n e s s to d e n s i t y r a t i o .  Aluminum i s one o f the c a r b i d e forming elements;  however, the g r a p h i t e f i b e r aluminum composites are expected t o be used s a f e l y at the p r a c t i c a l long time s e r v i c e temperature which i s much lower than the carbide formation temperature  (&gt;500\u00B0C).  \u000C8  Aluminum carbide formation on the surface of PAN  Type II* f i b e r s i n  pure aluminum composites produced by powder m e t a l l u r g i c a l process was f i r s t observed and measured by G. Blankenburgs technique.  (29), using a q u a n t i t a t i v e X-ray  F i g . 1 and 2 show the carbide formation of various temperatures  and various times. P.W Jackson (40) also studied PAN Type I * f i b e r s coated with aluminum by chemical vapour d e p o s i t i o n .  Tests on specimens held at 500\u00B0C f o r one day  exhibited no noticeable loss in room temperature strength of the coated f i b e r s . On the other hand, the apparent degradation of the coated f i b e r s at higher temperature than t h i s was recognized as shown in F i g . 3.  It was concluded that  the f i b e r degradation was caused by the chemcial attack of the f i b e r surface by aluminum at such high temperature. Such chemical reaction suggests good wetting between these m a t e r i a l s . The i n f i l t r a t i o n process was f i n a l l y applied to Thornel 50 graphite f i b e r 13% s i l i c o n aluminum a l l o y composites s u c c e s s f u l l y by R. Pepper, J . Upp, R. and E. Kendall (23) (25).  Rossi,  This process has been expected to be a p r a c t i c a l  f a b r i c a t i o n process because specimens f a b r i c a t e d by t h i s process e x h i b i t much higher values than any other f a b r i c a t i o n process and sometimes even higher values than the values according to the \" r u l e of m i x t u r e \" . f o r t h i s remarkable strength increase i s s t i l l  The real reason  unknown.  Although many kinds of high modulus graphite f i b e r s are being produced by manufacturers, only a few have been used for metal matrix composites.  They  can be c l a s s i f i e d i n t o some categories, of which properties are shown together with other r e i n f o r c i n g materials in Table 2. (Type II:  High strength type; Type I: High modulus type).  \u000C7 Vol % composite '.5r a n n e a l e d at 6 0 0 \u00B0 c \u2022  *\u2022  1.0  c  &lt;u c  s o  to 0 5  &lt; 500 550 600 Annealing Temperature  Fig.l  650 \u00B0C  The V a r i a t i o n of A l i n  5 wt%  ~0  5 10 Annealing  50 Time  100 hours  500  F i g . 2 . The V a r i a t i o n of Al^C^ i n 5 wt%  Graphite Fiber - Aluminum Composite a f t e r  Graphite Fiber - Aluminum Composite a f t e r  Annealing f o r 24 hr a t Various Temperatures  Annealing f o r Various Times at 600\u00B0C (29).  (Determined by Q u a n t i t a t i v e x-ray D i f f r a c t i o n ) (29).  \u000C10  400  \u2022-\u2022 300t  200  \u00A3  100  200 400 600 Processing Temperature  800 C\u00B0  IOOO  Fig.3. Strength of Al Coated Graphite Fibers a f t e r Heat Treatment f o r 1 Day at Various Temperatures (40).  60  /  *7  40  V  20 \u2022 fabricated in air o fabricated in argon  10  20  30 V  f  40  %  Fig.4. T e n s i l e Strength of the Al-Graphite Fiber Composites Fabricated by the Matrix F o i l Method. data i n Ref.30).  (Plotted from numerical  \u000C11  T a b 1 e  2.  Typical Properties of High Modulus Graphite Fibers Compared with  Other Reinforcing Materials.  Ultimate Tensile* Reinforcement  Strength x lO^p.s.i.  Tensile* modulus x 10 p . s . i .  Density lb/in.  3  Diameter u  Graphite Fibers Rayon-base Thornel 50 Thornel 75  2 7 5 - 320 3 5 0 - 385  44 - 55 7 0 - 80  0.06 0.065  6.6 6.0  PAN-base Type I (High modulus type)  2 2 5 - 275  55 - 6 0  0.072  7-9.7  325 -  375  32 - 38  0.063  7.6-8.6  Boron filament  400 - 500  55 - 60  0.092  100-150  Beryllium wire  150 - 200  35 - 40  0.066  100-250  Tungsten wire  550 - 600  48 - 5 2  0.7  Type II (High strength type)  Other Reinforcements  50-100  Aluminum whisker  4000  62  0.14  1-10  Silicon carbide whisker  3000  70  0.12  1-10  (* Measured on single fibers).  \u000C12  1-3.  Previous Work on Strength of Graphite Fiber Aluminum and A l l o y Composites A study on graphite f i b e r aluminum composites was reported by A. Morris  (30).  Specimens were f a b r i c a t e d from PAN Type II  using a l i q u i d phase hot pressing technique.  f i b e r s and aluminum f o i l s ,  The ultimate t e n s i l e strength  v a l u e s , c r , as a function of f i b e r volume f r a c t i o n s of these composites are r  p l o t t e d in F i g . 4.  The values are highly scattered and considerably lower  than the expected values from the \" r u l e of m i x t u r e \" . P. Jackson et a l - (2) f a b r i c a t e d PAN Type II  f i b e r aluminum composite  specimens by the chemical vapour deposition process, using T r i - i s o b u t y l aluminum.  The t e n s i l e strengths of these composites are shown in F i g . 5, f o r  various f a b r i c a t i o n c o n d i t i o n s , as a function of the f i b e r volume f r a c t i o n . In s p i t e of considerable e f f o r t to s a t i s f y the requirements of low p o r o s i t y , minimum chemical a t t a c k , minimum f i b e r breakage and uniform f i b e r d i s t r i b u t i o n , the t e n s i l e strengths of these composites were well below \" r u l e of mixture\" levels.  On the other hand, the t e n s i l e modulus of these composites were gener-  a l l y close to the expected values.  It was suggested that a f u r t h e r mechanism  was operating i n keeping strength l e v e l s down. Pepper eit aj_. (23) (25) f a b r i c a t e d samples by the i n f i l t r a t i o n technique as mentioned e a r l i e r .  A f t e r m u l t i p l e chemical washing, Thomel 50 graphite  f i b e r bundles were i n f i l t r a t e d i n a batch process with 13% s i l i c o n aluminum 3 alloy.  The mean t e n s i l e strength value of 106 x 10  p . s . i . was obtained with  28% f i b e r volume f r a c t i o n and t h i s value was unaffected by 20 thermal cycles between -193\u00B0C and 500\u00B0C.  As mentioned e a r l i e r , t h i s value compares favourably  with that expected from a \" r u l e of mixture\" c a l c u l a t i o n . In t h e i r f o l l o w i n g s t u d i e s , pure A l , Al-7Mg, Al-7Zn and A1 -13Si were used to f a b r i c a t e composites with Thornel 75 graphite f i b e r s .  alloys The r e s u l t s  \u000Cao,  ._\u2022 &lt;/)'  o \u00A9 * \u2022^x x  *  60  x\u00B0  x  7D  x&gt;  xo  40 A  CO H  V  20  \u2022 500 \u00B0  /  X  5lons/in  c  X 50 o \u00B0 C 5 t o n s / i n ihour a  A 600 Cl/2tonv\"in lh0ur o  /  o 600\"  to  20  1  C VOon/it?  50  %  f  Fig.5. T e n s i l e Strength of Graphite Fiber Aluminum  Composites  Fabricated by the Chemical Vapour Deposition Method  // / / / / //. *  100  CL  O  C/J  40  \u00B0/ / \u2022 / / / / / / / \u2022 / / / / \u2022 / r / 0  (2).  \u2022A  o  \u2022 \u2022  80  (A  n  I min  40  30  V  ._\u2022  I hour  a  A  \u2022  \u2022  220 \u2014-  \u2022 A! 3  o 6061 i  10  20  30  v  f  %  40  Fig.6. T e n s i l e Strength of Graphite Fiber Aluminum A l l o y Fabricated by the I n f i l t r a t i o n Method. data i n Ref.24).  Composites  (Plotted from numerical  \u000C14  of t h i s work are shown in Table 3.  The t e n s i l e strength values are again  lower than \" r u l e of mixture\" values and scattered very much. Further work on the composites of continuously produced i n f i l t r a t i o n composite strand was done by R. Pepper and R. Penty (24), using Al-13Si  (Al3),  Al-lOMg (220), and Al-IMg-O.6Si (6061) a l l o y s with Thornel 50 graphite f i b e r s . The t e n s i l e strength values are again lower than the \" r u l e of mixture\" values and s t i l l scattered in very wide region as shown in F i g . 6. These composites f a b r i c a t e d by i n f i l t r a t i o n process u s u a l l y possess higher ultimate t e n s i l e strength than the composites by other processes; however, the d i s t r i b u t i o n of the f i b e r s in the matrix of the i n f i l t r a t e d composites i s not uniform.  This non-uniformity seems to o r i g i n a t e in the hot pressing die con-  f i g u r a t i o n which involves the use of f i l l e r metal f o i l s among composite wires as shown in F i g . 7 (25). The f i b e r d i s t r i b u t i o n of these composites must be seen as a bundle structure.  1-4.  Purpose of Present Work The i n f i l t r a t i o n technique of aluminum a l l o y s developed by R.T, Pepper  et al_. led to remarkable progress in the f a b r i c a t i o n of graphite f i b e r aluminum composites. The major advantage of t h i s process includes low f a b r i c a t i o n c o s t s , minimum f i b e r damage, and p o t e n t i a l l y high strength.  The achievement of uniform  f i b e r d i s t r i b u t i o n i s d i f f i c u l t , making t h e o r e t i c a l modeling awkward. The strength values a c t u a l l y achieved are scattered in a very wide range and much lower than the t h e o r e t i c a l values c a l c u l a t e d according to the \" r u l e of mixture\".  \u000CTable 3.  Tensile Properties of Various Aluminum-Anoy-Thornel 75 Composites (25)  Strength  Matrix Composition  Specimen Condition  Volume Percent Fiber  Average MN/m  p.s.i.  \u201E , Number of Samples  a  ?  . Low Value (psi)  .... High Value (psi)  Average Modulus 3  2  (GN/m )  (psi)  Commerciallypure aluminum  As-infiltrated Pressed  32 35  68 65  99,000 95,000  8 7  65,000 85,000  116,000 104,000  178 147  25.7 21.3  Aluminum-7 w/o zinc  As-infiltrated Pressed  32 38  71 87  103,000 126,000  7 10  59,000 102,000  132,000 155,000  166 190  24.1 27.5  Aluminum-7 w/o magnesium  As-infiltrated  31  68  98,000  4  87,000  124,000  195  28.1  Aluminum-13 w/o silicon  As-infiltrated  22  55  80,000  7  73,000  88,000  165  23.8  \u000CFig.7. L i q u i d Phase Hot Pressing Die Configuration to Fabricate Specimens from the Al Graphite Fiber Composite Wire  (25).  Infiltrated  \u000C17  The reason f o r the d i f f e r e n c e between experimental r e s u l t s and theore t i c a l values has not yet been f u l l y understood.  A d i f f e r e n t mechanism from  the \" r u l e of mixture\" might be operating on the t e n s i l e f r a c t u r e of these graphite f i b e r aluminum composites.  The purpose of the present work i s to  i n v e s t i g a t e the mechanical behaviour of the composites and to make a theore t i c a l model which can c o r r e l a t e with t h i s  behaviour.  \u000C18  J!  11\u20141.  EXPERIMENTAL PROCEDURE  Preparation of Composite Specimens  Three kinds of aluminum a l l o y s were chosen as matrices of the composite. Their nominal compositions are tabulated in Table 4. The composites were f a b r i c a t e d by a unique method which was designed to obtain s t a t i s t i c a l l y or macroscopically homogeneous f i b e r d i s t r i b u t i o n i n the matrix.  The s t a t i s t i c a l homogeneity of f i b e r d i s t r i b u t i o n i s necessary f o r  the present work on f i b e r strengthening mechanism.  The o v e r a l l f a b r i c a t i o n  process in t h i s method i s o u t l i n e d in F i g . 8. Blended powders of each a l l o y composition were prepared from under 500 mesh powders of aluminum, s i l i c o n , magnesium,and copper shown in F i g . 9.  These  blended powders were mixed well together with denatured alcohol to make a powder suspended t h i n s l i p . Thornel 50 graphite yarns were washed i n b o i l i n g d i s t i l l e d water f o r about two hours to d i s s o l v e the P.V.A. coating f i l m applied by the manufacturer in order to r e i n f o r c e the yarns and avoid t h e i r degradation during handling. A f t e r the f i b e r s were dried and untwisted, they were bundled and the top of the bundles were glued with epoxy r e s i n .  The number of f i b e r s in each bundle  was varied from 76,000 to 228,000 depending on the f i b e r volume f r a c t i o n s which were desired in the f i n a l product.  The.number of f i b e r s in one bundle i s  l i m i t e d by geometrical f a c t o r s r e l a t e d to the diameter of the outer glass tube.  Too many f i b e r s prevent t h e i r free movement and adequate separation  during the subsequent powder penetration.  On the other hand, some bundles  were prepared from o r i g i n a l twisted yarns without untwisting them in order to obtain bundle structure f i b e r d i s t r i b u t i o n s in the matrices.  \u000CTable 4.  The Nominal Composition of Matrix A l l o y s  Alloy  Heat Treatment  601  T4 (Solution Treatment)  a t . 1.11 wt. 1.00  a t . 0.58 wt. 0.60  at. 0.11 wt. 0.25  at. 98.20 wt. 98.15  201 ,  T6 (Age Hardened)  a t . 0.46 wt. 0.40  at. 0.79. wt. 0.80  at. 1.92 wt. 4.40  a t . 96.84 wt. 94.40  7178  T6 (Age Hardened)  a t . 3.15' wt. 2.70  Density of Each Element, ^/cc  Table 5.  Mg %  1 .74  Si %  Cu %  Zn %  Al %  a t . 0.80 wt. 2.00  a t . 2.95 wt. 6.80  a t . 93.00 wt. 88.50  8 .96  7.14  2.70  2 .32  Properties of Thornel 50 Graphite Fiber  Tensile Strength* p . s . i . Tensile Modulus* p . s . i . Density 9 /  275 \u2014 320 x 10 44\u2014  55 x 10  6  1.66  Cc  Elongation at Break %  0.6  Equivalent Diameter y  6.6  No. of Fibers/ply  720  Plies/Yarn (* Measured on Single  3  2 Fibers)  U3  \u000C20  I Al. Cu. Si,Mg.POWDERS  THQRNEL50| 76O00~2280OO  fibers  WAS HED AND DRIED  ALCOHOL  total 3 0 gr \u2014 3 0 0 me sh  4 0 0 cc  BLENDED  UNTWISTED  BUNDLE  SLIP  I INTER PENETRATION-SEDIMENTATION  PROCESS  SEPARATION OF EXTRA POWDER 1 : VOLATILIZATION OF ALCOHOL AND SUBSTITUTION WITH C A M P H E N E LIQUID  P H A S E HOT P R E S S I N G \u2022 BURNING OFF OF ORGANIC MATERIALS A T I 5 0 \u00B0 C UNDER LOW PRESSURE  STEPWISE LOADING UP TO 6 0 0 p.s.i. AT HOT PRESSING TEMPERATURE IN H GAS 2  COOLING DOWN TO UNLOADING TEMPERATURE AND P R E S S U R E RELEASE TO 150 p.s.i.  MACHINING JTENSILE TEST PIECE  Fig.8. Flow Sheet of the Specimen F a b r i c a t i o n  Process  \u000C\u000C22  The f i b e r bundles were interpenetrated with the blended powder s l i p s using a device designed s p e c i a l l y f o r t h i s purpose. shown in F i g . 10.  This device i s schematically  The outer glass tube (A) i s f i l l e d with the s l i p a f t e r a  f i b e r bundle (B) i s connected to a rod (C) with a spring (D) and inserted i n t o a glass sheath (E) which moves up gradually from the bottom of the bundle to the top with v e r t i c a l v i b r a t i o n during the operation.  A d r i v e and v i b r a t o r  assembly at the top of the unit causes v e r t i c a l v i b r a t i o n of both the connecting rod and the glass sheath. The powder s t a r t s to interpenetrate i n t o the bundle and s e t t l e among the f i b e r s opened by these v i b r a t i o n a l movements of the rod and the sheath tube. The slow upward movement of the sheath tube  makes more complete i n t e r p e n e t r a t i o n  of the powder p a r t i c l e s among the f i b e r s p o s s i b l e .  A f t e r sedimentation of  the powder p a r t i c l e s proceeds, a d d i t i o n a l charge of s l i p i s poured i n t o the glass tube three or four times. Green composites which were produced by t h i s operation are pushed out from the outer glass tube.  These green composites contain more powder at the  boundary of the o r i g i n a l p l i e s than at the i n s i d e , because, even i f the p l i e s are untwisted, they s t i l l twisted form s l i g h t l y .  have a tendency to t w i s t back and keep the o r i g i n a l  To remove the extra powder p a r t i c l e s from these areas,  the green composites are transferred to a v i b r a t i n g boat containing a small amount of alcohol (see F i g . 11).  The d i s t r i b u t i o n of f i b e r s among the powders  becomes more uniform by t h i s separation treatment.  This process i s repeated  a few times, turning the specimens upside down u n t i l the amount of flowed out powder becomes very l i t t l e . The green composites are then transferred on to a f l a t p l a t e to be pressed into a rectangular s e c t i o n .  Alcohol was h a l f evaporated and Camphene was  i n f i l t r a t e d i n t o the green at around 50\u00B0C in order to r e i n f o r c e the g r e e n c o m -  \u000CC A M  M E C H A N I S M -  S P R I N G -  V I B R A T O R -  S P R I N G  -  OVER FLOW  R O D  C O A L U M I N U M  G L A S S  S H E A T H  C O N E C T I N G  A L L O Y  S U P  ( E )  S P R I N G  ( D ) -  &gt; suspended zone O U T E R  G L A S S  G R A P H I T E  T U B E  B U N D L E  ( B )  ( A ) \u2014 |  \u2014  Flg.10. S c h e m a t i c D i a g r a m o f the I n t e r p e n e t r a t i o n  Device  interpenetrating zone J. settled zone F I L T E R -  DRAIN  \u000C24  , ,-HOLE FOR THERMOCOUPLE  GRAPHITE UPPER RAM  GRAPHITE SPACER  |-Mo COVERED GRAPHITE DIE  025 \u00AB 0.225\"w  X0.073\"t  STEEL STOCK  F i g . 1 3 . Tensile Specimen  Fig.12. Hot Pressing Die Set  Geometries  \u000Cposite; (however, t h i s process i s not necessary i f h a l f dried green composites are handled very c a r e f u l l y in the f o l l o w i n g processes because such composites have enough strength for handling as long as they are wet).  The composites  were cut into short lengths to f i t the hot pressing die shown in F i g . 12, and pieces which contained defects were abandoned.  A l l of the numerical values  of factors in t h i s f a b r i c a t i n g process are tabulated in Table 6. These pieces of green composites were then hot pressed in f o l l o w i n g process.  The organic constituent was gradually evaporated under low pressure  at around 150\u00B0C in the die which was set in a hot pressing chamber.  The degree  of degassing was checked by a thermal gauge. A f t e r t h i s degassing treatment, pressure was applied stepwise at the rate of 25 psi/min. at the temperatures shown in Table 7, in hydrogen gas atmosphere u n t i l the maximum pressure 600 psi. was obtained.  The contents of the die were kept at these temperatures  under the pressure of 600 psi f o r one hour.  During t h i s p e r i o d , p a r t i a l  melting of the powder occurs forming a l i q u i d phase which e a s i l y i n f i l t r a t e s among f i b e r s .  Unreinforced a l l o y blanks were also f a b r i c a t e d in t h i s hot  pressing process from blended powders.  These blank specimens were used to  obtain a basis f o r f i b e r strengthening e f f e c t . These hot pressed composite or blank specimens were then machined to the shape of t e n s i l e t e s t specimens and heat treated to acquire d i f f e r e n t mechanical properties. 7.  The temperature and time f o r each a l l o y are also tabulated in Table  Steel stocks were attached to these heat treated specimens with Eastman  Kodak 910 in order to protect the specimens from the grips of the t e n s i l e t e s t machine, as shown in F i g . 13.  \u000C26  Table 6.  Numerical Values of C o n t r o l l i n g Factors i n Interpenetration Process  Weight of Blended Powders i n S l i p f o r one charge  30 gr  Volume of Alcohol i n S l i p f o r one charge  400 cc  Number  100 \u2014 3 0 0  of PI ies i n a Bundle  Moving Speed of Sheath  About 6 inch/hr  Frequency and Amplitude of Bundles V e r t i c a l V i b r a t i o n  60 c/s. 0.2 inch  Frequency and Amplitude of Sheath  60 c/s. 0.05 inch  Diameter of Sheath  9mm x 7 mm  Diameter of Outside Glass Tube  14mm  Table 7.  Hot Press and Heat Treatment  Hot Pressing Temperature and Time  601T4  201T6  7178T6  600\u00B0C, 1 hr  550\u00B0C, 1 hr  550\u00B0C, 1 hr  500\u00B0C  450\u00B0C  Pressure Releasing Temperature Solution Treatment Temperature and j j m  x 12 mm  e  Ageing Temperature and Time  r, o  520  n  C,30o n r  \u2022\u201E  mm  c  c 505 n  0  r  on C,30  450\u00B0C mm  160\u00B0C, 18 hr  nm\u00B0r 470  ic C,15  m  ,-\u201E mm  125\u00B0C, 28 hr  \u000C27  II-2.  Tensile Testing  Tensile tests at room temperature were c a r r i e d out with an Instron t e s t i n g machine at the cross head speed of 0.02 inch/min, using s e l f tightening g r i p s .  The t e n s i l e strength was determined f o r these specimens.  Due to the very small elongation of the composite specimens (under 1%),  it  proved to be very d i f f i c u l t to e s t a b l i s h the elongation at f a i l u r e . More accurate stress s t r a i n curves and l o n g i t u d i n a l e l a s t i c modulus of several t y p i c a l specimens were obtained using s t r a i n gauge on specimen surfaces.  11-3.  Microscopic  Observations  A f t e r the f r a c t u r e surfaces were ground and p o l i s h e d , photographs were taken at a magnification of 114 times. i s around  1/7  The area covered by these pictures  of the o r i g i n a l area on the specimen.  The volume f r a c t i o n  of f i b e r s i n each composite was c a l c u l a t e d using the numbers of the f i b e r s counted i n these pictures and a nominal value f o r f i b e r diameter  (6.6y).  Fracture surfaces and polished l o n g i t u d i n a l sections were observed with a scanning electron microscope. The surface of a 201-T6 composite specimen was polished p r i o r to t e n s i l e t e s t i n g and the deformation mode j u s t before f a i l u r e was observed by o p t i c a l microscopy.  \u000C28  11-4.  Micro Probe Analysis  The matrix composition of a l l specimens was analysed by means of micro probe a n a l y s i s .  Two counts at each of four points were c a r r i e d out  in the sections previously used to count the number of f i b e r s .  Regions  of the matrix which had been l i q u i d during the hot pressing were avoided in order to obtain representative values. The t e n s i l e t e s t data from specimens which showed too much deviation of composition from average values were excluded in order to improve the r e l i a b i l i t y of the data. The big s c a t t e r of composition among specimens i s due to the large density d i f f e r e n c e of each of the elements.  This was expected since blended  powders were used in the interpenetration process instead of a l l o y powders. The d e n s i t i e s of these elements are tabulated in Table 4.  \u000C29  i  III  111-1.  EXPERIMENTAL RESULTS  Fiber Volume Fraction  The volume f r a c t i o n of graphite f i b e r s in each specimen was c a l c u lated using pictures as shown in F i g . 14.  F i g . 15 shows the bundle structure  of the 7178 T6 a l l o y composite which was prepared from o r i g i n a l twisted yarns (1.5 turns/inch). The f i b e r volume f r a c t i o n of each specimen i s tabulated together with the number of p l i e s which were used to prepare each specimen i n Table 8. The average f i b e r volume f r a c t i o n i s shown p l o t t e d against the number of p l i e s per bundle in F i g . 16.  111-2.  Micro Probe Analysis of the Specimens  The matrix chemical composition of each specimen was analysed by micro probe measurement and c a l c u l a t e d using the \"Magic\" program.  Wide  s c a t t e r i n g of analysed composition was found as a n t i c i p a t e d e a r l i e r . In order to avoid u n c e r t a i n t i e s a r i s i n g from v a r i a t i o n s of matrix comp o s i t i o n , l i m i t s were established (see Table 9) f o r allowable compositions. A l l specimens l i s t e d in Table 8 l i e w i t h i n these established l i m i t s .  111-3.  Tensile Stress S t r a i n Curves  Most of the stress s t r a i n curves were obtained d i r e c t l y from the Instron recorder.  A l i m i t e d number of stress s t r a i n curves was obtained by  measuring the s t r a i n with s t r a i n gauges attached to the specimen surfaces.  \u000CFig.14. Cross Section of a Uniform Composite, 201T6, #29, 12.1% V , x f  Fig.15. Cross Section of a Bundle Structure Composite, 7178 T6, #39, 10.0% V~, x 62.  \u000C0201  0J5 h  o.ioh  0.05L  oL^ 0  !  I  !  !  100 200 Number of Plies in a Bundle  I  1  I  300  Fig.16. Relation between the Number o f P l i e s f o r the Interpenetration Process and the Fiber Volume Fraction i n Composites. correlation.  The dot-dash-line shows the ideal  \u000C32  Table 8.  Tensile Test Data of Specimens  Specimen Number No. Alloy of Plies  Fibre Volume Fraction, V f  Ultimate Composition,atomic% Tensile Strength Mg% SIX Cut Zn% crx103 p.s.i.  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4 601T4  0 0 0 0 100 100 150 150 200 200 200 200 250 250 250 250 250 250 300 300  0 0 0 0 4.3 7.8 \u20226.5 6.7 8.5 7.8 7.8 9.5 7.8 9.3 11.3 12.8 14.5 14.5 9.5 17.0  0.9 1.04 0.78 0.87 0.89 0.65 1.00 0.82 0.94 0.79 0.74 0.90 0.62 0.97 0.79 0.53 0.78 0.63 0.76 0.81  0.59 0.50 0.49 0.51 0.58 0.47 0.47 0.61 0.58 0.50 0.53 0.59 0.54 0.36 0.60 0.65 0.52 0.47 0.48 0.38  0.07 0.09 0.06 0.10 0.09 0.09 0.12 0.11 0.10 0.07 0.10 0.07 0.07 0.15 0.10 0.07 0.07 0.04 0.07 0.06  33.8 29.9 32.1 32.8 29.0 33.3 31.4 31.6 33.6 34.0 34.1 35.5 32.2 38.6 39.0 42.4 45.1 46.5 36.8 45.8  21 22 23 24 25 26 27 28 29 30 31  201T6 201T6 201T6 201T6 201T6 201T6 201T6 201T6 201T6 201T6 201T6  0 0 0 130 150 150 250 250 250 250 250  0 0 0 6.4 8.2 7.4 8.6 10.0 12.1 14.1 17.0  0.45 0.41 0.58 0.47 0.63 0.34 0.40 0.45 0.28 0.42 0.50  0.44 0.40 0.49 0.69 0.66 0.60 0.37 0.53 0.42 0.36 0.55  1.24 1.33 1.22 1.12 1.15 1.39 1.08 1.04 1.38 0.85 1.26  58.5 62.5 60.5 64.7 69.5 66.9 66.8 70.1 68.8 66.4 63.2  32 33 34 35 36 37 38 39 40 41 42  7178T6 7178T6 7178T6 7178T6 7178T6 7178T6 7178T6 7178T6 7178T6 7178T6 7178T6  0 0 0 0 250 250 250 250B* 250B* 250B* 250B*  0 0 0 0 9.6 12.8 13.8 10.0 12.7 16.4 19.0  1.06 2.79 1.49 1.06 2.31 2.62 2.38 1.31 1.08 1.47 2.16  (*B: Bundle Structure)  0.77 0.45 0.86 0.77 0.39 0.53 0.40 0.88 1.00 1.00 0.44  2.19 2.62 3.39 3.81 2.14 2.22 2.35 3.53 3.63 3.69 3.74  70.4 78.2 72.0 77.5 68.5 78.1 75.7 83.6 83.8 88.6 88.6  Strain Gauae Measurement \u20AC-..,C5,E  =18xl O^psi  6  {  fail \u00B0-50% E = l3.5xi0 psi  \u20AC  =  6  c  0^=49x10 psi cr =47xl03psi 6  f\u20ACfail = 0.63% \\ E =l53M0 psi 6  C  066=56x1 0,DSi 006=56x10 psi Obe=56x103psi J  OoiS=56x10 psi 3  \u000C60  Strain  \u00A3  %  Fig.17..Stress S t r a i n Curves of 601T4 A l l o y and the Composites  \u000CFig.18. Stress Strain Curves of 201T6 Alloy and the Composite  \u000C\u000C36  Table 9.  Allowable Limits of the Matrix Composition  Alloy  Mg, at %  S i , at %  Cu, at %  601  0.50-1.05  0 . 4 0 - a 60  0.04-0.14  201  0.20-0.60  0.30-0.70  1.00-1.40  7178  2.80-1.00  -  0.35-1.00  Zn, at %  2.10-3.90  The measured s t r a i n of composite specimens was always small ( 0.06%). The Young's modulus of composite specimens derived from these stress  strain  curves are shown i n Table 8. - These values are i n good agreement with the c a l c u l a t e d values according to \" r u l e of m i x t u r e \" . Some of these stress s t r a i n curves are shown i n F i g . 17, 18, and 19. The stress s t r a i n curve of 7178 T6 composite was not obtained because of the shortage of specimens due to the d i f f i c u l t y of c o n t r o l l i n g a l l o y composition.  Ill-4^  Ultimate T e n s i l e Strength  The v a r i a t i o n s i n ultimate t e n s i l e strength,  cr^, of each a l l o y com-  posites with f i b e r volume f r a c t i o n , V^, are shown in F i g . 20, 21, and 22. The ultimate t e n s i l e strength of unreinforced, blank, specimens, o~ ^ &gt; m  s  and the strength of the a l l o y at the breaking s t r a i n of the f i b e r s , CT^*, are also p l o t t e d i n these f i g u r e s as the points corresponding to zero f i b e r volume f r a c t i o n . The ultimate t e n s i l e strength of non-uniform, bundle s t r u c t u r e , 7178 T6 a l l o y composites i s also shown in F i g . 22 to be compared with uniformly  \u000C60 a/w=0  50  a/w =0.03 CJ&lt;-' a/w =0.05  40  a/w =0.10 Q.  b 30 x  Omuts  20 in 3  0 0  10  Fig.20. Tensile Strength of 601T4 Composites, and Theoretical Curves  15  \u000C\u000C100  a/w=0  BUNDLE  90  , o/v\u00BB=0.03 80  cx  muts a/w= 0.03  70  \u2022  60  :  O :  Om 10  15  Vf  UNIFORM  DISTRIBUTION  BUNDLE  STRUCTURE  20  (%)  Fig.2.2.. Tensile Strength of 7178 T6, Uniform and. Bundle .Structure Composites, and Theoretical Curves  \u000C40  d i s t r i b u t e d composites.  The ultimate t e n s i l e strength of each composite  calculated according to \" r u l e of mixture\" i s also exhibited i n these figures as a dot-dash l i n e . In a l l cases, the experimental values are lower than the \" r u l e of mixture\" l e v e l .  The discrepancy between the experimental and the \" r u l e of  mixture\" values becomes more evident as the f i b e r volume f r a c t i o n increases. This tendency is more prominent in 201T6 composites than 601T4 composites. The t e n s i l e strength of the bundle structure 7178 T6 composites i s higher than the uniform composites as shown i n F i g . 22.  This r e s u l t suggests  that some other mechanism, which i s d i f f e r e n t from the \" r u l e of m i x t u r e \" , i s operating to keep the strength of bundle s t r u c t u r e composites higher than the uniform ones. Other curves c a l c u l a t e d according to a model which w i l l be discussed l a t e r are a l s o shown i n these f i g u r e s .  HI-5.  Fracture Elongation  An attempt was made to obtain the f r a c t u r e elongation of specimens by measuring the distance between gauge marks on f r a c t u r e d specimen; however, the r e s u l t s were u n r e l i a b l e because of the d i f f i c u l t y i n measuring very small elongations and also because of the frequent f a i l u r e at specimen shoulders outside of the marks. was abandoned.  F i n a l l y , the measurement'by t h i s method  Only a few data about the elongation of these composite  specimens are a v a i l a b l e from the s t r a i n gauge t e s t data as shown in Table 8.  \u000C41  111-6.  Microscope Observations of Tested Specimens  Fractographic observations were made using a scanning e l e c t r o n microscope.  The p l a s t i c deformation of the matrix at the f r a c t u r e surface i s  prominent.  A small amount of f i b e r pull out was usually observed.  Small  voids, which are c h a r a c t e r i s t i c of p l a s t i c f a i l u r e , were observed somewhere on the ridges of the matrix as shown in F i g . 23 and F i g . 24. The f i b e r f a i l u r e zone which appeared on the polished 201T6 composite specimen surface was observed using o p t i c a l and scanning e l e c t r o n microscopy, a f t e r straining close to the f r a c t u r e point. zone surface i s shown in F i g . 25.  The t y p i c a l appearance of the  Within the zone, f i b e r s were broken into  small fragments as shown i n the scanning e l e c t r o n micrograph a t low m a g n i f i cation (x200), F i g . 26.  The s l i p l i n e s are observed at a higher magnification  (x 1000) i n the grains between f i b e r s as shown in F i g . 27. The i n t e r i o r s t r u c t u r e of t h i s specimen was examined by f u r t h e r p o l i s h i n g and etching.  A f i b e r f a i l u r e region around a specimen shoulder i s shown as  a marked area in F i g . 28.  In t h i s f i g u r e , the successive propagation of f i b e r  f a i l u r e from the r i g h t hand side to the l e f t hand side seems to be interrupted at a f i b e r f r e e (or low f i b e r density) region.  Etched grain boundaries,  and broken and s l i g h t l y t i l t e d f i b e r s were also observed as shown in Fig.29. Fractured f i b e r s can be seen at a considerably longer distonce from the f r a c t u r e surface in the low composite ( F i g . 30b).  composite ( F i g . 30a) than in the high  This same tendency i s also observed in the case of  201T6 composites of d i f f e r e n t V  f  values, 6.4% and 17% as shown in F i g . 31  (a) and (b). The broken fragments located c l o s e r to the f r a c t u r e surface have generally shorter lengths than the fragments located at a l a r g e r distance  \u000CFig.23. Fractured Surface of 601T4 Composite, #17, x  1000.  14.5% V  \u000C43  Fig.26. Scanning E l e c t r o n Micrograph o f F i b e r F a i l u r e Zone S u r f a c e Showing Broken F i b e r s , x 200.  Fig.27. Scanning E l e c t r o n Micrograph of Zone S u r f a c e Showing S l i p L i n e s , x 1000.  \u000CFiber Free Region  -Fiber F a i l u r e Zone-  Fig.28. Micrograph I n d i c a t i n g Propagation of the Fiber F a i l u r e p r i o r to F a i l u r e of the Specimen, Longitudinal S e c t i o n , x 32.  Fig.29. Longitudinal Section around the Zone Showing Matrix Grains, and Broken T i l t e d F i b e r s , NaOH S o l u t i o n Etch, x 180. -pi  \u000C45  (b) Fig.30. Longitudinal Sections of Fractured 601T4 Specimens, Showing the Difference in the D i s t r i b u t i o n of the Fracture Points in(a)Low V #9, and(b)High V Fibers, x 100.  f  f  (8.5%),  (14.5%), #17, Composites, Small C i r c l e s Showing Fractured  \u000C46  Fig.31. Longitudinal Sections of Fractured 201T6 Specimens Showing the Difference in the D i s t r i b u t i o n of the Fractured Points in a Low V V  f  f  (6.4%), #24,  and(b)High  (17%), #29, Composites, Small C i r c l e s Showing Fractured F i b e r s , x 100.  \u000C47  from the f r a c t u r e surface, as shown i n these p i c t u r e s . Scanning e l e c t r o n micrographs at low magnification (x50) were used to quantify the f i b e r f a i l u r e zone thickness.  The q u a n t i t a t i v e measure-  ments were made of the d i s t a n c e , B^, from the f r a c t u r e surface to the furthest  point at which broken f i b e r fragments could be observed.  r e s u l t s are given in Table 10.  The r e l a t i o n between B^ and the f i b e r volume  f r a c t i o n , V^, and Young's modulus E a l a t e r section I V - 2 - i v . these f a c t o r s .  The  c  was examined and t h i s i s discussed  F i g . 32 shows the v a r i a t i o n s of  values with  in  \u000C020 \\  Ec^Vf-Vfrninf XIO' Fig.32. Relation between  E , and V c  lb.  \u000CTable 10.  Data f o r Fiber Fracture Zone C h a r a c t e r i s t i c s  Q m u t s \" ^m* Alloy  ^uts Suts-^*  601T4  0.0446  201T6  + (  0.0385  E  V  f  V  f \" fmin V  c  x 1  \u00B0 P  s i  &lt; f f mV E  V  + E  E c (  V fmin V  ) 2  25.3  B  f  i  n  c  h  0.067  0.0224  12.68  0.17  0.085  0.0404  13.40  8.21  0.06  0.145  0.1004  15.80  1.57  0.02  0.086  0.0475  13.44  5.96  0.09  0.10  0.0615  14.00  3.70  0.05  0.17  0.1315  16.80  0.972  0.02  \u000C50  IV  IV-1.  DISCUSSION  Rule of Mixture  IV-1-i.  Rule of Mixture f o r Continuous Fiber Reinforced Materials  Since K e l l y and Tyson (41) proposed the so c a l l e d \" r u l e of mixture\" f o r the strength of composites, i t has been quite often used to evaluate the composite strength.  In t h i s r u l e , the stress on the composite specimen  O\" i s expressed as the summation of stresses which both f i b e r s and the matrix are supporting at the same amount of s t r a i n as shown i n F i g . 33. This r e l a t i o n can be obtained by assuming the same s t r a i n in f i b e r s and the matrix. (1) O\" : stress in a composite at a c e r t a i n s t r a i n  Where,  cri : stress in f i b e r s at the same s t r a i n cr : stress in matrix at the same s t r a i n m V.p : f i b e r volume f r a c t i o n V  : matrix volume f r a c t i o n ( =  m  1-V ) f  Dividing eqn. 1 by a s t r a i n , e , which i s smaller than the y i e l d s t r a i n of the matrix composite, E  i s obtained  E =cr/ c  \u20AC  =  ,  E where,  E E  the r u l e of mixture f o r Young's modulus of the  f  .  c  V  =  F  C  T  /  Vf  +  \u20AC  +  V  V cr/ m  \u20AC  m, E  = Young's modulus of f i b e r s = Young's modulus of the matrix  (2) (3)  \u000C\u2014; FIBER  s  /  / /  /  i  /  M 4)  CO  0\"c  Oh?  / /  |  I &gt;^ i  MATRIX  / /A,  c  COMPOSITE  A  y  c  VmO-  m  f ufs  Strain  \u20AC  F1g.33. Schematic Diagram of Stress Strain Curves of the Composite, the Fiber, and the Matrix, obtained according to \"Rule of Mixture\"  160  /  \u00A3.20 LO  80  Hj 40 3  02  0.4  o  UTS  \u2022  YS  Q6  0.8  Fig.34. Tensile Strength of Copper Reinforced with 5 mm Continuous B r i t t l e Tungsten Wires (41).  \u000C52  I f i t i s assumed that a l l the f i b e r s f a i l at the same time when they reach the f a i l u r e s t r a i n , the f a i l u r e stress \" r u l e of mixture\" c o n d i t i o n .  cr can s a t i s f y the  Consequently, the strength of the composite,  cr,, i s given i n the following expression as \" r u l e of mixture\" f o r the composite strength. cr = V cr_ + v cr * c f futs m m  v  f  where,  cCp^ rr * m  :  (4)  '  t e n s i l e strength of f i b e r s : flow stress of the matrix at the same s t r a i n with f i b e r fracture.  In the present work, the strength values c a l c u l a t e d according to t h i s equation have been discussed. When the volume f r a c t i o n of f i b e r s i s less than V .. , f i b e r s are crit thought to be broken successively, i . e . one a f t e r another, before the matrix f a i l s .  The maximum strength of such a composite i s expressed i n the  following equation, because only the matrix i s thought to support the load at the f a i l u r e point.  \u2022 \u00B0-muts &lt; \" V  &lt;&gt;  ]  where,  cr ^  s  5  : t e n s i l e strength of the matrix  The c r i t i c a l f i b e r volume f r a c t i o n , above which the \" r u l e of mixture\" condition i s s a t i s f i e d ( i f o v e r a l l f i b e r f r a c t u r e occurs), i s obtained from Eqn. 4 and 5. V  fcrit  =  &lt;Vor *)/(&lt;r m  f  -\u00B0- -cr *) m  m  .  (6)  Furthermore, the f i b e r volume f r a c t i o n has to exceed some value V  min  t 0  strengthen  t n e  composite.  This value i s obtained by s u b s t i t u t i n g  \u000C53  the cr i n Eqn. 4 w i t h e r ^ . V . mm  =  v  {cr - cr *)/{cr . -cr *) m m \u2022 futs m '  K  (7) '  F i g . 34 shows experimental data which were obtained by K e l l y and Tyson to prove \" r u l e of mixture\" f o r the strength of Cu~W wire composites.  IV-1-ii.  Strength of Discontinuous Fiber Reinforced M a t e r i a l s  In the case of discontinuous f i b e r s , the load on a composite i s thought to be transferred to the f i b e r s through the matrix.  The strength of a d i s -  continuous f i b e r composite becomes close to the strength of a continuous f i b e r composite as the aspect ratio,j2/d , (the r a t i o of f i b e r length and diameter) increases. There are three t h e o r e t i c a l works which can be distinguished from each other only by the d i f f e r e n c e of assumptions about the e l a s t i c i t y or the p l a s t i c i t y of the matrix (42, 43).  In t h i s s e c t i o n , only  Kelly-Tyson's  theory (91) i s described which i s based on the assumption of e l a s t i c f i b e r s and the p l a s t i c matrix, because t h i s assumption i s mostly a p p l i c a b l e in the case of metal matrix composites. Now consider the case where the matrix i s allowed to flow p l a s t i c a l l y . When a composite of discontinuous f i b e r s i s stressed i n a d i r e c t i o n along the f i b e r a x i s , d i f f e r e n t a x i a l displacements take place in the matrix and f i b e r s , and a large shear stress occurs at the end of the f i b e r s . F i g . 35 i s a model of a single discontinuous f i b e r in the c y l i n d r i c a l matrix.  The load i s transferred from the matrix to the f i b e r only by the  shear stress at the i n t e r f a c e , r  p z  , neglecting any stress t r a n s f e r across  the f i b e r ends which have small area.  The small increment of the l o a d ,  \u000Ci  54  Fig.35. Showing Notation Used in the Kelly and Tyson's Theory for the Discontinuous Fiber \"Composites (41).  Distance F1g.36. Expected Variation of Stress along a Fiber within a Plastic Metal Matrix (41).  Fig.37. Stress Distribution in the Discontinuous Tungsten Fiber Obtained by Means of Moire Technique. Applied Stress on the Composite i s -  and High (b) (44) (47).  Low (a)  (x: distance from one end of the fiber, d: diameter)  \u000CdP, due to the stress t r a n s f e r a t the small i n t e r f a c e area 2 \" r d z 77  Q  is  given by dP=  2Trr T dz 0  (8)  rz  Equation 8 integrates to  P = 277-rzr  (9)  For a p l a s t i c matrix which does not work-harden, If the i n t e r f a c e f a i l s ,  r i s constant.  r i s equal to the f r i c t i o n a l force per u n i t area  which the matrix exerts on the f i b e r as i t s l i d e s over the f i b e r . For work-hardening matrix, r depends on the s t r a i n i n the composite and i s thought to be i d e n t i f i e d with the ultimate shear strength of the matrix. Eqn. 9 means that the stress i n a f i b e r builds up l i n e a r l y from both ends as shown i n F i g . 36. The stress i n the f i b e r a t a distance z from the end, C  zz  , i s expressed by  ^ I f r ?  (10)  s u b s t i t u t i n g f o r P from Eqn. 9,  cr  2TZ  zz  / *| 1 \\  = - p \u2014  (11)  The s t r a i n i n the f i b e r cannot exceed the s t r a i n of the m a t r i x so that cr^ w i l l b u i l d up to the value i e n t l y long.  I f the stress i n the f i b e r ,  C T , provided the f i b e r i s s u f f i c f  cr\u201E, builds up to the f r a c t u r e  stress of the f i b e r c j f u t s , the f i b e r i s broken.  The c r i t i c a l f i b e r  length, T, f o r t h i s to take place i s given by  W  T= r  T  (12)  \u000C56  T/2 i s defined as the t r a n s f e r length. If  The value of T depends on  T .  T i s constant as in the case of a non-work hardening m a t r i x , T i s  also constant. If a condition of f i b e r length, i , &gt; T , of the composite occurs when of the f i b e r ,  c r  f  u t s  -  i s s a t i s f i e d , the f r a c t u r e  CTp increases to the ultimate t e n s i l e strength  The average t e n s i l e s t r e s s ,  at t h i s loading point  i s expressed as follows ?  f  ' \\  'o'\u00B0\"\"  d z  \u2022 \u00B0 * P - h )  &lt; 1 3 )  An equation which expresses the t e n s i l e strength of the discontinous f i b e r composites i s obtained, t r e a t i n g in continuous f i b e r composites. %  =  G  futs f V  Of as the stress i n the f i b e r s  From Eqn. 4 and 13,  \"V2\u00A3) cr* +  V  m  (14)  It i s seen from t h i s equation that the strength of a discontinuous f i b e r composite becomes c l o s e r to the strength of continuous ones i f the f i b e r length, 1  , i s much greater than the t r a n s f e r length, T/2-  Miura and Okuno's (44) study on the stress d i s t r i b u t i o n of a twodimensional Al-W wire composite by means of the M o i r e technique proved the appropriateness of the stress d i s t r i b u t i o n in f i b e r s in t h i s model. Their r e s u l t s are shown in F i g . 37. T, T,and discussions.  are used as important variables l a t e r in the f o l l o w i n g The Equation 14 i s used to express the strength of the  f i b e r f a i l u r e zone.  \u000C57  IV-2.  Propagative Fiber F a i l u r e Model f o r Graphite Fiber Reinforced Aluminum A l l o y Composites  IV-2-i.  Ultimate Tensile Strength of Homogeneously D i s t r i b u t e d Specimens  The composite t e n s i l e specimens of 601 T4 ( s o l u t i o n t r e a t e d ) , 201 T6 (age hardened) and 7178 T6 (age hardened) a l l o y s e x h i b i t e d great d i s crepancy between experimental strength values and values c a l c u l a t e d according to \" r u l e of mixture\" as shown in F i g . 20, 21, and 22.  The experimental  U.T.S. values of 601 T4 and 201 T6 a l l o y composites against f i b e r volume f r a c t i o n s appear to be on broad curves in s p i t e of great s c a t t e r i n g . The s c a t t e r of strength values in these experimental r e s u l t s might be due t o : a) m i s o r i e n t a t i o n of the f i b e r s with the specimen a x i s .  The f i b e r s might  not be aligned properly during the i n f i l t r a t i o n - s e d i m e n t a t i o n process, e s p e c i a l l y i n the case of low f i b e r volume f r a c t i o n composites. b) non-uniform f i b e r d i s t r i b u t i o n in the matrix and the contacts of neighboring f i b e r s which may act as defects. c) deviations of matrix chemical composition from average values.  The  ultimate t e n s i l e strength of the matrix i s dependent on i t s chemical composition.  The great density d i f f e r e n c e of each a l l o y element might  cause great d i f f e r e n c e of s e t t l i n g speed among the powders of each element in the i n f i l t r a t i o n - s e d i m e n t a t i o n process. f:  d) e r r o r in the measurement of f i b e r volume f r a c t i o n s .  The micrographs  which cover only one-seventh of the t o t a l specimen section were used to count the numbers of f i b e r s , so that e r r o r of a few percent i s unavoidable.  \u000C58  No q u a n t i t a t i v e measurements were done to estimate the c o n t r i b u t i o n s to the s c a t t e r from each one of the above f a c t o r s . General microscope observation revealed that the breakage of graphite f i b e r s occurs in a narrow region close to the f r a c t u r e edge of the s p e c i mens; however, the f i b e r s ' f a i l u r e i s in a b r i t t l e manner and the matrix f a i l u r e i s in a d u c t i l e manner as expected from the o r i g i n a l deformation c h a r a c t e r i s t i c s of each m a t e r i a l . This r e s u l t suggests that most of the p l a s t i c deformation of the matrix takes place in a l i m i t e d range from the f r a c t u r e surface, where f i b e r s are broken.  The propagation of a rather highly strained deformation  zone was observed in specimens being strained to some extent before the f a i l u r e as shown in F i g . 25.  The segments of broken f i b e r s were also obser-  ved in t h i s zone as shown i n F i g . 26, e t c .  The s t r a i n of the matrix in  t h i s zone i s higher than the matrix of the other part of the specimen as shown i n F i g . 27. From these r e s u l t s , i t may be said that such a highly strained zone propagates over the section of the specimen, breaking the f i b e r successively at the t i p f r o n t of the zone, during the t e n s i l e t e s t .  Such successive  f a i l u r e of f i b e r s in the matrix has been observed in boron filament r e i n forced aluminum composites by J . Steele et al_. (45).  In the present work,  the model proposed i s one i n which the ultimate t e n s i l e strength of the composites corresponds to the accelerated propagation of a f i b e r f a i l u r e zone. This f a i l u r e mode of composites i s quite d i f f e r e n t from the model which A. K e l l y et al_. adopted to e s t a b l i s h \" r u l e of mixture\" f o r large diameter f i b e r metal composite l i k e W wire-Cu composites.  In the model f o r  \" r u l e of m i x t u r e \" , a l l the f i b e r s in the composite are assumed to f a i l  at  \u000C59  once o r a t an i d e a l i z e d c o n d i t i o n .  The d i s c r e p a n c y i n the e x p e r i m e n t a l  s t r e n g t h v a l u e s and \" r u l e o f m i x t u r e \" v a l u e s may be m a i n l y due t o the d i f f e r e n c e between the a c t u a l p r o p a g a t i v e f a i l u r e mode and the i d e a l i z e d f a i l u r e mode i n \" r u l e o f m i x t u r e \" model. mode i s g o i n g t o be d i s c u s s e d pages,  The observed p r o p a g a t i v e f a i l u r e  i n mathematical e x p r e s s i o n  in  successive  i n order to give a p o s s i b l e explanation f o r experimental  IV-2-ii.  results.  Energy C r i t e r i a o f P r o p a g a t i v e F i b e r F a i l u r e f o r Homogeneously Distributed  Composites.  I t i s assumed f o r s i m p l i c i t y t h a t f i b e r s always deform i n an e l a s t i c manner and the m a t r i x changes  i t s d e f o r m a t i o n mode from e l a s t i c t o p l a s t i c  which does not i n c l u d e any work hardening e f f e c t when i t i s s t r e s s e d the y i e l d p o i n t , c x ^ , i . e . c r  = 0\"* m  in Fig.  beyond  38.  As a s i m p l e i l l u s t r a t i o n , c o n s i d e r a p l a t e specimen o f w i d t h w and u n i t t h i c k n e s s c o n t a i n i n g a small h a l f e l l i p t i c a l  zone i n which the f i b e r s have  f a i l e d i n t o small segments o f average t r a n s f e r l e n g t h , T  t  .  The zone  l e n g t h i s e x p r e s s e d as h a l f o f the l o n g d i a m e t e r o f an e l l i p s e , a , and the thickness  i s a l s o expressed as the s h o r t d i a m e t e r , 2b, i n F i g . 39.  The  s t r a i n o f the m a t r i x o u t s i d e t h i s h a l f e l l i p t i c zone i s expressed as e w i t h i n t h i s zone as the f i b e r f a i l u r e s t r a i n f \u20AC  on f i b e r s o u t s i d e o f t h i s zone i s e x p r e s s e d as the maximum s t r e s s  t  s  &gt;  ( f ts \u20AC  u  broken f i b e r segments  &gt; \u20AC  \u00B0^\"  ^  e  s  t  r  e  s  s  cr \u2022, which i s lower than  on the f i b e r fragments w i t h i n t h i s zone  average s t r e s s on the fragments i s ^Of^-  stress  u  and  Q  t s  -  The  The s t r e s s d i s t r i b u t i o n on  i n t h i s zone i s assumed as shown i n F i g . 40.  The  i s t r a n s f e r r e d from the m a t r i x t o t h e segment t h r o u g h a shear  stress,r,  a t the i n t e r f a c e o f m a t r i x and f i b e r segments.  The s t r e s s on the  \u000C60  201 'futs  r  H  W  Fig.39. Half E l l i p t i c Fiber-Failure Zone in a Composite  matrix  Strain \u00A3  *  Fig.38. Schematic Diagram of Stress Strain Curve of the Matrix and the Fiber  \u000C7\\  co CO  a&gt;  v\u2014  To  \\  tn  \\  i  7T  / \\  1  \\ / \\  / \\  / \\  / \\  T\\  / \\  TT  / \\  \\ /  Distance  Fig.40. Stress D i s t r i b u t i o n Change in a Fiber which i s Located in the Fiber F a i l u r e Zone,  s o l i d l i n e : before f a i l u r e ,  dot-dash-line: a f t e r f a i l u r e .  \u000C62 f i b e r b u i l d s up l i n e a r l y to the value ments.  \u00B0f t u  s  ^  rom  the ends o f the segT i s constant.  For a p l a s t i c matrix which does not work-harden \u00B0futs  =  T  T  uts  /  (15)  r  where r : r a d i u s of f i b e r s In other words, f i b e r s are assumed to be broken i n t o small segments which cannot be broken s h o r t e r than t h i s by the f u r t h e r p l a s t i c deformation of the matrix. The average s t r e s s i n a segment i n t h i s zone,  c r g &gt; i s given i n the e  f o l l o w i n g expression from Eqn. 13. ^seg  =  \u00B0futs /  (16)  2  Consequently, the s t r e n g t h of t h i s zone,0~ , i s obtained from Eqn. 14. z  V  \\  f ^uts  V  +  mV  &lt;&gt;  V  17  During a t e n s i l e t e s t , such small h a l f e l l i p t i c f i b e r f a i l u r e zones f i r s t s t a r t to grow a t some s t r e s s concentrated areas or d e f e c t i v e areas in a specimen as shown i n F i g . 41, d e v i a t i n g the a c t u a l s t r e s s - s t r a i n curve from a curve expected from \" r u l e o f mixture\".  Some amount of energy,  dW , z  i s r e q u i r e d to grow a zone from the shape o f a and b to a+da and b+db. During t h i s growth, the load does work,|dL|. A change of e l a s t i c energy, dU  , occurs i n the r e g i o n o u t s i d e of the f a i l u r e zone due to the shape  change of the zone. The e l a s t i c energy change may be e a s i l y understood i f we c o n s i d e r the f o l l o w i n g s p e c i a l l o a d i n g system shown i n F i g . 42. specimen s t r a i n e d to the s t a t e of e l o n g a t i o n ASL.  F i g . 42(a) shows the The area \u00A3j\" OYAC  \u000C63  Fig.41. Fiber F a i l u r e Zones in a  Fig.43. Schematic Diagram of Stress  Specimen  S t r a i n Curve at F a i l u r e Point  \u000C64  c o r r e s p o n d s t o t h e work which t h e l o a d d i d . I f t h i s specimen i s b r o k e n under t h e c o n d i t i o n o f f i x e d g r i p d i s t a n c e , t h e two s e p a r a t e d p i e c e s s h r i n k e l a s t i c a l l y and r e l e a s e e l a s t i c e n e r g y w h i c h c o r r e s p o n d s t o t h e hatched a r e a A A B C shown i n F i g . 4 2 ( b ) .  I f we s t r a i n a s p e c i m e n , ^ w h i c h a  f i b e r f a i l u r e zone has t r a v e r s e d t h e c r o s s s e c t i o n , t o g i v e t h e same amount of e l o n g a t i o n , A d, t h e l o a d i s i n c r e a s e d f o l l o w i n g t h e s t r e s s - s t r a i n c u r v e OY'ED i n F i g . 4 2 ( c ) .  C o n s e q u e n t l y , i f we assume t h a t t h e same f i b e r  f a i l u r e zone (which t r a v e r s e d t h e s e c t i o n ) o f t h i c k n e s s 2b, i s formed i n a specimen between f i x e d g r i p s , t h e l o a d i s l o w e r e d f r o m o r i g i n a l , L-j t o the l o a d , L^, which c o r r e s p o n d s t o Eqn. 17.  D u r i n g t h i s u n l o a d i n g under  the f i x e d g r i p c o n d i t i o n , t h e r e g i o n s o u t s i d e o f t h e zone s h r i n k e l a s t i c a l l y r e l e a s i n g energy which corresponds to the area A A E D i n F i g . 42(c).  In  the c a s e o f h a l f e l l i p t i c f i b e r f a i l u r e zone, some e l a s t i c e n e r g y w h i c h i s a c o m p l i c a t e d f u n c t i o n o f t h e shape o f t h e zone, a and 2b, i s r e l e a s e d from o u t s i d e o f t h e zone i n t h e same way as t h e c a s e o f F i g . 4 2 ( c ) . We l e t dW^. be t h e t o t a l f r e e e n e r g y change o f a specimen due t o t h e f o r m a t i o n o f h a l f e l l i p t i c f i b e r f a i l u r e zone. dW . t  = dl_ + dW  z  Then dW^ i s g i v e n by  + dU  (18)  When one o f t h e zones grows up t o a c r i t i c a l s i z e , t h e summation o f the potential energy c h a n g e of the lood.dL, and t h e e l a s t i c e n e r g y change, dU, can be b i g enough t o s u p p l y t h e f o r m a t i o n e n e r g y o f t h e z o n e , dW . z  The  e n e r g y b a l a n c e a t t h i s c r i t i c a l p o i n t i s g i v e n by dW  t  = dL + dW  2  + dU = 0  (19)  Once t h i s r e l a t i o n i s s a t i s f i e d , t h e zone can s t a r t t o grow q u i c k l y w i t h o u t any e x t r a e n e r g y s u p p l y from o u t s i d e o f t h e s p e c i m e n .  In o t h e r words, the  \u000C65  p r o p a g a t i o n o f the zone i s a c c e l e r a t e d beyond t h i s c r i t i c a l p o i n t which corresponds  t o the s t r o n g e s t s t a t e o f the s p e c i m e n , i . e . u l t i m a t e t e n s i l e  strength point.  A f t e r t h i s a c c e l e r a t i o n t a k e s p l a c e , the f i b e r s a r e  q u i c k l y broken i n t o s m a l l segments i n t h i s zone and a b r u p t l y lower supporting load.  the  The f r a c t u r e o f t h e m a t r i x o c c u r s s u c c e s s i v e l y r i g h t a f t e r  the t r a v e r s e o f the zone.  The f u r t h e r p l a s t i c d e f o r m a t i o n  l e a d i n g t o the  f a i l u r e o f the m a t r i x i s thought t o take p l a c e a t o n l y l i m i t e d l o c a l r e g i o n s around the broken f i b e r ends i n the zone.  The s t r e s s s t r a i n c u r v e around  the f r a c t u r e p o i n t may be shown s c h e m a t i c a l l y as i n F i g . 43, i f we exaggerate  the l o a d drop.  For s i m p l i c i t y , the t h i c k n e s s o f t h i s zone, 2b, i s assumed t o remain c o n s t a n t a f t e r the zone s t a r t s t o propagate  quickly.  The g r i p d i s t a n c e i s  a l s o assumed t o remain c o n s t a n t d u r i n g t h i s q u i c k p r o p a g a t i o n ,  because  the c r o s s head speed o f the t e s t i n g machine i s v e r y low (0.02  inch/min.)  compared w i t h the p r o p a g a t i n g Consequently,  s p e e d , ( i . e . the f i x e d g r i p c o n d i t i o n ) .  the l o a d does not do any work, so t h a t dL = 0.  e n e r g y change, dW , t  The t o t a l  has t o d e c r e a s e w i t h the i n c r e a s e o f zone l e n g t h , a ,  i n o r d e r t o propagate w i t h o u t any e x t r a e n e r g y i n c r e m e n t from o u t s i d e o f the specimen. propagate ^_t da  Then, the c r i t i c a l c o n d i t i o n a t w h i c h the zone can s t a r t to  spontaneously  \"  T h i s statement  d(U-rWz) da  i s s t a t e d as the f o l l o w i n g e x p r e s s i o n (20)  n _ u  i s o b t a i n e d by f o l l o w i n g the G r i f f i t h  (46) t h e o r y f o r an  e l l i p t i c crack in a b r i t t l e material. I f the l e n g t h o f zone exceeds the c r i t i c a l v a l u e a t which Eqn. i s j u s t s a t i s f i e d , elastic energy  released  20  i s more than s u f f i c i e n t  \u000C66  to p r o v i d e t h e i n c r e m e n t o f f r e e e n e r g y i n the zone, so t h a t t h e r a t e o f propagation i s accelerated. I t i s n e c e s s a r y t o d e s c r i b e U and W  z  i n terms o f t h e e x p e r i m e n t a l  parameters. (A)  The f r e e e n e r g y i n c r e m e n t i n t h e f i b e r f a i l u r e zone, W , 2  F i g . 40,  shows t h e s c h e m a t i c s t r e s s d i s t r i b u t i o n i n a f i b e r when i t i s broken i n t h e zone, where T  i s t h e s t r e s s t r a n s f e r l e n g t h o u t s i d e o f the zone.  Q  When  f i b e r s a r e broken i n t h e zone, t h e s t r e s s d i s t r i b u t i o n changes from t h e s o l i d l i n e t o t h e d o t t e d l i n e , r e l e a s i n g e x t r a e l a s t i c energy to t h e surrounding matrix.  On t h e o t h e r hand, work i s r e q u i r e d t o break t h e f i b e r s , and  s t r a i n t h e m a t r i x i n t h e zone.  W  z  can be d e r i v e d as the summation o f t h e s e  terms. a) The work p e r u n i t t h i c k n e s s , W , n e c e s s a r y t o deform t h e m a t r i x i n t h e zone from e ,  t h e s t r a i n i n t h e m a t r i x o u t s i d e o f t h e zone t o \u00A3 f  Q  u t s  \u00BB  the  fibre failure strain, is ww  m  =  Vv  m  T m V  cr*(\u20AC - \u20AC ) tuts o  2 qb(T  {  rrr 2 E  b) The work, W , f  \u20AC  futs  ;  (q  uts  -\u00B0-f &gt; 0  (21)  f  n e c e s s a r y to s t r a i n t h e f i b r e s i n t h e zone from \u20AC  1s  w f  _-IflZliSrtslV' f 4E  &lt;22)  Q  to  \u000Cc) The r e l e a s e d e l a s t i c e n e r g y from t h e broken f i b r e s , W , i s e x p r e s s e d as t h e d i f f e r e n c e between t h e e l a s t i c e n e r g y o f t h e f i b r e s b e f o r e t h e i r f a i l u r e and a f t e r t h e i r f a i l u r e . W  r  =  - bef W  where,  +  W  aft  &lt; &gt; 23  : e l a s t i c e n e r g y which t h e f i b r e s i n t h e zone p o s s e s s before breakage. W  aft  :  e  ^  i  a s t  e n e r g y which t h e fragments o f t h e same f i b r e s  c  i n t h e zone p o s s e s s a f t e r _  V  rr a b  f  4E Where,  T  Vf \u00B0 T  +  2E  f  =  0  breakage. ^  Q  f  =  r  Then, W bef  b e f  = =  V  f ~  T  Q  4 E  Woft  -  V  +  V  f  Vf o  +  f  ^  f  ^ts  b  T  2E cr f  ^  -  W  ^  (  Z  C7  3 f 0  (  f u t s  ) d z  a 2 | cr(z)\u20AC(z)dz  2 3  I  u t ?  2 T  uts  2 3  3 E r^  ^rl^-  f  +  +  3 E  \u2014^rJ  f  r 2 |  ,26)  2  5  )  \u000Cthen - bef+ w  w  vif  0  6  E  aft  T  uts ^fo  f  2E  \u00B0futs  (27)  (  d) The i n c r e m e n t o f s u r f a c e f r e e e n e r g y due t o t h e b r o k e n f i b e r ends i s n e g l i g i b l y s m a l l and w i l l be i g n o r e d below. The sum o f t h e s e e n e r g y terms must be W i n Eqn. 21. 2  w  z  = W =  +  m  W  _ ( f\u00B0 uts V  +  f  T  1  W  r  S ^ l A  ( H ^ b  3  ^0  2  f  v  f  7/0  b  V 7ra  \u00B0futs  m  +  +  bofhorfftsl  2E  f  j  ( 2 8 )  (B^) E l a s t i c e n e r g y U, r e l e a s e d f r o m t h e r e g i o n o f t h e s p e c i m e n . o u t s i d e the e l l i p t i c zone due t o t h e f o r m a t i o n o f t h e zone. I t i s i m p o s s i b l e t o d e r i v e an e x a c t e x p r e s s i o n f o r t h i s term l a c k i n g a m a t h e m a t i c a l a n a l y s i s o f t h e s t r e s s and s t r a i n i n t h e n e i g h b o u r h o o d o f the zone; however, i t may be p o s s i b l e t o d e r i v e an a p p r o x i m a t e e x p r e s s i o n f o r t h i s term by m o d i f y i n g G r i f f i t h e x p r e s s i o n f o r t h e e l a s t i c r e l e a s e d energy due t o t h e f o r m a t i o n o f a two d i m e n s i o n a l e l l i p t i c a l c r a c k o f l e n g t h 2a, i n an e l a s t i c specimen h e l d between r i g i d l y f i x e d g r i p s under s t r e s s , 0\"  \u000C69  H i s t h e o r y was l a t e r d i s c u s s e d by Knott etol(48) a n d i t was shown t h a t the assumption  o f r i g i d l y f i x e d g r i p s i s not e s s e n t i a l t o the G r i f f i t h  energy c r i t e r i o n f o r b r i t t l e f r a c t u r e . The same c r i t e r i o n i s o b t a i n e d i f the c r a c k p r o p a g a t i o n i s assumed t o o c c u r under c o n s t a n t l o a d . case o f f i x e d g r i p s , the e x t e r n a l f o r c e s cannot do work.  The  In t h e critical  l e n g t h o f t h e c r a c k above which i t can p r o p a g a t e s p o n t a n e o u s l y  i s then d e t -  ermined by t h e c o n d i t i o n dW where,  + dU = 0  c  (29)  c  dw\" : the f r e e energy r e q u i r e d f o r i n c r e a s i n g t h e l e n g t h o f a c  c r a c k from 2a t o 2a + da, dUc : t h e e l a s t i c energy r e l e a s e d s i m u l t a n e o u s l y i n t h e T h i s e l a s t i c r e l e a s e d energy i s g i v e n by t h e G r i f f i t h dUc =  . d { ^ V V + b )j  specimen.  expression  (30)  2  i n the c a s e o f p l a n e s t r a i n c o n d i t i o n ( f o r t h i c k s p e c i m e n ) where,  v i s Poisson's r a t i o  a n d E is Y o u n g ' s m o d u l u s , a n d b y  d u * - d { ^ \\ \u00B0 + &gt;) 2  &gt;  b2  t31  c  i n the case o f plane s t r e s s ( f o r t h i n specimens).  In f u r t h e r d i s c u s s i o n s ,  o n l y Eqn. 30 i s a d o p t e d , because t h e t e n s i l e t e s t p i e c e s i n the p r e s e n t work a r e t h o u g h t t o have r a t h e r l a r g e r t h i c k n e s s (1/3 o f the w i d t h ) . T h e r e l e a s e d e l a s t i c energy under t h e f i x e d g r i p c o n d i t i o n i s shown s c h e m a t i c a l l y as AOAC i n F i g . 44. On t h e o t h e r hand, i f t h e c r a c k p r o p a g a t e s w h i l e t h e l o a d i s kept cons t a n t , t h e l o a d does work,|dL|, shown a s D A D B E i n F i g . 44. The e l a s t i c e n e r g y i s i n c r e a s e d by the amount of d U during this c r a c k p r o p a g a t i o n . T h e total c  \u000C70  change in potential energy is a d e c r e a s e of m a g n i t u d e I d L l \u2014 dU^.. The du\u00A3 i s h a l f o f t h e dL, a s A B D E shown i n F i g . 44, so t h a t h a l f o f t h e e x t e r n a l work i s s t o r e d a s a d d i t i o n a l e l a s t i c energy o f t h e specimen, and t h e o t h e r h a l f i s a v a i l a b l e f o r i n c r e a s i n g t h e f r e e energy o f t h e c r a c k surface, dW \u00AB c  The c r i t i c a l c o n d i t i o n f o r t h i s c a s e i s e x p r e s s e d i n t h e  following equation. dL +  du'c + dWc =-dU'c + dWc =0  (32)  Now t h e r e l a t i o n s h i p between l o a d , L and e l o n g a t i o n , S, i s g i v e n by S = CL  (33)  where C i s a c o n s t a n t f o r g i v e n c r a c k l e n g t h , c a l l e d t h e c o m p l i a n c e o f t h e system.  As t h e change i n c r a c k l e n g t h , d a , tends t o z e r o , we may t r e a t C  as i d e n t i c a l f o r c r a c k l e n g t h s 2 o a n d 2 o + d o and w r i t e , dS = C dL  (34)  d U , dl/, a r e g i v e n by t h e f o l l o w i n g same e x p r e s s i o n s , u s i n g C. c  dll ~ - A 0 A C = 3 - i S dL = - i CLdL  c  2  (35)  2  dL - du'=-du'=-A0AB=-LdS + k2d S = - k2d S = -2k L d L ( 3 6 ) C C For an i n f i n i s t e s i m a l l y s m a l l amount o f c r a c k e x t e n s i o n , t h e d e c r e a s e i n s t o r e d e l a s t i c energy under t h e f i x e d g r i p s c o n d i t i o n i s i d e n t i c a l t o t h e d e c r e a s e i n p o t e n t i a l energy under t h e c o n s t a n t l o a d c o n d i t i o n . Eqn. 29 and 32 show t h a t t h e energy a v a i l a b l e f o r c r a c k p r o p a g a t i o n a t f i x e d g r i p s i s t h e same as a t t h e c o n s t a n t l o a d , so t h a t even i n t h e l a t t e r c a s e , we c a n use Eqn. 30 t o e s t i m a t e t h e energy a v a i l a b l e f o r t h e p r o p a g a t i o n . 2 2 dL - dU^= -dU^=-dU=-d j  7 7 1 1  (for plane s t r a i n condition)  \"^  ( a + b )} 2  2  (37)  \u000CA  B  0 Elongation  S  F i g . 4 4 . E l a s t i c L o a d i n g C u r v e s f o r C r a c k Lengths a and a + da.  \u000C72  When a c o m p o s i t e specimen f a i l s by t e n s i o n , ( i . e . c o m p l e t e l y s e p a r a t e d i n t o two p i e c e s ) , under t h e f i x e d g r i p c o n d i t i o n , e l a s t i c energy which c o r r e s p o n d s t o t h e a r e a A ABC i n F i g . 45 i s r e l e a s e d and t h e e n e r g y d OYAB i s s t o r e d i n t h e specimen.  Although, composites a r e not p e r f e c t l y  e l a s t i c m a t e r i a l , i f we a p p l y t h e G r i f f i t h  r e l e a s e d e l a s t i c energy expres-  sion f o r t h e i r e l a s t i c p r o p e r t y , an e x p r e s s i o n f o r t h e r e l e a s e d e l a s t i c energy u  d u r i n g t h e p r o p a g a t i o n o f a c r a c k i n a c o m p o s i t e i s g i v e n by  cc  7T(1- V )  _  where,  Z  Z  a/  ( 0  2 2, +  b  ( 3 8 )  y : Poissons r a t i o o f t h e composite cr\u201E : s t r e s s on t h e c o m p o s i t e E  c  : Young's modulus o f t h e c o m p o s i t e  When a f i b e r f a i l u r e zone t r a v e r s e s t h e c r o s s s e c t i o n o f a specimen h e l d between f i x e d g r i p s , t h e m a t r i x does n o t f a i l , b u t o n l y t h e f i b e r s a r e b r o k e n i n t o s m a l l f r a g m e n t s o f a v e r a g e t r a n s f e r l e n g t h , T ^ &gt; i n t h e zone. Then, t h e s t r e s s d r o p s down from t h e p o i n t A ( s t r e n g t h o f t h e t r a v e r s e d zone, C7j CT  tz  VfW  =  f : seg  +V0~*  mm  z  &lt; J c r m  ) &gt; t o E ( o~^ ). T h e z  , i s g i v e n by  = i c r , v  2 futs  r  f+  O - * V  m  m  ( 3;9 )  v  D u r i n g t h i s u n l o a d i n g f r o m A t o E , t h e r e g i o n o u t s i d e o f t h e zone s h r i n k s e l a s t i c a l l y and r e l e a s e s e l a s t i c e n e r g y which c o r r e s p o n d s t o t h e a r e a AADE.  The amount o f t h i s a r e a A A D E i s e x p r e s s e d i n t h e f o l l o w i n g f o r m u l a  i f t h e g r a d i e n t o f Y'E t o be  K f V  +  mV  E  ( i . e . , t h e Young's modulus o f t h e zone) i s assumed  \u000C73  F i g . 4 5 . S c h e m a t i c Diagram o f E l a s t i c Energy R e l e a s e d A ABC and A A D E , when t h e C r a c k and t h e F i b e r F a i l u r e Zone T r a v e r s e t h e C r o s s S e c t i o n  \u000C74  A A D E  ( \u00B0crm - Qfz ) E ,  -  (l- f fV  E  V  V  2  E  c  CT  crm  c  ( ! T f Ofut\u00AB) r \"2 ^crml E - - \u00A3 - v  where,  - AABC  m m ) cr E  f  E  )  f  A A D /~ -AABC  (40)  : composite strength before f i b e r f a i l u r e , c a l c u l a t e d according t o the \" r u l e o f mixture\".  When t h e h a l f e l l i p t i c f i b e r f a i l u r e zone has n o t y e t t r a v e r s e d t h e f u l l c r o s s s e c t i o n , t h e energy which corresponds In o r d e r t o m o d i f y t h e G r i f f i t h  t o AAD'E' i s r e l e a s e d .  expression for the e l a s t i c released  energy due t o t h e e l l i p t i c c r a c k , U , t o t h e e l a s t i c r e l e a s e d energy due c c  t o t h e h a l f e l l i p t i c f i b e r f a i l u r e zone, U, we s i m p l y assume t h a t U / U  cc  i s p r o p o r t i o n a l t o t h e r a t i o o f t h e t o t a l e l a s t i c energy r e l e a s e d when t h e c r a c k and zone c o m p l e t e l y t r a v e r s e t h e f u l l c r o s s s e c t i o n ( i . e . U / U = cc  AADE/AABC) . 2  U  .  U i s given by,  +b ) ( I v ^ )  iro-^qW  2  2 (  4  i  )  &lt;e4ff&gt; V E  E  \u00B0crm  Because o f t h e h a l f e l l i p t i c shape, U v a l u e s c o r r e s p o n d t o t h e h a l f o f U  cc  i n tJne c a s e o f  e  ^ ^ i p t i c shape c r a c k ( n e g l e c t i n g t h e e f f e c t o f t h e f r e e  surface o f the h a l f plane). Substituting 7/11\u20141/  U  0^  rrr]]  2x..  2  2  2  i  2 ~  iW-wJ+f-  - Ti \"- Vfr Ff ^) 8crc r m &lt;Ec ~ 8 c 7  E  V  E  L  c  ^IF+VU J  (42)  \u000C75  From Eqn.20,  W +U)/do= o , d i f f e r e n t i a t i n g Eqn. 28 and 42 by a,  d (  z  , which i s the s t r e s s i n f i b e r s o u t s i d e o f t h e f a i l u r e zone a t t h e a c c e l e r a t i n g p o i n t , i . e . the U.T.S. p o i n t o f t h e specimen, i s g i v e n by the f o l l o w i n g e q u a t i o n . 0-3+ I-*ZJh.  .+  ai  3T(l-i/?)  / 3o T w  4o T w  (2a  3o bLxl  2  V  uts  1  b 2ob x T^w T t w  3  2  f 2  u  T w  T w)  uts  uts  377(1-1/2) 4  where,  ^  5t\u00B0\"m+ ^  V  f  a  f  o  (  f  |  _ ^  )  \u00B0crm = \u00B0futs f + v  c  *  rm  _ m  nj^futs\"  S  v  f  2  2  m  Ofut s r  -^V E )  f  _2oi_  2  cr*  f u t s  c  3  ( E c  30  \"  f, V b o *, _ b _ 3  \\%~  E f  u?l  f  S  2  Tuts  \"fats  f  37T(l-v2) E Q2 (E ~l\\fE ) cr  2  2/  uts  E  0-  c  m  +  V  f  2  2 2/ o  3  ^  ^  (  OmVm  \u2022  ob  + \u2014  r  a.  )  +  V  m  0  ?  ;  j  , ,  2  2  ) +  (44)  (45)  The s t r e n g t h o f the c o m p o s i t e s can be c a l c u l a t e d from Eqn. 44, u s i n g t h e cr  fQ  v a l u e s o b t a i n e d from Eqn.  43.  \u000C76  IV-2-iii.  Improvement o f T e n s i l e S t r e n g t h i n B u n d l e S t r u c t u r e C o m p o s i t e s  G r a p h i t e f i b e r r e i n f o r c e d 7178 T6 age hardened a l l o y c o m p o s i t e s have non-uniform,  which  bundle s t r u c t u r e , f i b e r d i s t r i b u t i o n i n the m a t r i x as  shown i n F i g . 15 e x h i b i t e d h i g h e r u l t i m a t e t e n s i l e s t r e n g t h t h a n u n i f o r m l y d i s t r i b u t e d composites.  The r e s u l t was shown i n F i g . 22.  The i n c r e a s e o f  the s t r e n g t h may be a t t r i b u t e d t o the e x i s t e n c e o f t h e f i b e r f r e e r e g i o n s between f i b e r c o n c e n t r a t e d  bundles.  These f i b e r f r e e r e g i o n s can work as o b s t a c l e s a g a i n s t the o f the f i b e r f a i l u r e zone through two p o s s i b l e mechanisms.  propagation  The t h i c k n e s s o f  t h e h a l f e l l i p t i c f i b e r f a i l u r e zone may be e n l a r g e d from 2b t o 2h a t t h i s f i b e r f r e e r e g i o n consuming e x t r a energy as shown i n F i g . 46.  In a d d i t i o n ,  such f i b e r f r e e zone may a l s o a c t t o i n c r e a s e t h e c u r v a t u r e o f t h e  advancing  h a l f e l l i p t i c z o n e , t h u s d e c r e a s i n g t h e s t r e s s c o n c e n t r a t i o n a t the t i p . In the p r e s e n t c a s e , o n l y t h e f i r s t p o i n t w i l l be d i s c u s s e d . L e t the a v e r a g e f i b e r volume f r a c t i o n be  t h e a v e r a g e m a t r i x volume  f r a c t i o n V , the f i b e r f r a c t i o n i n a b u n d l e , v , the m a t r i x volume f r a c t i o n m  f  i n a b u n d l e , v , and the t o t a l number o f bundle g r o u p s , G, i n t h e c r o s s m  s e c t i o n o f w i d t h W and u n i t t h i c k n e s s .  Then we can o b t a i n the f o l l o w i n g  e x p r e s s i o n s from the g e o m e t r i c r e l a t i o n s shown i n F i g . 46. :  ;\u2022  v  V  f f  = K  n  2  (46)  ^  The number o f groups w h i c h t h e h a l f e l l i p t i c zone sweeps, G , z  i s g i v e n by  \u000C77  Fig.47. Schematic S t r e s s S t r a i n Curve f o r a Composite Showing E n e r g y R e l a t i o n s h i p s i n Eqn. 56.  \u2022  &lt;x\u2014-4  F i g . 4 6 . Geometry o f t h e F i b e r F a i l u r e Zone i n t h e Bundle S t r u c t u r e Composite D : t h i c k n e s s o f a hexagonal group d : thickness o f a f i b e r concentrated area 2b: t h i c k n e s s ; o f t h e zone ( s h o r t d i a m e t e r o f t h e h a l f e l l i p t i c zone) a : advanced d i s t a n c e o f t h e t i p o f t h e zone \u2022' ( l e n g t h o f t h e h a l f e l l i p t i c zone) 2h: t h i c k n e s s o f t h e zone i n f i b e r f r e e r e g i o n  \u000C78  G  z=-w-  (48)  G  The a r e a o f a g r o u p , A, i s g i v e n by A = ^ 8  D  2  (49)  Then D  (50)  2  The work p e r u n i t t h i c k n e s s , W g, m  necessary to deform the matrix  i n t h e b u n d l e s and f i b e r f r e e z o n e s f r o m f o l l o w i n g forms.  \u20AC t o \u20AC . i s d e r i v e d i n the o tuts \u00A3  I t i s t h o u g h t t h a t t h e m a t r i x i n t h e f i b e r f r e e zone can  be deformed, on a v e r a g e , up t o t h e f i b e r f a i l u r e s t r a i n ,  \u20AC  f  u t s  &gt; i n the  d i s t a n c e o f (h - b) from t h e edge o f t h e f i b e r f a i l u r e z o n e .  ._^!C3L| ^_v J  m  +  2(h-b)( -^)}  ,5,)  l  S u b s t i t u t i n g f o r w\" i n Eqn. 28 by t h i s \"W  , an e x p r e s s i o n f o r t h e  m  f r e e e n e r g y i n c r e m e n t i n t h e f i b e r f a i l u r e zone f o r a b u n d l e s t r u c t u r e composite i s o b t a i n e d . distribution of fibers).  (This corresponds to W f o r the case of a uniform z  Then, t h e f o l l o w i n g e x p r e s s i o n f o r c r  the c r i t i c a l c o n d i t i o n o f zone l e n g t h , o , i s o b t a i n e d .  f o  at  \u000C79  2JL_  T ts  b  T  w 2  u  377tl-z/ ) 2  4  2  (  :  2  u t s  Ob\"  w ' T  u t s  w  2  E f _ (Ec-^VfEf)  Qfuts^s, 3Q ut w  E -^-v E )  cr  c  f  f  4 a T w  2  f l T  S  2 c r  f m  m V  T  3  2  u t s  T  u t S  u t s  b T w  5  uts  w  2a b j &gt; T w 2 )  +  u t s  Tutsw'/^fo 3 TT(\\-V ) 2  4  2  K2  The s t r e n g t h o f b u n d l e s t r u c t u r e c o m p o s i t e s , r r . , c a n be c a l c u l a t e d from t h e f o l l o w i n g e q u a t i o n , u s i n g t h i s  *c  ~-  V C7 /1--\u00A7-)+ f  f&lt;  cr  f o  v a l u e s i n Eqn. 53.  Vf-^ti (-\u00A3\u2022) +  V cr * m  m  (54)  \u000C80  IV-2-iv.  E s t i m a t i o n o f V a r i a b l e s , b and T  uts  An a t t e m p t was made t o e s t i m a t e t h e a p p r o x i m a t e v a l u e s o f t h e t h e o r e t i c a l v a r i a b l e s , f i b e r f a i l u r e zone t h i c k n e s s , 2b, and s t r e s s t r a n s f e r l e n g t h i n t h e zone, T ^ / 2 ,  using scanning electron microscope  images o f t h e l o n g i t u d i n a l s e c t i o n . The T T  uts  d e  P  u t s  e n d s  o n  v a l u e s a r e i n d e p e n d e n t o f t h e f i b e r volume f r a c t i o n , V^. ly  on  tne  f i b e r - m a t r i x i n t e r f a c i a l s h e a r s t r e n g t h . The T  u t s  v a l u e can be e s t i m a t e d as a s l i g h t l y l a r g e r v a l u e than t h e s h o r t e s t l e n g t h of fragments n e a r  t h e f r a c t u r e s u r f a c e i n t h e h i g h e r V^ c o m p o s i t e s .  In  the c a s e o f t h e l o w e r V^ c o m p o s i t e s , t h e broken f i b e r segments a r e t i l t e d near  t h e f r a c t u r e s u r f a c e o f specimens as shown i n F i g . 3 0 ( a ) .  It is  more d i f f i c u l t t o o b t a i n t h e v a l u e s f r o m t h e m i c r o g r a p h s o f l o w e r V mens t h a n t h e h i g h e r V^ ones. tilted, fiber  f  speci-  The h i g h e r V^ c o m p o s i t e s have s t r a i g h t , non-  segments.  The b v a l u e s c a n n o t be o b t a i n e d d i r e c t l y from t h e m i c r o g r a p h s ,  b value  can be d e f i n e d as b = n T ^. (where n: t h e a v e r a g e number o f segments o f a broken f i b e r near t h e f r a c t u r e s u r f a c e ) . depend  The n v a l u e s a r e not c o n s t a n t and  on V^; however, i t i s a l s o d i f f i c u l t t o e s t i m a t e n v a l u e s i n t h e  m i c r o g r a p h s o f l o w e r V^. s p e c i m e n s , a g a i n because o f t h e same r e a s o n .  Conse-  q u e n t l y , f o r lower V^ s p e c i m e n s , t h e b v a l u e s c o u l d not be o b t a i n e d from t h e micrographs.  A t r e a t m e n t to o b t a i n b v a l u e s f o r a l l V  f  values using the b  v a l u e s o f t h e h i g h e r V^ c o m p o s i t e s i s d i s c u s s e d . As mentioned b e f o r e i n s e c t i o n I I I - 6 , t h e d i s t a n c e B^, between t h e f r a c t u r e s u r f a c e and t h e f a r t h e s t p o i n t , a t which broken f i b e r segments c o u l d be o b s e r v e d , was measured.  The r e l a t i o n between B  f  and V  f  and t h e Young's  \u000C81  modulus were examined and g i v e n by t h e f o l l o w i n g e x p r e s s i o n =K  2 B  E  \u00A3 'f fmin  (55)  ?  _,/  where, K: c o n s t a n t V  fnil  - i s t h e minimum f i b e r volume f r a c t i o n which must be exceeded n  s t r e n g t h e n t h e composite and e x p r e s s e d i n Eqn. 7.  Combining  to  Eqn. 55 and  7, the f o l l o w i n g e x p r e s s i o n i s o b t a i n e d .  (o  c- m\u00B0\"muts^^ K , 2 B = \u2014 ( cr - c r V  f  f  m C l t s  - o- ) m  (56)  I f we assume cr^* i s c o n s t a n t , t h e r i g h t hand s i d e o f t h i s e q u a t i o n i s constant.  T h i s means t h a t t h e e l a s t i c energy n e c e s s a r y t o i n c r e a s e t h e  s t r e s s i n t h e specimen from V cr t m  i s always c o n s t a n t . in Fig.  s  to the u l t i m a t e s t r e n g t h o f the  specimen  T h i s energy i s s c h e m a t i c a l l y shown as t h e hatched a r e a  47.  I f we assume t h e t h e o r e t i c a l a v e r a g e b v a l u e v a r i e s d i r e c t l y as B^, we can w r i t e the f o l l o w i n g e x p r e s s i o n  b  = \u00A3' 2  where, K':  E  c  ( V W  (57) 2  constant  The p r o p o r t i o n a l c o n s t a n t K' i n t h i s e q u a t i o n can be c a l c u l a t e d u s i n g the b values obtained f o r the higher  composites.  Then we can c a l c u l a t e  \u000Cb values f o r d i f f e r e n t V  f  u s i n g t h e K' c o n s t a n t .  The r e s u l t o f K'  c a l c u l a t i o n i s t a b u l a t e d i n T a b l e 11.  T a b l e 11.  Observed T  u t s  V a l u e s and C a l c u l a t i o n R e s u l t s o f K'  Specimen  V  601T4 #17  0.145  7 x 10\"  201T6 #31  0.17  2.5 x 1 0 \"  7178T6 #38  0.138  2.0 x 1 0 \"  f  inch  n  b inch  2.0  14.0 x l O \"  3  1.6  4.05 x 1 0 \"  3  2.4  4.8 x 1 0 \"  3  K (1 b \" i n c h ) 1  3  8.74  3  3  1  xlO\"  3  1 2  4.17 x 1 0 \"  1 2  1.98 x 1 0 \"  1 2  \u000C83  IV-2-v.  E v a l u a t i o n o f U l t i m a t e T e n s i l e S t r e n g t h o f Composites  by  P r o p a g a t i v e F i b e r F a i l u r e Model  The f i b e r f a i l u r e zones s t a r t t o grow a t t h e s t r e s s c o n c e n t r a t e d a r e a s or t h e d e f e c t i v e a r e a s i n t h e specimen.  They keep growing l a r g e r  g r a d u a l l y u s i n g the energy which i s p r o v i d e d from t h e a p p l i e d l o a d u n t i l one o f them becomes l a r g e enough t o s a t i s f y t h e energy c r i t e r i o n o b t a i n e d i n f o r m e r s e c t i o n s . Then t h e f i b e r f a i l u r e i s a c c e l e r a t e d and t h e s t r e n g t h o f t h e c o m p o s i t e d e c r e a s e s q u i c k l y . Such a p o i n t must c o r r e s p o n d t o t h e u l t i m a t e s t r e n g t h . o f the  specimen.  An attempt was made t o c a l c u l a t e t h e u l t i m a t e t e n s i l e s t r e n g t h o f c o m p o s i t e s from t h e e q u a t i o n s , w i t h t h e e s t i m a t e d v a l u e s o f zone t h i c k n e s s , b, and a v e r a g e t r a n s f e r l e n g t h T  u t s  i n t h e l a s t s e c t i o n . By f i x i n g t h e r a t i o  o f t h e zone l e n g t h , a \u00BB and t h e specimen w i d t h , w, a t 0, 0.03, 0.05 and  0.10,  t h e s t r e n g t h cr o f 601 T4 and 201 T6 c o m p o s i t e s were c a l c u l a t e d . In t h e case o f 7178 T6 c o m p o s i t e s , t h e \u00B0 / calculation.  Table 12\u201415  w  r a t i o s , 0, 0.03 were adopted f o r t h e  show n u m e r i c a l v a l u e s o f t h e parameters which were  nfecessary f o r t h e c a l c u l a t i o n s and the c a l c u l a t e d r e s u l t s . T h e s e r e s u l t s were superimposed  on t h e e x p e r i m e n t a l d a t a i n F i g . 20 \u2014 22.  The t h e o r e t i c a l c u r v e s r e s u l t i n g from the p r e s e n t model a r e i n v e r y much b e t t e r agreement w i t h t h e e x p e r i m e n t a l r e s u l t s t h a n i s t h e c a s e f o r t h e \" r u l e o f mixture\" curve, i n s p i t e of the considerable s c a t t e r of data. E s p e c i a l l y i n the c a s e o f t h e bundle s t r u c t u r e c o m p o s i t e s o f 7178 T6 a l l o y , the  improve-  ment c a n n o t be e x p l a i n e d by t h e \" r u l e o f m i x t u r e \" i n which t h e volume f r a c t i o n i s t h e o n l y one s t r e n g t h c o n t r o l l i n g parameter. The d i f f e r e n c e among e x p e r i m e n t a l d a t a r e p o r t e d by former workers  might  \u000CT a b l e 12. C a l c u l a t i o n o f S t r e n g t h o f 601T4 Composites  \u00B0/  = 0.17 w  Op  0 0.03 0.05 0.10  238.0 185.3 150.6 116.1 6  c  W P 1 O 3  / T  crksi  ksi  E xl0 psi S 1  uts  /w  V  f \" 0.15  CT ksi  244.5 204.9 183.4 145.0  V = 0.06  f  crksi  fo  55.4 46.3 42.15 35.3  V = 0.10 CJ ksi fo  52.0 45.8 42.6 37.1  261.0 251.1 244.6 230.1  f  c^ksi  Cf ksi  crksi  Q  42.3 41.0 40.2 38.4  273.0 272.5 272.1 272.0  33.3 33.0 32.4 32.0  16.8  16.0  14.0  12.4  65.9  60.3  46.2  34.9  0.67  0.90  2.85  (19.69V  0.02  0.028  0.089  (0.613*)  E = 50 x 1 0 p . s . i . , c r * = 18 x 1 0 p . s . i . 6  f  3  K' = 8.74 x 1 0 \"  1 2  inch  E = 10 x 1 0 p . s . i . cr = 32 p.s.i. 6  m  3  m u t s  3  Ib  x  _ 1  1 0  V  f m i n  w = 0.225 i n c h c r ^ = T = 7.0 x 1 0 i n c h  3 0 0  x 10  3  p.s.i.  _ 3  u t s  = 0.0446  t  ; g a u g e l e n g t h of t h e s p e c i m e n .  \u000CT a b l e 13. C a l c u l a t i o n o f S t r e n g t h o f 20TT6 Composites  V  f  =  0.15  \u00B0/  crksi  0 0.03 0.05 0.10  262.1 188.0 151.5 82.4  E xl0\u00B0psi c  b  /Tuts  f \" 0.125 CT ksi crksi  V  V  80.1 68.8 63.5 54.2  cr ksi  fo  269.2 226.3 201.8 150.8  f = 0.10  f o  75.6 70.0 66.9 60.8  f \" 0.06 cr ksi crksi V  cr ksi c  276.0 257.1 245.6 219.4  f o  70.8 68.6 67.3 64.4  285.4 281.0 279.3 281.6  62.2 61.8 61.5 61.2  16  15  14  12.5  85.8  79.5  73.2  63.1  1.07  1.67  3.09  22.37 0.249  0.012  0.0186 0.034 E = 50 x 1 0 p s i , E = 10 x 1 0 p s i , w = 0.225 i n c h cr ^ = Psj. C7 = 60 x 103 p s i , T = 2.5 x 1 0 - 3 ^ K' = 4.17 x 10\"'^ i n c h 3 lb-1 V = 0.0385 6  f  6  m  4 8  x  =  3 0 0  x  1  3 0  psi  10  m u t s  u t s  f m i n  00  \u000CT a b l e 14. C a l c u l a t i o n o f S t r e n g t h o f 7178 T6 Composites  f - 0.15 3 3 O- xl0\u00B0psi cTxlCTpsi v  u  /  x  w  fo  0 0.03  263.5 174.3  V  .  f  0.125  =  ;  3 crxlO psi  ^xlO^si  87.1 73.6  0.10 V 3 cr xl0 psi crxl0 psi  271.6 221.3  3  f o  83.0 76.4  278.7 261.6  78.3 76.2  E xl0 psi  16  15  14  crm x l 0 p s i  92.6  86.5  80.4  /Tuts  0.94  1.66  3.95  0.0084  0.015  0.035  6  c  3  r  b  b  /  E  = 50x10 p s i 6  f  \u00B0\"m* = 5 6 x l 0 p s i 3  V  f  \u2022 = 0.0581  Em = l O x l O p s i  w = 0.225 i n c h  6  cr  = 74.5x10 p s i  T  3  m u t s  K' = 1 . 9 8 x l 0 \"  1 2  inch lb\" 3  \u00B0 f u t s 300x10 = 2.0xl.0\" inch =  3  psi  3  u t s  1  ca  \u000CT a b l e 15. C a l c u l a t i o n o f S t r e n g t h o f 7178 T6 Bundle S t r u c t u r e Composites  V = 0.19  V  f  \u00B0/  7  &lt;^ xl0 psi  CTxIO p s i  0 0.03  281.6 203.6  98.9 83.7  o  E xl0 psi  V = 0.10 f  crxlO  Cr Xl0 psi 3  o  289.3 258.2  3  psi  3 crxlO psi  cr xl0 psi 3  Q  91.0 85.8  284.0 271.0  78.8 77.1  17.6  16  14  102.4  92.6  80.4  /Tuts  \u00B0-  0.94  3.95  b/  0.000225  0.0084  0.0035  6  c  ^  7  w  7  0.15  xl0 psi 3  c r m  b  w  E = 50xl0 psi 6  T ^ D  w =  c r * = 56xl0 psi  \"kits  m  = 300x10 psi 3  u t s  E = 10x10 psi 6  f  Cf  5 0  = 2xl0\" inch  3  3  u t s  = 0.85*  V f m  .  n  = 0.0581  0.225 i n c h =  74.5xl0 psi 3  K' = 1 . 9 8 x l 0 \" i n c h l b 1 2  3  _ 1  h= 1 2 x 1 0 ~ i n c h 3  CO  (  +  T h i s v a l u e was e s t i m a t e d from F i g . 15)  \u000C88  be a l s o u n d e r s t o o d by t h i s t h e o r y .  The c o m p o s i t e s f a b r i c a t e d by t h e  i n f i l t r a t i o n t e c h n i q u e showed much h i g h e r f i b e r s t r e n g t h e n i n g e f f i c i e n c y than by t h e o t h e r f a b r i c a t i o n t e c h n i q u e s , such as t h e c h e m i c a l vapour  depos-  i t i o n method.  ore  The c o m p o s i t e s f a b r i c a t e d by t h e i n f i l t r a t i o n p r o c e s s  thought to possess the bundle s t r u c t u r e d i s t r i b u t i o n i n e v i t a b l y . I t was found t h a t n o t o n l y t h e f a c t o r , V , used i n t h e \" r u l e o f f  m i x t u r e \" b u t a l s o o t h e r new f a c t o r s , such as E , t h e f i b e r d i s t r i b u t i o n , c  and t h e zone t h i c k n e s s , 2b,  have an i n f l u e n c e on t h e s t r e n g t h o f c o m p o s i t e s .  The 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 o f t h e m a t r i x , t h e f i b e r d i a m e t e r , and t h e f i b e r s p a c i n g a r e t h o u g h t t o c o n t r o l t h e zone t h i c k n e s s , 2b. L i s t e d below a r e some problems t h a t s t i l l r e m a i n , c o n c e r n i n g t h e p r e s e n t model. 1) The s h e a r s t r e s s , T , was assumed t o have t h e same v a l u e i n t h e r e g i o n o u t s i d e and i n s i d e o f t h e z o n e , because t h e work h a r d e n i n g e f f e c t o f t h e m a t r i x was n e g l e c t e d . The a c t u a l s t r e s s s t r a i n c u r v e o f t h e m a t r i x a l l o y does show work h a r d e n i n g . 2) The p r e s e n t e q u a t i o n s were o b t a i n e d , assuming t h e f i b e r f a i l u r e zone has a h a l f e l l i p t i c shape.  T h i s e q u a t i o n c a n n o t be a p p l i e d when t h e h a l f zone  thickness i s larger than the length ( b &gt; a ) . values are i n c o r r e c t .  The c a l c u l a t e d c u r v e s f o r l o w e r  These p a r t s a r e shown i n d o t t e d l i n e s .  3) Some a m b i g u i t y e x i s t s i n t h e method t o o b t a i n a v e r a g e t r a n s f e r l e n g t h , a v e r a g e number o f f r a g m e n t s and t h e zone t h i c k n e s s . 4 ) We c a n n o t o b t a i n t h e s t r e n g t h o f c o m p o s i t e s f r o m t h i s model u n c o n d i t i o n a l l y as l o n g as t h e e x a c t v a l u e s o f t h e zone l e n g t h , a \u00BB a t t h e a c c e l e r a t i n g p o i n t a r e not known.  \u000C89  IV-3.  C h a r a c t e r i s t i c s o f Powder S l i p I n t e r p r e t a t i o n Method  I t has been v e r y d i f f i c u l t t o f a b r i c a t e g r a p h i t e f i b e r r e i n f o r c e d metal c o m p o s i t e s b e c a u s e o f t h e s m a l l f i b e r d i a m e t e r .  I n t h e p r e s e n t work,  a v e r y u n i q u e p r o c e s s was d e v e l o p e d i n o r d e r t o make u n i f o r m l y d i s t r i b u t e d f i b e r c o m p o s i t e s ; however, some problems s t i l l r e m a i n .  The f o l l o w i n g c h a r -  a c t e r i s t i c s o f t h i s p r o c e s s c a n be p o i n t e d o u t . 1) Blended powders o f d i f f e r e n t e l e m e n t s c a u s e t h e s c a t t e r o f m a t r i x comp o s i t i o n i n t h e g r e e n c o m p o s i t e s , because each element powder has a d i f f e r e n t s e t t l i n g speed i n t h e p r o c e s s .  I f a l l o y powders c a n be u s e d , t h i s p r o b l e m  w i l l be s o l v e d . 2) E f f e c t i v e i n t e r p e n e t r a t i o n i s dependent on t h e r e l a t i v e s i z e s o f f i b e r s and powder p a r t i c l e s .  The more n e a r l y equal t h e s e d i m e n s i o n s a r e , t h e  h i g h e r t h e volume f r a c t i o n t h a t c a n be made. 3) T h i s p r o c e s s c a n be a p p l i e d t o f a b r i c a t e c o m p o s i t e s o f many o t h e r f i b e r metal systems because t h e w e t t i n g p r o p e r t y o f t h e two components  i s not  necessary f o r t h i s process. 4) T h i s p r o c e s s i s more s u i t a b l e t o p r o d u c e c o m p o s i t e s o f l a r g e s t r u c t u r a l components  than a r e t h e o t h e r p r o c e s s e s such a s i n f i l t r a t i o n and c o a t i n g ,  because t h e s i z e o f a f i b e r b u n d l e which i s p r e p a r e d p r i o r t o t h e o p e r a t i o n o f t h i s p r o c e s s , c a n be i n c r e a s e d w i t h o u t a n y p r o b l e m s .  In t h e case o f t h e  i n f i l t r a t i o n and c o a t i n g p r o c e s s e s , t h e s i z e o f a bundle f o r the o p e r a t i o n is limited.  The c o m p o s i t e w i r e s have t o be f i r s t f a b r i c a t e d p r i o r t o h o t  pressing t o f a b r i c a t e t h e composite structure. 5) On t h e o t h e r hand, i t i s d i f f i c u l t t o f a b r i c a t e t h e c o m p o s i t e s o f b u n d l e s t r u c t u r e w h i c h c a n be e x p e c t e d t o have h i g h e r s t r e n g t h t h a n u n i f o r m l y  \u000C90  d i s t r i b u t e d composites according to the present theory.  The  infiltration  p r o c e s s seems to have a g r e a t e r advantage t h a n t h e p r e s e n t p r o c e s s from t h i s p o i n t o f view.  \u000C91 V  SUMMARY AND CONCLUSIONS  1) A new f a b r i c a t i o n t e c h n i q u e has been d e v e l o p e d t o f a b r i c a t e g r a p h i t e f i b e r r e i n f o r c e d aluminum a l l o y c o m p o s i t e s o f 5% \u2014 17% f i b e r volume f r a c t i o n s , u s i n g metal powder s l i p a n d c o n t i n u o u s g r a p h i t e f i b e r s . 2) The s t r e n g t h o f t h e s e c o m p o s i t e s i s l o w e r than t h e \" r u l e o f m i x t u r e \" v a l u e as o t h e r workers have r e p o r t e d p r e v i o u s l y . 3) A \" f i b e r f a i l u r e zone\" p r o p a g a t i o n model has been proposed and v e r i f i e d by o b s e r v a t i o n s o f f i b e r f r a c t u r e b e h a v i o u r . 4) An e n e r g y c r i t e r i o n has been f o r m u l a t e d f o r t h e a c c e l e r a t e d p r o p a g a t i o n o f t h e zone i n t h e specimen, a n d t h e u l t i m a t e t e n s i l e s t r e n g t h o f t h e s e composites i s thought to correspond t o t h i s a c c e l e r a t i n g p o i n t .  The e x p e r -  i m e n t a l s t r e n g t h d a t a shows b e t t e r c o r r e l a t i o n w i t h t h i s p r o p a g a t i v e f i b e r f a i l u r e model t h a n t h e \" r u l e o f m i x t u r e \" model. 5) T h i s model s u g g e s t s t h e p o s s i b i l i t y o f i m p r o v i n g t h e s t r e n g t h by producing a bundle s t r u c t u r e f i b e r d i s t r i b u t i o n i n t h e composite i n s t e a d o f a uniform f i b e r d i s t r i b u t i o n . 6) T h i s model shows t h a t , i n a d d i t i o n t o f i b e r volume f r a c t i o n , t h e zone t h i c k n e s s i s a l s o a s t r e n g t h c o n t r o l l i n g f a c t o r . The zone t h i c k n e s s seems t o be r e l a t e d i n d i r e c t l y w i t h o t h e r f a c t o r s such a s t h e f i b e r d i a m e t e r , f i b e r s p a c i n g , d e f o r m a t i o n p r o p e r t y o f t h e m a t r i x , a n d t h e m a t r i x - f i b e r bond strength.  \u000C92  VI  SUGGESTION FOR FUTURE WORK  Some l i n e s o f f u t u r e i n v e s t i g a t i o n can be s u g g e s t e d f r o m t h e d i s c u s s i o n o f the p r e s e n t work. 1) In o r d e r t o o b t a i n t h e a v e r a g e zone t h i c k n e s s p r o p e r l y , t h e d i s t r i b u t i o n o f s t r a i n i n t h e m a t r i x has t o be measured. 2) A more e x a c t m a t h e m a t i c a l t r e a t m e n t f o r t h e e n e r g y change due t o t h e zone f o r m a t i o n has t o be performed. 3) A method t o o b t a i n t h e e x a c t a v e r a g e l e n g t h s o f f i b e r f r a g m e n t s i n t h e zone has t o be d e v e l o p e d . 4) The zone l e n g t h change d u r i n g t h e t e n s i l e t e s t has t o be measured i n o r d e r t o know t h e f a c t o r s which i n f l u e n c e i t .  \u000CREFERENCES  1. 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