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Numericalmodelsfo rnatural fibre compos i teswiths toch ast ic properties

b y

A m a n d e e p S i n g h V i r k

A t hes i s submi t t ed t o t he Un i ve rs i t y o f P l ymou t hin part ial fulfilment f o r the degree o f

D o c t o r o f P h i l o s o p h y

S c h o o l o f M a r i n e S c i e n c e a n d E n g i n e e r i n g

F a c u l t y o f S c i e n c e a n d T e c h n o l o g y

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This copy of the thesis has been supplied on condition that anyone who consults it isunderstood to recognise that its copyright rests with its author and that no quotationfrom the thesis and no information derived from it may be published without theauthor s prior written consent.

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AmandeepSingh YirkNumerical modelsfor natural fibrecomposites withstochast ic propertiesAbstract

Nat u ra l fibres are increasingly being considered as the reinforcement for polymermatrix composites as they are perceived to be sustainable being a renewabler e s o u r c e . However, they suffer from higher variability in mechanical properties andc o n c e r n s about their long-term durability in a moist environment.

In this study the physical properties of the jute fibres were cha rac ter ise d, the fibrelength distribution was determined and the fibre cr oss -sect ion wa s analysed usingdigital images. It was observed that the true fibre area followed a log-normaldistribution. The fibre area distribution for different geometrical shapes wase s f i m a t e d an d the error in the estimated area of ass um ed fibre cross -se cti on wa sa l so determined to a s s e s s the applicability of the assumed cross-section.

T h e mechanical properties of thejute technical fibres from a single batch from SouthA s i a were determined; fibre tensile tests were carried out at ten different gaugelengths between 6 mm and 300 mm and the Young's modulus, strain to failure andultimate tensile strengths were determined individually. A strong correlafion waso b s e r v e d between the fibre strength/fracture strain and fibre gauge length. It wasfound as the gauge length inc reases the fibre strength/f racture strain drops. Th e fibrefailure (Strength/Strain) was modelled using Weibull distribufion and three stafisticalm o d e l s were deve loped to relate the fibre strength/ fracture strain to the fibre ga ugel ength . Examination of tensile test data revealsthat the coefficient of variafion (CoV)for failure strain is consistently lower than the C o V for fracture st ress (strength), asthe failure strain is weakly influenced by the fibre cross-secfion.H e n c e , failure strainis the more consistent failure chterion and it is recommended to use failure strain asthe key design criterion for natural fibre composites in order to improve reliability inthe design of these materials.

Different authors have tried to model natural fibre reinforced polymer elastic modulusus ing micromechanical models and have suggestedthatfurther study should includefibre a n g l e and length distribution factors to improve the micromechani cal predict ion.

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T h i s t h e s i s further s e e k s t o va l i da t e a nove l me t hodo logy f o r t he p red i c t i on o f t het ens i l e m od u lus and s t reng th o f na t u ra l f ib re co m po s i t e s t h rough ca r e f u lcon s i de ra t i on o f ea ch o f t he pa ra m e t e r s i n t he ru l e of m i x tu res a l ong withcons ide ra t i on o f t he s t a t i s t i ca l va r i a t i on i nhe ren t i n re i n f o rcemen t s ex t rac t ed f romp lan ts .

The t ens i l e modu lus and s t reng t h o f jute f i b r e r e i n f o r c e d c o m p o s i t e s m a n u f a c t u r e df r o m w e l l c h a r a c t e r i s e d f i b r e s w a s m e a s u r e d e x p e r i m e n t a l l y . S i x w e l l e s t a b l i s h e dm i c r o m e c h a n i c a l m o d e l s w e r e u s e d to p r e d i c t t h e c o m p o s i t e e l a s t i c m o d u l u s . T w om i c r o m e c h a n i c a l m o d e l s w e r e u s e d t o p r e d i c t c o m p o s i t e s t r e n g t h . F o r b o t hm e c h a n i c a l p rope r t i es , t he i nc l us i on o f a f i b re a rea co r rec t i on f ac t o r t o accoun t f o rt he non -c i r cu l a r c ro ss -se c t i o n o f t he f ib re res u l t ed i n an imp rove d p red i c t i on o f t her e s p e c t i v e m e c h a n i c a l p ro p e r t ie s .

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Table of Contents1. Introduction 12. Experiments 2

2.1. Fibre physical characterisation 32.1.1. Fibre length dis tribu tion 32.1.2. Fibre cross-sectionalarea distribution 3

2.2. Fibre mechanical properties 62.3. Statist ical strength and fracture strain distribut ion 9

2.3.1. Weibu ll point estimate 92.3.2. Weak-Li nk Scal ing Model (WLSM) 92.3.3. Multiple Data Set (MDS) Weak-link scal ing mod el 112.3.4. Empiri cal Models 11

2.4. Jute/Epoxy Composite 132.4.1. Dyeing Fibres 132.4.2. Comp ari so n of tensile properties of un-d yed and dyed fibres 132.4.3. Manufacturing of co mpo si te plate 142.4.4. Mechanical properties of the co mpo si tes 152.4.5. Micro-structural characterisation 18

2.5. Modelling 262.5.1. Micromechanical model 26

3 . Results 303.1. Fibre physical characterisation 30

3.1.1. Fibre length distr ibutio n 303.1.2. Fibre true cross-section distribution 32

3.2. Fibre mechanical properties 393.2.1. Fibre mo du lus , strength and fracture strain 39

3.3. Statistical strength and fracture strain distribut ion 423.3.1. Weibul l point estimate 423.3.2. Weak-Li nk Scal ing Model (WLSM) 463 . 3 . 3 . Multiple Data Set (MDS) Weak-link sc ali ng model 483.3.4. Empiri cal Models 50

3.4. Jute/Epoxy Composite 533.4.1. Mechanical properties 533.4.2. Micros truc tural characterisatio n 53

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3.5. Modelling 543.5.1. Fibre area co rrecti on factor 543.5.2. Fibre orientation factor 583.5.3. IVIicromechanical model 59

4. Discussion 654.1. Fibre physical characterisation 65

4.1.1. Fibrecross-sectionalarea 654.2. Fibre mechanical properties 67

4.2.1. Coeffici ent of Variation (CoV) in fibre str ength and fracture strain674.3. Statistical strength and fracture strain distribution 76

4.3.1. Goodnessof fit test to evaluate the fibre failure models 764.4. Jute/Epoxy Composi te 80

5. Conclusion 836. Future Work 85References 87Appendices 96

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L i s t o f A p p e n d i c e sAppendix - A (o n C D R o m )A 1 - C o m p o s i t e s P a r t A : A p p l i e d S c i e n c e a n d M a n u f a c t u r i n g , 2 0 1 0 , 4 1 ( 1 0 ) , 1 3 2 9 -

1 3 3 5 .A 2 - C o m p o s i t e s P a r t A : A p p l i e d S c i e n c e a n d M a n u f a c t u r i n g , 2 0 1 0 , 4 1 ( 1 0 ) , 1 3 3 6 -

1 3 4 4 .A 3 - Jou rna l o f Na t u ra l F i b res , 7 (3 ) , 20 10 , 21 6 -2 28 .A 4 - M a t e r i a l s S c i e n c e a n d T e c h n o l o g y , 2 0 1 0 , 2 6 ( x ), y y y - z z z .A 5 - C o m p o s i t e s S c i e n c e a n d T e c h n o l o g y , J u n e 2 0 1 0 , 7 0 ( 6) , 9 9 5 - 9 9 9 .A 6 - C o m p o s i t e s P a r t A : A p p l i e d S c i e n c e a n d M a n u f a c t u r in g , N o v e m b e r 2 0 0 9 ,

4 0 ( 1 1 ) , 1 7 6 4 - 1 7 7 1 .

A 7 - M a t e r i a l s S c i e n c e a n d T e c h n o l o g y , O c t o b e r 2 0 0 9 , 2 5 ( 1 0 ) , 1 2 8 9 - 1 2 9 5 .

Appendix - B81 - M ax im i se t he l i ke l ihood me t hod8 2 - C o n f i d e n c e b o u n d s o n t he W e i b u l l p a r a m e t e r s8 3 - An de r son -Da r l i ng (AD ) go od ne ss o f fit t e s t8 4 - Co ef f i c ien t o f va r ia t ion fo r W e ib u l l D is t r ibu t ion8 5 - Dye ingjute f i b res8 6 - S p ec im en t ens i l e t es t resu l t s8 7 - S p e c i m e n a x i a l P o i s s o n ' s r at io8 8 - M i c rog ra ph f ib re vo l u me f rac t ion8 9 - M i c rog ra ph f ib re ang le d i s t r ibu t i on pa ram e t e rs8 1 0 - F ib re or ien ta t ion fac tor ca lcu la te d f rom f ib re d is t r ibu t ion811 - We ibu l l p lo ts o f raw da ta8 12 - W e ib u l l d is t r ibu t ion l it e ra ture su rve y

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List o f FiguresFigu re 1 : Ju t e f i b res s l i ver 2F i g u r e 2 : T y p i c a l fibre c r o s s s e c t i o n s 3F i g u r e 3 : T h e m a s k u s e d t o identify true fibre sh ap e and a re a o f f ib res sh o w n i nF igu re 2 4F i g u re 4 : C o nvex H u l l o f truefibre sh a p e fo r the f i b res in F igu re 2 5F i g u re 5 : Th e five s h a p e sfitted to the co nv ex hu l l o f the f i b res sh ow n in 6F i g u re 6 : S i ng l efibre mo u n ted o n atest ca rd 7F i g u re 7 : R e s i n i n fu s i o n a r ran g em en t 1 4F i g u re 8 : R e s i n i n fu sed p l a te : (a ) Sa m p l e l o ca t i o n ; (b ) S ch em at i c of t ens i l e tests p e c i m e n 1 6F i g u re 9 : ( a ) Typ i ca l jute fibre r e i n f o rc e d c o m p o s i t e s t r e s s - s t r a i n c u r v e ; (b ) T y p i c a la x i a l a n d t r a n s v e r s e s t ra i n g a u g e m e a s u r e m e n t f o r a s p e c i m e n 1 8F i g u r e 1 0 : L o c a t i o n o f m i c r o s c o p y s a m p l e s from a t yp i ca l test s a m p l e 1 9F i g u r e 1 1 : S a m p l e m i c r o g r a p h : (a ) T y p i c a l fibre v o l u m e fraction sa m p l e ; ( b) T yp i ca lfibreorientation s a m p l e 2 0F igu re 12 : F ib re vo lum e f rac t ion : (a ) Intensity h i s to g ram o f t h e sa m p l e ; ( b) B i n a ryi ma g e to es t i ma te vo l u m e fraction fo r the im ag e a t F igu re 11 a 21F i g u re 1 3 : So f twa re g ene ra ted F i b re vo l u m e fraction i m a g e 2 2F i g u re 1 4 : Typ i ca l b i na ry i ma g e to es t i m a te fibre orientation d e r i v e d from F i g u re l i b

2 3F i g u re 1 5 : Sc h e m at i c r ep rese n ta t i o n o f fibre ang l e es t i ma t i o n p ro c ed u re 2 4F i g u re 1 6 : H i s to g ram o f es t i m a ted F i b re an g l e distribution 2 5F i g u re 1 7 : So f twa re g en e ra ted F i b re ang l e distribution i m a g e 2 5F igure 18 : F ib relength distribution f o r 7 0 0jute f i b res 31F i g u re 1 9 : T ru e fibre c r o s s - s e c t io n a l a r e a distribution 3 3F i g u r e 2 0 : L o g - n o r m a l plot o f the ar ea d i s t r i bu t ions (106 f i b res) 34F i g u r e 2 1 : E r r o r in t h e a r e a m e a s u r e m e n t b a s e d o n a s s u m e d s h a p e 3 5F i g u re 2 2 : E r ro r i n t h e a rea m ea su r em en t o f t h e 2 f i b res i n F i g u re 2 , b a se d o na s s u m e d s h a p e a n d a n g l e o f m e a s u r e m e n t 3 7F i g u re 2 3 : E rr o r in t h e a r e a m e a s u r e m e n t b a s e d o n a s s u m e d s h a p e a n d a ll t h em e a s u r e d p o in t s 3 8F i g u r e 2 4 : B o u n d s of t h e e r ro r in t h e a r e a m e a s u r e m e n t b a s e d o n a s s u m e d s h a p e 3 9F i g u r e 2 5 : T y p i c a ljutefibre s t r ess -s t r a i n cu r v es ( a t a s t r a in rate of 0.01 min'^) 4 0

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F i g u r e 26: Tensile property variations with length: (a) Yo ung' s modulus; (b) ultimatetensile strength; and (c) fracture strain 42F i g u r e27: Weak link sca ling predict ion for strength 47F i g u r e28: Weak link scal ing predict ion for fracture strain 47F i g u r e29 : M D S Wea k link sca ling predic tion for strength 49F i g u r e 30 : M D S Weak- link scaling prediction for fracture strain 49F i g u r e 31 : Tensil e strength: (a) Weibu ll modu lus versus fibre lengthPijnear=-0.00196xibre ength+ 2.141 , =-OA 52ln fibre ength) + 3 AS7,Aff l is=300 (b) Characterist ic stress vers us fibre length

7 ^ ^ , = - 1 . 1 6 6 xi bre ength+ 4S1.S26 , T ^ - \ U.656\ n{fi brelength)+ m.63,71^^ \ 2A 3l^ ^ fibrelength 51F i g u r e 32: Fracture strain: (a) Weibu ll modulus versus fibre length 52F i g u r e 33: Apparen t fibre area distribution and True fibre are a distribution 55F i g u r e 34: Modi fied fibre area distribution 56F i g u r e 35: Experimental and predicted composite tensile modulus for individuals a m p l e s (solid points) and plate average (open points) 61F i g u r e 36: Experimental and predicted composite tensile Strength for individuals a m p l e s (solid points) and plate average (open points) 64F i g u r e 37: (a) Fibre strength against fibre diameter, (b) Fibre strength against meanfibre length 69F i g u r e 38: (a), Fibre fracture strain against fibre diameter, (b) Fibre fracture strainagainst mean fibre length 70F i g u r e 39: Strength and fracture strain C o V against fibre length 71F i g u r e 40: (a) Fibre strength against fibre diameter group, (b) Fracture strain againstfibre diameter group 73F i g u r e41: Strength and fracture strain C o V against fibre diameter 74

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L i s t o f TablesT a b l e 2 : M e a n Yo un g ' s m odu lus , f ib r e s tr eng th an d f r ac tu re s t r a i n ( s t and ar ddev ia t i ons i n pa r e n the ses ) 13T a b l e 3 : Sp ec i f i ed f ib r e ang le d i s t ri bu t ion f o r t he ima ge an d es t ima te d an g led i s tr ibu t ion by ang le me asu r em en t a lgo r i t hm 26T a b l e 4 : G e o m e t r i c M e a n f i b r e A r e a , G e o m e t r i c S t a n d a r d D e v i a t i o n a n d C o n f i d e n c ein terva l bou nd 32T a b l e 6 : Tw o -pa r am ete r W e ib u l l d i s t r ibu t i on pa r a me te r s f o r t ens i l e s t r eng th 4 3T a b l e 7 : Tw o -pa r am ete r W e ib u l l d i s t ri bu t ion pa r am ete r s f o r f r ac tu r e s t ra i n 44T a b l e 8 : Tw o pa r am ete r We ibu l l d i s t ri bu ti on pa r am ete r s : con f i den ce bound f o rs t reng th 45T a b l e 9 : Tw o pa r am ete r We ibu l l d i s t ri bu ti on pa r a me te r s : con f i den ce boun d f o rf rac tu re s t ra in 4 5T a b l e 10 : M D S we ak l ink W e ibu l l es t im a te s f o r s t r eng th 48T a b l e 11 : M D S we ak l ink W e ibu l l es t im a te s f o r f r ac tu r e s t r a in 48T a b l e 12 : Tens i le tes t resu l ts fo rjute fib re r e i n fo r ced Co m po s i t e 53T a b l e 13 : F ib r e vo lu me f r ac t ion a nd F ib r e An g le d is t r ibu f i on es f ima te d us in gm i c r o g r a p h s 5 4T a b l e 14 : F ib re o r ienta t ion d is t r ibu f ion fac to r 59T a b l e 15 : Ma t r ix an d F ib re e la s f ic prop er t ies 5 9T a b l e 16 : Su m m ar y o f da ta f rom exp er im en ts use d i n t he p r ed i c f i on and va l i da f i ono f t he m i c r o m e c h a n i c a l m o d e l s 6 0T a b l e 1 7 : P r e d i c t e d c o m p o s i t e m o d u l u s 6 2T a b l e 18 : E r r o r i n t he p r ed i c ted com po s i t e mo du lu s 62T a b l e 19 : W e ib u l l d i s t ri bu fi on s t r eng th pa r am ete r s 63T a b l e 20 : P r ed i c ted com po s i t e s t r eng th 63T a b l e 21 : E r r o r i n p r ed i c ted com po s i t e s t r eng th 65T a b l e 2 2 : M e a n , S tand ar d Dev ia f i on and C o V o f Yo un g ' s mo du l i , fib re s t r eng th an dfracture st rain at d i f ferent f ibre leng th 6 8T a b l e 2 3 : M e a n , S tand ar d dev ia f i on and C o V o f Yo un g ' s mod u l i , fib re s t r eng th an dfracture st rain at d i f ferent f ibre dia m ete r 72T a b l e 24 : Ty p ic a l C o V fo r m od u lus , s t reng th an d s t ra in to fa i lu re f rom the l i te ra tu re

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T a b l e 2 5 : A n d e r s o n - D a r l i n g G O F N f o r s t reng t h and f rac t u re s t ra i n (P o in t es t ima t es )7 6

T a b l e 2 6 : A n d e r s o n - D a r l i n g G O F N f o r s t reng t h a nd frac t u re s t ra in (W eak - l i n ksca l i ng ) 77T a b l e 2 7 : A n d e r s o n - D a r l i n g g o o d n e s s o f fit n u m b e r ( G O F N ) f o r We ibu l l t ens i l es t reng th an d f rac ture s t ra in a t d i f fe ren t f i b re leng th (E m pi r i c a l m od e l ) 79Tab le 28 : Repor t ed e l as t i c p rope r t i es o fjute f ib re re i n f o rced co m po s i t e 81T a b l e 2 9 : E s t i m a t e d c o m p o s i t e m o d u l u s u s i n g K r e n c h e l a n d E q u a t i o n 4 7 with t hee r ro r i n t he es t im a t ed m od u lus 82

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G l o s s a r yThe na t u ra l f i b res g l ossa ry cou ld bef o u n d ath t t p : / / w w w . t e c h . p l v n n . a c . u k / s m e / M A T S 3 2 4 / M A T S 3 2 4 A 9 F i b r e G l o s s a r v . h t m

Nomenclaturea S l o p eofW e i b u l l s h a p e p a r a m e t e ran P ropo r t i on off i b res m ak ing 6 a n g l e to app l i ed l oad

S i z e ofs u p e r - e l li p s e in x - a x e sb InterceptofW e i b u l l s h a p e p a r a m e t e rby S i z e ofs u p e r - e l li p s e in y - a x e sc S l o p eofWe ibu l l sca l e pa ram e t e rd InterceptofW e i b u l l s c a l e p a r a m e t e rEi C o m p o s i t e m o d u l u s infibre d i rec t ion (G P a )E2 C o m p o s i t e m o d u l u s ind i rec t ion t rans ve rs eto thefibre d i rect ion (G P a )E' C o m p o s i t e m o d u l u s ( G P a )Ef F i b r e m o d u l u s ( G P a )Err, M at r i x modu lus (GP a )1 F ib retest g a u g e length(mm)lo F ib re re f e rence length(mm)L F ib re length(mm)m S h a p e p a r a m e t e r fors u p e r - e l l i p s e inx - a x e sn S h a p e p a r a m e t e r fors u p e r - e l l i p s e in y - a x e sro Fibre rad iusR H a l fofinter-fibre s p a c i n gTA F ib re a reaTL F ib re length(mm)Vf F ib re vo l ume fraction

M at r i x vo l ume fractionXi C a r t e s i a n c o o r d i n a t eYi Car t es i an coo rd i na t ei8 W e i b u l l distribution s h a p e p a r a m e t e rPunear W e i b u l l distribution s h a p e p a r a m e t e r for l i nea r mo de lPtog W e i b u l l distribution s h a p e p a r a m e t e r for na tura l log m o d e lPMDS W e i b u l l distribution s h a p e p a r a m e t e r formultiple d a t a set mode l

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E r r o rin thee s t im a t e d s u p e r - e l l i p s e s h a p e p a r a m e t e rn W e i b u l l d i s tr ib u t io n s c a l e p a r a m e t e r ( M P a )Id F i b r e d i amete r d i s t r i bu t i on f ac to rni Le ng th d is t r ibu t ion fac to rHunear W e i b u l l d i s tr ib u t io n s c a l e p a r a m e t e rforl in e a r m o d e l ( M P a )iLog W e i b u l l d i s tr ib u t io n s c a l e p a r a m e t e rforna tu r a l logm o d e l ( M P a )HMDS W e i b u l l d i s tr ib u t io n s c a l e p a r a m e t e rformu l t i p l e da taset m o d e l(MPa)Ho F i b r e o r ienta t ion d is t r ibu t ion fac to rHp W e i b u l l d i s tr ib u ti o n p r e d i c t e d s c a l e p a r a m e t e r ( M P a )Hw W e i b u l l d i s tr ib u t io n s c a l e p a r a m e t e rforw e a k - l in k s c a l i n g ( M P a )B A n g l eofa p p l i e d l o a dto f ib resK F i b r e a r ea co r r ec t i on f ac to rA T L o g - n o r m a l d i s tr ib u t io n s c a l e p a r a m e t e r

Log -no r m a l d i s t ri bu ti on l oca t i on pa r am ete rPpparent G e o m e t r i c m e a nofap pa r en t f ib r e a r ea

G e o m e t r i c m e a nPN N o r m a l d i s t r i bu t i on l oca t i on pa r amete r ( mean)Ptrue G e o m e t r i c m e a noft rue f ib re a re aVf F i b r e P o i s s o n ' s rat ioVm Mat r i x P o i s s o n ' s rat io

R e i n f o r c i n g e f f i c i encya F i b r e s t r e n g t h ( M P a )Oc U n i d i r e c t io n a l c o m p o s i t e te n s i l e s tr e n g t h ( M P a )Ocu U l t im a t e c o m p o s i t e t e n s i le s tr e n g t h ( M P a )Of F ib re t ens i l e s t r eng th ( MPa)Ogeo G e o m e t r i c s t a n d a r d d e v i a t i o nON N o r m a l d is t ri bu t ion s ca le pa r am ete r ( s t and ar d dev ia t i on )Omax Mat r i x t ens i l e s t r eng th ( MPa)(Om)Ef M a t r i x s t r e s sats t ra i n e q u a ltof ib re fa i lu re s t ra in (M P a)(px F i b r e ang le ( deg r ee )r G a m m a func t ion

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AbbreviationCDF Cu m u la t i ve D i s t r ibu t i on Fu nc t i onCDFWLS Cumu la t i ve D i s t r i bu t i on Func t i onwith W e a k - L i n k S c a l in gCLAHE C o n t r a s t - L i m i t e d A d a p t i v e H i s t o g r a m E q u a l i z a t i o nCLSM C o n f o c a l L a s e r S c a n n i n g M i c r o s c o p eCoV Coe f f i c i en t o f V a r i a t i onCSA C r o s s - S e c t i o n a l A r e aFACF F i b r e A r e a C o r r e c t i o n F a c t o rGOFN G o o d n e s s O f F i t Numbe rUM L i n e a r Interpolation M o d e lMD S Mul t ip le Data Se tMLE M a x i m u m L i k e l ih o o d p a r a m e t e r E s t i m a t i o nNLIM Nat u ra l Loga r i t hm ic Interpolation M o d e lPDF P robab i l i t y Dens i t y Func t i onR o M | | R u l e o f M ix tureSD S t a n d a r d D e v i a t i o nW LSM W e a k - L i n k S c a l i n g M o d e l

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Acknowledgements

T h i s r e s e a r c h p r o je c t w o u l d n o t h a v e b e e n p o s s i b l e without t he suppo r t o f manypeop le .

T h e a u t h o r i s i m m e n s e l y th a n k f u l to h is s u p e r v i s o r s D r J o h n S u m m e r s c a l e s a n d D rW a y n e H a l l (Griffith Sc ho o l o f Eng ine er i ng - G r i ff it h U n i ve r s i t y ) f o r con t i nu ing l ya s s i s t i n g , suppo r t i ng and gu id ing h im th r oughou t t he r es ea r ch de g r e e . Th e au tho r isa l s o g r a t e fu l fo r th e c o m m e n t s o f D r M i g g y S i n g h o n t h e M P h i l - t o - P h D t r a n s f e rrepor t .

T h e a u tho r i s t hank fu l t o the U n i ve r s i t y o f P l ym ou th f o r t he f i nan c ia l as s i s t an ce t ocon duc t t h i s r ese ar c h . T he au tho r is a l so t hank fu l t o F in e T ub e s L td . f o r p r ov id i ng t heso f twar e f o r t he r esear ch .

T h e au tho r i s a l so g r a te fu l f o r ass i s t ance with ex pe r im en ts and /o r f o r he lp fu ld i s c u s s i o n s . T h e f o l lo w i n g p e o p l e s h o u l d b e a c k n o w l e d g e d : T e r r y R i c h a r d s , G r e gN a s h , N il m in i D i s s a n a y a k e , R i c h a r d C u l l e n a n d D a v i d H a r m a n . T h e o m i s s i o n o f a n yappr op r i a te name he r e does no t cons t i t u t e a de l i be r a te rebuff. The au tho r i s a l sog r a t e f u l t o a n o n y m o u s r e f e r e e s f o r t h e i r r e s p e c t i v e c o m m e n t s o n t h e m a n u s c r i p t s o fj ou r na l paper s .

T h e au tho r wo u ld l i ke t o o f fe r h i s g r a t i tude t o h is g r an dp ar e n t s . N a n a j i, pa r en t s a nds i s t e rs f o r t he i r l o ve , under s tand ing and encou r agement t h r oughou t my l i f e t ime . Theau tho r wou ld a l so l ike to thank A n u f o r t he a f fec t i on a nd supp o r t . S pe c ia l t han ks a l soto a ll h i s f r i ends wh o a lwa ys have b ee n t he r e . Ab o ve a l l t he au tho r w ou ld l i ke t othank G o d fo r eve r y th ing .

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Author's Declaration

At no time du r ing the reg is t ra t ion fo r the degree o f Doc to r o f Ph i losophy has theau tho r been r eg i s te r ed f o r any o the r Un i ve r s i t y awar d . Th i s s tudy has been f i nancedby Un i ve r s i t y o f P l ymou th .

R e l e v a n t s c i e n t i f i c s e m i n a r s a n d c o n f e r e n c e s w e r e a t t e n d e d . P a p e r s w e r ep r e s e n t e d a t s o m e o f t h e s e m e e t i n g s a n d s u b s e q u e n t l y r e f e r e e d a n d p u b l i s h e d i njou r na l s . Th e de ta i l s a r e on t he f o l low ing p ag es .

D a t e . . . . 6 . r . / . 2 . - . 2 . 0 i . Q

W or d coun t - 1700 0

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Training attended:Introduction to stat ist ics , Ju ly 2 00 8P resen t a t i on sk i l l s , 2009T r a n s f e r report, 2 0 0 9Introduction t o p rog ram ming i n C++ , S e p t e m be r 2 00 7Introduction to A b a q u s , S e p t e m b e r 2 0 0 7

Journal papers:1. J S u m m e r s c a l e s , N D i s s a n a y a k e , W H a l l a n d A S V i rk , A re v i e w o f b a s t f ib r e s

a n dtheir com pos i t es . P a r t 1: f ib res as re i n f o rcem en t s . Co m po s i t e s P a r t A :A p p l i e d S c i e n c e a n d M a n u f a c t u r i n g , 2 0 1 0 , 4 1 ( 1 0 ) , 1 3 2 9 - 1 3 3 5 .

2 . J S u m m er sc a le s , N D i s sa na ya ke , W Ha l l an d A S V i r k , A rev iew o f bas t f i b res an dtheir c o m p o s i t e s . P a r t 2 : c o m p o s i t e s . C o m p o s i t e s P a r t A : A p p l i e d S c i e n c e a n dM anu f ac t u r i ng , 201 0 , 41 (10 ) , 1 33 6 -13 44 .

3 . A S V i rk , W H a l l a n d J S u m m e r s c a l e s , P h y s i c a l c h a r a c t e r is a t i o n o f jute t e chn i ca lf ib res: fibre d im ens ions . Jou rna l o f Na t u ra l F i b r es , 7 (3 ), 201 0 , 2 16 -22 8 .

4 . A S V i r k , W H a l l and J S u m m er sc a le s , M o de l l i ng tens i l e p rope r ti es o f jute f i b res .M a t e r i a l s S c i en ce and Tec hno logy , 2010 , 26 (x ) , yy y - zz z ( co r rec t ed p roo f s n owon-l ine).

5 . A S V i r k , W Ha l l and J S u m m er sc a le s , Fa i l u re s t ra in as t he key des i gn c r it e r ionfo r fracture o f na tura l fibre c o m p o s i t e s . C o m p o s i t e s S c i e n c e a n d T e c h n o l o g y ,June 2010 , 70 (6 ) , 995 -999 .

6 . A S V i r k , W H a l l and J S u m m er sc a le s , M u l t i p le da t a se t (M DS ) weak- l i n k sca l i ngana l ys i s o f jute f ib r e s . C o m p o s i t e s P a r t A : A p p l i e d S c i e n c e a n d M a n u f a c t u r i n g ,N o v e m b e r 2 0 0 9 , 4 0 ( 1 1 ), 1 7 6 4 - 1 7 7 1 .

7 . A S V i r k , W H a l l and J S u m m er sc a le s , Te ns i l e p rope r ti es o f jute f i b res . Ma ter ia lsS c i e n c e a n d T e c h n o l o g y , O c t o b e r 2 0 0 9 , 2 5 ( 1 0 ) , 1 2 8 9 - 1 2 9 5 .

Conferences:1. ' B i o - C o m p o s i t e s ' - T h e N e x t G e n e r a t i o n o f C o m p o s i t e , 2 5 S e p t e m b e r 2 0 0 8 .2 . I C C M - 1 7 , 2 7 -3 1 J u ly 2 0 0 9 .3 . S A M P E / B C S A n n u a l S t u d e n t s ' S e m i n a r , 2 4 N o v e m b e r 2 0 0 9 , P r e s e n t e d p a p e r

"Fa i lu re s t ra in as the key de s ig n cr i te r ion fo r fracture of natura l fibre c o m p o s i t e s " .

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1. IntroductionN atu r a l f i b r es can be d i v i ded into t h r ee g r oups , vege tab le , an ima l o r m ine r a l f i b r es .V e g e t a b l e f i b r e s c a n b e w o o d (further s u b d i v i d e d into so f twood o r ha r dwood) o r non-w oo d (bas t , lea f o r se ed -ha i r ) f ib res [1 ]. B as t f ib res are tho se "ob ta ine d f rom the ce l ll a ye rs su r r ound ing t he s tems o f va r i ous p lan t s " [ 2 ] . N a tu r a l bas t f i b r es a r ei n c r e a s i n g l y b e i n g c o n s i d e r e d a s th e r e i n f o r c e m e n t f o r p o l y m e r m a t ri x c o m p o s i t e s a sthey a r e pe r ce i ve d t o be sus ta in ab le be ing a r ene wa b le r eso u r ce . N a tu r a l f i b r es , an dthe ir use as t he r e i n fo rcem en t i n co m po s i t e s , ha ve r ecen t l y bee n r ev iew ed by Jo h na n d Th om a s [3 ], H i ll an d H ug he s [4] an d Su m m e r s ca le s e t a l [5 , 6] (t he au tho r of t h i st hes i s i s a co -au tho r o f t he la t te r two r e fe r e nc es wh ich a r e i nc lud ed a t Ap pe nd ix A 1 -A 2 ) . N a tu r a l bas t f ib r es a r e co m p os e d p r imar i l y o f ce l l u l ose . Ce l l u l o se m ic r o f ib r il sh a v e a p o t e n ti a l Y o u n g ' s m o d u l u s o f - 1 4 0 G P a [7] w h i c h i s c o m p a r a b l e to that o fm a n - m a d e a r a m i d [ K e v l a r /T w a r o n ] f i b r e s at ~ 1 2 5 G P a . N a t u r a l f ib r e o f fe r s s o m eadd i t i ona l bene f i t s o f be ing a l ower cos t and l ower dens i t y ma te r i a l with h i ghe rspec i f i c m e c h a n i c a l p r o p e r ti e s . N a tu r a l f i b re s a r e n o n - h a z a r d o u s a n d n o n a b r a s i v ewh ich l eads t o f ewer hea l t h and sa fe t y i s s u e s . Us ing na tu r a l f i b r es as compos i t er e i n f o rc e m e n t c o u l d r e d u c e o u r d e p e n d e n c e o n n o n - r e n e w a b l e r e s o u r c e s , l o w e rp o ll u ta n t e m i s s i o n a n d g r e e n h o u s e g a s e m i s s i o n . B e i n g b i o d e g r a d a b l e n a t u r a lf i b r es can be d i sposed o f eas i l y a t t he end o f p r oduc t l i f e o r ene r gy can be r ecover edby i nc ine r a t i on o f t he compos i t e . Initial resu l ts f rom a quant i ta t i ve l i fe cyc lea s s e s s m e n t ha ve bee n p r e sen ted by D i ss an ay ak e e t . a l . [8 , 9 ]. Fu r the r m or e , na tu r a lf ib r e com po s i t e s p r es en t good aco us t i c i nsu la t i on p r ope r t ies [ 10 ].F o r the f ib r e r e i n fo r ced p o l ymer -m a t r i x co m po s i t e s , t he s t reng th an d s ti f fness o f t her e in fo r c i ng f i b r es dom ina te t he s t r uc tu r a l pe r f o r mance o f t he compos i t e . F i ve ma infac to rs in f luence th is f ib re cont r ibu t ion: the mechanica l (e las t ic ) proper t ies o f thef ib res ; the f ib re- res in in terac t ions ; the f ib re vo lume f rac t ion ; the f ib re o r ienta t ion andleng th o f the f ib res in the compos i te .H o w e v e r , na tu ra l fib res su f fer f rom h igh er var iab i l i t y in m ec ha ni ca l prop er t ies a ndc o n c e r n s ab ou t the i r long - t e r m du r ab i l i ty i n a mo is t env i r on m en t as they a bso r bmo is tu r e . The f i b r e p r oper t i es depend on t he cond i t i ons du r i ng g r owth , maturity a tha r ves t , fi b re p r o ces s ing t echn ique a nd t he m o is tu r e con ten t o f t he fi b r es . P o o rwet tab il it y a nd i nad equ a te ad he s ion be twe en na tu r a l f i b r es an d mat ri x l ead s t ounder -u t i li sa t ion o f t he fi b re po ten t i a l . T he non-u n i f o r m f ib r e c r oss -se c t i on ad ds t othe h igh var iab i l i t y in the f ib re mechanica l proper t ies .

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T h i s h i gh va r i ab i l i t y i n t he i r mechan i ca l p r oper t i es i s one o f t he f ac to r s cons t r a i n i ngthe w idespr ead use o f na tu r a l f i b r es as t he r e i n fo r cemen t i n po l ymer ma t r i xc o m p o s i t e s . M o r e ov er t h is va r iab i l it y l e ad s t o difficulty i n accu r a te l y p r ed i c t i ng t hec o m p o s i t e p r o p e r ti e s u s i n g e x i s t in g m i c r o m e c h a n i c a l m o d e l s . T h u s , a n i n - d e p t hunder s tand ing o f t he na tu r a l va r i a t i on o f t he fibre p h y s i c a l a n d m e c h a n i c a l p r o p e r t i e si s n e c e s s a r y f o r t hese f i b r es t o emer ge as a r ea l i s t i c a l t e r na t i ve t o syn the t i c fibrer e i n fo r cemen ts f o r s t r uc tu r a l compos i t es .T h e bas t f ib res wh ich are cur rent ly a t t rac t ing the most in teres t are f lax and hemp ( int e m p e r a t e c l i m a t e s ) o rjute and kenaf ( in t rop ica l c l imates) . Ju te is re la t i ve ly low-cos ta n d a c o m m e r c i a l l y a v a i l a b l e f ib r e . T h e i n v e s t ig a t i o n t h e r e fo r e a i m s to e n h a n c eunder s tand ing o f t he s tochas t i c na tu r e o f jute f ib res with r e f e r ence t o t he f ocus onna tu r a l fibre r e i n fo r ced po l ymer ma t r i x compos i t e s t r uc tu r es and t o impr ove t hep r ed i c t i on o f t he me cha n i c a l p r oper t i es ( spe c i f i ca l l y t ens i l e m od u lus a nd s t r eng th ) .

2. ExperimentsJu te t echn i ca l f i b r es from a s i n g l e s o u r c e i n S o u t h A s i a w e r e u s e d throughout thes tudy . Thejute f ib res were in s l i verform ( p r o c e s s e d u s i n g c a r d i n g t e c h n i q u e to g i v e9 0 - 9 5 % un iax ia l o r i en ta t i on ) as sh ow n in F igu r e 1. T he ph ys i c a l an d m ec ha n i c a lproper t ies o f the jute f ib r e s w e r e c h a r a c t e r i s e d . T h e n t h e f ib r e s w e r e u s e d t om a n u f a c t u r e c o m p o s i t e p l a t e s t o e v a l u a t e th e c o m p o s i t e p r o p e r t ie s a n d c o m p a r ee x p e r i m e n t a l p r o p e r t i e s with m i c r o m e c h a n i c s m o d e l p r e d i c ti o n s to a s s e s s theapp l i cab i l i ty o f the m od e ls .

Figure 1:Jute fibres sliver

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2.1. Fibrephysical characterisation2.1.1. Fibre length distributionT h e fibre length d ist r ibut ion of the jute f i b res in th is ba tch o f f i b res was de te rm ine dacc o rd i n g to IS O 6989 - 'M e t hod A ' [ 11 ]. Th e leng t hs o f t he f i b res we re m ea su re dand reco rded i nd i v i dua l l y unde r a light t ens i o n us i ng a s t ee l ru l e . Th e leng t h o f 70 0jute f i b res we re re co rd ed .2.1.2. Fibrec r o s s - s e c t io n a l area distributionT h e fibre c r o s s - s e c t i o n a l a r e a i s a n important phys i ca l p rope r t y wh i ch i s used t oca l cu l a t e t he fibre m ech an i ca l p rope r t i es (modu lus and s t reng th ) t he re fo re , ac cu r a t ede te rmin a t ion o f the fibre c r o s s - s e c t i o n a l a r e a i s important. The ' t r ue ' c ross sec t i ona la rea o f jute f i b res wa s de t e rm ined by d i g it a l imag e an a l ys i s o f 106 fibre c r o s s -s e c t i o n s . R a n d o m s a m p l e s o f jute t e chn i ca l f i b res we re cons i de red f o r t h i s s t udy .T h e fibre s a m p l e s w e r e b o n d e d t o a c e l l u l o s e a c e t a t e s h e e t u s i n g e p o x y a d h e s i v ef o r accu ra t e fibre a l i gnmen t and f o r ease o f fibre h a n d l i n g . T h e c e l l u l o s e a c e t a t es h e e t with th e jute f i b res was cas tinto potting c o m p o u n d ( e p o x y r e s i n ) . T h e c a s tjutefibre s a m p l e s w e r e g r o u n d flat i n s t ages on 180 , 240 , 400 , 600 , 800 , 1200 and 2500grit p a p e r a n d th e n s u b s e q u e n t l y p o l i s h e d with 6 p m a n d th e n 1 p m d i a m o n d . T h epotted s a m p l e w a s w a s h e d t h o r o u g h l y with d i lu te de te rgen t in an u l t rason ic ba thb e t w e e n e a c h s t a g e . T h e c r o s s - s e c t i o n o f e a c h fibre w a s e x a m i n e d u s i n g a nO l y m p u s L E X T O L S 3 0 0 0 C o n fo c a l L a s e r S c a n n i n g M i c r o s c o p e ( C L S M se r i a ln u m b e r - 6 E 2 3 0 1 3 ) a n d L E X T O L S i m a g e a n a l y s i s s o ft w a r e v e r s i o n 6 . 0 3 . F i g u r e 2shows t he c ross sec t i on o f t wo t yp i ca l t echn i ca l f i b res .

Figure 2: Typicalfibre c r o s s sections

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T h e d i g i t a l i m a g e s a c q u i r e d u s i n g C L S M w e r e p r o c e s s e d u s i n g M a t l a b R 2 0 0 8 a [1 2 -14 ] . The ou t l i ne o f t he f i b r e was se lec ted manua l l y t o l eave on l y t he a r ea o f i n t e r es t(fibre c r oss -s ec t i on ) i n t he d ig it a l ima ge ( se e F ig u r e 3 ) . T he f ib r e c r o ss sec t i ona l a r eaw a s ca l cu la ted us ing t he ve r t i ces [ 15 ] o f t he r ema in ing image .

1 ^A f^ a = - X ( x , , , + X , ) ( x , , - y^ ) (1)^ 1=1wh ere x , an d y , are co ord ina te v a l ue s o f i** ve r t i ce s an d N i s number o f ve r t i ces .T h e C a n n y e d g e d e t e c t io n m e t h o d [12-14] was used t o de tec t t he f i b r e edges , bylook ing f o r l o ca l max ima o f t he i n t ens i t y g r ad ien t o f t he o r i g i na l g r eysca le imagem a s k e d t o r e m o v e a r e a o f i n s i g n i f ic a n c e . A G a u s s i a nfilter w a s u s e d t o s m o o t h t h en o i s e i n t h e m a s k e d i m a g e t h e n a g r a d i e n t w a s c a l c u l a t e d u s i n g th e d e r i v a ti v e [ 1 2 -1 4 ]. T h e C a n n y E d g e I m a g e ( C E I ) w a s t r a n s l a t e d a n d r o ta t e d to a l ig n t he m a x i m u mf i b r e d i m e n s i o n with t he ho r i zon ta l ax i s ( x -ax i s ) o f t he coo r d ina te sys tem and t op l a c e t he o r i g i n o f t he coo r d ina te sys tem a t t he m idpo in t o f t he max imum f i b r ed i m e n s i o n . T h e m a x i m u m a n d m i n i m u m f i b re c o o r d i n a t e in o r d i n a t e (y - a x is ) w e r ed e t e r m i n e d . T h e f ib r e i m a g e w a s tr a n s l a t e d i n o r d i n a t e to m a k e t h e a b s o l u t e v a l u e o fm a x i m u m a n d m i n i m u m f ib r e c o o r d i n a t e s in y - a x e s e q u a l .

Figure 3 : T h e mask u s e d t o identifytruefibreshape a n d area o ffibres showni n Figure 2

T w o c i rc l e s w e r e o v e r l a i d o n t he C a n n y e d g e i m a g e . T h e m a x i m u m x- a n d y -coo r d ina te va lu es we r e us ed to de f i ne t he r ad ius o f t he ma jo r an d t he m ino r c i r c l es

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respec t i ve l y (No t e : the m ino r c i r c l e d i am e t e r i s f o rme d by t he p ro j ec t ed width of thefibre a t 90 deg rees t o t he max imum p ro jec t ed width of the f ibre: i t is not the smal lestc i r c l e wh i ch can be i nsc r i bed i n t he fibre image ) . The a rea o f t he ma jo r and m ino rc i r c l es w a s d e t e r m i n e d . T h e m a j o r a n d m i n o r c i r c le s a r e c o n c e n t r i c .A n e l l ip se wa s con s t ruc t ed t o bound t he ou te r sh ap e o f t he f ib re . The m ax im umcoord i na t e va l u es i n x and y ax es we re u se d t o de f i ne the sem i -m a jo r ax i s an d se m i -m ino r ax i s o f t he e l l i p se respec t i ve l y . The a rea o f t he e l l i p se was de t e rm ined .Th e co nv ex hu l l [16 ] i s the sh ap e form ed by a con t inuo us l i ne wh ich fo l lows as t ra igh t pa t h be t wee n ea ch p ro j ec t ion o f t he sh ap e su ch that n o c o n c a v e r e g i o nr e m a i n s . T h e c o n v e x h u l l w a s c a l c u l a t e d from th e C a n n y e d g e i m a g e to b o u n d t h eo u t e r s h a p e o f e a c h fibre a s shown i n F i gu re 4 . The convex hu l l a rea wasde t e rm ined us i ng E qu a t i on 1 [15 ] to cha rac t e r i se t he fibre a r e a .

Figure 4: ConvexHull o f truefibreshape f o r t h e fibres in Figure 2

A supe r -e l l i p se [ 17 ] was used t o b i nd t he ou t e r shape o f t he f i b re . The supe r -e l l i p seis de f in ed by Eq ua t ion 2 :

m nX y = 1by (2)whe re ax and byde f ine the s i z e o f t he supe r -e l l i p se , m and n de f i ne t he shape o f t hes u p e r - e l l i p s e . T h e p a r a m e t e r s dx , by, m a n d n a re a l l pos i t i ve numbe rs . The s i zep a r a m e t e r s , ax a n d b y o f t he sup e r -e l l i p se we re g i ven by t he ma x im um coo rd i na t eva lue i n a b s c i s s a and o rd i na t e re spec t i ve l y ( t hese a re t he d i ame t e rs o f t he ma jo rand m ino r c i r c l es and a re concen t r i c with t h o s e c i r c l e s ) . T h e s h a p e p a r a m e t e r s , m

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a n d n o f t he sup er -e l l i pse we r e es t im a ted f r om the f i b re co nv ex hu ll by m in im is i ng ' 'in E q u a t i o n 3 , u s in g t h e N e l d e r - M e a d s i m p l e x s e a r c h m e t h o d [ 18 , 1 9] :

N 1 -Yn

(3)w h e r e a x a n d b yde f i ne t he s i z e o f t he su pe r -e l l i p se . A / i s nu m be r o f ve r t i ces o f t hec o n v e x hu l l, x , an d y , a r e t he coo r d ina tes o f t he c on ve x hu ll ve r t i ce s . T he su pe r -e l l i p se pa r amete r s wer e es t ima ted f r om the f i b r e convex hu l l t o r educe t hec o m p u t a t i o n a leffort. T h e a r e a b o u n d b y t h e s u p e r - e l l i p s e w a s d e t e r m i n e d .F i g u r e 5 shows the f i b r e ma jo r and m ino r c i r c l e , e l l i pse , convex hu l l and super e l l i pseo v e r l a i d o n th e s a m e g r a p h .

2.2. Fibre mectianical propertiesThe t ens i l e p r oper t i es o f r e t t ed t echn i ca ljute f i b r es ( ex t r ac ted f r om a 127 mm w iderol l with a n a r e a weigh t o f 880 gm/m^ ) wer e a s s e s s e d . J u t e t e c h n i c a l f ib r e s w e r emoun ted on t es t ca r ds ( based on Gr a f i l Tes t 101 . 13 [20 ] ) with Devcon 2 Tone p o x y a d h e s i v e f o r e a s e o f h a n d l i n g a n d s u b s e q u e n t t e n s i l e t e s t in g . A s c h e m a t i cr ep r ese n ta t i on o f t he tes t ca r d i s sh ow n in F igu r e 6 . T e n ga ug e l eng ths w er ec o n s i d e r e d , i . e . 6 m m , 1 0 m m , 2 0 m m , 3 0 m m , 5 0 m m , 1 0 0 m m , 1 5 0 m m , 2 0 0 m m ,2 5 0 m m a n d 3 0 0 m m . T h e s e g a u g e l e n g th s w e r e s e l e c t e d b a s e d on a n a n a l y s i s o ft he f i b r e l eng ths f ound i n a r ep r esen ta t i ve samp le o f t he jute. T h e f i b r e s g a u g el e n g th s fr o m 6 m m t o 3 0 0 m m , c o v e r e d 9 3 . 3 % o f f ib r e l e n g t h s , e s t i m a t e d b y

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i n tegra t ing tlie f ib re leng t l i d is t r ibu t ion ( log-norm al ) prob ab i l i t y den s i ty fun c t ionb e t w e e n 6 a n d 3 0 0 m m ( s e e s e c t i o n 3 . 1 . 1) .A t l eas t one hundr ed i nd i v i dua l f i b r es wer e t es ted a t each o f f i ve gauge l eng ths ( 6m m , 1 0 m m , 2 0 m m , 3 0 m m , 5 0 m m ) a n d a t l e a s tfifty i nd i v i dua l f i b r es wer e t es ted a te a c h o f f i v e a d d i t i o n a l g a u g e l e n g t h s ( 1 0 0 m m , 1 5 0 m m , 2 0 0 m m , 2 5 0 m m a n d 3 0 0mm) . A t o ta l o f 785 f i b r e t es t s wer e ca r r i ed ou t . The f i b r e l eng th was measu r ed t o ana c c u r a c y o f 1 m m a t e a c h e n d . A b o v e 5 0 m m g a u g e l e n g t h f e w e r f ib r e s w e r etes ted b ec au se t hey a r e m o r e d i f fi cul t t o f ind w i t h in t he ma te r i a l supp l i e d an d i t w a so b s e r v e d that t he sp r ea d o f t es t r esu l ts dec r e a se d as the f ib r e ga uge l eng thi n c r e a s e d .

GaugeLength^ Fibre

M M/

M MT poxy

Figure 6 : S i n g l e f ibremounted o n atest c a r d

B e f o r e t es t i ng , measu r emen ts o f f i b r e ' d i amete r ' wer e t aken a t 1 mm in te r va l s a longthe f ib re leng th wi th in the window o f the tes t card fo r f ib res with gauge l eng th < 50mm and a t 10 mm in terva ls fo r f ib res with g a u g e l e n g t h s > 1 0 0 m m u s i n g a nO l y m p u s B X 6 0 M F o p t ic a l m i c r o s c o p e ( s e ri a l n u m b e r - 5 M 0 4 7 3 3 ) a n d a n a l y S I S i m a g e a n a l y s i s s o f t w a r e . T h e a p p a r e n t c r o s s - s e c t i o n a l a r e a o f e a c h f ib r e w a sc a l c u l a t e d fr om the m ea n f ib r e ' d i am ete r ' ass um ing a c i r cu la r c r o ss - se c t i o n .T a b l e 1 s h o w s th e m e a n fi b re d i a m e t e r s a n d s t a n d a r d d e v i a t i o n s a t e a c h f i b releng th .N o t e that "d iameters" here is used as in the tex t i le indus t ry to represent thechar ac te r i s t ic d im en s io n no r ma l t o t he p r i nc ipa l ax i s o f t he f i b re . T he t ext il e i ndus t r yno r ma l l y m ea su r e f ib r e f i ne ne ss as a l i nea r den s i t y i n d tex ( g r am s /10 km) . H ow ev er ,f or e n g i n e e r i n g p u r p o s e s in th e s t r e s s a n a l y s i s o f c o m p o s i t e s w e n e e d s t r e s s i n

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P a s c a l ( N / m ). Thec o n v e r s i o n b e t w e e n u n i ts is nots t ra i g h tf o rw a r d b e c a u s eof thediff iculty of ident i f y ing an a c c u r a t e d e n s i t y for the ( p rimar il y ) ce l l u l o se ma te r i a landt h e p r e s e n c eof l u m e n ( c e n tr a l v o i d s p a c e ) in thef i b r es . Ty p i c a l sec t i onof a f ibre iss h o w n in F i g u r e 2. C l e a r l y the f ibre is not c i r cu la r andt h u s the a s s u m p t io n of ac i r cu l a r c r o s s - s e c t i o n is t he r e fo r e one e r r o r s o u r c e in the c a l c u l a t i o n of them e c h a n i c a l p r oper t i es be low. The po ten t i a l e r r o r is l ikely to s c a l e withthem e a s u r e df ib re d iameter .

Table1:Meanfibrediameters(standarddeviations inparentheses).Fibre Length [mm] Diameter [fjm]

6 5 3 . 8 9 (13.96)10 5 7 . 6 7 ( 1 0 . 9 9 )2 0 61 . 06 ( 13 . 48 )3 0 5 9 . 5 9 ( 1 3 . 6 7 )5 0 6 1 . 5 3 ( 1 8 . 6 2 )

1 0 0 6 4 . 3 4 ( 1 3 . 9 3 )150 61 .07 (9 .91)2 0 0 6 4 . 5 9 ( 1 2 . 3 2 )2 5 0 6 2 . 6 7 ( 1 1 . 5 4 )3 0 0 6 1 . 2 3 ( 1 0 . 8 3 )A l l 60.16(13.79)

F ib re d i m e n s i o n m e a s u r e m e n t s a n d m e c h a n i c a l t e s ti n g w e r e c o n d u c t e d b e t w e e n16Apr i l and 19S e p t e m b e r 2 0 08 . Them e a n te s t te m p e r a t u r e s w e r e in the r a n g e1 1-2 0 C 1 ' ' C a n d 4 3 - 1 0 0 % R H 7 % .T h e f ib r e te s t s p e c i m e n s m o u n t e d in thet e s t c a r d s s h o w n in F i g u r e6 w e r e s e c u r e din the t ens i l e tes t m ac h ine g r i ps . T hetes t ca rd wa s cut at the m i d d l e with s c i s s o r sbefo re s tar t ing thet ens i l e t es t. T he t es t s we r e ca r r i ed out on anInstron 3 3 4 5 K 1 6 6 9u n i v e r s a l t e s t i n g m a c h i n e with anInstron 5 0 0 N l o a d c e l l ( m o d e l 2 5 1 9 - 1 0 4 , s e r i a ln u m b e r - 5 2 9 6 7 ) , a c c o r d i n g to G r a f i l m e t h o d 1 0 1 . 1 3 [20], m o d i f ie d to a c c o u n tfordi f fe rent f ib re leng ths at a cons tan t s t r a i n r a te of 0.01 min'^ for all the f ib r e ga ug eleng ths . The t es t isb r oad l y s im i l a rtoA S T M D 3 3 7 9 - 7 5[21].

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The l oad ce l l was ca l i b r a ted with dead we igh t s up t o 2N fo r ce : a cons i s ten t e r r o r o fl ess t han 5% o f t he t a r ge t f o r ce was obser ved .

2.3. Statistical strength an d fracture strain distribution2.3.1. Weibull po int estimateTh e na tu r a l an d ma n-m ad e f i b res exh ib it cons ide r ab le s t r eng th an d f a il u re s t r a i nvar ia t ions . W eib u l l s ta t is t i cs are o f ten us ed to ch ar ac te r ise the s ta t is t i ca l d is t r ibu t iono f t he f i b r e s t r eng ths . Rev iews have been pub l i shed o f We ibu l l s t a t i s t i c s f o r t heproba b i l i s t i c s t reng th o f ma ter ia ls [22 ], fo r the i r use in re la t ion to f i lam ent s t ren g ths[23] , fo r f rac tu re theor ies [24 ] and fo r f ib rous compos i te mater ia ls [25 ] . A b r ie fl it e ra tur e r ev iew o f t he W e ib u l l d i s tr i bu ti on i s g i ven i n Ap pe nd ix B 12 . The r e fo r e , t hefa i lu re s t reng th and fa i lu re s t ra in d is t r ibu t ion o f natu ra l f ib res were descr ibed by two-pa r amete r We ibu l l d i s t r i bu t i ons [26 -31 ] . The two -pa r amete r We ibu l l P r obab i l i t yDens i t y Func t i on ( PDF) [ 32 , 33 ] i s :

w h e r e p i s t h e s h a p e p a r a m e t e r ( W e i b u l l m o d u l u s ) , q is the s c a l e p a r a m e t e r(charac ter is t i c s t reng th o r s t ra in) and a is measured f ib re s t reng th . Note: o c a n b esubs t i tu ted by z fo r fa i lu re s t ra in th roughou t the fo l lowing text.Integration o f t he P D F y ie l ds t he two -p a r am ete r W e ib u l l Cu m u la t i v e D i s t r ibu t i onF u n c t i o n ( C D F ) :

Th e W e ibu l l d i s t ri bu t ion pa r am ete r s i n Eq ua t i o ns 4 and 5 f o r ea ch of t he t en ga ug el e n g th s w e r e c a l c u l a t e d u s i n g t h e M a x i m u m L i k e l i h o o d p a r a m e t e r E s t i m a t i o n ( M L E )method [34] de ta i l ed in Ap pe nd ix 81 .

2.3.2. Weak-Link S c a l i n g Model (WLSM)The f ib re tens i le proper t ies are l im i ted by the presence o f c r i t i ca l f laws. I f a f ib rec o m p r i s e s a ser ies o f e lements (o r l inks) then the s t reng th o f that f ib r e i s go ve r ne dby t he weakes t l i nk . A l onger f i b r e con ta ins mo r e l i nks t han a sho r te r one and t heprobab i l i ty o f a c r i t i ca l f law there fo re inc re as eswith f ib re leng th , resu l t ing in lo ng erf ib res hav ing a lowe r ten s i le s t ren g th (on ave rag e) [35 ] . Th is fo l lows f rom Gr i f f i th 's

(4)

OF{a ) = -e^ ^ (5)

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c r a c k theory [35 ]. Th e s t ren g th a t on e f ib re leng th is o f ten s ca le d to es t im ate thes t reng th a t a d i f fe rent leng th . T h i s ph ys ica l m od el i s o f ten re fer red to as the pr in c ip leo f 'weak l ink s c a l i n g ' . The re la t ionsh ip o f the s t reng th (o r fa i lu re s t ra in) to the f ib releng th i s mode l l ed us ing t he We ibu l l Cumu la t i ve D i s t r i bu t i on Func t i on with W e a k -L i n k S c a l i n g[26-31] ( C D F W L S ) :

F( - \ - exp (J> - \ - exp (7)

F ( o - ) = l - e x p

Equa t i on 7 can be s imp l i f i ed t o :n (8)

w h e r e q is the s c a l e pa r amete r ( cha r ac te r i s t i c s t r eng th ) f o r t he We ibu l l d i s t r i bu t i onwith weak - l i nk sca l i ng ( Equa t i on 6 ) , q is the s c a l e pa r amete r ( cha r ac te r i s t i c s t r eng th )fo r the s tan da rd W eib u l l d is t r ibu t ion (Eq ua t ion 5 ) , / i s the fib re tes t leng th an d ther e fe r ence l eng th Qw a s c h o s e n t o b e 1 m m f o r m a t h e m a t i c a l c o n v e n i e n c e .W e a k - l i n k s c a l i n g p r e d i c t i o n s a s s u m e t h e c h a r a c t e r i s t i c s t r e n g t h c a n b e s c a l e d f o ra n y f ib re leng th f rom a s ing le w ea k l ink (cha rac ter is t i c s t reng th ) po in t es t ima te a t ac h o s e n f ib r e t es t ga ug e leng th . Th e app r op r i a te l eng th , /, i s use d i n Eq ua t i on 8 t oy ie l d t he co r r espond ing p r ed i c t i on o f q. T o d i s t i n g u i s h b e t w e e n p r e d i c te d a n dm e a s u r e d e s t i m a t e s o f q , w e u s e qpf o r a p r ed i c ted va lu e and h en ce E qua t i on 8 isrewritten a s :

Hp 7n(9)

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2.3.3. Multiple Data Set (MDS) Weak-linkscal ing modelT h e relationship between the strength (and failure strain) distribution and the fibrelength can be improved when two or more strength or fracture strain data sets (atdifferent fibre lengths) are used to better estimate p and q^- These parameters fortwo or more strength or fracture strain data sets can be estimated by the M L Emethod [34]. T he P D F for Weibu ll distribution with weak link scaling is obtained fromthe derivative of the C D F (Equation 6)with respect to failure stress.

'O'/w(10)

T h e L ikelihood function for Multiple Data S e t s (MDS) Weibul l PD Fwith weak-links c a l i n g is given by:

;=1 r=l(11)

where / is data set number, k is the number of gauge lengths and /, is the fibre testg a u g e length for data set /.F o r computational convenience the log-likelihood function is u s e d :

A ( c 7 | A ^ , / o )= ZInn,In -I (12)T h e log-likelihood function is maximised by Newton's method [19, 36, 37] to estimatethe distribution parameters for the M D S weak -link sc ali ng mod el.

2.3.4. Empirical ModelsIn contrast to the phy sic al- based mod els (weak link s c a l i n g , MD S- we ak link scaling ),e m p i r i c a l or phenomenological models can be developed from a (limited) databaseof experimental results. A number of authors have used the principle of weak links c a l i n g in an attempted to fully charac teri se the statistical distribution of synthe tica n d natural fibre properties but have found limited s u c c e s s [26, 38, 39]. Therefore,two empirical models : (a) a linear and (b) a Natural Logarithmic Interpolation Mo de l(NLIM) for the estimation of the fibre properties were deve lop ed to better model theeffect of fibre length on the strength and fracture strain of a fibre.

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2.3.4.1. Linear InterpolationModel (LIM)F o rthe l i n e a r m o d e lthes h a p e p a r a m e t e r , p,and thes c a l e p a r a m e t e r , qin E q u a t i o n4 are r e p l a c e dby the li n e a r r e l a ti o n s h i p b e t w e e n thefibre l e n g t h g i v e nby E q u a t i o n13,3= aL^+bt ] = cL. + dw h e r e a and c are thes l o p e ,andbandd axethe i n t e r cep tfor thes h a p e p a r a m e t e r

a n d s c a l e p a r a m e t e r r e s p e c t i v e l y . isg i v e nfibre leng th .R e p l a c i n g the s h a p e p a r a m e t e r and the s c a l e p a r a m e t e r in thePDF by the l i nea rr e la t i onsh ip y i e l ds ,

(13)

/ ( - ) = aL.+b^cL,+d(J aLi+b-l

T h u s ,theW e i b u l l L i k e l i h o o d f u n c t i o nis.

L{a\ a,b,c,d)= YU=1 r i

aL^+bcL^+d) cL^ + d

(14)

(15)

A n dthe l og - l ike l i hood f unc t i onis.

1=1nXIn^aL.+b^^cL,+d IInr = l/ N O Z , + * - l(J.{cL^+dj - zr = l (16)

T h e p a r a m e t e r s for the l i nea r r e l a t i onsh ip are e s t i m a t e d by m a x i m i s i n g the log -l i ke l i hood f unc t i on us ing N ewton ' s me thod [19, 36, 37].T h e d i s t r i b u t i o n p a r a m e t e r s w e r e e s t i m a t e d for the s t r eng th and f rac tu re s t ra in bym a x i m i s i n gthe l og - l i ke l ihood f unc t i on for all of thee x p e r i m e n t a l o b s e r v a t io n sat all ofthefibre l eng ths .2.3.4.2. Natural-Logarithmic InterpolationModel(NLIM)A s in p r e v i o u s s e c t i o n , p a r a m e t e r s for a l o g a r i t h m i c r e l a t i o n b e t w e e n the s h a p ep a r a m e t e ror the s c a l e p a r a m e t e r and thefibre l eng th can be o p t i m i s e d .The l i nea rr e l a t i o n s b e t w e e n s h a p e ands c a l e p a r a m e t e r andfibre l eng th ares i m p l y r e p l a c e dwiththe l oga r i t hm ic f unc t i ons ,J3= ln{^)+bT]=\ n{L^) + d (17)

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T h e pa r am ete r s fo r t he na tu r a l l oga r i t hm ic r e l a t i onsh ip we r e a l so es t im a ted bymax im is i ng t he l og - l i ke l i hood f unc t i on us ing N ewton ' s me thod [19 , 36 , 37 ] f o r t hes t r eng th a nd f r ac tu r e s t r a i n expe r ime n ta l o bse r va t i o n a t a l l o f the f ib r e l eng ths .

2.4. Jute/Epoxy Composite2.4.1. Dyeing F i b r e sJu te fi b r es a r e p rimar i ly c om po se d o f ce l l u l os e [40 ] an d do no t sh ow up i nm i c r o g r a p h i c s a m p l e s w h e n c l e a r r e s i n is u s e d . P r o c i o n M X c o l d f ib r e re a c t i v e d y ew a s u s e d [40-42] t o co lou r t he f i b r es b lack f o l l ow ing t he p r oce du r e de sc r i b ed byM i l n e r [42] de ta i l ed i n A pp en d i x B 5 . T he ma t r ix w as p igm en ted wh i t e (g i v i ng aLum inos i t y Con t r as t Ra t i o o f 21 ) [ 43 ] . H igh con t r as t a l l ows d ig i t a l image ana l ys i s t oeas i l y sep ar a te t he f o r eg r oun d i n fo r ma t i on ( f i b res ) fr om the backg r oun d i n fo r ma t i on(mat r ix ) in mic rographs .

2 .4 .2 . C o m p a r i s o n o f tensile properties o f u n - d y e d a n d d y e d fibresA t ot a l o f 104 dye d fi b r es we r e t es te d ac co r d ing to G r a f i l me thod 101 . 13 [20 ], a t acon s ta nt s t ra in ra te o f 0 .01 min"^ to de ter m ine the e f fec t o f dy e in g on the f ib rem e c h a n i c a l p r o p e r t i e s . S i n g l e f ib r e s h a v i n g 5 0 m m g a u g e l e n g t h w e r e m o u n t e d o nthe t es t ca r ds . Th e ap par e n t f ib r e d iam ete r (p r o jec ted w id th ) wa s me as u r e d a t 1 mmin te r va l s a long t he f i b r e gauge l eng th t o ca l cu la te a mean appar en t f i b r e d iamete r .T h e c r o s s - s e c t i o n a l a r e a o f e a c h fi br e w a s c a l c u l a t e d f ro m t h e m e a n a p p a r e n t f ib r ed iam ete r a ssu m ing c ir cu la r c r o ss -se c t i o n t o ca l cu la te t he s t r es s i n t he fi b re . Th e t es tis b road ly s im i lar to A S T M D 3 3 7 9 - 7 5 [2 1]. T h e m e a n m e c h a n i c a l p r o p e r ti e s o f d y e df i b r es an d as - r e ce i ve d f i b r es ( 50 mm ga ug e leng th ) [ sec t i on 3.2.1] a r e s h o w n i nT a b l e 2 .

Table 2 :MeanY o u n g ' s m o d u l u s , fibrestrength an d fracturestrain(standarddeviations in parentheses).

Dyed Fibres As-received FibresElastic Modulus [GPa] 29.56 (7 .4 ) 2 8 . 3 (8.8)

Strength [MPa] 3 7 8 . 2 ( 1 5 1 ) 3 3 6 . 3 ( 1 3 2 )Strain to failure [ ] 1 . 1 6 ( 0 . 3 5 ) 1.11 (0.34)

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N o significant cinange in the mechanical properties of the fibres was observed afterd y e i n g as the variation in the properties was l e s s than one-third of a standardd e v i a t i o n . Wilcoxon rank sum test was performed on the fibre tensile test data. Thetest accepted the null hypothesis that the data is from independent samples havingi den t i ca l continuous distributions with equal medians [44].

2.4.3. Manufacturing of composi te plateC o m p o s i t e plates were manufactured by resin infusionwith a flow medium/distributorm e s h[45-49] using as received (three plates) and dyedjute fibres (one plate)with nopigment or white pigment respectively. The epoxy was pigmented white^ to improvethe contrast ratio between dyed fibres and the matrix. A flat glass plate was used asa mould. The reinforcement was laid directly on the pre-released mould surface. Toa c h i e v e uniform test plate thickness pre-released P e r s p e x sheet was placed on topof reinforcement fibres and space rs were inserted between the glass plate and theP e r s p e x sheet to control the composite plate thickness as shown in Figure 7.

Figure 7: Resin infusion arrangement

^ West System 501 White Pigmentfo r epoxy.14

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The pee l p l y was l a i d nea r t he r es in and vacuum l i ne ends t o ass i s t i n r emov ing t hec u r e d p l a te . T h i s a r r a n g e m e n t w a s b a g g e d a n d a v a c u u m o f 1 0 - 1 5 m b a r d r a w nbe fo r e t he r es in was i n fused a long t he f i b r e d i r ec t i on . The i n fus ion r es in sys tem wasS ico m in 810 0 ep oxy r es in an d S D 88 22 ha r d en er m ix ed i n 100 /31 ra ti o by w e igh t[50].Th e i n fused p la tes we r e cu r ed f or 24 hou r s a t am b ien t t em per a tu r e and t hen pos t -cu r ed a t 60 C fo r 16 hou r s i n an oven acco r d ing t o t he r es in manu fac tu r e r ' sr e c o m m e n d a t i o n .

2.4.4. Mechanical properties o f t h e c o m p o s i t e sTh e t ens i l e tes t sp ec im en s we r e m ac h in ed pa r a l l e l t o t he fi b re d i r ec ti on f r om ther es in i n fused p la tes as shown i n F igu r e 8a us ing a d iamond s l i t t i ng saw [6 ] . Thet e n s i l e t e s t s p e c i m e n s m a c h i n e d f r o m t h e p i g m e n t e d e p o x y p l a t e w e r e further u s e dto cha r ac te r i se compos i t e f i b r e vo lume f r ac t i on and f i b r e ang le d i s t r i bu t i on . Thes p e c i m e n en ds wer e r e i n fo r ced by g lu ing 45 g la ss f ab r i c / ep oxy end t abs a sshown i n F igu r e 8b , t o encou r age samp le f a i l u r e within the ga ug e leng th [51-54 ].

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(b)Figure 8 : Re s in infused plate: (a) Sample l o c a t i o n ; (b) Schematic o ftensiletest

s p e c i m e n

T h e te s t s p e c i m e n w a s m a c h i n e d a n d t e s t e d a c c o r d i n g to A S T M D 3 0 3 9 s t a n d a r d s[53] : 250 mm longwith 1 5 0 m m g a u g e l e n g t h , 2 5 m m w i d e a n d a v e r a g e t h i c k n e s s o f3 . 5 mm. B id i r ec t i ona l 0790 s t r a i n gauge was bonded on each su r f ace o f t hes p e c i m e n with c ya noa c r y la te adh es i ve acco r d ing to t he me thod de ta i l ed i n [ 55 ] t oeva lua te t he ax ia l s t r a i n and t r ansve r se s t r a i n du r i ng l oad ing . The s t r a i n gauger o se t te s ( V is h a y M i c r o - M e a s u r e m e n t s & S R 4 , C A E - 0 6 - 2 5 0 U T - 3 5 0 , 6 m m g a u g eleng th , 350 D gr id r es i s t a nce and 2.1 n om ina l ga uge f ac to r ) wer e bon ded a t t hecen t r e o f t he spec imen gauge l eng th as shown i n F igu r e 8b . E a c h s t r a i n g a u g e w a swi re d us in g th ree-w i re qu ar ter b r idge s t ra in ga ug e c i rcu i t to m e as ur e the s t ra in [56 ].T h e p r e p a r e d s p e c i m e n s w e r e t e n s i le te s t e d a t a m b i e n t te m p e r a t u r e ( 1 0 C a n d 7 0 %R H ) on an Instron 5 5 8 2 u n i v e r s a l t e s ti n g m a c h i n e with a n Instron l O O k N load ce l l( s e ri a l n u m b e r - U K 1 8 5 ) a t a c o n s t a n t c r o s s - h e a d s p e e d of 2 m m / m i n [5 3 ]. T h et e n s i le s tr a in i n t h e s p e c i m e n w a s m e a s u r e d with a 50 mm gauge l eng th Instron2 6 3 0 - 1 1 3 e x t e n s o m e t e r ( se r ia l n u m b e r - 7 7 ) . T h e a x i a l a n d t r a n s v e r s e st ra i n g a u g em e a s u r e m e n t s w e r e r e c o r d e d e v e r y s e c o n d until f a i l u r e us ing V i shay P3 s t r a i ni nd i ca to r and r eco r de r . A total o f 2 4 s p e c i m e n s w e r e te s t e d , 6 s p e c i m e n s f ro m e a c hp la te . 10 t es t samp les f a i l ed within t he gau ge l eng th ( S am p le 5b f a i l ed near t he t abtherefo re s t reng th a nd f rac tu re s t ra in resu l t a re d i sc ar de d fo r the sam pl e bu t the

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modu lus I s I nc l uded as t h i s pa rame t e r on l y depends on t he initial s l ope o f t he s t ressst ra in curve) .T h e te n s i l e m o d u l u s , s t r e n g t h a n d fa i lu r e s t r a in f o r e a c h s p e c i m e n w e r e c a l c u l a t e da c c o r d i n g t o A S T M D 3 0 3 9 a n d t h e re s p e c t i v e f a i lu r e m o d e s w e r e r e c o r d e d . T h eax ia l s t ra in range o f 1000 | j (0.1%) t o 3000 ( j (0 .3 %) was used t o ca l cu l a t e t ens i l emodu lus [ 53 ] . A t yp i ca l s t ress -s t ra i n cu rve with t h e c a l c u l a t e d m o d u l u s o v e r l a i d o nthe plot i s shown i n F i gu re 9a .T h e P o i s s o n ' s ratio w a s c a l c u l a te d from t h e r e c o r d e d s t ra i n g a u g e m e a s u r e m e n t sa c c o r d i n g t o A S T M E 132 -4 [57] s t an da rd . F i gu re 9b sh ow s a t yp i ca l ax i a l an dt r a n s v e r s e s t ra i n m e a s u r e m e n t f o r a s p e c i m e n . T h e f in a l P o i s s o n ' s ratio o f e a c hs p e c i m e n w a s c a l c u l a t e d a s t h e a v e r a g e o f t h e tw o p a i r e d st ra i n g a u g em e a s u r e m e n t s ( b o n d e d t o o p p o s i t e s u r f a c e s o f e a c h s p e c i m e n ) .

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14000

-I

-2000-4000-6000

1 1

A Axial Strain -1V Transverse Strain -1B Axial Strain- 2^ Transverse Strain- 2

A Axial Strain -1V Transverse Strain -1B Axial Strain- 2^ Transverse Strain- 2

A Axial Strain -1V Transverse Strain -1B Axial Strain- 2^ Transverse Strain- 2

11^^ ^^^^S^^

0.0 0.2 1.0 1.2.4 0.6 0.8Normalised Test Time

(b)Figure9: (a)Typicaljutef ibre reinforced composi te str ess-st rain curve; (b)

Typical axialandtransverse strain gaug e measurement for aspecimen

2.4.5. Micro-str uctural characterisationT h e fibre volume fraction and the fibre angle distributionof thejute fibre re inforcedc o m p o s i t e tensile test s p e c i m e n s were estimated using photomic rographict e c h n i q u e s [54]. Themicroscopy s a m p l e s were taken from e a c h tested spe cimenn e a rthefracture locationasshowninFigure 10. The composi te fibre volume fractionw a s estimatedbycastingthefibre volume fraction s a m p l e sin thepotting compo undp e r p e n d i c u l a rto thefibre a x i stoexaminethespecimen/fibrec r o s s - s e c t i o n . The fibrea n g l edistribution was estimated fromthesurfaces parallelto thefibre a x i s(i.e. thosein contact with the top andbottom mould surfaces). The s a m p l e swere ground flat ins t a g e s on 180, 240, 400 , 600, 800, 1200 and2500 grit paper. Thefibre vo lumefraction cast s a m p l e s were subsequently polished with 6 \im and then 1 pmd i a m o n d . The s a m p l e swere was hed thoroughly with dilute detergentin anultrasonicbath betweene a c h s t a g e .

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f . / ] Volume Fraction' ' j Angle Estimation

Figure 10: L o c a t io n o f m i c r o s c o p y s a m p l e s from atypicaltestsample

T h e p o l i sh e d s a m p l e s w e r e e x a m i n e d u s i n g O l y m p u s B X 6 0 M F o p t i c a l m i c r o s c o p e( se r ia l n u m b e r - 5 M 0 4 7 3 3 ) a n d a n a l y S I S i m a g e a n a l y s i s s o f t w a r e . T h e s a m p l em ic r o g r ap hs wer e cap tu r ed a t 100X an d SOX ma gn i f i ca t i on f o r f ib r e vo lum e f r ac t ionan d f ib r e an g le d i s t ri bu ti on r espe c t i ve l y . Mu l t i p le ima ge s we r e cap tu r ed und erp o l a r i s e d light (wh ich p rov ide d bet ter cont ras t ) an d then s t i t ched together tog e n e r a t e a c o m p o u n d m a c r o - i m a g e f ro m c o n t i g u o u s m i c r o g r a p h s . A n a l y s i s o f alarge area g ives a bet ter overa l l es t imate fo r bo th the f ib re vo lume f rac t ion and thef ib re ang le d is t r ibu t ion.T h e s a m p l e s i z e an a l y se d f o r t he vo lu m e f rac t i on an d fi b re an g le d i s tr ibu t ion we r e7 . 81 mm X 2 . 95 mm ( 11440 X 4324 p i xe l s ) and 27 . 60 mm X 12 . 16 mm ( 19900 X8764 p i xe l s ) r espe c t i ve l y . F igu r e 11a and F igu r e l i b sh ow t yp i ca l m ic r og r ap hs o f t hepo l i she d sam p le s f o r t he fi b re vo lu me f r ac t i on a nd t he fi b re a ng le d i s t ri bu t ioneva lua t i on r espec t i ve l y .

(a)

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(b)Figure 11: Sample m i c r o g r a p h : (a) T y p i c a l fibrevolumefraction s a m p l e ; (b )

T y p i c a l fibreorientation sample

T h e b lack f ea tu r es In t he m ic r og r aphs a r e jute f i b r e s . T h e a c q u i r e d d i g i t a l i m a g e sw e r e p r o c e s s e d us i n g M a t la b R 2 0 0 8 a [ 1 2 - 1 4 ] . T h e m i c r o g r a p h i m a g e s w e r ec o n v e r t e d t o 8-bit ( 0 -2 5 5 ) g r e y s c a l e i m a g e a n d t h e c o n t ra s t o f t h e g r e y s c a l e i m a g e sw a s e n h a n c e d by s c a l i n g t h e in t e n s it y v a l u e o f t h e i m a g e s othat it co ve r ed t he en t i r ed y n a m i c r a n g e .

2.4.5.1. Fibre volume fractionA n a l g o r i t h m f o r a u t o m a t i c f i b r e v o l u m e f r a c t i o n m e a s u r e m e n t w a s written i n M a t l a b .The r eg ion o f i n t e r es t on t he d ig i t a l image was se lec ted manua l l y and t he a r ea o f t hes e l e c t e d r e g i o n w a s r e c o r d e d . T h e m i c r o g r a p h d i g it a l im a g e w a s t h e n s m o o t h e du sin g 5 X 5 a v e ra g in g filter t o r educe no i se i n t he image . The i n tens i t y h i s tog r am o ft he se lec ted r eg ion o f t he imag e w a s ca l cu la ted [12 , 13 ] a f t e r sm oo th ing . F igu r e 12as h o w s the i n t ens i ty h i s tog r am fo r t he m ic r og r a ph sh ow n in F igu r e 11 a . Th e i n tens i t ywh ich f o r ms t he base o f t he va l l ey o f t he h i s tog r am [12 ] was se lec ted as t het h r e s h o l d i n t e n s i t y f o r c o n v e r t i n g g r e y s c a l e i m a g e t o b i n a r y i m a g e . T h e g e n e r a t e db i n a ry i m a g e w a s m o r p h o l o g i c a l l y c l o s e d ( d il a ti on f o l l o w e d b y e r o s i o n ) a n d o p e n e d( e r os ion f o l l owed by d i l a t i on ) t o r emove i n te r na l vo ids and sma l l f ea tu r es r espec t i ve l y[12 ] . The r esu l t i ng image i s shown i n F igu r e 12b . The f i b r e a r ea was ca l cu la ted f r omthe b ina r y image and f i b r e vo lume f r ac t i on was es t ima ted as t he r a t i o o f t he f i b r ea r e a to the area o f the recorded reg ion o f in teres t .

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5 0 0 0 0 0

4 0 0 0 0 0

^ 3 0 0 0 0 0 -c

u l 2 0 0 0 0 0 H

1 0 0 0 0 0 4

0.4 0 .6 0 .8N o r m a l i s e d In tens i t y

Figure 12 : Fibrevol ume fraction: (a) Intensityhistogram o f t h e s a m p l e ; (b )Binary image to estimatevolumefraction f o r t h e image at Figure 11a

T h e p re c i s i on o f t he vo l um e frac t ion es t ima t i on a l go r i thm w as i nves t i ga t ed us i ng aso f t wa re gene ra t ed ca l i b ra t i on image f o r wh i ch t he f i b re vo l ume f rac t i on was 18%.T h e c a l i b ra t io n i m a g e s h o w n i n F i g u r e 1 3 w a s g e n e r a t e d u s i n g A u t o C A D .

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Figure 13 : Software generated Fibrevolumefractio n image

The a lgo r i t hm es t ima ted t he f i b r e vo lume f r ac t i on f o r t he ca l i b r a t i on image t o be1 8 . 6 % . Th e ca l i b r a t i on imag e ha s r ogu e p i xe l s ( no i se ) wh i ch i nc r ea se t he f ib r evo lume f r ac t i on above t he t a r ge t va lue .

2.4.5.2. Fibre angle distributionT o e s t i m a t e t h e f i b r e a n g l e d i s t r i b u t i o n t h e m i c r o g r a p h i m a g e c o n t r a s t w a s e n h a n c e db y m e a n s o f C o n t r a s t - L i m i t e d A d a p t i v e H i s t o g r a m E q u a l i z a t i o n ( C L A H E ) [14, 58] .U s i n g a 3 5 0 X 3 5 0 p i x e l tile t o ca l cu la te t he i n t ens i t y h i s tog r am, C L A H E i m p r o v e st h e c o n t ra s t n e a r th e f ib r e m a t r ix i n t e r f a c e in th e m i c r o g r a p h i m a g e s . T h e e n h a n c e dcon t r as t n ea r t he fi b r e and t he ma t r i x he lps to d i s t i ngu i sh e a ch fi b r e in t he im ag et h u s g i v e s a b e t t e r b i n a r y i m a g e [ 1 2 ] . T h e e n h a n c e d m i c r o g r a p h i m a g e w a sth r esho lded us ing 70% o f t he i n t ens i t y spec i f i ed by O tsu ' s me thod [12 , 13 ] andc o n v e r t e d t o b i n a ry i m a g e . T h e b i n a r y i m a g e w a s m o r p h o l o g i c a l l y c l o s e d ( e r o s i o n /d i l a t i on ) and opened ( d i l a t i on / e r os ion ) t o r emove i n te r na l vo ids and sma l l f ea tu r esr espec t i ve l y [ 12 -14 ] . The r esu l t i ng image i s shown i n F igu r e 14 .

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Figure 14: Typicalbinary image toestimate fibreorientation derived fromFigure l i b

A 15 X 15 p i xe l a r r ay se ed point wa s pos i t i oned on a r egu la r g r id pattern on them ic r og r ap h im age . A se t o f 200 22 se ed po in ts wer e u se d f o r ea ch b ina ry imag e ( 213ho r i zon ta l X 94 ve r t i ca l seed po in t s with 9 3 p i x e l s s p a c i n g b e t w e e n e a c h s e e d pointo n both axes ) . The r esu l t i ng g r i d pattern is s imi lar to that sh ow n in F igu r e 15a . T hes e e d po in t s wh i ch comp le te l y l i e on t he fibre ( the sum of the intensi ty of them i c r o g r a p h p i x e l s m a s k e d b y t h e s e e d point is ze ro [i .e . b la ck = 0]) were se le c te dand ove r l a i d with a mask ing image t o cove r t he fibre in t he m ic r og r aph as sh ow n i nF igu r e 15b . The l eng th and t h i ckness o f t he mask ing image wer e 1000 X 15 p i xe l sr espec t i ve l y . The i n tens i t y o f t he p i xe l s enc losed by each mask ing image wass u m m e d a n d r e c o r d e d b e f o r e th e m a s k i n g i m a g e w a s r ot a te d in 1 d e g r e e s t e p sth r ough 180 de g r e es as sh ow n in F igu r e 15b . Th e fibre rotation ang le and t he sum o fi n tens i t i es wer e r eco r ded a t each s tep . The ang le with t he m in im um su m of i n t ens i tya t e a c h s e e dpointw a s a c c e p t e d a s t h e fibre ang le f o r that se e d p o in t.T h e ratio o f fibre a r ea enc losed by t he mask ing image t o t he total m a s k i n g i m a g ea r e a was ca lcu la ted fo r each seed po in t . I f the ratio o f t he a r eas was l ess t han 0 . 9then t he ang le es t ima ted f o r that s e e d point w a s d i s c a r d e d . T y p i c a l l y a b ou t 1 9 0 0da ta po in t s r em a in . A s show n i n F igu r e 15b se ed po in t s ' a ' , ' b ' an d ' c ' com p le te l y l ieon the fibre but the ratio o f a r ea f o r seed point ' a ' an d "c" w a s le ss than 0 .9 there fo ret h e y w e r e d i s c a r d e d .

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U U U u u U L LI U

ri ri rin ri

R e i n f o r c e m e n t F i b r e

- S e l e c t e d S e e d P o i n tn - D i s c a r d e d S e e d P o i n t

(a)

R e i n f o r c e n n e n t F i b r eM a s k i n g I m a g eM i n i m u m Intens i ty A n g l e

- S e l e c t e d S e e d P o i n t

(b)Figure 15 : Schematic representation o f fibreangle estimation procedure

F igu r e 16 shows the fibre ang le d i s t ri bu ti on es t ima ted f o r t he imag e show n i n F ig u r el i b ( g rey sca le ) an d F igu r e 14 ( b ina r y imag e) .T h e m e a s u r e d fibre an g les fo r t he sam p le im age f o l l owed a N o r m a l d i st r ibu t ion [ 32 ,33] . The normal probab i l i t y dens i ty func t ion is :

1 20-v (18)W h e r e q >x is the fibre a n g l e , PNi s t he m ea n fibre ang le (l oca t ion pa r am ete r ) a nd ONist he s tandar d dev ia t i on ( sca le pa r amete r ) .T h e m e a n a n d s t a n d a r d d e v i a t i o n (distribution parameters ) fo r the fibre a n g l e sm e a s u r e d o n e a c h m i c r o g r a p h w a s r e c o r d e d .

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500

400

oc(UZJ200

100 -

0

1i ^ ^ i Histogram

Average S D

-80 -60 -40 -20 0 20 40 60 80FibreA n g l e [Degree]

Figure 16 : Histogram o festimatedFibreangle distribution

T h e p r e c i s i o n of t h e fi br e a n g l e m e a s u r e m e n t a l g o r it h m w a s i n v e s t i g a t e d o ns o f tw a r e g e n e r a t e d i m a g e s fo r w h i c h t he fi b re a n g l e d is t ri b u ti o n w a s k n o w n . I m a g e swer e gener a ted us ing t he no r ma l f i b r e ang le d i s t r i bu t i on g i ven i n Tab le 3 . A t yp i ca ls o f t w a r e g e n e r a t e d i m a g e i s s h o w n i n F i g u r e 1 7 .

Figure 17 :Software generatedFibreangledi stribution image

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The es t ima ted f i b r e ang le d i s t r i bu t i on pa r amete r s f o r t he so f twar e gener a ted imagesar e g i ven i n Ta b le 3 fo r d if fe r en t r a t ios o f t he en c lo se d f ib r e a r e a t o t he m as k in gi m a g e a r e a . Th e f ib r e ang le d i s t ri bu t ion pa r am ete r s es t ima ted by t he f ib r e a ng lem e a s u r e m e n t a l g o r i t h m w e r ewithin 1 o f the spec i f ied f ib re ang le d is t r ibu t ion.

Table 3: S p e c i f i e d fibreangle distribution f o r t h e image a n d estimated angledistribution b yangle measurement algorithm

Specified Estimated Estimated Estimated EstimatedArea Ratio 0.80 0.85 0.90 0.95

Image Mean SD Mean SD Mean SD Mean SD Mean SD1 0 10.0 0 .44 11 . 20 0 . 48 11 . 19 0 . 47 11 . 23 0 . 43 11 . 282 5 10.0 5 .77 10 .86 5 .71 10 .84 5 .72 10 .86 5 .80 10 .923 0 20 .0 0 .22 2 0 . 4 3 0 . 27 20 . 44 0 . 25 20 . 42 0 . 28 2 0 . 4 84 2 20 . 0 2 . 20 20 . 28 2 . 10 20 . 25 2 . 15 20 . 30 2.17 2 0 . 3 05 5 20.0 5 .59 19 .86 5 .69 19 .87 5 .67 19 .90 5.78 19.91

S D - S t a n d a r d D e v i a t i o n

2.5. Modelling2.5.1. Micromechanical modelT h e e l a s t i c p r o p e r t i e s o f a c o m p o s i t e c a n b e p r e d i c t e d b y m i c r o m e c h a n i c a l m o d e l sb a s e d on t he p r oper t i es o f t he i nd i v i dua l cons t i t uen t m a te r i a l s o f t he com po s i t e an dthe i r geomet r i ca l cha r ac te r i s t i c s . Be t t e r p r ed i c t i on o f t he mechan i ca l p r oper t i es o fnatu ra l f ib re compos i tes wi l l he lp our unders tand ing o f the e f fec t o f the cons t i tuentson t he f ina l p r oper ti es o f t he ma te r i a l . Us ing m ic r o m ec ha n i c a l mod e l s , t he com po s i t ep r oper t i es can be op t im ised f o r a g i ven app l i ca t i on by va r y ing t he compos i t i on o f t hecom pos i t e . S i x d i ff e r en t m ic r o m ec ha n i c a l mod e l s w il l be use d t o p red i c t t he e las t i cmodu lus and two mode ls used t o p r ed i c t t he s t r eng th o f jute f ib re re in fo rce dc o m p o s i t e .

2.5.1.1. M odels for com posite mod ulus prediction2.5.1.1.1. T h e s i m p l e s t m i c r o m e c h a n i c a l m o d e l u s e d to p r e d i ct t h e c o m p o s i t ee las t i c modu lus pa r a l l e l t o t he p r i nc ipa l ax i s i s Ru le o f M ix tu r e {RoM\\). I t is a pa ral le ls p r i n g m o d e l b a s e d o n t h e a s s u m p t i o n that t he f ib r es and m at r ix w i ll exp e r i e nc e

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eq ua l s t ra in dur ing loa d in g in f ib re d i rec t ion. Th e RoM^i eq ua t ion [59 ] fo r the m od u lu so f a con t inu ous un id i re c t iona l f ib re com po s i te in the fib re d i rec t ion is ,

= Ej-Vj.+ EJ^ (19)W h e r e 7i s com po s i t e m od u lus i n f ib r e d i r ec t i on , Ef a n d Em a re f ib re an d ma t r ixm o d u l u s r e s p e c t i v e l y a n d W a n d Vm a r e fi b r e and m a t r i x vo lum e f r ac t i on .RoMn p r ov ides t he upper bound f o r t he compos i t e modu lus [ 60 ] when ,^ - " ^ ^ z (20)W h e r e Vm a n d Vfa re m at r ix a nd f ib re a x ia l P o i s s o n ' s ra t io respec t ive ly .

Th e com po s i t e m odu lus i n t he d i r ec t i on t r ans ve r s e to the fi b r e d i rec t i on i s g i ve n byRoMj^. T h i s s e r i e s s p r in g m o d e l a s s u m e sthat t he fi b r es and mat r i x exp er i e nc e t hes a m e s t r ess when t he compos i t e i s l oaded i n t he d i r ec t i on t r ansve r se t o t he f i b r es .T h e RoM eq ua t ion [59 ] i s ,

EfEE^= ^ (21)W h e r e i s compos i t e modu lus i n t he d i r ec t i on t r ansve r se t o t he f i b r es .RoM^ g i ve s t he l ower bound fo r t he com po s i t e mod u lus [ 60 ].

2.5.1.1.2. Halpin - T s a i . H a l p i n a n d T s a i [ 6 1 ] d e v e l o p e d a s e m i - e m p i r i c a l m e t h o dto p r ed i c t t he compos i t e p r oper t i es . Ha lp in -Tsa i me thod t r i es t o make a sens ib lei n te r po la t i on be tween upper and l ower bounds o f compos i t e p r oper t i es . Ha lp in -Tsa iequa t i on i s .

E'=E,W h e r e

(22)

Er-EriHT^ (23) * i s c o m p o s i t e m o d u l u s , Ef a n d Ema r e fi b re a nd m a t r ix m odu lus r es pec t i ve l y , Vf isf ib re vo lum e frac t ion an d ^ i s re in fo rc ing e f f i c ienc y (wh ich de pe nd s on f ib reg e o m e t r y , p a c k i n g a r r a n g e m e n t a n d l o a d i n g c o n d i t i o n ).

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Th e re in fo r c ing e f f i c iency f can be ca l cu la te d us ing Eq ua t i on 24 [62] f r om theexp er ime n ta l t es t r esu lt , wh er e co m po s i t e mod u lus , E* an d f ib r e vo lum e f r ac t i on , War e known.^ _E ,{E -E-V ,E (E^ -E ^ ^ ^ ^

Hi^ f- E )-V .{E,-E}a n d Vm i s ma t r ix vo lum e f rac t ion w h ic h is eq ua l to t-V /^ as su m in g a zer o vo id f rac t ion .V a l u e o f r e i n fo r cem en t e f f ic i ency , ^ ca n va r y f rom 0 t o . W h e n ^ = H a lp in -T sa ie q u a t i o n b e c o m e s RoM\\ and f o r ^ = 0 Ha lp in -Tsa i equa t i on i s r educed t o RoM^ [60].Th e h ighe r r e i n fo r cem en t e f f ic i ency , ^ s i gn i f i es that f ibre s are contr ibut ing to thecom pos i t e s t i f fness . Ha lp in -T sa i me thod o f fe r s t he adva n ta ge o f be ing s imp le ( ea syto use in de s ig n p r oce ss ) and o ff e r s mo r e e xa c t p r ed i c t ion bu t no r ma l l y r equ i r esemp i r i ca l da ta t o de te r m ine ^ .

2.5.1.1.3. C o x . T h e m o d u l u s f o r d i s c o n t i n u o u s f i b r e c o m p o s i t e c a n b e e s t i m a t e du s i n g C o x S h e a r - L a g m o d e l [6 3]. T h e RoM\\ i s mo di f ied by inc lud ing a leng th fac to r ,wh ich is a func t ion o f f ib re len g th , f ib re and m at r ix prope r t ies , f ib re geo m et ry an dp l a c e m e n t . T h e m o d i f i e d RoM\i e q u a t i o n [63-65] isE = n,E^V^ + EV^ (25)

= (27)

W h e r e n i is f ibre len gth d is t r ibut ion factor , / is f ibre len gth, Gm i s mat rix sh e arm o d u l u s , A f i s f i b r e c r oss sec t i ona l a r e a , roa n d R a re the f ib re rad ius and ha l f o fin ter - f ib re spac ing respec t ive ly .F o r squa r e an d hex ago na l f ib r e a r r ange me nt a nd f ib r e o f c i r cu la r c r o ss sec t i on t hef i b r e vo lume f r ac t i on i s g i ven by Equa t i on 28 and 29 r espec t i ve l y .

4R'f = ^ (28)

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T h i s model assumes the interface between fibre and matrix is perfect, fibre andmatrix responseiselastic andnoaxial forceis transmitted through thefibre end s.

2.5.1.1.4. Stiffnessof partially oriented compositec an beestimated by includingthe fibre orientation distribution factor by Krenchel [66] in the RoM\ equation.Th eresulting equation[65, 66] is ,

W h e r e r}o is fibre orientation dist ribution factor, an is the proportion of the fibresm a k i n g6n angleto theapplied l oad .

2.5.1.1.5. Stiffness of discontinuous fibre composite with partially orientatedfibrescan bepredictedbycombining Equation25 and 30 [63-66].E =rj ,rj.E,V^+E^V^ ^32^H o w e v e r , if qi is unity (for long fibres) this returns the same results as Krenchel( E q u a t i o n30).

2.5.1.1.6. The modulus of natural fibres has been reported to decrease withi n c r e a s i n g fibre diameter[67, 68]. Themodulusof composite reinforcedwith naturalfibrescan beestimated byequation propose dby Summerscaleset al [6]. TheRoM\\equation is extended to include a fibre "diameter" distribution factor, qa as inE q u a t i o n33:E^nMEfV,+EJ ^ (33)W h e n the fibres usedin the compositearewell characteri sedqa ca n betaken as 1i.e.themodulusof thebatchof fibres use dhasbeen mea su res independently.

2.5.1.2. Models for composite strength prediction2.5.1.2.1. Strength of the unidirectional (continuous fibre) composite can bepredicted byassumingall the reinforcing fibres have identical strength and the strainin the fibresand the matrix is equal during loading. If the fibre failure strain is l essthanthe matrix failure strain thenthecom posit e longitudinal tensil e strength (paralle ltothefibres)can beestimated using K elly-Ty son Equation34 [69],

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(34)W h e r e cr , i s un id i r ec t i ona l com po s i t e t ens i l e s t r eng th , i s f ib r e t ens i l e s t r eng th an d(cr^)^^ i s mat r ix s t ress a t the s t ra in equa l to fa i lu re s t ra in in the f ib res . Equat ion 34 isnot true fo r low f ib re vo lume f rac t ion , there fo re fo r low f ib re vo lume f rac t ion thec o m p o s i t e s tr e n g t h i s a p p r o x i m a t e d b y .

W h e r e c r ^ i s m ax im um mat r i x t ens i l e s