An Introduction to Composite Materials (Cambridge Solid State Science Series)

The tensile- shear interaction effects in the laminate as a whole depend on the set of ply orientations. The variation of the interaction ratio , TJ.II'.\' with t he angle

98

Elastic deformation of laminates 2

5.4 S tresses and distorlions

t

As these stra in s are imposed on all the plies, the strains within the kth ply (referred to the fibre direction of the ply, i. e. the Ik-direction) can be determ ined from Eqn (5. 15), which becomes

- - Single ply (0' ) -- ....... Angle-ply laminate (+30' 1_30' ) - - - - - Cross-ply laminate (0' /90 ' ) 0'160'1120' laminate

'<

~,

99

1::""

E lk ] [ E2k

,2 'iil ....

=

[ T' Lt>=k [

El"

E)"

1

( 5.32)

'"(n

'"( 12k

c:

.S: U OJ .... 0)

.5

The stresses in the ply are then obtained from these stra ins, using the (onaxis) stiffness matrix for the kth ply

0.5 J " . .1

0

,

,

'"

-0.5

O" lk [ 0"2k

,

,

IQ

20

30

40

50

Loading ang le,

60

__ ...

70

80

O" lk [ 0"2k

90

tP n

F ig,5. l0 Variation with load ing a ngle (p of the interaction ratio 1/"1",\ for a sin gle lam ina and for three laminates with different stack in g scquences, composed o f epoxy/ 50% glass fibre s. (The eq ual stress model was used for the tran sve rse Young's modulus of a la min a.)

balanced la minate. In a ll cases, the stacking orde r in which the plies are assembled does not enter into these ca lculations, although it may be important in determining the interlaminar co upling st resses (see bel ow).

[ O"X 0")"

~u

So me results fro m suc h a calcu latio n are shown in Fig. 5.11, which gives the variation of O" lk, as a ratio to the applied stress, with the a ngle tP 3 - - Single ply (0')

---

--

I;)~

......... Crossply laminate (0'/90' )

2

1;)"

- - - - - 0115/30/45/60175/90' lam inate

1.5

,

,

, j

0)

1::

U)

The internal stresses may be subdivided into ill-plalle stresses, wh ich can be calculated by the method s o utlined below, and ;nteriam;lIar or through-thickness stresses, which arise as a result of co nstraint effects and are more difficult to quantify. The approach to establishing illpl a ne stresses follows directly from the preceding trea tments. Refer rin g to Fig. 5.8, and assuming the stack of laminae to be subject to a stress state 0".0 0")" a nd T n " the laminate strains are esta blished from the transformed compliance tensor of the laminate.

T, ,.

(5.33)

Tn

'"'"

5.4.2 Stresses ill illdh'idual plies of a laminate

[SJ [ ::,]

[( 5. 10)1

1= [Cld T'] =JSg] 1

cS 'iil ....

,"( , "

E lk E2k

'"( 12k

2.5

[ :: ]

k

Thc stresses in the kth ply are rel a ted directly to the applied stresses by

-I

0

=

TI2k

I

...

1 [Cl [ 1

(S,] I )

0.5 0 -0.5 -I

0

10

20

30

40

50

Loading angle, tP

60

70

80

90

n

hg. 5.1.1 Variation with loadin g angle cP fo r a single lami na and fo r two la miflates 0 1 e poxy/ 50% g lass fibres , of the stress alk para llel to th e fibre axis in a p ly Ifl lllall y ol"le n ted ; 11 () 10 Ihe st ress ax is. T he stress is p lo tted a s a rati o to the ;q'pl led slress n, . (The eq ual slress mod el was used for the tran sve rse Young's Illodu lu s o r ;1 la lllina .)

100

Elastic deformation of laminates

5.4 Stresses and distortions

101

- - Along fibre axis (alk ) 1.5

------- -- Normal to fibre axis (au ) - - - - - Shear ('1"12)

•

••

o _

0.30"x

•• • -0.070"tJl

•

-0.5

Loading angle, cp

n

• •• •• •• ••

J

Fig.5.12 Variation with loading angle cP of the stresses alb a Zk and TI2k within the ply orien ted at 00 to the stress axis, for an epoxy/50% glass fibre 0/90 (erossply) laminate. The stresses are plotted as a ratIO to the applied stress . (The equal stress model was used for the transverse Young's modulus of a lamina.)

between the loading direction and the fibre direction in the 00 ply. This is shown for a single lamina and for two laminates. When more plies are present , so that the 00 ply being considered presents a smaller relative section , it bears a proportionately larger stress at cP = O. This is because the 00 ply is much stiffer than those at other orientations. Note also that the ply can be put into compression when it is oriented at a large angle to the loading direction. This is the result of a Poisson contraction effect. It can also be seen in Fig. 5.12, which shows all the stresses present in the 00 ply for a crossply (0/90) laminate. The compressive stress parallel to the fibres (in the I-direction), which has a magnitude of about 0.07 times the applied stress at cP = 90 0 , arises because the ply has a small natural Poisson contraction parallel to the fibres , but is being stressed by the larger natural contraction of the other ply (which is being loaded along its own fibre axis for q) = 90°). The stresses in the two plies are illustrated schematically in Fig. 5. 1:l for (/) () . (Those shown for the l)() ply arc the S;lllle ;I S Ihl' (Llta in

l'ig. 5.13 Schematic representation of the stresses alk and a2k within both plies of a 0/90 (crossply) epoxy/50% glass fibre lam inate, oriented so that the fibres are normal and parallel to an applied tensile stress ay.

F ig. 5. 12 for the 00 ply at cP = 90 0.) It is clear from Fig. 5.11 that the co mpressive stresses generated in this way within individual plies can in SO ll1e cases be surprisingly large. The details are sensitive to the Poisson 's ra tio of fibre and matrix, but - since (ceramic) fibres tend to have low v; dues (relative to both polymers and metals) - the composite Poisson's ra tio for contraction parallel to the fibre axis of a ply will always be less Ih;1I1 that in a transverse direction and hence an effect of this type is quite ge neral in composites based on these matrices.

5.4.3 COllplillK stresses alld .~ymmetric lamillates I"hrou g h- thickness (or 'coupling') stresses hetween the laminae are diffinill In (\escrilw ri!'ololl sly. Ilowever, they ;Ire important in practice and

102

Elastic deformation of laminates

can lead to significant distortions of the laminate. The general nature of the distortions can be illustrated by two examples. Consider the simple crossply laminate illustrated in Fig. 5.14. If thi s is loaded in uniaxial tension, as in (a), the difference in natural Poisson contractions of the two plies will cause the laminate to distort in the manner shown, and outof-plane stresses are needed to maintain the assembly flat. In addition to this, the transverse ply also exhibits a large through-thickness contraction as can be seen from the data in Fig. 4.IO(a). A similar effect is shown in Fig. 5.14(b), for a crossply laminate which has been heated. Because the thermal ex pansion coefficients are different parallel and normal to the fibre axis (see § I 0.1.2), the laminate is deformed and becomes saddleshaped. In thi s case it is assumed that Q2 > Q I , which is expected in view of the low thermal expansion coefficients ex hibited by (ceramic) fibres. The stresses which arise within such a laminate on changing the tempera ture can also cause microstructural damage - see § I 0.1.4. Distortions such as these can be reduced considerably if the arrangement of the plies is symmetric about the mid-plane of the laminate, i.e. if it has a mirror plane lying in the plane of the laminate. In symmetric laminates , the coupling forces large ly cancel out and the la minate as a whole will not distort, although there are still local stresses across the interlaminar boundaries. In addition, the use of many thin laminae rather than a few thick ones minimises the distortions and leads to a reduction of the local interlaminar stresses. The classification of laminae according to whether or nor the stacking sequence is balanced and / or symmetric is illustrated in Fig. 5.15 with some examples. There are many advantages in using balal/ced symmetric stacking sequences and this is common commercial practice. However, it should be noted that a laminate is often designed in the light of information about the expected in-service stress state. For example, with tubes to be subjected to internal or external pressure, unequal biaxial tension or compression will be imposed and ply angle sequences will be chosen with this in mind. The probable mode of failure , as well as the elastic deflections , may also need to be considered (see Chapter 8). Furthermore, the type and magnitude of permissible deflections and distortions will vary widely between different applications. It is therefore rather difficult to specify an optimum stacking sequence without detailed information about the performance requirem ents. This highlights the important concept o f designing the material and the compone nt simultaneollsly a recurrent theme when working with composites.

5.4 Stresses and distortions

103

expansions

contraction s

000 0

(b) (a)

Fig. 5. 14 Schematic illustration of how a crossply laminate will tend to distort as a result of through-thickness coupling stresses when subjected to (a) an external load and (b) a change in temperature. Unbalanced Asymmetric

0 90 -30 +30 -30 +30

l-"i g. ). 1)

Unbalanced Symmetric

m 90 0

-30 +30 +30 -30

Balanced Asymmetric

o 60 120 0 60 120 0 60 120

Balanced Symmetric

0 60 120 120 60 0 0 135 90 45 45 90 135 0

1·:.\: llIlpk, of slacking sequences whi c h result in laminat es c lassi fi ed \\' hcl hn Ihey :Ire halanced :Ind / o r symmet ri c.

: 1, '''1 11<1111 1' I"

104

Elastic deformation of laminates

6

References and further reading Chou, T. W. (1989) Flexible composites, J. Mater. Sci., 24 761 - 83 . Chou, T. W. and Kelly, A. (1980) Mechanical properties of composites, Ann. Rev. Mater. Sci. , 10 229- 59 Dieter, G. E. (1986) Mechanical M etallurgy. McGraw-Hill: New York GOl'don , J. E. (1978) Structures. Penguin: Harmondsworth , Middlesex Jones, R. M. (1975) Mechanics a/Composite Materials. McGraw-HIlI: New York Nye, J. F. (1985) Physical Properties 0/ Crystals Their Representation by Tensors and Matrices. Clarendon: Oxford

Stresses and strains in short-fibre composites

The previolls tll'O chapters are cOllcemed lI'ith the elastic behaviour

0/

colIIPosites cOlltaillillg jib res II 'hich are, ill efleet, in/initely long. The preparation

0/ composites

containing short .fihres ( or equiaxed part-

icles ) alloll's scope/or using a ll'ider range 0/ reinforcements and more l'ersatile processing ami forming rallies (see Chapter I! ) . Thlls , there is interest in understanding the distribution of stresses ll'ithin such a composite , and the COli sequences o/this for the stif/iless and other mechanical properties. In this chapter, hrie/ outlines are given O/tll'O Illlalytical approaches to this problem. III the shear lag model, a cl'lilldrical shape 0/ rein/orcemellts is assumed, alld the stress fields in jilne alld IIl(tlrix are simpli/ied so as to alloll' deril'ation

0/ straight/onmrd

allall'tical expressions/or the composite slll/iless. The Eshelhy method, on the other hand, is based on the assumption that the rein/ilrcement has an ellipsoidal shape ( ll'hich could rangeji-om a sphere to a cdinder or a plate ) . This alloll's derivation 0/ (Ill analytical solution 11'hich is more rigorous thall that o/the shear lag model, but 11'ith the penaltl' 0/ gl,(,(lter mathematical complexity. III the treatment gil'ell here, attelltioll is concelltrated

011

the principle o/the Eshelby approach; sources

lire suggested/or readers needing more mathematical details.

6.1 The shear lag model Ih: most widely used model describing the effect of loading an aligned short-fihre composite is the shear lag model, originally proposed by Cox ( 1952) and subsequently developed by others (Outwater 1956, Rosen 1%0, Dow 1963), which centres on the transfer of tensile stress from 1I1;ilrix to lihrc hy lllC;ln s or interracial shear stresses . The basis of the ,'alclllatioll s is sli oWIl sl' li"IIl;llicall y in l-'ig. 6. 1. In this dia gr;llll, reICrence In ')

106

Stresses and strains in short:fibre composites

111111111111

0

--: UR(X)~ :u (x) : ~

(b)

107

lines are drawn on a fibre and the s urrounding matrix, which are initially st raig ht and normal to the fibre axis. External loading is then applied (Fig. 6.1 (b» parallel to the fibre axis. The reference lines distort in the ma nner shown. Attention is concentrated on the shear distortions of the ma trix close to the fibre , represented schematically in Fig. 6.1 (c). The model is based on considering the shear stresses in the matrix and at the interface.

(a)

X=

6. I The shear fag model

r;..-

6.1.1 Stress ami straill distributiolls T he radial variation of shear stress in the matrix , T, (at a given axial dis tance x from the fibre mid-point) is obtained by equating the shear f'o rces on neighbouring annuli (with radii rh r 2 of length dx (see Fig. 6. I(c»

:

27rr1TI

dx

=

2m'2T2

dx

r2

. TI

T2

rl

The shear stress T in the matrix at any radius p is therefore related to that at the fibre / matrix interface (radius r) , Tj by

t

(c)

dp

•

R

p

r

T he strain field around the fibre can be defined in terms of the displacemellt u of the matrix in the x-direction , relative to the position for no applied stress (see Fig. 6.1). The increment of this displacement, du, on moving out from the fibre axis by dp , is determined by the shear strain "t, and hence by the shear modulus Gm du dp

Fig. 6.1 Schematic illustration of the basis of the shear lag model , showing: (a) unstressed system , (b) axial di splacements 11 introduced on applying tension parallel to the fibre and (c) variation with radial location of the shea r stress and strain in the matrix.

= "t =

T

Gm

=

Tj

Gm

(r)

P

For any given value of x , the difference between the displacement of the matrix at a radius R and that of the interface is given by a simple integra tion '/1/1 /

. "/

du

T-r

= - -'

Gm. T- r

ill

dp I (J

r

" In ( Rl r)

('Ill

(6. 1)

Stresses and strains in short~fibre composites

6.1 The shear lag model

The matrix strain is assumed to be uniform remote from the immediate vicinity of the fibre. The radius R represents some far-field location where this condition becomes operative. In a composite containing an array of fibres, the appropriate value of (R I I') is related to the proximity of neighbouring fibres and hence to the fibre volume fraction , f. The exact relationship between (R l r) and.l is dependent on the way the fibres are arranged: however, because (R l r) appears in a logarithmic term, the final result is relatively insensitive to the details of the fibre arrangement. An hexagonal array of fibres leads to

over most of the length of the fibre , and this, in turn , is close to the overa ll composite strain Cl

108

1

.l =

(2R~~/3)

(-;R)2 =2rJ3

I

1T

N

(6.6) Alt hough not rigorous, this represents a fairly good approximation; the tar- field matrix strain is shown in Fig. 6.I(b) as being approximately IInifo rm along (and beyond) the length of the fibre. The fibre strain t;lIld stress) builds up with distance in from the ends of the fibre. fi nally, the stress distribution in the fibre can be determined. Differentiation of Eqn (6.4) and substitution leads to 2

dx-

(6 .2)

---

(6.4)

1

The displacements themselves are unknown , but their differcntials arc related to identifiable strains. For the libre dur dx

=

E: _ ( I

~)

]I ~

(6.8)

. (nx) (nx) O"r = Erc l+Bs111h --; +Dcosh--;

(6.9)

O n applying the boundary conditions O"r = 0

at

x = ±L

where L is the fibre half-length, and writing the fibre aspect ratio (L l r) as this gives the solution

1",

(6.10)

T he variation of interfacial shear stress along the fibre length is derived , according to Eqn (6.3), by differentiating this equation, to give

and substitution in Eqn (6.1) leads to the result 2£111 (OR - ur) + 1/ )-' ln (1 / 1")

2EIII

O"r = Erc l [I - cosh(nxl r) sech(ns)]

£"1

dx = - (I

[

(6.3)

111 = ~

dO"r

(6.7)

Eq n (6.7) is a standard second-order linear differential equation with the so lution

Now , the variation of Tj with x is unknown a priori , but Eqn (6.1) can be used to relate it to displacements and hence to axial strains. It is assumed that there is no shear strain in the fibre and the interfacial adhesion is perfect (so that u, = Uf, the displacement of the fibre surface). The following relationship is used for the shear modulus of the matrix G

1'-

In which n is a dimensionless constant given by

The build-up of tensile stress in t]1e fibre O"r is determined from the distribution of interfacial shear stress. Referring to Fig. 6.1 (c), the basic force balance acting on an element of the fibre is

dx -

1

d O"r n-, = - dO"r - ErCI)

]

27rrd.\"Tj = _1fr 1dO"r 2rj dO"r

109

(6.S )

1:1'

The corresponding expression ror 1 Ill: l11alri x is less well defined . The difrcrential of 11/1 will ;Ipproxilllak 1J(l l!lc t;lr-field matri x slr;,ill , al ka st

Tj

. I (nx) = -nCI Er S1111 sech (ns) 2 r

(6.11 )

6.1.2 The stress transfer length

Fqllations (6.10) and (6.11) allow predictions to be made about the stress di st;'iblltion alon g the len gth of the fibre. An example is shown in Fig. 6.2(a) and (h) . Thi s shows the variations in fibre tensilc stress and interl"a ci;iI sil ear ., 1n'" :il o ll )' 1Il l" kll gl h 01" a f1hre in a composite of ali gncd

Stresses and strains in

110

short~ribre

6.1 The shear lag model

composites

80

(a) 70

"2 Q..

6.., ....

oD

60 - - Fibre aspect ratio, 50

.S

..,


b
CiJ

= 50

40 30

"

.;(


S

......... Fibre aspect ratio, S = 5

4:

.

20 10 0 -I

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 .6

0.8

Fractional distance from fibre centre, x /(r s )

10 8

"2 Q..

6.., u t~

6

f r

(b)

=50 .. . ...... Fibre aspect ratio, S =5

- - Fibre aspect ratio,

S

4 2

.S

'iii

..,

0


b
....

..,o:s

..c

-2 -4

Ul

-6 -8 -10 -I

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

I11

glass fibres in a polyester resin matrix. The curves are for two diITl' Il" 111 libre aspect ratios and the composite has been subjected to a tensik slr:1111 para llel to the fibres of 0.1 % (C l = 10- 3). The tensile stress is zero :11 I il l" lib re ends and a maximum in the centre. The interfacial shear stn.:ss is I.ero in the centre and a maximum at the ends. For the high aspect r:lti\) case (s = 50) , the fibre is long enough for the tensile stress to bui ld up until the fibre has a strain equal to that of the matrix and the compositc . T his gives rise to the plateau region of the fibre stress curve and a region of zero interfacial shear stress. (With continuous aligned fibres, all of thc co mposite is in this equal strain condition with respect to stress in the ax ia l direction - see §4.1.) There are regions of the fibre near the ends which are less heavily stressed than this central plateau region , so that the average fibre stress is lower than in a long-fibre composite subjected to the same external load. The reinforcing efficiency decreases as the fibre length is reduced , since this increases the proportion of the total fibre length which is not fully loaded. This behaviour leads to the concept of a stress trallsfer lellgth , over which the strain in the fibre builds up to the plateau (matrix) value. For the case shown , this length is about 10 fibre diameters (i.e . about 40% of the distance from the end to the mid-point of the fibre for the s = 50 c urve). Provided that the system remains fully elastic and there is no interfacial sliding, this value is dependent only on the elastic constants o f fibre and matrix. (With a stiffer matrix, such as a metal, the stress transfer length will be shorter, as a result of higher interfacial shear stresses.) For the low aspect ratio (s = 5) case shown in Fig. 6.2, the whole length of the fibre is only 5 fibre diameters, so that the stress in it does not build up to a plateau value. Such fibres are not providing very efficient reinforcement, because they carry much less stress than would lo nger fibres in the same system. A stress trallsfer aspect ratio, SI ' can be identified as that exhibited by fibres in which the peak (central) stress closely approaches the maximum possible (at which its strain is equa l to the value being imposed on the composite). From Eqn (6.10) , this will be the case when

0.8

Fractional distance from fibre centre, x /(r s) Fig. 6.2 Predicted vari a tions in (a) fibre ten sile stress and (b) interfacial shear stress a long the length of a glass fibre in a polyester/30°!., glass fibre composit e subject to an a xial te nsile strain of 10 " for two fibre aspect ratio s.

cosh(O) sech(ns)

«

I

Since cosh (O) I, the requirement is to set sech(lls) (= l / eosh(lls)) to a suitabl y low V:tilll' , ( 'iI()()sill g 0, 1 as hein g « I, the condition becomes

112

Slresses and slrains in

shorl~ribre

6.1 The shear lag model

composiles

113

cosh(ns) :::: 10 3

Sl

~­

n

(6. 12)

The value of n, obtained from Eqn (6.8), becomes smaller as the fibre / matrix modulus ratio rises, and as the volume fraction of fibres decreases. In general, however, the value does not vary widely; it is typically about 0.1 for polymer matrix composites and 0.4 for metal matrix composites. The corresponding values of s[ are therefore of the order of 30 and 7, respectively . (For ceramic matrix composites, the value of 11 is normally close to unity and hence s[ is small. However, the introduction of ceramic fibres into a ceramic matrix is normally done for purposes other than that of stiffening the material , so that efficient load transfer is not a primary objective.)

••••••••••

I

Bmm Fig. 6.4 Photoclastic fringe pattern (W ithcrs e / al. 1990) in and around a resin cylinder within a (mo re co mpli ant) resin mat ri x subjected to axia l stress. (The fringes represent co ntou rs of equa l shear stress.)

t t t t t t t t t t Fig. 6.3 An elastic deformation map obtained by a finite differcnce method (Tcrmonia 191;7). showing how an initiall y orthogonal grid aroLlnd a lihre end bccomes distorted on app lyin g an axial tensile stress. (The lihre/matri x stiffness r;ltio 40.)

The shear lag model has been shown to be qualitatively rea listic. For exa mple, using the finite difference method , Termonia (1987) illustrated how high matrix shear strains near the end of the fibre lead to a build-up o f fibre te nsile strain (see Fig. 6.3). Thi s has also been confirmed by rhotoe la stic observations. For example, the high-order fringes near the libre ends in Fig. 6.4 correspond to regions of high shear strain. However, quantita ti ve exam ina tion reveals discrepancies. The data in Fig. 6.5 were obtained (Galiotis el (I/. 1984) by mea s uring the loca l tensile strain a t dirrcrent roints a long a rolydiaeetylene fibre embedded in an epoxy matrix suhjcc tn l to :lll n tcl"Ilal tensile load. While th e gene ral arrearanec

114

Stresses and strains in

•

...

2.5

0

~

composites

0.99%

= 1.55% Em = 2.49% E

<=

0

1.5

...!l.l .D

u:

00

in which 17ro is given by substituting x = 0 in Eqn (6. 10) and 17m O is ta ken as EmCI (the average matrix stress). Thi s leads to an ex pression for 17c

m

o

o

_

00

17e -


0 0 0 0

o

Io.~

~

0.5

0

cp

•• • 0

•

115

6.1 Th e shear lag model

2

'@ b
Em =

short~fibre

•

• 2

3

•

.4

Distance along fibre (mm) Fig. 6.5 Stra ins in a po lyd iacety lene fibre embedded in a n epoxy mat ri x, measured by sh ifts in the Ra ma n resonance spectrum (Ga li otis et af. 1984), for three va lues of the macroscopic st rain applied to th e ma tri x. The fibre aspect ratio was about 200 and th e fibre/ matrix stiffn ess rati o was about 16.

of the c urves agrees well with the shear la g model for the higher imposed strains, there a re di screpancies. Notab le among these is that the fibre stress does not fa ll to zero at the e nd s. This is primari ly a resu lt of the transfer of tensi le stress across the fibre ends, which is neglected in the basic model and is re lat ively unimportant in composites with high fibre aspect ratios . This is briefly examined in the next section.

Cl

[Er ( I - sech(ns)) 2

+ Eml

_ -

Cl

E'

m

(6 .1 3)

a nd hence, using the new boundary conditions 17r = 17c a t x = ± L to so lve Eqn (6.9), a new ex pression is obtained fo r 17r, a nalogous to Eqn (6 . 10) 17r

= c dEI - (E1 - E,~)coShC:\') sech (ns)l

(6. 14)

In Fig. 6.6, pred iction s fro m this equa tion a re compared with those of the standard model (Eqn (6.10)) for (a) po lymer- and (b) meta l ma trix co mposites. It can be seen that the predicted stresses in the fibre are significant ly higher for the modified model, particularly nea r the fibre ends. Taking acco unt of fibre end stress tra nsfer naturally lead s to the fi bres carry ing more load, particularly for sho rt fibres. Thi s res ults in a n increase in the pred icted stiffness of the co mposite (see next secti on).

6.1.4 Prel/iction of s t~ffilCss T he basic results of the shear lag treatment can be used to predict the clas tic deformation of the composite. Consid er a section of area A taken norma l to the loading direction (in which a ll the fibres are a li gned), as shown in F ig. 6.7 . This section intersects individual fibres at random pos itions a long their length . The applied load can be expressed in te rms of the contributions from the two components

6.1.3 Tram/er of normal stress across fibre enlis Severa l attempts have been made (Fukuda and Cho u 1981 , Nardone and Prewo 1986, Cly ne 1989) to introduce corrections for the neglect of stress transfer across the fibre ends . Any attempt to account for the effect, while retaining the attractive simpli city of the shear lag a pproach , mu st involve postulating an ana lytica l expressio n for th e fibre end stress 17c . This mu st be an a rbitrary postula te, si nce there is no scope within the shear lag fra mework for a ny rigorous description of st resses beyond thc fibrc end. A n examp le is provided by th e suggcstio n (Clync 1989) tha t (To bc sc t cq ual to thc avcrage of the pcak fibre stress and th e remot e matrix st ress va lu es predicted hy th e stand a rd shear b g model

17IA = fA7Jr+( I - f)A7J m ". 17 1

= /7Jr+(I-/)7J m

(6. 15)

in which 7J r a nd 7J m are the volume-average stresses carried by fibre a nd matrix. This eq uation is often termed the ' Rule of Averages'. The average Ii bre stress is eva luated from Eqn (6 .1 0)

i

-17r _ Ercl - -L .

L

0

I;,I' E l(l

-

[

1 - COS h(nx/ r)] d.,' cosh (11.1') ta nh (II.I'))

-----'-----'11.1'

(6. 16)

116

Stresses and strains in

short~flbre

composites

6. J The shear lag model

(a)

117

""

(

60 ~

•

0..

6

(JI

50

... <1.l

.0

<..:: c::

'" '"<1.l '"

t:

40

30

]

><

--<

••••

20

10

o

'.

- - Standard shear lag, s = 10 ......... Modified shear Iag, s = 10 - - - - Standard shear lag, s = 2

. ~~,~,~,~~"g , -

j

."

I

" ""

j

2

1---------,

t:::::============::O::'&=':Q=":&=":B'='O:'='B:--=Q~'~'&;';~;";O~__~B'~'Q~"~&~~~~ o

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fractional distance from fibre centre, x /(r s)

:: ~ ~

i

400

r

350 300

r

.,.,

m:u=; · '··

(b) ;

l

~........... - - Standard shear lag, .I' = 10 ......... Modified shear lag, s = 10 - - - - Standard shear lag, s = 2 . . . . 0- . . . Modtfted shear lag, s = 2

250

r

200

r "o- ,,,,, o ..o •. o .. o- .-o ..

"' "

L

_ _ _ _ _ _ _ _ -.l

J-'ig. 6.7 Schematic illustration of a random section through a stressed shortlib re composite, showing how the section intersects individual fibres at various points along their length .

"

For the matrix , it is again conventional to resort to the assumption of a unifo rm tensile strain equal to that imposed on the composite Q. . -0 -. 0

" 0 - . '0 ·' 0. " 0 •• 0 ' -0

ISO -

-- -0 " 0- ' -0 "

0-

0'111

~

E l11 c l

(6.17)

-'0 " 1:) .

lOO

C ombi ning Eqns (6.15)- (6.17) gives the stress- strain relationship for the co mposite

50

o

~~~~~~~~~~~~~~~~~~~~~~

o

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(11 =C I

0 .9

. ] [/ . Er (tanh 1ns(nS) ) +( I -/)EI 11

(6. 18)

Fractional distance from fibre centre, x /(r s) Fig.6.6 Predicted variations in axial stress within (a) a glass fibre in a polyester/ 30% gla ss fibre composite and (b) a SiC fibre in an AI / 30% SiC fibre composile. 2. and Plot s a re a t a com posi te st ra in of 10 .\ for two fi bre aspecI ra Iios (.1' .I' 10). accordin g to the standard and modi lied shear la g ll111d cls.

The same procedure for the modified model , taking account of fibre end stress transfer. leads to (T I

(Er - E,;,) tanh (IIS)) F

.

-r"·1

(

+ 1- /

.

)EIIl

]

(6.19)

118

Stresses and strains in

shor t~rib re

composites

A linear increase in stress with increasing strain is predicted in both cases. The expressions in square brackets are the predicted Young's moduli for the two models . These equations ca n be tested b y making co mpari so ns with predictions from the (m ore rigorous) Eshelby mod el (see §6.2). For example, Fig. 6.8 shows the variation in composite Young' s modulus with fibre /matrix modulus ratio for two fibre as pect ratio va lues. It can be seen that the stand a rd shear lag model is inaccurate for low fibre as pect ratios. The predictions of the standard model look particularly unreliable when the fibre/ matrix modulus ratio is small. This suggests that the fibre end stress modification might be particularly useful for discontinuously reinforced metal matrix composites. Thi s is confirmed by th e data in F ig. 6.9 , whi ch co mpares prediction s (M c D a nels 1985) from the three model s with meas ured stiffnesses for particul a te MMCs. The standard shear lag model is clearly quite un suitable for ap plication to s uch material s. It may be noted from Eq ns (6. 18) and (6. 19) that, as expected , the stiffness approaches the limitin g (Rule of Mixtures) value as s becomes large eno ugh for tanh (ns)/ns to beco me negligible (<< I ). Since tanh (ns) cv I for ns ;;;:: 3, and ass umin g that 0.1 ca n be taken as « I , 10 SRM ;::o-

n

11 9

6.1 The shear lag model 30 kl

E

kl"

- - Standard shear lag 25

..... ... . Modified shear lag

.S 0;

- - - Eshelby

....

'"::l :;

20

-0 0

E >(

·c

IS

~

0;

..§ ~ 0'"

10

r

E 0

u

5 [.

l (a)

0 100

10

I

Fibre/matrix modulus ratio, Er / Em

- - Standard shear lag kl" 2.5

.. ....... Modified shear Jag

.2

- - - Eshelby

0;

....

B'"

2

::l

-0

o

E >( ·c

1.5

0;

..§

'I

~

J

'"o0.. 6.1.5 Ollset of illelastic behaviouI" Several phenomena ca n occ ur which cause depa rture fro m id ea l e las ti c behaviour. These include plastic deformat io n of the m a trix , debonding and s ubsequen t frictional sliding a t the interface , formation of cav iti es o r cracks in the ma trix (particularly at fibre end s) and fracture of fibres . These effects change the stress distribution and hence affect the stress strain curve. They are a lso relat ecllo th e o nset of failure and hen ce to the strength or the material. Detailed co nsideration or the ra c tors ill vo lvcd is prese nt ed in Ch~lpters X ~lIld 9 , hUl it is ~ Ippropriat c 11l: rl· t() n ;lll1ill c ;1

1

0..

( 6.20)

in which SRM is the fibre aspect rat io needed for the co mpo site modulus to app roach its m ax imum (R ule of Mi xtures) value. As noted ea rlier, va lues of 11 are typically around 0.1 for po lymer composites and 0.4 for those with metallic matrices. This suggests va lues fo r SRM o f about 100 and 25 for PMCs and MM Cs, respective ly. These can be regarded as target (m inimum ) aspect ratios when th e main objective is to maximise the load tra nsfer and hence the stiffness.

j

§

0.5

U

(b)

o L -____ J

~

__~~~~~~____~__~__~~~~u

10

100

Fibre/matrix modulus ratio, Er / Em

h g. 6.X Predicted composite/ matrix modulus ratio as a function of the fibre/ matrix modu lus ratio, for co mposites with 30% reinforcement and fibre aspect r;ltios or (1/) 30 and (h) 3. (Po isso n's ratios of fibre and matri x were taken as 0.3 and 0.2 respec ti ve ly.)

120

Stresses and strains in

shor t~fibre

composites

6.2 The Eshelby method

~:: ~ '2 160 0...

S~

:

~ :;

120 ....

"8

100

E

x

~


to de part from a linear plot. As an illu stra ti o n o f the use of Eqn (6.21), in a ty pical glass fibre reinforced polymer composite, with Tj ' = 20 MPa , the co mposite strain at the onset of inelastic behaviour is a bout 0.6% for long fibres (s'" 30) and abo ut 0.3% for short fibres (s'" 5). T he likelih ood of fibre fract ure takin g place before matrix yieldin g or interfacial sliding ca n a lso be exam ined. The peak stress in the fibre at the o nset of interfacia l slid ing or yielding is found from Eq n (6. 10) by setting \" = 0 and the composite strain to the value given by Eqn (6.2 1). This Icads to

Experimental - - Standard shear lag ..... .... Modified shear lag - - - Eshelby

140 :x

~

.-

~

x

.-

-W

80 ~;:-"""';:-:C;:;"':':"~:':":':"~:':"':':"':~_ ",,.. ... _ _.•_••_••_. _ _- -

~

60

2T'

(Jro

40 20

o

~~~~

o

____

0.1

~~-L~~~~

0.2

____~~L-~~~_

0.3

0.4

0.5

Fibre vo lume fraction.! Fig. 6.9 Compa ri so n between experimen ta l data (McDane ls 1985) and model predictions for th e stiffncss of A I/ SiC (particu late) co mposi tes produced by ex trusion. As the particlcs arc not truly cquiaxcd, and tcnd to bccomc a ligned during processing, an aspect ra ti o of 2 was Ll sed in th e prediction s.

simp le ex tens io n to the basic shea r lag theory designed to predict the onset of departure frol11 elas tic behaviour. The onset of matri x plasticity or interfacia l slidin g is expected to occur at the fibre ends, where the matrix shear stress is a maximum. A criti ca l interfacial shear stress Tj' can be specified for these processes . Substitu ti on ofTj' into Eq n (6.11), with x = L, gives the composite strain a t th e o nset of such inela sti c behaviour

=- ' n

[co th (ns) - cosech (ns)]

(6.2 1)

nEr

(6.23)

Sc hematic plots of this rela tionship are shown in Fi g. 6. 10, wh ich also gives an indication of the range of va lues expected for T j' in metallic and polymc ric matrices a nd for the fracture stress (Jr' ex hibited by ce ra mic fibres. (A dis ti nction must be drawn between thermosetting a nd thermoplastic polyme rs: the former are brittle and tend to ex hibit interfacia l sliding a nd / or mat rix microcracking, but not plastic yieldin g.) It is clear from this plot tha t, on increasing the load app lied to either type of composite, yieldin g or sl iding at the interface take place before fibres start to fracture. As the composite stra in is increased , yieldin g (or sliding) spreads a long the length of the fibre , raising the tensile stress in the fibre as the interf'acia l shear stresses increase. Fracture of fibres may then become possible and a simple treatment ca n be used to exp lore the limit of this effect. Ifit is ass umed that the interfacial shea r stress becomes uniform at Tj ' a long the length of the fibre , then a critical aspect ratio , s., can be identified, belo w which the fibre cannot undergo a ny further fracture. This correspo nds to the peak (centra l) fibre stress just a ttaining its ultimate strength (Jr', so that , by integra ting Eqn (6.3) a long the fibre ha lf-length

s - .(Jr'• -

h j' co th (ns)

12 1

(6.24)

2Tj'

It fo ll ows from this that a distribution of aspect ratios between s. and /2 is ex pected , if the composite is s ubj ected to a large strain. The value of .1' , ranges from over 100, for a polyme r composite with poor interfaci a l bo nding, to abo ut 2- 3 for a stro ng meta llic matri x.

.1',

This can be converted to an applied stress using Eqn (6. 18), lea ding to the ex pression (J I '

= -2Tj'

n~

[

/

.

. Er + ( I - /)E

I11 )

Er] co th (ns) -I -

m

(6.22)

This po int does no t co rrespond to a clearly ident ifi ab le co mposite yie ld stress, since yieldin g (or intcrf'acial slidin g) is o nl y taking pla ce ill a sma ll loca li scd rcgion. ll owcvcr. ~It thi s point the stress s tr ~ lill c ll rV\' wi ll st;lrt

6.2 The Eshelby method The method ha s it s ori ,ins ill some sim ple ' th o ug ht experiments' o utlined hy J . D . I':sll 'Ih y ill I Il l' I ()')()s (Eshc lhy 1957. Eshc lhy 1959). Thc power

6.2 The Eshelby method

Stresses and strains in short:/ibre composites

122

polymer matrix yielding r

..,/'...

,

polymer matrix debonding

~--------~~

/

r

metal matrix yielding .A.

} ,

metal matrix debonding

/ ------------~----------~" ,

-

- ----- - - - - ---- -

Typical <Jf , range for fibres

--- - - - - - - - -: - - - - - - -- - - - ---, -:- - - s = 100/ /. / / :

, s = 100'

:

' s = 10

/ / s=4 :'

:. / / :~\{ /. /~

/ / 1~

0.1

/

/

/ / // // //

/

: :

'

11-0.4 (metal matrix)

C onsider the sequence shown in Fig. 6.11 (a). The initial mesh spacings of matrix and inclusion (fibre) represent unstrained material. The heavier lines o f the inclusion denote its higher stiffness. For the case of differential thermal expansion on heating by 6.T, the imposed misfit strain, c r, is given by

'

100

The Eshelby method involves calculating the resultant stresses in the inclusion. These turn out to be uniform throughout the inclusion, provided it has an ellipsoidal shape. This is not as severe a limitation as it so unds, since many shapes approximate to ellipsoids. In particular, a sho rt fibre can be represented by a prolate ellipsoid having the same as pect ratio. Derivation of expressions for stresses and strains in composites, and hence of their elastic constants, involves some manipulation of te nsors. The treatment given here is highly abbreviated and readers interested in the details should consult other sources (Mura 1982, Taya 1988 , C lyne and Withers 1993). However, the model can be used without a full understand ing of the derivation.

6.2.1 A misfitting ellipsoid

0.01--¥:-----'L-r--r----+--,.----;-----r-~

10

123

1000

T i*

Critical interfacial shear stress (MPa) Fig. 6. 10 Plots of the depe nd ence of peak fibre st ress O"ro (at the onset of interfacial sliding or matri x yielding) o n the critica l shear stress for th e o nse t of these pheno mena T;- acco rdin g to Eqn (6.23) . Pl ots are show n for different as pect ratios, with 11 va lues typica l of polymer- and meta l-based composites. Also indica ted are typ ica l va lue ranges for fracture of fibres a nd for matrix yieldin g a nd interfacia l debo ndin g.

a nd versatility of the approach ha s only become widely appreciated quite recently. The original study concerned the hypothetical situation of a n infinite m a trix containing a stiff stress-free body. The body was removed and then ex perienced a change of size a nd shape (a ' stress-free' strain). The body was then subjected to surface forces so as to return it to its original dimensions a nd replaced in the hole from whence it came. The force s were then removed , allowing th e body to adopt a new sha pe by distorting the matrix. This , in effect, is what ha ppen s on heatin g a matrix containing a n inclu sion , when the two have different coefficients o f th ermal ex pan sio n. (Other exa mpl es arc prov id ed by a mart ensitic tr,lll sfo rmation o f th e inclusion o r a uniform plastic deformation o f the ma tri x.)

c

r

= 6.T( ar - am)

( 6.25)

noting that thermal expansivity a and strain c are both second-rank tensors. Since, in most cases, ar < a m, a decrease in temperature produces a change in inclusion sha pe, relative to the ma trix , as shown in Fig. 6.II(a) . After replacement of the inclusion in the cavity , it ado pts a new size and shape, distorting the surrounding matrix as it does so . This new size and shape represents a constrained strain, cc, relative to the original state. A key point here is that, /or the special case olan ellipsoidal shape, this strain is uniform in all parts of the inclusion. This must a lso be true of the stress, which can be written O"r

=

C

T'

Cr(c - c )

( 6.26)

where C r is the stiffness tenso r of the inclu sion (fibre) a nd the term in brackets rep resents the net strain rela tive to th e 's tress-free' state after cT' was imposed.

6.2.2 The eqllil'{t1ent /lOl1logeneolls ellipsoid The essence of th e Eshelhy method is to consider exactly the sa me ope ra tioll s Ix:in g C: IITinl (lIlt with ;In e llipso id havin g th e .IWJ/C c lastic cOJ/s/ml/s

124

S tresses and strains in short:flbre composites

6.2 The Eshelby method

as the matrix. Th is is shown in Fig. 6. 11 (b). The co mposite is now homoge neo us in terms of elasti c properties. It is possible to generate exactl y the sa me stra in a nd stress state (in bo th inclusion a nd surro undin g ma tri x) as arose with the rea l compos ite, provided tha t the imposed (s tress-free) stra in is the appropri a te one. Thi s is termed the transformation strain , ET. The schema tic shown in Fi g. 6. 11 (b) illustrates how choice of the a ppro pri a te va lue leads to th e co rrect fin a l stress sta te. The a ppro pri a te va lue of ET depend s o n, but differs fro m, the actua l mi sfit stra in ET '

125

c-.t--.

.....-f-

--

I--

E[{IIIIIII1l)

W

r'"

E =O ET = ET (ET~ C f ' Cm, S )

GIIIIIIID

I I I I I I I

11 I I I I I I I I I III VC---........ ~

S = S (V m, S)

E - E T*

--r'

I I

I I I I I I I I I I I I I I I I I I I I I

I I /1

I

j \ \

\ I

1 I /

I I

11\\\\\

\

\

I-

E = EC ( = SET)

/

I / / / I

I I

I

j \ \ \ \ I \

I I

IU\ \

Fig. 6. 11 (cant) I-

\ \\ \ I \

I

Jl/III /

E =E C

af =

CdE C ET*)

(a) Fig. 6. 11 Schemat ic illu stration of the stra ins duri ng the Es helby 'iPe rations of removing an inclu sion from the matrix, int roducin g a mi sfit strain E and repl acing it in the hole in the ma trix. The operations are shown fo r (a) the actual inclu si on and (b) an inclusion with the same elastic co nstant s as th e matrix which mu st be subjected to a different stra in ET in order to produce the same fi nal st ress sta te. (G rid spacin g depicts elast ic str~1 in. whi le the stiITlless is represe nted hy th e thick ness or th e grid lines. )

/ /

/

/11//1 I

I

ar =

CdE C - ET )

(b)

(O th er term s, such as the eigellStraill , E*, have bee n used by some a uth ors to denote the app ropriate transfo rm at io n strai n.) T he major co ntrib ut io n made by Es helby was to es tablish that the co nstrained stra in EC for thi s equivalellt homogeneous illclusion is uniq uely rel ated to the tra nsformation stra in by the expressio n (6.27)

IV hen: S', th e ' Eshelby tenso r', is a simpl e fun cti o n o f the ellipso id ax is r ~ ili o (i ,c. th e aspcc t r~ lti o o r th e fibre) and the Po isso n's rati o of the

Stresses and strains in short-jlbre composites

6.2 The Eshelhy method

matrix. (The symbol S is conventionally used for the Eshelby tensor; it is important to avoid confusion with S used to represent an elastic compliance.) Now, the stress in the homogeneous inclusion can be written

to support the hypothesis. It should be quite accurate, at least for volume fractions up to a practical limit for short-fibre composites of around 4050 % . The term 'meal/ stress' (a ), is now reserved for the volume-averaged stress in each component which contributes to this force balance

126

( 6.28) This is set equal to the stress for the actual inclusion given by Eqn (6.26) . SU,bstitution for cC from Eqn (6.27) a llows eT to be evaluated (for a given eT , elastic constants and ellipsoid as pect ratio) and the stress in the inclusion to be calculated.

6.2.3 The backgroul/d stress The above treatment was developed for an infinite matrix and is only applicable to a 'd ilute' composite, in which! is no more than, say, a few %. Extension to non-dilute composites is complicated by the fact that a given inclusion is affected by the m atrix stress field resulting from neighbouring inclusions. This changes the stresses in both matrix and inclusion. The change may be regarded as a 'backgro und stress' , ab, superimposed o n the stress state for a dilute case, a lthough various other expressions (such as ' image st ress ') are also used to describe this stress . Several authors (Tanaka and Mori 1970, Brown and C larke 1975, M ura 1982, Pedersen 1983) have contributed to the understanding and quantification of this effect, thereby increasing the usefulness of the Eshelby method enormously. It ca n be shown that, if the matrix stresses are integrated over the whole of the volume around a misfitting inclu sion , the result is zero. (This might have been expected from the symmetry of the situation , particularly on considering the example of a spherical inclusion.) The stress in the inclusion, however, is uniform and non-zero. This provides a means of eva luating the background stress in a non-dilute composite. For the equivalent homogeneous inclusion case, the background stress is expected to act uniformly throughout the composite, so that a force balance can be written ( 6.29) from which ab can be eva luated. For the case of the real composite, it is not clear how the inclusion experiences the background stress. The assumption is often made that the real and equivalent homogeneous inclusion s are subject to the same strain disturbance as a result of bein g in a matri x containin g oth er inclu sion s. This is the basi s o f the ' m('a"~/idd approximatio,,' ;lIld de lail ed arg um e nl s have heen pUI ro rw; lrd

127

( 6.30) T he mean stresses are indicative of load transfer: the inclusion carries a mean stress of the opposite sign (and different magnitude) to that in the matrix. From this point, some mathematical manipulations allow the mean strains of the two components, and hence of the composite, to be evaluated. For the case of a differential thermal expansion , the overall average composite strain , (equal to the sum of this mean strain and the strain from the normal expansion of the matrix) then determines the composite ex pansivity. (Thermal expansion is examined in § I 0.1.)

6.2.4 Composite

st~fjiless

T he analysis can be adapted to treat the case where the composite is subject to an applied stress, aA , and hence to predict the stiffness. The a ppropriate Eshelby operations are shown in Fig. 6. 12(a) and (b). The m isfit now effectively arises from the difference between the shapes which wo uld be adopted by the inclusion (fibre) and by the cavity in the matrix, if the two constituents were to be independently subjected to the applied stress. For the equivalent homogeneous composite, the appropriate transfo rmation strain eT is now dependent on aA , rather than on a pre-defined Er . Furthermore, the resultant strain is now the sum of the constrained strain cC and a strain arising directly from the applied stress cA (= Cl~laA). Similarly, the total stress in the inclusion is made up of a A a nd the contributi on introduced by the load tramfer between the consti t uents, represented by ar (as in the misfit stra in treatment). The same technique of equating the st resses in the actua l and equiva lent homogeneous inclusions ar

+ aA = =

;llld suhsliluliOIl or ,. . .

I

ror

,"_ C ,

Cr(ec

+ cA)

Cm(ec - eT + cA)

allows evaluation or eT as before.

(6.3 I)

128

S tresses and strains in

sh o rt ~flbre

6.2 The Eshelby method

composites

.... I""ii

.......1-

11""

} I I

1

I

I

I

/

I

\

\ \

\

(Jf

+

\

\

(J A

\

::s..y

J

7

(6.34)

lA

!\

...A

)0-...

1771/111 J I Ij I TI I I I

I \

'\ \

\ \ \ \ \ \ \\ 1\ \ T\ \ \ I

1

/ I

(6.33)

Cc[[C;;; I - /{ ( Cr - Cm)[S - f (S - !)] + Cm } - I (Cr - C,, )C,~ I Jr I

! /

V

(6.32)

The te rm 'mean stress, (a )', therefo re has a ra ther specia l mea nin g in Eshelby a na lysis, cl osely rela ted to the na ture of the load-sha ring betwee n the two co mpo nents: the vo lume-ave raged stress in the inclu sio ns is grea ter th a n tha t ove r th e co mposite as a wh o le by (a)f' whil e the volum e-ave raged ma tri x stress is less th a n the ove ra ll ave rage by (a)m (i.e. (a) m is nega tive). After so me ma thematical ma nipul a tio n , a n ex press io n ca n be deri ved fo r the stiffness tenso r o f the co mposite

in wh ieh I is t hl: i(k n tit Y tl: nso r.

1\1 " } / / } T,T \ -\ \ \ \ \ \ 1 I } / /7 I / I \ \ '\ \ \ \ 1 I } 1/ ! I /

aA

Fo r no n-dilute composites under a pplied stress , the fo rce ba la nce o f Eqn (6.30) still ho lds , wi th the p roviso th a t the m ea n stress (a) in each co mpo nent is no t now the actu a l vo lume-ave raged stress if, but is rat her a meas ure o f the difference betwee n thi s a nd a A

= (a)m + a A af = (a)f + a A

S = S (Vrn ' s)

TT \ \ \

= C r (E C + EA)

Schem at ic illust ra ti on of Es helby o pe ra ti o ns w ith a n a p p li ed s tress for (a) ac t ua l and (b) eq uiva le nt homoge neo Ll s co m posites.

am

Cm,S)

1

(a) Fi g. 6. 12

,

I I I I I I I l-rTTlll I I I I I I I

\ /

ET =ET« J A, C f

--

~

~

/

=

L..-=

1'-... .,..1 ""-.1

(EH] III [ID) .~~OE'

I I I I I I I -] I r I I I I I I I I I I I

0

E =

1 \

-

~

129

I

7

/ I

7-'

I

I

\ \

I

\

I \

\ \

\

(b) F ig . 6. 12 (Ca ll i)

T he engin ee rin g co nstants of th e co mposite ca n be deri ved from the stiffn ess tensor, which is best eva luated with a computer prog ra m . Ty pi ca l res ults a re sho wn in Fi g. 6. 13, which gives axial and tra nsverse sti ffness predi cti o ns fo r a po lyme ri c co mpos ite. Fi g. 6. 13(a) co nfirm s that , in th l: pr~ l c ti C; lhk vo lumc frac ti o n ra nge up to a bo ut 40- 50°/c), fibres with r;!irl y hi g h ;! SPlT I r;!tios
Ref erences and jilrther reading

Stresses and strains in short -f ibre composites

130

im provements in the stiffness. The transverse stiffness predictions in Fig. 6. 13(b) , however, show clearly that the aspect ratio ha s very little effect o n the transverse stiffness . This is the case for all compQsites.

60 . . . . . Fibre aspect ratio, s = I

(a)

..... ... . Fibre aspect ratio, S = 5

50

'2

- - Fibre aspect rat io, S = 100

0..

S kl

131

40

References and further reading

30

,)'hear lag treatments C lyne, T. W. (1989) A simple development of th e shear lag theory appropriate fo r composites with a rel a tively small modulus misma tch , Mat. Sci. & Eng . AI22 183- 92 Cox, H. L. (1952) The elasticity and strength of paper and other fibrou s materials, Brit. J. A ppl. Phy s. , 3 72- 9 I)o w, N. F. (1963) Study 0/ Stresses near Discontinuity in a Filament-rein/orced Composite M etal, GE Co ., Missile and Space div., Report No . R635D61 I: uk ud a, H . and Cho u , T. W. (198 1) An advanced shear la g model a pplicable to di scontinuo us fiber co mposites, J. Comp . Mat. , IS 79- 91 Na rdo ne, V. C. and Prewo , K. M. (1 986) On th e strength of di scontinuous silicon ca rbide-reinfo rced a luminium composites, S cripta M et. , 2043- 8 O utwater, J . O. (1956) The mech a ni cs o f pla stics reinfo rced in tension, M odern Plastics, 33 56- 65 Rose n, B. W . (1 960) M echa ni cs o f fibre strengthenin g, in Fibre Composite M aterials. B. W. Rose n (ed .) ASM : Meta ls Pa rk , Ohio, cha pter 3

'" :::I

:;

"0 0

E VJ

-co C

:::I

0

20

>10

0

0. 1

0

0.2

0.3

0.5

0.4

0 .6

0 .7

Fibre volume fracti o n , f

12

~

. . . . . Fibre aspect ratio, s

(b)

r

10 f-

- - Fibre aspect ratio, S = 100

'2 0..

S

kl

=I

.. .... ... Fibre aspect ratio, S = 5

8 '-

'" :::I

:; "0 0

E

6

r

VJ

-co C :::I

0

>2

0 0

0.1

0.2

0. 3

0 .4

0 .5

Fibre vo lume frac ti o n,

0 .6

0 .7

f

Fi g. 6. 13 Es helby pred icti o ns of th e Y o un g' s m od ~t1u s a s a fun c ti o n o f li brc vo lum e fr ac ti o n fo r glass fibres w it h as pect rati os 0 1 I, 5 a nd 100, III ;I n e pox y m a tri x fo r (a) a xi;l l and (h) tr;ln svc rse load in g .

Cshelb), m odelling Bro wn , L. M . a nd C la rk e, D. R . ( 1975) W o rk ha rd enin g du e to intern a l stresses in co mp osite materi a ls, Acta Me tall., 23 82 1- 30 C lyne, T. W. a nd With ers, P. J. ( 1993) An Introductioll to Me tal M atrix Composites. Ca mbrid ge U nive rsit y Press : Ca mbrid ge Es he lby, J. D . ( 1957) The d etermin a ti o n o f th e elas ti c fi eld o f a n ellipsoid a l inclusio n , a nd rela ted pro blem s, Proc. R oy. Soc ., A241 376- 96 Es he lby, J. D. ( 1959) The elas ti c fi eld o uts ide a n elli pso ida l inclusio n , Proc. Roy . Soc ., A252 56 1- 9 M ura , T. ( 1982) Microm echanics of Defects in Solids . M a rtinu s Nijh o ff Pcderse n, O . B. ( 1983) Therm oelas ti cit y a nd plasti cit y o f co mposites - I. M ea n fie ld t heo ry, Acta M etall. , 31 1795- 808 Ta na ka, K. a nd M o ri , T. ( 1970) The ha rd enin g o f crysta ls by no n-d efo rmin g pa rticl es a nd fibres, Acta M etall., 18 93 1- 9 Taya , M. ( 1988) M od elling o f ph ysica l pro perti es o f m eta llic a nd ce ra mic co mp osites; ge nera lised Es helby m od el, in Mechanical and Physical Behavio ur 0/ Me tallic and Ceram ic Composites. S. I. A nde rse n et al. (ed s.) Ri so Na t. La b .: Ros kilde, D enm a rk pp. 20 1- 3 1 W it hers, P. J ., Smith , A. N ., Ciy ne, T. W . a nd Stob bs, W . M . ( 1990) A p ho tog ra p hi c exam in a ti o n o f th e va lid it y o f the Eshelby app roac h to th e mod ellin g o f M M Cs, in Fi ll/(II/IJ/ellwl Relatiollships hetll"('ell Mic rostructural 111111 A/('cl/((I/im l I'mlll'rties A/etal Matrix CO /ll posites. M . N . Gungor ;lIHI i'. K . I i;l lV (l·d , . ) TM S pp . ~~) 40

or

132

S tresses and strains in

short~flb re

composites

7

General Carrara, A. S. and McGarry, F. l. (1968) Matri x and interfac ia l stresses in a di sco ntinuou s fibre co mposite model , 1. Comp. Mat ., 2 222~4 3 Galioti s, C, Youn g, R . l ., Ye un g, P. H . J. a nd Batchelder, D . N. ( 1984) The stud y of mod el pol ydiacetyl enejepoxy composites . Part I: The axia l strain in the fibre , 1. Mat. Sci., 19 3640~8 McDanels, D. L. ( 1985) Analysis o f stress~s tra in , fracture , a nd ductility of a luminium matrix composites co nt a inin g discontinuous silicon carbide reinforcement , Metall. TraIlS. , 16A II0 5~ 15 Termonia , Y. ( 1987) Theoretica l study of the stress tran sfer in single fibre co mposites, J . Mat. Sci., 22 504~8

The interface region

The preceding three chapters ha ve dealt lI'ith the elastic behaviour 0/ composites. Amollg the asslImptio/l.l' made ill IIIOst o/ these treatlllellts is that the illter/acial bond is 'perFect '. This means that there is no debonding, cracking or sliding

0/ allY

in/act , no elastic or illelastic processes

description. In practice , manl' importallt phenomella may take

place at the inter/ace, depending all its structure and the stresses gellerated there. These processes tend to promote plastic def ormation 0/ the matrix and can also inf7uence the onset alld nature of/ailllre. Before treating the strength andfi'acture beha viour o/composites ( Chapters 8 and 9), it is necessary to consider the illter/ace and examine hOIl' its response call be characterised and illfluell ced. III the present chapter, the meaning and meaSUrel1lell t

0/ bond streng th are described. 0/ inter/acial bOllds ill

fo llowed by an outline a/ the fo rmation

This is various

systems and a sllml1lary a/the techniques used to influence the bonding characteristics.

7.1 Bonding mechanisms 7.1.1 Adsorptioll alld wettillg

II' the s urfaces of two bod ies spo ntaneously come into intimate (atomic "call:) co ntact when they are brought close to each ot her (common ly with (ll 1l': of the bodies in liquid form) , then 'wetting' is sa id to have take n pla ce. Adhes ion is primarily caused by van der Waals forces , a lthoug h ot her types of bondin g may reinforce these . The occurrence of wetting ('a ll he trea ted lI sin g simple th ermod ynami cs, but in practice there ma y be (' hc lllical c han ges taki ng p la ce whieh arc tim e-dependent. Fig. 7. 1 illu str:t\cs so lid /so lid :rlld solid / liquid int e rfa ces . The so lid / solid con tac t area

In

7. J Bonding mechanisms

The interface region

134

a nd the interface surface energy . Interface surface energies are difficult to o btain (and may be innuenced by chemical reactions) , but they are freque ntly sma ller than the values for the phases being exposed to a ir . The surface energies of fibres and (liquid) matrices are generally known and systems where the former greatly exceeds the latter are likely to wet very easily. For example, glass hsv = 560 mJ m - 2 ) a nd graphite h sv = 70 mJ 111 - 2 ) are readily wetted by polyester hLV = 35mJ m - 2 ) and epoxy b LV = 43 mJ m - 2) resins, but polyethylene fibres hsv = 31 mJ m - 2 ) a re not. Lack of wetting is a problem for certa in metal matrix composites ~ Ind coatings on the fibres are used to improve this (see §7.3. 1).

(a)

(b)

135

'YLV

7.1.2 Interdijfilsioll alld chemical reactioll Fig. 7.1 (a) Isolated contact points leadin g to weak ad hesio n between tw o rigid ro ugh surfaces. (b) Contact a ngle a nd surface energies I for a liquid drop o n a so lid surface.

e

is limited to those regio ns where as perities to uch (Fig . 7.1 (a)) a nd the effective bo nd stre ngt h is very low unless extensive deformation is promoted to rem ove the as perities . Surface contamination ca n a lso rest rict the a rea of effective contact. For the liquid /so lid case , intimate co ntact ca n be establi shed providing the liquid is no t too viscous a nd a thermodynamic driving force exists. This is comm only expressed in terms o f surface energies, "'f , so that the 1V0rk of adhesioll , Wa , is a simple net sum , often termed the Dupre equation

Wa = "'fsv

+ "'fLV

- "'fSL

(7. 1)

The subscripts S, L and V refer to so lid , liquid and vapo ur, respectively . The vapour phase is com m on ly a ir. According to this eq uatio n, wetting is strongly favoured if the surface energies of th e two co nstituents are large and their interfacial surface energy is sma ll. In practice, however, a la rge va lue of the liquid surface energy inhibits the sprea ding of a liquid droplet. The eq uilibrium wetting o r COli tact allKle is dictated by the YOUIIK equatioll , obtained by a balance of hori zo nt a l forces , Fig. 7. I(b),

e

"'fsv

=

"'fSI.

+ "'f I.V cos 0

(7.2 )

It follow s that co mplet e wetting (0 0' ) occllrs ir th e slIri';lce e nergy or th e so li d is equ;!1 to or "re;iler th ;1I1 th e sum or th e liquid Si ll i';Il'l' e nCl'gy

Various types of diffusiona l process which promote adhes ion can take place at the interface. For example, Fig. 7.2(a) shows the diffusion of free c ha in ends at the interface between two polymers, which leads to cha in e nta nglements and a rise in the adhesive strength. This effect is employed in some coup lin g agents used on fibres in thermoplastic matrices (see ~i 7 .3.1) . Interdiffusion can also take place in non-polymeric systems, particu larly if it is accompan ied by a chemica l reaction. The adhesive stre ngth is dependent on the nature of the resultant interatomic bonds (and also on the stresses generated by the reaction - see below). Va rious types of chemical reaction may occur at the interface, either de liberately promoted or inadvertent. These can be represented , as in Fig. 7.2(d), by new A- B bonds being formed as a result of interfacial chem ica l reactio ns. These bonds may be cova lent, ionic, metallic, etc., and in many cases a re very strong. There are many examp les of the interfacial bond stre ngth being raised by loca li sed chem ica l reactions, but it is often observed that a progressive reaction occurs which results in the formation of a brittle reaction product. Carbo n fibres are prone to surface reactions with organic groups; the de la il s depend o n the manufacturing method s (e .g. see Scola 1974). An im po rta nt fea ture is the a ngle at which the ba sa l planes meet the free ';urface (see §2.1.1), as many reactions take pl ace preferentia ll y at th e edgcs of these pla nes. For example, the high-modulus PAN -based fibres h;!ve a thick sk in with ba sa l planes predominantly parallel to the surface; reaction takes place less readily tha n in carbon fibres with ba sa l pla nes Ilorma l to the surface a nd the fibres a re prone to co hesive failure as a res ult o r weak int er-plan e bonding. Heat treatme nt s pri or 10 co mposite

7.1 Bonding mechanisms

The interlace region

136

~/

("'~

(b)

///J/J'iJ//J//J/J///J'///' +++++++++++ -

-

-

-

-

-

-

-

-

-

If/T/T/T///T///7//77/7//

137

l"
7. 1.3 Electrostatic attractioll

(Cl_ Fig. 7.2 Interfacia l bonds fo rm ed by (a) mo lec ular entan gleme nt fo llowin g interdiffusio n, (b) elec trosta tic a ttrac ti o n, (c) ca ti on ic gro upsa t th e e nd of mo lecules a ttracted to a n a ni o nic surface, res ultin g in polymer o n entat lO n a t the surfa ce, (d) chemical reacti o n a nd (e) mec hani ca l keying.

fabri ca tion ra ise the bond st rengt h as a resu lt of ox ida ti o n o f the fibres and rem ova l of the surfa ce laye rs. T here arc seve ra l examples of bond strengt h en hanceme nt by loca li sed c hemical react io n in MM(,s and ('M('s . The ·S;ilriI™ . /l-: i1111 11ill:l lihre

11" the surfaces carry net electrical charges of opposite sign, as illustrated in Fig. 7.2(b), then a sustained adhesive force may resu lt. Th is effect is utilised in certain fibre treatments, as in the deposition of coupling agents o n glass fibres (see §7.3. 1). The surface may exhib it anion ic or cationic prope rties, depending on the oxide in the glass and the pH of the aqueous so lutio n used to apply the coupling agents. Thus, if ionic functional , i1anes are used, it is expected that the cationic functional groups will he attracted to an anionic surface and vice versa (Fig. 7.2(c)). Electrostatic forces are unlikely to constitute the major adhesive bond in a composite and they can readily be reduced , for example by dischargi ng in the presence of a strongly polar solvent, such as water.

7. 1.4 Mechallical key illg I"here may be a contributi o n to the strength of the interface from the surface roughness of the fibres if good wetting has occurred, as illustrated in Fig. 7.2(e). T he effects are much more sign ificant under shear loadi ng than for decohesio n as a res ult of tensile stresses. Some imp roved resistance to tensile fa ilure results if re-en trant angles are present and there is :In increase in strength under a ll types of loading as a co nseq uence of the Illcre;lsed arc:1 or co nt :lct .

138

The interface region

7.2 Experimental measurement of bond strength

139

7.1 .5 Residual stresses The nature of the interfacial contact is strongly influenced by the presence of residual stresses. These arise from a number of sources, such as plastic deformation of the matrix and phase transformations involving volume changes. There are also volume changes associated with the curing of thermosetting resins. One of the most important sources of residual stress is the thermal contraction which occurs during post-fabrication cooling. Since, for most composite systems, the fibre has a smaller thermal expansivity (thermal expansion coefficient) than the matrix , the resultant stresses are compressive in the fibre and tensile in the matrix. This arises because the matrix contracts onto the fibre and compresses it. The nature of the stress field is illustrated by Fig. 7.3. This shows the principal stresses (in axial, hoop and radial directions) for a long glass fibre surrounded by a tube of polyester resin , after cooling through 100 °C. (The stress field in a multi-fibre system , which cannot be predicted from the analytical model used to generate Fig. 7.3, is very simi lar - except near the outer surface.) In polymer matrix composites, some of the stresses generated by differential contraction are relaxed by viscoelastic flow or creep in the ma trix ./I n a metal matrix com posi te, the tem pera tu re cha nges d uri ng cooling are greater, and the matrices are usually quite resistant to creep and plastic now, so that the residual stresses are substantially higher. The normal stresses across the interface (radial stresses) are compressive. This is particularly relevant to the interfacial bonding, since thi s compression ensures that fibre and matrix are kept in close contact and increases the resistance to debonding and sliding. A relatively high interfacial shear strength results when the residual thermal stresses are large. However, the nature of the stresses is different if the fibre volume fraction is so high that the matrix becomes broken up into isolated regions, surrounded o n all sides by fibres. In this case , the matrix tends to contract away from the fibres , leading to a tensile normal stress across the interface!

7.2 Experimental measurement of bond strength The nature of the interfacial bonding affects the elastic and fractun; properties of the composite in a number of ways. Single-fibre experiments are used to obtain quantitativc infor mation about thc bond stren g th . Most measuremcnt s of bond st re ng th involve shea r dehond ing a nd slid in g (ol"ten usin g simpl e varian ts o f the she" r Ll g theory tn illtnprc t th c

50 Stress (MPa)

o

Radial & Hoop

AXiaJ7j-=---

Hoop

_\...!....------~ Radial

-50 Axial

-100

o

2

4

6 Radial distance

(~m)

J'i g . 7~3 Predicted elastic stress field in a long-fibre composite of polyester/ 35% I' lass ftbre, accordlllg to the analytica l coaxial cylinder model of Mikata and Taya (I lJX5 ), showing the effect of cooling through a temperature interval of 100 K. Prope rty data used in producing these plots arc given in Tables 2.2 and 2.5.

da ta), with little or no attempt to change the normal stress across the Inte rface. This is because of the difficulty in applying such stresses to a l·y lindrical interface in a contro lled manner. It might be argued that an IIlte rface exhibiting a high shear debonding stress would also be expected 10 offer strong resistance to a normal tensile stress. That this is not lIecessarily true can be seen from the fact that pronounced interfacial loughness is expected to raise the former (in shearing mode) while having litt le erf"cc t on the latter (in open ing mode). In relating data from intert:lc i:1i tes ts to th c m:ICI"oscopic behaviour of the composite, it should Iilnci"orc hc hOl"lll· III Illind Ihal Ihe two cases Illa y involve different

140

The inter/ace region

7.2 Experil11entall11easurel11ent

0/ bond strength

141

interfacial stress sta tes. Furthermore, some tests use artificial single-fibre 'composites' . The interfaces in these specimens may differ from those in the corresponding real materials , because different manufacturing techniques are used and there are different degrees of constraint in the absence of neighbouring fibres. Finally, while interpretation of test data is commonly carried out in terms of critical stress levels, a n energy-based (fracture mechanics) analysis (Kendall 1985 , Evans et al. 1990, Evans and Dalgleish 1993) may be more useful for some purposes see Chapter 9. Displacement of loading point

x=L

7.2.1 Single-fibre pIIlI-ollt test This has been extensive ly applied to polymer composites. A single fibre , half embedded within a matrix , is extracted under a tensile load. The load- displacement data can be interpreted using an adaptation of the shear lag theory (Lawrence 1972, Chua and Piggott 1985). A schematic illustration of the axial distributions of normal stress in the fibre and shear stress at the interface is given in Fig. 7.4. These distributions are shown at three stages of pull-out , corresponding to elastic loading up to debonding, propagation of the debonding front and subsequent pull-out by frictional sliding. Basic assumptions of the shear lag model (see §6.1.1) , such as no shear strain in the fibre and no transfer of normal stress across the fibre end , are retained in most treatments of this problem. It is conventional to assume that the peak in the load-displacement plot corresponds to the debonding event , occurring at an applied stress 0"0 ' see Fig. 7.4. The treatment parallels that in §6.1 , but with the ratio of the radius of the matrix to that of the fibre , Rl r, retained rather than written in terms of the fibre volume fraction . The analogous equation to Eqn (6.4) is therefore dITI"

dx

Em (UR - U,.) ( I + I/ m )r2 In (R l r)

(7.3)

The displacement conditions replacing Eqns (6 .5) and (6.6) arc now

dll,. ~ = (r

dll U d.\"

0

(jOfr

x

T.

x

x

hg. 7.4 Schematic stress distributions and load- displacement plot during the single-fibre pull-out test. The applied load generates an interfacial shear stress which has a peak near to the front surface. At some critical applied loa d ITo ' thi ~ shea r stress causes the Interface to debond. Debonding then spreads along the Interface and subsequent interfacial motion is by frictional sliding.

which corresponds to the interface being perfectly bonded (up until the de bonding point) and the matrix remote from the interface being uns trained. The second-order linear differential equation governing the va riation of ITr along the length of the fibre is now slightly simpler than Eq n (6.7)

(7.4 ) (7.5)

d 2 (1r d \" 2

1

li -

-0 (II'

r-

(7. 6)

7.2 Experimental measurement of bond strength

The inteljace region

142

lI1g the procedure to composites with a relatively stiff matrix . In general , It IS more convenient to use the push-out test described below.

where the dimensionless constant is given by

n

=

Er(\

(7.7)

+ vm ) In(R/r)

7.2.2 Sinxle-fibre pllsh-ollt and push-doJVn tests

Applying the boundary conditions O"r(O) = 0"0 (where the fibre emerges from the matrix) and O"f(L) = 0 (no stress transfer across the embedded fibre end) leads to

=

0"

0"

r

0

{Sinh[n(L - x)/r1} sinh(nL/ r)

(7.8)

The interfacial shear stress, according to the basic equation of the shea r lag model (Eqn (6.3)) , then becomes

T

r

dO" = -110"0 cos 2 dx 2

= -- -

and on applying this at x

=

from the peak fibre stress

(0"0 ' )

h[n(L -X)]cosech(nL)

=

coth(n L/ r) 2

(7.9)

r

0, the debonding shear stress,

MO ' T.

r

143

T.

is deduced

(7. 10)

Some variants of the basic model have been published. For example, Hsueh (1990) has incorporated the possibility of stress transfer across the fibre end and ensured that the load carried by the free fibre is balanced by that in the composite. This model leads to more complex equations , but the predictions are similar. In particular, the ratio of T , to 0"0' is usually very close to that for the ba sic shear lag treatment. Neither of these models takes account of the fact that the shear stress in the matrix should fall to zero at the free surface from which the fibrc emerges. It has been shown (Grande et al. 1988) that the build up to thc peak value typically takes place over a distance equal to about onequarter of a fibre diameter. This point is considered further in the next section. Models have also been devised for the frictional sliding part of the pull-out curve, but these are morc complex than for the dcbonding event. A largc data base now cxists for pull-out testing of polymcr matri x compositcs (e.g. see Favre 1989). Values for the debondin g shea r stress ge nerall y rail in th e range 5 100 M ra . There arc , howeve r, pr;( c lical difli c ulli es in specim en pre p;(r;Jlion ;Ind h;(ndlin g, p;(rli c lIl :lrl y wh 'n :Ippl y-

T hese tests are much easier to apply to pieces of actual composite than is the pull-out test. The basis of the method is the application of a compressive axial load to the top surface of an embedded fibre until debonding occurs. In the push-ollt test, the specimen is in the form of a thin slice with the fibre axis normal to the plane of the slice. The fibre become~ di splaced so that it protrudes from the bottom of the specimen. The sca nning electron microscopy (SEM) micrographs in Fig. 7.5 show SiC mo nofilaments that have been pushed out in this way. The test is easily a pplied to such large diameter fibres, but for fine (and strongly bonded) fi bres there can be difficulties in preparing and handling thin section specimens. In the pllsh-down (or indentation) test, the specimen is in bulk form and debonding is followed by the fibre frictionally sliding dow nwards over a certain distance, usually leaving a permanent displacement between the top of the fibre and the top of the matrix when the a pplied load is removed. The test was developed and analysed by Marshall (1985) and Marshall and Oliver (1987). Schematic illustrations of the stress distributions and load- displacement curve are shown in Fig. 7.6 for the push-out test. Note that the Po isson effect now raises, rather than lowers, the frictional sliding stress. Analyses have been based on a simple shear lag approach. Fo r example, Hsueh (1990) has presented models for both the de bonding and frictional sliding behaviour. More recently, finite eleme nt method (FEM) techniques have been applied , in order to examine the stress field in more detail (Kerans and Parthasarathy 1991 , Kallas et al. 1992, Kalton et al. 1994). An important conclusion from this work is that the shear lag models significantly overestimate the pea k interfacial shear stress, for a given applied tensile load. This is ill ustrated by the data in Fig. 7.7, which compares FEM and shear lag predictions of the shear stress distribution with experimental data from a ' macroscopic' push-out set-up, constructed from two photoclastic resi ns. The peak of the shear lag curve is apparently too high by a factor of three in this case. An erro r of similar magnitude is prcdic ted for th e pull-out test. In facl. sin ce Ill(: I: LM work has indicated that thc distribution of inlerf;lcial shear sl,,·s, 1,'IH ls 10 he more uniform than th e shear I;( g

7.2 Experimental measurement ol bond strength

The interlace region

144

145

Displacement of loading point

100llm

Fig.7.5 Scanning electron micrograph s (Watson and Clyne 1992) of a wedgcshaped Ti- 6AI-4V / 30% SiC monofilament specimen after slllgle-fibre push-out testing, showing (a) the top surface and (b) the underside near the thlll end of the wedge.

Fig. 7.6 Schematic stress distributions and load- displacement plot during the slI1gle-fibre push-out test. One difference from the pull-out test (see Fig. 7.4) is that the POlsson effect causes the fibre to expand (rather than contract), which raises (rather than offsets) the radial compressive stress across the interface due to differential thermal contraction.

models predict, it may be acceptable in many cases to take it as constant. A simple force balance can then be used to obtain T . from the stress a pplied to the fi bre

0"0" ?

0"0 , T(f'-

= T. L2rr.r 0"0-

... T.= ~

(7. 1 I)

where the fibre aspect rati o, .1", is given by it s length, L, divided by it s diameter. However, the FEM calcula ti ons have also hi ghli ghted the significance or thermal residual s tresses, which in m,IIlY PMC ' :Illd M MC '

sys tems will strongly affect the stress distributions. This effect, plus the e rrors introduced by using the simple analytical models , have probably been responsible for the rather poor agreement often observed between data from different tests. A variant of the basic push-out test has recently hee n developed (Kalton et al. 1994), involving externally applied tension in the plane or the specimen, which allows cxploration of the effects of Icnsilc s tresses Il()rtll;il to the interrace.

7.3 Control of bond strength

The interface region

146

147

(a)

L ID= 3.0 I

..

I- b- I

HID= 1.3 I1T = 0 K E!Em = l.45

x

x'

*


---------S--------~I

~en ....

•

O.l

..

0;

'u ~
.5

i

a

~ooo:ooo

o

en en

~ .c en

0°00°00 o 00 0 000 o 00 0 o 0 0

o

o

0.2

---- FEM prediction -- --- -- Shear lag prediction • Photoelastic data

(b)

•

00

0.8 Fractional distance along fibre, x l L 0.4

0

o ~

00 0 o 000 00 0 o 0 000 000 0 o 00

0.6

o 0 o o

0 0

00 0 0 0 0 0 o 00

o

0

00

0

000

Fig. 7.7 Compa ri son betwee n experimenta l (photoela stic) data and predictions from shear lag model and FEM modelling for the distribution of interfacial shear stress along the length of a fibre, starting from the top, during push-out of a resin ' fibre' in a matrix of a different resin (Ka lton et al. 1994).

o

l'ig . 7.8

o

00 0 0 0

Tests on unidirectional laminae to measure (a) intralaminar shear strength and (b) transverse tensile strength.

7.2.3 Other tests A method which has been used for MMCs and some PMCs is the socalled 'full-fragmentation ' technique. This procedure for deducing a shear strength involves embedding a single fibre in a matrix and straining the matrix in tension parallel to the fibre. The fibre fractures and fragments into a number of pieces. The aspect ratios exhibited by the resulting fibre segments are measured. Analysis is based on a cons tant T, with the Weibull modulus of the fibre taken into acco unt (see LePetitcorps et al. 1989). A related technique has been applied to thin ceramic coatings on a flat metallic substrate (Agrawal a nd Raj 1990). Tests are sometimes carried out on comp lete lam in ae, rather than focusing on particular fibres. For example, F ig . 7.8 s hows tests designed to measure intralaminar shear strength and tran sverse ten sile strength . Such mea s urem ent s ma y be use ful (for example, the y are required for the prediction of laminate failure see fiR. J), but it ca n he very diffi cu lt to relate th e d
7.3 Control of bond strength The interfacial bond strength can be influenced in a variety of ways . A hrief summary is given below of the main types of effect involved.

7.3.1 Coupling agents and environmental effects Ma ny coatings have been developed to improve the durability and Illechanical strength of the fibre- matrix bond and these are usually ter med coupling agents. A good example is provided by those used on g lass fibres , which often suffer from problems caused by pick-up of wate r. Some of the oxides in glass, such as Si0 2 , Fe203 and A1 20 3, i"o rm link s to hydroxyl groups during contact with water and these in turn fo rm hydrogen bonds to water molecules , so that glass picks up wate r very rapidl y. In lim e, thi s can leach out other species in the glass, Ilotabl y N
148

The inlet/ace region

of the water also reduces the wettability of the fibres , reducing "I from 500- 600mJm - 2 to 10- 20mJm - 2. In genera l, the interfacial shea r strength falls as polymer composites are exposed to water, altho ugh with some thermoplastic matrices an increase has been observed (Ga ur

el al. 1994). Coatings which function as coupling agents are designed to elimi na te the leaching effect and raise the effective "I value at least to about 4050 mJ m - 2. The primary function of the coupling agent is to provide a strong chemical link between the oxide groups on the fibre surface a nd the polymer molecules of the resin. A wide variety of commerci a l coupling agents have been developed (e.g. see Jones 1989), but the principles can be illustrated by the simple example shown in Fig. 7.9. This refers to the siialle cOllpiillg agellts, which have the genera l chemica l formula R- SiX 3 . This is a multifunctional molecule which reacts at one end with the surface of the glass and at the other end with the polymer phase. The X units represent hydrolysable groups such as the ethoxy group (- OC2H S)' The silane is hydrolysed to the corresponding silanol (see Fig. 7.9(a)) in the aqueous so luti on to which the fibres are exposed. These silanol molecules compete with water molecules to form hydrogen bonds with the hydroxyl groups bound to the fibre surface (Fig. 7.9(b)). When the fibres are dried , the free water is driven off and condensation reactions then occur, both at the silanol/ fibre junction and between neighbouring silano l molecules (Fig. 7.9(c)). The result is a polysiloxane layer bonded to the glass surface, presenting an array of R groups to the environment. This coating is water-resistant and can a lso form a stro ng bond to a polymer matrix. If the matrix is to be a thermosetting resin , then an R group is chosen which reacts with the resin during polymerisation , thus forming a permanent link . For a thermoplastic matrix , on the other hand, all the covalent links have been formed during manufacture of the polymer. However, choice of R with a fairly short chain which can interdiffuse with the chains of the matrix allows a strong bond to form . The efficiency of coupling agents in glass-fibre reinforced plastics (GFRP), and their effect on the behaviour of the composite, are evident in thc fracture surfaces shown in Fig. 7.10. These are from composites made by injection moulding of polypropylene containing short glass fibrcs , with and without a coupling age nt. It can be see n that the co uplin g agent promotes stro ng bonding, adhesion of a layer of plastic to the fibres and considerable plastic deformation of the matrix. A further point to note about the function of such coup lin g agents is the poss ibilit y or hond rormati o n bcin g reversibl e. Piu ·dd l' lll ; 1I11l (1974)

7.3 Control of bond strength (a)

R-Si X3

H 20

+

-

R-Si (OH)3

R

(b)

R

I

149 +

3HX

R

I

I

HO-Si-OH HO-Si-OH HO-Si-OH

1'(

)-......

____ H

...... o

'

I

0

J

"H ".0'

H'

,/

I

M

(c)

j~......H

"

H'

I

M

R

I

M

R

I

R

I

O-Si-O-Si-O-Si-O

I

o

I M

I

0

I M

I

0 I M

711lT/T///T//l (d)

~ POIYlller

~ Iletwork

R

I

R

I

R

I

O-Si-O-Si-O-Si-O

I

I

I

000

I

M

I

M

I

M

'llT/T///////// hg. 7.9 Silane coupling agents: (a) hydrolysis of an organo-silane to the correspo ndlllg silanol, (b) hydrogen bonding between hydroxyl groups of the silanol an d those attached to the glass surface, (c) polysiloxane bonded to the glass after co ndensa tion reactIons dUring drying and (d) bonding between the functional group R and the polymer matrix.

pro posed that movements at the interface could relax local stresses. This IS shown in Fig. 7.11. In the presence of sma ll quantities of water, surr,lCCS may be ablc to sl id e past each other without permanent bond failun: . Direct ev idence for this reversible bonding has been obta ined by hlllrier Tr
150

151

7.3 Control of bond strength

The interface region lill

I

R

I

O-Si-O

I

I

R H,O+

o

I I

H/

O-Si-O

H

o

0/

M

M

I

I

Ihl

IIIIIII

IIIIIII

IIIIIII

Polymer

Polymer

~;//;//m--

W/I/////02 -R

?

M

R

?

M

R

?

M

?

M

-W///ffml1 Glass

R

R

M

M

R

R

?

'

M

R

M

- V/;/////ff;/;l G lass

hg. 7. 11 (a) Mecha nisms of reversible bond formation associated with hydrolysis as proposed by Plueddemann (1974). (b) Shear displacements without permanent damage to the interfacial bond.

Fig. 7.10 SEM micro graphs of fracture surfaces from injectio n moulded polypropylene containing short glass fibres. A coupling agent was present on the fibres used to make specimens (c) and (d) but absent for (a) and (b). The hlghcl bond strength induced by the coupling agent is clear from the adherence of polymer to the fibres and the shorter pull-out lengths. (Co urtesy of Dow Corning Corporation.)

For MMCs, it is less common for the promotion of good bonding to bc necessary , because some local chemical reaction often occurs natura ll y during fabrication. There are, however, some systems in wh ich wetting is very poor and coatings have been used to improve this (e.g. scc Katzmann 1987). For CMCs, although various types of coating have been developed, these are rarely designed to improve wetting or adhesio n. Fibres are normally added to ceramic matrices in order to improve thc toughness and a relatively low debonding stress is usually preferrcd 10 promotc fri ctional slidin g during fibrc pull-out (see §7.3.2).

7.3.2 Toughlless-reducillg coatillgs I I is so metimes preferable to ensure that the fibre/ matrix interface has a low toughness, so that it debonds easily. This is usually the case when the ma in priority is to raise the toughness of the composite as a whole, by pro moting crack-deflection at the interface, often leading to subsequent ilb re pull-out by frictional sliding, which absorbs a substantial amount of l'nc rgy (see §9.2). As an example of this, consider the data shown in Fig. 7. 12. This shows the effect of carbon coatings, of increasing thickness, de posited on Nicalon™ fibres (§2.1.4) prior to incorporation in a SiC matrix. The interfacial shear strength is reduced progressively, but the macroscop ic toughness increases. Graphite is popular for this purpose in L'c ram ic syste ms and even thicker interface layers have been used in plali ar SiC laminates (see §9.2). Thc promotion of intc rFacial debondin g is often of over-riding impor1;IIlce for ce rami c systems, but it ca n a lso bc dcsirable in polymer composites , p;lrtinIi;lrl y with thcrmosctting matriccs. Unfortunatcly ,

7.3 Control

The interface region

152

10 1000 ,-..

'"

p...

I-

6

Composite toughness

n 0

I

3

"0 0 en

\->*

@"

~

0c:

bn c:
'"....

(IQ

::r ::I

100

Cb

en en

'"

Cl


..c:

'" 'u

I

0;

--0-

Interfacial shear strength

I

7? '3

~ ....

N ~

2:l .5 0.1 10 0

0.4

0.2

0.6

0.8

Coating thickness (~m) Fig. 7.12 Effect on mechanical properties of carbon coatings on Nicalon™ fibres in a SiC matrix. The plot is of measured interfacial shear strength (fro m push-down testing) and material toughness (from the area under the stress:-stra in curve during loading along the fibre axis) as a functIOn of coatll1g thIckness (Lowden 1993).

interfaces which readily debond can lead to poor transverse and shea r properties, and to poor resistance to environmental degradation such as water penetration. In MMCs, the contribution of fibre pull-out to the overa\1 toughness is usua\1y sma\1 and there is little incentive to encourage interfaces to debond readily.

7.3.3 IlIterfacial chemical reactioll alld clijjilsioll barrier coatillgs Interfacial reaction can be quite extensive in certain types of composite, particularly with meta\1ic matrices. It may occur during composite fabrication and under in-service conditions. There are many fibre /matrix combinations for which chemica l reaction is thermodynamically favoured. Extensive reaction is usually undesirable, since it tends to promote interfacial cracking and the reaction product itself is often a brittle ceramic o r intermetallic compo und (sce Chapter 9). Techniques for avoiding excessive rea cti on depend either on relativel y slow reaction ki!l(;ti cs o r O il the provi sion of some protective b ye r (dirrll sioll harrie r) .

0/ bond strength

153

Interfacial chemical reactions are of particular concern for titanium composites. Titanium and its alloys tend to react with most reinforcements and there is interest in their use at elevated temperatures (550~ 700 °C). Furthermore, in the case of titanium , the surface oxide film which usually protects such reactive metals tends to dissolve in the matrix at temperatures above about 600 QC. Titanium reacts with virtually all reinforcement materials during fabrication, which requires temperatures of at least about 850 °C. In particular, quite substantial reaction occurs J uring fabrication with SiC monofilaments (Martineau et al. 1984) , which is the most promising of the available long-fibre reinforcements I"or use in titanium. In general, fewer problems of interfacial reaction arise with aluminium , particularly during solid-state processing. Prolonged ex posure of SiC to an aluminium melt does cause chemical reaction but this is not a severe problem even for casting routes (see §11.2.2): Reaction problems can occur with magnesium alloys, although, since magnesium does not form a stable carbide, it is thermodynamically stable in contact with SiC. It does, however, tend to attack most oxides. Coatings designed to protect the fibre against interfacial chemical a ttack have been extensively studied for MMCs , particularly titanium, a nd, to some extent, for CMCs. These coatings must be thermodynamica lly stable, with a low permeability to migrating reactants and a high resistance to mechanical damage. The first condition severely limits the choice of materials, while the second effectively determines a minimum laye r thickness. Techniques such as Physical Vapour Deposition (PVD), C hemical Vapour Deposition (CVD) and sputter deposition tend to give coatings with a fine radial columnar grain structure, in which the grain bo undaries provide good through-thickness diffusion paths. In fact, most of the coatings developed hitherto for use in titanium M MCs are not thermodynamically stable in contact with Ti, but react re latively slowly and offer prolonged protection for the fibre itself. Carbon and duplex C/TiB 2 layers have been used. The micrograph in F ig. 7. 13 shows the interfacial region of a Ti/SiC composite, with a Cl Ti B2 coating on the fibre, after thermomechanical loading. In this case, so me reaction occurred during fabrication to form the monoboride TiB a nd debonding took place between this layer and the titanium ma;rix. '

7.3.4 The illterphase regioll The concep t of an int erl"ac ial reaction can bc genera lised to encompass variou s modiriC;llinll s In Ih e Ill ~ ltrix microstructure in the vicinity of th e

,11

The interface region

154

_ 1

References and further reading

2

155

~m

Fig. 7.13 SEM micrograph of a Ti / 30% SiC monofilament composite aftcr transverse loading with s uperimposed thermal cycling. The labe ll ed regions a rc I - SiC fibre, 2 - C inner coating, 3 - TiB2 outer coating, 4 - Ti B reaction layer, 5 - Ti- 6AI-4V matrix. (Courtesy of P. Feillard)

fibres. This idea has been extensively investigated for polymer composites. A summary of the effects involved has been presented by Hu ll (1994). Parts of the matrix which have been affected by the presence of the fibres are sometimes referred to as the illterphase region. Since the microstructural modifications often induce changes in mechanical behaviour, the presence of the interphase can have a strong effect on thc properties of the composite. An example is provided by the micrograph shown in Fig. 7.14. This shows a carbon fibre /PEEK composite in which a degree of crysta lli sation has occurred in the matrix surrounding the fibres. Nucleation o r spherulitic crystallites has occurred preferentially at the fibre surfaces. Such an increase in crystallinity tends to depress the fracture toughness and raise the tensile strength of the matrix. Furthermore, the vo lulll(; contraction accompanying crystallisation will set up residual stresses which may affect the behaviour (see §7.1.5). Stron g effect s can al so be produced in thermosetting matrices. It has bee n sho wn (Wri ght 1990) that the nature o f th e fibre surfa ce can affect th e curin g kine ti cs and c ross-link density o r nearby ma tri x. Surfa ce coatin gs or v
I ig. 7. 14 SEM micrograph (Barlow et al. 1990) of an etched section from a " a rbo n fibre / PEEK composite, showing partial crystallisation of the matrix around the fibres.

(li 7.3. 1) can modify the behaviour substantially. The proportion of the matrix affected in this way by proximity to the fibre surface can be ~ lI bs t a ntial , particularly in composites with high fibre contents.

References and further reading !\grawal , D . C and Raj , R. (1989) Measureme nt of the ultimate shear strength o f a m e tal ceramic inte rface , ACla M ewlI., 37 1265- 70 Ila r low, C. Y. , Peaco ck , J. A. and Windle, A . H. (1990) Relation ships between mi c ros tructures and fracture energies in carbon-fibre/ PEEK composites, C OIIlPOsiI I'S, 21 38 3- 8 ( 'app lc m a n, G. R ., Watts, J. F. and C1yne, T. W. ( 1985) Th e interfacial region in squ cczc infiltratcd co mposites co ntainin g D-alumina fibre in an ;i1uminiulll m;llri x, .J. M al. Sci., 20 2 159 68 ( 'li ua , P. and Pi ggo ll , M . R . ( 19X5) Th c gla ss fibre po lyme r inl e rfa ce: I Tli cnrcli L';i1 L"oll sid l'l"a lio ns !"o r sin g le librc pllll o ul tcsls, C OII/p . Sci. Tec h., 22 \ \ ..p

156

The inlet/ace region

Clegg, W. 1., Horsefall, I. , Mason , 1. F. and Edwards, L. F. (1988) The tensile deformation and fracture of AI- SaffiITM metal matrix composites, ACla Metall., 36 2151 - 9 Evans, A. G. and Dalgleish, B. J. (1993) The fracture resistance of metalceramic interfaces, Mal. Sci. & Eng. , 162A 1- 13 Evans, A. G ., Ruhle, M ., Dalgleish , B. 1. and Charalambides, P. G. (1990) T he fracture energy of bimaterial interfaces, Mal. S ci. & Eng., 126A 53- 64 Favre, J. P. ( 1989) Review of test methods and testing for assessment of fibrematrix adhesion , in Intet/acial Phenomena in Composile Materials 1989. F. R. 10nes (ed.) Butterworths: London pp. 282- 93 Gaur, U. , Chou, C. T. and Miller, B. (1994) Effect of hydrothermal ageing on bond strength , Composiles, 25 609- 12 Grande, D. H. , Mandell , J. F. and Hong, K. C. C. (1988) Fibre- matrix bond strength studies of glass, ceramic and metal matrix composites, J. Mal. Sci. , 23, 311 - 28 Hsueh , C. H. (1990) Evaluation of interfacial shear strength, residual clamping stress and coefficient of friction for fibre-reinforced ceramic composites, Acla M etall. Maler. , 38 403- 9 Hull , D. (1994) Matrix-dominated properties of polymer matrix composite materials, Mat er. Sci. & Eng. , A184 173- 83 Ishida , H. and Koenig, J. L. (1980) Hydrolytic stability of si lane coupling agents on E-glass studied by Fourier Transform infra-red spectroscopy, Proc. 35th SPI/ RP Ann. Tech. Conf , paper 23-A , Soc. Plas. Ind. Jones, F. R. (1989) lnterfacial aspects of glass fibre reinforced plastics, in Inter/aeial Phenomena in Composile Malerials 1989. F. R. Jones (ed.) Butterworths: London pp. 25- 32 Kallas, M. N. , Koss , D. A. , Hahn , H. T. and Hellman, J. R. (1992) lnterfacia l stress state present in a ' thin slice' fiber push-out test , J. Amer. Ceram. Soc. , 74 1585- 96 Kalton , A. F. , Ward-Close, C. M. and Clyne, T. W. (1994) Development of the tensioned push-out test for study of fibre/matrix interfaces, Composiles, 25 637--44 Katzman , H. A. (1987) Fibre coatings for the fabrication of graphite reinforced ma gnesium composites, 1. Mal. Sei. 22 144-8 Kendall , K. (1985) Fracture mechanics of interface failure , Mat. Res. Bull. , 40 167- 76 Kerans , R. J. and Parthasarathy, T. A. (1991) Theoretical analysis of the fiber pullout and pushout tests, J. Mal. S ci. , 27 3821 - 6 Lawrence, P. (1972) Some theoretical considerations of fibre pullout from an elastic matrix , J. Mal. Sei. , 7 1- 6 LePetitcorps, Y. , Pailler, R. and Naslain, R. (1989) The fibre /matrix interfacial shea r strength in titanium alloy matrix composites reinforced by SiC or B CVD filaments , Comp. S ei. & Tech. 35 207- 14 Lowden , R. A. (1993) Fiber coatings and the mechanical properties of a continuous fiber reinforced Si C ma tri x co mposite, in Designing Ceralllic Inlel/aces 11. S. D. Peteves (ed .) Co mm . o f Europ . C ommunities: Luxembourg pp. 157 72 MaI's ha ll , D . B. (1985) An indcntati o n meth od fo r mea surin g fibre ma tri x fri c ti o na l stresses in ce ra mi c co mposit es, .J. A 111(' ,. . C('rl/ll/ . Soc., 67 259 60 Marshall , D . B. and O li vc r. W . C. ( I<)X7) Mcas urcmc nt o f int c rfa c ia l mce hani cal pro pe rti es in lilx:r-n:info recd ec rallli c co mp os it es . .I . / I /I/{',. . ('('/'(1/ /1 .•<';0(, .• 711 542 X

References and fur/her reading

157

Ma rtineau, P. , Lahaye, M. , Pailler, R. ,. N a slain, R. , Couzi, M. and Cruege, F . (1984) SIC filament/ tItanIum matrIX composites regarded as model composites. Part rr: FIbre/ matrix reactions at high temperatures, J. Mal. Sci., 192731--48 Mi kata, Y. and Taya, M. (1985) Stress field in a coated continuous fibre composite subjected to thermomechanical loadings .J. Comp. Mat . 19 554-79 " Pl ueddemann , E. P. (1974) Mechanisms of adhesion through silane coupling agents, III Composile Mal erials, Vol. 6. E. P. Plueddemann (ed.) Academic: New York pp . 174-216 Sco la , D. A. (1974) High-modulus fibres and the fibre- resin interface in resin composites, in Composite Mat erials, Vol. 6. E. P. Plueddemann (ed.) Academic: New York pp. 217- 84 Ta kehashi , H. a nd Chou , T. W. (1988) Transverse elastic moduli of unidirectional fibre composites with interfacial debonding M elall. Trans . 19A 129- 35 " Wa tson, M. C. and Clyne, T. W. (1992) The use of single fibre push-out testing to explore II1terfaclal mechal1lcs III SIC monofilament reinforced Ti , Aela Metal!. Ma/er. , 40 135--45 Wri ght, W. W. (1990) The carbon fibre/epoxy resin interface - a review Compos. Poly m. , 3 231 - 57 '

8.1 Failure modes

0/ /ong~fibre

8

159

composites

"+B.~+ ~8(a)

Strength of composites

(b) The elastic behaviour o/Iong- and shorl~/ibre composites was described in Chapters 4 to 6. The stresses in the individual plies 0/ a laminate under an external load and the stress distributions alollg short fibres ll'ere examined. This informatioll is used to explore the ways in Ivhich a material sullers microstructural damage, leading to the ultimate/ailure 0/ a component. There are tll'O important aspects to this behaviour. Firstly, there is the de/lectioll , degree o/damage alld ultimate/ailure 0/ a component as a limction 0/ applied load. SecondlF , there are the processes ll'hich cause absorption o/energy ll'ithin a composite material as it is strailled. The laller determille the toughness o/the material and are treated in Chapter 9. In the present chapter, allelltion is cOllcentrated on predicting the applied stress at ll'hich dal1lage and /ailure occur. The treatment is oriented tOIl'ards long~/ibre materials and laminates, and, in particular, tOIl'ards polFmer-based composites. Most 0/ the prillciples appl) , equally to discontinuous reinforcement and other

EiiJ

~(C) ....... hg. 8. 1 Schematic illustration of how an arbitrary stress state in a lamina gives

IIse to failure as a result of exceeding critical values of (a) axial tensile stress Ulu , (b) transverse tensile stress

U2u

and (c) shear stress

T12u '

;It the fibre /matrix interface or primarily within the fibre. To predict the stre ngth of a lamina or laminate, values of the failure (ultimate) stresses 0 lu. (J2u and TI2u have to be determined.

types o/matrix. Some specific points concerning/ailure o/such systems are dealt ll'ith in Chapter 9.

8.1.1 Axial temile failure 8.1 Failure modes of long-fibre composites The application of an a rbitrary stress state to a unidirectional lamina can lead to fai lure by one or more basic failure processes. The three most important types of failure are illustrated in Fig. 8.1. Large tensile stresses parallel to the fibres . (JI. lead to fibre and matrix fracture, with the fracture path normal to the fibre direction. The strength is much lowe r in the transverse ten sion and shear modes and the compos ite fractures on surfaces parallel to the fibre direction when appropriate rr ~ or T I ~ streSSeS are app lied . In th ese caseS, I'r;lclurc 1ll;I Y OCCllr c ntirel y wilhill Ih l' 1ll ;ltri x, I "X

l J nde rstanding of failure under an applied tensile stress parallel to the libres is relatively simp le, provided that both constituents behave elastil'al ly and fail in a brittle manner. They then experie nce the same axial slrain and hence sustain stresses in the same ratio as their Young's moduli. Two cases can be identified , depending on whether matrix or fibre has Ihe lower stra in to failure, as illustrated in Fig. 8.2(a) and (b). In case (a), the matrix has the lower failure strain (E l11u < Eru)' For strains up to (mu ' th e composi te stress is given by the simple rule of mixtures

toI

I (I

f

)rrlll

[(4.2)]

• 1

160

.[0 rm. mm ... +..........• . .........• ........ . ..

..

Strength

X

0/ composites

~~~~~;e

X fr~~t~re

Ltij

) ( co mposite fracture

(Jru

0:

fmu

(Jru

............ .

(Jmu

a;nfu

cmu

Ca)

t

Eru

E ru

Cb)

Emu

_ - er [mu (J IllU

o;nfu

--_

-

(l·j·)(J _ _ 1l1U

mu

o CC)

f'

f

161

to rise and the load is now carried entirely by the fibres. F in al fracture occurs when the strain reaches Efu, so that the composite failure stress al u is given by /o-fu' Alternatively, if the fibres break before matrix cracking has become s ufficiently extens ive to transfer a ll the load to them, then the strength of the composite is given by al u = /O-rmu

+ (I

- ./)amu

.f"

==

O"m ll

aru - afmu

(Jfu

(J

composites

(S. I)

where afmu is the fibre stress at the onset of matrix cracking (E l = Emu). T he composite failure stress depends therefore on the fibre vo lume fractio n in the manner shown in Fig. S.2(c). The fibre volume fraction above which the fibres can sustain a fully transferred load is obtained by setting the ex pression in Eqn (S. I) equal to./o-ru , leading to

,

(J mu

o/ Iong~/ibre

8. J Failure modes

o f' Cd)

f

Fig. 8.2 Schematic plots for idealised lo ng-fibre composites with both component s behaving in a brittle manner. (a) and (c) refer to a system in which the fibre has a higher strai n to failure than the matrix and show respectively stress strain relationships (of fibre , matrix and composite) and dependence of composite fail ure stress on volume fraction of fibre. (b) and (d) show the sa me plots for the case where th e matrix has the higher strain to failure.

(S.2)

In case (b), shown in Fig. S.2(b) and (d) , Emu> Efu. The fibres fail first , at a composite stra in of Eru' Further straining causes the/ibres to break up into progressively shorter lengths and the load to be transferred to the I//(/ trix. This cont inues until a ll the fibres have aspect ratios below the c ritica l va lue (see §6.I.S). It is often ass umed in simple treatments that o nly the matrix is bearing a n y load by the time that break-up of fibres is co mp lete. Subsequent failure then occurs at an applied stress of ( I - ./)a mu . If matrix fracture takes place while the fibres are still bearing so me load , as shown in Fig. S.2(b), then the compos ite failure stress is al u = /o-fu

+ (I -

./ )amfu

(S.3)

where amfu is the matrix stress at the onset of fibre cracking. In principle, this implies that the presence of a small vo lum e fraction of fibres reduces the co mposite failure stress below that of the unreinforced matrix , as show n in Fig. S.2(d) . This occ urs up to a limitin g value/' given by setting the right-ha nd side of Eqn (S.3) eq ual to ( I - ./)amu ./" ==

0"11111 -

O"mfu

afu - amfu Above this strain, however, the matrix starts to undergo microcracking and this corresponds with the appearance of a 'knee' in the stress stra in curve, a s shown in Fig. S.2(a). The composite s ubsequently extends wi th little further increase in the applied stress . As matrix cracking continues, the load is tran sferred progress ively to the fIbres. If the strain does not reach (I'" during this st~ l ge, furthcr extens ion causes the cOlllposit e s tress

+ a mu

+ a mu

(S.4)

An example of the app lication of this analysis to rea l systems is given in Table S.I, for two polymer matrix composites and one ceram ic matrix co mposite. (It is unrealistic to apply this ca lculati on to ductile metallic ~ llld thermoplasti c matrices , which normally depart from linear ela stic hehavio ur we ll before failure occurs.) For the two cases where matrix failure OCCIIJ'S firsl, Ih L' vo lulll e fraction , I', above which the libres can

8.1 Failure modes

o

o o o

o o if)

0;

00

o

o

00

00 00 0-

~

00 00

or-

01

o 0'

r- 0 ON

010'

if) -

00 0

""

00 -

66

00 M

E

~ ;;..,

'" '" ~

'v;

o

P.-

E o

u

(l) ....

oD

t.=

U

r-

ollong~flb,.e

composites

163

l'arry all the load is about 11%. With the carbon/epoxy system, the fibres make a contribution to the failure stress at fibre contents above about )5% . Since most long-fibre composites have fibre volume fractions hetween about 30% and 70%, it IS evident that the fibre strength IS do minant 111 determining the axial strength of the composite and the product faru provides a reliable estimate of (/Iu (at least for polymer matrix composites). A lthough the above conclusion about estimation of (/Iu is often valid , Ihe o verall treatment represents a gross simplification. In reality, micro"racking of the matrix does not result in the matrix becoming completely IInloaded and fibres still carry some stress even after they have broken Illto short lengths. After the onset of damage at the 'knee', there is a ,'ha nge in the slope of the stress- strain curve, but it does not 111 fact Icduce to zero (cr. Fig. 8.2(a)). These effects arise because load is transkrred across the interface even after the fibre or matrix fractures. r urther assumptions in the above treatment are that the fibre strength I, a co nstant and that the fibres fail in isolation from each other. In fact , Illost types of fibre exhibit a range of strengths (see §2.2.4); the variability III strength is greater when the fibre displays a low Weibull modulus. I'hus, when a load is applied parallel to the fibres, the first failure is at Ihe weakest point. If the stress redistribution associated with this failure " no t sufficient to cause adjacent fibres to break , the applied stress Illcreases and further fractures occur randomly throughout the material , .IS il lustrated in Fig. 8.3. Several models have been proposed to treat this process , broadly falling into two groups. In the cumulative weakening I//ot/els , first proposed by Rosen (1965) , a random sequence of fibre Iract ures occurs with increasing load until the residual strength of the l"om posite across a section somewhere along the length is reached . This provides an upper bound on the strength , Sll1ce 111 practice there is a lendency for damage to become concentrated 111 certain regIons. In fibl'e break propagation models, suggested by Zweben and Rosen ( InO ), the initial failure sequence again involves fracture of individual Ilhres at weak points. As each fibre breaks , redistribution of stress occurs , kad in g to additional stresses on neighbouring fibres associated with a II)ca l stress magnification effect. Thus , there is an increased probability th;lt fracture will occur in closely adjacent fibres. This is illustrated by the , tress di stribution plots in Fig. 8.3, which show peaks where neighbourlilt: fibres have broken . T he mod e ls have limit ed predictive po wer, becau se the stress field ;IIO llIHI ;1 rr;l c tllrcd rihre is ,c ll siti vc to detail s o f int erfa ci;i1 stru cture (see

164

8.1 Failure modes

Strength oj" composites

F

oj"long~fib,.e

165

composites

Prediction of stress field s around cracks by the use of numerical modl'lIi ng has proved useful in exploring what is likely to occur after a fibre has frac tured . Pioneering work of this type was carried out by Cook and (io rd on (1964). They presented stress magnification contours around a crack tip, which are shown in Fig. 8.4. Thi s demonstrates that, while the "I stress parallel to the applied load is substantially magnified near the na ck tip (Fig. 8.4(a)), there is also a significant transverse 0"2 stress, in the direction of crack propagation (Fig. 8.4(b». Thi s transverse stress, which peaks at a point slightly in front of the crack tip, may cause debonding of ;1 lib re/matrix interFace in front of the crack , as shown in Fig. 8.4(e). This l'
"

"

(b)

(a)

"11 , '" 11 ,

Fig. 8.3 Schematic depiction of fibre fracture even ts according to the Zwebcn Rosen modcl of tensile failure in aligned long-fibre co mposites. The distribution of axial stress, and of local fibre strength (due to the presence of naws) along th..: lengt h is shown for the fibre marked F. (c )

Chapter 7) and to matrix yielding and fracture beh aviour, which can be influenced by a wide variety of factors . However, so me expected trends can be identified . Clearly, the relative importance of cumulative weakening and fibre break propagation depends on the Weibull modulus and the interfacial properties. For examp le, fibre break propagation becomes more impo rtant when the Weibull mod ulu s is hi gh, while randomly distributed Fracture eve nts predominate when there is a wide variation in local fibre strength, represe nted by a low Weibullm od ulu s. It may al so be noted that th e failure strength of the co mpos ite is ex pec ted to show much less variabilit y, and se nsiti vit y to speci men si/.e, than do th e co nstitu ent indi vi du ;i\ ribres (sce, ror exa mpl e, Ibtd or fand (/;tll; lriall Il) X'I).

"

"

(d)

(e)

l'ig. X.4 Stresses close to an elliptical crack tip predicted by numerical modelling (Coo k and Go rdon 1964). (a) Conto urs of ITI st ress, as a rati o to the applied str..:ss. (h) Contours of IT2 stress, in the transverse direction , as a ratio to the ;Ippli..:d str..:ss. (..:) GcoJllctry of ITI. IT2 and T strcsscs. (d) Crack tip approaching the lihre/matri x interfa ce. (c) Int crfa ci;t\ dcbondin g as a rcs ult or the IT2 stress, hluntin g th ..: crack.

166

ollong~fibre

8.1 Failure modes

Strength of composites

the crack from simply slicing through the fibres it encounters in the crack plane. This crack-blullting mechanism is important in raising the toughness of the composite. Criteria for such crack deflection are examined in §9.1 .2. Cook and Gordon's modelling was done for an isotropic, elastic continuum. More recent work has explored the significance of fibre /matrix stiffness mismatch , plastic flow in the matrix , interfacial sliding, etc. Fo r example, the work of He et al. (1993) is based on the geometrical mode l shown in Fig. 8.5. Predictions from this model are shown in Fig. 8.6. In

167

....~-.-"""

O. 2

-.--r-r--r-,....,-~.~•....,.-r..'""'~~

0-

(a)

0
c::

.S:?

0.15

..... - - - - - - -

~

....

c Cl)

u

c:

o

0.1

.'

u

'" '"

'" Cl)

.~ ~

............................. ..........-....,.._.- - - - - - - - - - , .

;'

/

•/

•••••• ,

--/=0.23 , ErlEm= I = 0.50, Er IE m = I ---/=0.50, E r IE m =2 ....• / = 0.50, Er I Em = IQ

.i .....,

~

005 . "

;,'~'"

0::

00

......... /

'

~,

Ol

Matrix

composites

0.05

,

0.1

I 0.15

,:

I

, ,

0.2

, ,

I

,

0.25

0.3

0.35

Relative sliding stress, or /a r

.... [ z=o o~

0
t::

025 ~

(b)

0.2

1

0

~ .... C Cl)

2R

0.15

I

l

u

c: 0

eeeee

e

u

'"'" '"

0.1

Cl)

b

Cl)

.~

~

Ol

0::

0.05

J 0

eeee eeeee Fig. X.5 ,,/. (1<)')3)

Schemat ic de pic tion o r the geo lllclrical arrangement dev ised hy lie r' l ror nUlll c ri eal Illodc llin g or th c stress li eld around ,I crackcd rihre in all IiC\, lgO II ,tI ' Irray . 1II1Ikr ,lxi,tI lo' ld .

j 0.2

0.4

0.6

Relative far field matrix stress,

0.8

am / a Ym

hg. X.6 Predictions from the model of He el al. (1993). The extra increment fTt or fibre stress in the direction of applied load , due to the presence of the n, lck in the neighbouring fibre, is plotted as a proportion of the stress carried by I he lihres remot e rrom the crack fTr. (a) Effec t of the shear stress T for interfacial \ Iiding adiacent to the crack. (b) Effect of the ratio between th e matrix stress Il'nlO te rrom Ihe n ,lck ,Ind th e Illatrix yield stress, for the C,ISC or Er! Em - 3, / 50'/:', and TlfT l 0.2.

168

Strength oj' composites

8. 1 Failure modes oj" IOl1g~/ibre composites

169

Fig.8.7 SEM micrograph of the fracture surface of an epoxy/ 60% carbon fibre lamina tested under axial tension. The fracture surface is relatively smooth a nd consists of a network of blocky o utc rops of fibres and resin at different levels.

Fig. 8.6(a), the peak stress in a fibre neighbouring the cracked one is plotted as a function of the interfacial shear stress, for two fibrc vo lume fractions a nd three fibre / matrix stiffness ratios. Several effects are apparent here. First ly, the stress concen tration is greater for Cl higher ratio of fibre stiffness to matrix stiffness. Secondly, stress co ncentration becomes negli gible as the interfacial shear strength falls to zero; this is expected, since the stress concen trati ng effect of thc crack is lost if stresses cannot be transmitted to the adjacent matrix. Finally, the stress concentration is reduced if the neighbouring fibre is further away (lower fibre content). A further effect is apparent in F ig . 8.6(b) . This shows the effect of plastic fl ow occurrin g in the matrix. At highcr values of thc ratio between the matrix strcss and its yicld strcss (i.c. more extens ive yieldin g), the stress concentratio n increases somcwhat. This is expected , since yie ldin g has a similar effcct to decrcasin g the matrix stiffness. The pract ical clTect or changing the interracia l shear s tre ngth is illu strated in h gs. X. 7 X. IO , which s how variou s rracturc sllrf;ll'l" frolll lon g-

Fig. 8.8 SEM micrograph of the fracture surface of a polyester/ 60'Yo glass fibre lamina tested under axial tension. Extensive fibre pull-out has occurred.

li bre composites subjected to axia l tension. For the epoxy/ca rbon fibre co mposi te shown in Fig. 8.7, the interfacia l bond strengt h was hig h, causing strong stress concentrat io n a nd a tendency for cracks to pass through large bundles of fibres without much deviation. For the polyes tcr/glass and epoxy/ Kevlar™ composites shown in Figs. 8.8 and 8.9, o n thc othcr hand, th e bond in g was weaker and ex tens ive fibre pull-out ha s occurred. Also evident in Fig. 8.9 is the tendency for Kevlar™ fibres to undcrgo necking and fibrillation (see §2. l .3). F ina ll y, Fig. 8.10 shows that thc prcsence of an adverse environmcnt (in this case hydrochloric a cid), which penetrates aiong thc advancing crack , can dr,llllatically redu ce the li hre sll l' Il I'th (rei;ltive to the interraci,1i s tren g th) and hence

170

8. 1 Failure modes of long~ribre composites

Strength of composites

17 1

I I)' X. IO

SEM mi crog ra ph of th e fracture surface of a po lyes te r/ 60% g lass lilm: 1.IIIl in
.llIse a highly planar fracture with a much red uced failure st ress. Til l' IIllporta nt question of the energy abso rpti on associated wit h dilTe n:llt t\pes of failure is exam ined in Chapter 9. For meta l-based co mposite lami nae, the situatio n is ra ther dirre rcnl ill'ca use the matrix usuall y has the capacity to deform plastically 10 'l' ,trains. A diagram a na logo us to Fig. 8.2(b) can be constructed, takin ' .IL'l'O unt of the ma trix work-hardening behaviour. Fig. 8. 11 shows 1;1) a schematic stress- strain curve a nd (b) predicted a nd ex pcri ill en l;iI liL' pen dence of fai lure stress on fibre volume fraction for compos it cs Illade up of tungsten wires in copper. The expe rimental data ;Ire in I'(lod agreement with si mple theory for thi s sys tem.

I

"'I'

8.1.2 Trallsverse tellsile failure F ig. 8.9 SEM mi c rog rap h o f th e fra c ture s urfa ce o f an e poxy/40"/., Kev la r™ 4<) fibre la min a tested und e r a xia l te ns io n . (b) is a hi g he r Illa g nifi cati o n view o r th e fibre ma rk ed X in (a) and s hows libr ill ation and k ink h; lIl d ro rlll ;ltion in th e lihre

at Y.

It IS not possible to make a simpl e estim a te of (J2u' co mparable with til e l·., tillla l(; o r (J ,u described in the previ o us secti on. The tran sv(; rse strel1 ,tll IS In llu enced by man y ra cto rs, such as the nature or th(; inll:rLtc i;iI hOlld 1111'" til(; lihn: di strihllti o n. Ihe presen ce of vo id s, etc. 111 ' (; 11 (;1'; iI , til '

'I 172

Strength of composites

8.1 Failure modes

ollong~f1bre

composites

173

70

(a)

x

..

60

--

-.

1.'-

0..

6

40

1."-

~/

'" Vl <1)

1..-

30

b

Ul

-- ----- ------- ------------ -

x

.

50

-;;;-

(a)

A I I

.'

.···B

,/

,...-

20 10

J: /:

l

00

2

3

4

5

6

Strain (%) 1200

'2 0..

6

1000

'" '"
b

'"

e

.2

;.§

Experimental data for Cu/W (Kelly 1966)

20

(b)

/ 15

600

'" '"

..J

200

00

/.'

6

400

/' 0.2

0.4


(l-j) O"mu

~.,

10

Ul

0.6

0.8

1.0

/.

Fig. 8. 11 Ax ia l tensil e testing of metal matrix laminae. (a) Idcalised stress- strai n plots for fibre , matrix and composite, a nd (b) a compa ri son between theory a nd experiment (Kelly and Macmillan 1986) for the dependence of fa ilure stress o n fibre vo lume fraction.

. )(

.'

f"

/'

5

Fibre volume fraction , f

'

..'

.' .' .' B

(.-"

b

(b)

..' ..

/

0..

c

'"

/

'2

~

.§

Ar

800

.

./'

/'

~

00

0.05

0 .1

0.15

0.2

0.25

0.3

Strain (0/0) I I)'. X. 12 Stress- strai n curves for (a) three unreinforced polyeste r resin s and (b) I.lIll inac bascd o n these resins, with 48 % glass fibre , tested in transverse tensi o n (Legg 1980) .

strength is less than that of the unreinforced m a trix , often sign ifi cant ly so, and the strain to failure can be eve n more dramatically red uced. A consequence of this is that the transverse plies in a c rossp ly laminate us uall y start to crack belore the parallel plies, even though th ey are less stilT and so carry less load sce }i lL~ .

l'h e influence of fibres on the transverse strength is illustrated by the npe rilllental data shown in Fig. 8.1 2, which compares transverse tensile ' tll'SS s tr;lin plot s ror laminae based on three polyester resins with the hl' lt :lviollr or tltl'Sl' 111:ltl'l"i :tl s ill the ullreillroreed rorm . Both the strengt h

174

8. J Failure modes of long~fibre composiles

Slrenglh of composites

175

and the stra in to failure have been markedly reduced by the presence of the fibres. This is largely due to the inherent tendency for high local stresses and strains to develop in the matrix (see Fig. 4 .3(b)). The fibrcs make little contribution to the strength. If the interfacial bonding is weak , then cracks tend to form at the interface and link up through highly stressed sections of matrix . A process of this type is illustrated in the micrographs of Fig. 8.13. If, on the other hand , there is strong resistance to interfacial decohesion , cracks will tend to form in the matrix close to the interface - where there is a concentration of stress and a high degree of triaxial constraint, so that matrix plasticity is inhibited. Alternatively, some fibres (such as carbon fibres) with a layered structure have little transverse strength and may fail internally. With a metallic matrix , broadly similar characteristics are exhibited . Plots are shown in Fig. 8.14 which illustrate the effect of interfacial bond strength in a titanium composite. These are PO;SSOIl plots, in which th e strain transverse to the loading direction is shown as a function of that parallel to the applied load . The gradient gives the Poisson ratio. The plots are for axial and transverse loading, before and after a heat treatment which raised the interfacial bond strength. Under axia l loading, the Poisson 's ratio ri ses slightly as inelas tic behaviour starts; this is due to plastic flow in the matrix (with an associated Poisson's ratio of 0.5) . Under transverse loading, however, there is a tendency for the Poisson's ratio to fall; this is a result of the interface opening up , which allows extension in the loading direction with little lateral contraction . After the heat treatment , this effect is very limited and failure occu rs at (I much reduced stra in . An estimate of the effect of the presence of the fibres on the transverse strength can be obtained by treating the fibres in the composite as a se t of cylindrica l ho les and consideri ng the reduction in load-bearing cross-secti on thus introduced. (This is not accurate, since the presence of evc n completely de bonded fibres would lead to a different stress distribution in the matrix than would be the case for holes, but th e approach is useful as a guide.) For a simple sq uare a rray of holes, consideration of the maximum reduction in matrix cross-sectional area leads to the following expression for the transverse strength of a lamina having a volume fraction f of fibres

0"211

=

0"11111

[I - 2 (;f") 1/2]

I:ig. !i.U

(R.5)

SEM microgra phs illustrating the propa gation of a transverse crack in a pol yes te r/gla ss lamin a (J o nes 1981).

Strength of composites

176 -2

i

:!

·1.6

~

-,A

1 l 1 1, l

.~ ·1.2

;" ~

-,

S

.5 -0.4

0...

~ ·1.6 gc -'A

S t>~

1

"::;; -0.6 ~

c

~

=-1 V

II

~ -0.2

O\f..-'---'-'---'-'---'-'---'-"-,-"-,-"-,-"-,-"-,-.LL.-'-'-l

-2 ',~~~~~~~~~~~~

~ ·1.8 ~

'"c

~ -0 .4

~

~ -0.2

'"~ '">~

( b) -

0

300

200

o 100

r:'"

-2 ~~~~~~~~~~~~

-1.8

.ll-'·6 l

Fibre vo lume fraction , f (%)

~ -'A ~

J"'~~ r~

400

~

= 0.03

;; (a)

Experimental (Prewo & Krieder 1972) o 6061/ BORSIC (As-Fab) • 6061 / BORSIC (T6)"

0'0

-,

:s -0.8

500

..c

.~ -1 .2

~

J

§ -0.6

-;C' -1.8

~

@ -0 .8

177

-2

~~~~~~~~~~~~

- 1.8

8.1 Failure modes of long:/ibre composites

{J

VI '

I Ig. H. IS Experimental data (Prewo and Kreid er 1972) for th e tran sverse tensile '.In: ngth of Al j BORSI C laminae as a function of fibre volume fra ction , compared with predictions from Eqn (8.5).

= 0.34

~ -0 .8

~

-0.6

~

~ -0.4 ~ ~

~ -0.2

o

r~

( d)

(c) .

8.1.3 Shear failure

L ·~~~~~~~~~~~W

o

10

LongilUdinal strain (m illistrai n)

12

10

12

Longitudinal strain (millistrain)

Fig.8.14 Experimental data (Wa tson a nd Clync 1993) fr om ax ia l and transverse tensile testin g of Ti 6AI 4VjSiC mo no filam ent laminae, plotted in th e form 01" latera l contraction as a function of extension in the load in g direc ti o n . Plots are for (a) ax ia l and (b) transverse load ing of as-fabricated composite, and for (c) axia l and (d) transverse loading a ft er a heat trea tme nt which rem oved the g rap hitic coa tin g o n the fibre s.

A comparison is shown in Fig. 8.15 between predictions from this equation and experimental data for AI(6061)/ Borsic composites. In this case. the model gives good agreement with experiment. There is interest in improving the transverse properties of laminae. particularly the failure st rain. Among the possibilities that have becn considered is the provision of a very compliant (e.g. rubber) la ye r on the fibre surface, so as to red uce the stra in loca li sat io n and co nstraint imposed on the matrix. Unfortunate ly. a layer suffi cient ly thi ck to increase sign ifi can tl y the transverse failure strain tends to have an adverse clTcct on ot her properlies. slIch as the shea r stiITIl l:ss.

\s with tensile fracture, shea r failure tends to occur on pla nes determined (he fibre direction. The six possible combinations of plane and shearIllg direct ion , and their indices, are depicted in Fig. 8.16. There are three "l'tS of equiva lent pairs. Normally, th ere is considerable resist a nce to the ilacture of fibres, so that the pa ir of modes denoted 72 1 and 73 1 are 1IIl like ly to occur. Of the other two pairs, involving sliding of fibres IIVl: r o ne a nother ei ther axially (712) or la terall y (7)2), it is not obvious \\ hcther one is inherently more likely than the other. However, when "o ll sidering the stressing of a thin lamina in the 1- 2 plane, stresses of llit: 712 type do not arise and only the magnitude of 712u is important. l\road ly speaking, this is affected by the same factors as the tran sve rse Il'lls il e st rength , because shear stresses and strain s become concentrated 111 thl: matrix betwee n fibres in a manner similar to that outlined above lor tens ile stresses and strain s. However, the details of thi s dependence ,11 e different. There is mo re scope for local matrix deformation to take 111;tcl: (wi th o ut cracking) under this type of stress and local stress concen11 ;I t ions a re relaxed more readi ly. No simple ;tll;il yli c; il l:x pression is available (0 predict the effect of fibre '() Ili ellt Oil TI']]" 1\t\ ;llll S ;llId DOrller (1967) have lIsl:d linite difTerence 11)

178

8.1 Failure modes of long-fibre composites

Strength of composites

179

10

.... 0

u

~

c

8

.~

§

=6 Q)

u

c 0

u Vl Vl Q)

4

b

Vl

.... o:S Q)

..c:

CI1

2

00

0.2

0.8

0.6

0.4

Fibre volume fraction, f I Ig. X.17 Shear stress concentration factor as a function of fibre volulllc fraepredicted by finite difference modelling for a square array of fibres (Adam s and Domer 1967).

11. 1I1,

Table 8.2 Typical experimental failure data for laminae based on therll/o.\·('1 malrices Composite 1"2 1

Fig. 8.16

1"3 1

Nomenclature and orientation of shear stresses acting within a n aligned fibre cOIllPosite.

methods to deduce how the shear stress concentration factor should vary with fibre volume fraction. The results are shown in Fig. 8.17 . Unless the fibre volume fraction is very high (when constraint on matrix deformation becomes severe) , this factor is quite close to unity and TI 2u is expected to have a value close to Tu for the matrix. This is broadly confirmed by the experimental data summarised in Table 8.2. It may be noted that there are both practical and theoretical difficulties to be overcome ill obtaining such data. These are outlined in §8.2.3.

8.1.4 Failure ill compressioll Failure in comp"es,\'ioll is dependent o n the wa y that thc IO ;ldin g is applied and , in parti c lll ;lr , on th e d eg ree of lateral eon s tr;linl. l lnd e r ;l xi;Ii

1'., lycster/ 50% glass I poxy/50% carbon WM) I poxy/50% KevlarT

Axial Transverse Shear strength strength strength al u alu Tl l u (MPa) (MPa) (MPa) 700 1000 1200

20 35 20

50 70 50

Axial Transverse failure I'ailun.: strain strain flu (,X» ( 2 11 ('Y. ,) 2.0 0.5 2.0

03 0.3 04

,,) m pression , there is a tendency for the fibres to huckle. Excepl at low Ilh re volume fractions, neighbouring fibres are constrained to bllckk: in Il hase, as shown in Fig. 8.18. Buckling results in compressive and tellsil' , tn.:sses across different parts of the fibre section, leading either to rra e IlIrc o r (if the fibre is one, such as Kevlar™, which can derorm s ig nili • ;lIlt ly) to local distortion. If buckling becomcs extensive, thell it will I .llIse general collapse, i.e . failure, of the specimen . A m o re common ty pe or railure occ urs fr o m the on set of' a loe;Ii bllck lIn g in s tability . A 1.;111. IWllt! or Illi sori e nted fibres Illa y rorm , a s illu str;lll'd II1 Jo'i g . X. II) . h)'III " X 10 s hows a rr;l e tllred c lrhon lihn: rl'inrorn'd

1I

180

Strength

0/ composites

8.1 Failure modes o/Iong:/ibre composites

181

I ig. 8. 19 Optical micrograph of a poli shed section of a carbon fibre/epoxy resin Jlu ltr uded specimen , showing a kink band formed in the compression zo ne of a four-point bend test specimen (Parry and Wronski 1981).

c/). Argon (1972) proposed that the compressive failure stress could be ex pressed as (8.6)

Fi g. 8.18 (a) T ensile and compressive stresses in a fibre due to in-pha se buck ling, leading to a kink zo ne. (b) Two planes offracturc formcd with brittlc carbo ll fibres . (c) Unfractured kink zo ne form cd with Kevla r™ 49 fibre s.

composite which has failed in thi s way in th e compression zonc 01" ;1 specimen loaded in four point bcndin g. Jclf and Flcck (1992) have shown that most failures und er axial co mpressio n arc o f thi s type . which they refer to as plastic micro/mc/dillg. Plastic dcformation or the matri x is neccssa ry for th c mcchani slll to initi a tc. The main ra ctors whieh inlluence the onset 0 (" this type 01" instahilit y. ; 1\) ~lrt rrom I"ihre co nt ent , ~ 1I"l: malri x shear yield slress. T YIIl' alld (;lver;l gc ) lihrc IlIi s; lli )' llIlI l' lll ;In gk ,

whcre 6. cjJ is in radians. It is assumed that the volume fraction of fibres is hig h enough for this type of failure to be likely. In this regime, the failure , t rcss is pred icted to be independent of fibre content. It is also assumed tha t the interfacial bond strength is high enough to ensure that debonding docs not initiate failure . T he data shown in Fig. 8.21, taken from a number of different studies, co nfirm that there is a strong correlation between matrix shear yield stress and composite fai lure stress in compression. The gradient of the bcst-fit line in this plot corresponds, via Eqn (8.6), to a value for 6.cjJ Il l" abo ut 3". This is physically reasonable, in that measured misalignIlle nts in nominally unidirectional co mposites often range up to about this lig ure. Errors in specime n grippin g geo mctry mi ght typically IlIvo lve l1li s~ di g nl1l l' lIt lip 10 '" I " . Furlhcrl1lore, th ere is sO lll e ev id ence

182

8. J Failure modes of long~flbre composites

Strength of composites

183

on on

~on ~

.2 ~

100

Q)

>

on

i3

0..

10

E

o

u

I

L-~~~~~~~~L-~~~~~~~~

0.1

10

Matrix shear yield stress,

100 TYm

1000

(MPa)

I'ig. 8.21 Compressive failure stress for a variety of unia xial composites, taken fro m several different studies, plotted against the shear yield stress of th e mat ri x (Jelf and Fleck 1992).

Fig. 8.20 SEM micrograph of the fracture surface of a carbon fibre/epoxy res in lamina a ft er a fibre buck lin g mode failure due to a lo ngitudinal compress ive stress. (a) Low ma gnifica tion view showing smoo th fra cture surfa ce. (h) Ili ghma gni fi ca t io n view show ing tensio n and com pressive fr,lct 1I re in , I singlc fi hrc . (I :win s ;Ind Potter Il) X() .

that composites showing particularly good alignment have excellent co mpressive strengths. It is, however, difficult to be precise about the reliability of Eqn (8.6), since it is unclear how large a group of fibres needs to be misa ligned by 6.q; in order for kink-band formation to in itia te . This is an area requiring further resea rch . A composite can also fail under axia l compression by macroscopic shear o n certain planes. This must involve fra cture or gross deformation of li bres, since no shear stresses a re developed on a ny planes parallel to the J"i bre axis. Thi s usually occurs at applied load s similar to those necessary to ca use axia l tensile failure , although modellin g of the mechanisms involved is di fficult. These compressive failure load s can be significantly reduced if fihre huckling occurs. This is genera lly favoured by measures which red uce the stiffness orthe matrix, such as heatin g or, with some polymer matri ces, prolo nged exposure to water. Reduction s in th e interfacial bond strength Ui 7.3) a lso tend to facilitate this. Fin a ll y, there is a significant depend cnce ' 1Il the fibre diameter. Large-diameter fibres a re much more rcs istant to huck lin g than fine fibres. This point was made in §2.2.2. The se nsiti vity to d i;ltlle ter can be inferred from the Eul er buckling formula given as Eqn (2 .1) and thc J"ihn: flexih ilit y data in Tabl e 2.4. An exa mple ofcxploitation o J" the cxce ll enl Il'sist:lnCl' oJ" largc-d iam cter fihrcs to buckling is givcn in :: 12.4, rei:! Iill ) ', 11 ) Ill ,' 11 \ , ' I ) i' horon 1110nofil:ll11cnts in go lJ" cluhs.

Strength of composites

8.2 Failure of laminae under o/raxis loads

In general, the stress- strain plot of a composite loaded under ax ia l compression is rather sim ilar to the corresponding tensile plot. Common ly, the initial gradient is slightly reduced and the failure stress somewhat lower under compression. These changes reflect the stra inin g and damage development which can occur under compression when fibres become mi sa li gned. These compression tests become meaningless if macroscopic (E uler) buckling of the specimen is a ll owed to occur. It is therefore necessary to ensure that the specimen has a low aspect ratio (length/diameter) and /or that suitab le anti -buckling guides are used. Application of the technique to thin lam in ates is described by Lagace and Vizzini (1988). Compression testing of MMCs has been used to obta in information, not only about damage development and failure mechanisms, but also co ncerning residual stresses in the matrix. Differences in yielding behaviour under tensile and compressive loading can be related to these residual stress levels. Some of the experimental aspects of such testing have been described by Kennedy (1989) . Further information can be obtained by load reversal tests, in which a specimen is plastically deformed in tension and then subjected to compressive loading (or vice versa). M MCs often exhibit a pronounced Bauscltillger eflect (easier yield ing after prior reversed plastic flow). Interest has centred mainly on discontinuously reinforced material. Interpretation of experimental data requires some care - see, for example, Taya et al. (1990). Under some circumstances, failure may occur under trallSI'erse compressioll, involving shear on planes parallel to the fibre axis . This is expected when a shear stress of the 732 type (see Fig. 8.16) reaches a critical value - which should be broadly similar in magnitude to 712u' It follows that this type of failure is expected at an applied normal compressive stress of about 271 2u, on planes inclined at 45° to the loading direction and parallel to the fibre axis. Experimental measurements are broadly consistent with this, although there is sometimes a dependence on interracial bond strength; for example, if a high bond strength leads to failure by shear yielding within a polymeric matrix , then thi s usually occurs less readily under compression than in tension .

\ Ive loads and through-thickness stresses are not present.) A number of /ililure criteria have been proposed. The main issue is whether or not the nitica l stress to trigger one mechanism is affected by the stresses tending In ca use the others - i.e. whether there is any interaction between the Illodes of failure.

184

8.2.1 Maximum stress criterioll

111 the simple maximum stress criterion , it is assumed that failure occurs \\ hen a stress parallel or normal to the fibre ax is reaches the appropriate nit ica l value, that is when one of the following is satisfied (ll 2: (llu

(8.7)

2: (l2u 712 2: 71 2u (l2

hH any stress system ((lx, (l, and 7,,0) applied to the lamina , evaluation of Ihese stresses can be carried out using Eqn (5.12)

[(5. 12)] 111 which [T] is given by Eqn (5 . 13) ?

[T] =

)

[, e

s-

s-

e

-es

cs

)

- 2" 2es ) e - s-?

1

[( 5.13)]

(e = cos c/>, s = sin c/» Monitoring of (ll , (l2 and 712 as the applied st ress is increased , allows the onset of failure to be identified as the point when one of the ineq ualIlies in Eqn (8.7) is satisfied. Noting the form of [ T ], and considering ap plied uniaxi al tension , the magnitude of (l,. necessary to cause failure CII1 be plotted as a function of loading angle c/> between stress axis and Ilbre axis, for each of the three failure modes. (ll u cos- c/>

a .nl

== --)-

(T

=--

8.2 Failure of laminae under off-axis loads

Failure of laminae subjected to arbitrary (in-plane) stress states can bc understood in terms of the three railure mechani sms (wilh dcJ'il1 e(\ va lu cs or (T I" . rr.,,, ; 111(\ 7 1 ~11 ) sho wn in Fi g. X. I. (It is ;ISS IIIlIUIIIl ;11 1;II I'l' L'Olllprcs-

185

(l 2u

xu

(T \ 11

sin 2 c/>

sin (/1cos rll

(8.8) (8. 9)

(R. IO )

Strength

186

0/ composites

8.2 Failure a/laminae under off-axis loads

The three curves are plotted in Fig. 8.22, using values of (Jll" (J2u and TI2u appropriate for a polyester/ 50% glass lamina (see Table 8.2). The solid line indicates the predicted variation of the failure stress as q; is increased , according to the maximum stress criterion. Typically, axia l failure is expected only for very small loading angles, but the predicted transition from shear to transverse failure may occur anywhere between 20° and 50°, depending on the exact values of TI2u and (J2u'

whe re (Jp, (Jq and (Jr are the principal stresses and (Jy is the yield stress tinde r uniaxial loading. I n plane stress ((Jr = 0) Eqn (8.11) red uces to (8.12) f'he von Mises criterion corresponds to yield occurring when the distortio nal (shape-changing) strain energy stored in the material reaches a cri tical value. This may be expressed ((Jp - (Jq)2

Various other attempts have been made to predict the failure of longfibre composites under combined stresses, particularly for the plane stress conditions applicable to individual plies in a laminate. A comprehensive review of the approaches adopted has been published by Rowlands (1985). Most trea tmen ts a re ba sed on ada pta tions of yield cri teria deve loped for metals. The most common yield criteria are those of Tresca and von Mises. The Tresca criterion corresponds to yield occurring when a critical value of the maximum shear stress is reached. This may be written as (8 .1 I)

I~

'2

6

....

,

on '"
~

,,,

•..••.•.• Axiat C a,,! cos 2t/!)

600 r

I

500 -

"0

400

0-

300

0-

-

I

- Shear (T'2u I cost/! sint/!)

I

- - Faiture stress

.~


(Jp)2

= 2(J~

(8 . 13 )

(8.14 ) Ada ptation of these criteria to describe failure of composites must take acco unt of the inherent anisotropy of a fibre composite and of the differences between the mechanisms of meta l yielding and of composite fai lure . rhe effect of anisotropy is also relevant to metals, since these are often hig hly textured and hence anisotropic in properties. The effect of anisotro py on yielding behaviour was treated in some depth by Hill ( 1950), who derived a modified von Mises yield criterion for metals with orthotro pic symmetry (having three orthogonal planes of symmetry). Under pla ne stress, this criterion may be written

(g .I S)

~

I

0'

'"

(J,i + ((Jr -

I "

0...

~

-

' I I

1./

'Vi c:

+ ((Jq

lJ nde r plane stress conditions, this becomes

8.2.2 Other failure criteria

:::

187

1

l

I

l 1

80

90

/

200

~

tOO

0

0

to

20

30

60 SO 40 Loading angle, i/J Cl

70

Fig. 8.22 Predicted dependence o n load in g angle IfI o r the applied s tress ror Ih e onset ordilTerentl;lilure modes ror a pol yes terfSO"" g la ss 1:llltill;J , :Iccordillg to th e maxilllulll s tress crill:rioll .

whe re (JI, (J2 and TI2 are the imposed stresses, referred to the orthogonal direc tions in the plane, and the material properties (JIY, (In, (Jw and T12)" ;Ire the mea s ured yield stresses , in tension and shear, when each is applied III iso lation . It has been common to adapt this criterion to failure of unidirectioll ;iI co mposites by replacing the yield stresses by the appropriate measured failu re stresses. Since s uch a composite is transversely isotropic, the Llil ure s tresses in the 2- and 3-directions are equal , so that th e co nditi OI1 red uces to

f'his fo rm ul;1 t ion was fi rst proposed by ;\ zzi a Ild Tsa i (1l)(lS) :111(1 bpt (X. I(l), COllllllOll ly kl1oWI1 :I S the Tmi- lIill criterioll , is widely quolcd ill l'o lllposil e Ic, 1hl\l\k , ;t llll I , IIlh'll lised ill 1:llll ill:ltc :lIl:llysis pro)! r:lII1 S (,l'l'

188

8.2 Failure

Strength of composites

§8.3). It defines an envelope in stress space: if the stress state (ai, a2 anti T12) lies outside of this envelope, i.e. if the sum of the terms on the kl"l hand side is equal to or greater than unity, then failure is predicted. T il l' failure mechanism is not specifically identified , although inspection of til e relative magnitudes of the terms in Eqn (8.16) gives an indication of til e likely contribution of the three modes. Care must be taken that til l' normal stresses are tensile, since the appropriate failure values may l e different in compression (particularly a2u) - see §8 .IA. The values of ai, a2 and T I 2 can be obtained, for a given anglc '/1 between fibre axis and applied stress ax is, from Eqn (5. 12). For a sin gk applied tensile stress a ql (= a x at an angle cp to the fibre axis) Eqn (8. 1Cl) can be written as

i

and this express ion gives the applied stress at which failure is predicted , as a function of loading angle cp.

i

"

I

I

'"

~

.~

300

~

"0

189

under o.fraxis loads 'i

i

i

i

o Experimental - - Maximum stress ........• Tsai·Hill

~ 400 ~

[

.~

0.. 200 0-

«

tOO ~

o (8. 17 )

0/ laminae

1

o

10

I, ,

,

20

30

40 50 Loading angle,

60

70

80

90

Ifi (')

lig. X.23 Dependence of the applied tensile stress for failure of epoxy/ 50% "arbo n laminae on loading angle . Experimental data (Sinclair and Chamis I')79) are shown , together with predictions from the maximum stress and TsaiH ill cri teria .

8.2.3 Experimental data for sillgle lamillae In Fig. 8.23 the predictions of the Tsai- Hill and maximum stress critc ri ;, a re shown, obtained using the sa me critical stress levels all!' a2u and T I 2,, ' These critical values were fitted to experimental data for epoxy/ SO'X, carbon laminae (obtained by cutting nat rectangular coupons at diffe rcnl a ngles to the fibre direction and testing them in tension). The experime ntal data fit rather better to the Tsai- Hill curve, notably for the specimc n at cp = 30° - which was observed to fail by both shear and transvc rse tension. It is appropriate at this point to note that there are certain difficu lti es in carrying out tests designed to obtain critical stress data for differen l failure modes. For example, there are severa l complications with the basic off-axis tensile test referred to above. One of these is illustrated by Fig. 8.24. This shows how tensile- shear interactions (see §S.2.2 and Fig. 5.6) can cause distortion of the lamina under load, such that thl' constraint imposed by the gripping system introduces complex, non-uni form stresses (Pagano and Halpin 1968). Furthermore, specimens ca n hc difficult to grip without slippage, edge effects may he important (sec ~i 8.3.3) and there is no freedom to select (T" (T 2 and T , : imkpendentl y so as to discriminate more elTectively hetween I; ,ilure crilni;1.

++ Unstressed

Ideal grips

Usual constraining grips

I:ig. X.24 Effect of grip constraint on the testing of laminac under off-axis loads, under conditions such that tensile- shear interaction strains are generated.

A more versatile testing procedure is to produce tubular specimens and test t hem under various combinations of applied stress. The performance or various types of composite tube and pipe (particularly under internal pressure) is or cOllllllercial importance and this topic is covered in ~igA. A

190

8.3 Strength o/Iaminates

Strength of composites

simple hoop-wound arrangement, with facilities for applying tension and / or torsion , is illustrated in Fig. 8.25. Any combination of (J2 and 71 2 can be applied , with zero hoop stress ( (JI ) ' For a thin-walled tube ofdiametcr d and wall thickness t (<< d) , the shear stress in the wall is obtained from the applied torque T (in N m) by the expression

-

70 ,.--.. 0;

2T nd 2 t

(8. 18)

60

6 N

~ <1.l

40

b

Vl

0;

(

(J1 )

(J2~1

2

I

( 7p ) 2

+ 71 2~1

=

<1.l

.s:::

1

50 [

.....

Using a grip section with a diameter larger than the tube under test, as shown in Fig. 8.25 , facilitates the application of a large torq ue, with less danger of slippage in the grips. Data are shown in Fig. 8.26 giving various combinations of (J2 and 7 12 which produced failure in epoxy/ 65% glass hoop-wound tubes. The fa il ure envelopes for the two criteria are simple and give very different predictions for this case. The maximum stress theory gives the two lines 71 2 = 712l! and (J2 = (J2u as limits and the Tsai- Hill condition, Eq n (8.16), reduces to

cr---:::-:...-=-------------

0...

en Vl

7p = - -

191

30

(I')

20

L

~

o

Experiment

- - Tsai-Hill, Eqn.(S.15)

10

-

0 0

-

- Maximum stress, Eqns(S.7)-(S.9)

10

20

30

Transverse tensile stress,

40 132

50

(MPa)

hg. 8.26 Comparison between experimental failure data (Knappe and Sc hne ider 1973), obtained by combined tension / torsion testing of hoop-wound l'[)t)x y/65 % glass-fibre composite tubes, and predictions from the maximum stress and T sai- Hill crite ria.

(8.19)

It is clear from the data in Fig. 8.26 that the Tsai- Hill criterion gives a more reliable prediction. Although the Tsai- Hill criterion is useful , it is not based on rigorous theory and it cannot be applied to all situations. Some of its limitations were highlighted by T sai and Wu (1971) , who proposed a more complex criterion involving both tensile and compressive failure stresses and an interaction parameter which must be measured under biaxial loading.

Lven with such modifications, however, the interactions between differc nt failure modes are bein g fitted to an assumed equation , rather than heing modelled on the ba sis of micromechanics. In fact , as pointed out by I la rt-Smith (1994) , much experimenta l data can be explained by establis hi ng the complete stress envelope for each possible failure mode , witho ut introducing any interactions between the modes. While this requires cx te nsive experimental data for each system of interest, it ma y be the best a pproach when reliable predicti o n of failure beha viour is essential.

8.3 Strength of laminates

Fi g. R.25

Sc hemati c illu slral io n o r how a hoo p-wo und lu hc is s uhj cc lcd

10

silllullan co ll s tors io ll
T he strength o f laminates ca n be predicted by an extension of the precedi ng secti o ns, utili sing the procedures set out in Ch a pter 5 for determin
Strength

192

0/ composites

8.3 Strength

considerably greater loads without catastrophic failure. Analysis of the behaviour beyond the initial , fully elastic stage is complicated by uncertainties as to the degree to which the damaged plies continue to bear some load . Nevertheless , useful calculations can be made in this regim e (although the major interest may be in the avoidance of any damage to the component).

8.3.1 Tellsile crackillg Consider first a crossply (0/90) laminate being loaded in tension alo ng one of the fibre directions. The stresses acting in each ply are show n schematically in Fig. 5.13 , for an epoxy/ 50% glass composite. Only transverse or axial tensile failure is possible in either ply , since no shear stresses act on the planes parallel to the fibre directions. The sequence of failure events when a laminate is loaded progressively in this way is illustrated schematically in Fig. 8.27. From the data for Elu and E2u shown in Table

•••• .... .... ....

(a)

(b)

(c)

Fig. 8 .27 Loadin g o f a eross pl y laminat e parall e l to o ne o r th e film; direc tion s. (a ) C ra c kin g o ftran svc rsc plies as (J~ reach es (J~ II ' (b) o nse t of crackin g para lkl to the fibrc s in a x ial plies as (J ~ (fro m inhibiti o n o f Po isso n co ntra c li o n) rC; lc hcs n ! 1I and (c ) Ilnal failur e a s (J I in a x i;" pli cs rea c hes n lll .

0/ laminates

193

2, it is clear that transverse plies fail first , as shown 111 Fig. 8.27(a), Ill-spite the fact that they are less highly stressed than axia l plies. Si nce most of the load is borne by the axial plies (about 85% for the "\am ple shown in Fig. 5.13), cracking of the transverse plies does not I'rca tly increase the iTl stress and the axial plies usually remain unda11I ;lged at this point. As the applied stress increases, the next type of d;lma ge to occur is often cracking parallel to the fibres in the axial piles - see Fig. 8.27(b). This is caused by the tensile iT2 stress, arising Il o m the resistance of the transverse plies to the lateral Poisson contrac11I1Il o f the axial plies. Figures 5.12 and 5.13 show that these stresses are .Ii)() u t 7% of the applied stress. Fo r purposes of comparing iTl and iT2 in the axial plies, a limiting case 1\ to assume that cracking of the transverse plies has made them stress11 Cl' in the loading direction, so that iTl in the axial plies reaches twice the ,Ippl ied stress. (All of the applied load is taken by the axial ply, which ,o mp rises one-half the thickness of the specimen.) Under these circum, lances , iTl is almost 30 times larger than iT2 within the axial plies. Il o wever, for the three polymer composites listed in Table 8.2, the 1;lt io (iTlu / iT2u) ranges from about 30 to 60. Hence , transverse cracks in Ille ax ial plies are frequently observed before the final failure depicted in I·ig. 8.27(c). (This is less likely with metallic matrices , for which the ratio "111/ (J2u is usually smaller and iT2 stresses are lower because of the lower I'oisson 's ratio of the matrix: similar arguments apply to ceramic-based lami nates.) A lthough the laminate depicted in Fig. 8.27(b) has not completely I rac tured , there is extensive microdamage. The network of cracks is \ lIch that the laminate is susceptible to the passage of gases and liquids Ihro ugh the walls, a point of particular relevance for pressure vessels. In ,Id di tion to concern about leakage , the ingress of certain fluids may ilaste n final failure - see, for example, the effect of dilute acid in Fig. 8.10 . A typical stress- strain curve for testing a crossply laminate is shown in I ig. 8.28 . Al so shown on this plot is the output from acoustic emission ('l\ui pment. These measurements are made with a piezo-electric transdu,n a ttached to the specimen surface, which picks up elastic waves trigl'L'red by the nucleation and growth of cracks. The onset of transverse n; lck in g is m a nifest both as an acoustic signal and as a 'knee' in the stress strain curve, cau sed by a reduction in the specimen stiffness as lilL' tran sve rse plies arc unload ed. This reduction in stiffness is most In arked at the beginnin g, but co ntinues o ver an appreciable ran ge o f stra in , a s th e cr;II.: ks ill til L' tra nsve rse pl y become more cl o sely spaced. K

194

Strength

0/ composites

8.3 Strength of laminates 2000

500

195

~

V>

Q)

== ;> Q) ~

'"

Q...

400

'0

1500 Q) ....

6 V> V> Q)

!:: V>

.1:l

E

::l

300

c::

1000

~ .;:;; c:: 200 Q) f-<

500 100

(;j

g '5 0'5 0

.g V>

::l 0

U

0.5

Fig. 8.28

I.5 Strain (%)

2

0 2.5

-<

Typical stress- stra in curve and acoustic emission output of a crossply lamin ate tested in uniaxial tension.

Eventually, however, the axial plies carry virtually all of the load a nd th e gradient of the plot becomes constant again. There are many factors which determine the details of how the transverse cracks develop . Discussions of some of these points can be found in Parvizi and Bai ley (1978) , A veston and Kelly (1980) and McCartney (1992).

8.3.2 Illterlamillar stresses Interlaminar stresses are also a source of damage in stressed laminates. II was pointed out in §5.4 that through-thickness coupling stresses ca use distortions and that these are reduced using ' balanced ' stackin g sequences. There are also interlaminar shear stresses operating to transfer load between laminae and these may give rise to interlaminar cracking. A common source of damage is the T e shear stresses, which arise frolll rotation of the individual laminae , as illustrated in Fig. 8.29 for an angle-ply laminate. This rotation was described in §5.2, durin g coverage of the tensile- shear interaction. For an applied tensile stress, the shea r strain resulting from the rotation is proportional to the interacti o n eOIll pli a nce S1 6 ' It follow s from Fi g . 5.6 that , for ply an gles (i.e . loadin' angles in Fig. 5.6) bel o w 60", th e fibre s tend to rotat e toward s th e s tress a xis, while for large r an g les th a n thi s the y tend to rotat e so a s to lie normal to it. Thi s is re llcc ted in th e d;(t ~1 prese nt ed in h ' . X.30, w hi c h

I I) ' . X.29 Schematic illustration of how axia l tensile stressing of an angle-ply 1.IIIIilla te generates a tendency towards rotation of the component plies, giving rise to interlaminar shear stresses .

', ho ws the result of computations carried out by Pipes and Pagano (1970) )'Iv ing the magnitude of Te as a function of ply angle rP for an angle-ply l'Il()x y/carbon laminate. The nature of the damage induced by these stres', ('S is illustrated by Fig. 8.31, which is a micrograph of a lateral surface of I he ty pe marked ABCD in Fig. 8.29. The matrix shear is evident from the dis pla cement of polishing marks across XX , while the tendency for outtll -p lane movement to occur is apparent from the difference in focus on "It her side of the crack.

8.3.3 Edge effects I he significance of the interlaminar shear failure represented in Figs. K.3 1 d e pends on the width of the specimen. Thi s type of dama ge IS in itiat ed on th e lat e ral ed ge surface , s ho wn a s ABeD in Fi g . 8.29 , and '; In p l;IY ~ I dOlllill ;lllt ro lc ill th e ove rall failure of narro w spec im e ns. Thi s ;-.; ,' l)

196

Strength

0/ composites

8.4 Failure

I.5

~ :.:

'" ~

I -

x

V>

A

V>

y~

.....

'"'"

..c: V>

.....

0.5 -

I

t

'"

0::

'" C

0

,

'"c:: 'E 0

.:::'" C
<)

cG

-0.5 0

10

20

30

40

50

Ply angle,

60

70

80

197

under internal pressure

,' 1kct is illustrated by the data in Fig. 8.32, which show the dependence "11 specimen width of the stresses at which transverse cracking and final 11 :lct ure occur for a ± 50 laminate . If the specimen is very narrow, Illtcria minar shear cracking becomes the dominant failure mode and It ca n occur at such a low applied stress that transverse cracking is , tl ll1p letely suppressed. These effects are most pronounced for the low pl y a ngles at which 7r: is predicted to be higher. In addition, the magIIllude of the fracture stress increases as q; is reduced and 0'1 increases Il' lat ive to 0'2 and 712' These two trends are apparent in the data shown III F ig . 8.33. Thus, care should be taken in specifying the specimen \I idt h for such tests. For certain ply angles, only very wide specimens .dlow the effect of interlaminar shear to be eliminated when interpreting Ih e res ults. This is one reason for the interest in testing of tubes, treated III ij 8.4.

"}

-t

0/ tubes

90

n

Fig. 8.30 Ca lculated interlaminar shear stress, as a function of ply angle 4;, an angle-ply carbon fibre/epoxy laminate (Pipes a nd Paga no 1970).

8.4 Failure of tubes under internal pressure

1'0 1

III addition to the industrial significance of the failure behaviour of COI11pos ite tubes under internal pressure , thi s mode of testing is a convenient

L

~

80

r

•

•

0...

, ;f

6 b'" 60 l ~

I-

V>

V>

c::

2:l

40 _••

"0 .~

0.. 0.

«

20

r.. 0

20~m

~ ~

0

>

~

Fig. 8.31 Optical micrograph o r a n interlaminar cra ck on th e cdgc r:lce (!\B( 'I) in Fig. X.29) o r a gla ss lihre/ po lycstc r rl:sin angk-pl y Ianlill:ll l' (.Ioncs I')X I).

,

,

VJ V>

,

0

o

0

t

0

0 0 ---0--

_

0

Onset of transverse cracking Complete fracture

0 .5

I

1.5

,I = 2

50"! 2 .5

Width of laminate, 2b (mm) I Ig. X.32 I:fl i.:c t (lr iL'st spcc im en width o n tcnsil e stress for transverse crackin g .llId co mpkiL' I'l:1 r lll rl' ill :ln gk -ply laminat es (cl) SO) o f polyester/SO'Y<, gla ss I'ihrl' (.Iones 19X I) .

198

Strength of composites

8.4 Failure of tubes under internal pressure

350

(a)

"1

300

J

199

p

]

.--. ro

0...

~

250

1 "

..c:

bn 200 c <1.)

~

'Vi c<1.) f-

(b)

~

b
150 100 50 0 0

----

/ifl = 45' 1

J I Ig. 8.34

/ifl 5

10

15

i

p

-.:J

= 55' 1

Schematic represe nt a tion of designs for tube test in g wit h (a) pure hoo p ((lA = 0) a nd (b) hoop plus ax ia l ((lA = (lH/ 2) load in g .

20

Width of laminate, 2b (mm) F ig. 8.33 Effect of tes t specimen width o n tensile fractu re streng th of polyes ter/ 50% glass fibre a ng le-ply laminates, fo r various ply an gles (J ones 198 1).

method of generati ng selected stress states in a lam in ate and avoidin g the co mplications of edge effects. Internal pressure P acts on the cy lindrica l surfaces of a tube to cause a hoop stress (TH in the tube wall a nd o n th e sea led ends to generate a n axia l stress (TA' Provided the wall thickness t is much less than the tu be radius r, these st resses are given by

Pr

(TH

=/ Pr 2t

(8. 20 ) (T H

(TA=-=-

2

(8.2 1)

Two modes of testing are commonly used, as depicted in Fig. 8.34. When the pressure is contained with rubber '0' rin gs, o n which th e tube is free to slid e, the axial stresses are elimin ated, so that (JA = O. When th e ends of the tube are sea led , the axia l stresses arc given by Eqn (8.2 1). When testing is to be continued beyond initi al cracking to IInal failure, ;1 rubber linin g is used to eliminate 1e;1!\;lge 01" prl'ss lll'isill l' nllid

8.4.1 Pure hoop loading

r\ngle-pl y tubes ca n be ma nufact ured using the fil ament winding Illelh ods o utli ned in § 11 . 1. 1. When tested under pure hoop loadin g, usin g Ih e se l lip of F ig. 8.34(a), the tube wa ll is subjected to the sa me loa ding co ndi lio ns as for testin g a flat lamina te under unia xial tension. The stresses ill Ihe component plies can be calculated using the methods of C hapt er 5. rhe fai lu re stresses can be predicted usin g a failure criteri o n a nd l1le;l \u red va lues of (Tlu, (T 12u and T2u' The resu lts show n in Fig. 8.]5, whi c ll lVe re obtai ned using the maximum stress criterion, demonstrate how Ih e \t resses in the plies, and the fai lure stress, va ry with pl y angle (/1 I"or Jlolyes ter/ 50% glass laminates. (The va lues used for (Jlu' (J2u and T I 'u ;I re those in Table 8.2.) The appearance of the failure stress plot (Fig. 8.35(b» is bro<1dl y, hlll 110t exactly, the same as that for a single lamina shown in Fi g. X.22 : IlI l' dilTe rences are due to the constraint imposed on each ply by th e prese llce Il l" the other ply. Simple est imates of the hoop stress at failure C<1n Ill' vn Ihcless be made by treating a single ply in isolation, so that Eqn (5. 12) G ill he used in conjunct ion with , for example, the Tsai Ilill crilerion to ,i Vl" Lq n (8.16): a simil ar procedure may also be emp loyed when both Iwnp ,llld axial stresses arc prese nt. I': xper imenl;d sludi es or th e stress strain beh<1viour or suc ll 11IIK'S 1l'; ld In tiat;1 or th l' rnl"111 "lI owll III Fi g. X.]6. The curves dep;lrl I"rolll illlli;d Illll'; lril y ;IS l'I;ll '''"I 1' PIPl'l"l'd\ (;1I1d poss ihl y alsn ;IS ; 1 rl'", dl or SOll1l'

8.4 Failure of tubes under internal pressure

Strength of composites

200

1.2 1 ' " [

1

.~_ E! '"

~

0.8

I"""

"

' "

I'"

I

, I '"

I

'

'I

'

L~ ~

[

'2 0...

I

6 '"'"..,

••••

........ ...

...... .,,;tr

- - Parallel to fibres,a,

."

.....•..• Normal to fibres, a 2 0.6 ----- Sh ear,T '2 04 ~ i.-_ _ _ _ _ _ _ _,~' -

0.2 ~

""

'"0.. 0 0

200

:r:

-,~: ' .......

,..../

300

b

,

/'

400

'~

(a)

100 ....... 2.5

~

o

~ ------

:-r...............

. ..... _-_ ....... ..

_.

.'

~

'

f----------/

~

700

-

600 ~

~,e

' "

..:! 500 ~

.•..•...• Transverse failure (a2 =( 2)

I

',4

400

, ,

,, ,

I

300 200 100

0

10

I

, I

, \

,,

,

, I

.. .

, ... -.:- - - - .............

20

I

I

- - - - - Shear failure (T 12 = T 12)

t

0

viscoelastic now of the resll1 matrix). The onset of non-linearity occurs ea rlier for the hi gher ply angles, as expected from Fig. 8.35(b). For q; greater than about 40°, initial damage is by transverse cracking, but this ca nnot happen for the lower va lues since 1J2 becomes negative - see Fig. X.3 5(a). The departure from linearity occurs in these cases as a result of shear displacements.

(b)

,-------------. _ _ Axial failure (a, = a , )

,

~

Fig. 8.36 Hoop st ress against hoop strain curves of polyester/ 50% glass fibre a ngle-ply laminated tubes tested in pure hoop loading (Spencer and Hull 1978) .

n

Ply angle, ~

f

3

Hoop strain (%)

_0.2 Lr--'-L.L.J--'---'....L.-..u....L-...c..L.L..L..-L.--'--'-'-L.L--'-....L.l---'-'---'-'-...L..C....L..L...L..L...L..L~ o 10 20 30 40 50 60 70 80 90

800

201

30

40

50

8.4.2 Combilled hoop alld axial 10 adillg

--

........... _................. -~ 60

Ply angle, ~

70

80

90

n

Fig. 8.35 Calculated variations of (a) the internal (ply) stresses, relative to an applied tensile stress and (b) the applied stress needed to cause different types of failure , as a function of the ply angl e 4) for an angle-pl y laminate of polyester/ 50 % gla ss fibre.

W hen tubes are tested with sealed ends, as shown in Fig. 8.34(b), the la minate is subject to biaxial tension , with tensile stresses of IJH in the hoo p direction and IJA = 1J1-I / 2 normal to this in the tube axis direction. Res ults for this case are shown in Fig. 8.37. The transverse stress 1J2 is no w always positive and significant, with the result that transverse c rac king is predicted to be the dominant initial failure mode for all ply an gles. These predictions may be compared with the experimental stress strain data sho wn in Fig. 8.38 . A s predi cted , initial failure occ urred hy tr;ln sw rsc c r;lckin g in all ca ses, cau sin g d eparture from li nc; lrit y at lo w s ll l" ~S l" s . Thi s wa s Illorc dcl ,lycd ror th e pl y an g le or

202 1.4

20]

8.4 Failure of tubes under internal pressure

Strength of composites

"-rT"T"""--'-,,T"T""---T'T-'--""T""-'T"T""---"T"T-,---,-r-r--r,,-,-,--,-,.,.-T"T""'"

500 1.2

~

i .!:2 0.8

'§

~ en

........• Normal to fibres (0)

en

~

400

en en

- - - - - Shear

0.6

o o

('l'1 2)

..'-..~ .• -...-•.- - - - - - - '

r/l

::r:

1T..... ····

.............

0.4

300

0-

..........................

:.: ~ .. ""- - ---

200 100

0.2

--------- ---

o

0.5

10

20

30

40

50 60 Ply angle, I/!

n

70

800 ~' , ~ 0...

6
~ C;j en en

~

(b)

'I

j

600

:

",J,

500 400 300

-0 .~

0. 200 0-

«

90

700


t:: en

80

100

I•

- - Axial failure (0'1 =0'1)

........• Transverse failure (0'2 = 0'2) ..

0

10

- - - - - Shear failure

('l'12

= 'l'12)

\'--------""'"'1__......

..................................

0

J

.------------'---.,,' ~

20

~

-1 J

1

.................. --:.:.:.:.- .....'.'...............J 30

40

50 60 Ply angle, I/!

n

70

80

90

Fig. 8.37 Calc ulated variations of (a) the ply stresses (relative to an applied tensile stress) and (b) the applied stress needed to cause different types of failure . as a function of the ply angle rjJ for an angle-ply laminate of polyester/ 50°!., g la ss fibre. In thi s case, there is a lso a seco nd applied tensile stress. normal to the I'irst and onc-ha lf its magnitude. a s would be the case for an internall y press uri sed tube.

2.5

3

3.5

4

Hoop strain (%)

·0.2 ~L-~~--LL~~~~~~~--~--~~~~~--LLL-~

o

2

1.5

I ig. 8.38 Hoop stress against hoop strain curves of polyester/ 50"!., gla ss li hl l' .l llgle-ply laminated tubes tested in combined hoop and axial loading (SpCIlL'l'1 and Hull 1978).

15', in agreement with Fig. 8.37(b). Prediction of the point oi' i'ill ;d i'ra ct ure is complicated by the unloading of damaged plies. and hy Ihe stress concentration effects of cracks. As a final topic in this CIt ; IP ler, a simple approach is outlined to describe the effects o r th e illiti ;d da mage on subseq uent behaviour.

8.4.3 Nettillg allalysis In netting analysis, it is assumed that only the fibres bear 10;ld . so th ;1i "e = 712 = O. Although this is not very realistic, it is broadly v;did ;d'il' I co nsiderable intralaminar and interlaminar cracking and shearin g h:l Vl' take n place. Analysis of this case cannot be carried out ri go rou sly. hut :111 ;tp proximate estimate can be made as follows. For an an gle-pl y i:tlllil1;til'. ea ch lamina is considered independently and the pl y angle 1/) t;lk en :I S t It l' :l ngle between the fibre axis and the x-axis (i.e. the hoop direction) . I is ill l' the inverse of Eqn (5.12) to relate applied and internal stresses. With IT, ITI" IT.. = ITII / 2 and IT 2 = 712 = 0 gives ITII -

IT 1

cos 2 (1)

204

References and./ilrther reading

Strength oj' composites

20S

There is potential for confusion, beca use Eqn (S.12) apparently s h ow ~ that 0" 1 is equal to O"H cos 2 rP + (O"H sin 2 rP) /2, which cannot exceed iTll ' This arises because a single ply cannot in fact support the hoop and axial stresses, while maintaining 0"2 = 0, without the other ply being pn.: sent to prevent failure in the transverse (2) direction . Accepting that th e treatment is simplified, it may be noted that there is a unique value 01" (/' for which Eqns (8.22) both apply

(8.:n) Thi s is so metimes called the ' ideal' anglc for an inte rnally pressurisl!d tube . If the netting analysis assumptions hold, then the fibres in laminaks with ply angles other than 3So must rotate towards this angle. This givl!s an insight into the behaviour noted in Fig. 8.38, which shows a m uch greater failure stress for thi s ply angle: in practice, rotation towards thi s angle for other laminates requires extensive matrix dam age and this is likely to stimulate fibre fracture . The analysi s can a lso be used to explore the stiffness after initi ;iI damage , and the final failure stress, for the rP = 3So case. Under nett in " analysi s conditions

11 1'. X.39 Photograph offinal failure region of an angle-ply tube tested in hia xial " ·II,io l1. A crack ha s formed pa rallel to one set of fibres and fibre fracture ha s occurred in the other set. (Hull 1'1 al. 1978) .

(8 .24 ) and , given that only the fibres contribute to the stiffness (El = f Er) , il follows that , since the axial, hoop and fibre strains are all equal in thi s special case, the hoop stiffness is given by

Erf

EH = - 1.5

(8. 25)

This has a value of about 2S- 30 GPa for polyester/ SO% glass, wh ich is in good agreement with the gradient of the rP = 35° plot in Fig. 8.38, a lk r the initial portion ; this may be compared with the va lue in the e las tic regime of about 90- 100GPa , obtained both experimentally and fro lll laminate analysis. Final failure occurred at O"H cv 500 MPa , indicatin g ;1 O"lu value of about 750 MPa and hence a fibre strength of about 1.5 G P;! . This is about one-half the strength of freshly drawn glass libres . T he appearance of the final fracture site is shown in Fi g . X.39. Estimaks this Iype can bc use ful , provid ed the implication s 01" the ne llin g ;In;ilysi s assumptions arc I"ull y apprec iated .

or

References and further reading \dams, D. F. and Dorner, D. R. (1967) Longitudinal shear loadin g or a unid irectional composite, J. Camp. Mal., I 4 17 \ I go n, A. S. (1972) Fraclure o( COl1lpOSiles. Academic: New York \'L's ton, J . and Kelly, A. (1980) Tensile first cracking strain and strength or hybr id composites and laminates, Phil. Tron s. Rot'. Soc, LOlldoll, A294 5 19 34 \//i. V. D. and Tsai , S. W. (1965) Anisotropic strength of composites, I ~·I/)/. Mechallics, 5 283 8 11.lldorf, S. B. and Gaffarian, R. (1984) Size effect and strength variabi lit y or unid irectional composites, 1111. J. Fraclllre, 26 11 3 23 ( (Io k. J . a nd GOI'don, J. E. ( 1964) A mechanism for the contro l or cr;lck propaga tion in all-brittle systems, Proc. Ror. Soc, LOlldoll , A2X2 50X 20 I Il lns. P. D. and Potter, R. T. (1980) Some observations on the n;lture or l, hIL' reinrorced pla stics and implications for structural design, I'hil. Froll s. Nol' , SOl. LOlldoll , A294 507 17 Il.lrt-Sm ith , L. .I . (1994) Should lihrou s composite railure modes he in\cl ;! 'In l or superimposed'). ( 'O/ll/}(} ,\';II'S, 24 5:\ 5 I k . M. Y .. I':vall s. i\ (; ;llld ( 'lIrtin . W. i\. (199:1) Th e lillilll ;ll e \cll sileslrell ,tll or III l'L d :111.1 "" 1:11111,' 11l; ltr ix co mposit cs. : lelll /III'III/! .. 41 X/ 1 X

207

Strength of composites

References and[urther reading

Hill , R. (1950) The Mechanical Theory ol Plasticity. Oxford University Press: London. Hull , D., Legg, M. J. and Spencer, B. (1978) Failure of glass/polyester filamcnt wound pipe, Composites, 9 17- 24 Jelf, P. M. and Fleck, N. A . (1992) Compression failure mechanisms in unidirectional composites, J. Comp. Mat., 26 2706- 26 10nes, M. C. L. (1981) PhD thesis, University of Liverpool Kelly, A. and Macmillan , N . H. (1986) Strong Solids. Clarendon: Oxford Kennedy, J. M. (1989) Tension and compression testing of MMC materials, in Metal Matrix Composites: Testing, Analysis and Failure Modes. ASTM STP 1032, W. S. 10hnson (ed.) pp. 7- 18 Knappe , W. and Schneider, W . (1973) The role of failure criteria in the fracture analysis of fibre- matrix composites, in Deformation and Fracture olHigh Polymers. H. H. Kausch et al. (eds.) Plenum: New York pp . 543 56 Lagace, P. A. and Vi zzini , A. J. (1988) The sandwich column as a compressivc characterisation specimen for thin laminates, in Composite Materials: Testing and Design. ASTM STP 972, J. D. Whitcomb (ed.) pp. 148- 60 Legg, M. J. (1980), PhD th esis, University of Liverpool McCartney, L. N. (1992), Theory of stress transfer in a 0° - 90° - 0° cross-p ly laminate containing a parallel array of transverse cracks, J. Mech. Phys. Sol. , 40 27- 68 Pagano, N. J. and Halpin, J. C. ( 1968) Influence of end constraints in the testing of aniso tropic bodies, J. Comp . Mats. , 2 18- 31 Parry, T. V. and Wronski, A. S. (1981) Kinking and tensile compressive and interlaminar shear failure mechanisms in CFRP beams tested in flexurc, J. Mat. Sci., 16439- 50 Parvizi, A. and Bailey, 1. E. ( 1978) On multiple transverse cracking in glass fibre epoxy cross-ply laminates, J. Mal. Sci., 13 213 1- 6 Pipes, R. B. a nd Pagano, N. 1. ( 1970) Interlaminar stresses in composite laminates under axial extension, J. Camp. Mat ., 4 538- 48 Prewo , K. M. and Krieder, K. R. (1972) The transverse tensile properties of boron fibre reinforced a luminium matrix composites, Metall. Trans. , 3 2201 - 11 Roscn , B. W. ( 1965) Mechanics of composite strengthening, in Fibre Composite Materials. ASM: Metals Park, Ohio chapter 3 Rowlands, R. E. ( 1985) Strength (failure) theories and their experimenta l correlation , in Handbook ol Composites, Vol. 3 - Failure Mechanics ol Composites. G. C. Sih and A. M. Skudra (eds.) Elsevier: Amsterdam pp. 71 - 125 Sinclair, 1. H. and C hami s, C. C. (1979) Fracture modes in off-axis fibre composites, in Proc. 34th SP!/ RP Tech. Cont:, paper 22A . Soc. of Plastics Industry: New York Spencer, B. and Hull , D. (1978) Effect of winding angle o n the failure of filament wound pipe, Composites, 9 263- 7 1 Taya, M., Lulay, K. E. , Wakashima, K. and Lloyd, D. 1. (1990) Bauschingcr effects in SiCp- 6061 Al composite, Mat. Sci. & Ellg. , AI24 103 11 Tsai, S. W. and Wu, E. M. (1971) A genera l thcory of strength for anisot ropic materials, J. COlllp. Mal. , 5 58 80 Watson , M. C. and ('Iyne, 1'. W . (1993) Reaction -induced changes in i ntcrfacia I a nd macroscopic mec ha nica I pro pert ies of SiC mOllllllla l11en t reini"on:cd titaniul11 , CIIIII/IIISill'S, 24 ~~~ X

E. M. (1974) Phenomenological anisotropic failure criterion , in Composite Mate rials Vol. 2 - Mechanics ol Composile Materials . G. P. Sendeckyj (cd.) Academic: New York pp. 353- 431 !lll'ben, C. and Rosen , B. W. (1970) A statistical theory of material strength with app lication to composite materials, J. Mech. Phys. So/., 18 189 206

206

\\ 11 ,

9. 1 Fracture mechanics

9 Toughness of composites

209

1I1l1eh higher than that in the bulk, (J oo . He derived the following expres"Ion fo r the stress at the tip of a crack of length c (or 2c if internal) and tip I;I(.Ii us r (9. 1)

The previous chapter covered factors aflectillg strength, Il'hich is related to the stresses at li'hich damage alldfailure occllr ill composites. In mall)' sitllations, the energy absorbed by the material under load is equally important. A tough material is olle/rJ/' II'hich large amounts 0/ energy are required to cause failure . In man)' loading conflgllrations. such as Irhen a component is strllck by a projectile, on I)' a/inite amoull t o/energy is available to causefailure. In other cases, such as II'ith load\' arising/i'on', temperature changes, 0111)' alinite degree o/strailllleeds to be accommodated ill order for the stresses to become small. In such situations, toughlless. rather than strellgth , is the ke)' propert)' determillillg li'hether the material is slIitable. III this chapter. a brief olltline is givel1 o/the basics o/Focture m echallics , with particlllar relerel1ce to the energetics

(~j'inter/acial

damage. This isfollOlI'ed b.1' an appraisal 0/

the sources of energr absorptioll ill composites. Fillalll' , slow crack groll,th in composites is examilled for cOllditiolls Il'here fast ./i'aClure is 1I0t energeticall.1' favoured.

9.1 Fracture mechanics The reader is referred to sources such as GOt'don (1978), Ashby and Jones (1980) a nd Ewalds and Wanhill (1984) for introductions to fracture mechanics. In this section, the treatment is abbreviated and oriented towards effects in composites.

9.1.1 Ba.\'ic concept.\' Fract Lire mecha nics has its roots in the work or I nglis ( 1(1 3), who demon strat ed th,lt th e stresses near a er,lck tip in a 1ll,lteri,ti IImkr lo,ld ,Ire ortC II

2()X

I hus, a circu lar hole (e = r) has a stress concentration Iactor of 3. While lit is is physically reasonable, the case of a sharp crack (I' --> 0) presents dtlTic ulties, in that the stress concentration, according to Eqn (9 . 1), heco mes very large. On this basis most structures should fail , under low a pplied loads, at the fine surface scratches which are a lm ost inevitahly present. This was contrary to engineering experience at the time, since IlIos t ar tefacts, particularly metallic ones, were able to function even whe n sma ll cracks and scratches were present. T he problem was resolved by the pioneering work of Griffith (1920) , who pointed out that a crack cannot propagate unless the energy of the ,ystem is thereby decreased. The ene rgy released when a crack advances ('(l llles from the associated release of stored elastic strain in the surroundIng materia l (p lu s any work done by the loading system). If this is insuflieient to counterba lance the energy absorbed in the material through the neation of new fracture surfaces and associated interna l damage or de rormation processes, then the crack cannot advance. In many materi;tis, efficient mechanisms for internal energy absorption are stimulated by Ihe high stresses at a crack tip, so that the energy balance for crack pro pagation is often unfavourable and they exhibit high toughness (res istance to fracture). This is particularly true for most metals, since Ihe dislocation motion which occurs is highly effective in this regard. Griffith considered brittle materials , such as g lasses, in which energy,Ibso rbing processes are not readily st imulated and the only significant l'llergy penalty of crack propagation comes from the new surface area of t he crack faces. He showed that the change in the sto red energy of a loa ded plate of unit thickness, caused by the introduct ion of an interior crack of length 2e, is given by (9.2) where IT( = IT",J is the applied stress and E is the Young's modulus. The ot her contribution to the overall energy change is that required to create the new s urface ;m:;I , whieh is positive and ha s a value of 4("'y, where "( is thL: sllrL ICL: ' lIlTl' V. Tit l' dqKndence or these two co ntributions on thc

2 10

Toughl/css of" COIII/)(ISi/cs

9. 1 Fracture mechanics

kll gth o r the crack is shown in Fig. 9. 1. Only cracks longer tha n a critica l lcngt h, e*, will grow spontaneo usly (with reduction in net energy) . Thi s c ritical length is found by differentiating the total energy with respect to crack length a nd setting the result eq ual to zero, leading to

I ab le 9. 1 Typical Fac/ure energy andFaclUre /oughness values fo r various materials. ( Af"/er Ashby and lanes / 980 )

2"( E e* = - ) (r rc

(9.3 )

Thi s approach was ex tended by Irwin ( 1948) to encompa ss toughcr materials. The surface energy term 2"( is supplemented by other con tribu tions to the energy absorbed in the vicinity of an advancing crack tip. For a given applied stress a nd pre-existing c rack size, an expression ca n be obtained from Eqn (9.3) for the energy release rate, G, which has units o r

I\ l.t lnia l

/'"/l'Illcrs , I'tI\y resins N\ lOll 6.6 I" ,I yp ro pylene

(9 .4 ) For fracture to occur, th is must exceed a critica l va lue, sometimcs referred to as the crack resistance, R. This critica l value represents th e total e nergy absorbed , per unit of crack advance area and is often termcd Gc, the critical energy release rate , or Factllre energy. Values of Gc a rc

Fracture energy Gc (kJ m - 2 )

Fracture to ughness

0. 1- 0.3 2- 4 8

0.3- 0.5 3 3

100- 1000 8- 30 lOO

100- 350 23-45 140

0.01 0.05 0.03

0.7 3 0.2

Kc (MPa JiTI)

111'1111.1'

IltttC A I \1 al loy tt tlld stee l

( ,Tlllllics .. ,,
J m- 2

211

, "llnete

Vlllll ral materials (crack ~ gra in ) 11')(ltiS (crack // gra in ) [,llIle

8- 20 0.5- 2 0.6- 5

1")(ltiS

( '1I111f1I1S ileS It hreg lass (gla ss/epoxy, planar rand o m Ilhn:s) \1 based particulate MMC SI(' lam inate (crack ~ la yers)

11 - 13 0.5- 1 2- 12

40- 100

42- 60

2- 10 5- 8

15- 30 45- 55

a

ttt

>-

~

Ql

c

llJ

Crack length c

Total energy of the system

2

U=_allC

2

E

F ig. 9. 1 Schemat ic plot o r th e two cont ribu ti o ns to the energy a ssociat ed wi th th e prese nce o r a cra ck in a brittle mate ri al, as a runctio n o r c ra c k le ng th . /\ c rack or le ng th c, o r large r wi ll grow spo ntan eo usly, with a redu e ti o ll in the to t;1I e ne rgy.

I.Ilr1y easy to obta in experimenta ll y. For exa mple, the work done in a 1(,llsion or bending test is given by the a rea und er a load-d isp lacement plo t a nd , provided this energy is all permanently absorbed in the speci111l:n, the fract ure energy is then found by simply dividing by the sectional ,m:a thro ugh which failure has occurred. The specimen is commonly prelI() tc hed so as to ensure that crack propagation occurs . Typical va lues of (;, are given in Table 9.1 for various materia ls. Tough (soft) meta ls have Irac ture energies of 100kJm - 2 or more, whereas a brittle materia l, such . IS gla ss, ca n have a value as low as 0.0 1 kJ m - 2 . Rea rranging Eqn (9.4), Iltc strcss necessary to cause spo nta neo us fracture in a component with a prc-ex isting crack of size e (2e if interna l) ca n be written as

rr ,

(9.5)

Toughness of composites

9.1 Fracture mechanics

This approach is particularly useful in practical terms, because atten tion is diverted from the complex problem of the precise nature of Ihe stress field close to the tip of a crack to a more global approach involvin ' macroscopic quantities which are measurable experimentally. However , there is still interest in the phenomena occurring locally near the crack tip. A useful link is provided between the energy and stress fie ld approaches by the concept of a stress illtensity factor, K. This parametc r, which largely evolved from the work of Irwin in the 1950s, can be expressed as

9.1.2 Interfacial fracture and crack deflection

212

K = CJvnc

(9.6 )

It therefore encompasses the effects of both the applied load and the preexisting crack size, with the relative weighting that these two parametcrs have in determining the value of G, the energy release rate (see Eqn (9.4)) . It characterises the severity of the stress field around the crack tip . A critical value can be identified, corresponding to the case where the associated value of G reaches Gc

Kc = CJ * V'nC Jl.(; = V!£G LV c

(9.7 )

This critical stress illfensity factor is often known as a Factllre toughness. Values are given in Table 9.1 for various materials. For tough materia ls, the fracture toughness can exceed 100 MPa JiTI, while a brittle materia l might typically have a value around I MPaJiTI. The usefulness of the stress intensity factor lies largely in the way it can be related to local crack tip features. For example, it can be shown that the si ze of the pla~, tic lOll/' ahead of the crack tip is related to the yield stress of the material by

ry : : :; ~ (~) 2 211: CJy

Similarly, the crack openillg displacemellt , 8, can be expressed as

8:::::;

(~) CJyE

(9.9 )

Such parameters are useful wh en considering how energy-absorbing processes might be stimulated in composite material s, sincc Ih cy a llow Ih L' scale of features of the cra c k lip 10 be relal cd 10 Ihe sca k or Ih e micr(lstru cl urc.

213

I "hk 9. 1 shows that a composite made from glass fibres and epoxy resin 11.1 \ a fracture energy comparable with those of metals (Gc rv 50 kJ m - 2 ) , , \ I'll though the constituents are both brittle (Gc rv 0.01-0.1 kJ m - 2). I illS high toughness of composites, which is very important in practica l il-llllS, is closely linked with interfacial effects. A first step in exploring Ill" is to analyse the conditions under which interfacial debonding, i.c . , 1,lck propagation along an interface between two different materials, '" L·urs . For a given loading configuration , the propagation of a crack ,d"llg a n interface between two constituents gives rise to an energy rclcasc 1,11l", G;, in much the same way as for the case when the crack is in a IlIllllogeneous material. Also , there is a critical value , G;c , an interfacial / fIIl'f llre energy, which G; must reach for the crack to propagate. Va lues of G;c are not as readily available as Gc values for homogeI H'OUS materials. There are several reasons for this. Firstly, the toughness "I an interface is sensitive to the way in which the interface was man IIlact ured , rather than being unique to the pair of constituents on cither '.lIk. A second reason is slightly more complex. Interfacial cracks oftc n I" opa gate under mixed mode loading conditions. This is in contrasl 10 a I Iack in a homogeneous material , which will always tend to advancc in a dlll:ct ion such that the stress field at the crack tip is purely tensile (mode I) A n interfacial crack , however, is constrained to follow a predelerm ined path. Depending on the loading configuration, thc stress IIl'id a t the crack tip may include a significant shear stress componenl .1\"Iing o n the plane of the interface (mode 11). In general , thc energy npe nded in debonding the interface is greater when there is a mode II IOlll po ne nt than for the case of pure mode I loading. Thi s compli cal es I ilL' experimental measurement of G;c . Not only can it bc diffi e ult 10 ('s la bli sh the exact stress field at the crack tip, but it may vary wilh posi lio n in the specimen , particularly for fibre / matrix intcrfaees . Th L' "llu;ll io n is further complicated if any residual strcsscs (e.g. see :: 10. 1. 1) are present. T hc proportion of opening and shearing modes al Ihe cra c k tip is oi"l e ll I IIa rac tc rised by means of the phase allgle, '1/) (psi). This is dc fined ill le rlll S Ill' Ihe mode I and mode II stress intensity factors, sho wn sehemali call y ill 1112.. 9.2.

,I,

214

9. 1 Fracture mechanics

Toughness of composites 10

(a)

~

N

E 0 u

cS ;:.. 0Jl ....
c::


'"

<.I::

Fi g. 9.2 Schematic depicti on of the stress fi eld at an interfacial crack. The phase angle 'I/J, used to characterise the crack tip loading mode, is defined in terms of th . stress intensity factors, K, and K II , for mode I and mode 11 loading. The va lue 0 1 'I/J ca n va ry from 00 (pure mode I, crack openin g) to 900 (pure mode 11 , shearin g).

c:a 'u

~ ....

9 0

8

la

7 6

5

l

1

0

4

• B,~n"",

3

0

2

0


:5

X

0

The va lue of 'if; is 00 for pure o pening ( KII = 0) and 90° fo r pure shear (K, = 0) . The phase angle ca n be esta blished fo r va ri o us loadin g a rran g~ ­ ments, a lth o ugh often thi s ca lcul a tion is no t a simple one (Rice 198X , Evans and Hutchin so n 1989, Wa ng a nd Suo 1990). Furthermore, '1/) is li kely to va ry with positi o n wh en the interface is no n-pla na r. The lim itcd experimenta l data currentl y ava il a ble suggest that the dependence of G'" on 'if; may be q uite signifi ca nt, dependin g o n the type o f system. Examp les a re shown in Fi g. 9.3 fo r pla nar interfaces in the fo rm of thin interlayers betwee n thi cker substrates. T hese da ta were obta ined using d ifferent tes ts and covering a wide ra nge of o pening and shea ring mode co nt ribution s. Increases in Gic appea r to res ult when the shea ring mode compo nent is increased . Thi s may be due, a t least pa rtl y, to more fri cti o na l wo rk bein I done immedi a tely behind the crack tip as asperi ties o n the crack flank s slide over each o ther. T he dependence o n 'if; a ppea rs to be stro nger wh en at least o ne of the co nstituent s ca n defo rm pl as ti ca ll y (see Fi g. 9.3( b» . One of the ma in reaso ns fo r interest in interfacia l to ughness co ncern s crack deflectioll. Fo r a composite to have a hi gh to ughness, a cra ck passing through the ma trix mu st be re pea tedl y de fl ected a t fib re/ma tri x interfaces, at least fo r ma teri a ls based on po lymer resins o r ce ra mi cs (sce §9.2). Ea rly wo rk by Coo k a nd Go rd o n ( 1964) on co nditi o ns fo r cra ck defl ecti o n was focused o n th e stress fi eld a head of a crack ti p. It W ~ I S po inted o ut tha t, when a crack approaches a fi bre in a com pos it e load ed pa ra ll el to the fi bre axi s, there is a tran sverse stress a head o f the crack tip . te nd in g to o pe n Lip th e in terface and hence to 'b lu nt' ~ I nd dcllcct the c r ~ l c k fro m en tering the libre UiX. I. I). Since th e pe
2 15

10

0

20

3-point bend delamination 3-point bend fract ure 4-point bend delamination

40

30

50

60

70

\

80

90

Phase angle, lI' n

250

(b)

~

N

E 0

0

200 u

0

C.)

;:.. 0Jl .... 150
0

c::


::l

U 100

0

'" <.I:: c:a 'u

'"

"....

0 0 0

50

0

Cb


0

0

c:

"Brazil nut"

I

0

10

20

30

40

50

60

70

\ 80

90

Phase angle, lI' n hg. 9.3

Ex perimental da ta for the interfacial frac ture energy Gic as a functi on

" I" the phase angle 'I/J which characterises the stress fi eld at the crack ti p. (a) Thin ( v S- IOpm) graphitic laye rs betwee n SiC sheets (Philli pps et al. 1993), and (b)

thin ( ~ 100 400 pm ) epoxy laye rs between alu miniu m bloc ks (Ak isanya and I:kck 1()92). Th L' lege nd re l"ers to the type of test used to obtain the data .

Toughness of composites

9.2 Contributions to work o/./i"acture

stress is always about 20% of the maximum axial stress, Cook a ll d GOI'don proposed that the interface debonds if its strength is less th ;11I about one-fifth that of the matrix. However, the 'strength' of an in terfaCl' is not well defined and is sensitive to the presence and distribution 0 1 flaws and to the method of loading. In view of the success of the G riff! t h treatment, it is clearly preferable to establish a criterion based o n till' energetics of crack propagation. Energy-based crack deflection criteria have been proposed by Ke nd; dl (1975) and by He and Hutchinson (1989) . Kendall considered two b loch bonded together and loaded in tension parallel to the interface, one of the blocks having a crack approaching the interface - see Fig. 9.4. He es ti mated the applied loads under which a crack would penetrate the o th er block or deflect along the interface, assuming that the crack requiri ng till" lower load would predominate. This produced the following criterion rO I deflection

IlIod uli and v is the Poisson 's ratio (taken as equal in both constituents). Ir and hr are taken as equal , corresponding approximately to a crack passing through the matrix between unbroken fibres in a typical compo\lte, then this critical ratio is around 20% for EI11 rv Er, falling to about I ()'Yo for EI11 « Er. He a nd Hutchinson 's analysis is based on considering whether the pcnetrating or deflecting crack gives a greater net release of energy. It IS mo re complex than Kendall 's model , but also yields a critical fracture l"ncrgy ratio of around 20% for the case of Em rv Er· For EI11 « Er , liowever, an increase (rather than a decrease) is predicted in this ratio. I'here are few experimental data available to validate either of these lI1ode ls. Kendall 's experimental work with rubbers did seem consistent wit h his predictions, but few systematic measurements have been perlormed with systems of practical interest. It is, however, clear that the Inter facial fracture energy must be appreciably lower than that of the Icinforcement if matrix cracks a re to be consistently deflected. Since (cera mic) fibres tend to have low fracture energies (see Table 9.1 and i:(). 2. 2), this means that interfaces of very low toughness are often IL'qu ired if crack deflection is essential. For ceramic matrix composites, In pa rticular, the retention of low interfacial toughness, through processi ng stages which tend to promote sintering and chemical reactions , rcpresents a major technological challenge.

216

Gic Gre

~ (hl11EI11 + hrEr) [ hrEr

J]

I 4n(1 - v-)

(9. 11 )

where Gic and Grc are the fracture energies of interface a nd uncracked block (fibre), hl11 and hr are thicknesses of cracked (matrix) a nd uncracked (fibre) blocks , EI11 and Er are the corresponding Yo un g's

t t

217

"m

9.2 Contributions to work of fracture Crack penetration

~

;\ relatively high toughness is essential for most engineering materials. ()n e of the advantages of composite materials is that there is often scope ror the promotion of energy-absorbing mechanisms in the material. It is Im po rtant to understand how these mechanisms a re controlled.

9.2.1 Matrix deformatioll

Crack deflection

Fi g.9 .4

Geollletry 0 1' c ra c k dell ec ti on at an int erl'acc (after KClltLlII ( 1<) 75 )) .

Wo rk of fracture data for some typical matrices are given in Table 9. 1. Mos t metallic matrices have a high toughness, mainly as a result of the l'x te nsive dislocation movement which occurs near a crack. Polymers, pa rtic ul a rly thermosets, and ceramics have low fracture energies. T he extent of matrix deformation during composite fracture may differ ;Ipp reciab ly from that in the same material when unrcinforced. The main l'IYec t is onc or ill Tcascd cOlIstraillt . so that th e matrix is unabl e to deform rn:cly hec llI sc il " SllllPlllltkd hy stilY and strOll g lihres. Thi s m : l y he

Toughness of composites

9.2 Contributions to work of fracture

partly a result of load transfer (§ 1.3), which reduces the magnitude of thl: matrix stresses. Of greater significance, however, is the tendency for triaxial stress states to be set up which inhibit plastic flow of the ma tri x. For example, when a region of matrix extends plastically, with associa ted lateral contraction, this contraction may be opposed by a surround in ' rigid cluster of fibres. This sets up transverse tensile stresses which red ucc the deviatoric (shape-changing) component of the matrix stress sta te. This in turn inhibits plastic flow , but may encourage cavitation a nd fracture. This type of effect accounts for the lower fracture energy shown in Table 9.1 for the AI-based MMC, when compared with unreinforccd AI. This loss of toughness can be minimised by eliminating reinforcemen l clusters and other inhomogeneities such as pores, debonded interfaces and cracked particles or fibres. Processing improvements for MMCs are currently aimed in this direction. It follows that a high interfacia l strength is desirable for MMCs and in most cases this is quite read il y achievable (§7.3). Only for relatively low-toughness metals (e.g. zinc) reinforced with long fibres is there any interest in a low interfacial toug hness , since in these cases toughening by fibre pull-out (§9.2.4) is preferable to retaining as much of the toughness of the matrix as possible (Vesce r ~ 1

Ihe gra in). In such cases, fibre fracture contributes up to a few kJ m ~ 10 Ihe o verall fracture energy. Metallic fibres can in principle make even larger contributions. Thus, even at a low volume content, steel rods in rcin fo rced concrete raise the toughness substantially, as we ll as enhancing Ihe tensile strength. Nevertheless, for most composites the fibres them selves make little or no direct contribution to the overall toughness.

218

et al. 1991).

2 19

9.2.3 Interfacial debol/dil/g Some interfacial debonding usually occurs during the fracture of a compos ite. If a crack is propagating normal to the direction of fibre alignme nt, debonding occurs provided the crack is deflected on reaching the interface. Debonding can also be stimulated under transverse or shear load ing conditions. Provided the value of the interfacial fracture energy, Uie , is known (see §9.1 .2), an estimate can be made of the contribution rro m debonding to the overall fracture energy. T he basis for such a calculation is shown in Fig. 9.5 for aligned short lib res. A simple shear lag approach is used . Provided the fibre aspect ra tio , s (= Lj r) , is less than the critical value, s. (= Uf ' j2Ti ,), see }i6. I.S , all of the fibres intersected by the crack debond and are

The constraint effects outlined above also operate with non-meta ll ic matrices. In these cases, however, there is in any event limited scope fo r energy absorption in the matrix and other mechanisms are often or greater significance. However, considerable improvements in the toug hness of PMCs have been achieved by increasing the toughness of the matrix , which can alter the micromechanisms of damage and increasc the associated energy absorption. Debonded interface

9.2.2 Fibre ./i'actltre Depending on the fibre architecture (Chapter 3) and loading configur~l ­ tion , component failure usually involves the fracture of fibres. The con tribution this makes to the fracture energy of the material is small for most fibres. Typical fracture energies for fibres of glass , carbon and SiC are only a few tens of J m - 2 . Some polymeric fibres are not completely brittle and undergo appreciable plastic deformation . An example of thi s can be seen in Fig. 2.7(d), which shows a fractured Kev lar™ fibre. Mosl natural fibres are in a similar category and fra ct ure o r Ih e I:e llul ose ribrcs mak es a signi li cant contributi on to the rr ~ l c lllr c CI1 LT)',y o r wood (across

r i = ri '

2L

.

2r~~ ..

2 L

hg.0.)

SChl'lll :llil' (li': l l'I": lck pa ss in g throu gh ~In aligned short - lihrc composite, sll(lll'll' I' 1111 ,'II :I(' I;1i dch()nding ~ Ind suhsequent lihre pull -out.

221

Toughness of composites

9.2 Contributions to work ofFacture

subsequently pulled out of their sockets in the matrix. The work done when a single fibre undergoes interfacial debonding can be written as

fro m their sockets in the matrix. Fracture surfaces illustrating various ex te nts of fibre pull-out are shown in Figs. 8.7- 8.10. Pull-out aspect ra ti os can range up to several tens or even hundreds. T he work done can be calculated using a similar approach to that in }i9. 2.3. Consider a fibre with a remaining embedded length of x be in g p ulled out an increment of distance dx . The associated work is given by the product of the force acting on the fibre and the distance it moves

220

(9. 12) where Xo is the embedded length of the fibre concerned on the side of the crack where debonding occurs (xo ::::; L ). To obtain the local work o f debonding for the composite, Ccd, this is summed over all of the fibres intersected by the crack. If there are N fibres per m 2 , then there will be (N dxo / L) per m 2 with an embedded length between Xo and (xo + dxo ). The total work done in debonding is therefore given by

Ccd

l

=

o

L

N dxo

- - 2nrx oC jc

L

(9 .1 3)

dU

Substitutin g this expression into Eqn (9.13) and integrating leads to

C cd

6.U =

{'o

lo

(9. 16 )

2nrXTj ' dx = nrX6Tj '

(9. 17)

T he next step is a similar integration (over all of the fibres) to that used to o bta in Eqn (9.13), leading to an expression for the pull-out work of fract ure, C cp

Ccp =

_ /2nrC jc L2 nr2 L 2

(2nrXTj ' ) dx

whe re Tj' is the interfacial shear stress, taken here as constant a long th e lengt h of the fibre. The work done in pulling this fibre out comp lete ly is there fore given by

The value of N is related to the fibre volume fraction , f, and the fibre radius, r (9 .14)

=

loo

'L

N dxo

2

- - nrXOTj'

L

(9. I X)

which , using Eqn (9.14) aga in for N , simplifies to

wh ich can be rearranged to (9. 15) Contributions from this mechani sm are relatively small. For examp le, if s = 50,/ = 0.5 and C jc = 10 J m - 2 , then C cd = 0.25 kJ m - 2 . If either the fibre aspect ratio or the interfacial fracture energy are much greater than these values (s > s* or C jc > C rc ), then it is probable that the fibres will fracture in the crack plane and little or no debonding will occur. Slight ly larger values are achievable in bending, when debonding can propaga te for long distances from the fract ure plane without causing the reinforcement to break. Recent work on ceramic laminates (Phillipps et al. 1993) ha s been based on this effect.

9.2.4 Frictio1lal slidi1lg a1ld fibre pull-out Potentially the most sign ificant source of fracture work for most fibre composites is interfacial frictional slidin g. Depending on the interfaci;d roughness, contact press ure and sliding distance , this process ca n absorh large quantilies or energy. The case of most inlcn.:sl is Pllll oul or ribres

f ·?

. S-'"Tj '

Ccp = -3-

(9 . 19 )

This contribution to the overall fracture energy can be la rge. I;or exa mple, taking f = 0.5, S = 50, r = 10 pm and Tj ' = 20 M Pa gives a va lue of abo ut 80 kJ m - 2 . Since (Jr' would typically be about 3 G Pa , tll . c ritical aspect ratio, s* (= (Jr / 2Tj ')' for this value of Tj', would be .1hOllt 75. Since this is greater than the actual aspect ratio, pull-out is expecll.:t! to occur (rather than fibre fracture), so the calculation sho uld he v. did . The pull-out energy is greater when the fibres have a large r diameter, ass uming that the fibre aspect ratio is the same. The potential for contribution to the toughness from libre pull -out is s ubstantial. However, the mechanism can apparently only opcrall.: wi lll re latively short fibres (s < s*). Co ntinuou s fibres arc expected to hI' '.Ik ill the crack plane, since there will a lwa ys be embedded len g th s Oil eilller side of the crack pIa ne which arc long enough for the s t ress ill the ri hrL' I () huild up sui'liL'iellll y to hre'lk it. (Interfacia l deholldill' C III s till oe 'lIr, perh.lps OV'r ; 111 :lppl(·t' I:lhk di sl.IIlU':, but it is clear frolll }i ') .2. "\ 111:11 1111.';

222

9.2 Contributions tu

Toughness of composites

is unlikely to make a major contribution to the fracture energy.) lJ SII I/ ' this argument, it appears that pull-out should not occur in long-li hi , composites. In fact , it is often highly significant in such materials. '111 , reason for this is related to the variability in strength exhibited by 111 0'. 1 fibres (see §2 .2.4). The effect of the variability in fibre strength, characterised by Ill, Weibull modulus, In , is depicted schematically in Fig. 9.6. Figure 9.(1( :1)

Ivork O/./i'(lC/III'('

"

\

,llt lWS how, in the case of a deterministic (single-valued) ril r ' sll l' ll )'111 1/11 (0), the probability, Pf , of the fibre breaking in the crack pl :lll l' wdl , \('ntually become 100% , while remaining zero elsewhere. For a rinll . \ .dlle o f In , however, (Fig. 9.6(b» not only are high values of Pr srm .: ad ,'I l'r a n appreciable distance on either side of the crack plane, but it is 1I 11\V possible for the fibre to break almost anywhere. Prediction of the 1I .Id ure energy becomes more difficult under these circumstances, since ,lll'llll nt should be taken of these probabilities in calculating the contri"lit io ns from different pull-out lengths. In fact , strictly, the effect of this ""trib ution of fracture probabilities should also be taken into account 111\ the short-fibre case. The details of such treatments need not concern 1I ~ here, but it may be noted that a low Weibull modulus always tends to 1.llse the pull-out work for long fibres, but is not necessarily beneficial for '.lio rt ones. Detailed mapping of the various contributions to the overall 1(lughness, for different Weibull moduli , has been undertaken by Wells

.111(.\ Bea umont (1988).

9.2.5 Eflects of microstructure

(a) m

=

00

--- 0;)

1

:

:

'

,

p,1 ...................~ 0-

(b)

0<m <

00

Fig.9 .6 Schcmatic depicti o n o f stress distribution and assoc i:il cd prohah ilil Y 0(" failure Pr along th e le ngth o fa lo ng lihrc brid gin g a Illalri x c r:,ck . (;1) i\ lixed lihrl' stre ngth ( Ill no) . a nd (h);1 lihre slre ng lh whi c h v; ,ri es ; d (l ll ) ' li s 1'lI g llI (Ii ni lt: Ill)

I here is considerable scope for controlling the fracture energy of compo~Ite s by altering various features of the fibre length , orientation and Inte rfacial characteristics. Examples of the effect of orientation are ~ho w n in Fig. 9.7. This shows measured fracture energies, obtained IIsin g the Charpy impact test. The fracture energy of the unidirectional l:im ina (Fig. 9.7(a» falls off sharply as the angle between the crack plane ,11ll! the fibre axis is reduced (loading angle increases) . This is largely heca use fibre pull-out becomes inhibited and fracture occurs parallel to t he fibre axis. The crossply laminate and woven cloth reinforced material ex hibit more isotropic behaviour. It is difficult to predict the dependence (In load ing angle in these cases, since complex interactions occur between the different plies and fibre tows. However, fracture always involves a co nsiderable degree of interfacial debonding, fibre fracture and fibre pullo ut, so that the toughness is always relatively high . This is an important at tri bute of long-fibre laminates. Fig. 9.7(b) shows that th e toughness of \o me metals can also be enhanced in this way. (The fracture energy of the ma trix in this case was about 90 kJ m- 2 ) Thi s fi gure also illustrates that good tou ghness can be obtai ned when the crack propa gation direc ti on lies in the plan e or th e laminate, provided the crack rront is no rmal to thi s

plane.

224

Toughness of composites

9.2 Contributions to work of fra cture

250 --0-

",'-"

-

200

~

'6

(a)

Unidirectional lamina, f = 0.67 Cross-ply laminate, f = 0.67 Woven 0/90 cloth, f = 0.47

2

C::/

150

;::.. Ol) .... <1.l

C <1.l

100

~

:::I

U ....

'"

[..I..,

50

o

o

10

20

30

40

50

60

Loading angle, l/J

n

70

80

90

225

Co ntrol over interfacial properties is often of critical importance. Thi s I ~ ill ust rated by the data shown in Fig. 9.S. This refers to a ceramic 11l;l te ria l composed of sheets of SiC, separated by thin graph itic inte rI. lyers . When loaded in bending so that a crack propagates through sucn'ssive sheets, deflection and extensive interfacial debonding occurs at (';Ich interlayer, provided the interfacial toughness is sufficiently low (see 1:1). 1.2). Depending on the loading geometry and specimen dimension , the lota l wo rk done, W, (= area under load- displacement plot) may exceed I he actual energy absorbed in the specimen (the excess being d issipa ted I hro ug h the loading system). Modelling of the interfacial cracking a ll ows I he true Gc to be established. It can be seen from Fig. 9.S that raisin g th e Inte rfacial toughness (by changing the processing conditions) , increa sed (;" even though the tota l work done was little affected. Further increa ses III Gc co uld not, however, be achieved in this way , since a tougher int er1;lce (higher G ic ) would inhibit crack deflection. In this case, although lou ghening comes only from interfacial debonding, the fracture surfa ce ,Ireas involved a re such that the fracture energy values are relatively hi gh Illr a ceramic material.

400

(b) 350 ,-..

'"6

8

300

2

'.:JU 250

I

Experimental data

Predicted scatter band for total work, W

o

;::..

Ol) 200 ....

-,

C

C,

<1.l <1.l <1.l

.... :::I U

'"

....

150

..oi (;; ~

100

[..I..,

--0-

50 0

I-

t 0

Loaded normal

10

plane of plies

Loaded parallel

10

plane of plies

~

:::I

1 j

10

20

30

40

50

Loading angle, l/J

60

n

70

80

4

Pred icted scaller band for Gc

'(;j [..I..,

,

90

Fig. 9.7 Ex perimental data for the fra cture energy Gc obtained using the C ha rpy Impact test, as a function of the loading angle 4> between the axis of th e test ba I' a nd the reference fibre direction. (a) Epoxy resin /glass fibre, loaded normal to th e la mina te plane (Harri s 1980) , and (b) crossply lamina tes of Z n 8A I I C lI /50"" ca rbo n fibre, loaded norm a l and parallel to the lam inate plan e (Vescera 1'1 (1/. 199 1)

7()

Interfac ial fracture energy , Cl e (J m ~ ) I Ig. I) .X Compari so n between predicted and observed dependence 0 1" lolal WIH k done IV o n inl erl"a cial I"r:lclure energy for SiC lamin at es co ntaillin g 'raphilll' IlIlcrla ye rs. Al so shl\w lI I ~ Ihl' pr<:di <: l<:d depend ence 0 1" Ih e tru e 1"1':1<: 1111'<: <: lI l' I)'Y (;, whi ch ill 1111 , (' : "1 ' " dlll l,'ldl 10 III L': ISur<: dircc ll y. (I' hillipps 1'1 ,d . 1')'iI)

9.3 Sub-critical crack growth

TOllghness 0/ composites

Fo r parti c ulate-reinforced composites, the energy absorption from dc bondin g and pull-out tend s to be small. In particulate MMCs, ~ I major concern is the retention of as much as possible of the inhercnl toughness of the matrix. Data are shown in Fig. 9.9 for the fracture energy of a n age-hardening aluminium alloy containing SiC particles, as a function of the tensile strength. As ageing progresses, the tensile strength ri ses and then falls (over-ageing). The fall in fracture energy during the initial rise in strength is expected on the grounds of reduced dislocation mobility . The failure of the toughness to rise again as over-ageing occurs is mon: difficult to explain, but may be owing to precipitates forming preferen tially at the interface, making it a preferred site for cavitation. Finally, evaluation of the energy absorbed can be particularly compl ex for laminates, for which contributions can come from interlaminar debonding as well as intrap ly processes. A detai led discussion of the various interactions which can occur, and the significance of microstructural characteristics, is given by Hull and Shi (1993).

9.3 Sub-critical crack growth When the rate of energy relea se during crack propagation (driving force) is lower than the critical value, spontaneous fast fracture does not occ ur . 10 9

'"

8

2

7

~

E

(.)U 6

;::.. OJ)

.....

t:


::: U ro .....

[..L.

9.3.1 Fatig.·. h)!' metals, fatigue failure is a n important topic which has been the subject of detailed investigatio n. Analysis is commonly carried out in ItTmS of the difference in stress intensity factor between the maximum and minimum applied load (flK). This is because, while the maximum va lue, Kmax, dictates when fast fracture occurs, the cyclic dissipation of ene rgy is dependent on flK. It is, however, common to quote the stress rlltio, R (= Kmin / Kmax), which enables the magnitudes of the K values to he established for a given flK. The resistance of a material to crack ex tension is given in terms of the crack growth rate per loading cycle (dc/ dN). At intermediate flK , the crack growth rate usually conforms to Ihe Paris- Erdogan relation (Paris and Erdogan, 1963)

dN

4


.....

l lnde r some circumstances, however, an existing crack may advance slow ly under this driving force. Since crack growth leads to an increased driv ing force (for the same applied load) , this process is likely to lead to :1Il accelerating rate of damage, culminating in conditions for fast fracture hei ng satisfied. There are two common situations in which such subrri tical crack growth tends to occur. Firstly, if the applied load is fluctuati ng in some way, local conditions at the crack tip may be such that a ,-; ma ll advance occurs during each cycle. Secondly, the penetration of a corrosive fluid to the crack tip region may lower the local toughness and allow crack advance at a rate determined by the fluid penetration kinetics ()J" chemical interaction effects. In both cases, as indeed with fast fracture , Ihe presence or absence of an initial flaw, which allows the process to Ini tia te , is of central importance.

~=

5


3 2

Over-aged

0 400

450

500

550

600

650

700

Tensile strength (MPa) Fi g.9.9 Experimental da ta fo r th e frac ture e nergy Gc as a fun cti o n o r Ihc te nsile stre ngth durin g prog ressive agein g o r a 7000-se ri es 1\1 a ll oy reinrorCL:d wilh I . .... o r SiC parti cles (Lewa nd ows ki cl al. I 'IX'» .

227

(3( flK )"

(9.20)

whe re {3 is a constant. Hence, a plot of crack growth rate (m/cycle) agai nst flK , with log. scales, gives a straight line in the Paris regime , wit h a gradient equal to n. At low stress intensities, there is a threshold , ~ Kl h' below which no crack growth occurs. The crack growth rate us ua ll y accelerates as the level for fast fracture , Kc, is approached. An a lterna ti ve way of presenting fati gue data is in the form of S/ NI" c urves, show in g the numbe r of cycles to failure (N r) as a function of the stress amp litud e (S ). Many materials show rapid crack grow th (Iow NI") when th e stress ~ lll1 plitude is hi gh. a cen tra l porti o n of decrea sin g S with risi ng NI . CO lT ' S P\ Hili ill )', 10 I he 1'<1ri s regim e, "nd ~ I fatigue filllit . wh ic h is

Toughness of composites

9.3 Sub-critical crack growth

a stress amplitude below which failure does not occur even after large numbers of cycles. This corresponds to a stress intensity below 6.Kth . Some of the features of fatigue can be illustrated by examining how the presence of particulate reinforcement affects the behaviour of a meta l. This is shown schematically in Fig. 9. 10. Below 6.Kth ' cracks are unable to grow at all. For AI/SiCp MMCs, 6.Kth is typically around 2-4 MPa JiTI, which is approximately twice that for unreinforced Al alloys (~ 1-2 MPa JiTI). A number of explanations for this have been proposed (Christman and Suresh 1988), including crack deflection at interfaces and a reduction in slip-band formation due to the particles. This beneficia l effect of the reinforcement in inhibiting the onset of fatigue cracks is a useful feature of such MMCs. However, for reasons outlined earlie r (§9.2.1 and Table 9. 1), MMCs have a lower fracture toughness than unreinforced metals, mainly as a result of the constraint imposed o n matrix plasticity. The Paris regime is usually short and the exponent 11 is often around 5- 6, which is higher than those typical of unreinforced systems (~4). Fast fracture is initiated at lower 6.K values than for the unreinforced metal. In practical terms, this means that MMCs can offe r improvements in fatigue performance compared with metals, providi ng that applied stress levels are low and /or flaw sizes are kept small. Contro l over processing of MMCs so as to eliminate inhomogeneities is importan t to ensure this.

While particulate reinforcement produces relatively minor mod ifications to the behaviour of the matrix , the presence of long fibres has a more pronounced effect. This is pa rticularly true for polymer composites, in which it is usually the type and orientation distribution of the fibres which is of most significance. The propagation of cracks through the matrix and along the interfaces usually dictates how fatigue progresses, but this is strongly influenced by how the fibres affect the stress distribution. The failure strain of the matrix is also important. A further difference from the particulate case lies in the distribution of damage. It is co mmon in long-fibre composites for matrix and interfacial microcracks to form at many locations throughout the specimen . Fibre bridgin g across matrix cracks often occurs, reducing the stress intensity at thl: crack tip. In contrast to this, fatigue crack growth in monolithic and pa rticle-reinforced materials usually involves a single dominant crad with a well-defined length.

228

229

The S / N r curves presented in Fig. 9.11 illustrate some features of t hl: fa tig ue performance of long-fibre reinforced polymers, when loa(kd a lo ng the fibre axis. Composites reinforced with stiff fibres , such as

1400 t200

Fast fracture

t/

10. 4 Q)

ti >~

MMC

10. 5

E

.§. ul~ 10. 6

rr Mth

"0"0

10. 7

2

3

s~ <>"p ~

eC>f·~.e

/ /

~"f"'"

~

~ 0...

6

................... .................. .... . .........

1000 K

OE 800

....

'" IU '"

b

'"

E ::I E .;( ro

600 400

::E

-

'"

-

- S-glass/epoxy

- - A I alloy (2024-T3)

""

-

- - - - - E-glass/epoxy --------- Boron/epoxy - - Carbon/epoxy

200 0

10 3 5

10

15

!';K(MPa . .J m)

Fig.9.10 Schematic depiction of fati gue crack growth rat e as a fun ction ,)1 applied stress intensit y factor 6.K, fo r a ty pi cal particle- reinforced MMC ;llld the co rrespondin g unreinforccd all oy, illu stratin g th e etTee t 0 1' reinl'orcemc nt (I ll th e I'ati guc res pon se.

10 ' Number of cyc les to failure , N,

1'1' '>.1 1 Ex perim ental .'> / N, plot s (!\garwa l and Broulm ;ln I <)XO), showin' th l' 1IIIIII her 0 1' cyc les to 1';lilllre dllrill g axial I'ati gue loadin g 01' unidiree li o ll ;ti 1,)tI F. 11I>ll' jl(l ly tll er cO lllposilL's. a ~ a 1'IIIlL'lio n 0 1' Ihe pe; lk ;Ippli ed l(lad . Th e stress r;ltl (l li was 0 . 1 ill ; llI e; l ~ l's /\ 1' (1 , It (lWIl IS a plo t I'or a ty pi c; ti IIllreinl'o reed ;tilllll i llllllll ;tll(l Y·

-230

Toughness of composites

9.3 Sub-critical crack growth

boron and carbon, show excellent fatigue resistance, being able to wilh stand alternating loads of around I GPa for very large numbers of cyck: ~ The fatigue performance of these materials is markedly superior to th; 11 of a typical aluminium alloy. With glass fibres , on the other hand, til l" lower stiffness of the fibre leads to reduced stress transfer, exposing till" matrix to larger stresses and strains. This causes progressive damage ;11 considerably lower applied loads than for the stiffer fibres. While the axial fatigue resistance of long-fibre composites tends to I l' very good, particularly with high stiffness fibres , performance is usua ll y inferior for laminates or under off-axis loading. This is illustrated by tl,.· plots in Fig. 9.12, which are for glass-reinforced polymer. The crossr ly and woven cloth laminates fail at appreciably lower loads than the uni directional material and show little evidence of a fatigue limit stress va lm' being identifiable. Damage to the transversely oriented regions starts ;11 low applied loads (see §8.1.2), transferring extra load and event ua ll y causing cracks to propagate into the axial regions. Nevertheless, I Ill' fatigue resistance of such materials compares quite well with that 0 1 many metals. Finally, the chopped strand mat (CSM) and dou. 1I

'llou lding compound (DMC) show relatively poor fatigue resistance. In 11 1(;se materials, fibres are misaligned and have relatively low aspect latios, particularly for the DMC. Broadly similar behaviour is exhibited by MMC laminates. Data in the lorm of S I Nr plots are shown in Fig. 9. 13(a). The best performance is shown by the unidirectional material, having the fibres parallel to the ,Ipp lied load. The performance of the other laminates can be rationalised hy calculating the range of stress to which the 0° ply (parallel to the ,Ipplied load) is subjected during loading. When this is plotted against IlIe number of cycles to failure (Fig. 9.13(b)), then the data for the difInent laminates fall on a common curve. This highlights the point that Ihe laminate does not fail until the fibres in the 0° ply become fractured. I'he fatigue properties can , however, become badly degraded if matrix nacks are not deflected at the fibre /matrix interface (Johnson 1993). A further point worthy of note with respect to the fatigue of composi tes is that the behaviour is often sensitive to the absolute values of the st resses being applied , rather than just the !::;'K range. In particular, the Int roduction of compressive stresses usually reduces the resistance to I"at igue. This is largely due to the axially aligned fibres having poor res istance to buckling (see §2.2.2). This results in damage to the fibres ;I nd the surrounding matrix and also allows larger stresses to bear on neighbouring, transversely oriented regions , accelerating their degradalio n. Aramid (e .g. Kevlar™) fibres are particularly prone to this effect , si nce they have poor resistance to compression. This is illustra ted by the data in Fig. 9.14, which show that the fatigue resistance of Kevlar™ com posites becomes poor for negative R values . (It should , however, he no ted that the effect is exaggerated by plotting the peak stress rather tha n the stress difference, which is larger for the lower R values.) Largedia meter monofilaments, on the other hand , tend to be resistant to buckling, leading to improved performance at lower R values. A final point concerns the frequency of cycling. For metals , this usually has little effect, but in polymer composites a higher frequency often has tens failure. Thi s is partly because of the viscoelastic response of po lymers. Matrix damage is more likely if the strain is imposed rapidly, :dlowin g no tim e for creep and stress relaxation . A second effect arises I"ro m the lo w thermal conductivity of polymers , particularly if the fibres are a lso poo r co nducto rs (gla ss, Kevlar™ , etc.) . Such composites tend to inc rease in tempe ra ture durin g fati gue loadin g, as a result of difficulties in diss ipalin· Ih e hl'a l 5 ne rated locall y by dama ge a nd vi scoe las ti c defo rIll ati o n. Thi s is ;'" "\' III II;ll l'd a l hi g h cyclin g I"reque nc ies. Sin cc Ih e streng th

__ 0°

- - - [W 1W )51s - - [(±45°,(00»)2 1s

-

- - - - - Woven cloth laminate --------- Chopped strand ma t - - - - - Dough moulding compound

~ 1000

6

""

--- --,

500

[

) -I

"-

""

--._- . . "-

"-

....

--

--- - ---

~,--­ ....

-

" ....

...

.. - .......... ....... -...... - ... -. -.. _- ..... -- .. -. -.... - ..... - ..... _......... -. .... - .......... . , - , - -7 - - 1- - - - -r - - - - .. --,.

o

2

3

- ..i - - -- _ ........... _~ _ _... L~_...l...-_

4

5

6

7

Log (N ,) Fi g. 9.1 2 Ex perim enta l SIN, plo ts (Ha rri s 1994). show in g th e num he r o i" eYl' lL-, to failurc durin g fati guc loadin g o r gla ss fibre/ po lyes ter co mpos it es with v; lri o ll s librc di stributi o ns. as a rUll cti o n o r th e peak appli ed IO; ld . T il e slress r; lti o I< (

"""""/""",,,,) IV;IS

0. 1 in:lll

':1 ' "

231

9.3 Sub-critical crock growth

Toughness of composites

232

-{)-W18 -

~ ~ -0

.~

-0---

[(0 ' )/±45' l,

_

[ 0 ' /90 ' 12, [0 ' / ±45 ' /90 ' 1,

233

800

1500

Q..

-o---R = 0.1

a. Ol

.><:

····0 ···

Ol

~

R =0.5

- - \1 - -

1000

500

R =0.01

•

_____ R = -0.3

200

_ R = - 0.5

(a) o

L-~-L~

-1

__L-~-L~__L-~-L~__L-~-L~__L-~~

2

3

4

5

6

7

R

Log (N f)

Cycles to fai lure (kilocycles)

3000

0

Fig.9. 14 Experimenta l S / Nr plots (Harris et al. 1990), showing the number of cycles to failure during axial fatigue loading of unidirectional Kevlar™ fibre co mposites, with various va lu es of the stress ratio R (= O"I11<1x/ O"I11;n) , as a fun c tion of the peak applied load. 0

• ~ 0..

~ on

(l)

....

of the matrix falls with increasing temperature, this tends to accelerate fa ilure.

0

2500

• •

2000

0

•

0

.0 C

(:;,

<=

1500

•

9.3.2 Stress corrosioll crackillg

0

(l)

OJ)

<=

r:

1000

on on

(l) ....

US

500

0

[0 ' 18

•

[(0 ' )/±45' l,

0

[0 ' /90' ]2,

•

[0 ' / ±45' /90 ' 1,

(b) 0

Fig. 9.13

0

10

40 30 20 Cycles to failure (kilocycles)

50

Ex perimental S / N r plots for severed Ti 15V 3A I 3er :lSn j35'i{, SCS-

6 SiC mon o filam e nt laminate co mposit es (J o hn so n 1993) . (a) Applied s tress a gain s t number of cycles to failure, and (b) the samc data plotted as the s tress ran ge experie nccd by th e tibres in th e 0 ' plies a gain st numhe r of L'YL'k s to failure .

Stress corrosion cracking is the term given to sub-critical crack growt h which occurs as a result of the effect of a corrosive nuid reac hin g the advancing crack tip . The micromechanisms respo nsible for this lowerill ' of the local toughness of the material vary widely between diflCrent mate rials and environment s. For example, in AI-based systems, th e presence of moi st or sa lt-laden air causes an acce leration of fatigue crack growth of between 5 10 times. A similar effect is obse rved ror parti culat . MMCs (O'owe and Hasson 1982). For long-fibre reinforced aluminilllll , the fibres give ri se to good retenti o n of fati gue resistance ulld er a xi;i1 loading, wherea s in th e tran sve rse direction th e fati gue resistancc is a ffected by th e corrosive ellvironment to a deg ree directl y rebt ed tll the proportil)1l pf Ill ;ltri x pccllpied by th e composi te (ILl ssoll ; 11)(1 ( 'rowc 19X7 ).

Toughness of composites

Re/erences and furth er reading

In the case of a luminium , stress corrosion cracking is usually due to the evolution of (atom ic) hydrogen at the crack tip , which is promoted by the presence of va ri ous liquid s. This tends to cause embrittlement by impeding disloca tion motion. In MMCs , a furt her factor is introduced in the form of strong traps for hydrogen at the ma trix / reinforcement interface, promoting cracking. Using straining electrode tests, Bernstein a nd Dollar ( 1990) showed that, whi le the strength levels of M M Cs were unaffected by testing in a hydrogen environment, the ductility was sharply reduced , irrespective of matrix ageing. This was attrib uted to strong trapping of hydrogen a t the interface, where voids (not seen in air-tested specimens) were observed. Fo r polymer composites, the behaviour of the matrix is often quite sensiti ve to the presence of fluid s. For examp le, the properties of many polymers are modified by absorpt ion of moisture. In many cases, this penetration occurs by diffusion through the matrix a nd is not con fin ed to access a lo ng the crack. Various features of such absorpt ion have been identified (S hen and Springer 1976). Commonly, water absorp ti on raises the toughness and stra in to fai lure of the matrix. Thus, the fatigue resistance of gla ss-reinforced epoxy ca n be rai sed by prior boiling in water (H arris 1994). However, water up take a lso tends to promote interfacial debonding, impa iring stiffness and strength, particularly under shear a nd transverse load ing. Some fluid s sha rpl y degrade the stre ngth of the fibres. An exa mple of this can be seen in Fig. 8. 10, wh ich shows the frac ture surface of a polyester/ 60% glass composi te tested in the presence of dilute hydroch loric acid . This attacks the fibre surface at the crack tip, reducing its strength considerably. Not o nl y does this impair the st rength of the composite, but the toughness a nd fatigue resistance are sharply reduced, si nce the fibres a ll break in the crack pla ne a nd pull-out work (§9.2.4) is virtually ze ro. The danger of such effects must a lways be borne in mind when glass-reinforced polymer composites a re to be used under loa d in chemically aggressive enviro nments , particula rl y mine ra l acid s (Pri ce a nd Hull 1987).

Bernstein , I. M. and D o ll a r , M . ( 1990) The effect of trapping o n hyd rogeninduced plasticity a nd fracture in s tructural a ll oys, in Hy drogen Ef/ects on Material Behaviour. N. R . Moody and A. W . Thompson (eds.) TMS pp.

234

References and further reading Agarwal , B. D . and Broutman , L. J. ( 1980) Alia /psis alld Per/rll"lll{{//c(' 0/ Filil'l" Composites. Wiley: New York Akisanya , A. R. and F leck , N. A . ( 1992 ) Brittle frac ture or adlll;sivc joint s, 111 / . l. Fract llre. 58 9] I 14 A s hby, M . F . and .l o nes , D . R . 11. ( 1<) l{O) 1:'lIg ill(,l'I" ;lIg flllI/ N;II/.\" . P<.: rgam o l1 : O xfo rd

235

703- 15 C hristman , T. and Suresh , S. ( 1988) Effects o f SiC reinforcement a nd ageing treatment o n fatigue crack grow th in an AI- SiC composite, Mat. Sci. &

Eng ., 102A 2 11 - 16 C ook , J . and Gordon , J. E . ( 1964) A m echa ni sm for the control of crack propaga tion in all-brittle system s, Proc. Roy . Soc., 282A 508- 20 C rowe , C. R. and Hasso n , D . F. (1982) Co rrosi on fatigue of SiC/AI MMC in sa lt-laden moi st a ir, in Proc. ICMSA 6. R. C. Gifkins (ed.) Pergamon pp .

859- 65 Eva ns, A . G . a nd Hutchinso n , 1. W. (1989) Effects of non-pl a na rity on the mi xed m o de fracture resista nce of b im aterial inte rfaces, Acta Metall.

Mat er., 37 909- 16 Eva ns , A. G. , Riihl e, M. , D a lgleish , B. J. a nd C hara lambides, P. G. ( 1990) Th e fra ct ure e ne rgy of bimater ia l inte rfaces, Mat. Sci. & Eng ., A126 53- 64 Ewa lds, H . L. a nd Wanhill , R. J. H. (1984) Fracture Mechanics. Edward Arno ld: Lo ndon G OI"don, J. E. (1978) Structures. Penguin: New York G riffith , A. A. (1920) Th e phenomena of rupture a nd flow in solids, Phil.

Trans. Roy ., Soc . A221 163- 97 H ,lrris, B. (1980) Engineering Composite Materials. Inst. of Metals: London Ha rris, B. ( 1994) Fatigue - glass fibre reinfo rced plastics , in Handb ook 0/ Polymer Fibre Composites, F. R. Jon es (ed.) Lon gman pp. 309- 16 H a r ri s, B. , Reite r, H. , Adam , T. , Dickson , R. F . a nd Fernando, G. (1990) Fatigue behaviour of carbon fibre reinforced pla stics, Composites, 21

232- 42 H asso n, D. F. and Crowe, C. R . (1987) F lex ura l fa ti g ue behavior of a ra mid reinforced AI 7075 la minate and Al 7075 Sheet in a ir a nd in sa lt-laden humid a ir, in Proc. ICCM 6, F. L. M a tth ews et al. (ed s.) Elsevier pp. 138-45 lie, M. Y. and Hutc hin so n, J . W. ( 1989) Crack defl ectio n at a n interface betwee n dissimilar and elas ti c m a teri a ls, Int . J. Solids S tructures, 25

1053- 67 11 ull , D. a nd S hi , Y. B. ( 1993) Damage mechani sm c ha racteri sa tion in composite damage to le ra nce in ves ti ga ti o ns, Composite St ructures, 23 99- 120 Ingli s, C. ( 19 13) St ress in a plate due to the presence of cracks a nd sharp corne rs, Trans . Inst. Naval Architects, 55 2 19- 30 I rw in , G. R. ( 1948) Fract ure dynamics , in Fracture of Meta/s. ASM pp. 152- 69 Jo il nso n, W. S. ( 1993) Damage development in tita nium m eta l m at ri x co mposites s ubjected to cycl ic load in g, Composites, 24 187- 96 K<.: nda ll , K. ( 1975) Transition betwee n co hesive and inte rfacial fa ilure in a laminate, Pro £". Ro.l". Soc. , 344A 287- 302 I c wandow s ki. J . J. , Liu , C. and Hunt, W. H. ( 1989) Effect of m a tri x mi c ros tru c ture and particle di s tributi o n on fract ure of an A l MMC, Mat. Se ;. (~ 1:·lIg .. A 1117 24 1 55 Pa ris, 1'. ;ll1d Lrd o gan , F . (1<)6 ] ) ;\ c riti ca l ana lys is o f c ra c k pro pa gation law s, .I . I/I/S;C 1:'lIgg , XS 521' :14

236

Toughness of composites

Phillipps, A. 1. , Clegg, W. 1. and Clyne, T. W. (1993) Fracture o f ceramic la minates in bending. Part 11 - Comparison of model predictions with experimental data , A cta M etal!. Mater. , 41 819- 27 Price, 1. N. and Hull , D. (1987) Effect of matrix on crack propagation during stress corrosion of glass reinforced composites, Comp. S ci. Tech. , 28 193- 210 Rice, 1. R. (1988) Elastic fracture mechanics concepts for interfacial cracks, J. Appl. M ech. , 55 98- 103 Shen , C. H. and Springer, G. S. (1976) Moisture absorption and desorption o f composite materials, J. Comp. Mat. , 102- 20 Suo, Z. and Hutchinson , J . W. ( 1990) Interface crack between two elastic layers, Int. J . Fracture, 43 1- 18 Vescera , F. , Keustermans, 1. P. , Dellis, M. A. , Lips, B. and Delannay, F. (1991) Processing a nd properties of Zn- AI matrix composites reinforced by bidirectional carbon tissues, in M etal Matrix Composites - Pro cessing , Microstructure and Properries. N. Hansen et al. (eds.) Ri s0 Nat. Lab.: Roskilde, Denmark pp, 719- 24 Wang, 1. S. and Suo, Z. (1990) Experimental determination of interfacial fracture toughness curves using Brazil nut sandwiches. A cta M etal!. Mat er., 38 1279- 80 Wells, 1. K . and Beaumont , P. W. R. (1988) The toughness ofa composite containing short brittle fibres, J. Mat er. S ci., 23 1274- 82

10 Thermal behaviour of composites

Th e beha viour oj" composite materials is ojien sensitive to changes in temperature. This arises j"or two main reasons. Firstly , the response oj" the matrix to all applied load is temperature-dependent and, secondly, changes in temperature can cause internal stresses to be set up as a result oj" diff erential thermal contraction and expansion oj"the tll'O constituents. Th ese stresses affect the thermal expansivity (expansion coefficient ) oj" the composite. Furthermore , significant stresses are normally present in the material at ambient temperatures, since it has in most cases been cooled at the end of the fabri cation process. Changes in internal stress state on altering the temperature can be substantial alld may strongly in/luence the response oj"the material to an applied load. Creep beha viour is affected by this, particularly under thermal cycling conditions. Finally , the thermal conductivity of composite materials is oj" interest , sin ce many applications and processing procedures involve heat floll' oj" some type. This property can be predicted/i-om the conducti vities oj"the constituents , although the situation may be complicated by poor thermal contact across the illterj"aces.

10.1 Thermal expansion and thermal stresses

10.1.1 Thermal stresses alld straim Da la fo r the therm a l expansion coefficients (et) of matrices and rell1I"orcemenl s, as a fun ction of temperature, are shown in Fig. 10.1. Po lymers and mcl,il s ge nerall y ex pand mo re than ce ramics. It can be ~ ec n Ihal Ih c dilTe n:ll ces in ex pa nsivil y betwee n fibre a nd matrix a re 1;1 rgc in 1ll ; IIl Y ·: ISl·S . Si li ce 1"; 1hr iGl1 io n a Im os l in cv ila bl y in vo lves co nso li d; llio ll ;11 n:I;11 iVl' ly 111 1' 11 IVlllpn: ll lm:s (sce C hapl er 11 ). co mpos il es lI sll;tll y ) \I

I

100

__ _

_

~-----

100

Fig. 10.1

4G m ( I

... - Alumina

300 Temperature CC)

400

+ vm )

Kp 6.a6.T

3 ( I - v m ) Km

p =

( 10.2)

( I + vm ) (Kp -- I) + 1 3( 1 - v m ) Km

...... .. . Silica

200

239

T he pressure in a spherical particle as a result of a misfit strain of 6.a 6.T ca n also be expressed analytica lly (Lee et al. 1980).

--AI -Ti - - - Epoxy - - - - - Polyester . . . . . SiC

..,.-., o x

10.1 Thermal expansion and thermal stresses

Thermal behaviour of composites

238

500

whe re G, v a nd K are shear modulus , Poisson's ratio and bulk modulus res pectively, a nd the subscripts p a nd m refer to particle a nd matrix , res pectively. The result of applyin g these equatio ns to the case of a SiC sphere in a titanium matrix after cooling through 500 deg K is shown in F ig. 10.2(a). This shows that a large compressive stress is developed in the rad ia l direction , but the hoop stress in the matrix is tensile . There is , therefo re, a large deviatoric (shape-cha nging) componen t to the stress sta te in the matrix near the sphere. In this case, the stresses are likely to cause yielding of the matrix . Equatio ns are also ava ilable (Lee et al.

Experimental data fo r the the rmal ex pa nsion coefficient s (expansivities) of va riou s re inforceme nts a nd m a trices .

contain significant differential thermal contraction stresses at ambient temperatures. In addition , for resin-based composites, the matrix undergoes shrinkage during curing, which represents a further contribution to differential contraction. Assuming the materia l to be effectively stress-free at some (high) temperature, Tesr , the stress sta te a t a lower tempe rature can be envisaged as arising from the fitting of a n oversized inclusion into all undersized hole in the matrix. The misfit straill is then 6.a 6.T, whe r' 6.T = Tcsr - To (ambient tempera ture) and 6.a = am - a r. For certain cases, the stress sta te which results from the mi sfit strain can be calculated analytically. The simplest situation is when the rein forcement is an isolated sphere in an infinite ma trix. The stresses in th e matrix are the same as those produced by a spherica l bubble o f radiu s 11 under a pressure P (= hydrosta tic stress, O"H of - P). Lame ( 1852) ri rst obtained the solution for the radial and hoop (ta ngenti a l) stresses, (T, and 0"0, in the matrix for thi s case

0",

=- 7 p(('

<0 ~ Vl Vl

~

( I () . I )

Hoop

100 ,'

- ------ ~ ---- - - ----- - ------

0

Cl..

<0

R'd;" & ~

~

Hoop :

Vl

~

(jj

-200

\!

-100

<75

&:

Radial Hoop

' / ,

_______ i~

0

Cl..

-100

-200

Radial

Ll T = - 500 K

\

O"my =100MPa -300

-300 0

(a)

Pa }

~

100

2 Radial distance, rla

3

0

(b)

2

3

Radial distance, rla

hg. 10.2 (:I) Ela sti c s trcss li c kl in an infinite matri x of titanium co ntaining a , p hcrica l parti c lc or Si( '. aft c r coo lin g throu gh 500 K . (b) New stress state a ft e r Inca l pla sti c il ow ( I ,L'L' /.( Ill. I')XO). ass Llmin g a matri x yie ld s tress o f 100 MPa , with Ill ) work h:lrLi c nin g.

I"

Thermal behaviour of composites

10.1 Thermal expansion and thermal stresses

1980) for prediction of the extent of the plastic zone and the associated changes in stress distribution. These are shown in Fig. 1O.2(b). It ca n bc seen that the plastic flow has caused substantial changes in the stress distribution, including a reduction in the hydrostatic pressure to which the particle is subjected. While the presence of neighbouring particles does affect the situation in a real composite, the fact that matrix plasticity often changes the stress state predicted for the elastic case remains true and should always be borne in mind. The elastic stress field for an infinitely long single fibre surrounded by ;1 cylindrical matrix (of finite radial extent) can also be obtained ana lyti ca ll y, although the equations involved are more complex. A typical result was shown in Fig. 7.3, for a glass/ polyester composite after coo lin g through 100 K. This stress field shows simi larities to the case of th e sphere, although, as might have been expected, the axial stresses become prominent for the fibre case. In a short-fibre composite, the stress field is more complex, particularly when the effect of neighbouring fibres is considered. In this case, recourse to numerical methods, with consid erable computational effort, is necessary to predict the stress field . However, the general nature of the stress distribution , and some idc;1 of the magnitudcs of the stresses, can be inferred from these simp lcr cases. It is a lso possible to use the Eshe lby method (§6.2) to predict th e volume-averaged matrix stresses resulting from differential thermal co ntraction. Before going on to examine how thermally induced stresses can affcc t inelastic deformation processes, it is useful to consider how the ela sti c accommodation of the strain misfit controls the overall coefficient 0 1" thermal expansion of the composite.

ness. Hence the presence of pores (of whatever shape, size a nd volume fract ion) does not affect the CTE, although they give rise to sharp reductions in stiffness .

240

The simplest case to treat is the axial expansion of a long-fibre compos ite. It can be seen from Fig. 7.3 that the axial stresses arc uniform in bot h constituents, which simplifies the problem. The slab model provides a good basis for this calculation, as it does for the Young's modulus Ui4. 1). There are two stages in the calculation, illustratcd in Fig. 10.3. Firstly , both constituents are allowed to expand freely. Then, they are co nstrained to have the same length in the axial direction, by the imposition of appropriate stresses. The final net expansion can then be treated as consisting of two terms; a natural expansion (that of the matrix is

......------.. t

2

~

~

_ _ _¥

" - - _ _ _ _ _--..1

1

I, ,r

U

1,---=========::::::: am L..I_____

---l

af

(b) :

Heat through !!.T . a",!!.T ~

Natura l thermal expansions

Bond constituents together

10.1.2 Thermal expallsil'ities Analysis of the stresses due to a change in temperature allows the coe lTi cient of thermal expansion (CTE) to be predicted . These stresses will have associated strains and the net effect of these on the length of the composite, in any given direction , can be calculated or estimated. This net Icn g t It change arising from the internal stresses is simply added to thc natur;t1 thermal expansion of the matrix to give the overall length changc ;Ind hence the composite expansivity. This simple view of thermal cxpa nsioll allows ccrtain poi nt s to bc idcntifi cd immcdiatc ly. For cxamplc, a porous matcrial, rcgarded as a composit e 01" vo id s in a matri x, does not develop any int e rn ;t1 stresses on heatin g ix:cau se th c ' inclll sion s' h;lvl' / L"rO stilT

241

Actual expansion, with internal stresses

I Ig. 10.) Sch cm ;lti c rcprcscntation or th c usc or th e slab model to derive an n prcss ion ror th c :I \ i:ll cx pan si v it y or a long- lIbrc compositc . (a) Slab dimcn \l Oll S, (h) inili:ll kll t' lh ill Ih e rihrl' dircCli(1I1 , (c) un con slrain cd Ih c rlllal cx pan sions " I Ih c I\\,(l l ·OII Slllll c· lll s, ;ll ld (d) linal dilll c nsio ns and inl c rn ;i1 slresses, IV hen IIII VII ;Il" I:l 1 hOlldill g is III:lilll:liIl Cd .

l O. 1 Thermal expansion and thermal stresses

Thermal behaviour o/, composites

242

usua lly co nsidered ) a nd a n a dditi o na l con tractio n or expa nsio n ca used by the intern a l stress. T h us

243

composites is tha t due to Schape ry ( 1968) , who used a n energy-based a pproach to o bta in the fo ll o wing ex press io n

( 10.3 ) ( 10 .8) where a c is the ex pa nsivity of the comp osite a nd tm is the elas tic stra in in. the m a tri x associa ted with the interna l stress (see F ig. 10 .3). The va lue 0 1 tm is fo und b y usin g two simple rela ti o nships, based o n a stra in balance a nd a fo rce ba la nce. T he straill balal/ce is o b ta ined by expressll1g the s um o f the two in te rna l stress-generated st ra in s as the d iffe rence between th e na tura l therm a l stra in s

( lOA )

(am - cvr)6.T = Er - Em

The forc e balallce is simpl y a requiremen t tha t, since there is no a ppli ed stress the two intern a l stresses mu st co unterba la nce each o ther whe n they ,:re co nverted to fo rces (multiplied by the rela ti ve secti o na l a reas)

( I - .l)rJ m + frJr = 0 ( l - f)Em Em + fErEr = O Co mbinin g Eqn s ( lOA ) a nd ( 10 .5) to d eri ve a n exp ress io n fo r

( I - f)Emf m + f EdEm + (am - ar) 6. T ] = 0 - FE r (cv m - cv r) 6.T m = (I - f)E m + fEr

".f

( 10 .5)

in which the ax ial ex pa nsivit y, a~" is th a t given by Eqn ( 10 .7) a nd th e Po isso n ra ti o V l 2c is o bta in ed fro m a simpl e rule o f mi xtures between those o f the co nsti t uents ( §4A). The predic ted depend ence of th ese two ex pa nsiviti es o n fibre co ntent , given by Eqn s ( 10. 7) a nd ( 10 .8), is shown in Fig. lOA fo r a po lymer ma tri x co mposite. A lso sho wn a re predi cti o ns o bta ined usin g the Es helby meth od (§6.2) . The co re of thi s technique in volves ca lcul a tin g the vo lume-ave raged stresses in bo th co nstituent s a s a res ult o f an im posed mi sfit stra in , so that it is rea dily ad a pted fo r th e predicti o n o f ex pa nsivities. The equa ti o n o bta ined (Clyne a nd Withers 1993) is a c = am - f{ (Cm - C r )[S - f(S - 1)] - Cm} - I Cr(Cl'r - Cl'm)

tm

- - Long- fibre, ax iat (Eshe lby) - - - 0 - - Long-fibre, transverse (Eshelby) Spheres (Eshe lby) . ..... .. . Long- fib re, ax ial (force balance) - '" - Long-fibre, transverse (Schapery)

( 10 .6) 60

x

Substituti o n o f thi s into Eqn (10. 3) a ll ows a n ex press io n to be o btaim:d fo r the (ax ia l) thermal ex pa nsivity of the composite

+ ac/' Er f) Em + fE r

50

;>

'Vi

( 10 .7 )

a m( 1 - f)E Ill

(I -

~

Co 40

f

Er (cv m - ar) a c = am - ( I - f ) E111 + f Er

( 10.9)

Since the ax ial force bal a nce is reli a ble fo r the sla b m od el (§4.1 ), this predicti o n should be quite accura te. (It is no t entirely ri go ro Lls, beca use differenti a l Poi sso n co ntracti o n stra ins a re neglected .) The expansivity in the tran sve rse directi o n, a nd the va lues fo r sho rtfibre a nd pa rti culate co mpos ites, a re mo re di ffi c ult to esta bli sh , SII1C<': the stresses a nd stra in s va ry with pos iti o n . Neve rth eless, a s with th e tr;ln sve rse stiffness, so me useful a pprox im at io ns can be mad <.:. A rev iew o r til l' mod els deve lo ped in th is a n.:a is giv<.: n b y Bowks a nd TOll1pkin s ( 19X(» . One o r th e l1lost sll cccssflll ror th e t r;lll sve rs<.: ex P ; I Il si vit V or IOIl).!,-ri h rl'

§

30

0X
(;i

...E

20


..c: f-

10

o LI~~~~~~_~L-~_~_~~__-l~~~~

o

0. 1

0.2

0.3

0.4

0.5

0.6

0.7

0 .8

0.9

Rei n fo rce m e nt vo lum e fractio n,f h g. 10.4 Pn:die led d e pcndenee 01' thnma l expa n siv ity on reinl'orce ment co n te nt 1,\1' g la ss I'ihres ; 11 1.1 s pheres in ;111 e pox y Illatr i.x, accord ing to the I'orce ba lan ce ( h pl ( I tl . 7 )) , Sl'Ii ;II" ' 1\' (hpl ( I1 U\)) and !-:she l hy (Lq 11 ( 10 .9)) mod e ls. T he c ur ves II'l' ll' ,d'I ; IIII (' d 11\ 111 1' 111 l' pl ll per ly (\ ;lla ill '\";Ihks 2.2 ;11 1(1 2.<;.

244

Thermal behaviour

0/ composites

where the stiffness, C, and expansivity, et, are here both tensors, as are the Eshelby tensor, S, (which depends on the aspect ratio of the fibre) and the identity tensor, f. Several features of the curves in Fig. lOA are worthy of note. Firstly, the force balance (Eqn (10.7» and Sehapery (Eqn (10.8» predictions agree well with the Eshelby model, confirming that they are quite reliable. Secondly, the case of spherical particles is quite well represented by a rule of mixtures (linear variation between the expansivities of the consti tu ents). Finally, the transverse expansivity of a (long-fibre) composite tends to rise initially as the fibre content increases. This occurs because, on heating the composite, axial expansion of the matrix is strongly inhibited by the presence of the fibres and the resultant axial compression of the matrix generates a Poisson expansion in the transverse direction , which more than compensates for the reduction effected in the normal way by the presence of the fibres, at least for low fibre contents. Plots such as those in Fig. lOA are of interest to engineers, since they allow the tailoring of expansivity via selection of constituents and reinforcement contents. However, it is important to note that these values are based on elastic behaviour. The associated internal stresses may become large, particularly if the temperature changes involved are substant ial, and under these circumstances the matrix is likely to undergo plastic now, or creep, which will alter the dimensional changes exhibited by the composite and make them difficult to predict.

10.1.3 Thermal cyclillg of ullidirectiollal composites

Large internal stresses can be generated when the temperature of a co mposite is changed. This often occurs during service, since temperature nuctuations of at least severa l tens of QC are likely even with compone nt s which arc not designed for high- (or low-) temperature use. The beh;l viour of composites during such thermal excursions is therefore of practical importance. Composites may respond to the associated intern;J1 stress changes in an inelastic manner. For example. dilatometry (length) measurements made on composites often exhibit significant hysteresis (i.e. the heating and cooling curves do not coincide). A schematic representation of the clTects or thermal cyclin g is presented in Fig. 10.5. This shows changes in axial matri x stress and COIl1 posite strain over a relati vely large temperature ran ge. r(lr ,I matrix pru ll l' to yieldin g (e.g. a ll1etal or;1 therll1opla stic) . Th e Illatri x is initi ;J1l y (pOillt A) t;lk ell ;IS h;IVill !-!-
10.1 Thermal expansion and thermal stresses

245

c::

.S? (J) c QJ

f(J) (J)

~

in x

·c


T

cti

'x

«

c 0 '00 (J)

~

D..

E

0

0

c

.~

Vi cti

'x

(b)

«

Temperature. T

F ig. 10.5 Schematic depiction of the variiltion in (a) matrix stress and (b) axia l stnlln of the specimen durll1g thermal cycling of an aligned fibre composit e.

(because of the coo ling cycle). On heating, this stress falls, becomes compressive and even tually causes yielding in compression (point B). A period of progressive plastic now then follows. On coo lin g (from point C), the matrix yields in tension at point D, before returning to A. The dilato metry traces (axial strain- temperature plots) are predicted to show hysteresis, but no net dimensional change. Experim enta l data for long-fibre compos ites (Kural and Min 1984, Wolff et al. 1985) are broadly consistent with this view. A simple ana lysis Or the dilatometer trace, proposed by Masutti et al. (1990), can be used to es timate the changing axial matrix stress in a long-fibre composite. Ass uming no interfacial sliding, fibre yielding, etc., thc axial strain of the composite must be equal to that of the fibres. which can be expressed :IS the sum or their natural thermal expansion and their elast ic strain ( I,

( 11

( 1 1 , \ ./ .

I IT II'

IldIT~ I' I 1:'1

IT1 ), )

( I () . I () )

246

10.1 Thermal expansion and Iherl11al slresses

Thermal behaviour of composites

Since the radial and hoop stresses in the fibre (0"2r and 0"3r) are relatively small compared with the ax ial stress (see Fig. 7.3) and the Poisson ratio 0 1" ceramic fibres fairly low (cv 0.2), the contribution from the transverse fibre stresses can be neglected. The axia l force balance (/O"lr + (I - f)O" lm = 0) can then be used to find the axia l st ress in the ma trix

247

3

c

.~

2

.~

I 0

e;J

f

0"1111 = ( l _ f)Er(etr6.T - Elc)

( 10.1 1) 0 300

Hence, O" lm is simpl y proportional to the difference, 6.f, between the nat ural thermal strain of the fibre and the measured strain of the com posite. The initi a l thermal stress in the matrix can be deduced by tak i n ~' the composite up to high temperature (cv 0.8Tm)' where the matrix stress will become ve ry sma ll , a nd running the fibre thermal expansion lin e bad from thi s region . This is illustrated in Fig. 10. 6, wh ich shows an expe ri mental dilatometer trace a nd the deduced mat ri x st ress hi sto ry. The latter is seen to be broadly of the form shown in Fig. 10.5. This simple procedure forms a convenient way of study in g the initi a l matrix stress, as we ll as the high-temperature characteristics. For examp le, Masutti el (11 . ( 1990) showed that a quench in liquid nitrogen , followed by heating to room temperature, generated a compressive residual stress of abo ut 25 M Pa in an AI /SiC composite, co mpared with a tensile st ress 0 1 50 MPa after heating and coo ling back to room temperature. The behaviour of short-fibre composites during thermal cyclin g i ~ rather simi lar to that described above, but there is more scope for vario us stress relaxation processes to operate. This is illustrated by the neutro ll diffraction da ta of Withers el al. (1987) presented in Fig. 10.7, wh icll shows the changin g axia l strains (a nd hence stresses) in both fibre alld matrix. The matrix stress history is sim il ar to those shown in Figs. 10. and 10.6, although the observa ti on of a significant drop in the l"ill :11 residual stress ove r an 8-h our period at room temperature is sugges ti w of cont inuin g stress relaxation (creep) processes. The scope for such st re ~' relaxation varies markedly with th e type of matrix , the fibre len gth alld the temperature range . When stress relaxation cannot occur easi ly. :111.1 the matrix is relativel y brittle, then matrix microeraekin g is lik ely. Thi s 1\ more common with laminali.:s than with unitlirec tion :i1 composit es su' }i I 0 l A.

500

700

900

T(K)

50

cu CL ~

.. .

E

_- .. _._---

\5 -50

(b) 300

500

700

900

T (K)

t Ig .. 10.6 (a) Expe rimenta l strain hi story (Ma sutli cl al. 1990) during thermal , \c llng and (b) deduced variat ion of matrix axial stress for an AI 3Mg/30% SiC IIIl1g- fibre composite. The matrix st ress is taken , Eqn (10. 11 ), as being propor111111 <11 to the difference between the mea sured composite strain and the natural lliermal expansion of the fibres, whi ch can be obtained by extrapo lation of the Illph-temperature data. The initial portions of the stress history curve, at the start "I' hea ting and of cooling, have grad ients correspond ing to elastic behaviour, indi cated by the dotted lines.

10.1.4 Thermal eye/ill/: of lamillates l'loh lem s associated with thermal stresses can be seve re with lamina tes. llil' rmal misrit stra in s now :Irise, not on ly between fibre and matrix , but 1,\' lwee n th e indi vidu;i1 plies or the laminate . For examp le, since a lamina 11\ 1I: i1l y h:ls: 1 mu ch 1:lrger ex pan sion eoel"l"ieient in the transverse direction lli: 111 in th e :I.xial direc tion , he: ltin g 01' :1 cross pl y 1:llllinate wi ll lead to the tl,I II S"l'I'Sl' 'XP:III Sillll ol" e: lch pl y IX'ill: stroll: ly inhihit ed by the presence "I thl' (l tli l' l pl y. Tlli s IS Il sl'l'td ill th c se nse Ih :lt it will 11l:lk e th e dim en.1. 11 1:11 l' Il :ln !'l" 111111i Iv" JlI OII Il IIII Vl' d :lllilllltll l' i, "llopic, hilt it does le;l((

248

10.1 Thermal expansion and thermal stresses

Thermal behaviour of composites )( SiC • SiC

c o

AI

Heating

• AI

Cooling

)C

Heating Cooling

- ,

249

siv ities (given by Eqn s ( 10.7) a nd ( 10.8) respecti vely). T he vo lume fractio n,.l, of the two co nstituents does not a ppear, since it is equa l to 0.5 for a cross pl y lamina te in whi ch the plies are o f equ a l thi ck ness a nd therefore ca ncels o ut in Eqn ( 10.6). T his st rain ca n be co nverted directly to a stress, since it a ri ses fro m the two pli es bein g forced to have the sa me length (a nd does no t include the free therm a l ex pa nsio n). The stress to which the tra nsve rse ply is subjected ca n therefore be ex pressed as

.~

:"

( 10. 12)

"-

~

-2

Cl

-3

Fig. 10.7 Neutron diffracti on measurement s (W ithers ('/ al. 1987) of the latticc' stra in variati on over a thermal cycle for Al co ntaining 5% ali gned SiC whi skcrs ill (a) rein fo rcement and (b) matri x. The observed stress relaxa ti on at th e end 0 1' the cycle took place over a peri od of 8 hours.

to intern a l stresses in the la minate . T hese ca n ca use the la min a te to d is to rt , so that it is no lo nger fl a t; a n illustra ti o n of the type of di sto rt 10 11 whi ch ca n occur was shown in F ig. 5. 14 (§5.4.3). However, eve n if such d isto rti ons do no t occ ur, the stresses whi ch arise on cha nging th e tem pera ture ca n cau se mi crostruct ura l da mage a nd im pa irment of pro perti es. T herma l stresses within a la mina te ca n be calcul a ted usin g th e numer ica l tech ni ques ou tlin ed in C hapter 5. Fo r the specia l case of a cross ply lam in ate, a simple a na lyt ica l approac h can be used. T he stresses ,ll1 d stra ins para llel a nd norma l to th e fibres in each ply ca n be fo und usill !! th e met hod o utlined in § 10. 1.2. Figure 10.3 is aga in appli cab le, but th . two co nstituent s a re now the two pli es (o ne o riented axia ll y a nd the o th el tra nsverse ly), rather th an the matrix a nd the fibres. The sa me equ a ti o ll s res ul t, but the mea ni ng of th e para meters cha nges sli ghtl y. Fo r exa mp le , ta kin g the tra nsve rse ply as the ' mat ri x' and the ax ia l one as th e ' fibres' , :1 mod ified versio n of Eq n ( 10.6) now gives the elas tic stra in in the trail S ve rse ply o n hea tin g thro ugh a tempe rat ure interva l f:::,. T

whcre

1~ 1 ' 1:'2

are th e a xial :l11d tral1 sverse

YOUI1 "'S

lIloduli ( live l1 hy hil i

(4 .1 ) :llld (4.7 ) res pec ti ve ly) :ll1d 111 , 11, :Ire tll , :t xi:ti :111(1 tr:II ISVl' rSl' n p:111

T he stress in the ax ia l pl y is equa l in magnitude to thi s a nd o pposite in sign. Plo ts o bta ined usin g Eqn ( 10. 12) a re presen ted in Fig. 10. 8. T hi s shows t he stresses in the tra nsve rse plies of two crossp ly la min ates, as a fu ncti o n or the vo lume fracti o n o f fibres, a ft er coo ling thro ugh a tempera ture in terva l of 100 deg K . Property da ta fo r the fi bres a nd the ma tri x we re tak en fro m Tab les 2.2 a nd 2.5. Seve ra l fea tures a re of in te rest in this

'2 0...

6

35

0' 30 25

.. ...... 20

.'

.....

.. .......

..'

'

,

15

10

•

......... Glass/epoxy cross ply - - Carbon/epoxy crossply

5 '/ 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Volume fraction of fibres in each plYJ I It!. IO .X Predict ed dependence on fibrc contcnt of thc thermal strcss in the 11:I IISVe rSe direc ti on wilhill each ply 01' Iwo cross ply lam inates, ari sing from a ll'111I)cr: ltllre dn:re:N' 01' lOO K. Th e pl ots WCi"e ohtain ed I'rolll Eqn ( 10. 12), II,i ll g li hre \1:11:1 I11 I :11l'" ) ) 1'0 1 I: gl:lss :111\1 carholl (IIM) and data I'm epoxy 1(" 111 III I :Ihle' :> . ~ .

2')0

i"i g un:. i\ p..:ak is observed o n increas ing the fibre content. This is ex p..:c ted, sin ce the stresses are due to the anisotropy in et; the difference between (VI and (V 2 has a peak at some intermediate value of/" (see Fig. lOA). Fo r the carbon fibre composite , the stress does not fall to zero asI approaches 100%. This is because the carbon fibres are inherently anisotropic in thermal expansivity (see Table 2.2); they show zero (or negative) ex pan sion axially on heating, but expand in the transverse (radial) direction s. Level s of the rmal stress are of interest for prediction of damage development. For a fibre content of rv 40- 70% , it can be seen from Fig. 10.8 that stresses of rv 25- 30 MPa are expected. This is approximately thc transverse failure stress for typical laminae (see Table 8.2). It follows that cooling through lOO K (from an initially stress-free state) would bc likely to cause damage in such a crossply laminate. This is consistent with experimental observations. For example, Fig. 10.9 is a section through a carbon fibre crossply laminate after cycling between 50 QC and - 50 QC showing a crack in the tran sverse ply. The progressive development o r s uch damage is illustrated by Fig. 10.10, which is a series of X-ray rad io graphs of the same material , taken afte r different numbers of therma l cycles. The progrcssive de vclopmcnt of cracks pa rallcl to thc fibres in each ply shows that the stresses generated during each cooling cyc le were close to the critica l level for cracking. Such damage is obvious ly of concern. It may be noted that the requirement to avoid such damage is not that the la minae should have a particul a r tran sve rse st ren gth leve l, but rather that they should be ab le to s ustain a particular transve rse strain without damage. There is thus interest in the development of po lymer formulations and manufacturing method s giving high transve rse

251

10.2 Creep

'f'/wmlll/ /w /illl'iollr of composites

.o..l..

150 cycles

2000 cych:s

1200 cycles

200 cycles

--

-...,.....

24{X) cydcs

I-~l

400cyc\cs

2600 cycles

600 cycles

3(x){) cycles

F ig. 10.10 Series o r X-ray radiographs (Jennings 1990) sho win g the developme nt o f mieroeraek s parallel to the fibres in a carbon fibre/bismaleimide e ross pl y laminate a fter thermal cycling between - 50 ne a nd + 50 C.

s trai ns to failure, pa rticularly for components subject to large tempera tu re fl uctua tion s.

10.2 Creep 10.2.1 Basics of matrix am/fibre bellll11iour

Fig . \0.9

Opt ical micro graph (.I c llllill gs \990) or lI1icrocr:lc\..ill .' III :1 c: lrhll ll lihrc/ hi slIlalcilllidc cro ss p\ y \:llIl ill :ll c :iI 'IL'I' I il l' I 111 :1\ l'yc llll l '

('r..:..: p is th..: tcrm used to describe progrcssive deformation of a material w h..:n s uhjecled 10 :1 Ul ll st; lnl load . This is o ft en undcsirable in engineerillg silll :llioII S, Sil l,',' I11 lim e 1:lrg..: slrain s Gin de velo p in a compo nent. Mos I 1l1 : 11 l' II : iI s ,, 1: 11 I 10 n lililil sig llii"i L':ln l L'l'CCP whc ll apprcL'i :lhlc IO:ld s

252

253

Thermal behaviour oj' composites

fO.2 Creep

(say, > 10 MPa) are imposed at temperatures greater than a bo ul 0.4- 0.5 T l11p (i .e. 40- 50% of their melting temperature in K). T hi s is essentially due to diffusional processes becoming activated, a ll ow in g molecular re-arrangements and associated straining. Thermose ttin g resins do not have well-defined melting temperatures, but they tend 10 become chem ica ll y and mechanica ll y degraded when subjected to rela tively modest temperature increases (~ 100- 200 K) above ambient. T lwy are fairly resistant to creep at room temperature. Thermoplastic po lymers , on the other hand, have well-defined glass transition (amorp ho us) and/or melting (crysta lli ne) temperatures, above wh ich they beco nlL' (viscous) liquids. These temperatures are in most cases re lative ly low (~400-600 K), so that they are prone to creep even at room tempera ture (~ 300 K). Meta ls in common use as structura l materia ls ranae in me ltin ' b point from ~ 600 K (lead) to ~ 1950 K (titanium). Hence, while mos l metals show little creep at room temperature, they virtually all beco me susceptible in the temperature range up to ~ 900 K (~ 600 QC), wh ich is required for many technological applications. Typically, the deformation history of a specimen during creep has th e form shown in Fig. 10.11. The details of the behaviour of various types or material have been summarised by Frost and Ash by (1982). Broad Iy, primary creep represents the setting up of some kind of microstruct ur:" balance, which is then maintained during the quasi-steady state of secondary creep, before breakdown begins as the tertiary regime is ente red. For example, this balance might be one in which dislocations are surmounting obstacles at a rate dictated by diffusional processes and th l' breakdown might represent irreparable microstructural damage such :IS

internal void formation. Interest commonly focuses on the strain rate d uring secondary creep, although properties such as the creep rupture strain may also be of concern. One of the major attractions of composite materials, particularly for thermoplastic and metallic matrices, is that the addition of certain reinfOl'cements can produce dramatic improvements in creep resistance. G lass typically softens at about 900- 1000 K (see Table 2.3), but is resistant to creep up to about 550 K. Furthermore, although polymeric fibres (e.g. Kevlar™) do not have good high-temperature properties, ceramics such as alumina (Tm ~ 2300 K) show little creep below 1100 K, while SiC a nd carbon are in general highly creep resistant to over 1200 K, depending on the details of their microstructure. In most creep situations with composite materials, such fibres can bc assumed to behave elastically, i.e. to experience no creep. The creep behaviour of the composite as a whole depends on load partitioning and constraint, which is in turn dependent o n geometrical factors and , in some cases, on the nature of the interface.

rupture secondary (stage 2) creep c

.~

' " tertiary (stage 3) creep

10.2.2 Axial creep of 10llg~fihre composites

Pro vided that the fibres remain elastic, treatment of this ca se is straightro rward. The initial strain, generatcd immediately the composite is loaded , is the applied stress over the Yo ung's modulu s of the composite, given by a rul e of mixtures (§4.1) (J

fO

= . . J Er + ( I - J ) Em

( 10.13 )

As creep occurs in thc matri x, the applied stress is progress ivel y transIC ITed to the fibres. As the fibres strain ela sticall y, the stress in them increases. The limit comes when the fibres carry all of the applied load. At thi s point , the strain of th e fibre s, and hence of the composite, is given by (J

(jj

f-x.

primary (stage 1) creep Time , t h g. 10.11

Sehclll:lli c dcpi cli on or a Iypi cal slr;lin hi slory dill ill )' nl'l' Jl dcrorlll ;I li oll 1I11dcr cO ll slall1 10; Id. sil olV ill' Iill' Sil'" , h sl" l" ""TP l:t ll·.

=I

Er

( 10.14)

Th e strain approac hes thi s valu e asymptotica lly, since the rate of creep of the matri x rail s o lT as the stress it carries decreases, a nd a stead y state is Ileve r set up . Fro m an engin ee rin g point o f view, thi s situ a ti o n is attracli ve, sill cc th e creep str:lill Gill never exceed a pre-d etermined level (which ill ge m:r:" is Slll :"I ), :ISS lIlll ill g th:lt th e ribres do not break. Ex perim ent a l rl's lill s al'l' l'O II , I, I" 111 1I' 111i llii , , illlpk :In :" ys is. ' :0 1' eX;llllp k. th e da ta in

254

Thermal behaviour of composifes

10.2 Creep

Fig. 10.12, from Endo ef al. (1991) , show the strain 111 an AI /40%SiC composite slowly approaching the calculated limit. It is possible to predict the rate at which the limiting strain is approached , from a knowledge of the creep characteristics of the matrix . McLean (1983) has shown that, for a matrix which exhibits power-law creep in the steady slate (d E/df = Aa") , the creep rate of the compos itc obeys the following equation

unreinforced material (for example, having a higher dislocation density in the case of metals). The presence of thermal resid ual stresses (§ I 0.1) may also have an effect, a lthou gh these tend to become sma ll if the composite is held at a (constant) elevated temperature.

"

Aa

dE df

10.2.3 TrallSI'erse creep and discOlltillllOIlS/Y reil!/orced composites

( ) I -

"

E

Eoo

( 10.15) [I

+

I

Er

( I - / )Em

] (I -

255

I)"

This equation can bc integrated to give the strain as a function of timc. In general , this gives fairly good agreement with experiment. In practice, however, the situation may be complicated by factors sLlch as the matrix in the composite differing in some way from the correspond in g

4

- - - -l>

The situation is very different if the composite is loaded transversely, or if the reinforcement is discontinuous (short fibres or particles). In this case , the composite is able to deform progressively and steady-state creep is often established. Creep rates are strongly influenced by the creep characteristics of the matrix. However, the cxtent to which the fibres relieve the matrix of load is important. For example, in a short-fibre composite under axial load, the load partitioning depends strongly on the fibre aspect ratio. This is illustrated by Fig. 10.13, which shows the volumea veraged stress in matrix and fibre for elastic loading of a polyester/50% glass composite, as a function of fibre aspect ratio. In this case, the stress in the matrix is reduced by a factor of about 5 as the fibre aspect ratio rises from I to 10.

2

o

8 --Fibre ........• Matrix

.......
0.5

2 0

10

20

30

40

50

60

70

80

90

Fig. 10.1 2 Experimental data (End o(' /ol. 1991) showin g th es traina s,ll"unc ti()11 of time, fo r a n A 1/40 vo l. o/" Si C (' N ica lo n™ ' ) long-li hre co m posi te, lo, llkd parallel to th e fihres. The applied stress was 2 6() MPa ,Ind th c tc mpc r;ltun; IV; I' 613 K. Al so sho wn arc th e in stant ;ln co us and limitin g str;lill s, ca icul ;ltcd I"nllll Eqn s (IO . Ll) and ( IO . I..!) using stilTll css v; liu cs .' iw ll ill T;lhks 2.2 ;llld 2.~.

........ '"

..........

100

Time (hours)

".

o

...............

... ... - .. -- ....... -. -.. ----- -- -.. -_...... _--

10

100

Fibre aspect ratio, S J-"i g. IO . I:l Prcdi c il'd depcnd c ncc on lihre aspec l ratio 01" th e volum e-a vera ged stress in c; ll'lI (·' "l stltll ,' II \. ;IS a r;ltio to th e appli cd stress, 1"01' a xial loadin g 01" POI Yl" l" 1 ~ tl "" 1' 1:" Id ",· CO l1lP()s it cs, ohtain cd usin g th e I':she lhy Illod el.

256

Thermal behaviour

0/ composites

Most experimental data for creep of short-fibre co mposites und er ax ia l load confirm a s trong se n sitivity to fibre aspect ratio. An exa mple is shown in Fig. 10.14, which gives minimum (steady-state) c reep ratcs for a n aluminium alloy reinforced with SiC pa rticles (aspect ratio '" I) or with SiC whiskers (aspect ratio '" 5). The creep rates of the whiske r materi a l a re lower b y about two o rd ers of ma gnitud e, at any given stress leve l. In many cases, h o wever, such sensiti vity to fibre aspect ratio is not just a co nseq ue nce of th e degrce of elas tic loa d tra nsfcr. During creep, thc st ress in the matrix is a lso dependent on the degree to w hich stress relaxation occurs. Thi s term is used to d esc ribe th e unl oading of the matrix by diffusive processes which allow it to change shape. These effects occ ur m ore rea dil y with low aspect ratio fibres, since diffu sio n distances a re shorte r. Another fac tor w hich is somctim es impo rtant in creep of discont inu o usly reinfo rced co mposites is the e ffect o f thermal resi dua l s tress. When the temperature is chan ged, there is a change in the level of thermal stress in the m at ri x (§ IO.I ). This may augment o r reduce the stresses from th e app li ed load and hence red uce or enh ance the creep rate. A noma lo usly high va lu es o f the act ivat ion energy and the st ress expo nent fo r c ree p, often obscrved for MMCs, have been exp la incd o n this basis (Nardo nc

10. 4 • o

10

A I (6061) 120% SiC whiskers AI (606 1) 120% SiC particles

5

~

.:!:.d)

c;;...

10. 6

P-

d) d)

10. 7

E E

10. 8

... u ::l C

~

10. 9

101 ~0

40

60

80

tOO

Stress (MPa) Fig. 10.14 Expcrimcnta l steady-s tat e creep rale dat:1 (Nieh 1')X4) fnr isolherlll; i1 creep a I 56 1 K of ;\ I allo y (6061) reinforced wil h ~ O vo l. "" o f SiC ' 11" rl iclcs nr Si( , whi skers.

10.2 Creep

257

an d Tien 1986, Jarry et a/. 1987). Furthermore, while thermal residual stresses tend to become rel axed a nd hence a re often of minor significance d uring iso thermal creep, they ca n be of prime importance during thermal cycling, when they a re continu o usly regenera ted. Under these circumsta nces , the creep behav iour can differ s ubsta nti a lly from that expected o n the basis of the de pend ence of creep rate on temperature observed d uring iso thermal testing . The effects outlincd above are illu strated by the plots in Fig. 10.15, which give strain hi story dat a o btained during thermal cycling creep of sho rt-fibre reinforced aluminium. The predicted plots were obta ined by us ing the Es helb y model (§6.2) to predict the stresses in the matri x from the applied load and as a co nsequence of the changes in temperature. Instantaneous strain rates were the n o bt a ined from the known depende nce o f mat ri x c reep rate o n stress and temperature. Also sh ow n, for the A I/ IO %A I2 0 , co mposite, are the meas ured and m od elled isoth erma l cree p ra tes of the co mposite at the diflilsional mean temperature of the the rmal cycle (Wu a nd Sherby 1984). This is the mea n te mpera ture of the im posed cycle, weighted for the temperature-dependence of the creep ra te. It ca n be see n that the average stra in ra te during cycling is co nsiderab ly grcatc r than thc isothcrmal ratc, rc n cctin g thc innucn cc of the thcrma l resid ua l stresses. I t is a ca use for co ncern th at s uch th erma l cycling ca n accelera te th e creep of MMCs , since th ere are ma ny technological a ppl icat io ns in w hich such co nditi o ns are imposed . Fina ll y, the interface so met im es plays a n importa nt role during creep or discon tinu ous ly reinforced materials. A n examp le is provided by th e transve rse loading of long-fibre reinforced titanium a ll oys, which has becn exam ined in some detail (Jan sso n et al. 1994, Newaz and Maj umdar 1994). Most of thi s material is prod uced current ly using SiC mono filaments with a graphi tic s urface coating. The coa tin g protects the fibrc from handling damagc and chcm ica l attack during fabrication , but does not a ll ow strong chem ica l bonding between fibre and matrix. At ;I mbient temperature, fibre / matrix co hesio n remains good as a result of the radial compressive thermal residual stress (§ IO . I). At elevated temperatu re (> 500 °C) , h owever, the radial stress tends to become tensile, allow ing thc interface to be opencd up by modest appli ed loads. When the crccp rate is a lso enhanced by thermal cycl in g, la rge voids can tkve lop at the interface. This is illu stra ted by F ig. 10. 16, which shows ; 1 micrograph of ;1 Ti composite after transverse loading under thermal cyclin g conditions . Such large interfacial vo id s lead to high (stage 3) c reep r:ltes :llld r:1I11d Oll 'i\'t or lTl'l' p rupture .

I 'h/ '/II/tll

1,/ )/1/1 '/111/1"

11/

clIlI/IJIIsil es

10.3 Thel"mal conduct ion

259

700 (a)

Q ~

600

...=s 'i;j ...

Cl)

0)

500

0..

E Cl)

f-<

400

c:

20

Iso thermal at T.

.~

b

.~

18

g c::

.~

,~

16

bVJ 0.. Cl)

... U

Modelled

14

0)

12

1400

1600

1800

2000 Time (s)

2200

2400

2600

Fig. 10.1 6 SEM mi crog ra ph (Clyne 1'1 af. 1995) of a po lis hed sec tion near the fract ure surface of a Ti- 6AI 4V j30 % SiC co mposite, after rupture und er tran sve rse load in g at a stress of 25 MPa with therm a l cycl in g (bet ween 400 c C a nd 700 QC, a t 100 QC per minute). The direction o f load in g is indi ca ted by the a rrows.

700

g

(b) 600

...=s

Cl)

...

'i;j Cl)

Temperature

500

0..

E

Cl)

f-<

400

c:

2

.~

b

E

1.5

g

10.3 Thermal conduction High th erm a l conductivity is useful in improvin g the resista nce of ma teria ls to thermal shock a nd avo idin g the d eve lopment of ' hot spots' durin g loca li sed heating . In o ther situa ti o ns, a low thermal conductivity can be be neficial in providing thermal in sulation. Before co nsiderin g conductivity levels in fibres , m a trices a nd composites, it is useful to review the mec hani sms of the rm a l conduction.

Measured

c::

.~ VJ

0.. Cl) Cl)

U

0.5

o

400

Modelled

Creep strain

600

800

1000 Time (s)

10.3.1 Heat trallS!'er mechallisms

1200

1400

F ig. 10 .1 5 Co mparison (Furness and C ly nc 199 1) bctwccn mcas ured and Ill o d cll ed In-cycle crce p s train hi s to ry for (a) Al j IO 'Y<, Safll l f M (s ho rt;\1 0 1'1'11")' I . " 1 l e dll( ( I) Al j10'Y< S '- I"tM ). - " a lii ,u nd er 20 MP" applied axi:1i 10
Heat n ows within a m a teri a l by the transmission of phonons (lattice vibra ti ons) and free electro ns (if present). Both of these carriers have a certain mean free path A between collisions (energy excha nge events) a nd ~I n avcr~ l ge ve locit y I '. The thermal conduct ivit y K is related to these pa ralll ell: rs h y :1 silllpk cqll:ltion derived fr om kinet ic theory

f:

( 10. 16)

I

l

_

260

Thermal behaviour

()l composites

10.3 Thermal conduct ion

where c is th e volume specific heat of the carrier concerned. 80th meta ls (electron-dominated) and non-metal s (phonons only) have zero conductivity at a temperature of 0 K (where c becomes zero in both cases), followed by a sharp rise to a peak and then a gradual fall as the tem perature is progressively increased . Th e rise reflects the increase in c towards a plateau value (rv 3 Nk, where N is Avogadro 's number and k is the 80ltzmann constant , with all th e vibrational modes of all the molecules ac tive). The fall is caused by the decrea sing A as th e greate r amplitude of lattice vibration causes more sca ttering of th e carriers. The ma ximum (low-temperature) value of A is dicta ted by atomic-sca le defects for electrons and by the ph ysical dimensio ns of th e specime n for phonon s. The average ca rrier velocity is in se nsitive to temperature in both cases. The phonon velocity (speed o f so und) is hig h in light , stiff ma te rial s. The mea n free path of a phonon is st ruc ture-se nsitive and can be very large in pure specimen s of hi g h perfection and large g rain size. Single crysta ls or material s like diamond a nd SiC ca n therefo re have very hi g h therma l conductivities. Some rece ntl y developed pitc h-ba sed carbon fibres also have very hi g h thermal co nduct iviti es (Kowalski 1987). With the exce ption of s uch cases, metals tend to have the hi ghest conductiv iti es, because electrons us uall y have a much la rger mean free path than phonons . This is substa ntiall y reduced by the presence of so lute a toms and va riou s defects which cause electron scattering. Polymers have no free electrons and low st iffn esses (low phonon ve locit y), so that cond uctivities are low . These trends a rc all apparent in the data shown in Table 10. 1.

10.3.2 C{)fu/llctiJ'ity

ol composites

The basic equat ion of heat flow may be written q = - KT'

261

Table 10.1 T/u' fI1w/ cOlldllctil'itr data fo r a rallge o/materia/s K (W m - I K I)

Material

at 300 K Diamond Graphite (parallel to c-ax is) Graphite (normal to c-ax is) Ag C LI C LI- 2%Ag C LI 2% Be C LI 40%Ni AI Ti Ti 6AI 4V SiC (single crystal) SiC (polycrystal) AI 2 0 , (singlc crystal) AI2 0 :1 (polycrystal) Epoxy resin Polyester rcsin Ny lo n

600 355 89 42 5 400 ~ 390 ~ 130 ~ 20 ~ 220 ~ 18

at 900 K

~

~8

100 10 50 ~ 100 ~ 5 30 0.2 0.5 0.2 0.24 0.2 0.25

40 180 ~ 18 ~ 12 ~ 70 5 30 ~ 30 3 10 ~

~

~

~

325 340 340

~ ~

The conductivity o f a composite ca n be predicted provided s uitable ass umpti o ns are made about the fl ow of heat through the co nstituent s, i.e. th e shape of the isot herm s. Several reviews are avai labl e coveri ng th e effective co nductivity of co mposites (Nielsen 1974, Ha le 1976, Hatta and Taya 1986). For hea t fl ow in lo ng-fibre composites, the slab m ode l (§4. 1) ca n be used. Axial a nd tran sve rse heat fl ow are re presented schema tica lly in Fig. 10.1 7. For th e ax ia l case, the thermal gradi e nt (spacing between the isot he rm s) is the sa me in eac h const ituent. The total heat flux is give n by the s um o f th e fl ows thro ugh fibre and matri x

( 10. 17)

( 10.1 9)

m - 2)

where q is the heat flu x (W arising from a the rmal gradient 1" (K m - I) in a material of thermal conduct ivit y K (W m - I~K I). It is important to distinguish K from the thermal diffusivity (/ (= A / C), whic h is the parameter determinin g the rate at wh ich a mate ri ;ti approaches thermal equi li brium. This appears in the unsteady diffusioll eq uation

in ih

These can be written in terms of co nducti vities a nd thermal grad ients

K l c T' = f KrT '

so that the composite co nductivity is given by a simple rule of mixtures 1\' 1,

uT"

( 10. I X)

+ ( I -- f)Km T'

I AI I ( I

I )Kill

( 10.20)

262

/0.3 Therlllal conduct ion

Thermal behaviour oj' composites

263

In fact , it represents a lower bound. Il becomes very unreliable when the fibres have a low conductivity; for in s ulating fibres , the composite is predicted to have zero conductivity even when the fibre volume fraction is small. More reliable treatments are available for the transverse conductivity. For example, the Eshelby method (§6.2) can be adapted to predict the conductivity of composites with reinforcement having any aspect ratio (Clyne and Withers 1993). The tensor equation obtained is

'___ _ _ _---' ~f

Kc = [ Kt~ 1 +.l{(Km - Kr)[S - I(S - !)] - Km} - I (Kr - Km )Km r l Isotherms

l

( 10 .23)

Isotherms

" ~,

': I ~ ---.....

:: :i• :ii• -J> :i iii • . ('Ij3

-i>c_ ! !::i~

which is closely analogous to Eqn (6.34) for stiffness. Hatta and Taya (1986) have shown that, for the transverse conductivity of a long-fibre composite, this reduces to the expression

: ~IC

-;> ~0(}\~: , " 1r 1-,- ,- , ...;.,____ :1-;> ..;.....;.I

,

I

I

,

I

,

I

2c

Fig. 10.17 Dcrivation of exprcssio ns for axial and transversc conductivity. SchematiC depiction of (a) thc slab modcl for lo ng-fibre composites, (b) axial heat now and (c) transve rse heat now.

The transverse conductivity, based on the slab model, is obtained by equatin g the heat nuxes through the two constit uents (Fig. 10.17(c))

Q2c

= K2c T: =

Q2f

= KrT( = Q2m = Km Tt~,

( 10 .2 1)

The thermal gradients in the two constituents are related to the overa ll average value by the expression

T:

=I

T(

+ ( I - I ) Tt~,

leadin g to the following express ion for the transvcrse conductivity

K2c =

1-I)-I

I +-(Kr

Km

( 10.22 )

This is analogous to the expression for the tr,lnsverse st illn ess, derived in ~l4.2. As with that derivation, the assumption that th e two ul nslilllcnl s li c ' in scrics' with e; lch olher ic;ld s to thi s ex press ion h -ill !' Il l' ptltlI ;lL·c lIr
+

= K

K

(b)

m

Km (Kr - Km )/. . Km + (I /-.' ) ( Kr - Km )/2

( 10.24)

Predictions obtained from these equations are shown in Fig. 10 . 18. The rule of mixtures expression gives a reliable prediction for the axial conductivity of long-fibre composites, but it can be seen that the slab model exp ression for transverse conductivity (Eqn (10.22)), which is completely unreliable for insulating fibres , also gives a large error when the fibres have high conductivity. For highly conducting fibres, the aspect ratio ha s little effect on the transverse conductivity and Eqn (10.24) provides a good approximation for most cases. The axial conductivity , on the other hand , is more sensitive to aspect ratio and Eqn (10.20) gives an ove restimate for short-fibre and particulate composites. An example is given in Fig. 10.19 of how such predictions can be used in exploring performance characteristics of different composite systems. This shows a map (Ashby 1993) of thermal expansivity against thermal co nductivity, for aluminium reinforced with silicon carbide or boron nitridc. The ratio K /rv. can be taken as a merit index for minimisation of thermal distortion during heating or coo ling. On this plot, the merit index is hi gh in th e bottom right and low in the top left. The data shown thererore intiic;llL' th ;ll the resistance of' aluminium to thermal distortion is illlpro vn l h y Ill" III L'tlrpor;ltion or silicon carhide, hut is impaircd if' horon 11I11I(k ". Id ,kd

264

0/ composites

Th ermal behaviour 3

/ 2.5

10.3 Thermal conduction

, ,... , ..

100~

____~~__L-~-U~____-L__~-L-L~LU~

I

/

..........

..

- - - - Axi al, s = = ._-- .. . _- Transverse, s = = -Axial, s= 3 - - - - - Transverse, s = 3 Spheres .... 0- .. Transverse, s = =. (Slab model) •

-

•

0

m,-;t/ (K/a)

(K/a) = 1 MW

, .' , ,.

.

0

0

265

'f

mol

~/

,

1000 series "" aluminium /

(Klow:aup)

0

=10 MW

,,'

aIIO~,/

X '7

~ ~

;iS '00

10

c

(\l

0..

." .'.'. '.'. '.'. '.

x

~,. ;:::,. ;:::.,.~,. ,. ,.

Q)

- - - - Axial, s = = Transverse, s = = - Axial , s = 3 - - - - - Transverse, S = 3 Spheres ........ 0

o

o

0

0

•

0

Cii

E

silicon carbide

Q)

..c I-

'----~~---I

o

0.2

0.4

0.6

"","",

0.8

(K/ a) = 100 MW m ol

Fibre volume frac tion , f Fi g. 10.1 8 Predi cted pl o ts o f co mposite co nducti vit y (rela ti ve to th a t o f th e ma tri x) as/ a fun cti o n o f fibre co ntcnt, fo r in sula tin g (Kr = 0) a nd highl y co nductlllg ( /( 1' = 10KI11 ) reinforcement s ha Vin g va ri o us as pect ra ti os . These curves werc o bt a in ed using thc Es helby meth od (Eqn (10.23 )), except fo r th e transve rse co nducti vit y o f a lo ng-fibre co mpos itc with hi ghl y co nducting fibres, in whi eh case th e pred icti o n o f Eqn ( 10.22), obtained fr om the slab model, is also shown.

10.3.3 Interfacial thermal resistance

The a bove ca lcul a ti o ns are based o n th e ass umpti o n that th ere is perfect therm a l co ntact betwee n fibre a nd ma tri x. In practi ce, thi s may be impa ired by th e prese nce of a n interfacia l la ye r of so me sort, o r by vo id s o r crack s in th e vicinity of the interface. Furtherm o re, eve n ill the a bse nce of such ba rri ers to hea t now, there may be so me loss or hea t tra nsfer effi ciency across the interface if th e carri ers a rc different in th e two co nstituents, as with meta l/ce ra mi c sys tems. Such a th ermal resista nce is cha racteri sed by a n interfacia l heat t/'{lllS/er coefficient or th erm a l co nducta nce, h (W m- 1 K I ), defin ed as th e pro po rti o nalit y co nsta nt betwee n th e hea t nu x thro ugh th e bo un dary a nd th e tcmpc rature dro p across it

10

Fi g. 10. 19 Predi cted map (Ash by 1993) of thermal expan sivit y n again st th crmal co nducti vity K fo r co mposites made up o f either silicon carbidc o r bOr(lIl nitride in an a luminium matri x. The di ago nal dotted lines represe nt co nst:lllt va lues of a merit ind ex, give n by K in, ta ken as indica ti ve o f the res istancc tn therm a l di storti on. The shaded areas, defin ed by upper a nd lowcr bo und s 0 1" tli e: two pa rameters, indicate the possibl e co mbina ti ons o f K and n cx pected 1"01" 1\ 1/ SiC a nd AI/ BN co mposites, depending o n the vo lumc fracti o n, shape a nd o ri e: n ta t io n di stributi o n o f the rcinfo rcemcnt.

Modellin g of the thermal co nductivity of co mpos ites in th e preS(; ll ce or such an interfa cia l resistan ce ha s been addressed by seve ral auth ors (Benveni ste and M il o h 1986, Hasselm a n a nd Jo hn so n 1987, Fadak : 111(1 Taya 199 1, Hasse lman (' t al. 1993). Ha sse lm a n and .I o hn so n dni ved ; 111 a na lyti ca l express io n fo r th e tra nsverse co nducti vit y of a lo ng- finr(; CO I11 pos ite.

I (K I

",',

( 10.2. )

1000

100

Thermal conductivity, K(W m -1 K -1 )

.

K,

I A'1I1

""111

rh A' 1

11 ( I "

III

I) "'. r/~

)

A'f

A',

"" 111 A'I

rh A'I

"" 111

1'/'

Il I1

( 10. 2() )

266

Th ermal behaviour of composites

/11 .3

where r is the radius of the fibres. The scale of the structure is now relevant because it determines how frequently interfaces are encountered by the heat flow. This scale effect is represented by the dimensionless ratio Kr/rh. Equation (10.26) can be used in conjunction with experimental conductivity data to estimate the interfacial conductance of a composite. For example, Fig. 10.20 shows measured conductivities over a range of temperature for a Ti-based long-fibrc composite, plotted as a ratio to that of the ma trix. Also shown are predictions from Eqn (10.26) corresponding to several h values, using the appropriate fibre rad ius ("'-' 50pm). These d ata are consistent with an h value of around 10 6 W m - 2 K - I. Although this figure represents relatively poor therma l contact for a fibre / matrix interface, it can be seen that in thi s system thc overall conductivity would not have been much greater had the inte rface been perfect (h = 00) . Thi s illustrates the size effect ; the largc diameter of the SiC monofilaments means that the magnitude of

o Experimentat - - Predicted, h =

:..::E

00

-- --- -- Predicted, h = 107 W m·2 K"

~u 1.2

- - - - - Predicted, h . . . . - Predicted, h

1.0

o

= 106 W m·2 K" =0

o

0' [ ____ _ ___________________ _____ _ __ _ _____ ___

_

: !---.-.-.--.----.-.-.-.-...---.-~-:J 100

200

300

400

500

600

700

Temperature CC)

or

Fig. 10.20 Plots (Gordon 1' / (ff. 1994) thermal cond ucli vil y rali n A: l A: again st temperature, 1'0 1' a Ti 6AI 4V all oy reinfo rced w ith ].'i':;', o f Si( ' 1l~'O II( ~" filament , wilh heat n ow tran sve rse 10 th e fihre :Ixis. Bolh cx pcrilll clll :iI d:II :1 :lIld pred icli olls from ":qll (Itl. ~(». h:lsed O il scver: iI v: illl cs for" (i ll W ill ' K 'l. : 11 \' show I!.

f'/ /(' fl/I!I/ Olllrlll('II!1II

interfacial resistance is not or III uch sig ni li Glncc (unless it heCO l1l eS very large indeed) . It may also bc noted that , although data in T:lhle 10 . 1 suggest that there is scope for rai sing the conductivity or Ti 6;\1 4V by addition of SiC, the monofilament in this composite ha s a re lative ly low conductivity ("'-' IOWm - 1 K - 1 ) , primarily as a consequence of its very fine grain size. Two additional points can be illustrated by some data for p
( 10.27)

where Kp is the conductivity of the particles. In Fig. 10.21(a), experimental conductivity data for a Ti/lO %SiC p composite have been plotted as ratios to that of the matrix , over the range of temperature . Also shown are prediction s from Eqn (10.27) corresponding to several h values, usin g the experimental conductivity values for matrix and reinforcement at the appropriate tempera tures , and a particle radius of 10 ~tln . These data are again consistent with an h value of a round 106 Wm - 2 K - I. In this case, howeve r, a consequence of the smaller size of the reinforcement , compared with the fibre composite data shown in Fig. 10.20, is that the ove rall conductivity is considerably lower than would be the case with perfect interfacial contact. Indeed , the conductivity is quite close to that expected with an ins ulating inte rface (h = 0). From th e data in Fig. 10.21(b), it is evident that the interfacial characte ri stics are different for the Ti/TiB2 particulate MMC. These plots show that the interfacial conductance is higher in this sys tem , by a factor o f about 10, givin g co nductivities close to those expected with perfect intcrl;tci: J1 co nt:l ct. The dilTerence between the two systems has been :I ttrihut ed « imd o ll cl If/. 11)94) to the thicker interracial reaction laye r in th e Ti /Si( ' l'(l1llj)() Slil:s ; 11 11 1 the het th ;lt th e reac tion involves a large Vo lt1lll l' rn lll l'tl ll ll , t(, IIl IIII I' It ) 1': III S,' illl ni': lc i;Ji cr; lc ks.

268

Thermal beha viour 0/ composileS

Re/erences and/ilrl"!:'r reading

References and further reading

1.5

(a)

:.:E ::;u

.s:" ~

1-0

C :~ tl

o Experimental - - Predicted , h =00

1.4

------- Predicted, h= 107 W m-2 K-'

1.3

- - - - - Predicted, h = 106 W m-2 K-' - - . . . Predicted, h =0

1.2 1.

:l "0

c::

o

<..l

E

o

o

~

0.9

tt)

0

0

0

0

0

000 00 0 0 00

0

0

- - - - - - 7:)- - - 0 - - 0 - - - - - - - - - - - - - - - - - ______ _

o

..c::

f-

o

...

o

--

'"

- - _ . - - - - - - -. - - - - - - - - - - - - - - - - - -

0.8

Temperature CC)

1.5

(b)

o Experimenlal - - Predicled, 00 -- --- -- Predicted, 107 W m-2 K-'

,,= ,,=

1.4

- - - - - Predicted , h = 106 W m-2 K-' . . - . . Predicted , h = 0

---------------------------- - - - - - - - - - - - - - - - -

0.8

- . - - - - - -. - -. - - -. - - -. - -- . - - - - _. - - - - _ . - - -

0.7

100

269

200

300

400

500

600

"

- .700

Temperature CC)

.Fi ~: 10. 2 1 PI.~t s (Go rd o n el 1994). of th e rmal conduClivit y ratio ''
ar

A shby, M. F. ( 1993) C riteri a for se lecting the compone nt s of compos ites, Acla M!:'lal/. Mal er., 41 13 13- 35 Benveni ste , Y. a nd Miloh , T. (1986) The e ffeeti vc th crma l co nduct ivity of co mposites wit h imperfec t thermal con tac t at co nst ituent interfaces, 1nl. J. Eng. Sei., 24 1537- 52 Bowles, D. E. and Tompkins, S. S. ( 1989) Pred icti o ns of coeffi cient s of therm a l ex pan sion for unidirec ti o na l composites. J. Comp . Ma t. , 23 370- 88 C1yne, T. W. and Withers, P. 1. ( 1993) An Introduction to Me lal Matrix COI1lPOSil!:'S. Cambrid ge Unive rsit y Press: Ca mbridge C lyne, T. W. , Feillard, P. and Kalton , A. F. ( 1995) Inte rfac ia l mec ha ni cs and m ac roscopic failurc in titanium-based composites, in Lite Prediction M!:'lhodolog.1'jin· Tilanillm Malri.\" Composiles. W. S. 10 hn son (ed.) AST M STP Endo, T., C hang, M. , Matsud a , N. a nd Matsuura, K . (1991) C reep behav io r o f SiC/ AI co mp osite at eleva ted temperature, in Metal Malrix Composiles Processillg, M icroslructure and Propl:'/'ti!:'s. N. Han sen el al. (cd s.) Ri s0 Nat. Lab.: Roskildc, Denmark pp. 323 8 Fada le, T. D. a nd Taya , M. ( 199 1) Effcc ti ve th crmal co nduct ivity of co mp os ites with fibre matrix d e bonding, .J. Mall:'/'. Sci. Lells., 10 682 4 Fros t, H . J . a nd Ashby, M . F. (1982) De/imllalioll Mechanism Maps Ih!:' Plaslicill' alld Cre!:'p ot Metals alld Ceramics. Perga m o n: O xford F urn ess, 1. A. G. a nd C lyne, T. W. (1991) Thermal cyclin g creep of sho rt fibre MM Cs measuremen t and m odelling of the strai n cycle, in Melal Mmri.\" COl1lposil!:'S Processing, Micro Sll'llClllre alld Properties. N . Han se n el al. (eds .) Ri s0 Nal. Lab.: Rosk ilde, Denmark pp. 349 54 Go rd o n, F. H. , Turner, S. P., Ta ylo r, R . a nd C lyne, T. W. (1994) The effect o f the inte rface o n the thermal co ndu cti vit y of Ti-ba sed composites, COll1posit es, 25 583 92 Hale, D. K. (1976) The physical properti es o f co mp osite m a teria ls, J . M al!:'r. Sci., 11 2 105-4 1 Hasse lma n, D. P. H. a nd John so n, L. F. ( 1987) Effective thermal co nductivit y of co mposites with interfacial thermal barrier resistance, .J. CO/llp . Mal. , 21 508 15 Hasse lm a n, D . P. H. , Donald so n , K . Y. a nd Thoma s, J . R. (1993) Effect ive th ermal co ndu cti vit y o f unia xial co mposite with cy lindrica ll y o rth ot ropic ca rb o n fibre s and interfacia l ba rri er res ista nce, .J. COl1lp. Mm. , 27 637 44 Halla, H. and Taya , M. ( 1986) Therm a l co ndu cti vity of coated fill er composites, J. App/. Pin's., 59 185 1- 60 Jansson , S. , Dalbello , D. J. and Leckie, F. A . (1994) Transverse a nd cycli c th erm a l loa din g o f the fib er reinfo rced m eta l m at ri x composite SCS6/Ti 15 3, Acla Mal er. Mewl/ ., 42 4015 24 .larry, P. , Louc, W. a nd Bouvai st, J. ( 1987) Rhe o logica l behaviour of Si C/ AI composit es. in Proi'. ICCM6. F. L. Matthews el al. (eds.) Elsevier: London pp. :U50 2. ]6 1 k nnin gs, T . M . ( I 'NO) Fh(,l'IIlIIl Faliglle o{ Car/JolI Film' l3i.l'lIl11leilllidl' ('1I 11I/III.I'il( '.I' I'll I ) IlI es is, li ni vers it y o r Ca mhrid gc K(lw;il s ki. I 1\1 ( I'i S / ) Nl'\I' IIl g h pni"ormance domeslica ll y produced carhon l'ihl' " . S / ,I/I '/I , I Zt)~\ (,I

270

Thermal behaviour of composites

Kural , M. H. and Min , B. K. (1984) Thc effects of matrix plasticity on the thermal deformatIOn of contllluous fibre graphite/metal composites, J. Comp. Mat. , 18519- 35 Lame, G. ( 1852) Le,cons sur la Theorie de nilasticite. Gautiers- Yillars: Paris Lee, 1. K. , Earmme, Y. Y. , Aaronson , H. I. and Russel , K. C. (1980) Plastic relaxation of the transformation strain energy of a mi sfit tin g spherical precIpitate: Ideal plastic behaviour, Metall. TraIlS. , IIA 1 837~7 Masutti , D. , Lentz, 1. P. and Delannay, F. (1990) Measurement of internal st resses and of the temperature dependence of the matrix yield stress in meta l matriX composites from therma l expansion curves, J. Mat. Sci. L ells. , 9 340- 2 Mclean , M. (1983) Directionally Solidified Materials/or High-Temperature S ervice. Metals Soc.: London Nardone, Y. C. and Tien, 1. K. (1986) On the cree p rate dependence of particle strength ened alloys , Scripta M et., 20 797- 802 Newaz, G . M . and Majumdar, B. S. (1994) A compari so n of mechanical response of MMC at room and elevated temperature, Comp . Sri. & Tech.

~ 8~OO

Nieh , T. G. ( 1984) Creep rupture of a silicon carbide reinforced alum inium composite, M etall. TrailS., ISA 139- 46 Nielson , L. E. (1974) The th erma l and electrica l cond uctivit y of two-pha se systems, Illd. Ellg. Chem. FUlldam. , 13 17- 20 Schapery, R . A. (1968) The rmal expansion coefficien ts of composi te ma terial s based on energy principles, J. Comp. Mat. , 2 380~04 Withers, P. 1., len sen, D . 1., Lilholt , H. and Stobbs, W. M . (1987) The eva luati on of Inte rnal stresses in a short fibre MMC , in Proc. ICC M6. F. L. Matthews t'l al. (eds.) Elsevier: London pp. 2.255- 2.264 Wolff, B.. K. , Min, B. K. and Kural , M. H. ( 1985) The rmal cycl ing of a ul1IdlrectlOnal graphite- magnesium composite, J. Mat. Sci., 20 1141 - 9 Wu, M. Y. and Sherby, O. D. (1984) Superpl as ticity in a silicon ca rbid e reIllforced a luminium allo y, Scripta Met., 18 773 6

'

11 Fabrication

All important aspect o/composite materials cOllcerns the techllolog), by Il'hich the)' are produced. Depelldillg and th e required architecture

011

the nature o/matrix alldfibre.

0/ the .fibre

distributioll . production at

reasonable cost alld lI'ith suitable microstructural qualit), can be a challellg illg problem. nent and productioll

111

most cases, mallu/acture

0/ the

0/

the .filial compo-

composite material are carried out at the

sam e time. This gives scope .for optimal./i'bre placement. but also demallds that the mechallical requiremellts

0/ the

application be II'ell

ullderstood and that the processing route be tailored acco rdingly. Fabrication proceduresfor most polymer composites are commerciall), and technically mature, II'hile II/ost

0/ those beillg applied to metal alld

ceralJlic composites are still under development. In manv such cases , commercial exploitation It'ill be depelldent

011

improved fabri catioll

efficiency.

Il.l Polymer composites Thc rc are many commercial processes for the manufacture of PMC co mponents. These may be su b-divided in a variety of ways, but hroa dly s peaking there are three m a in approaches to the manufacture or fibrc-reinforeed thermosetting resins and two distinct production met h o d s 1'0 1' thermopla s tic composites. These are briefly covered below unde r se parat c head in gs . A simp le overv iew of the start in g material s ;Ind appro ;lc h es ;Id op tcd to th e ir incorporation into components is .ivcn in I-'i ' . 11 . 1. 111 mos t cases, tile main microstruc tural objectives ;Irl' to e ll SIII \' liI :11 liI \' 1II11 \,S :11'\: we ll wetted , unirorml y di s trihut ed and

'11

272

//./ Po/rll/a cOII/IIIIsi/ l's

Fabrica I ion

A process more suited to automation , alth o ugh limitl.:d to I.:l.: rtain COIl1 ponent shapes, isfilamellt willdillg. Fibre tows , i.e . bundles of fihre s, arl.: drawn through a bath of resin , before being wound o nto a mandrel or former of the required shape. The equipment comprises (a) a creel stand , from which the fibre tows are fed under the required tension from a set of reels, (b) a bath of resin , through which the fibre tows pass via a set of guides, (c) a delivery eye, through which the fibres emerge, the position of which is controlled by a mechanical system and (d) a rotating mandrel , onto which the fibre tows are drawn. The key parameters are the fibre tension , the resin take-up efficiency and the winding geometry (Middleton 1994). There are a number of different designs for the movement of the delivery eye. One is shown in Fig. 11.2, incorporating a Fig. 11. 1 Schematic overview of the approaches employed in fabrication of po lymcr matrix compositcs.

correctly aligned. Pract ical considerations relating to capital cost , speed of production and component size and shape capability are often of paramount importance.

gantry for the traversing motion. In most cases, the eye motion and mandrel rotation systems are computer-controlled. Component shapes can be fairly complex, although they often exhibit a high degree of symmetry. There are also some limitations on the paths that the fibres take over the surface of the component. On any curved surface, there will be a tendency for the fibres to follow a geodesic path - i.e. the shortest one. This can cause problems with some shapes , since it may be difficult to ensure that fibres

11.1.1 Liquid resill impregllatioll routes

In the basic processing route, low-viscosity resin is impregnated into arrays of dry fibre. This can be achieved in several ways. The simplest IS a wet lay-up procedure in which the fibres , usually in the form of a mat on a polished/ormer or mould , are impregnated with resin by rolling or spraying. The resin and curing agent are mixed immediately prior to application. Curing usually takes place at ambient temperature . Recent developments include co-spraying of liquids so that mixing occurs as the liquids arc introduced to the fibres (Gotch 1994). The main advantages of the process lies in its versatility. Virtually any shape can be produced and capital costs are limited to that of the mould. The technique has trad itionally been used in a variety of small-scale operations, such as car body repair, commonly with glass fibres in the form of woven rovin u or chopped strand mat (see Figs 3.6 and 3.7, respectively). Larger-:ale operation is, however, increasingly common: for example , the techniqul.: is widely used in the boat-building industry for production or relcttivd y large craft such as minesweepers or up to ahout 50 m in len g th (Smith 1990). The main disadvanta gl.:s lil.: in thc dilTi c ult y of cns llrin! COlllpit:tc Impreg nation and thl.: 1;lhollr-intcn sivc naturc of ll1uch nl" ti ll' \\'() r~ .

h' . 11 .2

Pllol 01 ' 1;1pll " I'

;1 , '; 1111 ry-Iypc

1"i1;lmcn I wi ndi ng m
(l)l)-I ) .

11.1 Poly mer composites

I-i,hric{f I iOIl

cove r S()111e parI s of I he s urfa ce , o r li e in ccrtain orientations. It is, however, possible to (;IlSlIre Ih ;11 fibr(;s fo llow certain non-geodesic paths, provided thc delivery ey(; is Illov(;d appro priately and there is sufficien t friction between the tow and th(; lIll(krl yin g s urface (Wells and McAnulty 1987). F il ament winding is often used to produce high-performance com ponents and is obvious ly well su ited to simp le shapes such as tubes. Puitrus;oll is a process which is simil ar in some respects to filament winding. This a lso involves fibre tows being passed through a resin bath. In this case, the impregnated tows are then fed into a heated , tubular die, in which they become consolidated and the resin is cured. The die may have a relatively comp lex sectiona l shape. The composite is pulled from the die by a frictional extraction system . The process generates stock ma teria I, ra ther tha n fi nished com ponen ts, a nd the prod uct is si mi la r in that respect to plastic and metallic material produced by conventional extrusion. Another impregnation route involves injection of resin into a mould in which the fibres are placed in position. The resin is fed in under gravity or externa l pressure. There are several different types of machine. A commonly used process is that termed res;1l trallsfer mouldillg (RTM) , in which the fibres are enclosed in a die and pre-eatalysed resin , having a low viscosity , is injected into the mould at relatively low pressure. Cure occ urs within the mould, often assisted by heating. The mould is usually of metal , which gives good heat transfer and lasts for many moulding operations. A typical set-up is illustrated in Fig. 11.3, which shows the mould for production of a tennis racquet, with a carbon fibre preform a lready in place. To encourage good infiltration, the mould is sometimes evacua ted prior to injection of the resin. Relatively large mouldings , including many body parts for the automobi le industry, are made in this way.

11.1.2 Pressurised cOllsolidatioll of resill pre-pregs

This approach involves production of a pre-preg, which is a tape or shee t of fibres impregnated with resin. Pre-preg is manufactured by lay in g th e fibres and resin between sheets of silieonised paper or plastic film , which are pressed or rolled to ensure conso lidation and wetting out of the fibres, and then partially cured to produce a flexible aggregate. The process allows excellent alignment of the fibres in unidirectional laye rs. A com ponent is formed by stackin g up the layers of prc-preg in pr(;-d (; l(; rmined directions, conso lid ating them by pr(;SSllre, and finall y c urill . hy hL'ali ll '

Fig. 11.3

275

Photograph of equipment for resin transfer moulding production of a carbon fibre reinforced tenni s racquet (Blake 1989).

under pressure. The simplest arrangement involves a directional hydraulic press, which is used to apply pressure to a pair of matched mould ha lves. An alternative approach , l'aClIum mouldillg, involves enclosing the pre-preg (draped over a former) with a flexible, impermeable membrane. The enclosure is then evacuated , so that atmospheric pressure acts on the membrane and compresses the pre-preg onto the former. While the above processes are relatively quick and easy to carry out, they suffer from limitations in terms of the quality of the composite material produced and the size and shape of components that ca n be manufactured. For demanding applications, autoclave mouldillg is often pre ferred. This is sim ilar to vacuum moulding, but the enclosed composite assembly is placed into a large chamber which can be pressurised , typically up to ~ 10-20 atmospheres. Heating is applied during pressuris;ltion to cure the resin. In many cases, the temperature is first raised to an intermediate level , in order to reduce the viscosity of the resin and ensure th;lt all void s ;Ire removed (Jones 1994). Further heating then (; ns ur(;s Ihal Ih l' l"Ilrc is c0111]1lel(; . A schematic represcntat ion of the Ch;11l 'cs ill (cillp\,· r;llul \.', press ure ;llld r(;sin vis(;osit y during autoclave 1Il()l ild ill g is "hoW II I11 1;1)' II .cI. Th e proc(;ss Gill he appli ed to large

277

11.1 Polymer composites

Fahrim t ion Laminate

Gas

Gas pressure

viscosity Time Fig. 11.4

Schematic of an autoclave cure cycle (Jones 1994).

components. This is illustrated by Fig. 11.5(a), which shows a complete aeroplane wing being moved into a large autoclave . The p lane concerned is shown in Fig. 11.5(b).

11.1.3 COllsolidatioll of resill mouldillg compoullds The third approach for thermosetting resins is a variation of the prepreg approach. The intermediate products are sheet moulding compound (SMC) and dough mouldillg compoulld (OMC) , which are usua ll y based on polymer resins. To make SMC, resin containing thickening agents and particulate fillers such as calcium carbonate, is mixed with chopped fibres to form a slurry. This is then fed between two thermoplastic films and fed over a series of rolls, so that impregnation takes place and a consolidated sheet is produced. The fibres lie mainly in the plane of the sheet and the fibre volume fraction is usually in the range 15 to 40%. Sheets are typically between 3 and 10mm thick. OMCs are made by mixing together similar constituents to those in SMCs, but the mixing is carried out differently . Usually , a blender generating high shear rates is used for OMCs. This tends to break the fibres up into shorter lengths and also results in a more random orientation distribution. Both SMC and OMC are hot press moulded in closed moulds to produce the final product and, in the process , they undergo the final cure. SMCs are commonly processed into components by hot press or compression mouldin g. Thi s is don e by removin g the thermopla s ti c film s. cuttin g the s heets to s uitahle shapes and pla cin g them ill :1 Ill': 11 ("( I 11l (l lild .

(a)

(b) I-"i !.'.. 11 .5 (a) Photog raph of a large autoclave being used for production of the w i71 !.'. of a Beeeh St a rship aeroplane. (b) Photograph of a Beeeh Starship in night ~ (Blake 1989).

Similar ope r
27X

resin tran sfer moulding, but instead of using a pre-eatalysed resin which cures slowly, as with RTM , two components which react together quickly are mixed, together with some short fibres , and immediately injected into a mould. In all such processes, the presence of fibres raises the viscosity of the mixture dramatica ll y and strongly inhibits its flow, even when they are relatively short, so that the moulding operation may become im practica l unless the fibre content is low.

279

J J. J Polymer composites

Fahrication (a)

Solidified layer

,

... \

\ I

\

I

I

Gate

Vdocity profile in molten co re

Advancing flow front

J 1.1.4 b!iection moulding of thermoplastics Short-fibre reinforced thermoplastic materials are used to make com poncnts by il!iection moulding. This process is very widely applied to unreinforeed thermoplastics. Three polymers for which the process is very common are polypropylene, nylon and polycarbonate (see Table 2.5). The first two are semi-crysta lline with approximately 25- 50% crystallinity and the latter is amorphous. Polymer pellets are fed into thc hcatcd barrel of a chamber containing an Arehimedes screw for transporting the charge. The shear motion helps to homogenise and melt the polymer, which is then periodically injected into a mould. The same process is carried out using pellets containing short fibres, typically 1- 5mm long, intimately mixed and dispersed in the matrix. However, the comments in §11.1.3 about flow of a mixture becoming difficult when fibres are present also applies to these operations, and the volume fraction of fibres is usually no greater than about 10- 20%. Of interest for injection moulded components is the fibre orientation distribution. For example, it may be possible to arrange for the fibres to be oriented parallel to the direction of stress in parts of the component which will be heavily loaded in service. Fibre orientation is controlled by the nature of the flow field during filling of the mould. In general, fibres tend to become a ligned parallel to the direction in which the material is becoming elongated (provided the flow is not too turbulent). This is illustrated by the schematic diagram of the flow pattern during injection into a simple rectangular mould , and the correspo ndin g micro-radiographs, shown in Fig. 11.6. In this case, the fibres are well aligned in the outer layers of the moulding, but more randomly oriented towards the core. By predicting the !low behaviour under different injection COIlditions for specific componcnts, a dcgrcc of control OWl' Ihe fin:i1 fibrc oricntation pattern is oncll possihle.

Fig. 11.6 (a) Schematic diagram of the mould filling process during injection moulding, showing the deFormation of an initially square fluid elcment at successive positions of the advancing now Front (Folkes and Russell 1980). (b) Co nt act microradiograph of th e longit udinal section of a polypropylene/ 15% glass fibre injection moulding. (c) [overpage] Transverse section From the same moulding.

11. J.5 Hot press moulding of thermoplastics Long- or continuous-fib re reinforced thermopla stics are commonly used in IlIminatl'd structures. The fi rst sta ge in ma n u fact ure is the prod uction of a prc-preg by //Il'lt impregnation of the fihres, which may be in the form of CO il I i1111()II S :t1 il' 11,-(\ sit""I s, wovcn cloth, cte. The pre-prcg shec ts arc Ihell sl:1 '\..nllll Ill\' 1('111111('(\ ()I i,'I1I:llion s : 111(\ !tot pr('Hl'lI to form the final

2XO

Fabrication

11.2 Metal composites

281

difficulties and cost of production have been largely responsible for their usage being relatively limited. The diversity of approaches which have been attempted is rcflected in the complexity of the overview presented in Fig. 11.7. Several of the production routes represented in this figure are of limited commercial potential , since they are inherently expensive or cumbersome. Some of them may , nevertheless, find industrial use in due course, since there are certain applications for which a high manufacturing cost is justifiable. A basic distinction can be drawn, depending on whether or not liquid metal is involved. Broadly, the liquid metal routes are relatively cheap (apart from the spraying techniques, which have now been largely discarded), while the solid-state methods offer certain advantages at increased cost. In the classification given below, the first three routes involve contact between liquid metal and the ceramic, whereas in the following three there is no such contact.

11 .2.1 Squeeze i1!filtl'lltioll Fig. I 1.6

( COni )

product. Pre-pregs are available in many materials including polypropylene: polysulphone and polyether-ether-ketone (PEEK). Processing is carned out in a similar manner to that involved during pressurised consolidation of resin pre-pregs (§ 11.1.2). For thermoplastics, there is no requirement for the matrix to be cured and the temperature is usually set to the minimum necessary for the matrix to melt and flow sufficiently for consolidation to occur. However, even when fully molten, thermoplastic melts are much more viscous than uncured resins, since thermoplastics are already fully polymerised. Substantial heating and pressure are therefore necessary, and impregnation distances into fibre arrays arc always kept as short as possible.

The most common pressure-assisted solidification process for MMC production is squeeze i1!filtratioll. Liquid metal is injected into an assembly of short fibres, usually called a 'preform'. Common ly, the preform is designed with a specific shape to form an integral part of a finished product in the as-cast form (Feest 1988). Preforms are commonly fabricated by sedimentation of short fibres from liquid suspension, often using

11.2 Metal composites Production of MMCs is commercia lly less advanced than PMCs. Industrial exploitation of MMCs is still in its infancy and 1l1,IIure production technologies have yet to emerge. Nevertheless, Ih ere ha s heen considcrable research effort into fabrication aspects, parll y IH'('; III , C Ihe

h I' . I I . I

SI 11\' 11 1.111\

" "'1\

,,' woI' I he :1 pproachcs cm ploycd in ra hrica I ion oj' ""'1:11 IIlalrix composiles.

Fabrication

11 .2 Metal composites

short alumina fibres, such as 'SaffiITM' (see §2.1.5). The process can also be adapted for production of particulate MMCs (Klier et al. 1991). In order for the preform to retain its integrity and shape, a hinder (usually silica-based) is used. In most cases the fibres do not act as preferential crysta l nucleation sites during melt solidification (Mortensen and Cornie 1987). One consequence of this is that the last liquid to freeze, which is normally soluteenriched, tends to be located around the fibres . An example of this is shown in Fig. 11.8. Such prolonged fibre/ melt contact, under high hydrostatic pressure and with solute enrichment, tends to favour formation of a strong interfacial bond. Furthermore, oxide films cannot form because of the limited oxygen availability (Clyne and Mason 1987). This probably contributes to the high interfacial bond strengths commonly observed.

conventiona l processing equipment and is carried out on a continuous or semi-continuous basis. The process is in commercial use for particulate AI-based composites (Skibo et al. 1988) and the material produced is suitable for further operations such as pressure die-casting (Hoover 1991). Adaptation of conventional melt handling techniques allows the viscosity increases caused by the presence of up to about 25% of particulate to be accommodated. More problematical are sources of microstructura l inhomogeneity, including particle agglomeration and sedimentation in the melt and redistribution as a result of particle pushing by an advancing solidification front , which is still poorly understood and can be difficult to eliminate. A typical microstructure illustrating the effect of particle pushing during solidification is shown in Fig. 11.9. Stir casting usually involves prolonged liquid /ceramic contact, which can cause substantial interfacial reaction. This has been studied in detail (LJoyd 1989) for AI /SiC, in which the formation of AI 4 C 3 and Si can be extensive. This both degrades the final properties of the composite and raises the viscosity of the slurry, making subsequent casting difficult. The rate of reaction is reduced, and can become zero, if a Si-rich melt is used. The reaction kinetics and Si levels needed to eliminate it are such that casting of AI / SiC p involving prolonged melt holding operations is suited to conventional (high Si) casting alloys, but not to most wrought a lloys . This problem is one of the reasons for commercial interest in cast AI/ A1 2 0 3p , although this system has the penalty of slightly higher densities than AI/SiC.

282

11.2.2 Stir casting This technique involves stirring liquid metal with solid ceramic particles and then allowing th e mixture to so lidify . This can be done using fairly

283

11.2.3 Spray deposition

Fig. 11.8 Back-scattered SEM micrograph (Cl y nc ;Ind Withcrs I ()(iI) "r:ln ;\1 4.5 wt°!., Cuj SalTil lM composite, show in g coppcr-ri ch (light) rq~ i()ll s ;lI"llll lld the I·ihres.

Spray deposition was developed by Osprey Ltd (Neath, UK) as a method of building up bulk metallic material by directing an atomised stream of droplets onto a substrate. Adaptation to particulate MMC production by injection of ceramic powder into the spray has been extensively explored (Wi llis 1988), although with limited commercial success. One of the problems is inhomogeneous distributions of the ceramic particles. Ceramicrich layers approximately normal to the overall growth direction are ofte n seen. Porosity in the as-sprayed state is typically about 5- 15 % and seconda ry consolidation processing is needed. Thermal spraying involves injection of powder or wire into a hightemper;lture torch, from wh ich molten droplets emerge at high speed. Co mpared wi th III L'II :lhll11is;ltiol1 tec hniqu es, deposition rates (usually r~ 1 gs I) ; II L" ~ 1 (l\\' L"1 hilt p;lllick vcioc iti es (cv 2()() 7()() III si) arc higher.

Fa!Jric(f I iOIl

11.2 Meta! composites

285

Porosity levels in thermally sprayed deposits are at least a few % . Thermal spraying onto arrays of fibres to form MMCs has received some atte ntion (Clyne and Roberts 1995). An attractio n here, particularl y for titanium , is that no melt conta inment is necessary and there is o nl y very brief exposure to high temperatures. Provided the void content and distribution are such th at full consolidation could be effected with little further hea t treatm ent , this would a llow problems of excessive fibre/ matrix chemical reaction during processing to be avoided. Unfortunately, it has proved very difficult to spray onto fibre a rrays so as to produce MMCs with acceptably low void contents « 10%). Further problems arise from damage to fibre coa tin gs and uneven fibre distributions .

11.2.4 Powder b1elldillg alld consolidation Blending of metallic powder and cera mic fibres or particulate has the adva ntage of close co ntrol over the ceram ic content. Blending is followed by cold compaction , cann ing, evacuati on, degassing and a high-temperature co nso lid a ti o n stage such as hot isostatic pressing (HIP) or extrusion (Upadhyaya 1989). A feature of much powder route material is the presence of fin e oxide particles , usually present in AI-MMCs in the form of plate-like particles a few tens of nanometres thick , constituting abo ut 0.05- 0.5vol %, depending o n powder history and processing cond itions. This fine oxide tends to act as a dispersion strengthen ing agent and often has a strong influence on the matrix properties, particularly at high temperature. Microstructural changes occur during extrusion. Fibres become aligned parallel to the extrusion axis, but this is often accompanied by progressive fibre fragmentation. The degree of fibre fracture decreases with increasing temperature and decreasing loca l strain rate. Other microstruct ura l features of extruded MMCs include the formation in so me cases of ceramic-enriched 'bands' parallel to the extrusio n axis. The mechanism of band formation is sti ll unclear, but it appears to involve the concentration of shear strain in regions where ceramic particles or fibres accumulate.

Fig. 11.9 Microstructures (Lloyd 1991) or Al 7 wt% Si/20 vol. % SiC", (a) lI1Ves t~llent cast. (s low coolll1g) and (b) prcssure dic cast (rapid coo ling). At the

slowel coolIng late, S I ~ parti cles have becn pu shed into the intcrcl endritic regioll s by the glOwll1g dendnt es, GI U S ln~ severc clustcrill"C' . ~

11.2.5

D~ffilsion

honding offoils

Titan ium reinrorced with lon g ribres is commercially produced by placing ~I IT~l yS or rihn:s hetween thin metalli c foil s, often involving a filament windill g npn;iliplI . rllllnwnl hy hot pressing (Smith and Froes 1984). T he plPvn llll l' I' ;lItl:I('II\'(' I'or tit ;lniulll , sin ce it dissolves it s ow n oxide

286

Fabrication

11 .3 Ceram ic composites

287

at temperatures above about 700 DC and the rapid interfacia l chem ica l reaction (Martineau et al. 1984) caused by contact between liquid Ti and ceram ics is avoided. One of the m ain problems lies in avoiding excessive chem ica l reaction at this interface, which tends to lead to embrittlement. There is considerable interest in fibre coa tin gs designed to reduce these problems of interfacia l attack.

11.2.6 Physical vapour depositioll (PVD) Several PVD processes have been used in fabrication of MMCs (Eve rett 199 1). An evapo ratio n process used for fabrication of long-fibre reinforced titanium (Ward -C lose and Pa rtrid ge 1990) in vo lves co n tin uou s passage of fibre through an evacua ted cha mber, where condensa ti o n takes place so as to produce a relatively thick coating. The vapour is produced by directing a hi gh-power ("-' 10 kW) electro n beam so as to melt the end of a so lid bar feedstock. A wide range of a ll oys can be used; differences in evapo ratio n rate between different so lutes become compen sa ted by changes in compos iti on of the molten pool fo rmed on the end of the bar, until a steady state is reached in which the a ll oy co nte nt of the deposit is the same as that of the feedstock. There is littl e or no mecha nical disturbance of the interfacial region during processing. Composite fabricat io n is completed by assemb ling the coated fibres into a bundle and conso lid atin g by hot pressing o r HIPing. A very uniform distribution of fibres is produced, with fibre conte nts of up to about 80%. A typical coated fibre a nd consolidated titan ium composite are shown in Fig. 1l.10. Productio n ra tes are slow a nd the process is expensive, but it m ay neve rtheless have some co mmercial potential.

11.3 Ceramic composites

Fabrication of C M Cs presents severe techni ca l problem s. Thi s is large ly due to the brittle nature of ceramic ma trices, which mak es de formati o n processing difficult. Furthermore, the matrix is often unable to accommodate the volume changes associated with consolidation without c ra cks being formed. Thi s is particularly troublesome when fibres arc bcing incorporated , since these tend to resist the co ntractio n o f th e matri x as voids are eliminated. The need fo r hi gh temperatures (> 1000 "e) durin g most procedures a lso co ntributes to thc difficulties. Thcre is a limited n um bel' of processes which arc feasi blc for prod Ll et ion of ( ' M ( 's, and very few of th ese a rc in COJl1ll1cre i ~ JI Ll se. Th e oVLrv inv PI\'s,' lllLd in

Fig. 11 . 10 (a) A SiC mo nofi lament with a 351lm vapour-deposited laye r o r TiSA l 5V and (b) A Ti SA l 5V/80 vol. % Si C co mposite produced by HIPlI1g ora bundlc o r Ill o nori lamcnts with 8 1lm thi ck coatings (Ward-Close a nd Partridge 1990) .

Fi g. 11 . 11 rellcct s the limit ed cho ice ava il a bl e, a lth ough production de tail s ma y vary wide ly for differe nt materials and a pplica tions. Mo st proeetilll' 's ill vo lve startin ' materia l in th e form of ceram ic powders and thL' iirsl SL' ,tillll h 'Inw t'\lVl'I'S 'L'n c ral aspects o f how these a re handled.

2XX

F(fiJrica t ion

11.3 Ceramic composites

vapour infiltration Fig. 1l.11

Schematic overv iew of the approachcs employcd In fabrication of ceramic matrix composites.

Sections are then devoted to reactive processing methods and to layered composites, neither of which necessarily involve the presence of fibres. Finally, the fabrication of carbon/carbon composites is covered in a separate section, since this case diFFers from most other CMCs and the process is of commercial significance.

11.3.1 Powder-based routes Production of ceramic artefacts by powder-based routes is very well established in industrial practice. Green bodies are produced by cold compaction of fine powders, often with a binder of some type, and these are then sintered or hot pressed so that voids are eliminated by diffusive processes. In some cases a liquid is present during the consolidation, in which case void elimination is assisted by cap illary action and consolidation is much Faster. The problems arise when an attempt is made to introduce fibres into such operations . These difficulties are covered in detail by Phillips (1983) and Chawla (1993) . Fibres can be mixed with powder particles, For examp le by blending or by dragging a fibre tow through a suitab le slurry. However, subsequent conso lid a ti on is then strongly inhibited as the fibres resist the volume contraction or the matrix as it consolidates. Severe matrix cracking, of th e type shown in Fig. 11.12, usually results. In some cases, these con tra c ti oll strcsses can he at leas t partly olTsd by the imposition or a 1;11")',\ ' lJ \' dl(l ~ t ; ltic

1

289

CIn.

Fig. 11 .12 Photograph (Clegg 1992) of a composite made by sintering a powder compact containing dense Zr02- 3% Y 20 3 fibres in a matrix of the same material. Large c racks ha ve formed transverse to th e fibre direction, a long which tensile stresses developed in the matrix as consolidation occurred.

compressive stress, as in the hot isostatic pressing (H IP) process. However, this adds substantially to the processing costs and may not elimin ate cracking entirely. A possible way to resolve this problem is to employ a matrix which is partly , or completely, liquid at the consolidation temperature. This constrains the choice of matrix to those which are likely to show re la ti vely poor properties at elevated temperatures. Nevertheless , there has been interest in making CMCs in this way, using glass or glasscera m ic ma trices. Borosi Iica te glasses and cordieri le (glass-ceramic) have attracted particular attention. However, even when the consolidation can be achi eved without severe matrix crack in g, problems ofte n arise from differential thermal contraction. The misfit strain (see ~i 10.1.1) is often large for systems in which the matrix is fluid at high temperatures , since in such cases it is common for it to exhibit an appreciably highcr thermal expansivity than the fibres. An example of the co nsequences or such diFFerential thermal contraction is shown in Fig. 11 . l l . III .'cllcral , the diFFiculties and const raints on the manur;lctllrC o r rihll' ll' illi'orl'cd ccramics have strongly inhihited commercia l dcvcl()PlIlI'llt

2l)O

Fabricat ion

J J.3 Ceramic composites

291

reinforced MoSi 2 composite made in this way has been shown (Suzuki et al. 1993) to exhibit good creep resistance.

11.3.3 Layered ceramic composites

F ig. 11.13 Optical micrograph (Philiips 1983) of a composite made up of short carbon fibre s in a matrix of magnesia . Matrix cracking has occurred as a result of differential thermal contraction during cooling a fter fabrication.

//.3.2 Reactive processillg Several simil a r processes have been developed in which constituents are brought together under conditions such that a chemical reaction occurs while the mixture consolidates. In several such processes, liquid metal is introduced and progressively oxidises. For example, the directional oxi dation of aluminium is exploited in several processes patented under the 'X O' trade name. An attraction of such procedures is that, by makin g a suitable powder compact into which liquid metal is infiltrated , near-netshape forming is possible. Levels of internal stresses and porosity can be kept low by control over reaction kinetics, thermal gradients and liquid infiltra tion rates . In many cases, there is residual unreacted metal , but this can often be tolerated and may help to raise the toughness. Various composite systems have been developed , including several ba scd on intermetallic matrices (Stoloff and Alman 1990). The oxidation reaction s exploited by the Lanxide corporation have been covered by Newk irk 1' / al. (1987) and a n overview of th e materials produccd by thi s type or processing ha s been presen ted by U rq lIha rt (1991). SOllle good comb inati ons o r properties can be ac hi eved. For L·X; llll pk-. ;1 Si( '-

An attractively simple method of making relatively tough ceramic composites involves stacking thin sheets or tapes in the green state and consolidating these by a sintering operation. The procedures involved have been developed for SiC composites by Clegg and co-workers (Clegg et al. 1990, Phillips et al. 1994). Fine ceramic powder is blended with viscous polymer solutions and then rolled to produce tapes, typically about 200 ~1I1l thick. These tapes are stacked and sintered, after coating with thin (cv 5 J..lm) layers designed to provide weak interfaces which cause crack deflection (see §9. 1.2) and hence raise the toughness. Graphitic layers have proved effective, although they suffer from the disadvantage of very poor oxidation resistance at high temperature. The volume contraction on sintering, and subsequent cooling, ta kes place uniformly and does not res ult in significant internal stresses. Although the graphitic layers have a different expansivity from the SiC, they have a low modulu s and are apparently able to accommodate the shear associated with contraction without becoming damaged. The material is anisotropic, but this is acceptable for many applications (e.g. see § 12.8). The microstructure, shown in Fig. 11.14(a), is very similar to that of certain mollusc shells - see F ig. 11.14(b). The production process is relatively quick and cheap, since it does not involve either handling of fibres or application of pressure.

11.3.4 Carboll/carboll composites Carbon is an excellent high-temperature material , provided it is not exposed to oxidising environments. The excell ent mechanical stability of carbon at high temperature , particularly when suitable internal interfaces are present, has led to its use in several important applications (F itzer 1987), notably aircraft brakes (§ 12.9). Another consequence of th is sta bility, however, is that it cannot be sintered. There are two basic approaches to production of carbon /carbon composites. Both in volve the infiltrat io n o f a carbon-bearing fluid into the interstices between an array of carbon fIbre s. In both cases, the main concern is with achieving co mplete in III t ra t ion ill a rea so na bly sho rt time . F ull technical details ;m : ,ivcn hy MeAlli stcr ;Ind La chman ( 19lSJ). The two routes arc shown se ll 'nl ;lti e; tll y ill h g . 11 . 1 . Dllrin g liquid impreg nati o n. a pitch or resin

F(/hric({liOIl

References and further reading

293

is injected and then heated so that it decomposes to leave a carbon deposit. Chemical vapour impregnation (CVI), which can also be app lied to various other composite systems (Chawla 1993), involves injection of a suitable hydrocarbon gas, such as methane, together with hydrogen and nitrogen, which decomposes at the infiltration temperature to deposit carbon on the fibres. For both liquid and gaseous impregnation, severa l cycles of heating and cooling are necessary to complete the operation. Furthermore, it is common to set up a thermal gradient across the component, in order to encourage complete infiltration before the supply of fluid becomes choked off by closing of channels near to the source of fluid. For these reasons, processing is timeconsuming and costly.

F ig. 11.14 SEM micrographs (Clegg el al. 1995) of (a) a layered SiC/graphite composite, with the SIC layers about 200 ~lm thick and (b) a layered ca lcium ca rbonate! protell1 composite in the mollusc she ll pinctada margaritifera.

Liquid impregnation - thermosetting resin - pitch

Chemical vapour deposition - hydrocarbon gas (1000-1200 "C) ; repeat 1-3 times 1-5 times

composite I :ig. I I . I::;

Sc iJ e mal ic ove rvi ew (M c A lIi sle r 1994) o r mcthods oi" pr(ldllc l iOI1 o i" ca rh()l1 /c~ 1 rbu n co III posi tes.

References and further reading Becker, W. E. ( 1979) Reaclioll Injection Moulding. Van Nostrand Reinhold: New York Blake, J. (1989) Composite Materials. Hobsons Publishing: Cambridge Chawla , K. K. (1993) Ceramic Matrix Composiles. Chapman & Hall: London Clegg, W. J. (1992) The fabrication and failure of laminar ceramic composites, Acta Metall ., 40 3085- 93 C legg, W. J. , Kendall, K. , Alford , N. M. , Birchall , D. and Button , T. W. ( 1990) A simple way to make tough ceramics, Nature 347 455- 7 C lyne, T. W. and Mason , J. F. (1987) The squeeze infiltration process for fabrication of metal matrix composites, M etall. Trans. , 18A 1519- 30 Clyne, T. W. and Roberts, K. A. (1995) The influence of process parameters on consol id ation efficiency when forming titanium composites by spraying onto monofilaments, Acta Metall. Mat er., 43 254 1 50 Clyne, T. W. and Withers, P. J. (1993) All Introduction 10 Metal Matrix Composiles. Cambridge University Press: Cambridge Everett , R. K. ( 1991) Deposition technologies for MMC fabr ication , in Metal Matrix Composites: Processing and Intel/aces. R. K. Everett and R. J. Arsenau lt (eds.) Academic Press: Boston pp. 103 19 Feest , E. A. ( 1988) Exploitation of the metal matrix composites concept, M etals and Materials , 4 273- 8 Fitze r, E. (1987) The future of carbon-carbon composites, Carbon , 25 163- 90 Folkes, M. J. and Russell , D. A. M. (1980) Orientation effects during the flow of short fibre reinforced thermoplastics, Polymer, 21 1252- 8 G o tch , T. M . (1994) Spray deposition , in Handbook 0/ Polymer Fibre COlllflosites. F. R. Joncs (cd .) Longman: Harlow pp. 205- 9 Iloo ver. W . R . (1991) Di e ca stin g of Duralcan™ composites, in M etal Matrix COIII/}().\"it!'.\" /'roc('.\"sillg. Microstrllctllre alld Properties . N. Han sc n el al. (ed s.), RI '" N: lli o l1 :Ii I.aho rat o ry: D c nmark pp. 3X7 92 .I o nl's, I: I{ (1 ')1).1) A 11I Ol'I:IVl: mo uldin g, in /Ill/If/hook of'l'oll 'lIler Fi/m' ( '(/ III/ ,(/"t", I , I{ l" lll" (l' d .) I. o l1 g m:llJ: 1larlu w pp . !.I X 4.1

294

Fabrication

Klier, E. M ., M ortensen, A. , Co rnie , J. A. a nd F lemin gs, M. C. (199 1) Fabrication of cas t pa rticle-reinfo rced m etal s via press ure infiltration , J. Mat. Sci. , 26 2519~26 L1oyd , D. 1. (1989) The solidifica ti o n mi cros tructures o f particulate reinforced AI /Si C composites, Comp. Sci. & Tech. , 35 159~80 L1oyd , D. J. (1991) Factors influencin g the properties of particulate reinfo rced composites pro duced by m o lten metal mi xin g, in Metal Matrix Composites ~ Processing, Microstructure and Properties. N. Hanse n et al. (eds.) Ri s0 Nationa l Laborato ry: Denmark pp. 8 1 ~99 Martineau , P. , Lahaye, M. , Pa iller, R. , Naslain, R ., Co uzi, M. a nd C ruege, F. ( 1984) SiC filament / tita nium matrix comp osites rega rded as m odel co mposites. Part 2 : Fibre/ matri x chemical interact ions at hi gh temperatures, J. Mat . Sci., 19 27 49~7 0 McAllister , L. E. (1994) Ca rbon-carbo n co mposites, in Concise Encyclopaedia 0/ Composite Materials. A . Kelly (ed.) Pergamo n: Oxford McAllister, L. E. a nd Lachman , W. L. ( 1983) Multi-directio nal ca rbo n-carbon composites, in Fabrication 0/ Composites. A. Kell y and S. T . Mileiko (eds.) N o rth-Holl a nd: Amsterdam pp. 10 9~76 Middleton , V. (1994) F ilament winding, in Handb ook 0/ Polymer Fibre Composites. F. R. Jon es (ed.) Longman: H a rl ow pp . 154-60 M o rtense n, A. a nd Corn ie, J . A. ( 1987) On th e infiltra ti o n o f m etal matri x composites, Metall. Trans. , 18A 11 60~3 Newkirk , M. S. , Lesher, H. D., White, D. R. , Kennedy , C. R., U rquhart, A. W. and C laar, T. D. ( 1987) Preparation of Lanxide ceramic composite mate ri a ls: m a tri x formation by th e directed ox idat io n of m o lten m eta ls, Ceram. Eng. Sci. Proc. 8 879~95 Phillipps, A. J., C legg, W. J. and Clyne, T. W. ( 1994) The failure o f laye red ce ra mics in bending and tension, Composites 25 52 4~ 33 Phillips, D. C. ( 1983) F ibre-reinforced ce ra mics, in Fabrication 0/ Composites. A. K ell y and S. T. Mileiko (eds.) North-Holland: Amsterdam pp. 373--428 Skibo, M. , Morris, P. L. a nd L1oyd , D. J. (1988) Structure and properties of liquid m eta l processed SiC reinfo rced a luminium , in Cas t Reinforced Metal Composites. S. G. F ishm a n a nd A. K . Dhingra (eds.) ASM pp. 257~6 1 Smith , C. S. ( 1990) Design 0/ Marin e Structures in Composite Materials. E lsevie r: London Smith, P. R. a nd Froes, F. H. (1984) Developments in tit a nium met a l m a tri x co mpo sites, J . Metals , 36 1 9~26 Stoloff, N. S. a nd Alman, D. E. ( 1990) Inn ova tive processin g tec hniqu es for intermeta llic matri x composites, M RS Bulletin , 15 47~ 53 Suzuki, M. , Nut! , S. R. and Aiken , R. M . (1993) C reep behaviour of an SiCreinforced XD MoSi 2 co mposite, Mat. Sci. & Eng. , AI62 73~82 Upadhyaya, G. S. ( 1989) Powder metallurgy m etal matri x compositcs : a n ove rvi ew, Met. Mater. Process, I 2 1 7~28 Urquhart, A. W. (1991) N ovel reinforced ceramics a nd met a ls: a rev iew o f Lanxide's co mposite techno logies, Mat. Sci. & Eng. , A 144 75 82 Ward- C lose, C. M. and Partridge, P. G. ( 1990) A fibre coa tin g proccss for advanced metal matrix co mp osites, 1. Mat. Sc i., 2543 15 23 W ells, G . M. and M cA nult y, K. F . ( 1987) Co mputer-a id cd fil amc nt wi nd in g usin g no n-geodesic trajectorics, in hoc. ICCM6, Vo l.! . F . L. Matlh cws I't al. (eds .) Elsevier: London pp. 16 1 73 Willis, T. C. (1988) Spray dcpositi o n proccss fo r Illctailll ;ll rix cO lllposilc manufaclurc, MI'/ols olld MO/l'I'iol.l' , 44X 5 X

12 Applications

Composite materials are used in a very wide range 0/ industrial applications. In this chapter, the objective is to identify some 0/ the considerations in volved in commercial exploitation 0/ composites. This is done by means o/a/ew case studies and there is no attempt to present a systematic survey. The examples given cover a range o/composite type, engineering complexity, manu/acturing route, market si::e and competitive position relative to conve11lional materials. A t the beginning 0/ each case study, a list is given identifying the reasons/or preferring a composite to more conven tional engineering materials. Although the examples are spread over the .lidl range 0/ matrix types, the bulk 0/ the annual composite production 0/ around 10 million tonnes is currently inthe/OIm o/PMCs. At the start o/each example, a list is given

0/ the requirements 0/ the application. 12.1 Minesweeper hull • • • • •

low density ease of moulding to complex shape non-mag netic good resistance to corrosion and marine fouling good resista nce to fatigue and stress corrosion crack ing

Glass-reinforced plastic (GRP) is now very popul a r for various land and sea transport applications. While la rge ships are usuall y co nstructed in stee l, over 80 % of marine hull s less than abo ut 40 m in length are m ade ofGRP (Sm ith 1990). This is partly because fabricatio n in GRP is more econom ic for relatively smal l craft. Also of impo rtance, however, is the sco pe for ;1l' hi L'vin)2, weight reductions and eas ier maintenance. Addi li oll ;ili y, Ih t'Il' ; ll'l' l'L'I' i;lin app li cal ions in which the ma g netic,

296

Applications

Applications

~j -' . - -. ~;:;-~~ "' - ---. --.. - ... . .

~- .

-J:. '.

,

p

N ~.-

297

electrical or thermal properties of GRP are preferable to those of steel. An example is provided by minesweepers, which need to be non-magnetic in order to avoid activation of magnetic mines. A hull of this type is usually fabricated by contact moulding, using cold curing polyester resin and E-glass fibre , against an open female steel mould. The mould face is prepared by facing with epoxy, polishing and coating with a release agent. The glass fibre is placed against the mould , commonly in the form of a sequence of plies of chopped strand mat, woven rovings and unidirectional laminae (see §3.2.2). For a fairly large ship, such as a 40-m long minesweeper, the thickness of the layup is ~ 30-50 mm. This is too thick for placement and impregnation with resin to be carried out easily by manual methods. A semi-automated arrangement along the lines illustrated in Fig. 12.1 is normally uscd. Draping of the fibre lay-ups, mixing of resin and catalyst and pumpin g of the mixture to roller- impregnators is carried out on moveable gantries. Lay-up weights are typically ~ 2 kg m- 2 ; these can be quickly and fully wetted-out using this procedure. Production of a hull with these dimcnsions is completed in about 10 weeks (Smith 1990).

12.2 Sheet processing rolls • • • • •

(c) F ig. 12. 1 A ut o m a ted lay -up o f a la rge m a rin e hull in G RP (S mith 1990). (a) Secti o n thro ug h hull , sho win g ga ntry fo r place ment a nd wet-o ut o f glass libre m a ts. (b) Stiffe nin g ribs, fo rm ed ove r cores o f ex pa nded po lym e r fo am . (c l Sc hem a ti c pe rspec ti ve view of a min esweepe r hull un de r con stru ct io n .

high beam stiffness (ex: Young's modulus) high torsional stiffness (ex: shear modulus) low density good thermal stability good surface finish

High-performance rolls are needed for many processes involving the handling of thin sheets such as newsprint, plastic sheets, etc. In many cases, the rolls need to be relatively wide and so they need a high beam stifflless (= moment of inertia of section x Young' s modulus of material) in order to avoid excessive bending. A further requirement is that they can be rotated at high speed. This presents difficulties, since rolls with sufficient beam stiffness tend to be so heavy that they become unstable at high rotational speeds. Furthermore, roll speed cannot bc changcd quickly. In the past, steel rolls were employed , but the abovc rcquiremc nt s for li ght , stiff structures have Icd to thcir rcplacmcnt hy carhon fibrc /c po xy co mpos it es, produccd by filam c nt windin g Oi 11 . 1. 1) . A prohlc1l1 with Ih ' m;lnuf"a cture of suc h roll s I"rom tlhn.: co mposil cs ari ses I"rolll Ih l' I IlT d Ip l ho lh he ndin g and lo rsiollal slillll L'ss. T o rsioll ;iI

298

Applications

Applications

stiffness is needed to ensure that the rolls do not distort excessively when a torque is applied to make them rotate . Torsional stiffness is dependent on the shear modulus of the material. The dependence of (axial) Young's modulus, E , and shear modulus , G, on the winding angle, rP, for an angleply epoxy/65% carbon -fibre composite is shown in Fig. 12.2. A very low winding a ngle would result in both moduli being low. Peak shear stiffness occurs at rP = 45°, whereas the maximum in Young's modulus is at rP = 90°. At an angle of about 60°, values of E and G are about 70 GPa and 50 GPa respectively , which compare very favourably with steel (E '" 210 GPa , G '" 80 GPa) when account is taken of the fact that the composite has a density which is about one-fifth that of the steel. In industrial usage, weight savings of 75% have been achieved when replacing steel rolls. Rolls up to 9 m long and 0.35 m in diameter have been manufactured. The excellent surface finish required is obtained for the composite rolls by coating them with a thin layer of metal or rubber.

12.3 Helicopter rotor blade

• • • •

299

high beam stiffness (ex: Young's modulus) high torsional stiffness (ex: shear modulus) Iow density good fatigue resistance

The rotor blade of a helicopter provides a good example of a component requiring excellent specific stiffness. The blades act as aerofoils which generate lift. A typical rotor blade shape and rotor hub assembly configuration can be seen in Fig. 12.3. Composites have been used for rotor blades, and for other helicopter components, since the 1960s. Initial attractions of using composites included good fatigue resistance as well as specific stiffness. Full use has also been made of the scope for tailoring the elastic properties via control of the fibre arch itecture and improved aerodynamic blade designs have emerged by stress and fluid

300 - - Young's modulus (HM fibre) ,---.

250

. . . . . Young's modulus (HS fibre)

~

0.

8

.•..•... - Shear modulus (HM fibre)

'"

200

""E

ISO

:l

-a 0

.~

AD

'" ~

\il

100

SO .....

.... .. '

..............

.......

10

20

30

40

SO

Winding angle, cp

60

70

80

90

n

Fig. 12.2 Predicted dependence of Young' s m odu lus and shear m o dulu s o n the wind in g angle, for a tube of angle-pl y epoxy/65% ca rbon (!-I M) IIbre composite, loaded in bending or torsion. A Y o un g's m od ulu s pl o t is also s hown for usa ge of high-stren g th (!-IS) carbon IIbres. The plots were ob tain ed by the method s out lin ed in ~i 5.3. 1 . usin g property data g iven in Tahles 2.2 :Ind 2.5. The IO:ldi ll' :llI g lc for this gCO ll1cl ry is g iven hy (l)O' 'M .

I:ig . 12 ..1

I'IHlto g r:lpli 01':1 Wes tland !\gusta E lflOI helicopter. (Courtesy or Wes t 1:lnd Ilclicoptns) .

300

Applications

Applications

dyn a mics modelling, utilising the anisotropic properties of the material (Holt 1994). A particular problem arises with helicopter blades from the combination of forward and rotational motion. Since the forward velocity of the aircraft may be up to about 100 m S- I and the linear speed of the rotating blade, even at its tip, is often little more than 200 ms - I, the airspeed of the blade during the advancing part of the rotational cycle is often substantially greater than that during the retreating phase. If the pitch angle of the blade were the same during each part of the cycle, then the uplift would vary substantially on the two sides of the aircraft and it would be tipped over. Compensation for this effect is achieved by altering the pitch angle of each blade during every rotation. Further changes in pitch angle are used to alter direction during manoeuvring. It is therefore very important that the blades have adequate torsional stiffness, since they must respond quickly and faithfully to pitch-angle changes imposed at the rotor hub. The beam stiffness of the blade must a lso be high , to ensure that the tip does not lag behind during rotation or flap under its own weight excessively. From this point of view, the requirements are similar to the rolls in § 12.2, but the blade presents much greater complexity in terms of section al shape, loading configuration and fatigue performance. The construction of a typical blade section is shown in Fig. 12.4. The necessary torsional stiffness is provided by the carbon fibres at rv ± 45° to the blade axis (see §12.2). The carbon and glass fibres aligned parallel to the blade axis provide the beam stiffness necessary to minimise lag and flap. This construction also confers exce llent fatigue resistance.

12.4 Golf driving club

± 45 ' carbon

Titanium erosion shield

Elecnical heater mat

fibre/epoxy ~

± 45 ' carbon fibre/epoxy

Fig. 12.4

/

'NomexTM' (epoxy) honeycomb

Unidirectional glass-carbon/epoxy hybrid

Schematic section through a typical composite construction for a heli copter rotor blade. (Courtesy of Westlancl Helicopters .)

• • • •

301

high beam stiffness (ex Young's modulus) high torsional stiffness (ex shear modulus) Iow density high strength

There are many applications in sports goods with a requirement for stiff, slender beams. These include fishing rods, tennis racquets, skis, surfboards and go lf clubs. Polymer-based composites hold a dominant share in all such markets. The sports goods market is one in which small improvements in component performance often justify significant increases in material or manufacturing costs. The golf club (driver) provides a good example of a competitive market with a high premium on performance. As with the previous two case studies, there is a need for the golf club shaft to have good stiffness both in bending and in torsion. High torsional stiffness is very important, since this ensures that the shaft does not rotate significantly under the torque imposed when the club head strikes the ball; any such rotation would introduce an error in the direction of flight of the ball. The requirements for bending stiffness are slight ly more complex; some bending of the shaft on impact with the ball may be beneficial, since subsequent straightening can increase the contact time between club head and ball and hence increase the momentum transfer. Designs with a massive club head and slender shaft (see Fig. 12 .5(a» favour increased length of drive. However, in a shaft of low beam stiffness, the axial stresses induced during contact with the ball will be higher and the danger of damage or fracture correspondingly greater. The axia l strength of the shaft therefore becomes a key issue. In view of these requirements, hybrid fibre architectures (i.e. involving more than one type of fibre) have been developed. An example of a typical construction is shown in Fig. 12.5(b). In this case, the club shaft has several layers of carbon fibre laminae at relatively high angles to the shaft axis. These confer high torsional stiffness. Among the axial laminae are some reinforced with boroll monofilaments . These have a similar stiffness to carbon (HM), but higher strength (see Table 2.2). In particular, they have a high compressive strength , and are more resistant to bucklin g than most fibres because of their large diameter (rv 100pm). Th ese libres improve the fracture strength on the tensile side of the shal"i :Ind , parli clll:lrl y, redu ce the dan ger of failure by kink-band formalion ()Il Ih c cOlllprcss ivc sid e. Thi s o\"\"set s the di sadvanta ges of hi gher dc nsil y ;111(\ ("1l , 1 I II 11I l" hlll"O n lihrt.:s (Bu c k 1992 ).

302

Applications

A ppiic(f / ions

303

12.5 Racing bicycle • • •

high stiffness good fatigue resistance low density

Both DURALCAN and Aerospace Metal Composites (AMC) have developed bicycle frames for commerci a l sale. The DURALCAN AI(6061)/ I 0%A1 2 0 3p composite material is used in the ' Stumpjumper M2 ' mountain bike manufactured by Specialized Bicycle Components Inc. , while AMC AI(2124) /20%SiC p material is used for the frame of Raleigh racing bikes. Both models have been successfully tested in extensive sports trials. In the former case, tubing of cv I .S-mm wall thickness is extruded and rejoined under pressure a round the tube mandrel. The tube sections are then MIG fusion welded , using conventional techniques. In the case of the AMC frame , the material is made by a powder route and then the tubing is adhesively joined. The application of both types of joining procedure to MMCs is described by Ellis et al. (1994) . In addition to improved specific stiffness, both frames have exceptionally good fatigue endurance, as a result of the enhanced value of 6.K th (see §9.3.1) compared with unreinforced material. F a ti gue data for welded tubes have confirmed that the performance of particulate MMCs is superior to that of corresponding unreinforced aluminium (Harrigan 1994). A photograph of the Stumpjumper bicycle in action is shown in Fig. 12.6. Some of the manufacturing procedures involved in producing this bicycle a re given by Klimowicz (1994).

12.6 Diesel engine piston • • • •

good wear resistance high thermal stability high-temperature strength go od thermal conductivity

Thi s appli cation represents a major early s uccess in the industrial use

or MM Cs. Production in Japan has been increasing steadily over the past Fig. 12.5 (a) Photograph of the head of a golf club driver. (b) Schematic secti o n throu gh a hybrid carbon fibre/ boron monofilament construction for a golf club sha ft. (Co urtesy of Textron Specia lity Materials).

se veral years a nd no w runs to millions of units annually. Originally, a Ni ca st iro n (Ni -res ist™) insert wa s used in the ring area of alumin ium pi ston s in o rd e r to preve nt seizure of the pi s ton ring with the top ring g roove
304

Applications

305

Applications

Fig. 12.7 Photograph (Feest 1988) of a diesel engine piston , showing the region of fibre reinforcement (darker area) in the land (support region) of the groove for the piston ring.

Fig. 12.6 Photograph of the Stumpjumper M2 bicycle in action. The frame is made of Duralcan ™ AI(6061 )/ 10% A1 2 0 3p ' (Courtesy of Duralcan).

thus reducing the weight by 5- 10%. This was achieved by squeeze casting (Rohatgi 1991) into an alumina preform to produce a selectively reinforced com ponen t - see Fig. 12.7. rn sta ndard tests , wea r was red uced by over four times and seizure stress doubled relative to the unreinforced Al alloy. This was combined with four times the thermal conductivity of the Ni-resist ™ insert. Another important factor is the thermal fatigue life (Myers and Chi 1991), which is limited by cracking between the ring groove and the piston itself, or by dimensional instability. The AI 2 0 3f insert outperforms both the Ni-resist™ insert and the base alloy. The fibre content employed represents a compromise between improved wcar and sc izurc resistance, combined with good machineability re lative to th e Ni resi s t™, and acce ptabl y small det e ri o ration in i';lti gul: s trl: ngtil
thermal conductivity relative to the unreinforced alloy. Homogeneously reinforced pistons cast from AI / 20% SiC p are also being developed (Rohatgi 1991) .

12.7 Microelectronics housing

• • •

•

thermal expansion matched ('" 8 10- 6 K -

1)

low density high thermal conductivity (> 120Wm - 1 K electrically conducting

1)

Microelectronic devices often require highly stable environments. For example, microwave radar and communication systems need to be ho used so that they are shielded from stray fields and are mechanically, th e rmally ancl electri cally stable. Kovar™ (high-Ni steel) or brazed steel / mo lybd enulll hOll sin gs have been used to s upport ceramic (AI 2 0 3) s ubs tr ~ lt c l: kL'lro lli l' p:lcb ges . T hese materials give fairl y close matchin g of til l: I'II1:11 n : p:lIl , ivity wit h :i1ulllina (X 10 (, K I). Il o weve r, they arc de nse

Applications

307

Applicatiol/S

306

and have low thermal conductivities. This latter point has become a major drawback as progressive increases in component density have led to a greater need for effective heat dissipation . Possible solutions (Thaw et al. 1987, Premkumar et al. 1992) include use of AI/SiC p or AI/ boron fibre MMCs. The predicted dependence of thermal conductivity (§10.3.2) and thermal expansivity (§ 10.1.2) on ceramic content for these two systems is shown in Fig. 12.8 . Mapping of these values onto a plot of conductivity against expansivity, as shown in Fig. 12.8(c) for a ceramic volume fraction of 50%, allows identification of materials with good resistance to thermal distortion (high K /a ). It can be seen that the SiC particulate reinforcement is most effective. Packaging weight is also reduced significantly (> 65 %), and machining and brazing distortion minimised « ± 50 ~Lm) . In add ition , if the housing is cast it is possible to leave an unreinforced region on top of the side walls to aid

200 I

AI

~

-; E

particle

o

~ C :~

u::l

~

Alf50%SiC

150

Increasing

c=:>

Alf50%B fibre 100

c:

0

'"E

" f-

50

(a)

.<:

(c)

0

..,I

:::

x

12.8 Gas turbine combustor can • •

high-temperature strength good thermal shock resistance

• • •

Iow density good oxidation resista nce stable up to cv 1450 QC

Components at the hot (exhaust) end of a gas turbine are often exposed to very high temperatures and severe thermal shock conditions. The com bustor can shown in Fig. 12.9 provides a good example. A fuel /cur mixture is fed through the large hole in the end of the can and more air is aspirated through the first set of holes along the sides. The combustion temperatures along the periphery of the flame can reach 1500 QC, a.nd the sides of the can rapidly become heated to temperatures close to thI S level (Schneider et al. 1990). At such temperatures , most metals are molten,

resIstance

to thennal distortion TM

.",

u

welding of the cover onto the housing. Such components can be cast in particulate MM Cs with thin-section walls (Hoover 1991).

25

25

20

AIO 23 15

Kovar (high Ni steel)

00

10

Thennal expansivity

5

(1(- 1 x

0 10- 6 )

20

~ C

:~

c:

"c.><

'""E "

.<:

f-

5

(b) 0 0

10

20 30 40 50 Volume fraction (%)

60

70

Fig. 12.8 Predicted dependence of (a) thermal conductivity and (b) therm a l expa nsivity on ceramic content for aluminium reinforced with SiC particles or boron monofilaments (long fibres). These plots were obtained by th e methods described in § 10.1.2 and 10.3 .2, respectively, usin g data in Tables 2.2 and 2.5. Al so show n (c) is an Ashby map 0; 1.2) of the two properties, indicating wlH:n: selec ted unreinforced and cO IllPosit e mat e rial s arc loca ted .

sys tcm, I ;'Ig, . I ')-. () I'h o t( )I', .r;111h o r a co mbu s tor can for a ga s turbin e cx. hau sJt Cl ) 111a (iL o r a la y,' I,'!! S I( "/g l;lphit e co mpo sit e mat e rial. (Court esy o i W . .. egg

Applications

AppliC(f/iol1s

309

very soft or, in the case of refractory metals such as molybdenum , prone to rapid oxidation. Some ceramic materials retain good strength and oxidation resistance under these conditions, but are likely to crack under the stresses generated by the high thermal gradients. The can shown in Fig. 12.9 was fabricated from SiC layers (~200 Ilm thick) , separated by thin (~ 71lm) graphitic interlayers (see § 11.3.3). The toughness, and the resistance to thermal shock, is raised considerably by the presence of the interfaces, at which any through-thickness cracks become deflected (Phillipps et al. 1993).

12.9 Aircraft brakes • • • • • • •

good thermal stability and thermal shock resistance Iow density good strength at high temperature high thermal capacity high thermal conductivity good frictional characteristics good wear resistance

Aircraft brakes require a particularly demanding set of properties. During an emergency landing or aborted take-off, a very large amount of energy must be absorbed by the brakes in a short time without disintegration or seizure. A typical construction is based on multiple rotating and sta tionary disks - see Fig. 12.10. Friction between these disks can raise them to average temperatures of 1500 °C, with transient surface temperatures of up to 3000 °C (Chawla 1993). The disk material must therefore have excellent thermal shock resistance and high-temperature strength. Good thermal conductivity is essential to avoid overheating of the disk surfaces. Carbon has good conductivity and high-temperature stability. Solid graphite is a candidate material, which is much cheaper than carbon /carbon composites, but the strength and toughness of the composite is markedly superior. The disks are made using the infiltration techniques outlined in § 11.3.4. The weight of aircraft brakes is also significant. Typically there a rc eight brakes on a large civil airliner and, when made with a conventional construction (steel against frictional material), these toge ther weigh over 1000 kg. An equivalent set of carbon/carbon brakes weighs less than 7.00 kg. Weight reductions of seve ral hundred kilograms rep resent hi g hly slgndicant savings ill fuel over the liretime or the aircr:iI ·1. Currcntl y,

Fig. 12.10

Photograph of the carbon /carbon brake assembly used on the Boeing 767 (Chawla 1993).

carbon/carbon brakes are used in most military aircraft and increasingly in civil airliners. Concorde is equipped with carbon/carbon brakes, as are the Boeing 767, 777 and several Airbus models. The major factor inhibiting use of carbon/carbon composites for applications demanding more prolonged exposure to high temperature is their lack of oxidation resistance, although work is continuing on the development of suitable coatings (Savage 1988).

References and further reading Buck, M. (1992) Boron liber: the st ren gth to compete, Mats. Eng. , 14-15 Chawla , K. K. (1993) Ceram ic Mall'ix Composites. Chapman & H a ll: London Donomoto, T., Funatani, K. , Miura, N. and Mi yaki , N. (1983) Ceramic Fibre Reil1/orced Piston jar High Performance Diesel Engines. SAE paper 830252 Ellis, M. B. D., Gittos, M. F. and Threadgill, P. L. (1994) Joining aluminiumbased metal matrix co mposi tes, Materials World, 2 415 17 Feest , E. I\.. (1988) Exploitation of the metal matrix composites concept, Afl,/{/Is lIlId MlIterillls, 4 273 8 I \:irri g:l11 , W . ( '. (1l)()4) I:ati gue testing welded joints for r / M aluminium I1lall i \ C(lIllpmilL" s, .I . All'lllls, 4 (July) 5~ :I

310

Applications

Holt, D. (1994) Helicopter applications of composites, in Concise Encyclopaedia 0/ Composite Mat erials. A. Kelly (ed.) Perfamon: Oxford pp. 125- 9 Hoover, W. R. ( 1991) Die casting of Duralca n M composites, in Metal Matrix Composites - Processing, Microstructure and Properties. N. Hansen et al. (eds.) Ris0 Nat. Lab.: Roskilde, Denmark pp. 387- 92 Klimowicz, T. F. ( 1994) The large sca le commercialisa tion of aluminium matrix composites, 1. M etals, 42 (November) 49 53 Myers, M. R. , and C hi , F. (1991) Factors Affecting the Fatigue Performance 0/ Metal Matrix Composites/or Diesel Pistons. SAE paper 910833 Phillipps, A. J. , Cl egg, W. J . and Clyne, T. W. ( 1993) Fracture behaviour of ceramic la minates in bending, Acta Me tall. Mater., 41 805- 25 Premkumar, M . K. , Hunt, W. H. a nd Sawtell , R. R. (1992) Aluminium composite materia ls for multichip mod ules, 1. Metals, 40 (July) 24-8 Rohatgi, P. (199 1) Advances in cast MMCs, Ad\!. Mats. & Procs. , 137 39-44 Savage, G. ( 1988) Ca rbo n-carbon composite materia ls, Metals and Mat erials, 4 544-8 Schneider, G . A., Nickel , K. G . and Petzow, G. (1990) Thermal shock and corrosion of SiC - a combustion chamber model case stud y, in Th e Physics and Chemistry 0/ Carbides, Nitrides and BOI·ides. R. Freer (ed .) Kluwer Academic Publishers: New York pp . 387- 401 Smith , C. S. ( 1990) Design 0/ Marin e S tructures in Composite Mat erials. Elsevier: London Thaw, c., Minet, R., Zemany, J . a nd Zweben , C. ( 1987) MM C microwave packaging components, 1. M etals, 35 55

Appendix Nomenclature

Parameters

A

(m 2 )

a a a

(m) ( m? s - I)

(- )

C C- I c c d E e

f G G h h h I

K K Kc K L M 111

N

(Pa) (Pa - I ) (JK- I m - 3 ) (m) (m) (ra)

(- ) (- ) (J m- 2)

cross-sectional area direction cosine radius of sphere thermal diffusivity stiffness (tensor o f 41h rank) compliance (tensor of 41h rank) volume specific heat crack length fibre or particle di a meter Young's modulus relative displacement reinforcement volume fraction strain energy relea se rate shear modulus spacing between fibres

(ra) (m) height (m) I 2 (W m - K - ) heat transfer coefficient unit tensor (identity matrix) - see Eqn (6.34) (- ) (ra)

bulk m od ulus stress intensity factor critical stress intensity factor (fracture toughness)

(MPaJiTI) (MPa JiTI) (W m- I K - I) thermal conductivity fibre ha lf-length (m) bending moment (m N) Weibull mod ulu s ( ) I) !\vog: ldro' s Ilumher (mole "\ \1

\ 12 N N

Appendix ( ) (m 1)

11

(- ) (- )

p

(Pa)

P Q

(- )

11

q R R r S S S s

T T'

U

u V v W W

(J mole - I) (W m - 2)

(- ) (m) (m)

(- ) (- ) (Pa)

(- ) (K) (Km - I) (m) (s) (J) (m) (m 3 ) (m S- I) (kg) (J m - 2 )

\11

(- )

Cl'

(K - I)

~

(- )

8

(m)

E

(- )

cp

CO)

'Y 'Y

i

(- ) (J m - 2 ) (S - I)

17

(- )

r;,

(m - I)

A

(m)

()

(0)

number of loading cycles number of fibres per unit area dimensionless constant - see Eqn (6.8) stress exponent pressure probability activation energy heat flux stress ratio far-field radial distance from fibre axis radius of fibre or tube compliance tensor Eshelby tensor stress amplitude during fatiguc fibre aspect ratio e2L/ d = L/ r) a bso lute tem pera tu re thermal gradient ply or wall thickness time work done during fracture displacement in x-direction (fibre axis) volume velocity weight work of adhesion weight fraction thermal expansion coefficient relative change in volume crack opening displacement strain loading angle (between fibre axis and loading direction) shear strain surface energy shear strain rate interaction ratio curvature m ea n free pa t h wctting anglc

N omenc/ature v

(- )

p

(kg m - 3 )

p T

(m) (Pa) (Pa)

~

CO)

a

Poisson's ratio density distance from fibre axis stress shear stress phase angle (mode mix)

Subscripts

0 I 2 3

initial x-direction (along fibre axis) y-direction :-direction

A

applied ; axis cllr buckling coated composite critical debonding fibre end effectively stress-free failure fibre (reinforcement) frictional sliding global glass transition hoop hydrostatic interfacial liquid ma trix melting point pull-out particle matrix notation integers principal strcss notations radial SII rvi v;iI

a b c c c d e esf f f fr g g H H

L m m p p p,q

p, q, r S

313

3 14

RM

AI' pe 11 di.r

Author index

th trans u u v x ,y ,~ Y

Rule or Mixtures stress transfer threshold transverse failure (ultimate tensi le) uncoated vo lume Cartesian coordinate directions yield (0 .2% proof stress often taken)

El

hoop

The work of the author mentioned is cited on the page(s) indicated.

*

critica l (e .g. debonding or fracture)

Aaronson , H. I. 239, 270 Abis, S 22, 37 Adams, D. F. 177, 179, 205 Agarwa l, B. D. 229, 234 Agassant , 1. F. 50, 59 Agrawal , D. C. 146, 155 Aiken , R. M. 291 , 294 Akisa nya , A. R. 2 15, 234 Alford , N. M. 35 , 37, 291 , 292, 293 A lm an , D. E. 290, 294 Argon, A. S. 18 1, 205 Arsenault, R. 1. 53, 59 As hby, M. F. 5, 8, 208 , 2 11 , 234, 252, 263, 269 Avestoll , 1. 194, 205 Azzi, V. D. 187, 205

S uperscripts ax C T T* tr

ax ia l constra ined transforma tion mi sfit transverse

Bader, M. G . 22, 37 Bailey, 1. E. 194, 206 Bansal, R. C . 36 Barlow, C. Y. 155 Batchelder, D. N. 11 3, 114, 132 Batdorf, S. B. 165, 205 Beaumont, P. W. R. 223, 236 Becker, W. E. 277, 293 Bennett , S. C. 12, 36 Benveniste, Y. 265, 269 Bernstein, I. M . 234, 235 Bircha ll , J. D . 20, 22, 35, 37, 29 1,292,293 Black, W. B. 36 Blake, 1. 275, 293 Bouvaist, 1. 257, 269 Bo wlcs, D. E. 269 Brinlr up. A . 67. 77 Ilroull11:1n . I. . .I . :!2'i . D4 Ilrowll . I .. M , 17(,. 111

Bruills, P. F. 37 Buck, M. 30 1, 309 Bunge, H. 1. 5 1, 59 Bunsell , A. R. 37 Butto n, T. W. 35, 37, 29 1, 292, 293 Cappleman , G, R. 22, 37 , 137, 155 Carrara, A , S, 132 Cham is, C. C. 66, 77, 189, 206 Chang, M, 254, 269 C hara la mbides, p , G, 140, 156, 235 Chaw la , K, K, 288 , 293, 308, 309 Chi, F, 304, 3 10 Cho rley, E, M, 65, 77 C hou, C. T. 148 , 156 Chou , T. W , 8, 93 , 104, 11 4, 131 , 157 Christman , T. 228, 235 Chua, p, 140, 155 C laar, T. D, 290, 294 C larke, D, R, 126, 13 1 C legg, W, 1, 35, 37, 137, 156,2 15, 220, 225, 23~ 289, 29 1, 292, 293, 307, 308 , 310 C lyne, T. W, 22, 35, 37, 65, 74, 77, 11 3, 114, 123, 13 1, 137, 143, 144, 145, 155, 156, 157, 176, 206, 2 15, 220, 225, 236, 243, 263, 258, 26~ 26~ 282, 285, 29 1, 293, 308, 3 10 Cogswell , F, N, 34, 38 Cook , 1, 165,205,2 14, 235 Co rnie, 1, A. 282, 294 Co uzi, M, 19, 29,37, 137, 153, 157, 286, 294 Cox, H, L. 105, 13 1 Crowc, C. R, 233, 235 C rucgc, F , 19.29,37. 1J7, 153, 157, 286, 294 C urrcy . .\ , D, I. X C'urlin . W, A, 1('(,. 167.205

1 15

!I ll/I/O!" illdnI):ilhcllo , I) . .I. 257, 2(1) Dal gle is h, B . .I. 140, 156, 235 Daniel, I. M. 65, 77 Darlinglon , M. W. 48, 49 , 59 Dclannay, F. 224, 236, 245, 247, 270 Dellis, M. A. 224, 236 DelMonle, J. 36 Desarmot, G. 36 Devour, M. G. 14, 36 Dhingra , A. K. 2 1, 37 Dieter, G. E. 78, 80, 104 Dinwoodie, J. M. 36 Dobb, M. G. 18, 37 Dollar, M. 234, 235 Donaldson, K. Y. 269 Donnet, J-P. 36 Donomoto, T. 303, 309 Dorner, D. R. 177, 179, 205 Dow, N. F. 105, 13 1 Earmme, Y. Y. 239, 270 Edwa rd s, L. F. 137, 156 Elli s, M. B. D. 303, 309 Endo, M. 14, 36 Endo, T. 254, 269 Erdogan , F. 227, 235 Eshe lby, J. D. 121, 131 Evans, A. G. 140, 156, 166, 167, 205, 214, 235 Everett, R. K. 286, 293 Ewa ld s, H. L. 208 , 235 Ewins , P. D. 182, 205 Fada le, T. b. 265, 269 Favre, J. P. 142, 156 Feest, E. A. 293 Feesl, E. A. 305, 309 Feillard, P. 154, 259, 269 Filzer, E. 35, 38, 291 , 293 Fleck, N. A. 180, 183 , 206,215, 234 Flemings, M. C. 282, 294 Folkes, M. J. 59, 279, 293 Froes, F. H. 285,294 Frost, H. J. 252, 269 Fukuda, H. 114, 131 Funatani, K. 303, 309 Furness, J. A. G. 258, 269 Gaffarian, R. 165, 205 Gagin, L. V. 36 Ga li o li s, C. I 13, I 14, 1.12 Gaur, U 148. 156

Gillham, J. K. 38 Gitlos, M. F. 303, 309 Gordon, F. H. 266, 269 Gordon , J. E. 88, 104, 165, 205, 208, 214, 235 GOlch, T. M. 272, 293 Gra nde, D. H. 142, 156 Griffith, A. A. 209, 235 Guigon , M. 36 Hahn , H. T. 143, 156 Hale, D. K. 26 1, 269 Halp in , J. C. 66, 77, 188, 206 Hamada , T. 36 Hannan~ D. J. 35, 38 Harrigan, W. C. 303, 309 Harris, B. 224, 234, 235 Hart-Smith, L. J. 19 1,205 Hasegawa , Y. 19,37 Hasselm an , D. P. H. 265, 269 Ha sso n, D. F. 233 , 235 Halla , H. 26 1,263,269 Hayashi , J. 19, 37 He, M. Y. 166, 167,205,216, 235 Hellman , J. R. 143 , 156 Hill , R. 187,206 Holt , D. 300, 310 Hong, K. C. C. 142, 156 Hoover, W. R. 283, 293, 307, 3 10 Horsefall , I. 137, 156 Hsueh , C. H. 142, 143, 156 Hubert, P. A. 22, 37 Hull , D. 154, 156,201 , 203,205,206, 226, 234, 235, 236 Hunt , W. H. 226, 235,306, 310 Hutchinso n, J. W. 2 14, 216, 235, 236 Ingli s, C . 208, 235 Irwin , G. R. 2 10, 235 Ishida , H. 149, 156 Jaffe, M. 37 Ja nsso n, S. 257, 269 Jarry, P. 257, 269 Jelf, PM. 180, 183, 206 Jennin gs, T. M. 250, 25 I , 269 Jenscn, D. J. 246, 248, 270 Johnson, 1) . .1 . 12, I X, 36, 37 .I o il11 son, L. F. 265, 2(1) .I 0 h11 S0 11 , W . 12, 14,3(, .I0il11 S() I1 , W. S 2-'2, 2.15 .IoIlCS, I) . R. 11 . 2()X , :> 11 , _'1· 1

Author index Jones, F. R. 148, 156, 275, 293 Jones, M. C. L. 175, 196, 197, 198 ,206 Jones, R. M. 77, 93, 104 Jones, R. S. 37 Juul Jenscn , D. 51 , 59 Kalla s, M. N. 143, 156 Kalton , A. F. 143, 145, 156, 259,269 Katzman , H. A. 150, 156 Kays, A. O. 34, 38 Kelly, A. 8, 93,104, 172,194, 205, 206 Kendall , K. 35, 37, 140, 156, 2 16, 235, 291 , 292, 293 Kennedy, C. R. 290, 294 Kennedy, J. M. 184,206 Kerans, R. J. 143, 156 Keustermans, J. P. 224, 236 Klicr, E. M. 282, 294 Klimowicz, T. F. 303, 3 10 Knappe, W. 19 1,206 Koenig, J. L. 149, 156 Koss, D. A. 143, 156 Kowalski, I. M. 13, 269 Koyama , T. 14, 36 Krieder, K. R. 177, 206 Kristiansen, K. 5 I, 59 Kural , M. H. 245, 270 Lachman , W. L. 291 , 294 Lagace, P. A. 184, 206 Lahaye, M. 19, 29, 37,137, 153, 157, 286, 294 Lame, G. 238, 270 Lawrence, P. 140, 156 Le Petitcorps, Y. 19, 37, 146, 156 Leckie, F. A. 257, 269 Lee, J. K. 239, 270 Legg, M. J. 173, 205,206 Lentz, J. P. 245,247, 270 Lesher, H. D. 290,294 Lewandowski, J. J. 226, 235 Lilholt, H. 51 , 59, 246, 248, 270 Limura , M. 19, 37 Lips, B. 224, 236 Liu , C. 226, 235 Ll oyd , D. J. 184, 206, 283, 284, 294 Louc, W. 257, 269 Lowden, R. A. 152, 156 Lulay, K . F. 11'4,206

Mandell , J. F. 142, 156 Marloff, R. H. 65, 77 Marsh , P. 37 Marshall, D. B. 143 , 156 Martineau , P. 19,29, 37, 137, 153, 157, 286, 294 Mason , J. F. 137, 156,282, 293 Masulli , D. 245, 247, 270 Matsuda , N. 254, 269 Matsumoto, T. 36 Matsuura , K. 254, 269 McAllister, L. E. 291 , 292, 294 McAnulty, K. F. 274, 294 McCartney, L. N. 194, 206 McDanels, D. L. 2 1, 37, 11 8, 132 McGarry , F. J. 132 McGinley, P. L. 48 , 49, 59 Mclean , M. 254, 270 Middleton, V. 273,294 Mikata , Y. 139, 157 Miller, B. 148, 156 Miloh , T. 265, 269 Min, B. K. 245, 270 Minet, R. 306, 3 10 Miura , N. 303 , 309 Miyaki , N. 303, 309 Mockford , M. J. 20, 37 Mohr, J. G. 36 Monteiro , H. A. 38 Moreton, R. 14, 36 Mori , T. 126, 131 Morri s, P. L. 283, 294 Morten se n, A. 282, 294 Mura , T. 123, 126, 131 Mu zzy, J. D. 34, 38 Mye rs, M. R. 304, 310 Nardone, V. C. 114, 131,256, 270 Naslain , R. 19, 29,37, 137, 146, 153, 156, 157, 286, 294 Newaz, G. M. 257, 270 Newkirk , M _ S. Nickel , K. G. 307, 3 10 Nieh , T. G. 256, 270 Nielson, L. E. 261, 270 Nishida, T. 36 NUll , S. R. 29 1,294 Nye, J. F. 77, 78 , 8 1, 82, 104 Oherlin , A . 14,36

M;!Clllill:lIl , N. 11

17.' ,

~ ()(,

M;!j'"'HI :II , Il S .'\/ , '/11

317

Ok:lIlIlIr;! , K. 1'), 37

OliveI', W . C. lel \ 1'(,

318

Author index

Omori , M. 19, 37 Otani, S. 13, 36 Outwater, J. O. 105, 131 Ovland , S. 51 , 59 Pagano , N. J. 188, 196, 206 Pai ller, R. 19, 29 , 37, 137, 146, 153, 156, 157,286,294 Paris, P. 227 , 235 Parry, T. V. 18 1, 206 Partha sarathy, T. A. 143, 156 Partridge, P . G. 286 , 287, 294 Parvizi, A. 194, 206 Peacock, J. A. 155 Pedersen , O. B. 126 , 131 Penn ington , D. 54, 59 Peters, D. M. 8, 36 Petzow, G. 307, 3 10 Phillipps, A. J. 2 15, 220, 225 , 236, 29 1, 293 , 308, 3 10 Phillips, D. C. 38 , 288 , 290, 294 Phillips, L. N. 12, 36 Pi ggott , M . R. 140, 155 Pi go tt , G. H. 20, 37 Pinto , P. J. 20 , 37 Pipes, R. B. 196, 206 Plueddemann , E. P. 148, 15 1, 157 Potter, R. T. 182, 205 Pra tlen , N. A. 55, 59 Premkumar, M. K. 306, 3 10 Prewo, K. M. 114, 13 1, 177, 206 Price, J. N. 234, 236 Proctor, B. A. 36 Puck , A. 65 , 77 Raj , R . 146, 155 Rice, J. R . 2 14, 236 Ro berts, K . A . 285 , 293 Rodda , E. G. 14, 36 Rohatgi , P. 304, 310 Rosen, B. W. 105, 13 1, 163 , 164, 206,207 Rowe, W . P. 36 Rowland s, R. E. 186, 206 Ruhle, M . 140, 156, 235 Russel , K. C. 239, 270 Russell , D. A. M. 59, 279, 293 Sajiki, Y. 36 Sato, M . 19, 37 Savagc, G. 309, 3 10 Sav ille, B. P. 18, 37 Sawtc ll . R. R. 306. 3 10

Schapery, R. A. 243 , 270 Schneider, G . A. 307, 310 Schneider, W . 191 , 206 Scola, D. A. 135, 157 Shahan i, R. A. 56, 59 Shen, C. H. 234, 236 Sherby, O. D. 257, 270 Shi , Y. B. 226, 235 Simon, G. 37 Sinclair, J. H. 189, 206 Skibo, M. 283 , 294 Smith , A. N. 11 3, 13 1 Sm ith , C. S. 294 Sm ith , C. S. 295 , 3 10 Smith , G. R . 48 , 49, 59 Smith , P. R. 285, 294 Spencer, A. 66 , 77 Spencer, B. 20 I, 203 , 205, 206 Springe r, G. S. 234, 236 Sta nley, D. R . 20, 37 Sto bbs, W. M . 11 3, 131 ,246, 248, 270 Stoloff, N. S. 290, 294 Suo, Z . 2 14, 236 Suresh, S. 228 , 235 Suzuki , M. 29 1, 294 Takehashi , H . 157 Tanaka , K . 126 , 13 1 Taya , M. 123 , 13 1, 139, 157, 184, 206 , 26 1, 263, 265 , 269 Taylor, R. 266, 269 Termonia , Y. 11 2, 11 3, 132 Thaw, C. 306, 3 10 Thomas, J. R . 269 Th o mas, K . L. 8, 36 Threadgill, P. L. 303, 309 Tibbetts, G. G. 14, 36 Tien , J. K. 256 , 270 Tompkins, S. S. 269 Tsai , S. W . 66, 77 , 187, 190, 205 , 206 Turner, S. P. 266, 269 Upadhyaya. G. S. 294 Urquhart , A. W . 290,294 Vesce ra . 1-'. 224, 23 6 Vincent , M . 50. 59 Vizzini , A . .I . 1 ~4 . 206 Wak:i s ililll a. K. 1 ~4 . 20(, Wall g, .I . S. 2 14, 2, (, W; II ,il ill . R . .I . 11. : OX, .) l ~

Author index Ward-Close, C. M. 143, 145, 156, 286 , 287 , 294 Watson , M. C. 144, 157, 176,206 Watt , W. 12, 14, 36 Watts, J. F. 137, 155 Weeton , J. W. 8, 36 Weibull , W. 27 , 36 Wells, G. M. 274, 294 Wells, J. K. 223 , 236 White, D. R. 290 , 294 Willis, T. C. 283 , 294 Windlc, A. H. 155 With ers, P. J. 35, 37, 5 1, 59, 65 , 77 , 11 3, 123 , 131 , 243, 246, 248 , 263,269 , 270 , 282 , 293 Wolfenden , A. 67, 77

Wolff, B. K . 245, 270 Wolla, J. M. 67, 77 Wood hams, R. T. 38 Wright , W. W. 154, 157 Wronski , A. S. 18 1, 206 Wu , E. M. 190 , 206, 207 Wu , M. Y. 257 , 270 Yakima , S. 19, 37 Yeung, P. H. J. 11 3, 114, 132 Young, R. J. 11 3, 114, 132 Zema n ~ J. 30~

3 10 Zweben , C. 163 , 164, 207,306 , 3 10

319

Subj ect index

SUbject index

aco ustic em issio n 193 , 194 acti va ti on energy 256 adso rpti on 133 aircraft brakes 13, 29 1, 308 a lumin a fibres 2 1, 26 1 aluminium 34, 56, 11 6, 153, 177, 234 a lumin osi li ca te fibres 2 1 angle-ply laminates 44, 95, 97, 196, 197, 199, , 200, 203 anisotropy I, 2, 10, 32 Hooke's law 78 lami nate elastic co nsta nts 95, 96 thermal co nduct ivit y 13 thermal expa nsivi ty 13 Young' s modulus 2, 11 APC_2rM 34 a ramid fibres 9, 16, 17, 22, 24, 25, 27 Archimedes principle 55 Archimedes screw 278 asbestos fibres 20, 22 Ashby map 6, 7, 306 aspect ratio o f fibres 23,50, 109, 110, Ill , 11 8, 12 1, 125, 130, 220 a utoclave mo ulding 275, 276, 277 average fibre length 53 averages, rule of 115 Avogadro's number 260 background stress 126 balanced lamin ates 97, 194 balanced symmetric la minat es 102 Ba uschinge r effect 184 beam stiffness 297 Beech Starship 277 bendin g or lib rcs 26 hicyck J OJ

binder 282 biological matcrials I boa t building 272 Boltzmann 's constant 260 bond strengt h 133, 138, 147, 184,2 19 bonding mechani sms 133 bone I, 2 11 boron nitride 263, 265 borosilicates 34, 289 braiding 39, 46 , 272 buckling of fibres 23, 24, 179, 180, 182, 183 , 23 1 bulk modulus 74, 239 C-glass fibre 15 calcium ca rbo nate 276 carbon fibre H M (h igh modulus) 11 , 249, 298 HS (hi gh strength) 11 , 298 mesophase pitch 13 ox idati on 22 PAN (polyacry lo nitrile) 12, 14, 135 pyrolitic deposition 14 surface energy 135 surface react ions 135 turbostratic 10, 12 carbon/carbon composi tes 9. 13,22,35, 288, 29 1, 292 cavities 55, 56, 17 1. 2 18, 253. 257 cel lul ose I, 11 , 17 cement 35 Charpy impac t tes t 2D chemical res istan ce 33 chcllli c; d vapour dc position ( 'VI») Il) . I q c hcl llic: d vapo ,,,- illliltratio" 7XX. : '):

chopped stra nd mat 43 , 46, 48 , 78, 230, 272, 297 clay 2 1 clustering 42 , 283, 284 co-spraying 272, 28 1 coa tings o n fibres 137, 147, 148, 15 1, 152, 176, 286 coax ia l cylinder model 139 coefficient of therma l expansio n 3, 4, 11 , 13, 15, 3 1, 237,240 co ld drawing 56 collagen I comp li a nce tensor 8 1, 88 compress ion moulding 272 compressive strength 23 , 178, 18 1, 183, 184 conca tcna tion 87 concrete 35, 2 11 const rain ed strai n 123 constrai nt 179 con tact angle 134 cordieri te 289 cost 2 1 cot ton 17 coupling agents 16, 135, 137, 147, 148 co uplin g stresses 101 c rack: blunting 166,2 14 deflection 35, 2 13, 2 16, 228 opening displacement 2 12 propagation 21 1, 2 13, 227, 233 resistance 2 10 creel sta nd 273 creep resistance 25, 138, 23 1, 246, 25 1, 257, 29 1 critica l fibre aspect ratio 12 1 critical stress intensity factor 2 12 cross-link density 154 cross- linkin g 30 crossp ly lam in ate 44, 95, 97, 99, 100, 172, 19 1, 192, 223, 230, 247, 250 cubic symme try 82 cum ul at ive weakening models 163 curi ng kinetics 154 cu rin g shrink age 32, 138, 238 dela mination 57, 58 density 3, 4. 5, 6, 11 , 3 1, 55 deviato ric strain 65 diamond 17. 26 1 di c castin g 2X3. 2X4 di esel e nginc pi stoll 10 <;

321

differential thermal contractio n 122, 238, 240,289 diffusion barrier coati ngs 152 diffusion bonding 285 diffusional mean temperature 257, 258 dilatometry 244, 246 dilute composite 126 directional ox id at ion 290 di sloca ti ons 20, 234, 255 di spersion strengthening 285 dough moulding compound 23 1, 276 Dupr!: eq uat ion 134 DURALCAN 303 E-glass fibre 15, 25, 249 edge effects 195 eigenstra in 125 Einstei n summ ation convention 80 elastic deformation 60 electros tatic forces 137 energy relea se ra te 210 engineerin g elasti c co nstants 82, 88 enginee rin g shear strain 79 env ironmenta l degradation 152, 169, 234 epoxy resin s 30, 32, 33, 58, 72, 89, 113 , 130, 135, 168, 170, 188, 191 , 196,249, 26 1. 298 eq ua l stra in condit ion 6 1, 66 eq ua l stress co nditi o n 62 equiva lent homogeneou s ellipsoid 123 , 125 e rosion shi eld 300 Eshe lby model 62, 67, 68, 71 , 72, 105 , 11 8, 121 , 240, 243,255, 257, 263, 264 Es helby tenso r 125, 244 ethoxy group 148 Eu ler buckling 23, 183, 184 evaporation 286 extrusion 53, 120, 28 1, 285 fabrication 27 1 failure: compressive 23, 178, 18 1, 183, 184 criteria 185, 190, 199 lo ng- fibre composites 158 particulate co mposi tes 226, 228, 233 pressurised tubes 197, 198 probability 27, 29 short-fibre composi tes tran sverse 17 1. 174, 184 li lti guc crac k grow th 22 7. 233. 303 rati g uc limit 227.230 lihre hreak pro pa gatioll Ill odel s 1(,\

111

Subject index libre aspect ra ti o 23 , 50, 109, 110, Ill , 11 8, 12 1, 125, 130, 220 bending 26, 179 , 182 bridging 229 clu stering 42 coatin gs 137, 147, 148, 15 1, 152, 176,286 damage 16 diamete r 26 di stribution 171 end stress 114 nexibility 24, 26 fracture 218 length di stribution 53 misa li gnment 42, 59, 69, 180, 18 1 o rientation distribution 48,49, 50, 5 1, 52, 59,278 preform s 34, 48 pull-out 30, 150, 152, 169, 218 , 220, 221 spacings 41 s urface roughness 137 surface trea tments 16 to ws 273 fibre s a lumina 2 1 a lumin osilicates 2 1 aramid 9, 16, 17, 22, 24 asbestos 20, 22 boron 10, 11 , 19, 183,3 01, 302 ca rbon 2, 6, 9, 11 , 12, 13, 14,22, 24, 250 I 35, 154 cell ul ose I I, 12, 17, 2 18 FP'" 10, 11 , 2 1 glass 2, 6, 9, 14, 15, 24, 135 Kev lar'" 10, 11 , 16, 17, 18, 23,24,25, 169, 218,23 1,23 3 mon o filament s 19, 25, 27, 29 , 176, 183, 23 1, 287, 30 I Nica lon'" 10, 11 , 19, 151 , 152,254 orga ni c 16, 17, 21 8 polyacrylonitrile 12 polydiacetylene I 13, I 14 polyethylene 16 Saffil™ 10, 11 , 22 , 136, 258 SiC 10, 11 , 19,20,25,29, 176 Si 3N4 19 staple 10 whiskers 9, 10, 11 , 19, 20, 27, 5 1 fibrillation 25, 169 , 170 filament wind in g 25, 199,272 , 27], 2X5, 297 finit e diffe re nce meth od ( 1; f)M ) 11 2, 113, 177, 179

finite element method (FE M ) 67 , 143 , 146 fi shing rod 30 I n ax 11 , 17 n ex ibility of fibres 24, 26 force balance 242 , 246 Fourier tran sform I R spectrosco py 149 Fo uri er tra nsfo rm 5 1 fracture : energy 2 10, 2 17, 223 , 224 data 2 11 mecha nics 140, 208 to ughness 2 11 fri ctional sli ding 142 full fragmentation test 146 gas turbin e comb usto r ca n 307 geodesic path 273 glass ceram ics 34, 289 glass fibres 2, 6, 9, 14 a to mi c structure 15 C-glass 15 co mpositi on 15 E-gla ss 15, 25 S-glass 15 stiffness 15, 16 s urface energy 135, 148 surface treatm ent 16 thermal properties 15 water pi ck-up 147 glass transiti o n temperature 34, 252 glucose 17 go lf club 301 ,302 graphite interla yers 35, 29 1 turbostratic 9, 10 gree n bod y 288 Griffi th cracks 209 Halpin- T sa i ex press io ns 66, 67, 6X, 7 1, 7'2, 75,88,89,90,9 1,92 hea t di storti o n 32, ]3 heat trans fer 259, 260 helicopt o r rotor blad e 299 hemi -cellul ose 17 hide I Hooke's la w 7X hoo p stress I ')() , I')X , 2() I, 23X ho t isos t:!ti c press in g ( 1111') 57, 2x , :OX (" 2X7, 2X9 ho t press l1l o llldin g 27'1 hyhrid lihre CO I" lllI l' ll o n 1II 1 hydroc hl oric I (,'I , I ll , '\"

:Il''''

hyd rogen 234 hydrostatic stress 74, 238 hydroxya patite I hysteresis 244 idea l angle 204 identity ten so r 128 image stress 126 indentati o n testin g 143 injecti o n mouldin g 148,272,278,279 interactio n compli a nce 92, 194 interacti o n ra ti os 88, 89, 90, 97 , 98 interdiffu sio n 135 interfa cial: adhesion 106, 132 bond strength 13 3, 138, 147, 152, 171 , 174, 184, 2 19, 225, 282 c hemica l reaction 135, 152, 286 fracture energy 2 13, 2 15, 219, 225 frictional sliding 142 hea t tra nsfer coeffi cient 264 shea r stress 108, 109, 110, 167, 168 slidin g 120 st resses 138 structure 133 testin g 140, 146 interlamina r stresses 98, 194, 195, 196 interphase region 153 in versio n relat io nships 95 isotropic material 82 jute 17 Kevlar'" fibre 10, 1" 16, 17, 18, 23, 24, 44, 169 , 170, 179 , 2 18, 231, 233, 253 ki nk bands 24, 27, 179 , 18 1,3 01 Kirchoff ass umpti ons 93 ' knee' in st ress- stra in plot 160, 163, 193, 194 kn ittin g 25 , 39, 46 Kova r' M 305 ladder polymer: 12 laminae 40, 78 laminates: 43 , 78, 93 a ngle-p ly 44, 95, 97 halanced ')7 balanced sy mmetric 102 crossply 44, 95, 97 , 247, 249, 250 dcl'"llination 57 hOI press in g 27<') Sla cki ng selfll ence 44 si resses 100

323

symmetric 44, 101 thermal cycl ing 247 Lanxide processes 290 layered structures 35, 288, 29 1, 307 length of fibres 53 lignin I , 17 load tra nsfer 127 , 2 18, 230 loading a ngle 94, 185 magnesia 290 magnesium 34, 137, 153 marten siti c tran sfo rm a ti on 122 matri x notation (tenso rs) 80, 8 1 matrix: constraint 42, 2 17 crackin g 160, 16 1 infiltration 55 plasticit y 120, 122, 174, 2 17 viscosity 59 yield stress 120 work hardenin g 17 1 max imum stress criterio n 185, 188, 199 mea n fi eld a pprox imatio n 126 mea n stress 127 , 128 m eas urement o f bond strength 138 mechanica l keyin g 137 melt n ow 32 melt impregna ti o n 279 melting tempera ture 33 , 252 meniscus curvature 43 merit index 5, 6, 7, 263, 265 mesophase pi tch 13 methyl trichlorosilane 19 mica 2 1 microcracking 163 mi croelectronics housing 305 microfibrils 17 mi crospheres 2 1 minesweeper 272 , 295 misalignment of fibres 42, 59, 69 , 180, 18 1 misfit stra in 123,238,243,247 mi xed mode loading 2 13, 2 14 mixtures , rule of 62, 11 8 m od ifi ed shear lag model 114 m o llu sc shell s 29 1 m o lybdenum di silicide 29 1 mon ofi la ments 10, 11 , 19,25,27, 29, 176, 183, 23 1, 302 neck in g of fibres 169 nettin g analysis 2()] ne ut ron diffra ctio n 5 1,2411, 24X

324

Subject index

Ni-resist '" 304 Nica lon '" fibre 10, 11 , 19, 15 1, 152,254 N o mex™ 300 nylo n 31 , 33, 34, 2 11 , 26 1, 278 o rr-ax is ela stic co nsta nt s 83 o ff- ax is loading 184 optica l bire rrin gence 65 o rga nosil a nes 16 o rientati o n o r fibres 48, 49, 50, 5 1, 52, 59, 278 o rth o tropi c sy mm etry 8 1, 82, 83 O sprey process 283 os teones I Pa ri s~ E rd ogan rela ti o n 227 pa rticle a gglo merati o n 283 pa rticle pushin g 283 , 284 pa rtic ul a te MM C 11 8, 120,226, 228,233, 256, 267, 282 pa rti cul a te reinro rcement 9, 20 , 276 P EE K 3 1, 33 , 34, 154, 280 pe rnuoro- I-meth yl deca lin 57 ph ase a ngle 213 , 214, 2 15 ph o no ns 259 phot o-degrada tion 23 ph o toe la sticity 65, 11 3, 146 ph ys ical vapo ur depositi o n (PVO) 153, 286 pi ezo-electri c tran sduce rs 193 pitch a ngle 300 pl a ne stress 83 p las tic mic ro buc klin g 180 pla stic zo ne 21 2 ply 43, 78 Poi sso n co ntracti o n 100, 143, 192,24 2,244 Po isson pl o ts 174 Po isso n's rati o compos ite 62 , 7 1, 73, 75 , 76, 174, 243 fibre 11 , 2 17,246 matrix 31 , 125, 2 17, 239 lamin a 92 laminate 95 , 96 po le fi gure 5 1, 52 pol yca rbo na te 278 po lycarbo sil a ne (PCS) 19 polydia cetylene fibres 11 3, 114 polyester resin s 30, 32, 33, 56, 67 , 110, 116, 135, 139, 17 1, 173, 175, 186, 196, 197, 200, 261 po lyeth ylene fibres 16, 135 po lypro pylene 31 , 33 , 49, 148, 150, 27X, 2XO po lys il ox an e 14X

po lysulph o ne 280 po lyurethane rubber 90 po lyv inyl ace ta te 16 po rosit y 57, 2 18, 240 powder blending 285 prero rm s 34, 48, 28 1 pre-preg 274 press ure die cas tin g 283 propert y ma ps 5, 6, 7 pull-o ut o r fibres 30, 150, 152, 169, 2 18,220, 22 1 pultru sio n 18 1, 272, 274 Ra ma n reso na nce spectrum 114 reacti o n injection mo uldin g 277 reacti ve processin g 288, 290 reciprocal relati o n 73 rel a ti ve di spl acement tenso r 79 residua l stresses 3, 7, 32, 138, 144, 154 , 2 13, 248, 255 resin tra nsrer mo ulding 274, 275 Reuss model 64 ro pes 23 rota ti o n tenso r 79 rot a ti o na l sy mme try 97 rubber 176 rule o r ave rages 11 5 rule o r mi xtures 62 , 11 8 S~g l ass

fi bre 15 pl ots 227 , 229, 232, 233 Sa ffil 'M fibre 10, I I, 22 , 136, 258, 282 Scha pery equati o n 24 3 shea r di sto rti o ns 93 shea r n ow 59 shear lag model 105, 106, I 16, 140 shear mo dulu s 69, 7 1, 72, 89, 239 , 29X sheet mo uldin g co mpo und 276 shot 2 1 sil a nes 19, 137, 148, 149 sila nol 148 sili ca 2 1 sili ca tes 2 1 sili co n ca rbid e fibres 10, 11 , 19,72 rract ure energy 2 1 I mo no filam e nt 10, 11 , 1<) , 2.'i, 2(), X'), I I, 176, 257, 2(,(" 2X7 p" rt ides 2.'i6, 2(, 7 powdc r 2') I st rll CIlIl"e 17 \V lli ' ~l' " I ') , ~ f) , 'i I, ,' ,IX S~ N

Subject index silico n nitride fibres 19 single-fibre pull-o ut test 140 sin gle-fibre pu sh-down test 143 sin gle-fibre push- o ut test 143 sinterin g 288 size (fibre surrace trea tment) 16 ski s 30 I slab model 61 , 69, 241 , 261 , 264 slip band s 228 specific hea t 260 spheruli tes 154 spinn eret 17 spra y depositi o n 283 sputter depositi o n 153 squeeze infiltrati o n 28 1, 304 stackin g sequence 4 3, 44 sta ple fibres 27 sta ple ya rn 46 stereogra m 5 1, 52 stereogra phic proj ecti o n 5 1, 52 stirrness a ni so tro py 2, 11 bulk 74 shear 69, 7 1, 72 sho rt-fibre composites I 15 tenso r 79 , 123 transve rse 130 stir ca stin g 28 1, 282 stra in ba lance 242 stra in energy relea se rate 2 10 stra in tenso r 79 stra inin g electrode tests 234 strength compress ive 23, 178, 18 1, 183 , 184 fibres 22, 26 inte rracial shea r 152 laminates 19 1 lo ng-fibre co mposites 160 matrices 31 shea r 177 ten sile 3, 4 , 12, 226 tran sverse 17 1, 174, 184 va ri a bilit y 27, 29 stress-rree strain 122 stress co rrosion cracking 233 ex po nent 256 intensit y racto r 2 12 rel axa ti o n 23 1, 246, 248, 256 te nso r 79 tran sfe r a spect rati o I11 t r;IIl sll:r k ngt h 109

325

stresses backgro und 126 co ncentra ti o n 163, 165 , 168 , 177, 178,209 co upling 10 1 fibre 11 0, 114 hoop 190, 198, 20 I, 238 image 126 intensit y racto r 2 13, 227 interracia l no rm a l 138 interracia l shea r 108, 109, I 10, 168 interla min a r 98, 194, 195 , 196 mea n 127 ra ti o 227 , 229 residual 3, 7, 32, 144, 255, 256 therm a l 5, 138, 144,237,244, 248,249, 255, 256, 257 thresho ld 227 thro ugh-thi ckness 98, 101 tran sve rse 165 tri ax ia l 2 18 sub-criti ca l c rack growth 226 surrace energy va lues 135, 148 surface treatment o r fibres 16, 147 surfboa rd 301 surviva l pro ba bilit y 27 sy mmetric laminates 101 , 102 symmetrica l matrices 82 symme trica l stackin g sequence 44 sy mmetry S I cubi c S2 o rtho tropic 8 1, 82, 83 ro tati o na l 97 tra nsversely isotro pi c S I, 82 ta ic 2 1 teeth I temperature: dirrusio na l mea n 257, 258 glass transition 34 hea t di storti o n 32, 33 meltin g 33 tenni s racquet 274, 275, 301 tensile strength 3, 4, 12 te n s il e~s h ea r interacti o ns 84, 194 tensio n a nd torsio n (co mbined) 1<)0 tensio ned pu sh-o ut test 14 5 ten so r compli a nce 8 I ide ntit y 128 rank 7<), ID rdati vc di spla ce lll c,,1 7<) st r; ,i" 7')

SlIhj('('/ illdex IL:nsor (i'olllilllll'd) stress 79 stiffness 79, 123 transformed 86, 87, 88 tenso ri a l shear stra in 79 texture 2, 51 thermal conductance 264 co nductivity 3, 4 , 11 , 13 , 3 1, 23 1, 259, 265 , 268, 306, 308 cycl in g 244, 246, 247, 250, 257 diffusivity 260 di sto rti on 263 , 306 expansivity 3, 4, 11 , 13, 15, 3 1, 237 , 240 , 242 , 265 , 306 fati gue 304 resista nce 264 shock 259, 307, 308 sprayin g 28 1, 283, 285 stresses 5, 138, 144, 237 , 249,255 thermopla stics 2, 4, 31, 32, 33, 34, 137, 148, 252 , 271 , 278 thermosets 2, 4, 30, 3 1, 33, 138, 148, 179, 252 , 271 thin-walled tube 190 throu gh-thick ness stresses 98 timber 17 titanium diboride 268 titan ium 34, 35, 72 , 89, 137, 153 , 174, 176, 232, 239 , 257,26 1,267,285,286,300 torque 190 , 19 1 torsion 190, 19 1, 297 toughness 3, 4, 30, 35, 152, 208 transformati on matrix 85 tra nsform ed compli a nce ten so r 86 transformed stiffness tensor 87 transverse: conducti vity 265 cracking 193 , 197 , 20 I, 289 therma l conductivi ty 262 therma l expansivi ty 242 stiffness of fibre co mposite 62 , 130 tra nsversely isotropic sy mmetry 8 1, 82, 83 Tresca yield c riterio n 186 triclinic crystal 82 Tsa i- Hill fa ilure criterio n 187, 188, 190, 199 tungsten 19 turbos tratic graphite 10

twin s 20 ultrasonic sca nning (C-sean) 57, 58 vacuum mo ulding 275 va n der Waal s fo rces 10, 24, 133 vi nyl esters 30, 32 viscoelastic now 138, 20 I , 23 1 voids 55, 56, 17 1, 2 18, 253, 257 voigt model 62 vo n M ises yield crite ri on 186 warp 46 water a bso rpti on 33, 183, 234 weakest link theory 27 wear resista nce 35 weaving 25, 39, 46, 272 weft 46 Weibull mod ulu s 27,28,29, 146, 163, 164, 222 weight fracti o n 39 wetting 133, 134 whiskers 9, 10, 19, 20, 27 width effects 198 winding angle 298 wood 1, 2 fracture energy 2 18 timber 17 tracheids 2 work of a dhesion 134 work o f fracture 2 1 I, 2 17 woven cloth 46, 223 , 230, 279 wove n roving 46, 47, 272, 297 X-ray radiography 48 , 250, 25 1 yield cri teria 186 Young equa ti o n 134 Young's modulus: ax ia l 60, 6 1 anisotropy 2, I I, 16 composi tes 60, 6 1, 62, 6X, X9, I ~O fibres 3, 4, 5, 6,7, 10, 12, 14 , 16,23, lam in ae 89 la min a tes 95, 96 matrices 3 1 tran sve rse 62, 66, (,X

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