Acta metall, mater. Vol. 41, No. 7, pp. 2097-2104, 1993 0956-7151/93 $6.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ,~ 1993 Pergamon Press Ltd

ROLE OF COATINGS IN AXIAL TENSILE STRENGTH OF LONG FIBRE-REINFORCED METAL-MATRIX

COMPOSITES

Z H E N H A I XIA Department of Metallic Materials Science and Engineering, Hebei Institute of Technology,

Tianjin 300130, P.R. China

(Received 15 May 1992; in revised form 22 December 1992)

Abstraet--A model that includes effects of coatings, interfacial behavior, plastic deformation of matrix and reaction zone has been developed for analysis of the behavior of metal-matrix composites reinforced by coated fibres. The stress concentration caused by the cracks in the reaction zone was calculated from the model and the results show that it can be relaxed by choosing coatings of low modulus and controlling interfacial strength to a relatively low value. The predictions of the model in composite strength are consistent with that from the Griffith relationship with strong interfacial bonding and the experimental results for SiC-coated C/AI composites. Both the theory and the experiment show that high strength of the composites can be maintained by SiC coating even when there is a serious reaction at the coating/matrix interface.

Zusammenfassung--Ein Modell, das die Einfl~sse der Schichten, das Grenzflfichenverhaltes, Reaktionzone und der plastischen Verhindern der Matrix behandelt, wird ffir die Mechanischenverbalten-Analysen der Metall-Matrix-Verbundwerkstoffen mit den geschlichteten Fasern beschrieben. Die Spannungskonzentra- tionen in den Fasern, die Risse in der Reaktionzone geherbeifuhrt, wurden berechnet, und die Ergebnisse zeigen, dal3, er wird mit den Schichten niedrigen Moduls und den Faser/Schichte-Grenzfl/ichen relativen niedrigen Festigkeit erchlafft. Die Voraussagen des Modells in der Festigkeit der Verbundwerkstoffen sind konsequent mit Griffith-Relation in der Beschaffenheit der festen Grenfl/ichenfestigkeit und dem experimentellen Material von Aluminium-Matrix-Verbundwerkstoffen mit Siliziumkarbidgeschlichten Kohlefasern.

1. INTRODUCTION

Most metal-matrix composites are non-equilibrium systems. During fabrication and service of the com- posites at high temperature, chemical reaction usually occurs at the fibre/matrix interface, resulting in prop- erty degradation. The chief reason for the degrada- tion is the formation of microcracks in reaction zones which lead to the premature failure of the fibres during loading. The strength of the composites as a function of the thickness of the reaction zones was well predicted by Ochiai [1] and Metcalfe [2].

The problem of interphase reactions and accom- panying property degradation can be rectified by intro- ducing coatings that can reduce the growth of reac- tions between the fibre and the matrix. In addition to serving as diffusion barriers, an important role played by the coatings would be to arrest the impinging cracks and contain the damage [3, 4]. In most coat- ings of current interest, the reaction zone is still formed at the coating/matrix interface due to diffusion through the coatings and/or the reactions between the matrix and the coating. Nevertheless, by means of the crack-arresting characteristic of the coatings, it is possible to prevent the cracks from the partially

reacted outer layer into the fibres by proper choice of the coatings and control of the strength of the inter- face between the fibres and the coatings. However, analysis of the effects of the coatings, interfacial behavior and reaction zone is made difficult by the lack of the methods.

In the present paper a model has been proposed for analyzing the behavior of composites reinforced by the coated fibres. Effects of the coatings, interfacial behavior, matrix yielding and reaction zone on the stress concentrations and strength of the composites are predicted. Comparisons are made between the results from the model and the Griffith relationship and the experimental data on SiC-coated C/A1 composites to examine the validity of the model.

2. DEVELOPMENT OF THE MODEL

We consider a composite consisting of a volume fraction Vf of uniformly-distributed and aligned fibres coated with a coating of thickness h and a reaction zone of thickness R at the matrix/coating interface. The composite is assumed to be composed of aligned cylinders in which a central core of the coated fibre is surrounded by a reaction zone and a

2097

2098 ZHENHAI XIA: ROLE OF COATINGS IN METAL-MATRIX COMPOSITES

Ring \ Matrix yielding crack zone

Fig. I. Schematic representation of the model, a unity cylinder in the composites, consisting of a core of coatcd

fibre, reaction zone and outer matrix layer.

matrix layer, as shown schematically in Fig. 1. The thickness of the matrix layer is given by

df ]/2 _ 1) -- h. (1) 6 = ~-(v?

We assume that thcrc is a ring crack in the reaction zone. As the cylinder is subjcct to an overall stress to the cylindrical axis, the crack would result in stress concentrations depending on the thickness of thc reaction zone, propcrties of the coating and matrix, and intcrfacial bonding strength. Wc assumc that shear stresses in the fibre and rcaction zone can be neglected and the matrix and coating play a role only as the strcss-transfcr media. With this ncglcction, thc displacements of thc fibre and reaction laycr could be given as a function only of x [5], the axial coordinate parallel to the fibre. The effect of thermal residual stress is also neglcctcd in the analysis.

Because of largc clastic shear stress concentrations in the matrix and the coating at thc root of the crack and relatively weak interracial strength and matrix strength of most compositc materials we anticipate that thcrc will be nonelastic effects such as interracial debonding and plastic dcformation in the matrix at the root of the crack. Wc therefore include in the model three regions: Region A where intcrfacial debonding occurs and the shear stress in thc matrix is higher than its shear yicld strcngth, Region C whcrc neither interracial debonding nor plastic deformation of the matrix occurs, and Rcgion B where there arc two cases: one (Casc I) that intcrfacial dcbonding occurs but the matrix does not yield and thc other (Case II) that the matrix yields but no dcbonding occurs. Noting that axial dimensions of the interracial debonding and matrix yielding zones as a and b, respectively, Region A covers the region of 0 ~< x ~< a (a < b) or 0 ~< x ~< b (a > b), Region B thc region of a <~x <~b(a < b) or b <~x <~a(a > b)and Region C the region of b ~< x (a < b) or a ~< x (a > b). In regions where the interfacial debonding occurs, the

shear stress in coating is assumed to be constant and equal to zs. In the stage of elastic shear deformation of the matrix, its shear stress is given by

Xm = Mm Y (2)

and in the stage of plastic shear deformation it is given by

"r m = flMmy + (1 -- fl)Zy (3)

where fl is the slope of the shear strain stress-strain curve in plastic deformation, normalized with respect to Mm. The coating deforms only elasticly in shear before fracture.

Noting the displacements of the fibre Uf and the reaction layer Ur, the shear stresses of the coating and matrix are given by

Z c = (U r -- uf)nc/h (4)

"rm=CUr--~ffx)Mm/~ (~'m ~ "Cy) (5)

( - f o ~ ) ~m; U,---~x Mm~/~+(l--~)~, (~m>~,) (6)

and the stress concentration factor for the fibre are given by

N =dUf dx ' (7)

Combining (1)-(6), we have the equations of equilibrium as follows:

For Region A

S _ d2Ur . f/$f " ~ -1-/taf'~ s = 0 (8)

d2Ur SrEr"~x2 -- ~(df+ 2h)z, -- rg(df--l- 2h + 2R)

x [ - ~ - ~ ( U r -af t° "~ -~f x ) + ( 1 - f l ) z y ] = 0 . (9)

For Case I of Region B

S _ d2Uf f/£f-~-]-+ Itdf'~ s ---- 0 (10)

d2Ur S~E,---~T -- lt(df + 2h)*,

n(df+ 2 h + 2R)Mm (~f~o x ) + 6 \El -Ur =o. (ii)

For Case II of Region B

Sf_ d2Uf . rcdrMo /~ f . -~ .t- T (U, -- Vf) = 0 (12)

SrErd~--~ ~ n(df+ 2h)Mc Uf)

._ . f fM.{ . af~

+(1 - fl)Zy} = 0. (13)

ZHENHAI XIA: ROLE OF COATINGS IN METAL-MATRIX COMPOSITES 2099

For Region C

d2Uf Me S f E f - - ~ + x d f - - - ~ - (O r - Uf) = 0 (14)

d2U~ 2h) ~ ( U r - - U r ) SrEr -d-~x2 - ~r (dr +

+ I r ( 4 + 2h + 2R) (~--~ ) X - - U r M m = 0 . ( 1 5 )

The solutions of equations (8) and (9) for Region A are

A 2z, Ur =d--~rx2 + Al x + A2 (16)

UO = A 3 e x p ( - x /~ tx)

+ A4 exp(x//-~tx) + ~-~ x

~(df + 2h)x s (1 -- fl)6 flMm(df+ 2h + 2R) flMm ry (17)

where

=V.~Mm(df+ 2k + 2 R ) ] '/2. t L SrE, 6

The solutions of equations (10) and (1 l) for Case I of Region B are

2 z s 2 U ~ = d ~ X + n i x + n 2 (18)

U~ = B 3 e x p ( - tx) + B4 exp(tx)

6(dr + 2h)% (19) + x - M m ( d r + 2 h + 2 R ) "

The solutions of (12) and (13) for Case II of Region of B are

U~ = B~ exp(--01x) + B~ e x p ( - 02x)

+ B~ exp(01 x) + B~ exp(02x)

+ f x t i M m (20)

U~ = B~ W l exp(--01 x) + B~ W 2 exp ( -02x )

+ B~ WI exp(01x) + B~ W 2 exp(02x)

af~ (1 - f l) '~y (21) + --~f x f lM~

where

nclfM~ n(d r + 2h)Mc P = S - - - ~ ' q = S ,E,h

Ol = {p + q + fit 2 + [(P + q + fit2) 2 - 4flpt2]m}/2

02 = {p + q + fit 2 - [(p + q + fit2) 2 - 4flpt2]l/2}/2

W , = 1 + ( f i t 2 - - 02)/q

W 2 = 1 + ( f i t 2 - - 0 2 ) / q .

The solutions of equations (14) and (15) for Region C are

U c = Cl VI exp(-- ~q x) + C 2 V2 exp( -~2x )

+ C 3 V1 exp(al x) + C4 V2 exp(a2x)

-{- O'f°° X (22) Ef

U c = Ci exp(-- CtlX) + C2 exp ( - a2x )

+ C3 exp(~l x) + C4 exp(a2x) + trr~ x (23) er

where

V 1 = 1 + (t 2 -- ot2)/q

V2 = 1 + (t 2 -- ot~)/q

a, = {p + q + t 2 + [(P + q + t2) 2 _ 4pt2]'/2}/2

~t 2 = {p + q + t 2 _ [(p + q + t2) 2 _ 4pt2]m}/2.

The boundary conditions are

dUe" x = 0 , Uf A=0 , - - = 0 (24)

dx

As dU c dU c af~

x --, oo, dx dx Ef (25)

We also require that displacements and forces be continuous at Boundaries A-B and B-C, i.e.

For Case I (a > b), at x : a ,

U B = U c, U~ = UCr, dU~/dx = dUC/dx

and dUBr/dx=dUC/dx, (26)

and at x = b,

U ~ = ~ A Uf, Ur = UBr, dUk /dx = dU~/dx

and d U g / d x = d U ~ / d x . (27)

For Case II (a < b). at x = a ,

U A = U~, U A = UBr, dUA/dx = dU~/dx

and d U A / d x = d U ~ / d x , (28)

and at x = b,

U~ = U c, Ur _ U c, dU~/dx = dU~/dx

and d U ~ / d x = d U C / d x . (29)

In addition, the shear stresses in the coating at x = a and in the matrix at x = b should be equal to the interfacial failure strength in shear and matrix yielding strength, respectively

a t x = a ,

at x = b,

h U~ - Ur a = - - zi (30) Mc

6 b vp = - - ~y (31) + -~- af~.

Mm ~r

2100 ZHENHAI XIA: ROLE OF COATINGS IN METAL-MATRIX COMPOSITES

These boundary conditions are used to determine the integration constants in equations (16)-(23).

The calculations of stress concentrations and stress distributions were carried out on an IBM AT personal computer using properties of the constituent phases based on SiC-coated C/A1 composites listed in Table 1.

3. RESULTS AND DISCUSSION

3. I. Stress states during loading

At low stress levels, only Region C where Zm < Zy and z¢ < zi exists. With increasing stress level, Region B arises and grows. Whether Case I or II occurs in the region is dependent of ra and Zy and stresses in the coating and matrix. When stress in the coating at x = 0 is the first to reach z i, Case I arises; otherwise, Case II occurs. With further increasing stress level, Region A arises, where both interracial debonding and matrix yielding takes place.

Figure 2 shows the effects of interfacial strength and thickness of the reaction zone on the regions during loading. At the low interracial strength, interfacial debonding occurs before matrix yielding while a large interfacial strength leads to yielding before debonding, Fig. 2(a). With increasing the interfacial strength, the critical stresses resulting in debonding with yielding increase but the critical yielding stress becomes constant when yielding takes place before debonding.

(a)

2.01.6 ---- R = 1 ~tm A ~ ~ _ _ _ _ B - 1 I _

1.2 -- / ' " " / /

C

~" / / Debonding ~" , / / / Yielding

0.4 ; / D

e, 0 400 800 1200

Interfacial strength (MPa)

Z (b) > 2.0 --

"~i = 140 MPa

1.2

0.8

_

0.4 -- Region C t

0 0.5 1.0 1.5

Thickness of reaction zone 0xm)

Fig. 2. Stress states of the composites as functions of (a) interfacial bonding strength and (b) thickness of reaction

zone.

7

E P

P

~q

z

1.5

1,0

0.5

R = 0.5 ktm 1 of~ = 740 MPa x i = 300 MPa 2 of~ = 2000 MPa

~ ~ 2

\.1 N

_ 2 -~I~ \ Xc/'r i

~. 1 I '~X\ - - - Xm/~Y j l \ , ,

\

- i j ?" -".l i i 0 4 8 12 16 20 24

x 0tm)

Fig. 3. Variations of N, ~¢/~i and Tm/Ty as a function of x.

With increasing the thickness R, debonding and yielding occur at lower stress level at any interfacial strength, Fig. 2(b).

3.2. Stress concentrations

Figure 3 shows the stress concentration factors for fibre N, coating ~c/Z~ and matrix Zm/"Cy as a function of x at the given stress levels. It can be seen that at low stress level (Region C and Case II of Region B), the factors of N, rc/Ti and 17m/Ty have the maximum values at x = 0 and decrease with an increase of x. When increasing stress leads to interfacial debonding, with an increase of x, the value of N increases up to a maximum value at a point of x = a, and beyond the point it decreases quickly. The value of z¢/zi is equal to rs/Z~ in the region of debonding and beyond this region it decreases from one to zero while 27rn/'l~y decreases from the maximum value at x = 0 to zero.

It is of interest to determine the locations and values of the maximum stress concentrations because they influence the fracture mechanisms and strength of the composites. The maximum stress concen- tration in fibres is located at x = a. Figure 4 shows

30

of~ = 2000 MPa

}20

~ lo

R = 0 . 5 gm

0 500 1000

Interfacial strength xi (MPa)

Fig. 4. Debonding length as a function of interfacial strength.

ZHENHAI XIA: ROLE OF COATINGS IN METAL-MATRIX COMPOSITES 2101

16 I 1.5

1.4

1.3 Z

1.2

1.1

1.0

o f ~ = 2 0 0 0 M P a /

R = l~tm / / \ ~ _ / e

_ Debonding ~ , , region / / / /

-- / / / / 3 = 0 . 5 / / /

_ R = 0 . 2 Nd/'/,'f"" Non-debonding

_ ~ " region

~ ' 1 I I I 400 800 1200 1600

lnterfacial strength x i (MPa)

Fig. 5. Maximum stress concentration factor for fibres N=~ as a function of interfacial strength.

the interfacial debonding length a as a function of 17 i. As zi--~ 0, the length becomes infinite and with an increase of r i, it decreases up to zero. This implys that with a strong interfacial bonding, the composites would fail in a brittle manner while a weak interfacial bonding results in debonding and fibre-pullout in the fracture process of the composites.

The maximum stress concentration factor for fibres Nm~ as functions of stress level and interfacial bond- ing strength was calculated and the results show that it remains constant during the initial loading. After the interfacial debonding takes place, Nm~ ~ decreases with an increase of the stress level. If the matrix yields before debonding, Nmax increases with increasing stress after yielding till the debonding occurs. This indicates that the interfacial debonding can relax the stress concentration in fibres and the matrix yielding increases it.

Figure 5 shows the effects of the interfacial bonding strength and thickness of the reaction zone on Nm~. With an increase of interfacial bonding strength, Nm~, increases up to the maximum value at any h, while increasing h leads to an increase of Nm~ x at any z~. Therefore, decreasing the interfacial strength

1.3 m

of® = 2000 MPa - - - h = 2.0 p,m R=0.5 p,m - - h= 1.21xm

1.2 - - - - ~ - . . . . . - -

z° 1.1

1.0 I t I 50 100 150

Coating modulus M c (GPa)

Fig. 6. Maximum stress concentration factor for fibres as a function of coating shear modulus.

can effectively decrease the stress concentration in fibres. However, too low a strength can cause too long debonding length which decreases the efficiency of fibre-reinforcement.

Figure 6 shows the effects of the shear modulus and thickness of the coating on Nmax. With an increase of Me, Nma ~ decreases for a low interfacial strength, and it increases in the case of strong interfacial bonding. For a middle interfacial bonding, Nma ~ increases with increasing of Me up to a maximum value and beyond this point it decreases. Therefore, decreasing the shear modulus of the coating, especially in the range of Me < 70 GPa, is of benefit in relaxing the stress concentration in fibers.

4. VERIFICATION OF THE MODEL

4.1. Comparison with the results from the Griffith relationship

In principle, the failure of the fibre coated with the partially consumed coatings can be predicted from an energetic standpoint, as similarly as in the analysis of [1]. The energy required to break the coated fibres at the root of the cracks in the reaction zone would be GfSf+ GcSc. When the coating/fibre interracial bonding is not strong enough to prevent interracial debonding, other energy-absorbing mechanisms such as debonding, pullout etc. take place in the fracture process. Therefore, the total energy absorbed is not necessarily related to the energy of (GfSf+ GcS¢). Following this line of reasoning the critical parameter that defines the onset of the fibre fracture is stress in the fibres.

It has been shown that the cracks in the reaction zone result in stress concentration in fibres. This will lead to the premature failure of the fibres during load- ing, decreasing the composite strength. According to the criterion of the maximum stress, we calculated the fibre failure strength as a function of thickness of

, ~ 1.2 ~- - - - From Griffith relationship From present model

~ 0.8 x i = 100 MPa

.~'m~ ~ "~ca 0.40'6 i \" ~ ~N°n-deb°nding~

,-[~ 0.2 . . . . . . . . . . 0 z

I I I 0 1 2 3

Thickness of reaction zone R(gm)

Fig. 7. Comparison of relations between normalized fibre failure strength O'f/O'~ and thickness of reaction zone predicted by present model and the Griffith relationship.

2102 ZHENHAI XIA: ROLE OF COATINGS IN METAL-MATRIX COMPOSITES

Table 1. Properties of the constituent phases used in calculations

Phase Parameter Value Reference

Fibre Ef 390 GPa Gf 3.98 J/rn 2 I T] df 6 ,urn [111 a ~' 2200 MPa vf 0.4

Matrix zy 40 MPa fl 0.1 [61 M m 27 GPa [8]

(SIC) coating E~ 360 GPa Me 150 GPa G¢ 14.5 J/m 2 [9] h 1.2#m

Reaction layer (AI4Ca) Er 297.5 GPa [81

Interface z s 10 MPa [10]

the reaction zone, as shown in Fig. 7. The strength decreases with increasing the thickness at any xi. With a low interfacial bonding strength, high fibre failure strength results. When the fibre/ coating interface bonding is strong enough to avoid debonding, the fibre reaches its lowest value in strength.

In the case of non-debonding, the fibre strength can be simply predicted from the Grittith relation- ship

af = (32)

where

E = (ErSf + EcSc)/(Sf + Sc) (33)

G = (GrSf + GcSc)/(Sr+ S¢). (34)

Numerical constants used in the calculations are shown in Table 1. The normalized fibre failure

strength as a function of R is also shown in Fig. 7 and is seen to agree with that from the model under the same condition.

4.2. Comparison with the experimental results for SiC-coated C/AI composites

Xia et al. [12] studied the behavior of A356 matrix composites reinforced by PAN-I carbon fibres coated with a duplex-coating of 0.4 #m nickel and 1.2 #m silicon carbide. Figure 8 shows the measured tensile strength of the composites as a function of thickness of the reaction zone. They found that three regions are included: Region I where the composites have high strength with a duplex pullout fracture mode in which fibres are pulled out from the coatings and the coatings are pulled out from the matrix, Region II where the strength decreases with increasing thickness of the reaction zone and only fibre-pullout from the coatings is observed in the fracture surfaces, and Region III where the strength is very low and the fracture is brittle.

According to the model, which of fracture mech- anisms acts is dependent of the interfacial bonding strength and the strength of the composites is deter- mined by the stress concentration in fibres. With a low zi, interfacial debonding may occur. Because the maximum stress concentration in fibres is located at x = a, the fibres would fail at this point, forming the pullout fracture surfaces in the fracture process of the composites. When the interfacial bonding is strong enough to avoid debonding during loading, the cracks in the reaction zone directly propagate through the fibre/coating interface into the fibres to form the planar fracture surfaces because the maxi- mum stress concentration in fibres is located at the root of the cracks.

I 1 . 2 - - Ii \

1.0 I \ Present model Io Experimental data [12]

"~= 0.8

o / . ! ~ 0.6 - -

"~ ~ I i II

~ 0.4

1,7-) = 0.2 --

I I t I I 0 0.2 0.4 0.6 0.8 1.0

R e a c t i o n z o n e t h i c k n e s s to c o a t i n g t h i c k n e s s ( R / h )

Fig. 8. Comparisons between normalized measured composite strength a~/ROM as a function of reaction-zone thickness to coating thickness and that predicted by the present model and the Griflith relationship. I = R e g i o n , I , II = Region If, and III = Region III. ROM = tensile strength calculated f r o m

equation (35) at crf= o'~'.

ZHENHAI XIA: ROLE OF COATINGS IN METAL-MATRIX COMPOSITES 2103

The s t rength of the composi tes as a funct ion of the thickness of the react ion zone has been calculated f rom the model by applying the rule of mixtures

O'cu = O'f Vf-Jt - ¢rm(1 - - V f ) . ( 3 5 )

In the calculat ion, the bond ing s t rength of the fibre/ coat ing interface is assumed to be cons tan t and equal to 140 M P a [11] in Region I, and in Region II, due to diffusion and reaction, it increases linearly f rom 140 M P a to a value at which the interfacial debonding is jus t avoided. The predicted results show in Fig. 8 indicate a m a x i m u m disagreement of abou t 20% in the range of compar ison , which is reasonable.

Next, the results calculated f rom equat ions (32) and (35) were compared with the exper imental data. Only in Region III do the calculated results agree with the exper imenta l data.

Noticeably, the results br ing to l ight an i m p o r t a n t fact tha t high s t rength of the composi tes is main- ta ined by SiC coat ing even when there is a serious interracial react ion between the mat r ix and the coat- ing. This role of the coat ing has been well predicted by the present model.

5. CONCLUSIONS

A model has been developed for analysis of the effects of coating, plastic de fo rmat ion o f the matrix, interracial behav ior and react ion zone in tensile behav ior of f ibre-reinforced composites. The results of the present analysis are consis tent with those ob ta ined f rom the analysis based on the Griffith relat ionship with s t rong interface bonding. The pre- dict ions of the model agree with the exper imental da ta on SiC-coated C/AI composites. Both the theory and exper iment show tha t h igh s t rength of C/AI composi tes is ma in ta ined by SiC coat ing even there is a serious react ion at the coa t ing /mat r ix interface. The ma in results f rom the model are as follows:

(1) The f ibre/coat ing interracial debond ing relaxes the stress concen t ra t ion in fibres while the mat r ix yielding increases it.

(2) The lower the coat ing/f ibre interfacial bond ing s t rength is the lower the stress concen t ra t ion in fibres becomes, bu t too low an interracial s t rength decreases the efficiency o f f ibre-reinforcement.

(3) The tensile s t rength of the composi tes can be improved by choos ing the coat ings of low shear modu lus and control l ing the coat ing/f ibre interracial bond ing s t rength to a relatively low value.

REFERENCES

1. S. Ochiai and Murakami, J. Mater. Sci. 14, 831 (1979). 2. A. G. Metcalfe and M. J. Klein, Composite Materials,

Vol. 1 (edited by A. G. Metcalfe). Academic Press, New York (1974).

3. T. Erlurk, V. Gupta, A. S. Argon and J. A. Cornie, ICCM-VI, Vol. 2, pp. 2. 156-2. 160 (1987).

4. R. R. Kiechke and T. W. Clyne, Mater. Sci. Technol. 6, 145 (1990).

5. C. Zweben, Engng Fract. Mech. 6, 1 (1974).

6. S. Ochiai and K. Osamura, Metall. Trans. 19A, 1491 (1988).

7. D. D. Hinbeault, R. A. Varin and K. Piekarski, Metall. Trans. 20A, 165 (1989).

8. H. Nayeb-Hashemi and J. Seyyedi, Metall. Trans. 20A, 727 (1989).

9. I. Merkel and U. Messerschmidt, Mater. Sci. Engng 151, 131 (1992).

10. W. H. Hunt Jr, Interfaces in Metal-Matrix Composites (edited by A. K. Dhingra and S. G. Fishman), pp. 3-26. TMS (1986).

11. Zhenhai Xia, Acta Mater. Comp. Sinica 10, No. 1 (1993).

12. Zhenhai Xia, Yaohe Zhou and Zhiying Mao, Mater. Sci. Prog. 5, 448 (1991).

A P P E N D I X Nomenclature

a = interracial debonding length A = region of debonding and yielding

A t , A 2 , A 3 , A 4 = integration constants b = matrix yielding zone width B = region of debonding or yielding

Bl, B2, B3, B 4 = integration constants B~, B~, B], B~ = integration constants

C = region of neither debonding nor yielding Cj, C 2, C 3 , C 4 = integration constants

df = fibre diameter E c = Young's modulus of coating Ef = Young's modulus of fibre E r = Young's modulus of reaction layer

E m = Young's modulus of matrix G c = critical strain energy release rate for the

coating Gf = critical strain energy release rate for the

fibre h = thickness of coating

M c = shear modulus of coating M m = shear modulus of matrix

N = stress concentration factor for fibre N ~ = maximum fibre stress concentration factor

R = thickness of reaction layer ROM = composite tensile strength calculated from

rule of mixtures as trf = a? Sc = cross-sectional area of coating Sf = cross-sectional area of fibre Sr = cross-sectional area of reaction layer Uf = displacement of fibre Ur = displacement of reaction layer Vf = fibre volume fraction x = coordinate parallel to fibre /] = slope of matrix shear stress-strain curve

after yielding r = shear strain of matrix 6 = thickness of matrix layer in a unity cylinder

af~ = stress in fibre at x = oo af = fibre failure strength

a~' = fibre strength a m = matrix stress at x = ~ when fibre failure

occurs a~u = composite strength Zm = shear stress in matrix % = shear stress in coating • y = yield strength of matrix ~s = frictional shear stress at interface post-

debonding z i = interracial bonding shear strength

Super scip t :

A = Region A B = Region B C = Region C

A M 41/7~L

2104 Z H E N H A I XIA: R O L E O F C O A T I N G S I N M E T A L - M A T R I X C O M P O S I T E S

Subscript: c = coa t ing f = fibre

m = mat r ix r = react ion p roduc t i = interface