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Effects of fibre volume fraction on the stress transfer in fibre pull-out tests
E f f e c t s of f i b r e v o l u m e f r a c t i o n on t h e stress t r a n s f e r in f i b r e pull - o u t tests JANG-KYO KIM; LIMIN ZHOUt, S.J. BRYAN t and YIU-WlNG MAI t ('Australian National Universityf University of Sydney, Australia) Received 11 October 1993; revised 2 December 1993
The elastic stress transfer taking place across the fibre/matrix interface is analysed for the fibre pull-out test by means of both micromechanics and finite element (FE) analyses. A special focus has been placed on how fibre volume fraction, Vf, affects the interface shear stress fields in the model composites containing both single and multiple fibres. In a so-called 'three-cylinder model', where a fibre, a matrix and a composite medium are coaxially located, the constraint imposed on the central fibre due to the surrounding fibres is properly evaluated. It is shown in the FE analysis that the differences in stress distributions between the composite models containing single and multiple fibres become increasingly prominent with increasing Vf. The principal effect of the presence of surrounding fibres in the multiple-fibre composite model is to suppress effectively the development of stress concentration near the embedded fibre end and thus eliminate the possibility of debond initiation from this region for all Vf considered. This is in sharp contrast to the single-fibre composite model, in whichthe interfacial debond can propagate from the embedded end if Vf is larger than a critical value. These findings are essentially consistent with the results from micromechanics analysis on the same specimen geometry. The implications of the results for the practical fibre pull-out test as a means of measuring the interface properties are discussed. Key w o r d s : fibre/matrix interface; fibre pull-out test; single-fibre composite model; three-cylinder composite model; finite element analysis Phenomena of vital importance in fibre composite technology are the stress transfer across the fibre/matrix interface and the interface debond process, particularly in the neighbourhood of a fibre break or matrix crack, Theoretical analyses dealing with the stress states and the debond propagation in various loading geometries are essential to understanding how and to what extent the interface properties influence the mechanical/structural performance and fracture behaviour of the composites 1. A significant effort has been put into proper characterization of the interface properties, and a number of experimental techniques have been devised to evaluate the interactions at the interface region which include the physicochemical analyses of the surface properties, optical/electron microscopies of various kinds, mechanical testing and non-destructive testing. Among the various mechanical testing techniques the fibre pull-out test has become one of the most popular single-fibre composite testsL From both the stress-transfer and fracture mechanics viewpoints, improved micromechanics analyses3-5 have been developed for the single-fibre pull-out test, taking
into account the several different boundary conditions and the instability criteria during the debond process. The theory compares favourably with the experimental data for several different composite systems containing polymer and ceramic matrices and some cementitious composites6. In addition to the analytical micromechanics models, a finite element method has been successfully adopted to investigate the effects of the compliant fibre coating on the stress distributions before and during the debond process in the fibre pull-out test 7. In the light of the foregoing studies, it is identified that most micromechanics theories are developed based on a shear lag model of single-fibre composites where the external stress is applied to the fibre end with either of the matrix top or bottom being fixed, while the matrix cylindrical surface is invariably assumed to be stress-free (Fig. l(a)). This assumption leads to an unacceptably high applied stress required to initiate/propagate interface debonding when the radial dimension of the matrix is similar to that of the fibre (i.e. for a high fibre volume fraction, Vf). In other words, the application of the conventional models to practical composites is limited to
0010-436119410710470-06 O 1994 Butterworth-Heinemann ktd 470
COMPOSITES. VOLUME 25. NUMBER 7. 1994
a
d
b
d
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~
~"
Fibre Matrix
>>
L
I 1 z
[
.-T';'," 7 ,:-.; ",,'-7- :.. I,"l I /' .// /, J. i / / /I / ' / . ~
i I1,"
i
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, ,_.. _, , / / _' . 7 ~--,-,s / is'-. - ,~ "-."'
•
. . . .
Composite
.
: .... :
medium
....
' I-i-" ............ ~
~
- - i--4
b
2B Fig. 1 Schematic illustrations of (a) the single-fibre pull-out test, (b) the fibre pull-out test of a three-cylinder composite and (c) the finite element model of a three-cylinder composite
those with a very small Vr where any effects of interactions between neighbouring fibres are completely neglected. In the present study, as a part of a continuing project on composite interfaces, a three-cylinder composite model is developed to accommodate properly the limitation of the shear lag model of the single-fibre pullout test. The micromechanics analysis and finite element method are employed in parallel to validate the results obtained from each model,
MICROMECHANICS THREE- CYLINDER
ANAL YSIS OF A COMPOSITE
_
dz
(1)
(2)
where y = a2/(b 2 - a 2) and 7~ = b2/( B2 - b2). B is the outer radius of the composite medium (Fig. l(b)). The superscripts represent the coordinate directions, and the subscripts f, m and c refer to fibre, matrix and composite medium, respectively. Equilibria of the interface shear stresses, the shear tractions on the cylindrical surface and the axial stresses in the constituents require d~r(Z)
To analyse the stress transfer in the fibre pull-out test of a multiple-fibre composite, the specimen is treated as a three-cylinder composite where a fibre is located at the centre of a coaxial shell of the matrix which in turn is surrounded by a trans-isotropic composite medium with an outer radius B (Fig. l(b)). The fibre and matrix are assumed to b e perfectly elastic and isotropic, and their radii a and b are related to the fibre volume fraction Vf = a2/b% which is the same as that of the composite medium, Here z is the direction parallel to the fibre axis, and r and 0 respectively are the radial and tangential coordinates, When the fibre is subjected to an external stress, or, at the loaded end (z = 0), while the matrix and composite medium are fixed at the embedded end (z = L), stress transfers from the fibre to the matrix and in turn from the matrix to the composite medium via the interface shear stresses, r~(z) and rb(Z), respectively. Following the previous analyses for the fibre pull-out test of a singlefibre composite 3,4 and for the fibre push-out test of a multiple-fibre composite 8, the relations between the interface shear stresses, r~ and rb, and the axial stresses, o-~f, O'=mand ~ , respectively in the fibre, matrix and composite medium, are obtained. The mechanical equilibrium conditions between the external and internal stresses require cr = ~ ( z ) + 17/o-~(z) + V r - - ~ ( z )
_ 2b fz 1 ~m(Z) a2 J0 rb(z) dz --- ~ ( z ) +
cr
2
r,(z)
(3)
a
dcr~(z) _ a27/[ara(z) 2 dz - brh(z)]
(4)
Based on the Lame solution, the relationship between the interface shear stresses is taken as rb(z) = b7/2 r,(z) (5) aT/~ where 7/2 = a 2 / ( B 2 a 2 ) • The additional radial stress, q,(z), acting at the fibre/matrix interface, which is caused by Poisson contraction of the fibre when subjected to an axial tension, is obtained from the continuity of the tangential strain at the interface -
q,,(z) = - o ~ v f V f k l o Z f ( 2 ) + [ v m V f k l -- 27/(Vm -- C~Vc)](Ym(Z)
--
Vrkt[0~(1 - vr) + 1 + Vm + 27/] -- 47/2
(6)
where ~ = Em/Er, oq = Em/Ec and k~ = 1 + 27 - Vm + ~l(1 + 27/~ + vc). E and v are Young's modulus and Poisson's ratio, respectively. Combining Equations (1)(6) yields a second-order differential equation for the fibre axial stress d2o.~(z) dz 2 A3cr~(z)= A4cr (7) The coefficients A3 and A4 are complex functions of the entselasticproperties and geometric factors of the constitu-
COMPOSITES.
NUMBER
7.
1994
471
~(z)_ a
-- ?'Vm(?'2k4 -- 2~Vf?')] (8) 2fl { ~ } A4 = ~ ?'2 1 [?'vmk4 - (0~vf q-- ?'Vm)Vfk2] (9) where ?'Em
(10)
Vm)[2?'(B/a)21n(b/a)-
fl = a?'2(1 +
k2=~r{Vrk,[o:(2-
vf)+ 1 + 2 ? ' +
1]
aN/~l 2
sinh(x/~lL )
(17) The coefficients A1 and A2 are given by A, = 2[~(1 - 2kvr) + ?'(1 - 2kvm)] (1 + Vm)[2?'Mln(b/a)- a 2]
(18)
A2 _ 7"(1 -- 2kvm) (19) A, ~1 - 2kvr) + 7(1 - 2kvm) where k = (~vr + ?'Vm)/[~(1 -- Vf) + 1 + Vm + 2?'].
Vm] -- 4?' 2} FINITE ELEMENT (FE) ANAL YSIS
2?'
k 3 = ~ff
(V m --
k 4 = (vm -
vmk 1
0~1Vc ) - -
0q Vc)[~(l -
vf) + 1 + 2?' + Vm] -- 2?'Vm
The solution for Equation (7) is subjected to the bounddry conditions o'er(0) = cr and o'~f(L) = 0, assuming an unbonded cross-section for the embedded fibre end. Therefore, the solutions for the fibre axial stress, cry(z), matrix axial stress, o-~(z), and interface shear stresses, ra(Z) and rb(Z), normalized with the applied stress o-, are obtained
(A4+ )
Recognizing the significance of the stress concentration at the broken fibre ends, increasing efforts are being directed towards employing numerical methods. The use of the finite element method (FEM) in particular allows a more accurate description of the interactions between neighbouring fibres in practical composites containing fibres of large volume fraction, and especially of the interface shear stress fields near the singularity. In addition, the specific end geometries as well as the continuously varying mechanical properties across the interface region of finite volume (i.e. the interphase) can be properly taken into account.
\
o-~(z) _ ~A3 1 sinh[x~3(L-z)] + osinh(x~3L)
~
sinh(x/~3z )
A__Z" A3
(11) ( A4) - ?'2,1 + A33,
o'Z(Z)
cr
A4 + 1) s i n h [ ~ 3 ( L - z)] + - ?'2 (A33
sinh ( ~ 3 z )
sinh(~3L) (12) r,(z)
a~3
(A4+) ~ ~A3 llcosh[x/~3(L-z)]-..3 o s h ( ~ 3 z )
-
cr
2
sinh(x/~3L ) (13)
rb(Z)
_
cr
b?'2xA/-~33(A~ + 27'1
1)c°sh[x/~A3(L-z)]-~3sh(x~33z) sinh(x~3L) (14)
The corresponding equations for the single-fibre composite model are taken from an earlier analysis 3 __ (A~ a~(z) = +1 sinh[x/r~(L-z)] + ~sinh(x/-~z) A2 cr sinh(x/r~L) -41 (15)
a~(z) t7
I `42 + 1)sinh[x/~l(L- z)] + ~ s i n h ( x ~ V ) ~A1
Az -- ? ' Z - -
?'
sinh(~L) (16)
472
COMPOSITES
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7 . 1994
Both the composites containing single and multiple fibres are considered for the present FE analysis. The geometry, the loading method and the boundary conditions are selected to represent those of the actual experimental technique (Fig. l(c)), which are analogous to those used in the corresponding micromechanics analyses. A mesh has been created of the isometric eightnoded quadrilateral elements for the axisymmetric loading geometry of a two-dimensional model using the FE program 'Strand 6 '7. The total number of elements is 480, with 20 and 24 elements in the longitudinal and lateral (or radial) directions, respectively. This ensures sufficient resolution and thus accuracy of the results while maintaining a reasonable time needed for computation. A uniformly distributed constant stress, a = 100 MPa, is applied to the partially embedded fibre at the surface (z = 0). The boundary conditions are imposed such that the bottom surfaces of the matrix and composite medium are fixed at z = 2L, and the axis of symmetry (r = 0) is fixed where there is no displacement taking place. Therefore, the global default freedom is set to allow displacemerits for 0 < z < 2L and 0 < r < B in the axial and radial directions, respectively. Specific results are calculated for an SiC fibre glass matrix composite 3 with elastic constants Ef = 400 GPa, E m = 70 GPa, vf = 0.17 and Vm = 0.2. A constant embedded fibre length L = 2.0 mm and constant radii a = 0.2 and B = 2.0 considered with matrixmmradiusb. All stressmmarevalues reported hereinVaryingare normalized with the external stress, or, and negative signs for the interface shear stress are omitted for simplicity. RESULTS
The stress distributions along the axial direction shown in Fig. 2 are obtained from the FE analysis for the com-
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....
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Fibre V o l u m e Fraction, Vf Fig. 3 Interface shear stress distributions as a function of fibre volume fraction, Vf, obtained from FEM calculations: (©, -) single-fibre composite; ( A , A, - - ) three-cylinder composite model O ,
~
0.3
,
,
,
,
,
,
~o 0.2 ~ ~:
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. ~
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-
-
b
.~
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-
/',,
0 i 0
"~
~
Vf = 0.03 /
".. ~ ~ ~..... i ~ i ~ . , 0.2 0.4 0.6 0.8 Axial Distance, z/L
1.0
Fig. 2 (a) Fibre axial stress and (b) interface shear stress distributions along the axial direction obtained from FEM calculations for two fibre volume fractions, Vf= 0.03 and 0.6: ( . . . . . . ) singlefibre composite; ( ) three-cylinder composite model
nally with Vr, and the magnitude of the 1ss at the loaded end is always greater than that at the embedded fibre end. This ensures that, if the fibre is loaded continuously, debonding is always expected to initiate at the loaded fibre end for all Vf, based on the shear strength criterion of interface debonding. However, for the single-fibre composite model, the ISS at the embedded fibre end increases rapidly, whereas that obtained at the loaded fibre end decreases with increasing Vr. Therefore, there is a critical fibre volume fraction above which the magnitude of the former stress exceeds that of the latter, leading to debond initiation from the embedded fibre end in preference to the loaded fibre end. The approximate critical value Vf ~ 0.26 is determined based on the superimposed curves of the data points in Fig. 3. The predicted results based on the micromechanics
posites with two extremes of fibre volume fraction, Vr = 0.03 and 0.6. The three-cylinder composite model predicts that both fibre axial stress (FAS) and interface shear stress (ISS) decrease from a maximum near the loaded fibre end towards zero at the embedded fibre end. An increase in Vf (and thus the stiffness of the composite medium) slightly increases both the maximum ISS and the stress gradient, without altering the general trend of the stress fields. For a small Vr, the stress distributions in the single-fibre composite model are equivalent to those of the three-cylinder model. In sharp contrast, the stress fields change drastically in the single-fibre composite model when Vf is large, such that the fibre axial stress does not diminish to zero at the embedded fibre end and the stresses in the central portion of the fibre length are almost constant. More importantly, the ISS displays two peaks at the ends of the fibre, the one at the embedded end being greater than the other at the loaded end for Vr = 0.6 (Fig. 2(b)).
analysis are essentially similar to those obtained in the FE analysis for the identical properties of the composite constituents. The FAS calculated from Equations ( l l ) and (15) and the lss from Equations (13) and (17) are plotted for the axial direction in Fig. 4. As V~.increases, the gradient of the FAS increases for the three-cylinder composite model, but for the single-fibre composite model it decreases over the whole embedded fibre length except near the embedded end (Fig. 4(a)). This results in completely different trends for the ISS distributions between the two composite models. For the threecylinder model, the maximum ISS at the loaded fibre end increases steadily, while that at the embedded end is always close to zero. Conversely, the single-fibre composite model predicts that the maximum ISS obtained at the embedded end increases rapidly with increasing Vf, whereas that at the loaded end remains a constant, finite value. These observations are further manifested in plots of the ISSs obtained at both ends of the fibre in Fig. 5 where a critical Vf ~ 0.15 is obtained, above which the location of debond initiation would change from the loaded end to the embedded end of the fibre.
The pronounced effect of Vr is summarized in Fig. 3, where the characteristic interface shear stresses obtained at the ends of the fibre are plotted as a function of Vf from F E M calculations. It is clearly demonstrated for the three-cylinder model that these stresses vary only margi-
One of the major differences between the results obtained from the FE and micromechanics analyses is in the relative magnitude of the stress concentrations. In particular, the maximum lss values at the loaded and embedded fibre ends tend to be higher for the micromechanics
COMPOSITES.
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7. 1994
473
o~ 1.2(~ a 1 ,
composite cylinder was fixed at z = L and z = 2L in the micromechanics and FE models, respectively. More importantly, the fibre was assumed to be stress-free in the
Increasing Vf
~
0.8 _ "";':';~iiiiiiii~'iiiiii~i~i~iiii--"-'"----------- .............................."'"",
micromechanics embedded mesh near end the loaded in the model FE and model, and embedded perfectly Further ends refinement bonded of the fibre atof the in
u.
0.6.
041
the FE analysis would promote significantly the stress concentrations towards singularity at these regions.
0
z
0
"....
.
. 0.2
0
.
. 0.4
". . . .
. 0.6
In the light of the findings presented in the preceding section, it is seen that in the three-cylinder composite model the surrounding composite medium acts as a stiff annulus to suppress the development of ISS at the embedded fibre end by constraining the radial boundary of the matrix cylinder. This in turn ensures that for all Vr the maximum ISS always occurs at the loaded fibre end where the interface debond propagates inwards. The maximum ISS tends to increase slightly with increasing Vr, allowing debond initiation at a low external stress.
0.8
Axial Distance, alL 0.8 b
0.6 'coo o
Increasing Vf
I~
0.4
II
E "o m 0.2 .-~ E z
"' iiii . ~....
00
0.2
0.4 0.6 0.8 Axial Distance, z/L
Fig. 4 (a) Fibre axial stress and (b) interface shear stress distributions along the axial direction obtained from the micromechanics analysis for different fibre volume fractions, Vf = 0.03, 0.3 and 0.6
(dashed and solid lines as in Fig. 2)
In contrast, the single-fibre composite model predicts that the ISS concentration becomes higher at the embedded end than at the loaded end if Vris greater than a critical value, suggesting the possibility of debond initiation at the embedded fibre end in a so-called 'two-way debonding' phenomenon. This phenomenon, peculiar to the single-fibre composite model, has been studied theoretically9,1° as well as experimentally for a relatively stiff fibre embedded in a soft matrix (e.g. a polyurethane matrix" or silicone resin t2 with glass rods) which can satisfy the criterion given by Equation (20). The criterion for debond initiation at the embedded end in preference to the loaded end is derived from the shear strength criterion (i.e. ri(0) < zq(L) in Equation (17)) A2 _
1I
(/)
m
,"
A~
,"
1
0.8
~o~
/
,,"
"
J
(20)
1 - 2kvr
7 > ~1 - 2kvm ~ oc
"'""" 0.6
Equation (20) is essentially the same as equations proposed previously ~°,~3,and is found to be independent of embedded fibre length, L, while insensitive to both Vrand Vm. In other words, the relative magnitudes of the fibre volume ratio, 7( = a2/(b 2 - a2)), and the Young's modulus ratio, ~ ( = Em/Ef), control the two-way debonding phenomenon in the single-fibre pull-out test.
_.E 0.4
~_;_::~::"_:'_i~iii.~___
•~- 0 . 2 -~ .... o z 00
Em_be!d__edEn?
-. . . . . . . . . /~ , 012
014
).6
F i b r e V o l u m e F r a c t i o n , Vf Fig. 5 Interface s h e a r s t r e s s e s a s a function of fibre volume fraction, V, obtained from the micromechanics analysis (dashed and solid lines as in Fig. 2)
analysis than for the FE analysis for large Vf (compare Figs 2(b) and 4(b)). This results in a slightly lower critical Vr required for the transition of debond initiation in the micromechanics model than in the FE model of singlefibre composites. All these observations appear to be associated with the slightly different boundary conditions used in these models. The matrix bottom of the
474
7(1 - 2kvm) < _ 1 2kvr) + 7(1 - 2kvm) 2
and thus
o
"o m
7(1 -
COMPOSITES.
NUMBER 7. 1994
As pointed out in a recent study ~3,the conventional shear lag model based on a single-fibre composite is inadequate for modelling a composite with a high Vr. A modified analysis has been presented '3 that takes into account the radial dependence of the axial stresses in the constituents so that the unacceptably high ISS at the embedded fibre end can be avoided. From the experimental viewpoint, to measure the relevant fibre/matrix interface properties the fibre volume fraction in single-fibre pull-out tests is always very low (i.e. Vr < 0.01). This implies that testing with these specimens has the fundamental limitation of generating interface properties that are only valid in the comparative sense for a given set of specimen geometry/ size and testing conditions which seldom represent those
o f practical composites o f large Vf. In this regard, the use o f multiple-fibre composite specimens (made from real
composites or from model composites with a regular fibre arrangement for the surrounding composite medium) can eliminate such a limitation of the single-
3 4
fibre pull-out test. Details o f the experimental technique have yet to be developed, although significant difficulties are envisaged in specimen preparation with the current technology. It is w o r t h noting that thin specimens prepared from real composites have been successfully employed in microindentation tests 14, to which the theoretical results presented in this paper are also closely related,
5 6 7 8
CONCLUDING
REMA RKS
The present paper has examined the effects o f Vr on the stress transfer across the fibre/matrix interface in the fibre pull-out test. As an alternative to the conventional single-fibre composite, a three-cylinder composite model was developed by means o f both micromechanics analy-
9 lO
sis and the finite element m e t h o d which represents m o r e
accurately those practical composites with large Vf.
11
Within the limitations o f the available data, the para-
metric study has demonstrated that a 'two-way debonding' phenomenon occurs in the single-fibre composite model when the fibre volume ratio, 7, is greater than the Y o u n g ' s m o d u l u s ratio, ~, o f the composite constituents. In the three-cylinder composite model, however, d e b o n d i n g is always expected to initiate at the loaded fibre end regardless o f Vf. These findings were not specifically c o m p a r e d with the relevant experimental results; such results will be the subject o f forthcoming publications.
REFERENCES 1 Kim, J.K. and Mai, Y.W. 'Interfaces in composites' in Materials Science and Technology, Vol 13 edited by T.W. Chou (VCH, Weinheim, 1993) pp 229-289 2 Kim, J.K., Zhou, L.M. and Mai, Y.W. 'Techniques for studying composite interfaces' in Handbook of Advanced Materials Testing
12 13 14
edited by N.P. Cheremisinoff(Marcel Dekker, New York, 1994)in press Kim, J.K., Baillie, C. and Mai, Y.W. 'Interfacial debonding and fibre pull-out stresses: part I. A critical comparison of existing theories with experiments' JMater Sci27 (1992) pp 3143 3154 Zhon, L.M., Kim, J.K. and Mai, Y.W. 'Interfacial debonding and fibre pull-out stresses: part I1. A new model based on the fracture mechanics approach' J Mater Sci 27 (1992) pp 3155-3166 Zhou, L.M., Kim, J.K. and Mai, Y.W. 'On the single fibre pull-out problem: effect of loading methods' Compos Sci Techno145 (1993) pp153-160 Kim,J.K., Zhou, L.M. and Mai, Y.W. 'lnterfacial debonding and p~ll-out stresses: part III. Interfacial properties of cement matrix composites' J Mater Sci 28 (1993) pp 3923-3930 Kim, J.K., Lu, S.V. and Mai, Y.W. 'Interfacial debonding and pull-out stresses: part IV. Influence of interface layer on stress transfer' J Mater Sci 29 (1994) in press Zhou,L.M. and Mal, Y.W. 'A three-cylinder model for evaluation of sliding resistance in fibre push-out test' in Ceramics Adding the Value edited by M.J. Bannister (CSIRO, Melbourne, 1992) pp Ill3 Ill8 Banbaji, J. 'On a more generalised theory of the pull-out test from an elastic matrix. Part I--Theoretical consideration' Compos Sci Techno132 (1988) pp 183 193 Leung,C.K.Y. and Li, V.C. 'A new strength-based model for the debonding of discontinuous fibres in an elastic matrix' J Mater Sci 26 (1991) pp 599645100 Betz, E. 'Experimental studies of the fibre pull-out problem' J Mater Sci 17 (1982) pp 691-700 Gent, A.N. and Liu, G.L. 'Pull-out and fragmentation on model fibre composites' J Mater Sci 26 (1991) pp 2467 2476 Hsueh,C.H. 'Embedded-end debond theory during fibre pull-out' Mater Sci Eng A 163 (1993) pp L1 L4 Grande, D.H., Mandell, J.F. and Hang, K.C.C. 'Fibre-matrix bond strength studies of glass, ceramic and metal matrix composites" J Mater Sci23 (1988)pp 311-328
AUTHORS
J a n g - K y o Kim, to w h o m correspondence should be addressed, is with the D e p a r t m e n t o f Engineering, Faculty o f Engineering and I n f o r m a t i o n Technology, Australian National University, Canberra A C T 0200, Australia. The co-authors are with the Centre for A d v a n c e d Materials Technology, D e p a r t m e n t o f Mechanical Engineering, University o f Sydney, Sydney N S W 2006, Australia.
COMPOSITES. N U M B E R 7 . 1994
475
The elastic stress transfer taking place across the fibre/matrix interface is analysed for the fibre pull-out test by means of both micromechanics and finite element (FE) analyses. A special focus has been placed on how fibre volume fraction, Vf, affects the interface shear stress fields in the model composites containing both single and multiple fibres. In a so-called 'three-cylinder model', where a fibre, a matrix and a composite medium are coaxially located, the constraint imposed on the central fibre due to the surrounding fibres is properly evaluated. It is shown in the FE analysis that the differences in stress distributions between the composite models containing single and multiple fibres become increasingly prominent with increasing Vf. The principal effect of the presence of surrounding fibres in the multiple-fibre composite model is to suppress effectively the development of stress concentration near the embedded fibre end and thus eliminate the possibility of debond initiation from this region for all Vf considered. This is in sharp contrast to the single-fibre composite model, in whichthe interfacial debond can propagate from the embedded end if Vf is larger than a critical value. These findings are essentially consistent with the results from micromechanics analysis on the same specimen geometry. The implications of the results for the practical fibre pull-out test as a means of measuring the interface properties are discussed. Key w o r d s : fibre/matrix interface; fibre pull-out test; single-fibre composite model; three-cylinder composite model; finite element analysis Phenomena of vital importance in fibre composite technology are the stress transfer across the fibre/matrix interface and the interface debond process, particularly in the neighbourhood of a fibre break or matrix crack, Theoretical analyses dealing with the stress states and the debond propagation in various loading geometries are essential to understanding how and to what extent the interface properties influence the mechanical/structural performance and fracture behaviour of the composites 1. A significant effort has been put into proper characterization of the interface properties, and a number of experimental techniques have been devised to evaluate the interactions at the interface region which include the physicochemical analyses of the surface properties, optical/electron microscopies of various kinds, mechanical testing and non-destructive testing. Among the various mechanical testing techniques the fibre pull-out test has become one of the most popular single-fibre composite testsL From both the stress-transfer and fracture mechanics viewpoints, improved micromechanics analyses3-5 have been developed for the single-fibre pull-out test, taking
into account the several different boundary conditions and the instability criteria during the debond process. The theory compares favourably with the experimental data for several different composite systems containing polymer and ceramic matrices and some cementitious composites6. In addition to the analytical micromechanics models, a finite element method has been successfully adopted to investigate the effects of the compliant fibre coating on the stress distributions before and during the debond process in the fibre pull-out test 7. In the light of the foregoing studies, it is identified that most micromechanics theories are developed based on a shear lag model of single-fibre composites where the external stress is applied to the fibre end with either of the matrix top or bottom being fixed, while the matrix cylindrical surface is invariably assumed to be stress-free (Fig. l(a)). This assumption leads to an unacceptably high applied stress required to initiate/propagate interface debonding when the radial dimension of the matrix is similar to that of the fibre (i.e. for a high fibre volume fraction, Vf). In other words, the application of the conventional models to practical composites is limited to
0010-436119410710470-06 O 1994 Butterworth-Heinemann ktd 470
COMPOSITES. VOLUME 25. NUMBER 7. 1994
a
d
b
d
" I
~
~"
Fibre Matrix
>>
L
I 1 z
[
.-T';'," 7 ,:-.; ",,'-7- :.. I,"l I /' .// /, J. i / / /I / ' / . ~
i I1,"
i
L I i I
.r
,_
2b
, ,_.. _, , / / _' . 7 ~--,-,s / is'-. - ,~ "-."'
•
. . . .
Composite
.
: .... :
medium
....
' I-i-" ............ ~
~
- - i--4
b
2B Fig. 1 Schematic illustrations of (a) the single-fibre pull-out test, (b) the fibre pull-out test of a three-cylinder composite and (c) the finite element model of a three-cylinder composite
those with a very small Vr where any effects of interactions between neighbouring fibres are completely neglected. In the present study, as a part of a continuing project on composite interfaces, a three-cylinder composite model is developed to accommodate properly the limitation of the shear lag model of the single-fibre pullout test. The micromechanics analysis and finite element method are employed in parallel to validate the results obtained from each model,
MICROMECHANICS THREE- CYLINDER
ANAL YSIS OF A COMPOSITE
_
dz
(1)
(2)
where y = a2/(b 2 - a 2) and 7~ = b2/( B2 - b2). B is the outer radius of the composite medium (Fig. l(b)). The superscripts represent the coordinate directions, and the subscripts f, m and c refer to fibre, matrix and composite medium, respectively. Equilibria of the interface shear stresses, the shear tractions on the cylindrical surface and the axial stresses in the constituents require d~r(Z)
To analyse the stress transfer in the fibre pull-out test of a multiple-fibre composite, the specimen is treated as a three-cylinder composite where a fibre is located at the centre of a coaxial shell of the matrix which in turn is surrounded by a trans-isotropic composite medium with an outer radius B (Fig. l(b)). The fibre and matrix are assumed to b e perfectly elastic and isotropic, and their radii a and b are related to the fibre volume fraction Vf = a2/b% which is the same as that of the composite medium, Here z is the direction parallel to the fibre axis, and r and 0 respectively are the radial and tangential coordinates, When the fibre is subjected to an external stress, or, at the loaded end (z = 0), while the matrix and composite medium are fixed at the embedded end (z = L), stress transfers from the fibre to the matrix and in turn from the matrix to the composite medium via the interface shear stresses, r~(z) and rb(Z), respectively. Following the previous analyses for the fibre pull-out test of a singlefibre composite 3,4 and for the fibre push-out test of a multiple-fibre composite 8, the relations between the interface shear stresses, r~ and rb, and the axial stresses, o-~f, O'=mand ~ , respectively in the fibre, matrix and composite medium, are obtained. The mechanical equilibrium conditions between the external and internal stresses require cr = ~ ( z ) + 17/o-~(z) + V r - - ~ ( z )
_ 2b fz 1 ~m(Z) a2 J0 rb(z) dz --- ~ ( z ) +
cr
2
r,(z)
(3)
a
dcr~(z) _ a27/[ara(z) 2 dz - brh(z)]
(4)
Based on the Lame solution, the relationship between the interface shear stresses is taken as rb(z) = b7/2 r,(z) (5) aT/~ where 7/2 = a 2 / ( B 2 a 2 ) • The additional radial stress, q,(z), acting at the fibre/matrix interface, which is caused by Poisson contraction of the fibre when subjected to an axial tension, is obtained from the continuity of the tangential strain at the interface -
q,,(z) = - o ~ v f V f k l o Z f ( 2 ) + [ v m V f k l -- 27/(Vm -- C~Vc)](Ym(Z)
--
Vrkt[0~(1 - vr) + 1 + Vm + 27/] -- 47/2
(6)
where ~ = Em/Er, oq = Em/Ec and k~ = 1 + 27 - Vm + ~l(1 + 27/~ + vc). E and v are Young's modulus and Poisson's ratio, respectively. Combining Equations (1)(6) yields a second-order differential equation for the fibre axial stress d2o.~(z) dz 2 A3cr~(z)= A4cr (7) The coefficients A3 and A4 are complex functions of the entselasticproperties and geometric factors of the constitu-
COMPOSITES.
NUMBER
7.
1994
471
~(z)_ a
-- ?'Vm(?'2k4 -- 2~Vf?')] (8) 2fl { ~ } A4 = ~ ?'2 1 [?'vmk4 - (0~vf q-- ?'Vm)Vfk2] (9) where ?'Em
(10)
Vm)[2?'(B/a)21n(b/a)-
fl = a?'2(1 +
k2=~r{Vrk,[o:(2-
vf)+ 1 + 2 ? ' +
1]
aN/~l 2
sinh(x/~lL )
(17) The coefficients A1 and A2 are given by A, = 2[~(1 - 2kvr) + ?'(1 - 2kvm)] (1 + Vm)[2?'Mln(b/a)- a 2]
(18)
A2 _ 7"(1 -- 2kvm) (19) A, ~1 - 2kvr) + 7(1 - 2kvm) where k = (~vr + ?'Vm)/[~(1 -- Vf) + 1 + Vm + 2?'].
Vm] -- 4?' 2} FINITE ELEMENT (FE) ANAL YSIS
2?'
k 3 = ~ff
(V m --
k 4 = (vm -
vmk 1
0~1Vc ) - -
0q Vc)[~(l -
vf) + 1 + 2?' + Vm] -- 2?'Vm
The solution for Equation (7) is subjected to the bounddry conditions o'er(0) = cr and o'~f(L) = 0, assuming an unbonded cross-section for the embedded fibre end. Therefore, the solutions for the fibre axial stress, cry(z), matrix axial stress, o-~(z), and interface shear stresses, ra(Z) and rb(Z), normalized with the applied stress o-, are obtained
(A4+ )
Recognizing the significance of the stress concentration at the broken fibre ends, increasing efforts are being directed towards employing numerical methods. The use of the finite element method (FEM) in particular allows a more accurate description of the interactions between neighbouring fibres in practical composites containing fibres of large volume fraction, and especially of the interface shear stress fields near the singularity. In addition, the specific end geometries as well as the continuously varying mechanical properties across the interface region of finite volume (i.e. the interphase) can be properly taken into account.
\
o-~(z) _ ~A3 1 sinh[x~3(L-z)] + osinh(x~3L)
~
sinh(x/~3z )
A__Z" A3
(11) ( A4) - ?'2,1 + A33,
o'Z(Z)
cr
A4 + 1) s i n h [ ~ 3 ( L - z)] + - ?'2 (A33
sinh ( ~ 3 z )
sinh(~3L) (12) r,(z)
a~3
(A4+) ~ ~A3 llcosh[x/~3(L-z)]-..3 o s h ( ~ 3 z )
-
cr
2
sinh(x/~3L ) (13)
rb(Z)
_
cr
b?'2xA/-~33(A~ + 27'1
1)c°sh[x/~A3(L-z)]-~3sh(x~33z) sinh(x~3L) (14)
The corresponding equations for the single-fibre composite model are taken from an earlier analysis 3 __ (A~ a~(z) = +1 sinh[x/r~(L-z)] + ~sinh(x/-~z) A2 cr sinh(x/r~L) -41 (15)
a~(z) t7
I `42 + 1)sinh[x/~l(L- z)] + ~ s i n h ( x ~ V ) ~A1
Az -- ? ' Z - -
?'
sinh(~L) (16)
472
COMPOSITES
. NUMBER
7 . 1994
Both the composites containing single and multiple fibres are considered for the present FE analysis. The geometry, the loading method and the boundary conditions are selected to represent those of the actual experimental technique (Fig. l(c)), which are analogous to those used in the corresponding micromechanics analyses. A mesh has been created of the isometric eightnoded quadrilateral elements for the axisymmetric loading geometry of a two-dimensional model using the FE program 'Strand 6 '7. The total number of elements is 480, with 20 and 24 elements in the longitudinal and lateral (or radial) directions, respectively. This ensures sufficient resolution and thus accuracy of the results while maintaining a reasonable time needed for computation. A uniformly distributed constant stress, a = 100 MPa, is applied to the partially embedded fibre at the surface (z = 0). The boundary conditions are imposed such that the bottom surfaces of the matrix and composite medium are fixed at z = 2L, and the axis of symmetry (r = 0) is fixed where there is no displacement taking place. Therefore, the global default freedom is set to allow displacemerits for 0 < z < 2L and 0 < r < B in the axial and radial directions, respectively. Specific results are calculated for an SiC fibre glass matrix composite 3 with elastic constants Ef = 400 GPa, E m = 70 GPa, vf = 0.17 and Vm = 0.2. A constant embedded fibre length L = 2.0 mm and constant radii a = 0.2 and B = 2.0 considered with matrixmmradiusb. All stressmmarevalues reported hereinVaryingare normalized with the external stress, or, and negative signs for the interface shear stress are omitted for simplicity. RESULTS
The stress distributions along the axial direction shown in Fig. 2 are obtained from the FE analysis for the com-
1.2 1
e
a
t.D
" '"-.. ~ ,
0.3
/t
oo ~
,
L o a d e d End
/
2 0.6 .--
/
v, = 0.6
0.4
~
....
Vf~O,O
g
z I
I
I
I
o.2
I
I
L
I
I
L
o.4
I
I
L
L
I
o.6
I
L
I
o.8
L
I
1.o
Axial Distance, z/L
Z
-
o.1
/
~ E o
0.2
o o
~
-
3
~ ."
•
~ ~ ~ ~ ~
/ ,~....
E m b e d d e d End
0
o
o:2
o'.4
o'.6
Fibre V o l u m e Fraction, Vf Fig. 3 Interface shear stress distributions as a function of fibre volume fraction, Vf, obtained from FEM calculations: (©, -) single-fibre composite; ( A , A, - - ) three-cylinder composite model O ,
~
0.3
,
,
,
,
,
,
~o 0.2 ~ ~:
A ~ . "~. vf = 0.6
. ~
0.1
-~o z
-
-
b
.~
"~-
-
/',,
0 i 0
"~
~
Vf = 0.03 /
".. ~ ~ ~..... i ~ i ~ . , 0.2 0.4 0.6 0.8 Axial Distance, z/L
1.0
Fig. 2 (a) Fibre axial stress and (b) interface shear stress distributions along the axial direction obtained from FEM calculations for two fibre volume fractions, Vf= 0.03 and 0.6: ( . . . . . . ) singlefibre composite; ( ) three-cylinder composite model
nally with Vr, and the magnitude of the 1ss at the loaded end is always greater than that at the embedded fibre end. This ensures that, if the fibre is loaded continuously, debonding is always expected to initiate at the loaded fibre end for all Vf, based on the shear strength criterion of interface debonding. However, for the single-fibre composite model, the ISS at the embedded fibre end increases rapidly, whereas that obtained at the loaded fibre end decreases with increasing Vr. Therefore, there is a critical fibre volume fraction above which the magnitude of the former stress exceeds that of the latter, leading to debond initiation from the embedded fibre end in preference to the loaded fibre end. The approximate critical value Vf ~ 0.26 is determined based on the superimposed curves of the data points in Fig. 3. The predicted results based on the micromechanics
posites with two extremes of fibre volume fraction, Vr = 0.03 and 0.6. The three-cylinder composite model predicts that both fibre axial stress (FAS) and interface shear stress (ISS) decrease from a maximum near the loaded fibre end towards zero at the embedded fibre end. An increase in Vf (and thus the stiffness of the composite medium) slightly increases both the maximum ISS and the stress gradient, without altering the general trend of the stress fields. For a small Vr, the stress distributions in the single-fibre composite model are equivalent to those of the three-cylinder model. In sharp contrast, the stress fields change drastically in the single-fibre composite model when Vf is large, such that the fibre axial stress does not diminish to zero at the embedded fibre end and the stresses in the central portion of the fibre length are almost constant. More importantly, the ISS displays two peaks at the ends of the fibre, the one at the embedded end being greater than the other at the loaded end for Vr = 0.6 (Fig. 2(b)).
analysis are essentially similar to those obtained in the FE analysis for the identical properties of the composite constituents. The FAS calculated from Equations ( l l ) and (15) and the lss from Equations (13) and (17) are plotted for the axial direction in Fig. 4. As V~.increases, the gradient of the FAS increases for the three-cylinder composite model, but for the single-fibre composite model it decreases over the whole embedded fibre length except near the embedded end (Fig. 4(a)). This results in completely different trends for the ISS distributions between the two composite models. For the threecylinder model, the maximum ISS at the loaded fibre end increases steadily, while that at the embedded end is always close to zero. Conversely, the single-fibre composite model predicts that the maximum ISS obtained at the embedded end increases rapidly with increasing Vf, whereas that at the loaded end remains a constant, finite value. These observations are further manifested in plots of the ISSs obtained at both ends of the fibre in Fig. 5 where a critical Vf ~ 0.15 is obtained, above which the location of debond initiation would change from the loaded end to the embedded end of the fibre.
The pronounced effect of Vr is summarized in Fig. 3, where the characteristic interface shear stresses obtained at the ends of the fibre are plotted as a function of Vf from F E M calculations. It is clearly demonstrated for the three-cylinder model that these stresses vary only margi-
One of the major differences between the results obtained from the FE and micromechanics analyses is in the relative magnitude of the stress concentrations. In particular, the maximum lss values at the loaded and embedded fibre ends tend to be higher for the micromechanics
COMPOSITES.
NUMBER
7. 1994
473
o~ 1.2(~ a 1 ,
composite cylinder was fixed at z = L and z = 2L in the micromechanics and FE models, respectively. More importantly, the fibre was assumed to be stress-free in the
Increasing Vf
~
0.8 _ "";':';~iiiiiiii~'iiiiii~i~i~iiii--"-'"----------- .............................."'"",
micromechanics embedded mesh near end the loaded in the model FE and model, and embedded perfectly Further ends refinement bonded of the fibre atof the in
u.
0.6.
041
the FE analysis would promote significantly the stress concentrations towards singularity at these regions.
0
z
0
"....
.
. 0.2
0
.
. 0.4
". . . .
. 0.6
In the light of the findings presented in the preceding section, it is seen that in the three-cylinder composite model the surrounding composite medium acts as a stiff annulus to suppress the development of ISS at the embedded fibre end by constraining the radial boundary of the matrix cylinder. This in turn ensures that for all Vr the maximum ISS always occurs at the loaded fibre end where the interface debond propagates inwards. The maximum ISS tends to increase slightly with increasing Vr, allowing debond initiation at a low external stress.
0.8
Axial Distance, alL 0.8 b
0.6 'coo o
Increasing Vf
I~
0.4
II
E "o m 0.2 .-~ E z
"' iiii . ~....
00
0.2
0.4 0.6 0.8 Axial Distance, z/L
Fig. 4 (a) Fibre axial stress and (b) interface shear stress distributions along the axial direction obtained from the micromechanics analysis for different fibre volume fractions, Vf = 0.03, 0.3 and 0.6
(dashed and solid lines as in Fig. 2)
In contrast, the single-fibre composite model predicts that the ISS concentration becomes higher at the embedded end than at the loaded end if Vris greater than a critical value, suggesting the possibility of debond initiation at the embedded fibre end in a so-called 'two-way debonding' phenomenon. This phenomenon, peculiar to the single-fibre composite model, has been studied theoretically9,1° as well as experimentally for a relatively stiff fibre embedded in a soft matrix (e.g. a polyurethane matrix" or silicone resin t2 with glass rods) which can satisfy the criterion given by Equation (20). The criterion for debond initiation at the embedded end in preference to the loaded end is derived from the shear strength criterion (i.e. ri(0) < zq(L) in Equation (17)) A2 _
1I
(/)
m
,"
A~
,"
1
0.8
~o~
/
,,"
"
J
(20)
1 - 2kvr
7 > ~1 - 2kvm ~ oc
"'""" 0.6
Equation (20) is essentially the same as equations proposed previously ~°,~3,and is found to be independent of embedded fibre length, L, while insensitive to both Vrand Vm. In other words, the relative magnitudes of the fibre volume ratio, 7( = a2/(b 2 - a2)), and the Young's modulus ratio, ~ ( = Em/Ef), control the two-way debonding phenomenon in the single-fibre pull-out test.
_.E 0.4
~_;_::~::"_:'_i~iii.~___
•~- 0 . 2 -~ .... o z 00
Em_be!d__edEn?
-. . . . . . . . . /~ , 012
014
).6
F i b r e V o l u m e F r a c t i o n , Vf Fig. 5 Interface s h e a r s t r e s s e s a s a function of fibre volume fraction, V, obtained from the micromechanics analysis (dashed and solid lines as in Fig. 2)
analysis than for the FE analysis for large Vf (compare Figs 2(b) and 4(b)). This results in a slightly lower critical Vr required for the transition of debond initiation in the micromechanics model than in the FE model of singlefibre composites. All these observations appear to be associated with the slightly different boundary conditions used in these models. The matrix bottom of the
474
7(1 - 2kvm) < _ 1 2kvr) + 7(1 - 2kvm) 2
and thus
o
"o m
7(1 -
COMPOSITES.
NUMBER 7. 1994
As pointed out in a recent study ~3,the conventional shear lag model based on a single-fibre composite is inadequate for modelling a composite with a high Vr. A modified analysis has been presented '3 that takes into account the radial dependence of the axial stresses in the constituents so that the unacceptably high ISS at the embedded fibre end can be avoided. From the experimental viewpoint, to measure the relevant fibre/matrix interface properties the fibre volume fraction in single-fibre pull-out tests is always very low (i.e. Vr < 0.01). This implies that testing with these specimens has the fundamental limitation of generating interface properties that are only valid in the comparative sense for a given set of specimen geometry/ size and testing conditions which seldom represent those
o f practical composites o f large Vf. In this regard, the use o f multiple-fibre composite specimens (made from real
composites or from model composites with a regular fibre arrangement for the surrounding composite medium) can eliminate such a limitation of the single-
3 4
fibre pull-out test. Details o f the experimental technique have yet to be developed, although significant difficulties are envisaged in specimen preparation with the current technology. It is w o r t h noting that thin specimens prepared from real composites have been successfully employed in microindentation tests 14, to which the theoretical results presented in this paper are also closely related,
5 6 7 8
CONCLUDING
REMA RKS
The present paper has examined the effects o f Vr on the stress transfer across the fibre/matrix interface in the fibre pull-out test. As an alternative to the conventional single-fibre composite, a three-cylinder composite model was developed by means o f both micromechanics analy-
9 lO
sis and the finite element m e t h o d which represents m o r e
accurately those practical composites with large Vf.
11
Within the limitations o f the available data, the para-
metric study has demonstrated that a 'two-way debonding' phenomenon occurs in the single-fibre composite model when the fibre volume ratio, 7, is greater than the Y o u n g ' s m o d u l u s ratio, ~, o f the composite constituents. In the three-cylinder composite model, however, d e b o n d i n g is always expected to initiate at the loaded fibre end regardless o f Vf. These findings were not specifically c o m p a r e d with the relevant experimental results; such results will be the subject o f forthcoming publications.
REFERENCES 1 Kim, J.K. and Mai, Y.W. 'Interfaces in composites' in Materials Science and Technology, Vol 13 edited by T.W. Chou (VCH, Weinheim, 1993) pp 229-289 2 Kim, J.K., Zhou, L.M. and Mai, Y.W. 'Techniques for studying composite interfaces' in Handbook of Advanced Materials Testing
12 13 14
edited by N.P. Cheremisinoff(Marcel Dekker, New York, 1994)in press Kim, J.K., Baillie, C. and Mai, Y.W. 'Interfacial debonding and fibre pull-out stresses: part I. A critical comparison of existing theories with experiments' JMater Sci27 (1992) pp 3143 3154 Zhon, L.M., Kim, J.K. and Mai, Y.W. 'Interfacial debonding and fibre pull-out stresses: part I1. A new model based on the fracture mechanics approach' J Mater Sci 27 (1992) pp 3155-3166 Zhou, L.M., Kim, J.K. and Mai, Y.W. 'On the single fibre pull-out problem: effect of loading methods' Compos Sci Techno145 (1993) pp153-160 Kim,J.K., Zhou, L.M. and Mai, Y.W. 'lnterfacial debonding and p~ll-out stresses: part III. Interfacial properties of cement matrix composites' J Mater Sci 28 (1993) pp 3923-3930 Kim, J.K., Lu, S.V. and Mai, Y.W. 'Interfacial debonding and pull-out stresses: part IV. Influence of interface layer on stress transfer' J Mater Sci 29 (1994) in press Zhou,L.M. and Mal, Y.W. 'A three-cylinder model for evaluation of sliding resistance in fibre push-out test' in Ceramics Adding the Value edited by M.J. Bannister (CSIRO, Melbourne, 1992) pp Ill3 Ill8 Banbaji, J. 'On a more generalised theory of the pull-out test from an elastic matrix. Part I--Theoretical consideration' Compos Sci Techno132 (1988) pp 183 193 Leung,C.K.Y. and Li, V.C. 'A new strength-based model for the debonding of discontinuous fibres in an elastic matrix' J Mater Sci 26 (1991) pp 599645100 Betz, E. 'Experimental studies of the fibre pull-out problem' J Mater Sci 17 (1982) pp 691-700 Gent, A.N. and Liu, G.L. 'Pull-out and fragmentation on model fibre composites' J Mater Sci 26 (1991) pp 2467 2476 Hsueh,C.H. 'Embedded-end debond theory during fibre pull-out' Mater Sci Eng A 163 (1993) pp L1 L4 Grande, D.H., Mandell, J.F. and Hang, K.C.C. 'Fibre-matrix bond strength studies of glass, ceramic and metal matrix composites" J Mater Sci23 (1988)pp 311-328
AUTHORS
J a n g - K y o Kim, to w h o m correspondence should be addressed, is with the D e p a r t m e n t o f Engineering, Faculty o f Engineering and I n f o r m a t i o n Technology, Australian National University, Canberra A C T 0200, Australia. The co-authors are with the Centre for A d v a n c e d Materials Technology, D e p a r t m e n t o f Mechanical Engineering, University o f Sydney, Sydney N S W 2006, Australia.
COMPOSITES. N U M B E R 7 . 1994
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