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The behavior of ceramic matrix fiber composites under longitudinal loading
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Composites Science and Technology 46 (1993) 105-113
THE B E H A V I O R OF CERAMIC MATRIX FIBER COMPOSITES U N D E R LONGITUDINAL LOADING I. M. Daniel, G. Anastassopoulos Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208, USA
&
J.-W. Lee Composite Materials, Samsung Shipbuilding and Heavy Industry Co., Chang Won City, South Korea (Received 14 October 1991; revised version received 3 December 1991; accepted 27 January 1992) Abstract Failure mechanisms were studied by reflection light microscopy in a unidirectional silicon carbide/glassceramic composite loaded in longitudinal tension. The material behaves linearly up to the point where the first transverse matrix cracks appear. Thereafter, it undergoes a rapid stiffness decrease corresponding to matrix crack multiplication and saturation. These matrix cracks increase in density with applied stress up to a limiting level of 28 cracks/ram or a minimum crack spacing of 361zm (O.O014in), which corresponds approximately to two fiber diameters. Fiber breaks and fiber debonding, which start before matrix crack saturation, continue until final failure. In the last stage the material exhibits quasi4inear behavior with small stiffness variation. Experimental results of crack density and stress/strain behavior were compared with predictions based on a modified shear lag analysis.
carbide (SIC) and mullite fibers. Matrices used successfully include glass, glass-ceramic, silicon carbide, silicon nitride, and aluminum oxide. Composites consisting of glass-ceramic matrices, such as lithium aluminosilicate (LAS) and calcium aluminosilicate (CAS), reinforced with silicon carbide yarn or monofilament have been developed. 1-3 These materials can be fabricated in both unidirectional and multidirectional laminate form, and can attain strengths of 1000 MPa (140 ksi) and ultimate strains of over 1%. Fracture toughness, as determined by a notched beam method, is more than five times that of the monolithic ceramic. 2 Brittle-matrix composite materials behave nonlinearly and inelastically beyond a relatively low stress threshold. Their overall behavior under load is intimately related to the micromechanisms of inelastic deformation and failure that develop and interact with each other. In these materials the failure strain of the matrix is lower than that of the fibers and damage initiates with the development of multiple matrix cracking. 4'5 This is accompanied by partial debonding at the fiber/matrix interface and fiber fractures, which strongly depend on the properties of the interface. This type of behavior has been observed in glass ceramics reinforced with SiC fibers4-7 and cement reinforced with glass or steel fibers. 8'9 The micromechanics of stress transfer and fracture of brittle-matrix composites is being studied analytically by many investigators. T M Although the various failure mechanisms are known, their relative magnitude, exact sequence and quantitative effect on overall behavior vary from case to case. Of great importance is the influence of constituent properties, including fiber, matrix and interphase region, and the processing residual stresses, on the failure process and overall behavior. This paper describes experimental results obtained
Keywords: ceramic-matrix composites, failure mechanisms, micromechanics, matrix cracking, fiber debonding, shear-lag analysis INTRODUCTION The application of ceramic materials to hightemperature structures is hampered by their inherent limitations of brittleness, low strain to failure, low tensile strength, and low fracture toughness. These limitations are somewhat overcome by reinforcing the brittle ceramic materials with high-strength fibers. The resulting composite materials display appreciable ductility, higher strength and higher fracture toughness than the monolithic matrix materials. Promising reinforcements include carbon, silicon
Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd.
105
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I . M . Daniel, G. Anastassopoulos, J.-W. Lee
on failure mechanisms and overall behavior of a SiC/CAS unidirectional composite under longitudinal tensile loading. Experimental results are compared with predictions of a recently presented modified shear lag analysis. 14 EXPERIMENTAL P R O C E D U R E
The material investigated was SiC/CAS, calcium aluminosilicate glass ceramic reinforced with silicon carbide fibers, manufactured by Corning Glass Works. The fiber is silicon carbide yarn known as Nicalon (Nippon Carbon Co.). The fiber is available in continuous length tows with an average diameter of 15 #m. The composite material was obtained in the form of 8 ply and 24 ply unidirectional plates. The composite material was characterized first to obtain average physical and mechanical properties. Transverse to the fiber cross sections were examined under the microscope and photomicrographs were taken and analyzed. Unidirectional specimens instrumented with strain gages were tested under longitudinal and transverse tensile loading in a servohydraulic testing machine to determine average mechanical properties. Specimens used for the study of failure mechanisms under the microscope were 4.62x0.76cm (1-82 x 0-30 in) coupons, 0.Smm (0.02in) thick, cut from the edge of a thick 0° composite plate. (The width of the coupon, 0.76 cm, was in the thickness direction of the composite plate.) These coupons were polished on one side and instrumented with strain gages on the other side. They were tabbed with 2.54cm-long (1.0in) glass/epoxy tabs. These tabs were bonded to the specimen only over a length of l c m (0.4in) from each end and extended beyond the ends of the specimen. A special loading fixture was designed and built for loading the specimens under a reflection optical microscope (Fig. 1). Load is applied and controlled by means of a pneumatic cylinder. The specimen with the grips attached is mounted onto a reaction frame attached to one end of the pneumatic cylinder. One end of the specimen is connected to the moving piston while the other end is reacted at the other end of the reaction frame through a strain gage load cell. The entire assembly, including air cylinder, reaction frame, specimen with grips and load cell, is suspended with counterweights from a movable upright frame. Thus, any part of the loaded specimen can be moved to the stage of the microscope without any weight or force being exerted on the microscope. These specimens were loaded in steps with still photomicrographs taken at every step and videotapes recorded during loading. In one case a specimen was loaded continuously and a videotape record was taken for real time observation of the initiation and propagation of the various failure mechanisms.
Fig. 1. Fixture and setup for loading composite specimens under the microscope. EXPERIMENTAL RESULTS Material characterization Some basic properties of the matrix and fiber constituents obtained from the literature and from previous tests are given in Table 1. Examination of photomicrographs of transverse cross sections showed that the fiber diameter ranges between 8 and 20 #m with a median value of 15/~m. The fiber distribution is nonuniform with an average
Table 1. Constituent material properties
Property Maximumuse temperature (°C (°F)) Fiber diameter(~m) Density(g/cm3) Coefficientof thermal expansion (10-6/°C (10-6/°V)) Elastic modulus (Gaa (106psi)) Tensile strength (MPa (ksi))
CAS matrix3
SiC fiber1'2'5
1350(2460)
1300(2370)
-2.8 5.0 (2.8)
15 2.6 3-1 (1.7)
98 (14-2)
163(23-6)
124 (18) (flexural)
1930(280)
The behavior of ceramic matrix fiber composites under longitudinal loading Table 2. Measured properties of SiC/CAS unidirectional composite
Property
Value
Fiber volumefraction, Vt Ply thickness, t (mm (in)) Longitudinal modulus, E l (GPa (Msi)) Transverse modulus, E2 (GPa (Msi)) In-plane shear modulus, Gt2 (GPa (Msi)) Major Poisson'sratio, v~2 Longitudinal tensile strength, FIT (MPa (ksi)) Transverse tensile strength, F2T(MPa (ksi)) Longitudinal ultimate tensile strain, e~T Transverse ultimate tensile strain, ezr "
0.39 0-38 (0.015) 121 (17.6) 112 (16.2) 52 (7.5) 0-18 393 (57) 55 (8) 0"0084 0-0005
fiber volume fraction, Vf, of 0-39. A low degree of porosity was measured (Vv = 0.01). Basic properties obtained from macromechanical tests of the unidirectional SiC/CAS composite are tabulated in Table 2. Macromechanical behavior Figure 2 shows a typical longitudinal stress/strain curve to failure obtained from a test in the servohydraulic testing machine. This curve displays several characteristic features which are related to the failure mechanisms and failure process. The initial linear region AB corresponds to the elastic linear behavior of the material prior to any significant microfailures. The measured modulus and the one predicted by the rule of mixtures are:
E1 (measured) = 125 GPa (18.1 Msi) E1 (predicted) = 123 GPa (17.9 Msi) At an applied stress of approximately 207MPa (30 ksi) (Point B) there is a marked departure from linear behavior corresponding to transverse matrix cracking and to fiber/matrix debonding. Region BC is
500 v. ,f"
400 -
300-
/
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,/
,/
j . -, ; ' " D
...,"
1:5
f" C
B .............. ,f
200It
/ 100-
/ ,,'A i
0
0.0
i '0.2
O.
0.6
O.
1.0
Strain, e, (~)
Fig. 2. Stress/strain curve for [08] SiC/CAS specimen under uniaxial tensile loading.
107
believed to correspond to transverse crack multiplication up to a saturation density to be discussed below. This is followed by a quasi-linear region CD in which no further significant matrix damage takes place, and a final slightly steeper linear region DE. Fiber fractures and debonding which start before transverse matrix crack saturation continue until final failure. The terminal modulus (both secant and tangent) is E'I = 52 GPa (7.6 Msi) Assuming that the entire load is carded by the fibers in this last region, the predicted terminal modulus is 63.5 GPa (9-2 Msi). The difference may be due in great part to the fact that some of the fibers are already damaged. Failure mechanisms The micromechanical behavior and failure mechanisms were studied by testing and observing specimens under the microscope. No cracking was observed at applied stresses below 100MPa (14.5 ksi). The first isolated transverse matrix cracks appeared above this stress level, but crack density started increasing sharply above a stress of 180 MPa (26 ksi) corresponding to a strain of el = 1-14 x 10 -3 which is very close to the ultimate tensile strain of the bulk matrix material. Figure 3 shows typical photomicrographs with transverse matrix cracking increasing in density with applied stress. The transverse crack density was measured with the microscope along three longitudinal zones of the specimen and the average crack density recorded. A plot of applied stress versus normalized crack density was superimposed on the average stress/strain curve for the material (Fig. 4). The crack density, t, was normalized by multiplying it by the fiber diameter, 2rf. This density increases rapidly between 180MPa (26 ksi) and 275 MPa (40 ksi). It reaches a saturation level of 28 cracks/mm, or a minimum crack spacing of 36/tm (0-0014 in), at an applied stress of approximately 275 MPa (40ksi). Thus, the minimum crack spacing appears to be on the order of two fiber diameters. Isolated fiber breaks were observed before saturation of transverse matrix cracking and increased in frequency as the matrix cracks reached their maximum density (Fig. 5). Most fiber breaks occurred at a short distance, one to four fiber diameters, from the nearest matrix crack, possibly as a result of some limited debonding causing a local stress rise at the end of the debonded length. As the load was increased the fiber cracks opened wider indicating further debonding and sliding. At a stress level of approximately 345 MPa (50 ksi) no further matrix cracking or fiber fracture is observed. It is likely, although not possible to observe in the microscope, that the next failure process
I.M. Daniel, G. Anastassopoulos, J.-W. Lee
108
Fig. 3. Typical photomicrographs showing initiation and multiplication of transverse matrix cracks under longitudinal tensile loading.
500
consists of nearly total fiber debonding leading to linear behavior in the last stage D E up to total failure (Fig. 2).
400 o o
/
o
~'~ 3 0 0
o
./ ,./
THEORETICAL ANALYSIS
6 ~'200
o f" o o
I"
o
/ /'
100-
o
,,
w
0.0
- ....
Streu-Strain
o o o o
Stress
v~.
Crack
Density
, , , , , , , ,. , , , , , , ~ , , r 0.2
0.4
0.8
Strain, e, (~) Normalized Crack Density
0.8
1.0
(2Xr~)
Fig. 4. Stress/strain and stress versus crack density curves for [08] SiC/CAS specimen under uniaxial tensile loading.
In materials with low interfacial shear strength, such as the SiC/LAS system, it can be assumed that debonding occurs immediately after matrix cracking and that load transfer is effected via the interface frictional stress. 6'7 However, as further experimental evidence shows, the S i C / C A S system studied here has appreciably higher interracial shear strength. For this reason a model with elastic stress transfer was used. Under longitudinal loading for fully bonded fibers, the axial strain is the same in both fiber and matrix. As the load increases, the phase with the lower ultimate strain, in this case the matrix, will fail by
The behavior of ceramic matrix fiber composites under longitudinal loading
109
Fig. 5. Photomicrographs illustrating fiber crack opening between matrix cracks. transverse cracking (Fig. 6a). In the vicinity of a matrix crack the axial stress in the matrix is relieved (Fig. 6c) while the axial fiber stress is increased and a highly localized interracial shear stress develops (Fig. 6b). Matrix cracking increases in density up to a point where the interracial shear stress exceeds the fiber debond strength and debonding initiates while the maximum axial stress in the matrix remains below its tensile strength. A modified shear lag analysis was conducted for a cylindrical element of matrix with a single fiber between two matrix cracks (Fig. 7). 14 The stress/strain behavior is linear up to an applied stress level where the matrix stress (or strain) reaches the ultimate value of the matrix tensile strength (or strain). This stress level is given by E1
Oa = Em (FmT -- O'm) where era = applied stress in composite o . . = residual stress in matrix
(1)
FreT = matrix tensile strength Em = matrix modulus E1 = longitudinal modulus of composite Above this applied stress level matrix cracking and a redistribution of stresses take place. The average axial stresses in the matrix and fiber and the interracial shear stress have the following distributions:
[gmOa Orm)
Ef of=~l
(
1
(2)
cosh (-- -
Em Vm \ 2 1 + E,-----~f cosh(-~)
oa
L M. Daniel, G. Anastassopoulos, J.-W. Lee
110
Axial stress in fiber ~a
ttItttttttt? ?tt?
inteffacial shear stress in fiber
#
(b) BIB
6 S a
StreSS
"/,//4
i wl
(
atrix
I1'1 I / t / /i / ¢ ///// .////
, //// zl //j / / ~/// z / ~/// i/t
~a
(a)
(c) IF~. 6. Matrix cracking and local stress distributions in longitudinally loaded specimen. sinh(~-rex) Z'i(X) -----ocrfVm2vf~--E-~I
Orm)
(od) cosh ~-
where 1 m =~Gf + ~ m
(4)
× [~ × Vf(rf --r.) (4
where o , = average axial stress in matrix of = average axial stress in fiber ~i = interfacial shear stress Orm = axial residual stress in matrix o~f = axial residual stress in fiber El, Em= fiber and matrix moduli, respectively Vf, Vm -- fiber and matrix volume fractions, respectively x = axial coordinate measured from crack face l = crack spacing The parameter tr is obtained as 14
~
2 =
- -
Arf
E, x
-
-
EfEm Vm
r,
l
I - Vf ]
Gf, Gm= fiber and matrix shear moduli, respectively rf, rm -- fiber and matrix radii in model, respectively In this range of deformation, prior to any fiber debonding, the matrix crack spacing I is obtained from eqn (2) by setting x = l/2 and am --- FreT
Em°a + ElO__rm_ 1 = 2c~c°sh-1 [ E m o a +
El(arm
--
FreT) }
(7)
The above relation gives the matrix crack spacing as a function of applied stress up to the point of fiber debonding initiation.
(5) r
I
(6)
l
Fig. 7. Cylindrical element of composite with a single fiber and two matrix cracks.
The behavior of ceramic matrix fiber composites under longitudinal loading The maximum interfacial shear stress ~ at the crack location is calculated from eqn (4) by setting x = 0.
v,t-VF +°=/Em°a) t a n h - ~
ri(0) =°¢rf Vm2
In the bonded area (d -< x -< 1 - d)
(8) (2Vfrfd
When the above shear stress reaches the value of the interracial shear strength F~ debonding occurs and increases with applied stress. The interfacial shear strength F~s is not an easily measurable quantity. An estimate of its relative value can be obtained from crack density measurements alone. For materials with relatively low matrix tensile strength and sufficient interracial shear strength, it can be assumed that debonding starts after matrix crack saturation, i.e., there is no more matrix cracking once debonding starts. These conditions can be expressed as follows: /min
orrn = FreT at X = 2
(9)
and ri(0) = Fiis at
x = 0
(10)
which, when used with eqns (2) and (4), yield Fis _ V m o ~ f
Fmr
cotha'lmin
Vf 2
The above gives a relative magnitude of the interracial shear strength F~ in terms of the matrix tensile strength. The extent of debonding, d, is calculated as follows (Fig. 8): In the debonded area (0 -< x < d and 1 - d -< x -< l) the redistributed stresses are
2Vfrfx Om = - -
(12)
Vmrf
of =
Oa
21~fX
v,
rf
(13)
l'i = ~'f
(14)
where rf is the interracial frictional stress in the debonded area (which was not accounted for in Ref. 14).
Debon~ng P
Crack Face
L. l
q"\ V~
( EmVm
+ \ E1Vf
..J -!
Fig. 8. Geometry and loading of cylindrical model with partial debonding at the interface.
E m
)coshoc(l/2-x)
E l O'a--Orm c o s h a ' - ~ - - - d )
(15)
2rfd) cosh oc(l/2 - x) Oa -- Off - - -rf / cosh
ot(1/27-~
(16)
2Vf rrd] sinh oc(1/2- x) a'rfVm Em 2Vf [ E l O'a "~"O'rm cosh d) (17) The debonded length, d, is calculated by assuming that the interfacial shear at the end of the disbond is equal to the interfacial shear strength. Setting ri(d) = F~s and x = d in eqn (17) we obtain an implicit relationship for d which can be solved by iteration techniques. However, if friction is neglected, i.e. if rf = 0, an explicit expression can be obtained for d:
1-- j
(11)
4
111
(18)
where 2Fis Vf E1 = -¢x'rf Vm Emma + ElOrrm
(19)
The average axial strain in the damaged composite, which is taken to be equal to the average fiber strain, is obtained by calculating the average stress in the fiber from eqn (16) and dividing it by the fiber modulus. Thus, a stress/strain relationship can be predicted for all stages of damage up to failure.
COMPARISON OF PREDICTED A N D EXPERIMENTAL RESULTS
The theoretical analysis discussed before, subject to the validity of assumptions, can predict transverse matrix crack density, debonded fiber length and average strain in the composite as a function of the applied stress, in terms of geometrical, elastic and strength parameters of the fiber and matrix. These include fiber volume fraction, fiber and matrix moduli and strengths and interfacial shear strength. The matrix tensile strength was obtained by observing the applied stress at matrix crack initiation and taking into consideration a calculated value of 55 MPa (8ksi) for the axial residual stress in the matrix (Or,). The radial compressive residual stress at the fiber matrix interface was calculated to be approximately 6 0 M P a (8.7ksi). For a frictional coefficient of # =0.15, this would yield an interface
112
1. M. Daniel, G. Anastassopoulos, J.-W. Lee
Normalized Debonded Length (2d/l) Normalized Crack Density ( 2krf ) 0.0 0.z 0.4 0.6 0.B 1.o 500 j. ]
i /
+-/ /
j'+/
+
CX,300
-I
/o,
°+ ....-
"/'/'"/"/:
/
/
/
SUMMARY AND CONCLUSIONS
,.J
I° /
100 Io / -1 / I /
0.0
+-----Expertm,tn~aStress--Str~ - ~ ~
P r e d i c t i o n of S t r e s s - S t r a i n of S t r e s s vs. C r a c k Density S.tress vs. _Debond.~mg ..
~ -- P r e ~
0.2
0.4
0.8
The debonded fiber length is expressed as a fraction of matrix crack spacing. Debonding starts when the matrix crack density reaches the maximum level and increases up to approximately 75% of the crack spacing (Fig. 9). It was not possible to observe and monitor the extent of debonding experimentally.
O.B
1.0
Strain, ~, (m) Fig. 9. Stress/strain, stress versus matrix crack density and stress versus debonded length curves for SiC/CAS [08] unidirectional composite. frictional stress of rf = 9 MPa (1-3 ksi), a relatively low value. The interfacial shear strength as calculated from eqn (11) was approximately 221MPa (32 ksi). Although this value may appear high, it is in substantial agreement with experimental results obtained by indentation tests on SiC/CAS material.15 In the predictions a matrix tensile strength of 159 MPa (23 ksi) and an interfacial shear strength of 221 MPa (32 ksi) were used and the residual stresses and friction were neglected. The predicted stress/strain, stress versus crack density and stress versus debonded length curves are shown in Fig. 9 and compared with experimental results. The predicted stress/strain curve agrees well with the experimental one initially, but it deviates from it at higher stresses. Possible factors causing this discrepancy are: (1) a degree of random fiber misalignment, '4 (2) some uncertainty of the true effective values of tensile matrix and interfacial shear strengths and (3) the neglect of residual thermal and interface frictional stresses. The fiber misalignment also may explain why the measured strength of the composite is much lower than the predicted value of 814MPa (118 ksi) based on the constituent properties. The predicted crack density increases rapidly between 211MPa (30.6ksi) and 221MPa (32ksi) when it reaches the saturation level of 29 cracks/mm or a minimum crack spacing of 34 ~m (0-00134in) (Fig. 9). The experimental results show matrix cracking starting earlier at an applied stress of 104MPa (15 ksi) and reaching a saturation crack density of 28 cracks/mm or a minimum crack spacing of 36 ~m (0.00141 in). The apparent early cracking is due primarily to the tensile residual stress in the matrix and to the statistical nature of its strength.
The behavior of a unidirectional ceramic matrix composite under longitudinal tensile loading was studied. The material investigated was SiC/CAS, calcium aluminosilicate glass ceramic reinforced with silicon carbide fibers. Unidirectional specimens were loaded under the microscope in a specially designed fixture. Failure mechanisms were observed in real time and recorded by photomicrography and video photography. The initial failure consists of transverse matrix cracks increasing in density with applied stress up to a limiting level of 28 cracks/mm or a minimum crack spacing of 3 6 # m (0-00140in), which corresponds approximately to two fiber diameters. Experimental results were compared with predictions from a modified shear lag analysis. The analysis predicts a minimum crack spacing of 34/~m (0.00134 in) and fiber debonding initiation at the point of maximum crack density. It also predicts a complete stress/strain curve to failure. This curve agrees well with the experimental one for the most part, but it deviates from the experimental results at higher stresses. This discrepancy may be due to a variety of factors, including fiber misalignment, uncertainty about constituent and interface properties and the unknown effects of residual stresses and interfacial friction.
ACKNOWLEDGEMENTS The work described here was sponsored by the Air Force Office of Scientific Research (AFOSR). We are grateful to Lt Col. George Haritos of the A F O S R for his encouragement and cooperation, to Mr David Larsen of Corning Glass Works for supplying the material and to Mrs Yolande Mallian for typing the manuscript.
REFERENCES 1. Prewo, K. M. & Brennan, J. J., Silicon carbide yarn reinforced glass matrix composites. J. Materials Science, 17 (1982) 1201-6. 2. Brennan, J. J. & Prewo, K. M., Silicon carbide fiber reinforced glass-ceramic matrix composites exhibiting high strength and toughness. J. Materials Science, 17 (1982) 2371-83.
The behavior of ceramic matrix fiber composites under longitudinal loading 3. Larsen, D. C. & Adams, J., Coming Glass Works, private communication. 4. Daniel, I. M., Anastassopoulos, G. & Lee, J.-W., Failure mechanisms in ceramic-matrix composites.
11.
Proc. of SEM Spring Conf. on Experimental Mechanics, 5.
6. 7. 8.
9. 10.
29 May-1 June, 1989, pp. 832-8. Daniel, I. M., Anastassopoulos, G. & Lee, J.-W., Experimental micromechanics of brittle-matrix composites. Micromechanics: Experimental Techniques, AMD-Vol. 102, ASME Winter Annual Meeting, San Francisco, CA, December 1989, pp. 133-46. Marshall, D. B. & Evans, A. G., Failure mechanisms in ceramic-fiber/ceramic matrix composites. J. American Ceramic Society, 611 (May) (1985) 225-31. Marshall, D. B., Cox, B. N. & Evans, A. G., The mechanics of matrix cracking in brittle-matrix fiber composites. Acta Metall., 33(11) (1985) 2013-21. Aveston, J., Cooper, G. A. & Kelly, A., The properties of fiber composites. Conference Proceedings, National Physical Laboratory, IPC Science and Technology Press, Ltd, Surrey, UK, 1971, pp. 15-26. Stang, H. & Shah, S. P., Failure of fiber-reinforced composites by pull-out fracture. J. Materials Science, 21 (1986) 953-7. Aveston, J. & Kelly, A., Theory of multiple fracture of
12. 13.
14.
15.
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fibrous composites. J. Materials Science, 8 (1973) 352-62. Greszczuk, L. B., Theoretical studies of the mechanics of the fiber-matrix interface in composites. Interfaces in Composites, ASTM STP 452, American Society for Testing and Materials, Philadelphia, PA, USA, 1969, pp. 42-58. Hsueh, C.-H., Analytical evaluation of interfacial shear strength for fiber-reinforced ceramic composites. J. American Ceramic Society, 71 (1988) 490-3. McCartney, L. N., New theoretical model of stress transfer between fiber and matrix in a unidirectionally fiber-reinforced composite. Proc. R. Soc. Lond., A425 (1989) 215-44. Lee, J.-W. & Daniel, I. M., Deformation and failure of longitudinally loaded brittle-matrix composites. Composite Materials: Testing and Design, (Tenth Volume), ASTM STP 1120, Glenn C. Grimes, Ed. American Society for Testing and Materials, Philadelphia, PA, USA, 1992, pp. 204-221. Grande, D. H., Mandell, J. F. & Hong, K. C. C., Fiber-matrix bond strength studies of glass ceramic and metal matrix composites. J. Materials Science, 23 (1988) 311-28.
Composites Science and Technology 46 (1993) 105-113
THE B E H A V I O R OF CERAMIC MATRIX FIBER COMPOSITES U N D E R LONGITUDINAL LOADING I. M. Daniel, G. Anastassopoulos Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208, USA
&
J.-W. Lee Composite Materials, Samsung Shipbuilding and Heavy Industry Co., Chang Won City, South Korea (Received 14 October 1991; revised version received 3 December 1991; accepted 27 January 1992) Abstract Failure mechanisms were studied by reflection light microscopy in a unidirectional silicon carbide/glassceramic composite loaded in longitudinal tension. The material behaves linearly up to the point where the first transverse matrix cracks appear. Thereafter, it undergoes a rapid stiffness decrease corresponding to matrix crack multiplication and saturation. These matrix cracks increase in density with applied stress up to a limiting level of 28 cracks/ram or a minimum crack spacing of 361zm (O.O014in), which corresponds approximately to two fiber diameters. Fiber breaks and fiber debonding, which start before matrix crack saturation, continue until final failure. In the last stage the material exhibits quasi4inear behavior with small stiffness variation. Experimental results of crack density and stress/strain behavior were compared with predictions based on a modified shear lag analysis.
carbide (SIC) and mullite fibers. Matrices used successfully include glass, glass-ceramic, silicon carbide, silicon nitride, and aluminum oxide. Composites consisting of glass-ceramic matrices, such as lithium aluminosilicate (LAS) and calcium aluminosilicate (CAS), reinforced with silicon carbide yarn or monofilament have been developed. 1-3 These materials can be fabricated in both unidirectional and multidirectional laminate form, and can attain strengths of 1000 MPa (140 ksi) and ultimate strains of over 1%. Fracture toughness, as determined by a notched beam method, is more than five times that of the monolithic ceramic. 2 Brittle-matrix composite materials behave nonlinearly and inelastically beyond a relatively low stress threshold. Their overall behavior under load is intimately related to the micromechanisms of inelastic deformation and failure that develop and interact with each other. In these materials the failure strain of the matrix is lower than that of the fibers and damage initiates with the development of multiple matrix cracking. 4'5 This is accompanied by partial debonding at the fiber/matrix interface and fiber fractures, which strongly depend on the properties of the interface. This type of behavior has been observed in glass ceramics reinforced with SiC fibers4-7 and cement reinforced with glass or steel fibers. 8'9 The micromechanics of stress transfer and fracture of brittle-matrix composites is being studied analytically by many investigators. T M Although the various failure mechanisms are known, their relative magnitude, exact sequence and quantitative effect on overall behavior vary from case to case. Of great importance is the influence of constituent properties, including fiber, matrix and interphase region, and the processing residual stresses, on the failure process and overall behavior. This paper describes experimental results obtained
Keywords: ceramic-matrix composites, failure mechanisms, micromechanics, matrix cracking, fiber debonding, shear-lag analysis INTRODUCTION The application of ceramic materials to hightemperature structures is hampered by their inherent limitations of brittleness, low strain to failure, low tensile strength, and low fracture toughness. These limitations are somewhat overcome by reinforcing the brittle ceramic materials with high-strength fibers. The resulting composite materials display appreciable ductility, higher strength and higher fracture toughness than the monolithic matrix materials. Promising reinforcements include carbon, silicon
Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd.
105
106
I . M . Daniel, G. Anastassopoulos, J.-W. Lee
on failure mechanisms and overall behavior of a SiC/CAS unidirectional composite under longitudinal tensile loading. Experimental results are compared with predictions of a recently presented modified shear lag analysis. 14 EXPERIMENTAL P R O C E D U R E
The material investigated was SiC/CAS, calcium aluminosilicate glass ceramic reinforced with silicon carbide fibers, manufactured by Corning Glass Works. The fiber is silicon carbide yarn known as Nicalon (Nippon Carbon Co.). The fiber is available in continuous length tows with an average diameter of 15 #m. The composite material was obtained in the form of 8 ply and 24 ply unidirectional plates. The composite material was characterized first to obtain average physical and mechanical properties. Transverse to the fiber cross sections were examined under the microscope and photomicrographs were taken and analyzed. Unidirectional specimens instrumented with strain gages were tested under longitudinal and transverse tensile loading in a servohydraulic testing machine to determine average mechanical properties. Specimens used for the study of failure mechanisms under the microscope were 4.62x0.76cm (1-82 x 0-30 in) coupons, 0.Smm (0.02in) thick, cut from the edge of a thick 0° composite plate. (The width of the coupon, 0.76 cm, was in the thickness direction of the composite plate.) These coupons were polished on one side and instrumented with strain gages on the other side. They were tabbed with 2.54cm-long (1.0in) glass/epoxy tabs. These tabs were bonded to the specimen only over a length of l c m (0.4in) from each end and extended beyond the ends of the specimen. A special loading fixture was designed and built for loading the specimens under a reflection optical microscope (Fig. 1). Load is applied and controlled by means of a pneumatic cylinder. The specimen with the grips attached is mounted onto a reaction frame attached to one end of the pneumatic cylinder. One end of the specimen is connected to the moving piston while the other end is reacted at the other end of the reaction frame through a strain gage load cell. The entire assembly, including air cylinder, reaction frame, specimen with grips and load cell, is suspended with counterweights from a movable upright frame. Thus, any part of the loaded specimen can be moved to the stage of the microscope without any weight or force being exerted on the microscope. These specimens were loaded in steps with still photomicrographs taken at every step and videotapes recorded during loading. In one case a specimen was loaded continuously and a videotape record was taken for real time observation of the initiation and propagation of the various failure mechanisms.
Fig. 1. Fixture and setup for loading composite specimens under the microscope. EXPERIMENTAL RESULTS Material characterization Some basic properties of the matrix and fiber constituents obtained from the literature and from previous tests are given in Table 1. Examination of photomicrographs of transverse cross sections showed that the fiber diameter ranges between 8 and 20 #m with a median value of 15/~m. The fiber distribution is nonuniform with an average
Table 1. Constituent material properties
Property Maximumuse temperature (°C (°F)) Fiber diameter(~m) Density(g/cm3) Coefficientof thermal expansion (10-6/°C (10-6/°V)) Elastic modulus (Gaa (106psi)) Tensile strength (MPa (ksi))
CAS matrix3
SiC fiber1'2'5
1350(2460)
1300(2370)
-2.8 5.0 (2.8)
15 2.6 3-1 (1.7)
98 (14-2)
163(23-6)
124 (18) (flexural)
1930(280)
The behavior of ceramic matrix fiber composites under longitudinal loading Table 2. Measured properties of SiC/CAS unidirectional composite
Property
Value
Fiber volumefraction, Vt Ply thickness, t (mm (in)) Longitudinal modulus, E l (GPa (Msi)) Transverse modulus, E2 (GPa (Msi)) In-plane shear modulus, Gt2 (GPa (Msi)) Major Poisson'sratio, v~2 Longitudinal tensile strength, FIT (MPa (ksi)) Transverse tensile strength, F2T(MPa (ksi)) Longitudinal ultimate tensile strain, e~T Transverse ultimate tensile strain, ezr "
0.39 0-38 (0.015) 121 (17.6) 112 (16.2) 52 (7.5) 0-18 393 (57) 55 (8) 0"0084 0-0005
fiber volume fraction, Vf, of 0-39. A low degree of porosity was measured (Vv = 0.01). Basic properties obtained from macromechanical tests of the unidirectional SiC/CAS composite are tabulated in Table 2. Macromechanical behavior Figure 2 shows a typical longitudinal stress/strain curve to failure obtained from a test in the servohydraulic testing machine. This curve displays several characteristic features which are related to the failure mechanisms and failure process. The initial linear region AB corresponds to the elastic linear behavior of the material prior to any significant microfailures. The measured modulus and the one predicted by the rule of mixtures are:
E1 (measured) = 125 GPa (18.1 Msi) E1 (predicted) = 123 GPa (17.9 Msi) At an applied stress of approximately 207MPa (30 ksi) (Point B) there is a marked departure from linear behavior corresponding to transverse matrix cracking and to fiber/matrix debonding. Region BC is
500 v. ,f"
400 -
300-
/
.J
,/
,/
j . -, ; ' " D
...,"
1:5
f" C
B .............. ,f
200It
/ 100-
/ ,,'A i
0
0.0
i '0.2
O.
0.6
O.
1.0
Strain, e, (~)
Fig. 2. Stress/strain curve for [08] SiC/CAS specimen under uniaxial tensile loading.
107
believed to correspond to transverse crack multiplication up to a saturation density to be discussed below. This is followed by a quasi-linear region CD in which no further significant matrix damage takes place, and a final slightly steeper linear region DE. Fiber fractures and debonding which start before transverse matrix crack saturation continue until final failure. The terminal modulus (both secant and tangent) is E'I = 52 GPa (7.6 Msi) Assuming that the entire load is carded by the fibers in this last region, the predicted terminal modulus is 63.5 GPa (9-2 Msi). The difference may be due in great part to the fact that some of the fibers are already damaged. Failure mechanisms The micromechanical behavior and failure mechanisms were studied by testing and observing specimens under the microscope. No cracking was observed at applied stresses below 100MPa (14.5 ksi). The first isolated transverse matrix cracks appeared above this stress level, but crack density started increasing sharply above a stress of 180 MPa (26 ksi) corresponding to a strain of el = 1-14 x 10 -3 which is very close to the ultimate tensile strain of the bulk matrix material. Figure 3 shows typical photomicrographs with transverse matrix cracking increasing in density with applied stress. The transverse crack density was measured with the microscope along three longitudinal zones of the specimen and the average crack density recorded. A plot of applied stress versus normalized crack density was superimposed on the average stress/strain curve for the material (Fig. 4). The crack density, t, was normalized by multiplying it by the fiber diameter, 2rf. This density increases rapidly between 180MPa (26 ksi) and 275 MPa (40 ksi). It reaches a saturation level of 28 cracks/mm, or a minimum crack spacing of 36/tm (0-0014 in), at an applied stress of approximately 275 MPa (40ksi). Thus, the minimum crack spacing appears to be on the order of two fiber diameters. Isolated fiber breaks were observed before saturation of transverse matrix cracking and increased in frequency as the matrix cracks reached their maximum density (Fig. 5). Most fiber breaks occurred at a short distance, one to four fiber diameters, from the nearest matrix crack, possibly as a result of some limited debonding causing a local stress rise at the end of the debonded length. As the load was increased the fiber cracks opened wider indicating further debonding and sliding. At a stress level of approximately 345 MPa (50 ksi) no further matrix cracking or fiber fracture is observed. It is likely, although not possible to observe in the microscope, that the next failure process
I.M. Daniel, G. Anastassopoulos, J.-W. Lee
108
Fig. 3. Typical photomicrographs showing initiation and multiplication of transverse matrix cracks under longitudinal tensile loading.
500
consists of nearly total fiber debonding leading to linear behavior in the last stage D E up to total failure (Fig. 2).
400 o o
/
o
~'~ 3 0 0
o
./ ,./
THEORETICAL ANALYSIS
6 ~'200
o f" o o
I"
o
/ /'
100-
o
,,
w
0.0
- ....
Streu-Strain
o o o o
Stress
v~.
Crack
Density
, , , , , , , ,. , , , , , , ~ , , r 0.2
0.4
0.8
Strain, e, (~) Normalized Crack Density
0.8
1.0
(2Xr~)
Fig. 4. Stress/strain and stress versus crack density curves for [08] SiC/CAS specimen under uniaxial tensile loading.
In materials with low interfacial shear strength, such as the SiC/LAS system, it can be assumed that debonding occurs immediately after matrix cracking and that load transfer is effected via the interface frictional stress. 6'7 However, as further experimental evidence shows, the S i C / C A S system studied here has appreciably higher interracial shear strength. For this reason a model with elastic stress transfer was used. Under longitudinal loading for fully bonded fibers, the axial strain is the same in both fiber and matrix. As the load increases, the phase with the lower ultimate strain, in this case the matrix, will fail by
The behavior of ceramic matrix fiber composites under longitudinal loading
109
Fig. 5. Photomicrographs illustrating fiber crack opening between matrix cracks. transverse cracking (Fig. 6a). In the vicinity of a matrix crack the axial stress in the matrix is relieved (Fig. 6c) while the axial fiber stress is increased and a highly localized interracial shear stress develops (Fig. 6b). Matrix cracking increases in density up to a point where the interracial shear stress exceeds the fiber debond strength and debonding initiates while the maximum axial stress in the matrix remains below its tensile strength. A modified shear lag analysis was conducted for a cylindrical element of matrix with a single fiber between two matrix cracks (Fig. 7). 14 The stress/strain behavior is linear up to an applied stress level where the matrix stress (or strain) reaches the ultimate value of the matrix tensile strength (or strain). This stress level is given by E1
Oa = Em (FmT -- O'm) where era = applied stress in composite o . . = residual stress in matrix
(1)
FreT = matrix tensile strength Em = matrix modulus E1 = longitudinal modulus of composite Above this applied stress level matrix cracking and a redistribution of stresses take place. The average axial stresses in the matrix and fiber and the interracial shear stress have the following distributions:
[gmOa Orm)
Ef of=~l
(
1
(2)
cosh (-- -
Em Vm \ 2 1 + E,-----~f cosh(-~)
oa
L M. Daniel, G. Anastassopoulos, J.-W. Lee
110
Axial stress in fiber ~a
ttItttttttt? ?tt?
inteffacial shear stress in fiber
#
(b) BIB
6 S a
StreSS
"/,//4
i wl
(
atrix
I1'1 I / t / /i / ¢ ///// .////
, //// zl //j / / ~/// z / ~/// i/t
~a
(a)
(c) IF~. 6. Matrix cracking and local stress distributions in longitudinally loaded specimen. sinh(~-rex) Z'i(X) -----ocrfVm2vf~--E-~I
Orm)
(od) cosh ~-
where 1 m =~Gf + ~ m
(4)
× [~ × Vf(rf --r.) (4
where o , = average axial stress in matrix of = average axial stress in fiber ~i = interfacial shear stress Orm = axial residual stress in matrix o~f = axial residual stress in fiber El, Em= fiber and matrix moduli, respectively Vf, Vm -- fiber and matrix volume fractions, respectively x = axial coordinate measured from crack face l = crack spacing The parameter tr is obtained as 14
~
2 =
- -
Arf
E, x
-
-
EfEm Vm
r,
l
I - Vf ]
Gf, Gm= fiber and matrix shear moduli, respectively rf, rm -- fiber and matrix radii in model, respectively In this range of deformation, prior to any fiber debonding, the matrix crack spacing I is obtained from eqn (2) by setting x = l/2 and am --- FreT
Em°a + ElO__rm_ 1 = 2c~c°sh-1 [ E m o a +
El(arm
--
FreT) }
(7)
The above relation gives the matrix crack spacing as a function of applied stress up to the point of fiber debonding initiation.
(5) r
I
(6)
l
Fig. 7. Cylindrical element of composite with a single fiber and two matrix cracks.
The behavior of ceramic matrix fiber composites under longitudinal loading The maximum interfacial shear stress ~ at the crack location is calculated from eqn (4) by setting x = 0.
v,t-VF +°=/Em°a) t a n h - ~
ri(0) =°¢rf Vm2
In the bonded area (d -< x -< 1 - d)
(8) (2Vfrfd
When the above shear stress reaches the value of the interracial shear strength F~ debonding occurs and increases with applied stress. The interfacial shear strength F~s is not an easily measurable quantity. An estimate of its relative value can be obtained from crack density measurements alone. For materials with relatively low matrix tensile strength and sufficient interracial shear strength, it can be assumed that debonding starts after matrix crack saturation, i.e., there is no more matrix cracking once debonding starts. These conditions can be expressed as follows: /min
orrn = FreT at X = 2
(9)
and ri(0) = Fiis at
x = 0
(10)
which, when used with eqns (2) and (4), yield Fis _ V m o ~ f
Fmr
cotha'lmin
Vf 2
The above gives a relative magnitude of the interracial shear strength F~ in terms of the matrix tensile strength. The extent of debonding, d, is calculated as follows (Fig. 8): In the debonded area (0 -< x < d and 1 - d -< x -< l) the redistributed stresses are
2Vfrfx Om = - -
(12)
Vmrf
of =
Oa
21~fX
v,
rf
(13)
l'i = ~'f
(14)
where rf is the interracial frictional stress in the debonded area (which was not accounted for in Ref. 14).
Debon~ng P
Crack Face
L. l
q"\ V~
( EmVm
+ \ E1Vf
..J -!
Fig. 8. Geometry and loading of cylindrical model with partial debonding at the interface.
E m
)coshoc(l/2-x)
E l O'a--Orm c o s h a ' - ~ - - - d )
(15)
2rfd) cosh oc(l/2 - x) Oa -- Off - - -rf / cosh
ot(1/27-~
(16)
2Vf rrd] sinh oc(1/2- x) a'rfVm Em 2Vf [ E l O'a "~"O'rm cosh d) (17) The debonded length, d, is calculated by assuming that the interfacial shear at the end of the disbond is equal to the interfacial shear strength. Setting ri(d) = F~s and x = d in eqn (17) we obtain an implicit relationship for d which can be solved by iteration techniques. However, if friction is neglected, i.e. if rf = 0, an explicit expression can be obtained for d:
1-- j
(11)
4
111
(18)
where 2Fis Vf E1 = -¢x'rf Vm Emma + ElOrrm
(19)
The average axial strain in the damaged composite, which is taken to be equal to the average fiber strain, is obtained by calculating the average stress in the fiber from eqn (16) and dividing it by the fiber modulus. Thus, a stress/strain relationship can be predicted for all stages of damage up to failure.
COMPARISON OF PREDICTED A N D EXPERIMENTAL RESULTS
The theoretical analysis discussed before, subject to the validity of assumptions, can predict transverse matrix crack density, debonded fiber length and average strain in the composite as a function of the applied stress, in terms of geometrical, elastic and strength parameters of the fiber and matrix. These include fiber volume fraction, fiber and matrix moduli and strengths and interfacial shear strength. The matrix tensile strength was obtained by observing the applied stress at matrix crack initiation and taking into consideration a calculated value of 55 MPa (8ksi) for the axial residual stress in the matrix (Or,). The radial compressive residual stress at the fiber matrix interface was calculated to be approximately 6 0 M P a (8.7ksi). For a frictional coefficient of # =0.15, this would yield an interface
112
1. M. Daniel, G. Anastassopoulos, J.-W. Lee
Normalized Debonded Length (2d/l) Normalized Crack Density ( 2krf ) 0.0 0.z 0.4 0.6 0.B 1.o 500 j. ]
i /
+-/ /
j'+/
+
CX,300
-I
/o,
°+ ....-
"/'/'"/"/:
/
/
/
SUMMARY AND CONCLUSIONS
,.J
I° /
100 Io / -1 / I /
0.0
+-----Expertm,tn~aStress--Str~ - ~ ~
P r e d i c t i o n of S t r e s s - S t r a i n of S t r e s s vs. C r a c k Density S.tress vs. _Debond.~mg ..
~ -- P r e ~
0.2
0.4
0.8
The debonded fiber length is expressed as a fraction of matrix crack spacing. Debonding starts when the matrix crack density reaches the maximum level and increases up to approximately 75% of the crack spacing (Fig. 9). It was not possible to observe and monitor the extent of debonding experimentally.
O.B
1.0
Strain, ~, (m) Fig. 9. Stress/strain, stress versus matrix crack density and stress versus debonded length curves for SiC/CAS [08] unidirectional composite. frictional stress of rf = 9 MPa (1-3 ksi), a relatively low value. The interfacial shear strength as calculated from eqn (11) was approximately 221MPa (32 ksi). Although this value may appear high, it is in substantial agreement with experimental results obtained by indentation tests on SiC/CAS material.15 In the predictions a matrix tensile strength of 159 MPa (23 ksi) and an interfacial shear strength of 221 MPa (32 ksi) were used and the residual stresses and friction were neglected. The predicted stress/strain, stress versus crack density and stress versus debonded length curves are shown in Fig. 9 and compared with experimental results. The predicted stress/strain curve agrees well with the experimental one initially, but it deviates from it at higher stresses. Possible factors causing this discrepancy are: (1) a degree of random fiber misalignment, '4 (2) some uncertainty of the true effective values of tensile matrix and interfacial shear strengths and (3) the neglect of residual thermal and interface frictional stresses. The fiber misalignment also may explain why the measured strength of the composite is much lower than the predicted value of 814MPa (118 ksi) based on the constituent properties. The predicted crack density increases rapidly between 211MPa (30.6ksi) and 221MPa (32ksi) when it reaches the saturation level of 29 cracks/mm or a minimum crack spacing of 34 ~m (0-00134in) (Fig. 9). The experimental results show matrix cracking starting earlier at an applied stress of 104MPa (15 ksi) and reaching a saturation crack density of 28 cracks/mm or a minimum crack spacing of 36 ~m (0.00141 in). The apparent early cracking is due primarily to the tensile residual stress in the matrix and to the statistical nature of its strength.
The behavior of a unidirectional ceramic matrix composite under longitudinal tensile loading was studied. The material investigated was SiC/CAS, calcium aluminosilicate glass ceramic reinforced with silicon carbide fibers. Unidirectional specimens were loaded under the microscope in a specially designed fixture. Failure mechanisms were observed in real time and recorded by photomicrography and video photography. The initial failure consists of transverse matrix cracks increasing in density with applied stress up to a limiting level of 28 cracks/mm or a minimum crack spacing of 3 6 # m (0-00140in), which corresponds approximately to two fiber diameters. Experimental results were compared with predictions from a modified shear lag analysis. The analysis predicts a minimum crack spacing of 34/~m (0.00134 in) and fiber debonding initiation at the point of maximum crack density. It also predicts a complete stress/strain curve to failure. This curve agrees well with the experimental one for the most part, but it deviates from the experimental results at higher stresses. This discrepancy may be due to a variety of factors, including fiber misalignment, uncertainty about constituent and interface properties and the unknown effects of residual stresses and interfacial friction.
ACKNOWLEDGEMENTS The work described here was sponsored by the Air Force Office of Scientific Research (AFOSR). We are grateful to Lt Col. George Haritos of the A F O S R for his encouragement and cooperation, to Mr David Larsen of Corning Glass Works for supplying the material and to Mrs Yolande Mallian for typing the manuscript.
REFERENCES 1. Prewo, K. M. & Brennan, J. J., Silicon carbide yarn reinforced glass matrix composites. J. Materials Science, 17 (1982) 1201-6. 2. Brennan, J. J. & Prewo, K. M., Silicon carbide fiber reinforced glass-ceramic matrix composites exhibiting high strength and toughness. J. Materials Science, 17 (1982) 2371-83.
The behavior of ceramic matrix fiber composites under longitudinal loading 3. Larsen, D. C. & Adams, J., Coming Glass Works, private communication. 4. Daniel, I. M., Anastassopoulos, G. & Lee, J.-W., Failure mechanisms in ceramic-matrix composites.
11.
Proc. of SEM Spring Conf. on Experimental Mechanics, 5.
6. 7. 8.
9. 10.
29 May-1 June, 1989, pp. 832-8. Daniel, I. M., Anastassopoulos, G. & Lee, J.-W., Experimental micromechanics of brittle-matrix composites. Micromechanics: Experimental Techniques, AMD-Vol. 102, ASME Winter Annual Meeting, San Francisco, CA, December 1989, pp. 133-46. Marshall, D. B. & Evans, A. G., Failure mechanisms in ceramic-fiber/ceramic matrix composites. J. American Ceramic Society, 611 (May) (1985) 225-31. Marshall, D. B., Cox, B. N. & Evans, A. G., The mechanics of matrix cracking in brittle-matrix fiber composites. Acta Metall., 33(11) (1985) 2013-21. Aveston, J., Cooper, G. A. & Kelly, A., The properties of fiber composites. Conference Proceedings, National Physical Laboratory, IPC Science and Technology Press, Ltd, Surrey, UK, 1971, pp. 15-26. Stang, H. & Shah, S. P., Failure of fiber-reinforced composites by pull-out fracture. J. Materials Science, 21 (1986) 953-7. Aveston, J. & Kelly, A., Theory of multiple fracture of
12. 13.
14.
15.
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fibrous composites. J. Materials Science, 8 (1973) 352-62. Greszczuk, L. B., Theoretical studies of the mechanics of the fiber-matrix interface in composites. Interfaces in Composites, ASTM STP 452, American Society for Testing and Materials, Philadelphia, PA, USA, 1969, pp. 42-58. Hsueh, C.-H., Analytical evaluation of interfacial shear strength for fiber-reinforced ceramic composites. J. American Ceramic Society, 71 (1988) 490-3. McCartney, L. N., New theoretical model of stress transfer between fiber and matrix in a unidirectionally fiber-reinforced composite. Proc. R. Soc. Lond., A425 (1989) 215-44. Lee, J.-W. & Daniel, I. M., Deformation and failure of longitudinally loaded brittle-matrix composites. Composite Materials: Testing and Design, (Tenth Volume), ASTM STP 1120, Glenn C. Grimes, Ed. American Society for Testing and Materials, Philadelphia, PA, USA, 1992, pp. 204-221. Grande, D. H., Mandell, J. F. & Hong, K. C. C., Fiber-matrix bond strength studies of glass ceramic and metal matrix composites. J. Materials Science, 23 (1988) 311-28.