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On whisker toughening in ceramic materials
Acta metall, mater. Vol. 39, No. 11, pp. 2509--2517, 1991
0956-7151/91 $3.00 + 0.00 Pergamon Press pie
Printed in Great Britain
ON WHISKER T O U G H E N I N G IN CERAMIC MATERIALS M. BENGISU, O. T. INAL and O. TOSYALI New Mexico Institute of Mining and Technology, Department of Materials and Metallurgical Engineering, Socorro, NM 87801, U.S.A. (Received 8 November 1990; in ret,ised form 8 May 1991)
Al~ract--Toughening achieved by incorporation of whiskers to ceramic matrices was calculated based on an approach that combines individual contributions from effective toughening mechanisms. The relative importance of each mechanism, among others, was found to substantially vary depending on the selection of whisker and composite parameters. Analysis of the effect of various parameters showed that whisker strength, pullout length, debond length, and volume fraction of whiskers are the most important parameters in whisker toughening. Very good agreement was found between predictions made by the combinatory approach and experimental toughness values of AI20~/SiC, composites. R/~mr~. On a calcul6 le durcissement dfi ~i rincorporation de trichites dans des matrices de c~ramiques, en se basant sur une approche qui combine les contributions individuellesdes divers m~'canismeseflicaces de durcissement. L'importance relative de chaque m6canisme par rapport aux autres varie de faqon importante suivant le choix des param~tres du trichite et du composite. L'analyse de l'effet des diff6rents param6tres montre que la r6sistance mc~canique des trichites, leurs longueurs d'arrachement et de d~coh6sion, et leur fraction volumique sont les param6tres les plus importants dans le durcissement par trichites. L'accord est tr6s bon entre les pr6visionsde I'approche combinatoire et les valeurs exl~rimentales de la duret~ des composites AlzO3/SiC~mhi,,. Znt~mm~nfMsung--Der H~irtungseffektdurch Einlagerung von Whiskern in keramisches Material wird auf der Basis einer N~iherungberechnet, die die einzelnen Beitnige der wirkenden H~rtungsmechanismen kombiniert. Die Wichtigkeit der einzelnen Mechanismen im Vergleich zu den anderen variiert betrichtlich je nach Auswahl der Whisker- und VerbundmateriaI-Parameter. Aus der Analyse des Einflusses verschiedener Parameter geht hervor, dab Festigkeit, Ausziehl/inge, Abl6sel/inge und Volumanteil der Whisker die wichtigstenParameter der Festigkeitsteigerungdurch Whiskereinlagerungsind. Die mit dieser Kombinations-N~iherungerhaltenen Voraussagen und die experimentell erhaltenen Z/ihigkeitswertevon AlzO3/SiCw-Verbundmaterialienstimmen sehr gut iiberein.
1. INTRODUCTION The possibility of increasing the toughness of ceramic materials to levels where they can be used as engineering components has focussed attention on various toughening methods. Significant toughening has been achieved in many ceramics by the incorporation of whiskers [I-5]. Whisker toughening is as effective as ZrO2 toughening at ambient temperature and, in contrast to ZrO z toughening, the amount of toughening remains relatively constant at high ( > 1000°C) temperatures. Toughness values can further be increased by the incorporation of both ZrOz and whiskers and, in some cases, synergistic toughening has also been observed [2, 6]. Although toughness levels in fiber reinforced ceramic matrix composites (CMCs) are the highest among all ceramics [7, 8], whisker reinforcement offers advantages of easier fabrication and a higher degree of isotropy in material properties due to the smaller aspect ratios involved. To make the best use of whisker toughening it is of primary importance that the effect of fabri-
cation and material parameters are correctly predicted. Recent models of [lecher et al. [9] and Liu et al. [10], to some extent, have shown the effect of some of these parameters on toughening. The aim of this paper is to discuss, and further develop, available models that describe whisker toughening and to identify important toughening parameters.
2. DISCUSSION OF PROPOSED TOUGHENING MECHANISMS Observations in whisker reinforced CMCs, so far, suggest the possibility of five toughening mechanisms to be operative. These mechanisms are crack deflection [10--12], crack bowing [11], microcracking [13, 14], whisker pullout [9, 15], and crack bridging [9, 16-191. Microcracking is not uncommon in whisker reinforced ceramics. Li and Bradt [20] have shown various possibilities for thermal microcracking in mullite and Si3N~ matrix/SiC whisker (SIC,,) corn-
2509
2510
BENGISU et a/.: WHISKER TOUGHENING IN CERAMIC MATERIALS
posites. Microcracking was observed in AI203/SiC w composites by Angelini et al. [21] and Ruhle [14], and in TZP/SiCw composites by Claussen et al. [22]. Although microcracking is likely to occur in whisker reinforced ceramics, toughening due to microcracking in non-transforming ceramics is shown to be insignificant by recent theoretical and experimental analyses. Evans and Faber's [23] model of microcrack induced toughening is based on the concept that microcracking leads to dilatations in a material which produces compressive stresses on the macrocrack. Their model predicts toughness increases of only 10%. Another model by Dolgopolsky et al. [24] considers two opposing effects due to microcracking. One effect is the reduction of crack growth resistance due to linking of microcracks and the other effect is crack shielding by nucleated microcracks. Karbhari's [25] calculations, based on the application of their model to a SiC fiber/glass composite, predict a 3.1% decrease in the composite cracking stress in the case of crack amplification due to microcracks (i.e. linking of microcracks) or a 3.5% increase in the case of crack shielding by microcracks. Again, the effect of microcracking on crack resistance is not significant. In light of these observations, the toughening contribution due to microcracking in whisker reinforced ceramics can be assumed to be negligible at best. Crack bowing originates from resistant second phase species in the path of a propagating crack. According to Faber and Evans' calculations, the highest toughening by this mechanism is achieved in the case of discs with high aspect ratios whereas the lowest toughening is achieved in the case of rods. The crack bowing theory suggests that toughening is mainly a function of obstacle penetrability (related to obstacle toughness and strength, and coherency of obstacle/matrix interface), volume fraction of the second phase, and obstacle size. It follows that whiskers are theoretically less effective in bowing a crack front than for example metallic inclusions or discs since whiskers have rod-like shapes and they are easier to penetrate by the crack due to their low fracture toughness. Indeed, direct evidence of crack bowing in whisker reinforced ceramics is generally not available except for crack pinning observations in AI203/SiC . composites during in situ TEM straining experiments [21]. Therefore, it can be argued that crack bowing is not a major toughening mechanism in whisker reinforced ceramics, although it should not be ruled out completely. Crack pinning observations suggest that a minor contribution is possible, though crack bowing theory is not fully established that it can be used to calculate this contribution with any confidence. The remaining mechanisms are the ones that lead to the observed toughness increases in whisker reinforced CMCs in the present authors' opinion; i.e. whisker pullout, crack deflection, and crack bridging.
2.1. W h i s k e r pullout
Becher et al. [9] calculated whisker pullout toughening considering the work done by sliding whiskers. The toughness increase was determined to be AGp = ( 4 V : ~ Pp)/(3Ewr 2)
(I)
where Ti is the whisker/matrix interfacial shear strength, Vf is the whisker content, lp is the pullout length, Ew is the elastic modulus of whiskers, and r is the whisker radius. For plane strain conditions Gc = K~,(I - v~)/Ec
(2)
where vc is the Poisson ratio of the composite and E, is the composite elastic modulus. The strain energy release rate and the fracture toughness of the composite can be divided into two terms Gc = Go + AGp
(3)
f~ = fo + AK,
(4)
where Go is the strain energy release rate and Ko is the toughness of the matrix and AGp and AKp are pullout contributions to each term, respectively. Combining equations (2) to (4) yields (Ko + AK,) 2 = GcE~/(I - v~)
(5)
from which AKp can be derived as AKp = [GeEc/(I
- r e2) ]
I/2-
K0.
(6)
Other contributions related to whisker pullout, i.e. toughening by whisker/matrix debonding and whisker fracture, are found to be negligible from our calculations based on the analysis of Wells and Beaumont [26]. Although their equations are based on a brittle polymer/short brittle fiber composite system, they can conveniently be applied to whisker reinforced CMCs as well. Whisker pullout is frequently observed in whisker reinforced CMCs although in many cases pullout lengths are limited to l-2/~m [27-30]. Becher et al. [9] measured the average pullout lengths as 1.2-2.0 #m in AlzO3/SiCw, 0.4--0.8 #m in mullite/ SiCw, and 0.2-0.4/zm in glass/SiCw composites, where the whiskers were untreated. Pullout toughening can be promoted by increasing the pullout lengths. This can be done by preventing the formation of interaction layers between the matrix and the whiskers and using whiskers with smooth surfaces to prevent mechanical keying. Chemical interaction layers have been observed by careful examination of whisker/matrix interfaces. Thin amorphous layers have been observed in Si3 N(/SiCw composites formed due to chemical reaction at high fabrication temperatures [31, 32]. Similarly, high temperature oxidation or reaction was found to produce a surface scale on SiC whiskers in alumina matrices, sometimes to result in mullite interfaces [33]. When the oxygen coraent on SiC whiskers is high, the resulting composite toughness is lower [9, 34].
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS Tiegs [35] demonstrated that fracture toughness of an A1203/20 vol.% SiC, composite can be increased from 4.7 to 6.5 MPa m ~/2 by carbon coating of the whiskers. The presence of carbon results in a weak interfacial bond which enhances pullout and possibly other mechanisms such as crack deflection and bridging. Homeny et al. [19] observed that the presence of SiO2 promotes whisker/matrix bonding which prevents whisker pullout. Surface treatment of the whiskers in H2/Ar atmospheres activated pullout and increased the composite toughness. These results imply that increased pullout lengths achieved by surface treatment result in higher toughening. 2.2. Crack deflection
Crack deflection is commonly observed in whisker reinforced ceramics [2, 5, 21, 36] regardless of the matrix type. Liu et al. [10] modeled toughening by crack deflection for whisker reinforced CMCs with whiskers randomly oriented in two dimensions to account for whisker orientation in hot pressed composites. Liu et aL compared their predictions to experimental results from the literature. They state that their results appear to agree with experimental values. The present authors have compared recent data from the literature [2, 4, 9, 15, 27, 37] and found no such agreement as can be seen in Fig. 1. We believe that the attribution of all observed toughening, in whisker reinforced ceramics, to only crack deflection overemphasizes this mechanism. Carter and Hurley
2511
[12] measured crack deflection angles in MoSiz/SiC, composites and reveal that a 20voi.% whisker addition to the matrix increases median deflection angles from 7.2 ° to only 13.0, while the toughness was increased from 5.3 to 8.2 MPa m 1/2 (about 50%). Seshadri et al. [38] showed, by computer simulation, that significantly lower toughening should be expected from crack deflection than originally suggested by Faber and Evans [1 l] since the deflection angles are overestimated by their model. The analytical model suggests that the main crack is deflected at all angles evenly up to a maximum of 90°, ignoring a large contribution of zero angle, i.e. undeflected cracks. To account for this discrepancy we have calculated deflection toughening for maximum average deflection angles ranging from $~.~/5 to ~ . The total strain energy release rate for cracks tilted and twisted by rods is given as [11] (G)rod=(~/2)(G)T +(~/2)(G) t
(7)
where ( G ) r and ( G ) ~ are strain energy release rates due to tilting and twisting of the crack front, respectively, ( and ~ are ratios of undeflected to deflected crack front lengths. Liu et al. determined the strain energy release rate due to twist of the crack front as CI C1 Co t'~;2
x {2v sin 2 ~ + cos 2 ~bcos2(Q.)/2) legend
x [1 + 2 sin2((2)/2)]} 2
solid l i n e - Liu et al. ~0 •-
+ {sin ~ cos ~b cosZ((2)/2)
T i e g s 3s
o - S h a w a n d F a b e r ~s
× [cos:((2)/2) - 2v] + sin2((2)/2)
• - l i o e t al. 4
x [3 cos2((2 )/2) - 2v]}2) d01 d0z da db.
~t- Claussen and Petzow 2 , ~ - S m i t h et al. 3~ o-
(8)
B e c h e r et al. 9
Similarly, the tilt contribution is calculated from
12
C1 t't C~ t'~
o
10
0
x (cos4((I)/2) d01 d02 da ~ )
i,1
(9)
where G m is the strain energy release rate of the matrix, a and b are relative locations where rods are intercepted by the crack plane, ( 2 ) is the average tilt angle, v is Poisson's ratio, 01 and 02 arc the angles between rod consecutive axes and the fracture plane, and ~ is the twist angle given by -- tan- I[(a sin 01 + (1 - b) sin 02)/A']
(10)
and A' is given by A'=[A/H-acos01+(1 0
10 whisker
20 content
30
40
50
(vol.%)
Fig. 1. Comparison of experimental toughening data in AIzO3-SiC. composites to Liu et al.'s predictions.
- b) cos 02l.
(ll)
Since 4~ is a function of a, b, 0t, and 02, it cannot be varied independently inside the multiple integrals. Therefore, in order to vary 4~ and calculate toughening at different deflection angles, Faber and Evans'
2512
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS
model for a composite containing spherical particles has been used. The results were multiplied by a factor of 2.857, calculated from the results of Faber and Evans for deflection toughening by rods and spherical particles, which compensates for the difference in the respective cases. Faber and Evans derived the ratios of strain energy release rates of undeflected and deflected cracks as
toughening phenomenon in whisker reinforced ceramics by primarily attributing toughening to the crack bridging mechanism. They derived toughening as dK ~ = {[(/(0)2 + 4EcVf(a'[)ZlDs/ v~)E'] ~:2 -
(l -
K0}/2 (16)
where K0 is the matrix toughness, E c and E* are composite and whisker elasticmoduli, respectively,Vf is the volume fraction of whiskers, v is the Poisson ( G ) / G m = l/O~,,J ° [sin O/(sin • + cot ~)] ratio of the composite, and/DE is the debond length. Becher et al. calculated whisker toughening in x [(k~)2 + (k[):] d~ + I / O ~ alumina and mullite/SiC, composites from equation (16) and compared their predictions to their experx (cot q~ cos ~/sin 2 • + c o t O) imental results. In their calculations a whisker strength of I0 GPa has been used and this is significantly higher than 6.89 GPa, the average whisker X [(1/* f:k iTde0 z + (I/¢~::k~d*)2] d¢ ~ strength quoted for the same whiskers by the manu(12) facturer[5].W e have recalculatedwhisker toughening from equation (16) using u~' = 6.89GPa and other where k[ and k~ are local stressintensityfactors for parameters given in Appendix I. Figure 2 compares the tiltedand twisted portions of the crack, respect- Becher et al.'s and the present authors' results.It is ively; these are detailed elsewhere [II]. seen that the present results are about one half of The maximum twist angle is given by what Becher et al. calculated for each material for differentIDs/r ratios.Since Becher et aL showed their • . ~ = sin - l(2r/A) (13) predictions to be very close to their experimental where A can be calculated from [39] results, while a more realisticcalculation results in half of theirpredictions,itisconcluded that the crack A / r ~_ (e4Vr/V~~ x 1/2 e -~ d x . (14) bridging contribution to whisker toughening was J4 Vt overestimated. Experimental evidence thus supports In the present claculations, ¢~m,~ derived from the idea that crack bridging is not the only important equations (13) and 04) is divided by 5 and equation toughening mechanism, as discussed earlier. (12) is solved for each of these five angles. The a AI203 ,o w = 10 GPa, ~ = 5 toughening increment is calculated from the followb AI203 ,Ow = 10 GPII,~ = 3 ing equation GC/G m = G m / ( G ) .
(15)
Results from these calculations are used in total toughening calculations to be detailed later. 2.3. C r a c k bridging
Crack bridging phenomena have been observed in various composites and are well documented in rubber toughened polymers [40], cermets [41, 42], and recently in coarse grained A1203 [43--46]. In single phase ceramics with coarse grains, toughening is hypothesized to occur by two mechanisms. The first mechanism is frictional interlocking of opposing fracture surfaces as the main crack tends to propagate and the crack opening displacement (COD) increases. The second mechanism involves the formation of ligamentary bridges. The fracture resistance is increased by this phenomenon since propagation of the crack can only occur if these ligaments are fractured. Whiskers and fibers can act as crack bridging sites just like coarse grains do. Experimental studies by TEM and SEM confirm the existence of crack bridging in many whisker toughened composites including AIzO3/SiC. [9, 16, 18], mullite/SiC, [9], and giass/SiC, [9]. Becher et al. [9] treated the
8
7
c
Mulllle, ow = 10 GPa, ~ - 2
d
AI20s ,aw - 6.69 GPa.[$ - 6
•
Mulllte, o w . 10 GPa,~,. 1
f
AI2 O3 ,ow - 6.89 GPa.[~ - 3
g
Mulllte, a w - 6.69 GPa,~ = 2
h
Mullite, o w - 6.89 GPa.J$ - 1
I
f'~"
6
/°
A I'q
°
E
.~-
.~...~.
b
f f°
/
#.4 v
/" f "
..$
/" ./
-'l
•
3
"0
. .
f"
~.°
2
1
).Z.z /
h
0.10 0.20 0.30 0.40 Whisker vol. fraction (%)
Fig. 2. Comparison of crack bridging toughening for ,r. = 6.89 GPa (present study) and o,, = 10 GPa (Becher et
a/. [9]) in AIzOs/SiC,, and muilite/SiC, composites.
2513
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS 3. A COMBINATORY APPROACH TO
WHISKER TOUGHENING
\
Our previous discussions suggest that whisker toughening results from the combination of mainly three mechanisms, namely whisker pullout, crack deflection, and crack bridging. An attempt will be made here to define whisker toughening as the additive contributions from these mechanisms. It is assumed that these three mechanisms do not interact either positively or negatively since occurrence of either phenomenon could not affect any parameter of the aforementioned toughening models to a large extent. It is also assumed that whiskers are uniformly distributed in the matrix and their mechanical and physical properties do not deviate significantly from average values used in the calculations. An imaginary crack path undergoing all three phenomena is shown in Fig. 3. This figure shows that, during crack propagation and opening, all of these processes can occur simultaneously. Recent observations of crack propagation with SEM support the feasibility of this approach [18]. Based on these assumptions, total toughening upon whisker reinforcement may be found by simply adding up the individual contributions since fracture toughness is an additive term for a crack subjected to mode I loading dKto~t = dKpullout + d K ~ o o + dKb~n s.
(17)
Three parameters were used to derive theoretical toughening curves for AI203/SiC,, composites; whisker radius, pullout length (taken to equal the bridging length with the assumption of limited pullout), and whisker tensile strength. A fourth parameter, intrinsic to the crack deflection model, is the deflection angle. The maximum deflection angle was divided to 5 and total toughening was calculated for five different angles, the smallest being ~m~/5 and the largest being ~m~. Parametric values selected for conditions practically possible were r = 0 . 3 x 10-6m
and
/Da=/po=i0-6m, 2xl0-6m, e ~ ' = l x l03MPa, 5 x l03m,
0.6x 10-6m, and and
5x10-6m, 10x 103MPa.
Calculated results for whisker contents up to 50% and for five deflection angles are presented in Fig. 4 for a typical composite. Individual contributions as well as total toughening results calculated using the parameters given above for three volume fractions are shown in Table 1. The results show that, among all the parameters, whisker volume fraction, whisker strength and pullout length have the largest effect on total toughening. From Table 1, it is seen that increasing the whisker strength from I to 5GPa with all other parameters constant, increases total toughening by 25-50% while increasing the whisker strength from 5 to 10 GPa increases total toughening
-\
~ -[
crack deflection
]
/
J
\
Fig. 3. Typical processes occurring during crack propagation in a whisker reinforced ceramic. by 40-55%. These numbers suggest that high whisker strengths can effectively improve toughening. Pullout or debond lengths seem to be even more important than whisker strength. An increase in the pullout and debond length from 1 to 2 #m results in 20-70% increase in total toughening depending on the selection of other parameters, while an increase from 2 to 5/zm almost triples total toughening values (Table 1). This shows that increasing pullout and debond lengths as much as possible is an extremely important task to the composite designer. In contrast to whisker strength and pullout length, doubling of the whisker diameter does not result in any increase in total toughening values according to our calculations; instead, a slight decrease occurs. This decrease results from the decreased contribution of whisker pullout mechanism (Table 1). The bridging contribution remains constant when the whisker diameter is increased. This result conflicts with Becher et al.'s [9] experimental results as well as their interpretation of their own model. Becher et al.'s model does not, in fact, predict an increase in toughening when whisker radii are increased unless the debond length increases with increased whisker radii. The experimental results suggest that the debond length increases when the whisker radius is increased, which leads to an increased toughening. Therefore, whisker radius should be regarded as an indirectly effective parameter. Earlier studies generally show that whisker toughening increases with increasing whisker content [l, 2,4, 5, 30, 36]. Present toughening calculations agree with this trend. According to present calculations, the volume fraction of whiskers should be increased to levels which fabrication conditions allow, in order to achieve the highest composite toughness. Figure 4 also shows that increasing the crack deflection angle can further improve toughening, although the effect seems minimal compared to the effect of other parameters, especially for high whisker strengths and pullout lengths. Comparison of individual toughening contributions (Table l)
2514
BENGISU et al.: O 0
WHISKER TOUGHENING IN CERAMIC MATERIALS
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B E N G I S U et al.:
2515
W H I S K E R T O U G H E N I N G IN C E R A M I C M A T E R I A L S
Table I. Toughening results for different combinations of selected parameters • = 0.3 pm I~ll = l ~ = 1 p m o~' = I GPa
r - 0.3pm I pm ¢~' = 5 GPa
r = 0.3 pm Ito = 1~o- I pm ~ ' = 10GPa
1 ~ = Ipo -
V,
dKpo°
dK~,
dK~¢
dK,.
dK~
dK~
dK~
dK~
dK~
dK~
dK~
dK~
0.10 0.25 0.50
0.20 0.50 0.99
0.03 0.07 0.16
1.87 2.08 2.39
2.13 2.65 3.53
0.20 0.50 0.99
0.54 1.22 2.19
1.87 2.08 2.39
2.62 3.79 5.57
0.20 0.50 0.99
1.61 3.18 5.29
1.87 2.08 2.39
3.69 5.76 8.66
r = 0.6pm lr~e = Ipo = 1 pm o~" = I GPa
r = 0.6~m IDe = Ipo = 1 pm ~ ' = 5 GPa
• = 0.6pm 1~ - 1 pm ~ ' - 10 GPa
IDI ~ =
Vr
dKw
dK~
dK~
dK~,
dK~
dK~
dK~
dKu~
dK~
dK~,
dK~
dK~,
0.10 0.25 0.50
0.11 0.28 0.54
0.03 0.07 0.16
1.91 2.15 2.54
2.05 2.50 3.24
0.11 0.28 0.54
0.54 1.22 2.19
1.91 2.15 2.54
2.57 3.64 5.27
0.11 0.28 0.54
1.61 3.18 5.29
1.91 2.15 2.54
3.63 5.61 8.37
r = 0.3 I~m It,ll = Ipo = 2 pm a~' = 1 GPa
r = 0.3 ~m /De = 1~ = 2/am ~" = 5 GPa
r = 0.3 ~m ID!! = I~ = 2 ~m ¢~' = 10GPa
Vf
dKpo
dKb,
dK~
dKto~
dKpo
dKe,
dK~.r
dKtot
dKpo
dKe,
dK~
dK~
0.10 0.25 0.50
0.92 2.08 3.79
0.05 0.14 0.29
1.87 2.08 2.39
2.85 4.30 6.47
0.92 2.08 3.79
0.96 2.01 3.46
1.87 2.08 2.39
3.75 6.17 9.63
0.92 2.08 3.79
2.60 4.89 7.91
1.87 2.08 2.39
5.40 9.05 14.08
• = 0.6/Jm IDB= Ipo = 2 pm a~' = I GPa
• = 0.6/sin loll = Ipo = 2 pm ~ ' = 5GPa
r = 0.6pro l~s = Ipo= 2 ~m ¢~'- 10GPa
Vr
dKpo
dKb,
dKd,.f
dK~oL
dKpo
dKb,
dKdd
dKux
d/~
dKb,
dK~
dK~,
0.10 0.25 0.50
0.32 0.77 1.51
0.05 0.14 0.29
1.91 2.15 2.54
2.28 3.06 4.34
0.32 0.77 1.51
0.96 2.01 3.46
1.91 2.15 2.54
3.18 4.93 7.50
0.32 0.77 1.51
2.60 4.89 7.91
1.91 2.15 2.54
4.82 7.81 11.96
• = 0.3 pm loll = lpo = 5 pm ¢~ = I GPa Vf 0.10 0.25 0.50
dKpo 6.73 12.39 19.77
dKb, 0.13 0.32 0.65
dK~.¢ 1.87 2.08 2.39
• = 0.3 pm lb, = lw = 5 pm o~' = 5 GPa dKto, 8.74 14.78 22.81
dKpo 6.73 12.38 19.77
r = 0.6 pm 5/~m o~' = 1 GPa
dKb, 1.89 3.67 6.03
dK~r 1.87 2.08 2.39
• = 0.3/~m IDll= Ipo = 5 pm a~' = l0 GPa dK~ 10.50 18.13 28.19
dKi,o 6.73 12.38 19.77
• = 0.6 prn Ipo = 5 pm a~" = 5 GPa
/De = lpo =
dKb, 4.64 8.34 13.15
dK~ 1.87 2.08 2.39
dK~ 13.25 22.80 35.31
• = 0.6 pm In, = Ip. = 5 pm o~' = 10GPa
loll =
vr 0.10 0.25
dKpo 2.63 5.30
dK~ 0.13 0.32
dK~ 1.91 2.15
dK,oL 4.66 7.77
dKpo 2.63 5.30
dKb, 1.89 3.67
dKed 1.91 2.15
dK~ 6.43 11.12
dKpo 2.63 5.30
dKb, 4.64 8.34
dK~ 1.91 2.15
dKtoL 9.18 15.79
0.50
8.91
0.65
2.54
12.10
8.91
6.03
2.54
17.48
8.91
13.15
2.54
24.60
'All toughening results in MPa m ~'~.
shows that there are certain parametric combinations w h e r e o n e m e c h a n i s m is p r e v a l e n t a m o n g o t h e r s . At the lower end of debond length, pullout length, a n d w h i s k e r s t r e n g t h v a l u e s , c r a c k deflection c o n t r i b u t i o n s a r e t h e largest. T h e effectiveness o f t h e c r a c k b r i d g i n g m e c h a n i s m is i n c r e a s e d relatively as t h e w h i s k e r s t r e n g t h i n c r e a s e s . Similarly, t h e effectiveness o f t h e p u l l o u t m e c h a n i s m is i n c r e a s e d relatively as t h e pullout length increases. At intermediate pullout lengths, debond lengths, and whisker strength v a l u e s (e.g. In, = lro = 2 p m a n d or, = 5 G P a ) all t h r e e m e c h a n i s m s h a v e s i m i l a r c o n t r i b u t i o n s to t o t a l t o u g h e n i n g . W h e n p u l l o u t l e n g t h s a r e i n c r e a s e d u p to 5/~m, our calculations show that pullout toughening b e c o m e s t h e m o s t effective t o u g h e n i n g m e c h a n i s m . A s t h e v o l u m e f r a c t i o n is i n c r e a s e d , t h e relative importance of one mechanism over others becomes m o r e p r o n o u n c e d f o r a n y o f t h e selected p a r a m e t r i c combinations.
A d d i t i o n a l t o u g h e n i n g c u r v e s were g e n e r a t e d for /De ffi 1.6 p r o , r ffi 0.4 p m , a n d o~' ffi 6.89 G P a in o r d e r to c o r r e l a t e t h e o r e t i c a l r e s u l t s to B e t t e r et al.'s e x p e r i m e n t a l d a t a o n A I 2 0 3 / S i C , c o m p o s i t e s . Figure 5 shows that the predicted results agree very well w i t h t h e i r e x p e r i m e n t a l results. T h i s i m p l i e s that the combinatory approach used herein could also be a p p l i e d to o t h e r w h i s k e r r e i n f o r c e d C M C s . T h i s a p p r o a c h c a n be u s e f u l in o p t i m i z i n g p r o c e s s a n d m a t e r i a l p a r a m e t e r s a n d a l s o in d e s i g n i n g n e w whisker reinforced composites with maximum toughn e s s v a l u e s . F u r t h e r s t u d y , u s i n g statistical m e t h o d s , w o u l d be v a l u a b l e f o r m o r e precise p r e d i c t i o n s since available whiskers have a wide range of physical and mechanical properties. The works of Kageyama and C h o u [48] o n s h o r t fiber r e i n f o r c e d C M C s a n d S u t c u [49] a n d T h o u l e s s a n d E v a n s [50] o n fiber r e i n f o r c e d c e r a m i c s utilized statistical m e t h o d s to d e t e r m i n e fracture toughness, although whisker reinforced
2516
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS
AIO
B 6
t--
5. o . (whisker strength): whisker strength should be as high as possible since higher strengths provide significant improvement in toughening. 6. loe (debond length): this parameter should also be increased as much as possible by decreasing whisker/matrix attachment. 7. ¢~ (deflection angle): deflection angles should be maximized by controlling whisker/matrix interface characteristics. Acknowledgement--The study was supported by ATAC/
O
0.10
0.20
0.30
0.40
0.50
W h i s k e r vol. fraction (Vf)
Fig. 5. Comparison of predictions (squares) made by the combinatory approach for /~ = 4 (lab= 1.6/~m), ow = 6.89GPa, and r = 0.4#m and experimental results (bars) in AI~O3/SiCwcomposites [9]. CMCs, to be specific, have not yet been modeled in statistical terms. More complicated composites, where both whisker and ZrO 2 (or other toughening agents) are used, require further consideration. Finally, it must be remembered that the high temperature performance of whisker reinforced ceramics is quite different from that under ambient temperatures where the present approach should not be used. Since one of the major potential application areas of whisker reinforced CMCs is high temperature use, this issue also requires further consideration. 4. CONCLUSIONS Comparison of predictions of a combinatory approach to whisker toughening with experimental results in A1203/SiC . composites are encouraging since there is good agreement between the two. Our calculations reveal that whisker pullout and crack bridging are the most effective mechanisms especially when significant pullout and whisker debonding occurs. The following key points can be extracted from the present study as a guide for optimum use of the attendant parameters to achieve maximum toughening. 1. ~i (interface shear stress): some resistance to slip is required, but this cannot be too high; otherwise no pullout will occur. Lower shear stresses are preferable in order to increase pullout and debond lengths. 2. lpo (pullout length): pullout lengths should be maximized for maximum toughening. This can be achieved by minimizing the compressive stress acting on the whiskers, decreasing mechanical clamping by using smooth surface whiskers, and coating the whiskers to prevent chemical reaction and decrease interracial friction. 3. r (whisker radius): smaller whisker radii provide higher toughening. 4. Vt (whisker volume fraction): the amount of whiskers should be as high as processing allows.
Los Alamos under contract LANL 9XT 96721 RI and is gratefully acknowledged. REFERENCES
I. P. F. B~her and G. C. Wei, J. Am. Ceram. Soc. 67, C-267 (1984). 2. N. Claus.sen and G. Petzow, 13th Int. Conf. on Science of Ceramics (edited by P. Odier, F. Cabanncs and B. C.ales), pp. C1-693. Les Editions de Physique, Les Ulis Cedex, France (1986). 3. S. T. Buljan, A. E. Pasto and H. J. Kim, Am. Ceram. Soc. Bull. 68, 387 (1989). 4. S. Iio, M. Watanabe, M. Matsubara and Y. Matsuo, J. Am. Ceram. Soc. 72, 1880 (1989). 5. K. P. Gadkare¢ and K. C. Chyung, Am. Ceram. Soc. Bull. 65, 370 (1986). 6. P. F. Becher and T. N. Tiegs, J. Am. Ceram. Soc. 70, 651 (1987). 7. C. A. Anderson, P. B. Antolin, A. S. Fareed and G. H. Shiroku, Whisker and Fiber-Toughened Ceramics (edited by R. A. Bradley, D. E. Clark, D. C. Larscn and J. O. Stiegler), p. 209. ASM International, Metals Park, Ohio (1989). 8. L. Heraud and P. Sprier, Whisker and Fiber-Toughened Ceramics (edited by R. A. Bradley, D. E. Clark, D. C. Larsen and J. O. Stiegler), p. 217. ASM International, Metals Park, Ohio (1989). 9. P. F. Becher, C. H. Hsueh, P. Angelini and T. N. Tiegs, J. Am. Ceram. Soc. 71, 1050 (1988). 10. H. Liu, K. L. Weisskopfand G. Petzow, J. Am. Ceram. Soc. 72, 559 (1989). II. K. T. Faber and A. G. Evans, Acts metall. 31, 565 (1983). 12. D. H. Carter and G. F. Hurley, J. Am. Ceram. Soc. 70, C-79 (1987). 13. J. P. sing,h, K. C. Goretta, D. S. Kupperman, J. L. Routbort and J. F. Rhodes, Adv. Ceram. Mater. 3, 357 (1988). 14. A. G. Evans, Ceramic Microstructures '86, Role of Interfaces (edited by P. A. Pask and A. G. Evans), Vol. 21, p. 775. Plenum Press, New York (1987). 15. M. C. Shaw and K. T. Faber, Ceramic Microstructures '86, Role of Interfaces (edited by P. A. Pask and A. G. Evans), Vol. 21, p. 929. Plenum Press, New York (1987). 16. M. Ruhle, B. J. Dalgleish and A. G. Evans, Scripta metall. 21, 681 (1987). 17. M. G. Jenkins, A. S. K. Obayashi, K. W. White and R. C. Bradt, J. Am. Ceram. Soc. 70, 393 (1987). 18. G. H. Campbell, M. Ruhle, B. J. Dalgleish and A. G. Evans, J. Am. Ceram. Soc. 73, 521 (1990). 19. J. Homeny, W. L. Vaughn and M. K. Ferber, J. Am. Ceram. Soc. 73, 394 (1990). 20. Z. Li and R. C. Bradt, Whisker and Fiber-Toughened Ceramics (edited by R. A. Bradley, D. E. Clark, D. C. Larsen and J. O. Stiegler), p. 289. ASM International, Metals Park, Ohio (1989). 21. P. Angelini, W. Mader and P. F. Becher, Advanced Structural Ceramics (edited by P. F. Becher, M. V.
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS
22.
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37. 38.
Swain and S. Somiya), Vol. 78, p. 241. Materials Res. Soc., Pittsburgh, Pa (1987). N. Clausscn, K. L. Weisskopf and M. Ruhle, Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Has,~lman and F. F. Lange), Vol. 7, p. 75. Plenum Press, New York (1986). A. G. Evans and K. T. Faber, J. Am. Ceram. Soc. 67, 255 (1984). A. Dolgopolsky, V. Karbhari and S. S. Kwak, Acta metall. 37, 1349 (1989). V. M. Karbhari, Scripta metall. 23, 1207 (1989). J. K. Wells and P. W. R. Beaumont, J. Mater. Sci. 23, 1274 (1988). M. L. Vaughn, J. Homeny and M. K. Ferber, Ceram. Engng Sci. Proc. 8, 848 (1987). A. A. Morrone, S. R. Nutt and S. Suresh, J. Mater. Sci. 23, 3206 (1988). N. D. Corbin, C. A. Wilkens, V. K. Pujari, R. L. Yeckley and M. J. Mangaudis, Whisker and FiberToughened Ceramics (edited by R. A. Bradley, D. E. Clark, D. C. Larsen and J. O. Stiegler), p. 131. ASM International, Metals Park, Ohio (1989). H. Kodama, T. Suzuki, H. Sakamoto and T. Miyoshi, J. Am. Ceram. Soc. 73, 678 (1990). V. K. Sarin and M. Ruhle, Composites 18, 129 (1987). R. F. Davis, C. H. Carter Jr, S. R. Nutt, K. L. Moore and S. Chevacharocnkul, Ceramic Microstructures '86, Role of Interfaces (edited by P. A. Pask and A. G. Evans), Vol. 21, p. 897. Plenum Press, New York (1987). W. M. Kriven, G. Van Tendeloo, T. N. Tiegs and P. F. Becher, Ceramic Microstructures '86, Role of Interfaces (edited by P. A. Pask and A. G. Evans), Vol. 21, p. 939. Plenum Press, New York (1987). T. N. Tiegs, P. F. Becher and L. A. Harris, Ceramic Microstructures '86, Role of Interfaces (edited by P. A. Pask and A. G. Evans), Vol. 21, p. 91 I. Plenum Press, New York (1987). T. N. Tiegs, Whisker and Fiber-Toughened Ceramics (edited by R. A. Bradley, D. E. Clark, D. C. Larsen and J. O. Stiegler), p. 105. ASM International Metals Park, Ohio (1989). R. Hayami, K. Ueno, I. Kondou, N. Tamari and Y. Toibana, Tailoring Multiphase and Composite Ceramics (edited by R. E. Tressler, G. L. Messing, C. G. Pantano and R. E. Newnham), Vol. 20, p. 663. Plenum Press, New York (1986). S. M. Smith, R. O. Scattergood, J. P. Singh and K. Karasck. J. Am. Ceram. Soc. 72, 1252 (1989). S. G. Seshadri, M. Srinivasan and K. M. Keeler, Ceram. Engng Sci. Proc. 8, 671 (1987).
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39. P. P. Bansall and A. J. Ardell, Metallography 5, 97 (1972), 40. S. Kuntz-Douglas, P. W. R. Beaumont and M. F. Ashby, J. Mater. Sci. 15, 1109 (1980). 41. R. C. Leuth, Fracture Mechanics of Ceramics (edited by R. C. Bradt, D. P. H. Hasselman and F. F. Lange), Vol. 2, p. 791. Plenum Press, New York (1974). 42. V. D. Krstic, P. S. Nicholson and R. G. Hoagland, J. Am. Ceram. Soc. 64, 499 (1981). 43. P. L. Swanson, C. J. Fairbanks, B. R. Lawn, Y. W. Mai and B. J. Hockey, J. Am. Ceram. Soc. 70, 279 (1987). 44. Y. W. Mai and B. R. Lawn, J. Am. Ceram. Soc. 70, 289 (1987). 45. D. R. Clarke and K. T. Faber, J. Phys. Chem. Solids 48, 1115 (1987). 46. R. Knehans and R. Steinbrech, J. Mater. Lett. 1, 327 (1982). 47. D. W. Richerson, Modern Ceramic Engineering. Marcel Dekker, New York (1982). 48. K. Kageyama and T. W. Chou, Sixth Int. Conf. on Composite Materials (edited by F. L. Matthews, N. C. R. Buskell and J. M. Hodgkinson), Vol. 2, p. 260. Elsevier Applied Science, London (1987). 49. M. Sutcu, J. Mater. Sci. 23, 928 (1988). 50. M. D. Thouless and A. G. Evans, Acta metall. 36, 517 (1988).
APPENDIX I
Parametric Values used for Crack Bridging Toughening Calculations in AI2O j/SiC~ and Mullite / SiCw Composites AI203/SiCw a , = whisker fracture strength = 6.89 GPa [5] E,~ = elastic modulus of the matrix = 400 GPa [9] Ew = elastic modulus of whiskers = 689 GPa [5] vm = Poisson's ratio of the matrix = 0.24 [45] v,, = Poisson's ratio of whiskers = 0.14 [45] K~ = fracture toughness of the matrix = 2.5 MPa m ~'2 [9] r = whisker radius = 0.4 x 10 -6 m [9] fl = ldb/r = 3 and 5 [9] Mullite/SiC,, Em= 210 GPa [9] vm= 0.27 [20] K.m = 2 MPa m t'~ [9] [3 = lab/r = 1 and 2 [91
0956-7151/91 $3.00 + 0.00 Pergamon Press pie
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ON WHISKER T O U G H E N I N G IN CERAMIC MATERIALS M. BENGISU, O. T. INAL and O. TOSYALI New Mexico Institute of Mining and Technology, Department of Materials and Metallurgical Engineering, Socorro, NM 87801, U.S.A. (Received 8 November 1990; in ret,ised form 8 May 1991)
Al~ract--Toughening achieved by incorporation of whiskers to ceramic matrices was calculated based on an approach that combines individual contributions from effective toughening mechanisms. The relative importance of each mechanism, among others, was found to substantially vary depending on the selection of whisker and composite parameters. Analysis of the effect of various parameters showed that whisker strength, pullout length, debond length, and volume fraction of whiskers are the most important parameters in whisker toughening. Very good agreement was found between predictions made by the combinatory approach and experimental toughness values of AI20~/SiC, composites. R/~mr~. On a calcul6 le durcissement dfi ~i rincorporation de trichites dans des matrices de c~ramiques, en se basant sur une approche qui combine les contributions individuellesdes divers m~'canismeseflicaces de durcissement. L'importance relative de chaque m6canisme par rapport aux autres varie de faqon importante suivant le choix des param~tres du trichite et du composite. L'analyse de l'effet des diff6rents param6tres montre que la r6sistance mc~canique des trichites, leurs longueurs d'arrachement et de d~coh6sion, et leur fraction volumique sont les param6tres les plus importants dans le durcissement par trichites. L'accord est tr6s bon entre les pr6visionsde I'approche combinatoire et les valeurs exl~rimentales de la duret~ des composites AlzO3/SiC~mhi,,. Znt~mm~nfMsung--Der H~irtungseffektdurch Einlagerung von Whiskern in keramisches Material wird auf der Basis einer N~iherungberechnet, die die einzelnen Beitnige der wirkenden H~rtungsmechanismen kombiniert. Die Wichtigkeit der einzelnen Mechanismen im Vergleich zu den anderen variiert betrichtlich je nach Auswahl der Whisker- und VerbundmateriaI-Parameter. Aus der Analyse des Einflusses verschiedener Parameter geht hervor, dab Festigkeit, Ausziehl/inge, Abl6sel/inge und Volumanteil der Whisker die wichtigstenParameter der Festigkeitsteigerungdurch Whiskereinlagerungsind. Die mit dieser Kombinations-N~iherungerhaltenen Voraussagen und die experimentell erhaltenen Z/ihigkeitswertevon AlzO3/SiCw-Verbundmaterialienstimmen sehr gut iiberein.
1. INTRODUCTION The possibility of increasing the toughness of ceramic materials to levels where they can be used as engineering components has focussed attention on various toughening methods. Significant toughening has been achieved in many ceramics by the incorporation of whiskers [I-5]. Whisker toughening is as effective as ZrO2 toughening at ambient temperature and, in contrast to ZrO z toughening, the amount of toughening remains relatively constant at high ( > 1000°C) temperatures. Toughness values can further be increased by the incorporation of both ZrOz and whiskers and, in some cases, synergistic toughening has also been observed [2, 6]. Although toughness levels in fiber reinforced ceramic matrix composites (CMCs) are the highest among all ceramics [7, 8], whisker reinforcement offers advantages of easier fabrication and a higher degree of isotropy in material properties due to the smaller aspect ratios involved. To make the best use of whisker toughening it is of primary importance that the effect of fabri-
cation and material parameters are correctly predicted. Recent models of [lecher et al. [9] and Liu et al. [10], to some extent, have shown the effect of some of these parameters on toughening. The aim of this paper is to discuss, and further develop, available models that describe whisker toughening and to identify important toughening parameters.
2. DISCUSSION OF PROPOSED TOUGHENING MECHANISMS Observations in whisker reinforced CMCs, so far, suggest the possibility of five toughening mechanisms to be operative. These mechanisms are crack deflection [10--12], crack bowing [11], microcracking [13, 14], whisker pullout [9, 15], and crack bridging [9, 16-191. Microcracking is not uncommon in whisker reinforced ceramics. Li and Bradt [20] have shown various possibilities for thermal microcracking in mullite and Si3N~ matrix/SiC whisker (SIC,,) corn-
2509
2510
BENGISU et a/.: WHISKER TOUGHENING IN CERAMIC MATERIALS
posites. Microcracking was observed in AI203/SiC w composites by Angelini et al. [21] and Ruhle [14], and in TZP/SiCw composites by Claussen et al. [22]. Although microcracking is likely to occur in whisker reinforced ceramics, toughening due to microcracking in non-transforming ceramics is shown to be insignificant by recent theoretical and experimental analyses. Evans and Faber's [23] model of microcrack induced toughening is based on the concept that microcracking leads to dilatations in a material which produces compressive stresses on the macrocrack. Their model predicts toughness increases of only 10%. Another model by Dolgopolsky et al. [24] considers two opposing effects due to microcracking. One effect is the reduction of crack growth resistance due to linking of microcracks and the other effect is crack shielding by nucleated microcracks. Karbhari's [25] calculations, based on the application of their model to a SiC fiber/glass composite, predict a 3.1% decrease in the composite cracking stress in the case of crack amplification due to microcracks (i.e. linking of microcracks) or a 3.5% increase in the case of crack shielding by microcracks. Again, the effect of microcracking on crack resistance is not significant. In light of these observations, the toughening contribution due to microcracking in whisker reinforced ceramics can be assumed to be negligible at best. Crack bowing originates from resistant second phase species in the path of a propagating crack. According to Faber and Evans' calculations, the highest toughening by this mechanism is achieved in the case of discs with high aspect ratios whereas the lowest toughening is achieved in the case of rods. The crack bowing theory suggests that toughening is mainly a function of obstacle penetrability (related to obstacle toughness and strength, and coherency of obstacle/matrix interface), volume fraction of the second phase, and obstacle size. It follows that whiskers are theoretically less effective in bowing a crack front than for example metallic inclusions or discs since whiskers have rod-like shapes and they are easier to penetrate by the crack due to their low fracture toughness. Indeed, direct evidence of crack bowing in whisker reinforced ceramics is generally not available except for crack pinning observations in AI203/SiC . composites during in situ TEM straining experiments [21]. Therefore, it can be argued that crack bowing is not a major toughening mechanism in whisker reinforced ceramics, although it should not be ruled out completely. Crack pinning observations suggest that a minor contribution is possible, though crack bowing theory is not fully established that it can be used to calculate this contribution with any confidence. The remaining mechanisms are the ones that lead to the observed toughness increases in whisker reinforced CMCs in the present authors' opinion; i.e. whisker pullout, crack deflection, and crack bridging.
2.1. W h i s k e r pullout
Becher et al. [9] calculated whisker pullout toughening considering the work done by sliding whiskers. The toughness increase was determined to be AGp = ( 4 V : ~ Pp)/(3Ewr 2)
(I)
where Ti is the whisker/matrix interfacial shear strength, Vf is the whisker content, lp is the pullout length, Ew is the elastic modulus of whiskers, and r is the whisker radius. For plane strain conditions Gc = K~,(I - v~)/Ec
(2)
where vc is the Poisson ratio of the composite and E, is the composite elastic modulus. The strain energy release rate and the fracture toughness of the composite can be divided into two terms Gc = Go + AGp
(3)
f~ = fo + AK,
(4)
where Go is the strain energy release rate and Ko is the toughness of the matrix and AGp and AKp are pullout contributions to each term, respectively. Combining equations (2) to (4) yields (Ko + AK,) 2 = GcE~/(I - v~)
(5)
from which AKp can be derived as AKp = [GeEc/(I
- r e2) ]
I/2-
K0.
(6)
Other contributions related to whisker pullout, i.e. toughening by whisker/matrix debonding and whisker fracture, are found to be negligible from our calculations based on the analysis of Wells and Beaumont [26]. Although their equations are based on a brittle polymer/short brittle fiber composite system, they can conveniently be applied to whisker reinforced CMCs as well. Whisker pullout is frequently observed in whisker reinforced CMCs although in many cases pullout lengths are limited to l-2/~m [27-30]. Becher et al. [9] measured the average pullout lengths as 1.2-2.0 #m in AlzO3/SiCw, 0.4--0.8 #m in mullite/ SiCw, and 0.2-0.4/zm in glass/SiCw composites, where the whiskers were untreated. Pullout toughening can be promoted by increasing the pullout lengths. This can be done by preventing the formation of interaction layers between the matrix and the whiskers and using whiskers with smooth surfaces to prevent mechanical keying. Chemical interaction layers have been observed by careful examination of whisker/matrix interfaces. Thin amorphous layers have been observed in Si3 N(/SiCw composites formed due to chemical reaction at high fabrication temperatures [31, 32]. Similarly, high temperature oxidation or reaction was found to produce a surface scale on SiC whiskers in alumina matrices, sometimes to result in mullite interfaces [33]. When the oxygen coraent on SiC whiskers is high, the resulting composite toughness is lower [9, 34].
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS Tiegs [35] demonstrated that fracture toughness of an A1203/20 vol.% SiC, composite can be increased from 4.7 to 6.5 MPa m ~/2 by carbon coating of the whiskers. The presence of carbon results in a weak interfacial bond which enhances pullout and possibly other mechanisms such as crack deflection and bridging. Homeny et al. [19] observed that the presence of SiO2 promotes whisker/matrix bonding which prevents whisker pullout. Surface treatment of the whiskers in H2/Ar atmospheres activated pullout and increased the composite toughness. These results imply that increased pullout lengths achieved by surface treatment result in higher toughening. 2.2. Crack deflection
Crack deflection is commonly observed in whisker reinforced ceramics [2, 5, 21, 36] regardless of the matrix type. Liu et al. [10] modeled toughening by crack deflection for whisker reinforced CMCs with whiskers randomly oriented in two dimensions to account for whisker orientation in hot pressed composites. Liu et aL compared their predictions to experimental results from the literature. They state that their results appear to agree with experimental values. The present authors have compared recent data from the literature [2, 4, 9, 15, 27, 37] and found no such agreement as can be seen in Fig. 1. We believe that the attribution of all observed toughening, in whisker reinforced ceramics, to only crack deflection overemphasizes this mechanism. Carter and Hurley
2511
[12] measured crack deflection angles in MoSiz/SiC, composites and reveal that a 20voi.% whisker addition to the matrix increases median deflection angles from 7.2 ° to only 13.0, while the toughness was increased from 5.3 to 8.2 MPa m 1/2 (about 50%). Seshadri et al. [38] showed, by computer simulation, that significantly lower toughening should be expected from crack deflection than originally suggested by Faber and Evans [1 l] since the deflection angles are overestimated by their model. The analytical model suggests that the main crack is deflected at all angles evenly up to a maximum of 90°, ignoring a large contribution of zero angle, i.e. undeflected cracks. To account for this discrepancy we have calculated deflection toughening for maximum average deflection angles ranging from $~.~/5 to ~ . The total strain energy release rate for cracks tilted and twisted by rods is given as [11] (G)rod=(~/2)(G)T +(~/2)(G) t
(7)
where ( G ) r and ( G ) ~ are strain energy release rates due to tilting and twisting of the crack front, respectively, ( and ~ are ratios of undeflected to deflected crack front lengths. Liu et al. determined the strain energy release rate due to twist of the crack front as CI C1 Co t'~;2
x {2v sin 2 ~ + cos 2 ~bcos2(Q.)/2) legend
x [1 + 2 sin2((2)/2)]} 2
solid l i n e - Liu et al. ~0 •-
+ {sin ~ cos ~b cosZ((2)/2)
T i e g s 3s
o - S h a w a n d F a b e r ~s
× [cos:((2)/2) - 2v] + sin2((2)/2)
• - l i o e t al. 4
x [3 cos2((2 )/2) - 2v]}2) d01 d0z da db.
~t- Claussen and Petzow 2 , ~ - S m i t h et al. 3~ o-
(8)
B e c h e r et al. 9
Similarly, the tilt contribution is calculated from
12
C1 t't C~ t'~
o
10
0
x (cos4((I)/2) d01 d02 da ~ )
i,1
(9)
where G m is the strain energy release rate of the matrix, a and b are relative locations where rods are intercepted by the crack plane, ( 2 ) is the average tilt angle, v is Poisson's ratio, 01 and 02 arc the angles between rod consecutive axes and the fracture plane, and ~ is the twist angle given by -- tan- I[(a sin 01 + (1 - b) sin 02)/A']
(10)
and A' is given by A'=[A/H-acos01+(1 0
10 whisker
20 content
30
40
50
(vol.%)
Fig. 1. Comparison of experimental toughening data in AIzO3-SiC. composites to Liu et al.'s predictions.
- b) cos 02l.
(ll)
Since 4~ is a function of a, b, 0t, and 02, it cannot be varied independently inside the multiple integrals. Therefore, in order to vary 4~ and calculate toughening at different deflection angles, Faber and Evans'
2512
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS
model for a composite containing spherical particles has been used. The results were multiplied by a factor of 2.857, calculated from the results of Faber and Evans for deflection toughening by rods and spherical particles, which compensates for the difference in the respective cases. Faber and Evans derived the ratios of strain energy release rates of undeflected and deflected cracks as
toughening phenomenon in whisker reinforced ceramics by primarily attributing toughening to the crack bridging mechanism. They derived toughening as dK ~ = {[(/(0)2 + 4EcVf(a'[)ZlDs/ v~)E'] ~:2 -
(l -
K0}/2 (16)
where K0 is the matrix toughness, E c and E* are composite and whisker elasticmoduli, respectively,Vf is the volume fraction of whiskers, v is the Poisson ( G ) / G m = l/O~,,J ° [sin O/(sin • + cot ~)] ratio of the composite, and/DE is the debond length. Becher et al. calculated whisker toughening in x [(k~)2 + (k[):] d~ + I / O ~ alumina and mullite/SiC, composites from equation (16) and compared their predictions to their experx (cot q~ cos ~/sin 2 • + c o t O) imental results. In their calculations a whisker strength of I0 GPa has been used and this is significantly higher than 6.89 GPa, the average whisker X [(1/* f:k iTde0 z + (I/¢~::k~d*)2] d¢ ~ strength quoted for the same whiskers by the manu(12) facturer[5].W e have recalculatedwhisker toughening from equation (16) using u~' = 6.89GPa and other where k[ and k~ are local stressintensityfactors for parameters given in Appendix I. Figure 2 compares the tiltedand twisted portions of the crack, respect- Becher et al.'s and the present authors' results.It is ively; these are detailed elsewhere [II]. seen that the present results are about one half of The maximum twist angle is given by what Becher et al. calculated for each material for differentIDs/r ratios.Since Becher et aL showed their • . ~ = sin - l(2r/A) (13) predictions to be very close to their experimental where A can be calculated from [39] results, while a more realisticcalculation results in half of theirpredictions,itisconcluded that the crack A / r ~_ (e4Vr/V~~ x 1/2 e -~ d x . (14) bridging contribution to whisker toughening was J4 Vt overestimated. Experimental evidence thus supports In the present claculations, ¢~m,~ derived from the idea that crack bridging is not the only important equations (13) and 04) is divided by 5 and equation toughening mechanism, as discussed earlier. (12) is solved for each of these five angles. The a AI203 ,o w = 10 GPa, ~ = 5 toughening increment is calculated from the followb AI203 ,Ow = 10 GPII,~ = 3 ing equation GC/G m = G m / ( G ) .
(15)
Results from these calculations are used in total toughening calculations to be detailed later. 2.3. C r a c k bridging
Crack bridging phenomena have been observed in various composites and are well documented in rubber toughened polymers [40], cermets [41, 42], and recently in coarse grained A1203 [43--46]. In single phase ceramics with coarse grains, toughening is hypothesized to occur by two mechanisms. The first mechanism is frictional interlocking of opposing fracture surfaces as the main crack tends to propagate and the crack opening displacement (COD) increases. The second mechanism involves the formation of ligamentary bridges. The fracture resistance is increased by this phenomenon since propagation of the crack can only occur if these ligaments are fractured. Whiskers and fibers can act as crack bridging sites just like coarse grains do. Experimental studies by TEM and SEM confirm the existence of crack bridging in many whisker toughened composites including AIzO3/SiC. [9, 16, 18], mullite/SiC, [9], and giass/SiC, [9]. Becher et al. [9] treated the
8
7
c
Mulllle, ow = 10 GPa, ~ - 2
d
AI20s ,aw - 6.69 GPa.[$ - 6
•
Mulllte, o w . 10 GPa,~,. 1
f
AI2 O3 ,ow - 6.89 GPa.[~ - 3
g
Mulllte, a w - 6.69 GPa,~ = 2
h
Mullite, o w - 6.89 GPa.J$ - 1
I
f'~"
6
/°
A I'q
°
E
.~-
.~...~.
b
f f°
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/" f "
..$
/" ./
-'l
•
3
"0
. .
f"
~.°
2
1
).Z.z /
h
0.10 0.20 0.30 0.40 Whisker vol. fraction (%)
Fig. 2. Comparison of crack bridging toughening for ,r. = 6.89 GPa (present study) and o,, = 10 GPa (Becher et
a/. [9]) in AIzOs/SiC,, and muilite/SiC, composites.
2513
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS 3. A COMBINATORY APPROACH TO
WHISKER TOUGHENING
\
Our previous discussions suggest that whisker toughening results from the combination of mainly three mechanisms, namely whisker pullout, crack deflection, and crack bridging. An attempt will be made here to define whisker toughening as the additive contributions from these mechanisms. It is assumed that these three mechanisms do not interact either positively or negatively since occurrence of either phenomenon could not affect any parameter of the aforementioned toughening models to a large extent. It is also assumed that whiskers are uniformly distributed in the matrix and their mechanical and physical properties do not deviate significantly from average values used in the calculations. An imaginary crack path undergoing all three phenomena is shown in Fig. 3. This figure shows that, during crack propagation and opening, all of these processes can occur simultaneously. Recent observations of crack propagation with SEM support the feasibility of this approach [18]. Based on these assumptions, total toughening upon whisker reinforcement may be found by simply adding up the individual contributions since fracture toughness is an additive term for a crack subjected to mode I loading dKto~t = dKpullout + d K ~ o o + dKb~n s.
(17)
Three parameters were used to derive theoretical toughening curves for AI203/SiC,, composites; whisker radius, pullout length (taken to equal the bridging length with the assumption of limited pullout), and whisker tensile strength. A fourth parameter, intrinsic to the crack deflection model, is the deflection angle. The maximum deflection angle was divided to 5 and total toughening was calculated for five different angles, the smallest being ~m~/5 and the largest being ~m~. Parametric values selected for conditions practically possible were r = 0 . 3 x 10-6m
and
/Da=/po=i0-6m, 2xl0-6m, e ~ ' = l x l03MPa, 5 x l03m,
0.6x 10-6m, and and
5x10-6m, 10x 103MPa.
Calculated results for whisker contents up to 50% and for five deflection angles are presented in Fig. 4 for a typical composite. Individual contributions as well as total toughening results calculated using the parameters given above for three volume fractions are shown in Table 1. The results show that, among all the parameters, whisker volume fraction, whisker strength and pullout length have the largest effect on total toughening. From Table 1, it is seen that increasing the whisker strength from I to 5GPa with all other parameters constant, increases total toughening by 25-50% while increasing the whisker strength from 5 to 10 GPa increases total toughening
-\
~ -[
crack deflection
]
/
J
\
Fig. 3. Typical processes occurring during crack propagation in a whisker reinforced ceramic. by 40-55%. These numbers suggest that high whisker strengths can effectively improve toughening. Pullout or debond lengths seem to be even more important than whisker strength. An increase in the pullout and debond length from 1 to 2 #m results in 20-70% increase in total toughening depending on the selection of other parameters, while an increase from 2 to 5/zm almost triples total toughening values (Table 1). This shows that increasing pullout and debond lengths as much as possible is an extremely important task to the composite designer. In contrast to whisker strength and pullout length, doubling of the whisker diameter does not result in any increase in total toughening values according to our calculations; instead, a slight decrease occurs. This decrease results from the decreased contribution of whisker pullout mechanism (Table 1). The bridging contribution remains constant when the whisker diameter is increased. This result conflicts with Becher et al.'s [9] experimental results as well as their interpretation of their own model. Becher et al.'s model does not, in fact, predict an increase in toughening when whisker radii are increased unless the debond length increases with increased whisker radii. The experimental results suggest that the debond length increases when the whisker radius is increased, which leads to an increased toughening. Therefore, whisker radius should be regarded as an indirectly effective parameter. Earlier studies generally show that whisker toughening increases with increasing whisker content [l, 2,4, 5, 30, 36]. Present toughening calculations agree with this trend. According to present calculations, the volume fraction of whiskers should be increased to levels which fabrication conditions allow, in order to achieve the highest composite toughness. Figure 4 also shows that increasing the crack deflection angle can further improve toughening, although the effect seems minimal compared to the effect of other parameters, especially for high whisker strengths and pullout lengths. Comparison of individual toughening contributions (Table l)
2514
BENGISU et al.: O 0
WHISKER TOUGHENING IN CERAMIC MATERIALS
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B E N G I S U et al.:
2515
W H I S K E R T O U G H E N I N G IN C E R A M I C M A T E R I A L S
Table I. Toughening results for different combinations of selected parameters • = 0.3 pm I~ll = l ~ = 1 p m o~' = I GPa
r - 0.3pm I pm ¢~' = 5 GPa
r = 0.3 pm Ito = 1~o- I pm ~ ' = 10GPa
1 ~ = Ipo -
V,
dKpo°
dK~,
dK~¢
dK,.
dK~
dK~
dK~
dK~
dK~
dK~
dK~
dK~
0.10 0.25 0.50
0.20 0.50 0.99
0.03 0.07 0.16
1.87 2.08 2.39
2.13 2.65 3.53
0.20 0.50 0.99
0.54 1.22 2.19
1.87 2.08 2.39
2.62 3.79 5.57
0.20 0.50 0.99
1.61 3.18 5.29
1.87 2.08 2.39
3.69 5.76 8.66
r = 0.6pm lr~e = Ipo = 1 pm o~" = I GPa
r = 0.6~m IDe = Ipo = 1 pm ~ ' = 5 GPa
• = 0.6pm 1~ - 1 pm ~ ' - 10 GPa
IDI ~ =
Vr
dKw
dK~
dK~
dK~,
dK~
dK~
dK~
dKu~
dK~
dK~,
dK~
dK~,
0.10 0.25 0.50
0.11 0.28 0.54
0.03 0.07 0.16
1.91 2.15 2.54
2.05 2.50 3.24
0.11 0.28 0.54
0.54 1.22 2.19
1.91 2.15 2.54
2.57 3.64 5.27
0.11 0.28 0.54
1.61 3.18 5.29
1.91 2.15 2.54
3.63 5.61 8.37
r = 0.3 I~m It,ll = Ipo = 2 pm a~' = 1 GPa
r = 0.3 ~m /De = 1~ = 2/am ~" = 5 GPa
r = 0.3 ~m ID!! = I~ = 2 ~m ¢~' = 10GPa
Vf
dKpo
dKb,
dK~
dKto~
dKpo
dKe,
dK~.r
dKtot
dKpo
dKe,
dK~
dK~
0.10 0.25 0.50
0.92 2.08 3.79
0.05 0.14 0.29
1.87 2.08 2.39
2.85 4.30 6.47
0.92 2.08 3.79
0.96 2.01 3.46
1.87 2.08 2.39
3.75 6.17 9.63
0.92 2.08 3.79
2.60 4.89 7.91
1.87 2.08 2.39
5.40 9.05 14.08
• = 0.6/Jm IDB= Ipo = 2 pm a~' = I GPa
• = 0.6/sin loll = Ipo = 2 pm ~ ' = 5GPa
r = 0.6pro l~s = Ipo= 2 ~m ¢~'- 10GPa
Vr
dKpo
dKb,
dKd,.f
dK~oL
dKpo
dKb,
dKdd
dKux
d/~
dKb,
dK~
dK~,
0.10 0.25 0.50
0.32 0.77 1.51
0.05 0.14 0.29
1.91 2.15 2.54
2.28 3.06 4.34
0.32 0.77 1.51
0.96 2.01 3.46
1.91 2.15 2.54
3.18 4.93 7.50
0.32 0.77 1.51
2.60 4.89 7.91
1.91 2.15 2.54
4.82 7.81 11.96
• = 0.3 pm loll = lpo = 5 pm ¢~ = I GPa Vf 0.10 0.25 0.50
dKpo 6.73 12.39 19.77
dKb, 0.13 0.32 0.65
dK~.¢ 1.87 2.08 2.39
• = 0.3 pm lb, = lw = 5 pm o~' = 5 GPa dKto, 8.74 14.78 22.81
dKpo 6.73 12.38 19.77
r = 0.6 pm 5/~m o~' = 1 GPa
dKb, 1.89 3.67 6.03
dK~r 1.87 2.08 2.39
• = 0.3/~m IDll= Ipo = 5 pm a~' = l0 GPa dK~ 10.50 18.13 28.19
dKi,o 6.73 12.38 19.77
• = 0.6 prn Ipo = 5 pm a~" = 5 GPa
/De = lpo =
dKb, 4.64 8.34 13.15
dK~ 1.87 2.08 2.39
dK~ 13.25 22.80 35.31
• = 0.6 pm In, = Ip. = 5 pm o~' = 10GPa
loll =
vr 0.10 0.25
dKpo 2.63 5.30
dK~ 0.13 0.32
dK~ 1.91 2.15
dK,oL 4.66 7.77
dKpo 2.63 5.30
dKb, 1.89 3.67
dKed 1.91 2.15
dK~ 6.43 11.12
dKpo 2.63 5.30
dKb, 4.64 8.34
dK~ 1.91 2.15
dKtoL 9.18 15.79
0.50
8.91
0.65
2.54
12.10
8.91
6.03
2.54
17.48
8.91
13.15
2.54
24.60
'All toughening results in MPa m ~'~.
shows that there are certain parametric combinations w h e r e o n e m e c h a n i s m is p r e v a l e n t a m o n g o t h e r s . At the lower end of debond length, pullout length, a n d w h i s k e r s t r e n g t h v a l u e s , c r a c k deflection c o n t r i b u t i o n s a r e t h e largest. T h e effectiveness o f t h e c r a c k b r i d g i n g m e c h a n i s m is i n c r e a s e d relatively as t h e w h i s k e r s t r e n g t h i n c r e a s e s . Similarly, t h e effectiveness o f t h e p u l l o u t m e c h a n i s m is i n c r e a s e d relatively as t h e pullout length increases. At intermediate pullout lengths, debond lengths, and whisker strength v a l u e s (e.g. In, = lro = 2 p m a n d or, = 5 G P a ) all t h r e e m e c h a n i s m s h a v e s i m i l a r c o n t r i b u t i o n s to t o t a l t o u g h e n i n g . W h e n p u l l o u t l e n g t h s a r e i n c r e a s e d u p to 5/~m, our calculations show that pullout toughening b e c o m e s t h e m o s t effective t o u g h e n i n g m e c h a n i s m . A s t h e v o l u m e f r a c t i o n is i n c r e a s e d , t h e relative importance of one mechanism over others becomes m o r e p r o n o u n c e d f o r a n y o f t h e selected p a r a m e t r i c combinations.
A d d i t i o n a l t o u g h e n i n g c u r v e s were g e n e r a t e d for /De ffi 1.6 p r o , r ffi 0.4 p m , a n d o~' ffi 6.89 G P a in o r d e r to c o r r e l a t e t h e o r e t i c a l r e s u l t s to B e t t e r et al.'s e x p e r i m e n t a l d a t a o n A I 2 0 3 / S i C , c o m p o s i t e s . Figure 5 shows that the predicted results agree very well w i t h t h e i r e x p e r i m e n t a l results. T h i s i m p l i e s that the combinatory approach used herein could also be a p p l i e d to o t h e r w h i s k e r r e i n f o r c e d C M C s . T h i s a p p r o a c h c a n be u s e f u l in o p t i m i z i n g p r o c e s s a n d m a t e r i a l p a r a m e t e r s a n d a l s o in d e s i g n i n g n e w whisker reinforced composites with maximum toughn e s s v a l u e s . F u r t h e r s t u d y , u s i n g statistical m e t h o d s , w o u l d be v a l u a b l e f o r m o r e precise p r e d i c t i o n s since available whiskers have a wide range of physical and mechanical properties. The works of Kageyama and C h o u [48] o n s h o r t fiber r e i n f o r c e d C M C s a n d S u t c u [49] a n d T h o u l e s s a n d E v a n s [50] o n fiber r e i n f o r c e d c e r a m i c s utilized statistical m e t h o d s to d e t e r m i n e fracture toughness, although whisker reinforced
2516
BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS
AIO
B 6
t--
5. o . (whisker strength): whisker strength should be as high as possible since higher strengths provide significant improvement in toughening. 6. loe (debond length): this parameter should also be increased as much as possible by decreasing whisker/matrix attachment. 7. ¢~ (deflection angle): deflection angles should be maximized by controlling whisker/matrix interface characteristics. Acknowledgement--The study was supported by ATAC/
O
0.10
0.20
0.30
0.40
0.50
W h i s k e r vol. fraction (Vf)
Fig. 5. Comparison of predictions (squares) made by the combinatory approach for /~ = 4 (lab= 1.6/~m), ow = 6.89GPa, and r = 0.4#m and experimental results (bars) in AI~O3/SiCwcomposites [9]. CMCs, to be specific, have not yet been modeled in statistical terms. More complicated composites, where both whisker and ZrO 2 (or other toughening agents) are used, require further consideration. Finally, it must be remembered that the high temperature performance of whisker reinforced ceramics is quite different from that under ambient temperatures where the present approach should not be used. Since one of the major potential application areas of whisker reinforced CMCs is high temperature use, this issue also requires further consideration. 4. CONCLUSIONS Comparison of predictions of a combinatory approach to whisker toughening with experimental results in A1203/SiC . composites are encouraging since there is good agreement between the two. Our calculations reveal that whisker pullout and crack bridging are the most effective mechanisms especially when significant pullout and whisker debonding occurs. The following key points can be extracted from the present study as a guide for optimum use of the attendant parameters to achieve maximum toughening. 1. ~i (interface shear stress): some resistance to slip is required, but this cannot be too high; otherwise no pullout will occur. Lower shear stresses are preferable in order to increase pullout and debond lengths. 2. lpo (pullout length): pullout lengths should be maximized for maximum toughening. This can be achieved by minimizing the compressive stress acting on the whiskers, decreasing mechanical clamping by using smooth surface whiskers, and coating the whiskers to prevent chemical reaction and decrease interracial friction. 3. r (whisker radius): smaller whisker radii provide higher toughening. 4. Vt (whisker volume fraction): the amount of whiskers should be as high as processing allows.
Los Alamos under contract LANL 9XT 96721 RI and is gratefully acknowledged. REFERENCES
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BENGISU et al.: WHISKER TOUGHENING IN CERAMIC MATERIALS
22.
23. 24. 25. 26. 27. 28. 29.
30. 31. 32.
33.
34.
35.
36.
37. 38.
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APPENDIX I
Parametric Values used for Crack Bridging Toughening Calculations in AI2O j/SiC~ and Mullite / SiCw Composites AI203/SiCw a , = whisker fracture strength = 6.89 GPa [5] E,~ = elastic modulus of the matrix = 400 GPa [9] Ew = elastic modulus of whiskers = 689 GPa [5] vm = Poisson's ratio of the matrix = 0.24 [45] v,, = Poisson's ratio of whiskers = 0.14 [45] K~ = fracture toughness of the matrix = 2.5 MPa m ~'2 [9] r = whisker radius = 0.4 x 10 -6 m [9] fl = ldb/r = 3 and 5 [9] Mullite/SiC,, Em= 210 GPa [9] vm= 0.27 [20] K.m = 2 MPa m t'~ [9] [3 = lab/r = 1 and 2 [91