SiC nanocomposites

Computational Materials Science 22 (2001) 155±168

www.elsevier.com/locate/commatsci

Strengthening mechanisms in Al2O3=SiC nanocomposites Giuseppe Pezzotti a, Wolfgang H. M uller b,* b

a Kyoto Institute of Technology, Sakyo-ku, Matsugasaki, 606-8585 Kyoto, Japan Department of Mechanical and Chemical Engineering, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK

Received 8 March 2001; accepted 18 April 2001

Abstract A strengthening mechanism merely arising from internal (residual) microstresses due to thermal expansion mismatch is proposed for explaining the high experimental strength data measured in Al2 O3 =SiC nanocomposites. Upon cooling, transgranular SiC particles undergo lower shrinkage as compared to the surrounding matrix and provide a hydrostatic ``expansion'' e€ect in the core of each Al2 O3 grain. Such a grain expansion tightens the internal Al2 O3 grain boundaries, thus shielding both weakly bonded and unbonded (cracked) grain boundaries. It is shown that the shielding e€ect by intragranular SiC particles is more pronounced than the grain-boundary opening e€ect eventually associated with thermal expansion anisotropy of the Al2 O3 grains, even in the ``worst'' Al2 O3 -grain cluster con®guration. Therefore, an improvement of the material strength can be found. However, a large stress intensi®cation at the grain boundary is found when intergranular SiC particles are present, which can produce a noticeable wedge-like opening e€ect and trigger grain-boundary fracture. The present model enables us to explain the experimental strength data reported for Al2 O3 =SiC nanocomposites and con®rms that the high strength of these materials can be explained without invoking any toughening contribution by the SiC dispersion. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Nanocomposites; Strengthening; Toughening; Eshelby technique

1. Introduction Achievement of remarkably high strength in brittle materials is a recurrent topic in materials science which, based on earlier success with strengthening of glass, have also been pursued in ceramic materials. Although in the majority of cases the achieved results have been marginal, substantial progress has been made in strengthening conventional polycrystalline ceramics using approaches involving quenching and thermal treatments to induce compressive surface stresses [1]. For example, a polycrystalline Al2 O3 , quenched in silicon oil, was found to achieve nearly GPa strength levels according to an optimized quenching procedure, also involving external glazing [2,3]. The principal advantage of inferring a compressive surface layer treatment is the decreased penetration and local shielding of surface damages which leads to the strength improvement.

*

Corresponding author. Tel.: +44-131-451-3689; fax: +44-131-451-3129. E-mail address: [email protected] (W.H. M uller).

0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 8 5 - 9

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Niihara and coworkers [4,5] have shown that polycrystalline Al2 O3 dispersed with a low fraction of submicrometer SiC particles can also achieve a strength level of GPa. Remarkable residual stresses could be measured in the bulk phases constituting this composite, both by X-ray di€raction [6] and piezospectroscopic techniques [7]. However, no clear evidence of compressive stress ®elds more pronounced at the specimen surface as compared with the bulk microstructure could be obtained. From a fracture mechanics point-of-view, when either surface or internal residual stresses are involved, it is not possible to explain the strength data of brittle solids by merely invoking the Grith law [8]. In other words, high strength can be found despite the ceramic materials being inherently brittle. Although the phenomenological disagreement of the fracture behavior with the Grith law is an important common feature between surface-stressstrengthened Al2 O3 and bulk-stress-strengthened Al2 O3 =SiC nanocomposite, di€erent micromechanisms should be considered for explaining the fracture behavior in both the above cases. A questionable interpretation of the strengthening e€ect in Al2 O3 =SiC nanocomposites has recently been given by Ohji et al. [9] in terms of a near-tip crack-face bridging mechanism. A strengthening mechanism merely arising from residual stresses is proposed in this paper instead, which is considered to be more plausible. Experimental evidences have been provided elsewhere [10], which enable us to rule out near-tip bridging in the nanocomposite and, concurrently, to substantiate the presence of remarkably high (bulk) residual stresses. From a phenomenological point-of-view, it is also found that such bulk residual stresses inhibit intergranular fracture propagation and almost suppress spontaneous microcracking upon cooling, which are both phenomena arising from the thermal expansion anisotropy of Al2 O3 . Microcracking strongly depends on grain size, the larger the size the more pronounced the cracking, and triggers fracture at low strength levels [11±13]. In the Al2 O3 =SiC nanocomposite system, anisotropy-related thermal stresses in the matrix overlap with a residual stress ®eld due to the thermal expansion mismatch between the matrix and the dispersoid. This work presents a theoretical model, which quanti®es the e€ect of both the internal residual stress ®elds on the macroscopic fracture strength of the composite polycrystal. The analysis is aimed to clarify the role on fracture strength of ®ne SiC dispersoids that are either embedded in the Al2 O3 matrix grains or segregated at grain boundaries. The main concept here is that, despite di€erent strengthening mechanisms may be operative in bulk-stress-strengthened Al2 O3 =SiC as compared to surface-strengthened Al2 O3 , the strengthening e€ect is an extrinsic one and does not arise from any inherent toughening.

2. Theoretical stress analysis 2.1. Internal stresses from thermal anisotropy of the matrix Thermal microstress problems in polycrystalline ceramics have previously been treated [11,14,15] according to a theoretical procedure proposed by Eshelby [16]. A grain element is ®rst extracted from the surrounding polycrystalline matrix and allowed to undergo an unconstrained shape change. Then surface forces are applied to achieve shape compatibility and, after reinserting the grain element into the original matrix cavity, interface forces are applied that provide continuity of stress across the interface. The Eshelby method is applied in Fig. 1(A) to the case of a monolithic Al2 O3 polycrystal, which shows a marked thermal expansion anisotropy. 1 The e€ect of thermal expansion anisotropy on the internal microstress distribution is ®rst evaluated by considering a hexagonal Al2 O3 grain embedded in an isotropic Al2 O3 matrix. This approach has previously 1 Elastic anisotropy of Al2 O3 is neglected instead, according to previous analyses of microscopic thermal stresses in polycrystalline ceramics [11,15].

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157

Fig. 1. Application of the Eshelby procedure to determine the sign of residual microstresses acting on Al2 O3 grain boundaries: stresses from thermal expansion anisotropy of the matrix (A), and stresses from thermal expansion mismatch between matrix and dispersoids in the case of transgranular dispersoid location (B) and intergranular dispersoid location (C).

been described in details by Evans [11] and Evans and Clarke [15], thus only its salient parts are summarized hereafter. For simplicity, the orthogonal directions of maximum and minimum thermal expansion in the grain extracted from the matrix are chosen in-plane and along the main directions x and z, respectively (cf., Fig. 1(A)). Then the non-vanishing eigenstrains in x and z direction between the extracted grain and the matrix cavity are: exx ˆ ezz

Dax DT ;

ˆ Daz DT ;

…1† …2†

where (cf., [11]) Dax ˆ ax haim and Daz ˆ haim az with ax;z and haim being the linear expansion coecients of the grain and of the encompassing (isotropic) matrix, respectively; and DT > 0 is the range of temperature over which di€usional relaxation at grain boundaries is not allowed in polycrystalline Al2 O3 . To achieve dimensional conformity between the extracted grain and the matrix, the grain must be subjected to distributed surface forces which give rise to stresses within the grain:

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hEi …Dax mDaz †DT ; m2 hEi rzz ˆ …Daz mDax †DT ; 1 m2

rxx ˆ

1

…3† …4†

where hEi ˆ 420 GPa and m ˆ 0:25 are the (average) Young's modulus and Poisson's ratio of the isotropic matrix, respectively. 2 The stresses acting on the grain boundary arise from reinserting the previously extracted grain into the matrix. For doing so, application of distributed forces (i.e., equivalent in magnitude and opposite in sign) at the interface is required to achieve stress continuity. Now the superposition principle is applied to the formulae for the stresses resulting from a point force (cf., [21, Section 38; 11]). Then the stress ®eld generated at any arbitrary point …x; z† by a stress, p, homogeneously distributed along a grain-boundary line arbitrarily placed and inclined at an arbitrary angle, w, can be expressed as: Z p cos w nˆa z ‡ n sin w rxx …x; z† ˆ 2 4p x cos w† ‡ x2 ‡ z2 nˆ0 n ‡ 2n…z sin w ! 2…1 ‡ m†…n cos w x†2  1 m dn; …5† n2 ‡ 2n…z sin w x cos w† ‡ x2 ‡ z2 Z p cos w nˆa z ‡ n sin w rzz …x; z† ˆ 2 4p x cos w† ‡ x2 ‡ z2 nˆ0 n ‡ 2n…z sin w ! 2 2…1 ‡ m†…n cos w x†  …3 ‡ m† ‡ 2 dn; …6† n ‡ 2n…z sin w x cos w† ‡ x2 ‡ z2 Z p cos w nˆa n cos w x rxz …x; z† ˆ 2 4p x cos w† ‡ x2 ‡ z2 nˆ0 n ‡ 2n…z sin w ! 2 2…1 ‡ m†…z ‡ n sin w†  1 m‡ 2 dn; …7† n ‡ 2n…z sin w x cos w† ‡ x2 ‡ z2 where n is an abscissa taken along the selected grain boundary of length, a. Assuming, for simplicity, that Dax ˆ Daz ˆ Da, Eq. (3) results in a tension p ˆ p1 ˆ rxx ˆ hEi=…1 ‡ m†DaDT . Provided that the elastic and thermal properties of the polycrystal are known, Eqs. (5)±(7) were integrated numerically using a simple Simpson algorithm to ®nd the residual (thermal) microstress distribution along any grain-boundary line selected within the polycrystalline network. 2.2. Internal stresses from thermal expansion mismatch The presence of a secondary dispersed phase showing a thermal expansion mismatch with the matrix gives rise to a further residual stress component in the polycrystalline array. For the nanocomposite studied in this paper, the SiC dispersoid has a cubic phase, thus its thermal expansion is equal with respect to the main crystal directions and thermal expansion anisotropy will be neglected. It is also expected that the 2 Strictly speaking, Eqs. (3) and (4) hold in this simple form only for penny-shaped inclusions (cf., [17, p. 82]). In general, the shape of the inclusion enters the equation for the stresses in its interior through shape parameters. Moreover, it is questionable if a simple Eshelby type of solution (i.e., a homogeneous solution) exists in the case of a hexagon. First studies on stresses within angular inclusions are available (cf., [18±20]) that indicate that this is not the case and that the stresses may vary within the inclusion. This becomes also evident by some of the photoelastic experiments performed by Mura [19,20]. However, in this study we refrain from taking such complications into account.

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transgranular or intergranular location of the SiC dispersoids play an important role in the residual stresses acting on internal grain boundaries. First, the e€ect of transgranularly located dispersoids is treated by considering an additional hydrostatic stress component generated inside the matrix grains, which overlaps the stresses due to the matrix thermal expansion anisotropy. Although micrographs of the Al2 O3 =SiC nanocomposite [22] usually reveal a matrix microstructure in which several ®ne SiC particles are transgranularly embedded, a simpli®ed model considering one single dispersoid placed within each matrix grain is adopted here (Fig. 1(B)). Nevertheless, for maintaining the actual volume fraction, Vf , of the dispersoid t phase unchanged, an equivalent diameter, deq , is calculated for the transgranularly (index `t' embedded particle, according to the following procedure. A three-dimensional (regular) array of regular (polyhedral) matrix grains with an edge, a, is considered. The mean (index `m') intercept length, dm , and area, Am , of a matrix grain cut by a random plane are, respectively, given by [3, p. 92]: dm ˆ 0:545a;

…8†

2

Am ˆ 0:4a ;

…9†

Eliminating the edge length, a, from Eqs. (8) and (9) and invoking the coincidence between volume and areal fractions, the average number of dispersoids (index `d') nd , entrapped within one matrix grain can be expressed by:  2  2 Vf 6
0:4 dm dm nd ˆ 6Am 2 ˆ Vf  2:572Vf ; …10† dd dd pdd p0:5452 where dd is the actual average particle size of the dispersoid and, idealizing them to be of spherical shape, pdd2 =6 being their mean intercept area (cf., [23, p. 90]). Equating the transgranular areal fraction covered by the dispersoid phase in the real material to that of the equivalent (single) dispersoid of the model (cf., Fig. 1(B)), one obtains:  2  1=2  1=2 p  t 2 2 dd 2nd 2
2:572Vf 1=2 t d ˆ nd ) deq ˆ dd ˆ dm  1:31Vf dm : …11† 4 eq 3 2 3 3 Then, the radial di€erence in length accumulated upon cooling between a matrix grain with and without the embedded (equivalent) dispersoid can be calculated as: t Dlr ˆ deq …haim

haid †DT ;

…12†

where haid is the coecient of thermal expansion of the SiC dispersoid phase. This is also the radial expansion obtained upon extracting the grain out from the matrix array (cf., Fig. 1(B)). Neglecting di€erences between the elastic constants of the dispersoid and the surrounding matrix the Eshelby procedure is applied again to achieve dimensional conformity between the extracted grain and the surrounding matrix. With inferring compressive distributed surface forces along the grain boundaries, a hydrostatic stress ®eld is generated within the matrix grain, which should be proportional to the volume fraction of SiC dispersoid 2 t and, thus, to …deq =dm † (cf., Eq. (11)): 3 3 The formulae for an elliptic cylinder shown in [17, p. 80] were used here. Moreover, note that, strictly speaking, this relation holds under the assumption that there is no di€erence between the elastic constants of the dispersoid and the surrounding matrix. As it was demonstrated in [24,25] a ®rst-order correction can be obtained by multiplying the equation by the factor n ˆ …1 ‡ a†=…1 2b ‡ a†, 0 0 where a and b are the two Dundurs parameters, i.e., a ˆ …E0 E‡ †=…E0 E‡ †, E0 =‡ ˆ E =‡ =…1 m2 =‡ † (for plane strain) and 0 b ˆ 0:5…E =……1 ‡ m †…1 2m †† E‡ =……1 ‡ m‡ †…1 2m‡ †††=…E0 ‡ E‡ †, ) and + referring to the Young's moduli and Poisson's ratios of the dispersoid and the matrix, respectively. By insertion of the following sti€ness data, E‡ ˆ 420 GPa, m‡ ˆ 0:25, E ˆ 390 GPa, m ˆ 0:16, we conclude that n ˆ 0:82 and consequently will neglect, for the present case, a moderate in¯uence of sti€ness mismatch on the stresses.

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rr ˆ

Vf

hEi …haim 2…1 m2 †

haid †DT ˆ

1 hEi 1:716 2…1 m2 †



t deq dm

2 …haim

haid †DT :

…13†

t Taking p cos w ˆ rxx ˆ rzz ˆ rr ˆ 1=1:716h Ei=‰2…1 m2 †Š…deq =dm †2 …haim haid †DT ˆ p2 ; Eqs. (5)±(7) can be numerically integrated to provide the stress distribution due to matrix/dispersoid thermal expansion mismatch along any selected trajectory within the polycrystalline network. Image analysis results collected on Al2 O3 =SiC nanocomposites revealed [10] that, by increasing the added volume fraction of SiC dispersoids (i.e., Vf > 5%), an increasing fraction is trapped at grain boundaries after densi®cation. If a strong bonding is achieved at the Al2 O3 =SiC interface, which guarantees stress continuity, the lower shrinkage upon cooling of intergranular SiC particles (as compared with the Al2 O3 matrix) will make them act like grain-boundary wedges and trigger grain-boundary fracture. Fig. 1(C) shows a matrix-grain array accommodating a SiC dispersoid at triple-grain junctions. The Eshelby procedure is again applied and a tensile stress ®eld is found on the adjacent grain boundaries. Taking the diameter of an intergranular SiC particle equal to the average dispersoid size (i.e., dd ˆ 0:3 lm), the maximum wedge-opening (index `w') displacement on a grain boundary is given by:

Dlw ˆ ddt …haim

haid †DT

and the corresponding opening stress on the microcrack is: rw ˆ

hEi …haim 2…1 m2 †

…14† 4

haid †DT ;

thus giving a grain-boundary opening stress, p cos w ˆ rxx ˆ rzz ˆ rw ˆ hEi=‰2…1 p3 , acting on a portion of grain boundary length equal to dd =2.

…15† m2 †Š…haim

haid †DT ˆ

2.3. Overlapping stress ®elds on critical grain array The highest tensile stress that can develop in a polycrystalline array with thermal expansion anisotropy occurs for connecting grains with closely similar orientations. The ``worst'' local array con®guration is considered to trigger catastrophic fracture in the polycrystal, which originates from the ``weakest'' grain boundary as the fracture origin. Previous studies by Evans [11] and Evans and Clarke [15] indicated that a hexagonal grain array is liable to maximum stress intensi®cation at grain boundaries because its characteristic inclination angle, w ˆ 30°, maximizes the stress magnitude according to Eqs. (5)±(7). This array is represented in Fig. 2, in a local con®guration, which was considered to be a typical worst multigrain con®guration for polycrystalline Al2 O3 . This typical array con®guration is also used here to display the predicted stress distribution along the grain boundary AB, which experiences the maximum tensile stress. It is our intention to extend the stress analysis of thermal expansion anisotropy in a single-phase Al2 O3 aggregate to the case of a nanocomposite polycrystalline array, in which dispersoids are transgranularly lot cated (with each matrix grain containing one SiC dispersoid of diameter deq (Fig. 3(A))) or intergranularly located (with a dispersoid of size, dd , at each triple-grain junction (Fig. 3(B))). The stress magnitude acting on the weakest grain boundary AB of length a can be calculated according to Eqs. (5)±(7). Clearly, the residual stress component of interest here is rxx , which is perpendicular to the grain boundary AB and, thus, triggers fracture. Upon integration, Eq. (5) allows to estimate the overall stress distribution along AB, provided that the grain-boundary pressures/stresses p1 , p2 and p3 are known (from Eqs. (4), (13) and (15), respectively). 4

See the remarks and references in the previous two footnotes.

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161

Fig. 2. Critical grain ``cluster'' con®guration in the Al2 O3 matrix. This grain con®guration yields the maximum normal tension along the central facet AB, which is thus considered to be the fracture origin when an external tensile stress ®eld is applied to the polycrystal.

Fig. 3. Schematic of a grain-boundary microcrack with the related stress ®eld acting on it in the case of: (A) overlapping residual microstresses arising from matrix thermal expansion anisotropy …hrxx ita † and thermal expansion mismatch by transgranularly located SiC particles …hrxx itm †; and, (B) residual wedge-opening microstress …hrxx iw † due to a single SiC particle located at triple point.

Fig. 4 shows the stress distributions calculated along the grain boundary AB, 5 which arise from the thermal expansion anisotropy (index `ta') of the Al2 O3 matrix array, hrxx ita …p ˆ p1 †, from the thermal expansion mismatch between Al2 O3 and SiC phases (both in the case of transgranularly (index `tm') located dispersoids, hrxx itm …p ˆ p2 †, and from the intergranularly located dispersoids, hrxx iw …p ˆ p3 †. The stress component hrxx itm is displayed for two typical volume fractions, Vf ˆ 5 and 10 vol% of SiC. Stress distributions are plotted as a function of the distance from the triple-grain junction, n=a, normalized by the grain-edge length, a. In the calculation of the stress components hrxx ita and hrxx itm residual stress contributions are included from the grain boundary AB and its neighboring boundaries AA0 , AA00 , BB0 and BB00 (cf., Fig. 2). These two residual stress distributions experience a singularity at the triple point that decreases very rapidly with the abscissa n=a. The character of this singularity has previously been discussed in details by Evans [11].

5 In the calculation the following values were assumed, which are pertinent to the case of Al2 O3 =SiC nanocomposites: Dax;z ˆ 3:7  10 7 C 1 , haim haid ˆ 4:0  10 6 C 1 , dm ˆ 3 lm and dd ˆ 0:3 lm. The elastic stress-free temperature on cooling has been experimentally determined by Sergo et al. [26] as DT ˆ 1180°C.

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Fig. 4. The normal residual stress ®eld along the central facet AB for the grain con®guration shown in Fig. 2.

It is important to note that, if the thermal expansion mismatch, …haim haid † > 0, then p2 < 0 and a compressive stress component is generated by the matrix/dispersoid thermal mismatch on all the adjacent grain boundaries. Thus, a compressive stress component hrxx itm < 0 is also available on the weak grain boundary AB, which may counteract the tensile stress, hrxx ita arising from thermal anisotropy of the matrix grains. The magnitude of this boundary-closure stress component depends on the volume fraction of dispersoid, Vf , the larger Vf the larger p2 (cf., Eqs. (11) and (13)). However, it has been shown in previous experimental work [10] that, at relatively large SiC volume fractions (i.e., Vf > 10%), the assumption of a mere transgranular location of the dispersoids breaks down and a conspicuous fraction of SiC particles remains trapped at grain boundaries. An SiC particle trapped at grain boundary (e.g., at a triple point) in the Al2 O3 grain array will involve a wedge-opening mechanism, which produce a tensile stress distribution, hrxx iw on the grain boundary AB. The additional tensile stress component due to the wedging e€ect by a SiC particle located at a triple point of the grain boundary AB is also plotted in Fig. 4. In principle, also other SiC particles located at neighboring triple points should contribute to the stress distribution on the grain boundary AB. However, a calculation of these additional stress contributions by neighboring intergranular SiC particles (performed according to Eqs. (5) and (15)) showed values <2.5% of the hrxx iw calculated by only considering one SiC particle located at the triple points of AB. 3. Fracture analysis 3.1. Grain-boundary closure e€ect by transgranular dispersoids The thermal stresses given by Eqs. (5)±(7) are intensi®ed at pre-existing microstructural defects which, in ceramics, occur primarily at grain boundaries, especially at triple-grain junctions (cf., Figs. 3 and 4). An estimate of the stress intensi®cation e€ect on such defects can be obtained knowing the prior stress distribution, r…n†, along the crack trajectory [27]:  1=2 Z ‡a 1 a‡n KI ˆ r…n† dn; …16† 1=2 a n …pa† a which is valid for a through crack of length 2a, with n being the distance from the crack center. This approach [11] has been shown to re¯ect the behavior of a triple-grain edge crack with a negligible error,

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163

Fig. 5. Stress intensity factors acting on a grain-boundary microcrack (shown in Fig. 3) are plotted as a function of relative crack length. These stress intensities correspond to the residual microstress distributions shown in Fig. 4.

provided that the crack length, 2a, is in the range 0 < 2a=a < 0:2. 6 The stress intensity factor calculated according to Eq. (16) should be regarded as an upper-bound solution for triple-point defects in the matrix. The stress ®eld, r…n†, on a grain-boundary microcrack in the nanocomposite grain array (e.g., the grain boundary AB in Fig. 2) can be expressed as the sum of four distinct stress components, as follows: r…n† ˆ hrxx iex ‡ hrxx ita ‡ hrxx itm ‡ hrxx iw ;

…17†

where hrxx iex is the external (remote, index `ex') tensile stress ®eld. The latter three stress distributions are those discussed in Section 2.3 and plotted in Fig. 4. According to the stress superposition principle, each stress component can be introduced in Eq. (16) and, upon numerical integration, the correspondent stress intensity factors, Kex , DKta , DKtm , and DKw can be calculated, which are related to both the external and the residual stress ®elds. Some details of how to deal with the singularity involved during the numerical integration of Eq. (16) can be found in [28]. In the present case an elementary Simpson algorithm was used to obtain the following results. The calculated stress intensity components concerning the residual stress distributions are plotted in Fig. 5, as a function of the normalized crack length, 2a=a. These stress intensity components should be added to the stress intensity, Kex , due to the external load. It should be noted that, if the SiC dispersoid is only located within the Al2 O3 grains, the residual stress ®eld component, hrxx itm developed on the grain boundary is of a compressive nature. Thus, it gives rise to a DKtm < 0, which promotes the ``closure'' of the defect and, in turn, counteracts the crack opening e€ect arising from both hrxx iex and hrxx ita . It is also noteworthy that 5 vol% SiC can produce a DKtm larger that the maximum DKta achievable in Al2 O3 (cf., Fig. 5). This can be considered to be the actual origin of the strengthening e€ect experimentally found in polycrystalline Al2 O3 added with a low fraction of SiC nanosized dispersoids. 3.2. Grain-boundary wedging e€ect by intergranular dispersoids Fig. 5 also shows the stress-intensity component, DKw , which arises from the wedge-opening e€ect operated by the intergranularly located SiC particles. According to Eq. (15), this stress intensity factor is given by [29]: 6

Strictly speaking this formula holds for a crack in an in®nite, elastically homogeneous medium.

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DKw ˆ

hEi…haim 2…1

haid †DT  p 1=2 ua m2 † 2

…18†

thus being positive and acting to open the grain-boundary microcrack. In this equation 0 < u ˆ 2a=a < 1 denotes a dimensionless factor, the ``relative crack length'', that characterizes how much of the grain boundary is cracked. As in Section 2.2, the grain boundary has been estimated to be of length dd . As can be seen from Fig. 5, the e€ect of the wedge-opening mechanism can markedly a€ect the fracture behavior of the Al2 O3 polycrystal, since the magnitude of DKw is suciently large to annihilate the microcracking closure e€ect DKtm . In addition, the grain-boundary wedge-opening e€ect by the SiC intergranular dispersoid is additive to DKta due to thermal anisotropy of the matrix grains. Thus, it can be argued that the positive e€ect on strength of transgranularly located SiC dispersoids can be completely annihilated by the presence of dispersoids trapped at grain boundaries. It should be also noted that usually the SiC dispersoids of larger size are selectively left at grain boundaries, while the smaller ones more easily enter the Al2 O3 grains during densi®cation. For this reason, using the average dispersoid size, dd , for calculating DKw from Eq. (18) allows only to estimate a lower boundary value for the actual stress intensi®cation acting on the grain-boundary microcrack. 3.3. Strengthening and weakening e€ects Let us assume, in ®rst approximation, that the typical grain boundary microcrack triggering catastrophic fracture, is of the same size, 2a, both in monolithic Al2 O3 and Al2 O3 =SiC nanocomposites. 7 In absence of a rising R-curve behavior [10], the fracture strength of the Al2 O3 matrix (superscript `m'), rm f , and that of the nanocomposite (superscript `c'), rcf can be simply related to the critical stress intensity factors Kcm and Kcc , respectively. For a grain-boundary microcrack (superscript `gb'), it follows: Kcgb Kcgb

DKta ˆ rm f Y …2a†

1=2

ˆ Kcm ;

…DKta ‡ DKtm ‡ DKw † ˆ

1=2 rcf Y …2a†

…19† ˆ

Kcc ;

…20†

where Y is a morphological factor and Kcgb is the intrinsic critical stress intensity factor for grain-boundary crack propagation, i.e., a material constant that can be calculated from the grain-boundary fracture energy, cgb , according to the equation (for plane strain):  1=2 hEicgb gb : …21† Kc ˆ 1 m2 According to Eqs. (19) and (20) the strengthening e€ect resulting from the SiC dispersoids can be expressed by considering a common morphology (i.e., the same factor Y) for both the monolithic and nanocomposite materials: Drf rcf rm K c K m DKtm ‡ DKw f ˆ ˆ c m c ˆ …gb† : m m rf rf Kc Kc DKta

…22†

If a typical value for the intrinsic fracture energy of the Al2 O3 grain boundary is assumed, for example cgb ˆ 1 J=m2 , the strengthening e€ect expected in the nanocomposite (as compared to the pure matrix) can be calculated from the stress-intensity factors plotted in Fig. 5 as a function of the relative crack length, u, 7 Evidence has been provided by other authors [30,31] that the addition of nanosized SiC dispersoids to a Al2 O3 matrix statistically involves a reduction of the inherent ¯aw size. Hence, the present assumption of constant size for the inherent ¯aw in the polycrystalline network, independent of the SiC addition, leads only to an estimate of a lower bound of the strengthening e€ect.

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165

Fig. 6. Strengthening and weakening e€ects in Al2 O3 =SiC nanocomposites calculated from Eq. (22) as a function of the relative crack length.

by using Eqs. (21) and (22). The predicted strengthening e€ect is shown in Fig. 6 for both the 5 and 10 vol%added nanocomposites. As seen, a remarkably high strength can be expected for the nanocomposite provided the location of the SiC dispersoids is merely transgranular, the higher the fraction of SiC the higher the strength. The strength improvement is increasing with increasing the intrinsic ¯aw size at grain boundary. The increase is rapid for small intrinsic ¯aws, primarily in the range, 0 < u < 0:05, however, for greater fractions becomes relatively crack-length independent. However, the strength behavior of the nanocomposite will drastically change if, in addition to transgranularly located SiC particles, isolated particles are left at grain boundaries. A weak strengthening e€ect can be only expected if 0 < u < 0:05, while a strength even signi®cantly lower than that of the pure matrix should be expected for relatively large grain-boundary microcracks.

4. Discussion The majority of the investigations on Al2 O3 =SiC nanocomposites have shown a clear change in fracture mode from intergranular to transgranular, induced by the dispersion of the SiC nanoparticles within the Al2 O3 matrix [10,30,31]. This behavior was explained by Levin et al. [32] as a consequence of the tensile residual stress ®eld, which develops upon cooling in the Al2 O3 matrix because of the thermal expansion mismatch. The tensile residual stress in the Al2 O3 grains tends to de¯ect the propagating crack towards the transgranularly located SiC particles. Also SiC particles located at grain boundaries were found to de¯ect cracks into the grain interior [33], provided that the grain boundary is strongly bonded and does not act as fracture origin. However, an SiC particle located at a triple point, in presence of wedge-like grain-boundary defect or weakly bonded grain boundary, will exert an opening e€ect (by thermal expansion mismatch) on the microcrack with increasing its potentiality as fracture origin. Our present model is basically in agreement with the interpretation given by Levin et al. [32] and Jiao et al. [33] regarding the change of fracture mode in the nanocomposites. However, emphasis is given here to the e€ect of residual microstresses on grain-boundary strength and, in particular, on the closure e€ect by transgranularly located SiC particles on weakly bonded or partly cracked grain boundaries. Here, the observed change of fracture mode is considered to be the phenomenological proof that internal microstresses strengthen the grain boundaries of the Al2 O3 matrix. At high temperature, as a consequence of grain-boundary migration and grain growth, SiC particles are incorporated within the grains, which already underwent thermal expansion. During cooling, SiC dispersoids located within the matrix grains

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undergo a smaller shrinkage than the matrix, thus producing an internal ``core expansion'' of each Al2 O3 grain due to thermal expansion mismatch (cf., Fig. 1(B)). This e€ect tightens all grain boundaries and, consequently, shields weak boundaries and eventual grain boundary microcracks. It is noteworthy that, despite the low SiC fraction added, the magnitude of such a grain-boundary ``tightening'' e€ect can completely overcome the grain-boundary microcracking e€ect arising from the marked thermal anisotropy of the Al2 O3 matrix. On the other hand, the presence of a SiC particle at a weak boundary or boundary microcracking can lead to a noticeable wedge-opening e€ect on grain boundaries (cf., Fig. 1(C)), which may completely annihilate the grain-boundary closure e€ect by transgranular dispersoids. Clearly, the high strength in the nanocomposite will always depend on the reduced probability of ®nding weak grain boundaries or grainboundary microcracks acting as a fracture origin. Upon matrix grain growth, the presence of weak grain boundaries can be statistically reduced [34] but, in monolithic Al2 O3 , a larger grain size will also involve larger linear displacements, Dl, due to thermal expansion anisotropy (cf., Eqs. (1) and (2)) and, thus, microcracking. In the nanocomposite grain growth enables incorporation of SiC particles within the Al2 O3 grains, the lower expansion of which in the grain core may compensate for the microcracking e€ect associated with matrix grain growth. An optimum grain size of the Al2 O3 matrix could be found as a result of these two competing phenomena. This may explain the strength improvement reported by Niihara and Nakahira [35] and Zhao et al. [36] after annealing (i.e., matrix grain growth) of the nanocomposite. It is thus emphasized that further strength increase upon annealing is not incompatible with our strengthening model based on residual stresses. Niihara [37] empirically postulated two main factors for explaining the high strength of Al2 O3 =SiC nanocomposites, ®rst, the re®nement of the microstructural scale from the order of the matrix grain size to that of the SiC interparticle spacing, thus reducing the critical ¯aw size, and, second, the occurrence of a toughening phenomenon due to the SiC particles on the submicrometer scale, i.e., their acting in the very neighborhood of the crack tip. This argument lead to controversial opinions in the literature. Experimental analyses [38,39] have shown that critical ¯aws in nanocomposites, although smaller than in the non-reinforced matrix, are much larger than the interparticle spacing of the nanophase. On the other hand, Sternitzke et al. [30] and Perez-Rigueiro et al. [31] characterized the critical defects by quantitative fractography in both monolithic Al2 O3 and Al2 O3 =SiC nanocomposites added with various volume fractions of SiC dispersoid. These researchers found that the critical defect was signi®cantly smaller in the nanocomposites, regardless of the added SiC volume fraction. In this study, no evidence is provided to disprove the e€ect on strength of either a matrix grain re®nement or a reduction of the inherent defect size. However, according to our present model, residual stress arguments seem to fully suce to explain the strength improvement in the nanocomposite. Thus, the reduction of inherent defect size is considered to only provide a secondary contribution to the strength increase. A comparison of the experimental strength data [10] with the predictions based on Fig. 6 shows that the 100% strengthening e€ect experimentally found for the 5 vol%-SiC nanocomposite can be indeed achieved, provided that the critical defect size is P 0.3±0.5 lm. For the nanocomposite containing 10 vol% SiC (which also contained a larger fraction of intergranular SiC particles), experimental data and theoretical predictions indicate no strength improvement at typical defect sizes P 0.3 lm, thus showing the internal consistency of the model and supporting its validity. The ``new design concept'' proposed by Niihara [30] was based on the postulated existence of a toughening mechanism operative on a scale smaller than that of the material microstructure. Ohji et al. [9] applied this concept to discuss the high strength of the Al2 O3 =SiC nanocomposite in terms of a toughening e€ect by crack-face bridging, operated by the SiC dispersoids. However, we have provided evidences to rule out the ``nanoscale'' toughening e€ect according to the results of near-tip R-curve and crack-opening displacement characterizations [10]. Alternatively our model enables us to explain the me-

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chanical behavior of the Al2 O3 =SiC nanocomposite merely by invoking a strengthening mechanism based on thermally induced residual stresses, i.e., without any toughening e€ect involved. This model is proposed as the appropriate counterpart to the questionable concept of ``nanotoughening,'' suggesting that the high strength measured in Al2 O3 =SiC nanocomposite is an extrinsic phenomenon and, similar to surface-stress strengthened Al2 O3 , is not an inherent material property. 5. Conclusion A theoretical model is presented which envisages a strengthening e€ect in Al2 O3 =SiC nanocomposites to merely originate from bulk residual microstresses. First, stress distributions at grain boundaries induced by both thermal expansion anisotropy of the Al2 O3 matrix and thermal expansion mismatch between matrix and SiC dispersoids have been computed for the ideal case of an array of regular hexagons. Then, the corresponding stress intensity factors, as related to the above residual stress distributions, have been computed assuming the presence of a triple-point defect as the critical ¯aw origin for fracture in the Al2 O3 polycrystal. The analysis indicates that transgranularly located SiC particles can produce a hydrostatic residual stress ®eld which tightens all the internal grain boundaries, thus also shielding possible grainboundary defects. On the other hand, SiC particles trapped at grain boundaries show a wedge-opening e€ect, which weakens the grain boundaries of the polycrystal. Finally, the strength increase in the nanocomposite, as compared to that of the pure matrix, has been computed for both the cases of transgranularly located and trans/intergranularly located SiC dispersoids. Good agreement was found between the theoretical predictions and the experimental strength data. The most important outcome of the present analysis is that strengthening in Al2 O3 =SiC nanocomposites can be explained without invoking any inherent toughening mechanism. High strength can be achieved because of residual microstresses, despite such microstresses are distributed in the bulk microstructure rather than being localized at the surface of the polycrystalline sample. The high strength of nanocomposites should be regarded as an extrinsic material property, similar to the case of a polycrystalline Al2 O3 strengthened by surface-localized compressive stresses. References [1] H.P. Kirchner, Strengthening of Ceramics, Marcel Dekker, New York, 1979. [2] H.P. Kirchner, R.E. Walker, D.A. Platts, Strengthening alumina by quenching in various media, J. Appl. Phys. 42 (10) (1971) 3685±3692. [3] H.P. Kirchner, R.M. Gruver, R.E. Walker, Strengthening alumina by glazing and quenching, Bull. Am. Ceram. Soc. 47 (9) (1968) 798±802. [4] K. Niihara, A. Nakahira, Strengthening of oxide ceramics by SiC and Si3 N4 dispersions, in: V.J. Tennery (Ed.), Proceedings of the 3rd International Symposium on Ceramic Materials and Components for Engines, The American Ceramic Society, Westerville, OH, 1988, pp. 919±926. [5] K. Niihara, A. Nakahira, Strengthening and toughening mechanisms in nanocomposite ceramics, Ann. Chim. Sci. Mater. 16 (4±6) (1991) 479±486. [6] J. Otsuka, S. Iio, Y. Tajima, M. Watanabe, K. Tanaka, Strengthening mechanism in Al2 O3 =SiC particulate composites, J. Ceram. Soc. Jpn. 102 (1) (1994) 29±34. [7] G. Pezzotti, V. Sergo, K. Ota, O. Sbaizero, N. Muraki, T. Nishida, M. Sakai, Residual stresses and apparent strengthening in ceramic-matrix nanocomposites, J. Ceram. Soc. Jpn. 104 (6) (1996) 497±503. [8] G.R. Irwin, Fracture, in: Handbuch der Physik, vol. 6, Springer, Berlin, Germany, 1958, pp. 551±590. [9] T. Ohji, Y.K. Jeong, Y.H. Choa, K. Niihara, Strengthening and toughening mechanisms of ceramic nanocomposites, J. Am. Ceram. Soc. 81 (6) (1998) 1453±1460. [10] L.P. Ferroni, G. Pezzotti, J. Am. Ceram. Soc., to be published. [11] A.G. Evans, Microfracture from thermal expansion anisotropy. I Single phase systems, Acta Metall. 26 (1978) 1845±1853.

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