MECHANICS MATERIALS ELSEVIER Mechanics of Materials 30(1998)111-123 Toughening mechanisms of nano-composite ceramics lai Tan, Wei Yang Department of Engineering Mechanics, Tsinghua Unicersity, Beijing 100084, China Received 9 October 1997: received in revised form 3 March 1998 abstract Recent experiments showed that nano-composite ceramics with particles distributed within the matrix gras g s along the grain boundaries can acquire high toughening. The toughening of alumina ceramics with dispersed silicon carbide nano-particles are studied in the present paper. Three toughening mechanisms are identified: switching from the intergranular cracking to the transgranular one by nano-particles along the grain boundaries, fracture surface roughening by zigzag crack path perturbed by the internal stresses of nano-particles within the grains, and shielding by clinched rough surfaces near the crack tip. For different volume fractions of nano-particles, the estimated gross toughening agrees with the experiments. o 1998 Elsevier Science Ltd. All rights reserved Keywords: Toughening mechanisms; Fracture; Ceramics; Nano-particles 1. Introduction stresses in the material. The residual stresses in fluence the path of crack propagation. Levin et al. Many attempts have been made to improve the (1994)measured the average distribution and inherently brittle ceramics. Recent experiments by fluctuation of micro-strain in the matrix of Al O3/ Izaki et al.(1988) and Niihara and coworke nano-SiC composites by X-ray diffraction method (Niihara, 1991; Niihara and Nakahira, 1990: Nil- While the matrix toughness remains unchange ha aL, 1993, 1994; Sawaguchi in nano-composite ceramics(Zhao et al., 1993) Sasaki et al., 1992) showed that fracture toughness other toughening mechanisms exist. One approach of nano-composite ceramics, in which nanometer is to steer the crack to propagate along the path of sized second particles are dispersed within the ce- higher toughness. For ceramics, the toughness of ramic matrix, can be greatly enhanced grain boundaries is lower than that within the The substantial toughening of nano-composite grains. Thus the fracture pattern in conventional tes many related studies. Sawa- ceramics are mainly intergranular. Fo or nano- guchi et al. (1991) discovered an inter/intra type of composite ceramics, nano-particles along the grain nano-composite ceramics that induces transgran- boundaries tend to switch the intergranular frac ular fracture. Thermal mismatch between the ture to the transgranular one, and consequently nano-particles and the matrix generates residual toughen the composites. The other approach is to gain gross toughening by local weakening. The residual stresses generated by nano-particles pro- mote crack curving in ceramics, thus the crack du.cn grows along a wavy path. The global toughness is 0167-663698/S-see front matter c 1998 Elsevier Science Ltd. All rights reserved PI:S0167-6636(98)00027
Toughening mechanisms of nano-composite ceramics Honglai Tan, Wei Yang * Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Received 9 October 1997; received in revised form 3 March 1998 Abstract Recent experiments showed that nano-composite ceramics with particles distributed within the matrix grains and along the grain boundaries can acquire high toughening. The toughening of alumina ceramics with dispersed silicon carbide nano-particles are studied in the present paper. Three toughening mechanisms are identi®ed: switching from the intergranular cracking to the transgranular one by nano-particles along the grain boundaries, fracture surface roughening by zigzag crack path perturbed by the internal stresses of nano-particles within the grains, and shielding by clinched rough surfaces near the crack tip. For dierent volume fractions of nano-particles, the estimated gross toughening agrees with the experiments. Ó 1998 Elsevier Science Ltd. All rights reserved. Keywords: Toughening mechanisms; Fracture; Ceramics; Nano-particles 1. Introduction Many attempts have been made to improve the inherently brittle ceramics. Recent experiments by Izaki et al. (1988) and Niihara and coworkers (Niihara, 1991; Niihara and Nakahira, 1990; Niihara et al., 1993, 1994; Sawaguchi et al., 1991; Sasaki et al., 1992) showed that fracture toughness of nano-composite ceramics, in which nanometer sized second particles are dispersed within the ceramic matrix, can be greatly enhanced. The substantial toughening of nano-composite ceramics stimulates many related studies. Sawaguchi et al. (1991) discovered an inter/intra type of nano-composite ceramics that induces transgranular fracture. Thermal mismatch between the nano-particles and the matrix generates residual stresses in the material. The residual stresses in- ¯uence the path of crack propagation. Levin et al. (1994) measured the average distribution and ¯uctuation of micro-strain in the matrix of Al2O3/ nano-SiC composites by X-ray diraction method. While the matrix toughness remains unchanged in nano-composite ceramics (Zhao et al., 1993), other toughening mechanisms exist. One approach is to steer the crack to propagate along the path of higher toughness. For ceramics, the toughness of grain boundaries is lower than that within the grains. Thus the fracture pattern in conventional ceramics are mainly intergranular. For nanocomposite ceramics, nano-particles along the grain boundaries tend to switch the intergranular fracture to the transgranular one, and consequently toughen the composites. The other approach is to gain gross toughening by local weakening. The residual stresses generated by nano-particles promote crack curving in ceramics, thus the crack grows along a wavy path. The global toughness is Mechanics of Materials 30 (1998) 111±123 * Corresponding author. E-mail: yw-dem@mail.tsinghua.edu.cn. 0167-6636/98/$ ± see front matter Ó 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 9 8 ) 0 0 0 2 7 - 1
H. Tan, H. Yang Mechanics of Materials 30(1998)111-123 improved by the increasing of fracture surfaces and by the shielding of the clinched rough crack O--experimental point calculated curve 2. Fracture toughness 1.5 1.0 AlO3 /nano-SiC ceramics are fabricated by di ect coagulation casting technology( Gauckler and Graule. 1992). The sintering temp ratures are 1650°C,1650°C,1700°C,1750°C,and1750C when the nano-particles have volume contents of %. 5%6.10%. 15% and 20%. The relative densities of the specimens all exceed 99%. The average size of matrix grains and nano-particles is 2 um and 80 nm, respectively. Representative mechanical Fig. l. Toughening effect of the nano-composite ceramics with respect to the volume fraction of the dispersed nano-particles properties are listed in Table 1. For comparison, pure AlO3 ceramics are made with the same ma- trix grain size. Denote as Gcm and Gc the critical energy release rates of the nano-composite ce- 3. Microscopy observations ramics and the comparison Al,O3 ceramics. The latter is measured as Go=33. 8 J The frac- 3.1. Fractography observations ture toughness is measured by three-point-bending specimens, with length 24 mm, height 6.0 mm and The fracture surface of a three-point be thickness 4.0 mm specimen is examined under a field emission The toughening factor a is defined as scanning electronic microscope. Fig. 2 compares x=C/C-1, the fracture surface of nano-composite ceramics and that of the pure AlO3 ceramics. Fig. 2(A) which varies with the volume fraction of nano- shows that the fracture surface of Al,O3 ceramics particles, Vr. The experimental o versus Vr curve is consists of many smooth facets of grain size. shown in the cycles of Fig. I for AlO3/nano-Sic Fig. 2(B)shows that the fracture surface of nano- composites. The curve peaks at a critical volume composite ceramics is zigzag within the matrix fraction of added nano-particles. Volume fractions grains, due to the residual stress fields of the nar higher or lower than this critical value will reduce particles within the grain. From Fig. 2(B), few the toughening effect. For AlO3/nano-Sic ce- debonded nano-particles or holes can be found ramics, the highest toughening occurs at about along the crack surface, indicating strong cohesion between the nano-particles and the matrix anical parameters of matrix and nano-particles AlO,(matrix) Thermal expansion coefficient x=843×10-K oung's modulus Em=402 GPa Ep=450 GPa V=0.2
improved by the increasing of fracture surfaces and by the shielding of the clinched rough crack surfaces. 2. Fracture toughness Al2O3/nano-SiC ceramics are fabricated by direct coagulation casting technology (Gauckler and Graule, 1992). The sintering temperatures are 1650°C, 1650°C, 1700°C, 1750°C, and 1750°C when the nano-particles have volume contents of 0%, 5%, 10%, 15% and 20%. The relative densities of the specimens all exceed 99%. The average size of matrix grains and nano-particles is 2 lm and 80 nm, respectively. Representative mechanical properties are listed in Table 1. For comparison, pure Al2O3 ceramics are made with the same matrix grain size. Denote as Gnm C and Glm C the critical energy release rates of the nano-composite ceramics and the comparison Al2O3 ceramics. The latter is measured as Glm C 33.8 J mÿ2. The fracture toughness is measured by three-point-bending specimens, with length 24 mm, height 6.0 mm and thickness 4.0 mm. The toughening factor a is de®ned as a Gnm C =Glm C ÿ 1; 1 which varies with the volume fraction of nanoparticles, Vf. The experimental a versus Vf curve is shown in the cycles of Fig. 1 for Al2O3/nano-SiC composites. The curve peaks at a critical volume fraction of added nano-particles. Volume fractions higher or lower than this critical value will reduce the toughening eect. For Al2O3/nano-SiC ceramics, the highest toughening occurs at about Vf 10%. 3. Microscopy observations 3.1. Fractography observations The fracture surface of a three-point bending specimen is examined under a ®eld emission scanning electronic microscope. Fig. 2 compares the fracture surface of nano-composite ceramics and that of the pure Al2O3 ceramics. Fig. 2(A) shows that the fracture surface of Al2O3 ceramics consists of many smooth facets of grain size. Fig. 2(B) shows that the fracture surface of nanocomposite ceramics is zigzag within the matrix grains, due to the residual stress ®elds of the nanoparticles within the grain. From Fig. 2(B), few debonded nano-particles or holes can be found along the crack surface, indicating strong cohesion between the nano-particles and the matrix. Table 1 Mechanical parameters of matrix and nano-particles Al2O3 (matrix) SiC (nano-particle) Thermal expansion coecient am 8:43 10ÿ6K-1 ap 4:45 10ÿ6K-1 Young's modulus Em 402 GPa Ep 450 GPa Poission's ratio mm 0:23 mp 0:17 Fig. 1. Toughening eect of the nano-composite ceramics with respect to the volume fraction of the dispersed nano-particles. 112 H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123
H. Tan, W. Yang/ Mechanics of Materials 30(1998)111-123 1日.日kU AMRAY 8日13 (A) Ceramics without added particles 16.8 kV AMRAY (B)Nano-composite ceramics Fig. 2. Comparison of the fracture surface: (A) ceramics without added particles: (B)nano-composite ceramic 3. 2. SEM image of the crack path pecimen by an alloy string stained with Sic par- ticles. The specimen ligament is comparable to the Cracks formed in a brittle material usually specimen thickness. The two edges of the sp ause a catastrophic failure. In order to get stable are slightly tilted: one side is higher and the other cracks in ceramics specimen for in situ electron side is lower. During the in situ test, compressing nicroscopies, a double side-cracked specimen is force is applied to the top and the bottom edges of devised. As shown in Fig 3, two sharp cracks of the specimen. The tilted edges cause a bending the same length are sawed into the two sides of the moment, the higher side is compressed and the
3.2. SEM image of the crack path Cracks formed in a brittle material usually cause a catastrophic failure. In order to get stable cracks in ceramics specimen for in situ electron microscopies, a double side-cracked specimen is devised. As shown in Fig. 3, two sharp cracks of the same length are sawed into the two sides of the specimen by an alloy string stained with SiC particles. The specimen ligament is comparable to the specimen thickness. The two edges of the specimen are slightly tilted: one side is higher and the other side is lower. During the in situ test, compressing force is applied to the top and the bottom edges of the specimen. The tilted edges cause a bending moment, the higher side is compressed and the Fig. 2. Comparison of the fracture surface: (A) ceramics without added particles; (B) nano-composite ceramics. H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123 113
l14 H. Tan, H. Yang Mechanics of Materials 30(1998)111-123 3.3. Distribution of nano-particles in the matrix Nano-particles may be distributed in three 0.8mm patterns: the intra-type with nano-particles dis- persed within the matrix grains, the inter-type with nano-particles dispersed along the grain bound aries, and the intra/inter-type with nano-particles 6mm dispersed both along the grain boundaries and within the matrix grains. Experimental results showed that the intra/inter-type possesses the highest toughness and the intra-type the lowest (Sawaguchi et al., 1991). The analysis in Section 4 Fig 3 Double side-cracked specimen for SEM observation will explain these experiments: the nano-particles along the grain boundaries steer the crack to propagate into the matrix grains, while the nano- particles within the grains may lead the trans- wer ed. The crack normal to the lower granular crack to take a wavy path. Transgranular side grows under a local tensile stress. The initial fracture is unlikely to be induced by the intra-type crack should be sharp so that a small compressing ceramics, so the nano-particles inside the grain force by the loading stage in SEM can drive it. As have little effect on the fracture event along the the gap between the specimen edges and the folder grain boundaries. For the inter-type and the intra/ narrows, the crack driving force declines. The inter-type ceramics, transgranular fracture is in- stress intensity factor decreases with the propaga tion of the crack, and stable crack growth in the duced by the nano-particles along the boundary. The intra/inter-type ceramic lower side of the specimen is obtained toughens by the nano-particles within the grains theg. 4 shows the clinched crack surfaces near he tip. The crack propagates in a wavy path by 3.4. In 3.4. Influence of nano-particles on the crack path influence of the nano-particles. The stress field at the zigzag crack tip is inherently mixed mode, ig. 5 shows the TEM image(H-800)of a crack though the specimen is loaded externally by pure path under the influence of nano-particles. The mode I. The partial locking of the crack surfaces crack in the thin film specimen is obtained by provides shielding to the crack tip field ig. 4. Clinched rough surface near the crack Fig. 5. TEM image reveals the influence of the ceramics sustains a remote mode i loading on the crack path
lower side is lifted. The crack normal to the lower side grows under a local tensile stress. The initial crack should be sharp so that a small compressing force by the loading stage in SEM can drive it. As the gap between the specimen edges and the folder narrows, the crack driving force declines. The stress intensity factor decreases with the propagation of the crack, and stable crack growth in the lower side of the specimen is obtained. Fig. 4 shows the clinched crack surfaces near the tip. The crack propagates in a wavy path by the in¯uence of the nano-particles. The stress ®eld at the zigzag crack tip is inherently mixed mode, though the specimen is loaded externally by pure mode I. The partial locking of the crack surfaces provides shielding to the crack tip ®eld. 3.3. Distribution of nano-particles in the matrix Nano-particles may be distributed in three patterns: the intra-type with nano-particles dispersed within the matrix grains, the inter-type with nano-particles dispersed along the grain boundaries, and the intra/inter-type with nano-particles dispersed both along the grain boundaries and within the matrix grains. Experimental results showed that the intra/inter-type possesses the highest toughness and the intra-type the lowest (Sawaguchi et al., 1991). The analysis in Section 4 will explain these experiments: the nano-particles along the grain boundaries steer the crack to propagate into the matrix grains, while the nanoparticles within the grains may lead the transgranular crack to take a wavy path. Transgranular fracture is unlikely to be induced by the intra-type ceramics, so the nano-particles inside the grain have little eect on the fracture event along the grain boundaries. For the inter-type and the intra/ inter-type ceramics, transgranular fracture is induced by the nano-particles along the grain boundary. The intra/inter-type ceramic further toughens by the nano-particles within the grains. 3.4. In¯uence of nano-particles on the crack path Fig. 5 shows the TEM image (H-800) of a crack path under the in¯uence of nano-particles. The crack in the thin ®lm specimen is obtained by Fig. 3. Double side-cracked specimen for SEM observation. Fig. 4. Clinched rough surface near the crack tip when the nano-composite ceramics sustains a remote mode I loading. Fig. 5. TEM image reveals the in¯uence of the nano-particles on the crack path. 114 H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123
H. Tan, W. Yang/ Mechanics of Materials 30(1998)111-123 l15 pressing a needle lightly on the film. The induced direction. The local mode I and II stress intensity crack path clearly follows the scattering of nano- factors, denoted as Ki and Ki, can be written as particles. The residual stresses generated by the (Cotterell and Rice, 1980; Sumi et aL., 1983, 1985) nano-particles alter the crack direction for slightly tilted crack k(0)=cos3(0/2)K1-3sin(0/2)cos2(0/2)K KI(O= sin(0/2)cos (0/2)KI+ cos(0/2) 4. Toughening mechanisms In 4.1. Mechanism 1: Transgranular fracture induced by The error of the above formulas is less than 5% provided闭≤40°. For the special case of mode I We first analyze the toughening by increasing remote loading, the 5% error can be retained even the extent of transgranular fracture. For the se- if 10 <90, which is larger than the maximum lection of crack paths, our development here is tilting angle for intergranular fracture (58.5%),as similar to the discussion of Sumi (1989, 1992)on will be shown by the analysis in the sequel. The he kinked fracture along the degraded zone of an energy release rate for the tilt crack to advance imhomogeneous material. Denote as GEb and Ga along 0 angle, G, can be expressed as the fracture energy of the grain boundary and the fracture energy of the lattice(without the influence of nano-particles). Usually GC is considerably less than Ga. Under mode I loading, a crack will The competition between intergrana (5) penetrate into the grain and grow transgranularly transgranular fracture relies on the relative values if it is approximately normal to the grain boun- of G/Gc and Ge/Gl. For the case of remote dary. Thus, a portion of the fracture path would mode I loading, as can be compared with the ex perimental results listed in the previous sections, particles. For the Al, O, ceramics with a density of one can define a characteristic angle that the fraction of intergranular fracture, hence- o =2 arccos(Ga/da) 99.5%. a detailed fra forth denoted as f, is about 65%(McColm, 1990) and thisf value is insensitive to grain sizes Intergranular fracture occurs when Bo E 0, o]and transgranular fracture occurs when Bo E 0o, T/2 The overall toughness G for the Al,O3 ce For a random grain boundary orientation, the ramics can be estimated by a surface average of the definition of leads fracture energy f=26/兀 ⑦=f鹦+(1-fc Then we have Take the coordinate xI along the macroscopic crack direction. Denote the mode i and ii stress cos intensity factors for the main crack as Ki and Ku f +f cos!(fr/4) The energy release rate for a crack to extend along the xi direction is I-f+f cos!(r/4) (K2+K) (3) Accordingly, the value of Ga and G can be duced from the measurable values Gc and f. where E and v are the Youngs modulus and the Numeric calculation gives gc=270 J m- and Poisson's ratio, respectively. Suppose the grain GC=465J m-2 boundary ahead of the crack tip forms an angle 0 Nano-particles along the grain boundaries may with the x, direction For intergranular fracture to steer the crack into the matrix grains. Fig. 6(A) occur, the crack will tilt an angle of 0 with the xI shows the case when nano-particles are absent
pressing a needle lightly on the ®lm. The induced crack path clearly follows the scattering of nanoparticles. The residual stresses generated by the nano-particles alter the crack direction. 4. Toughening mechanisms 4.1. Mechanism 1: Transgranular fracture induced by nano-particles We ®rst analyze the toughening by increasing the extent of transgranular fracture. For the selection of crack paths, our development here is similar to the discussion of Sumi (1989, 1992) on the kinked fracture along the degraded zone of an imhomogeneous material. Denote as Ggb C and Gla C the fracture energy of the grain boundary and the fracture energy of the lattice (without the in¯uence of nano-particles). Usually Ggb C is considerably less than Gla C. Under mode I loading, a crack will penetrate into the grain and grow transgranularly if it is approximately normal to the grain boundary. Thus, a portion of the fracture path would be transgranular even for ceramics without nanoparticles. For the Al2O3 ceramics with a density of 99.5%, a detailed fracture path analysis indicated that the fraction of intergranular fracture, henceforth denoted as f, is about 65% (McColm, 1990), and this f value is insensitive to grain sizes. The overall toughness Glm C for the Al2O3 ceramics can be estimated by a surface average of the fracture energy: Glm C fGgb C 1 ÿ f Gla C: 2 Take the coordinate x1 along the macroscopic crack direction. Denote the mode I and II stress intensity factors for the main crack as KI and KII. The energy release rate for a crack to extend along the x1 direction is G 1 ÿ m2 E K2 I K2 II; 3 where E and m are the Young's modulus and the Poisson's ratio, respectively. Suppose the grain boundary ahead of the crack tip forms an angle h with the x1 direction. For intergranular fracture to occur, the crack will tilt an angle of h with the x1 direction. The local mode I and II stress intensity factors, denoted as Kh I and Kh II, can be written as (Cotterell and Rice, 1980; Sumi et al., 1983, 1985) for slightly tilted crack, Kh I h cos3 h=2KI ÿ 3sin h=2 cos2 h=2KII; Kh II h sin h=2 cos2 h=2KI cos h=2 1 ÿ 3sin2 h=2KII: 4 The error of the above formulas is less than 5% provided |h| 6 40°. For the special case of mode I remote loading, the 5% error can be retained even if |h| 6 90°, which is larger than the maximum tilting angle for intergranular fracture (58.5°), as will be shown by the analysis in the sequel. The energy release rate for the tilt crack to advance along h angle, Gh, can be expressed as Gh 1 ÿ m2 E Kh2 I Kh2 II : 5 The competition between intergranular and transgranular fracture relies on the relative values of Gh=Ggb C and Gh=Gla C. For the case of remote mode I loading, as can be compared with the experimental results listed in the previous sections, one can de®ne a characteristic angle h0 2 arccos Ggb C =Gla C 1=4 : 6 Intergranular fracture occurs when h0 2 0; h0 and transgranular fracture occurs when h0 2 h0; p=2. For a random grain boundary orientation, the de®nition of f leads to f 2h0=p: 7 Then we have Ggb C cos4 f p=4 1 ÿ f f cos4 f p=4 Glm C ; Gla C 1 1 ÿ f f cos4 f p=4 Glm C : 8 Accordingly, the value of Gla C and Ggb C can be deduced from the measurable values Glm C and f. Numeric calculation gives Ggb C 27.0 J mÿ2 and Gla C 46.5 J mÿ2. Nano-particles along the grain boundaries may steer the crack into the matrix grains. Fig. 6(A) shows the case when nano-particles are absent. H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123 115
H. Tan, W. Yang Mechanics of Materials 30(1998)111-123 The probability of transgranular fracture can be estimated from the area percentage occupied by nano-particles along a strip covering the grain boundary. Assuming that nano-particles distribute homogeneously inside the matrix grain and along the grain boundary strip, one finds that this area als Vr. A crack particle-free grain boundary can extend trans- Fig. 6. Nano-particles on grain boundary may steer the crack granularly by a probability of (1-f). with the presence of nano-particles, the probability for transgranular fracture raises to(1-f +fv). The main crack extends intergranularly along the The toughening by mechanism I(transgranular grain boundary, since the fracture resistance of the fracture induced by nano-particles) can be ex- grain boundary is lower than that of the grain pressed as lattice. In Fig. 6(B), nano-particles scatter along the grain boundary. The strong cohesion between 1=fr (12) the nano-particle and the matrix enables the crack o make a turn within the nano-particle and to which is proportional to vr enter the matrix grain. The subsequent trans- granular fracture proceeds along the direction to 4.2. Mechanism 2: Wavy fracture surface maximize the mode I stress intensity factor Denote the critical energy release rate for the The electron microscopies showed that cracks debonding of a nano-particle as Gcb. Suppose the within the matrix grain take a wavy path. The crack tip reaches the nano-particle, and the angle wavy crack path enlarges the of the crack between the main crack(along xI direction)and surface and thus improves the toughness, as dis the matrix/nano-particle interface is 0. Denote the cussed earlier by Rubinstein(1990)for the energy release rate as Ga for the crack to advance toughness Increase by wavy crack path. We instead along xI direction, and GB for the crack to advance adopt a slice model shown in Fig. 7 to describe along the matrix/nano-particle interface. They are quantitatively the wavy crack path. A 3-D trans ated by(see eqs. (3H5)) granular crack with the macroscopic direction GB=GA coS"(0/2) along xi is influenced by nano-particles Cut a slice n perpendicular to the average front of the By Griffith fracture criterion, transgranular frac- maincrack Nano-particles A,B, C, and d etc in ture can be induced if which requires Gago Ga cos*(0/2) (11) where Ga refers to the lattice toughness of SiC for xI the case when the crack cuts into the nano-particle and the lattice toughness of Al2O3 for the cas when the crack cut into the matrix. Better tough ening requires that Gc be greater than GC and comparable to Ga. Therefore a strong SGB should be maintained in the fabrication of nano-com- Fig. 7. Slice model of crack extension in nano-composite ce-
The main crack extends intergranularly along the grain boundary, since the fracture resistance of the grain boundary is lower than that of the grain lattice. In Fig. 6(B), nano-particles scatter along the grain boundary. The strong cohesion between the nano-particle and the matrix enables the crack to make a turn within the nano-particle and to enter the matrix grain. The subsequent transgranular fracture proceeds along the direction to maximize the mode I stress intensity factor. Denote the critical energy release rate for the debonding of a nano-particle as Gngb C . Suppose the crack tip reaches the nano-particle, and the angle between the main crack (along x1 direction) and the matrix/nano-particle interface is h. Denote the energy release rate as GA for the crack to advance along x1 direction, and GB for the crack to advance along the matrix/nano-particle interface. They are related by (see Eqs. (3)±(5)) GB GA cos4 h=2: 9 By Grith fracture criterion, transgranular fracture can be induced if GA Gla C > GB Gngb C ; 10 which requires Gngb C > Gla C cos4 h=2; 11 where Gla C refers to the lattice toughness of SiC for the case when the crack cuts into the nano-particle, and the lattice toughness of Al2O3 for the case when the crack cut into the matrix. Better toughening requires that Gngb C be greater than Ggb C and comparable to Gla C. Therefore a strong SGB should be maintained in the fabrication of nano-composite ceramics. The probability of transgranular fracture can be estimated from the area percentage occupied by nano-particles along a strip covering the grain boundary. Assuming that nano-particles distribute homogeneously inside the matrix grain and along the grain boundary strip, one ®nds that this area percentage equals Vf. A crack encountering a particle-free grain boundary can extend transgranularly by a probability of 1 ÿ f . With the presence of nano-particles, the probability for transgranular fracture raises to 1 ÿ f fVf. The toughening by mechanism 1 (transgranular fracture induced by nano-particles) can be expressed as a1 fVf Gla C ÿ Ggb C Glm C ; 12 which is proportional to Vf. 4.2. Mechanism 2: Wavy fracture surface The electron microscopies showed that cracks within the matrix grain take a wavy path. The wavy crack path enlarges the area of the crack surface and thus improves the toughness, as discussed earlier by Rubinstein (1990) for the toughness increase by wavy crack path. We instead adopt a slice model shown in Fig. 7 to describe quantitatively the wavy crack path. A 3-D transgranular crack with the macroscopic direction along x1 is in¯uenced by nano-particles. Cut a slice P perpendicular to the average front of the maincrack. Nano-particles A, B, C, and D etc. in Fig. 7. Slice model of crack extension in nano-composite ceramics. Fig. 6. Nano-particles on grain boundary may steer the crack into the matrix grain. 116 H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123
H. Tan, W. Yang /Mechanics of Materials 30 (1998)111-123 l17 Fig. 7 are particles either on or near slice Il. Those particles perturb the crack path x(x1)formed in Denote the lattice resistance to a wavy crack luring the transgranular fracture as Ga. Assum ing that the surface energy per unit area of the matrix is the same for a wavy or a flat crack, we /G出=/7=1+x2 (13) where I is the arc length of the zigzag crack z(x, and I is the projected length of z(xi) on the xI direction. The ratio between the arc length and the projected length of a zigzag crack determines the toughening due to wavy fracture surface. We can get I/I by tracing the crack propagating path z(x1) Fig 8. Residual shear stress along the crack plane generated by A nano-particle affects the crack propagation a nano by its residual stress field In a 2-D model the dual stress field can be expressed in a polar co- Crack propagation path under the influence of ordinate system as a single nano-particle is simulated in Fig. 9. The r≤dp/2 external loading KipP is set to be the critical stress d=(42)(e-e),r>4/2 intensity factor in transgranular fracture which path is calci where dp denotes the diameter of the nan(14) lated by solving an integral equation( Cotterell and cles and the mismatch pressure pe is given by method (Tan, 1996). The two methods give the same results as shown in Fig. 9 (15) Fig 9 shows that the crack begins to tilt when its distance to the nano-particle becomes less than The material is sintered at high temperature. In the R,. Denote the tilted height of the crack by the following calculation, the lowest temperature nano-particle as hilt. The tilting angle ep is defined without the generation of residual stresses in the as cooling process, Trelax, and the room temperature, Troom, are chosen as 1200oC and 20C, respectively 0p= arctan(hilt/Rp), The influence of a single nano-particle can be which is a function of d, the vertical distance of the described by two parameters: affecting distance Rp nano-particle to the crack plane. The tilt angle 6 and tilt range Op. Tilting of a crack is mainly of the crack is plotted in Fig. 10 with respect to d caused by shear stress, so Rp can be determined The maximum value of ep is denoted as Op, which through the shear stress acting along the crack is 25.8 from calculation front. Fig 8 plots the curve of non-dimensional We now consider the crack propagation path shear stress dres d/presd versus the non-dimen Inder the influence of many nano-partio sional distance r/dp calculated from Eq (14), simplify the 3-D wavy crack propagation, we as- where d represents the vertical distance of the sume that the segments composing the crack front nano-particle to the crack extension line. The advance independently. Thus one only needs to shear stress decays quickly and Rp can be taken simulate the trace of a representative crack seg approximately as 4dp. The nano-particles with ment. In Fig. 11, a segment of crack front is distance to the crack tip greater than Rp have no passing through a nano-particle O. The current influence on the crack propagation propagating direction lies on the T-plane and the
Fig. 7 are particles either on or near slice P. Those particles perturb the crack path v x1 formed in slice P. Denote the lattice resistance to a wavy crack during the transgranular fracture as G~la C . Assuming that the surface energy per unit area of the matrix is the same for a wavy or a ¯at crack, we have G~la C=Gla C ~l=l 1 a2; 13 where ~l is the arc length of the zigzag crack v x1, and l is the projected length of v x1 on the x1 direction. The ratio between the arc length and the projected length of a zigzag crack determines the toughening due to wavy fracture surface. We can get ~l=l by tracing the crack propagating path v x1. A nano-particle aects the crack propagation by its residual stress ®eld. In a 2-D model the residual stress ®eld can be expressed in a polar coordinate system as rres pres ÿeheh ÿ erer; r 6 dp=2; dp=2r 2 eheh ÿ erer; r > dp=2; ( 14 where dp denotes the diameter of the nano-particles and the mismatch pressure pres is given by pres am ÿ ap Trelax ÿ Troom 1 mm= 1 mpEm 1 ÿ 2mp=Ep : 15 The material is sintered at high temperature. In the following calculation, the lowest temperature without the generation of residual stresses in the cooling process, Trelax, and the room temperature, Troom, are chosen as 1200°C and 20°C, respectively. The in¯uence of a single nano-particle can be described by two parameters: aecting distance Rp and tilt range Hp. Tilting of a crack is mainly caused by shear stress, so Rp can be determined through the shear stress acting along the crack front. Fig. 8 plots the curve of non-dimensional shear stress rres 12 d2=presd2 p versus the non-dimensional distance r=dp calculated from Eq. (14), where d represents the vertical distance of the nano-particle to the crack extension line. The shear stress decays quickly and Rp can be taken approximately as 4dp. The nano-particles with distance to the crack tip greater than Rp have no in¯uence on the crack propagation. Crack propagation path under the in¯uence of a single nano-particle is simulated in Fig. 9. The external loading Kapp I is set to be the critical stress intensity factor in transgranular fracture which equals EGla C= 1 ÿ m2 p . The crack path is calculated by solving an integral equation (Cotterell and Rice, 1980), or simulated by the ®nite element method (Tan, 1996). The two methods give the same results as shown in Fig. 9. Fig. 9 shows that the crack begins to tilt when its distance to the nano-particle becomes less than Rp. Denote the tilted height of the crack by the nano-particle as htilt. The tilting angle hp is de®ned as hp arctan htilt=Rp; 16 which is a function of d, the vertical distance of the nano-particle to the crack plane. The tilt angle hp of the crack is plotted in Fig. 10 with respect to d. The maximum value of hp is denoted as Hp, which is 25.8° from calculation. We now consider the crack propagation path under the in¯uence of many nano-particles. To simplify the 3-D wavy crack propagation, we assume that the segments composing the crack front advance independently. Thus one only needs to simulate the trace of a representative crack segment. In Fig. 11, a segment of crack front is passing through a nano-particle O. The current propagating direction lies on the p-plane and the Fig. 8. Residual shear stress along the crack plane generated by a nano-particle. H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123 117
H. Tan, W. Yang/ Mechanics of Materials 30(1998)111-123 3 3 2 e 1 crack crack particle…3- propagating propagating Fig 9. The crack propagation path under the infuence of a current crack front is T. Subsequent advance of age sense(namely they will not affect the zigzag of the crack segment is influenced by all nano-parti- the crack), as justified by Figs. 8 and 9(C)and 10 cles near point O. Draw a cylindrical surface of When the crack cuts through o, under the in- radius Rp, with the axis perpendicular to the I- fluence of the nano-particles within Q2aff, it will tilt plane at point O. Draw two planes T and T2 toward the easiest propagating direction. Experi through I with angles Op and -Op to the T-plane. ments(see Fig. 5)showed that a crack traces nano- The region enclosed by the cylindrical surface and particles along its path. The crack tilts toward a the oblique planes TI and y define an affecting nano-particle in afr by a gradual process: the in- region Qaff. As the first order approximation, we fluence of that particle is gradually enhanced as the regard that the nano-particles within Saft have crack tilts toward it, and the crack is eventually near-range and directional influence on crack drawn to the nano-particle. Thus, nano-particles in propagation. Whereas the nano-particles outside Qaff provide candidate tilting directions. The diffi Qar affect the crack propagation only in an aver- culty of tilting is measured by the energy release rate along the specific direction. Denoting the tilting angle for the crack to link a nano-particle Q as Bo, the energy release rate for the tilted crack under remote mode I loading KIPP +[cos2(0/2)sn(/2)kP+km13},(17) are de i and ii stress intensity factors at the tilted crack tip, by the re- sidual stresses generated by all nano-particles. In Eq(17), we neglect the tearing stress intensity factor Klif caused by wavy crack front and those nano-particles in Qaff offset from the slice II. The crack propagates along the direction to maximize Go. When the crack reaches the selected nano- Fig 10. Crack tilting angle versus the height of the nano-par- particle, another affecting region Safr is defined, ticle above the crack and the analysis repeats
current crack front is C. Subsequent advance of the crack segment is in¯uenced by all nano-particles near point O. Draw a cylindrical surface of radius Rp, with the axis perpendicular to the pplane at point O. Draw two planes p1 and p2 through C with angles Hp and )Hp to the p-plane. The region enclosed by the cylindrical surface and the oblique planes p1 and p2 de®ne an aecting region Xaff. As the ®rst order approximation, we regard that the nano-particles within Xaff have near-range and directional in¯uence on crack propagation. Whereas the nano-particles outside Xaff aect the crack propagation only in an average sense (namely they will not aect the zigzag of the crack), as justi®ed by Figs. 8 and 9(C) and 10. When the crack cuts through O, under the in- ¯uence of the nano-particles within Xaff, it will tilt toward the easiest propagating direction. Experiments (see Fig. 5) showed that a crack traces nanoparticles along its path. The crack tilts toward a nano-particle in Xaff by a gradual process: the in- ¯uence of that particle is gradually enhanced as the crack tilts toward it, and the crack is eventually drawn to the nano-particle. Thus, nano-particles in Xaff provide candidate tilting directions. The di- culty of tilting is measured by the energy release rate along the speci®c direction. Denoting the tilting angle for the crack to link a nano-particle Q as hQ, the energy release rate for the tilted crack under remote mode I loading Kapp I is GQ 1 ÿ m2 E cos3 hQ=2Kapp 1 Kres 1 2 n cos2 hQ=2sin hQ=2Kapp I Kres II 2 o ; 17 where Kres I and Kres II are the mode I and II stress intensity factors at the tilted crack tip, by the residual stresses generated by all nano-particles. In Eq. (17), we neglect the tearing stress intensity factor Kres III caused by wavy crack front and those nano-particles in Xaff oset from the slice P. The crack propagates along the direction to maximize GQ. When the crack reaches the selected nanoparticle, another aecting region Xaff is de®ned, and the analysis repeats. Fig. 10. Crack tilting angle versus the height of the nano-particle above the crack plane. Fig. 9. The crack propagation path under the in¯uence of a nano-particle. 118 H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123
H. Tan, W. Yang /Mechanics of Materials 30 (1998)111-123 l19 local propagating direction particle KlK rack segme Fig. Il. Crack propagation under the influence of many nano-particles. The local crack plane has At a prescribed value of Ve, a random distri- ticles, the transgranular fracture will cut through bution of nano-particles can be generated by a the Al2O3 matrix; and for densely distributed computer. We use this particular distribution to nano-particles, the crack will link the particles al- simulate the crack propagation path under the most direct ahead. The largest attainable zigzag of external loading KipP. Define one propagating step crack surface is reached in the intermediate level of as one inter-particle advance of the crack tip particle volume fraction, as suggested by Fig 9 number of 10 propagating steps were calculated and the statistical result showed that the ratio of 43. Mechanism 3: Clinching at the crack tip /I stabilized. The toughening by mechanism 2 (wavy fracture surfaces) is shown in Fig. 12, the Experiments(Fig 4)showed that the upper and oughening factor a] depends on the particle vol- lower surfaces near the crack tip clinch together ume fraction Vf. The calculation shows that the even under mode I loading. The clinched crack toughening factor o] takes the peak value of 0.39 surfaces shield the crack tip, thus toughen the at Vr=9. For sparsely distributed nano-par- material Under remote mode I loading, the crack will take a straight line macroscopically. due to the continuous tilting of the crack tip, however the local stress intensity factors at the crack tip is of mixed mode. A clinching region is formed by the surface waviness and by the local mixed mode near the zigzag crack tip. The description of the clinching region includes the average half height h of the rough surface, the clinching degree fi and the constitutive law of the hypothetical clinching medium The crack propagating path is simulated to haracterize the self-locking. In nano-composite ceramics, both transgranular and intergranular fracture exist. For intergranular fracture, the crack extension along the grain boundary hardly exceeds the distance of adjacent nano-particles, while a g. 12. Toughening caused by the wavy fracture surface. crack can run transgranularly through a matrix
At a prescribed value of Vf, a random distribution of nano-particles can be generated by a computer. We use this particular distribution to simulate the crack propagation path under the external loading Kapp I . De®ne one propagating step as one inter-particle advance of the crack tip. A number of 106 propagating steps were calculated and the statistical result showed that the ratio of ~l=l stabilized. The toughening by mechanism 2 (wavy fracture surfaces) is shown in Fig. 12, the toughening factor a2 depends on the particle volume fraction Vf. The calculation shows that the toughening factor a2 takes the peak value of 0.39 at Vf 9%. For sparsely distributed nano-particles, the transgranular fracture will cut through the Al2O3 matrix; and for densely distributed nano-particles, the crack will link the particles almost direct ahead. The largest attainable zigzag of crack surface is reached in the intermediate level of particle volume fraction, as suggested by Fig. 9. 4.3. Mechanism 3: Clinching at the crack tip Experiments (Fig. 4) showed that the upper and lower surfaces near the crack tip clinch together even under mode I loading. The clinched crack surfaces shield the crack tip, thus toughen the material. Under remote mode I loading, the crack will take a straight line macroscopically. Due to the continuous tilting of the crack tip, however, the local stress intensity factors at the crack tip is of mixed mode. A clinching region is formed by the surface waviness and by the local mixed mode near the zigzag crack tip. The description of the clinching region includes the average half height h of the rough surface, the clinching degree fL and the constitutive law of the hypothetical clinching medium. The crack propagating path is simulated to characterize the self-locking. In nano-composite ceramics, both transgranular and intergranular fracture exist. For intergranular fracture, the crack extension along the grain boundary hardly exceeds the distance of adjacent nano-particles, while a Fig. 12. Toughening caused by the wavy fracture surface. crack can run transgranularly through a matrix Fig. 11. Crack propagation under the in¯uence of many nano-particles. The local crack plane has an angle htip with the main crack propagating direction x1. H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123 119
H. Tan, W. Yang /Mechanics of Materials 30 (1998)111-123 ain several times of the inter-particle distance. So For a given volume fraction Vf, a distribution the clinching mechanism dominates in transgran- of nano-particles can be randomly generated, then ular cracking a zigzag crack path be simulated. The ith genera- We simulate the transgranular fracture path tion provides the ith crack configuration. Under z(x1) in the nano-composite ceramics by the af- remote mode I field, the crack surface displace fecting region model stated above. Set the current ments which allow overlapping of the upper and crack tip at xi=0. The average half height of the lower surfaces is simulated in Fig 14. The clinch avy crack is estimated by ing degree at point xi is defined as the ratio of the Cix(zi-zl dx i length occupied by the contacted crack surfaces in cone ti neighborhood of xI. For the ith crack uration, Fig 14 indicates the contacted where Z is the average vertical position of the crack crack surfaces sections s(x1), s(x1),., in that in-1,o. In order to obtain a steady value of h, range. In the interval Kx1-Ax1/2, x1+Ax,/2 the the value of I should far exceed the average inter- total length of contacted surfaces is particle distance(which is in the range of hundreds 5,(x1)=E,s(1), thus the clinching degree of this of nanometers and depends on the volume fraction configuration at point xi is si(xi)/Ax. Repeating of nano-particles ). We take 1=400 m in the cal- the above process for N random generations of culations to follow. The average half height of the nano-particles and averaging the outcomes, one clinching region is shown in Fig. 13 for different Vr derives the clinching degree at point xI as values. As Vf increases, h rapidly climbs up and reaches the maximum 0.76 um at Vr=2%. After that peak, Vr remains at a range of 0.6-0.7 um f(x)=A皿∑s() With the influence of nano-particles, the crack In the calculations, we take Ax as 2 um, compa- will zigzag even under remote mode I loading. The rable to the size of the matrix grain crack tip field is controlled by mixed mode stress The fi(n1) curves obtained are shown in Fig. 15 intensity factors. The near tip mode mixity leads to Vr=2%, 10% and 25%, respectively. They are partial contact of the rough crack surfaces. The furnished by averaging the data from N degree of crack surface contact is described by a calculations. The dashed line in the figure function fi(x1), and varies with the distance to the ents the clinching due to the zig crd crack tip by intergranular fracture, the shielding effect in that case is small and contributes little to the toughening. When Vf= 10%, the clinching effect of the transgranular crack surface is high, so is the AAAA ith crack configurati Fig 14. Clinching of the crack surface at the crack tip. The 13. Half height of wavy crack versus the volume fraction of dashed line in the figure is the zigzag crack without deforma- tion. the solid line shows the crack surfaces after deformation
grain several times of the inter-particle distance. So the clinching mechanism dominates in transgranular cracking. We simulate the transgranular fracture path v x1 in the nano-composite ceramics by the affecting region model stated above. Set the current crack tip at x1 0. The average half height of the wavy crack is estimated by h R 0 ÿl jv vl ÿ vj dvl l ; 18 where v is the average vertical position of the crack in ÿl; 0. In order to obtain a steady value of h, the value of l should far exceed the average interparticle distance (which is in the range of hundreds of nanometers and depends on the volume fraction of nano-particles). We take l 400 lm in the calculations to follow. The average half height of the clinching region is shown in Fig. 13 for dierent Vf values. As Vf increases, h rapidly climbs up and reaches the maximum 0.76 lm at Vf 2%. After that peak, Vf remains at a range of 0.6±0.7 lm. With the in¯uence of nano-particles, the crack will zigzag even under remote mode I loading. The crack tip ®eld is controlled by mixed mode stress intensity factors. The near tip mode mixity leads to partial contact of the rough crack surfaces. The degree of crack surface contact is described by a function fL x1, and varies with the distance to the crack tip. For a given volume fraction Vf, a distribution of nano-particles can be randomly generated, then a zigzag crack path be simulated. The ith generation provides the ith crack con®guration. Under remote mode I ®eld, the crack surface displacements which allow overlapping of the upper and lower surfaces is simulated in Fig. 14. The clinching degree at point x1 is de®ned as the ratio of the length occupied by the contacted crack surfaces in the Dx1 neighborhood of x1. For the ith crack con®guration, Fig. 14 indicates the contacted crack surfaces sections s1 i x1;s2 i x1; . . . ; in that range. In the interval x1 ÿ Dx1=2; x1 Dx1=2, the total length of contacted surfaces is si x1 P j s j i x1, thus the clinching degree of this con®guration at point x1 is si x1=Dx1. Repeating the above process for N random generations of nano-particles and averaging the outcomes, one derives the clinching degree at point x1 as fL x1 1 Dx1 limN!1 1 N XN i1 si x1: 19 In the calculations, we take Dx1 as 2 lm, comparable to the size of the matrix grain. The fL x1 curves obtained are shown in Fig. 15 at Vf 2%, 10% and 25%, respectively. They are furnished by averaging the data from N 1200 calculations. The dashed line in the ®gure represents the clinching due to the zigzag crack caused by intergranular fracture, the shielding eect in that case is small and contributes little to the toughening. When Vf 10%, the clinching eect of the transgranular crack surface is high, so is the Fig. 13. Half height of wavy crack versus the volume fraction of nano-particles. Fig. 14. Clinching of the crack surface at the crack tip. The dashed line in the ®gure is the zigzag crack without deformation, the solid line shows the crack surfaces after deformation. 120 H. Tan, W. Yang / Mechanics of Materials 30 (1998) 111±123