《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_residual stress in Si3N4 laminate

Availableonlineatwww.sciencedirect.com SCIENCE Acta materialia ELSEVIER Acta Materialia 53(2005)289-296 www.actamat-journals.com Apparent fracture toughness of Si3 N4-based laminates with residual compressive or tensile stresses in surface layers M. Lugovy a,* V. Slyunyayev a, N. Orlovskaya b, G. Blugan, J. Kuebler M. Lewis d Institute for Problems of Materials Science, 3 Kchizhnouski St, 03142 Kiev, Ukraine Drexel Unirersity, Philadelphia, US.A EMPA, Duebendorf, Switzerland nirersity of Warwick, Coventry, UK Received 18 December 2003: received in revised form 17 September 2004: accepted 20 September 2004 Available online 28 October 2004 Abstract The effect of macroscopic residual stresses on the fracture resistance and stable/unstable crack growth in Si3N4/Si3N4-30 wt% TiN layered ceramics has been investigated. The laminates were manufactured using rolling and hot pressing techniques. An appar- ent fracture toughness Kapp was calculated as a function of the crack length parameter d= Y(a)a-for the laminates with residual compressive or tensile stresses in the top layers. The toughness increases in the layers with a compressive stress with increasing crack length, and it decreases in the layers with a tensile stress as the crack continues to grow. An explanation for the experimentally meas- ured and calculated Kapp values is proposed. The existence of the threshold stress and the stable/unstable crack growth conditions is 004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: A. Layered structures; B. Fracture toughness; Modeling: C. Crack; Residual stress 1. Introduction cation in layers with a bulk residual compression [41 phase transformation of zirconia grains under loading Ceramic matrix composites have a broad range of [5], and compositional gradient [6-8 industrial applications. They have been extensively used The mismatch of thermal expansion coefficients be- as structural components in order to improve the tween different layers inevitably generates thermal resid mechanical, thermal and chemical performance of engi- ual stresses during subsequent cooling of layered neering devices. However, despite a high hardness, an ceramics with strong interfaces [9]. The relative thick excellent oxidation resistance, and high temperature sta- ness of different layers determines the relative magni- bility, ceramics are inherently brittle. One of the strate- tudes of compressive and tensile stress, while the strain gies to decrease brittleness and improve composite mismatch between the layers dictates the absolute values performance is through the design of ceramic laminates of the residual stresses 1]. In recent years, a number of papers have been pub- A residual compression of layers results in laminates lished on laminates with weak interfaces for crack toughening, which is a crack shielding phenomenon [10] deflection [2], surface compressive stress [3], crack bifur- It has been shown that a residual compression of 500 MPa in a surface layer of a three-layered alumina-zirco- Corresponding author. Tel :+38 44 457 4890: fax: +38 44 296 nia composite can increase the fracture toughness by a 1684/380444442131. factor of 7. 5(up to 30 MPa m")for crack lengths equal E-mail address: lugovy @viptelecom net (M. Lugovy to the surface layer thickness [3]. This mechanism is 1359-6454$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved doi: 10. 1016/j. actamat. 2004.09.0

Apparent fracture toughness of Si3N4-based laminates with residual compressive or tensile stresses in surface layers M. Lugovy a,*, V. Slyunyayev a , N. Orlovskaya b , G. Blugan c , J. Kuebler c , M. Lewis d a Institute for Problems of Materials Science, 3 Kzhizhnovski St., 03142 Kiev, Ukraine b Drexel University, Philadelphia, USA c EMPA, Duebendorf, Switzerland d University of Warwick, Coventry, UK Received 18 December 2003; received in revised form 17 September 2004; accepted 20 September 2004 Available online 28 October 2004 Abstract The effect of macroscopic residual stresses on the fracture resistance and stable/unstable crack growth in Si3N4/Si3N4–30 wt% TiN layered ceramics has been investigated. The laminates were manufactured using rolling and hot pressing techniques. An appar￾ent fracture toughness Kapp was calculated as a function of the crack length parameter a˜ = Y(a)a1/2 for the laminates with residual compressive or tensile stresses in the top layers. The toughness increases in the layers with a compressive stress with increasing crack length, and it decreases in the layers with a tensile stress as the crack continues to grow. An explanation for the experimentally meas￾ured and calculated Kapp values is proposed. The existence of the threshold stress and the stable/unstable crack growth conditions is discussed.
2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: A. Layered structures; B. Fracture toughness; Modeling; C. Crack; Residual stress 1. Introduction Ceramic matrix composites have a broad range of industrial applications. They have been extensively used as structural components in order to improve the mechanical, thermal and chemical performance of engi￾neering devices. However, despite a high hardness, an excellent oxidation resistance, and high temperature sta￾bility, ceramics are inherently brittle. One of the strate￾gies to decrease brittleness and improve composite performance is through the design of ceramic laminates [1]. In recent years, a number of papers have been pub￾lished on laminates with weak interfaces for crack deflection [2], surface compressive stress [3], crack bifur￾cation in layers with a bulk residual compression [4], phase transformation of zirconia grains under loading [5], and compositional gradient [6–8]. The mismatch of thermal expansion coefficients be￾tween different layers inevitably generates thermal resid￾ual stresses during subsequent cooling of layered ceramics with strong interfaces [9]. The relative thick￾ness of different layers determines the relative magni￾tudes of compressive and tensile stress, while the strain mismatch between the layers dictates the absolute values of the residual stresses. A residual compression of layers results in laminates toughening, which is a crack shielding phenomenon [10]. It has been shown that a residual compression of
500 MPa in a surface layer of a three-layered alumina–zirco￾nia composite can increase the fracture toughness by a factor of 7.5 (up to 30 MPa m1/2) for crack lengths equal to the surface layer thickness [3]. This mechanism is 1359-6454/$30.00
2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.09.022 * Corresponding author. Tel.: +38 44 457 4890; fax: +38 44 296 1684/380 44 444 21 31. E-mail address: lugovy@viptelecom.net (M. Lugovy). Acta Materialia 53 (2005) 289–296 www.actamat-journals.com

associated with the closure stress field behind the crack tapes. Two compositions of layers were used: (1) Si3N tip, which is similar to a crack bridging phenomenon (MIl, Stark, Germany), (2)Si3N4-30 wt% TiN (grade ening by the residual compressive stress is determined 2 wt% Al2O3 and 5 wt%Y2O3 additiver was used with acting in non-layered ceramics [11]. However, a tough- C, Stark, Germany). The silicon nitride by the overall crack length [12], while the toughening The mixtures of various compositions were milled in by the crack bridging depends only on the length incre- the ball mill for 5 h. The average grain size of the milled ment of the moving crack Crack bridging can provide powders was about I m. Crude rubber (4 wt%)was increased toughening by a factor of approximately 2 added to the mixture of powders as a plasticizer through [11]. Therefore, the residual compressive stresses can a 3% solution in petrol. The powders were then drie provide a significant increase in a ceramic's toughness leaving 2 wt% residual amount of petrol in the mixture &e Although fracture toughness of layered composite can After sieving powders with a 500 um sieve, granulated cause of the superposition of different effects like stress mill with 40 mm rolls was used for rolling. The velocity shielding and intrinsic properties of the structure, such of rolling was 1.5 m/min. The working pressure was as grain size, composition, interfaces, etc. In fractur about 10 mPa to obtain a relative of mechanics, both residual and applied stresses are usually 64%. The thickness of green tapes was 0.4-0.5 mm and included in the crack driving force. However it can be the width was 60-65 mm. Green tapes were stacked to- ful to consider residual stresses as part of the gether to form the desired layered structures and cera- ance. Thus, in laminates with residual compressive stress, mic samples were prepared by hot pressing the stacked the higher resistance to failure results from a reduction of tapes. The hot pressing was performed at 1820C and crack driving force rather than from an increase in the 30 MPa for 45 min without a protective atmosphere intrinsic material resistance to crack extension[12] Both monolithic and layered samples were produced A number of the symmetrical layered structures have The monolithic samples were fabricated from stacked already been considered in relevant publications [3,5, 10]. tapes of the same Si3 N4 or Si3N4-30 wt%TIN composi- However, real laminates show some asymmetry of their tions Layered samples were prepared using two different architecture due to random deviations during the fabri- designs. The first type of specimen was with the outer lay cation process. Sometimes the asymmetrical layered ers in residual compression (Si3 N4 layers); the second structure is designed to meet specific engineering type of specimen was with the outer layers in residual equirements. Another problem is that only a few tension(Si3 N4-30 wt% TiN layers). The laminate param authors have considered the effect of elastic moduli mis- eters, such as composition and thickness of layers match between layers on toughening and fracture and calculated bulk residual stresses are presented in behavior [13, 14]. It was shown that in order to obtain Table 1. The deviation in the measured thickness of the the higher resistance to failure, the tensile layer should layers was about 5%. Though both monolithic and lay be made as stiff as possible (i.e. a high elastic modulus), ered samples have been nominally prepared from materi- whereas the compressive layers should be as compliant als of the same grades using the same manufacturing as feasible (i.e. a low elastic modulus)[14]. However, techniques, in reality the samples were prepared from d the conditions of stable or unstable crack growth in ferent batches of similar compositions and therefore cer- ceramic laminates have not been considered in [13, 14]. tain variations between impurities, defects, and other The effect of the residual stress on the apparent frac- parameters were expected to exist ture toughness and crack growth in non-symmetric The specimens for mechanical tests pared by Si3N4-based layered composites is analyzed in this machining the hot pressed tiles. Standard mor bars of study. Special attention is paid to analytical modeling dimensions 50 x 4 x 3 mm were surface ground to the to estimate the fracture toughness as a function of crack specification stated in EN843-1 The bars were also cham length in laminates having different elastic moduli of lay- fered along the long edges with a chamfer angle at 45to a ers. The validity of the method is examined by a com- dimension of0. 12 +0.03 mm. The fracture toughness was parison of calculated and measured fracture toughness measured by the single edge V-notch beam(SEVNB) values. Crack propagation behavior for laminates with technique [15, 16]using Eq (1). The V-notches with a tip the residual compressive or tensile stress in top layers radii of an order of 10-15 um were made in the specimen is investigated as well. with a diamond saw, followed by a stainless steel blade notching, and finally a diamond abrasive to obtain a sharp tip for the notch. The elastic modulus was measured 2. Experimental by a standard for ending techniqu The coefficients of thermal expansion(CTE) of the The manufacturing steps of Si3N4-TiN based lami- monolithic materials were measured using 50 mm long nates included (a) ball milling of powders in certain pro- MOR bars with a Baehr Dil 802 dilatometer from room portions;(b) rolling of thin tapes;(c)hot pressing of temperature to 1 100C in a nitrogen/hydrogen atmos

associated with the closure stress field behind the crack tip, which is similar to a crack bridging phenomenon acting in non-layered ceramics [11]. However, a tough￾ening by the residual compressive stress is determined by the overall crack length [12], while the toughening by the crack bridging depends only on the length incre￾ment of the moving crack. Crack bridging can provide increased toughening by a factor of approximately 2 [11]. Therefore, the residual compressive stresses can provide a significant increase in a ceramic
s toughness. Although fracture toughness of layered composite can be measured experimentally, it is an apparent value be￾cause of the superposition of different effects like stress shielding and intrinsic properties of the structure, such as grain size, composition, interfaces, etc. In fracture mechanics, both residual and applied stresses are usually included in the crack driving force. However it can be use￾ful to consider residual stresses as part of the crack resist￾ance. Thus, in laminates with residual compressive stress, the higher resistance to failure results from a reduction of crack driving force rather than from an increase in the intrinsic material resistance to crack extension [12]. A number of the symmetrical layered structures have already been considered in relevant publications [3,5,10]. However, real laminates show some asymmetry of their architecture due to random deviations during the fabri￾cation process. Sometimes the asymmetrical layered structure is designed to meet specific engineering requirements. Another problem is that only a few authors have considered the effect of elastic moduli mis￾match between layers on toughening and fracture behavior [13,14]. It was shown that in order to obtain the higher resistance to failure, the tensile layer should be made as stiff as possible (i.e. a high elastic modulus), whereas the compressive layers should be as compliant as feasible (i.e. a low elastic modulus) [14]. However, the conditions of stable or unstable crack growth in ceramic laminates have not been considered in [13,14]. The effect of the residual stress on the apparent frac￾ture toughness and crack growth in non-symmetric Si3N4-based layered composites is analyzed in this study. Special attention is paid to analytical modeling to estimate the fracture toughness as a function of crack length in laminates having different elastic moduli of lay￾ers. The validity of the method is examined by a com￾parison of calculated and measured fracture toughness values. Crack propagation behavior for laminates with the residual compressive or tensile stress in top layers is investigated as well. 2. Experimental The manufacturing steps of Si3N4–TiN based lami￾nates included (a) ball milling of powders in certain pro￾portions; (b) rolling of thin tapes; (c) hot pressing of tapes. Two compositions of layers were used: (1) Si3N4 (M11, Stark, Germany), (2) Si3N4–30 wt% TiN (grade C, Stark, Germany). The silicon nitride was used with 2 wt% Al2O3 and 5 wt% Y2O3 additives. The mixtures of various compositions were milled in the ball mill for 5 h. The average grain size of the milled powders was about 1 m. Crude rubber (4 wt%) was added to the mixture of powders as a plasticizer through a 3% solution in petrol. The powders were then dried, leaving 2 wt% residual amount of petrol in the mixture. After sieving powders with a 500 lm sieve, granulated powders were dried to 0.5 wt% residual petrol. A roll mill with 40 mm rolls was used for rolling. The velocity of rolling was 1.5 m/min. The working pressure was about 10 MPa to obtain a relative tape density of 64%. The thickness of green tapes was 0.4–0.5 mm and the width was 60–65 mm. Green tapes were stacked to￾gether to form the desired layered structures and cera￾mic samples were prepared by hot pressing the stacked tapes. The hot pressing was performed at 1820 C and 30 MPa for 45 min without a protective atmosphere. Both monolithic and layered samples were produced. The monolithic samples were fabricated from stacked tapes of the same Si3N4 or Si3N4–30 wt% TiN composi￾tions. Layered samples were prepared using two different designs. The first type of specimen was with the outer lay￾ers in residual compression (Si3N4 layers); the second type of specimen was with the outer layers in residual tension (Si3N4–30 wt% TiN layers). The laminate param￾eters, such as composition and thickness of layers, and calculated bulk residual stresses are presented in Table 1. The deviation in the measured thickness of the layers was about 5%. Though both monolithic and lay￾ered samples have been nominally prepared from materi￾als of the same grades using the same manufacturing techniques, in reality the samples were prepared from dif￾ferent batches of similar compositions and therefore cer￾tain variations between impurities, defects, and other parameters were expected to exist. The specimens for mechanical tests were prepared by machining the hot pressed tiles. Standard MOR bars of dimensions 50 · 4 · 3 mm were surface ground to the specification stated in EN843-1. The bars were also cham￾fered along the long edges with a chamfer angle at 45to a dimension of 0.12 ± 0.03 mm. The fracture toughness was measured by the single edge V-notch beam (SEVNB) technique [15,16] using Eq. (1). The V-notches with a tip radii of an order of 10–15 lm were made in the specimen with a diamond saw, followed by a stainless steel blade notching, and finally a diamond abrasive to obtain a sharp tip for the notch. The elastic modulus was measured by a standard four-point bending technique. The coefficients of thermal expansion (CTE) of the monolithic materials were measured using 50 mm long MOR bars with a Baehr Dil 802 dilatometer from room temperature to 1100 C in a nitrogen/hydrogen atmos- 290 M. Lugovy et al. / Acta Materialia 53 (2005) 289–296

Lugory et al. Acta Materialia 53(2005)289-296 Table I Geometrical characteristics of Si3 N/Si,N 30 wt% TiN layered materials Layer imens of type 2 Composition Layer thickness Residual stress tion Layer thickness Residual stress MPa) 752 Si3Na30 wt%TIN 852 30 wt% TiN 618 l86 142 23456789 l81 -225.7 sssss Si3Na30 wt%TIN 6l1 Si3Na-30 wt% TIN 1844 145 -227.3 Si3Na30 wt%TIN 6l1 Si3Na-30 wt% TIN 626 182.7 143 -228.8 Si3Na30 wt%TIN 616 47.1 Si3Na-30 wt% TIN 620 14 230.9 Si3Na-30 wt% TiN 733 Thickness(um) 45 4000 The average residual stress in layer. phere at a pressure of 10 Pa to avoid oxidation and disso- toughness Klc calculations because of a significant sim- ciation of the materials. An analysis of the macrostructure plification of the procedure. The most appropriate coor of the specimens was made with an optical microscope. dinate origin is on the tensile surface of the sample 3. Measurements of the CTes, the Ei and the ke of monolithic ceramics The values of the thermal expansion coefficient B of the si3 N4 and Si3Na-30 wt% TiN monolithic ceramics a function of temperature are presented in Fig. 2. The CTEs of both compositions are a linear function of the temperature, and the Cte of the Si3N4-30 wt% TiN is higher than the Cte of the Si,N4 over all investi- gated temperature ranges. The temperature dependence B [K be asB×10° 101+3.62797×10-3 T and B×10°=1.979406+302463 x 10-'Tfor the Si3N4 and the Si3 N4-30 wt%TiN, respec tively. The accuracy of such measurements is about 15%. Surface under tension Si3N4-30 TiN monolithic samples were measured to be 308 and (b) 323 GPa, respectively. Mean values of the intrinsic frac x ture toughness of monolith materials, measured by SEVNB, are approximately the same for both the Si3N4 and the Si3 N4- 30 wt% TiN compositions, about 4+ 1 MPa m".These measured values of CTEs, E and K(o were used in all calculations 4. Calculation of the apparent fracture toughness A weight function analysis has been used to estimate the apparent fracture toughness in laminates with resid- ual stresses [3, 13, 17, 18]. A schematic presentation of a two-componen Fig. 1(a), where ti is the thickness of a ith layer, w is Surface under tension the total thickness of the specimen, b is the width, and Fig 1. Scheme of a two-component multilayer specimen: (a)n Nis the total number of layers. The choice of coordinate of layers and layer boundary coordinates: (b) an analyzed system is of great importance to the apparent fracture location in a layered sample

phere at a pressure of 10 Pa to avoid oxidation and disso￾ciation of the materials. An analysis of the macrostructure of the specimens was made with an optical microscope. 3. Measurements of the CTEs, the Ei and the K1c (i) of monolithic ceramics The values of the thermal expansion coefficient b of the Si3N4 and Si3N4–30 wt% TiN monolithic ceramics as a function of temperature are presented in Fig. 2. The CTEs of both compositions are a linear function of the temperature, and the CTE of the Si3N4–30 wt% TiN is higher than the CTE of the Si3N4 over all investi￾gated temperature ranges. The temperature dependence of b [K1 ] can be presented as b · 106 = 1.09274 · 101 + 3.62797 · 103 T and b · 106 = 1.979406 + 3.02463 · 103 T for the Si3N4 and the Si3N4–30 wt% TiN, respec￾tively. The accuracy of such measurements is about 15%. Young
s moduli of the Si3N4 and the Si3N4–30 wt% TiN monolithic samples were measured to be 308 and 323 GPa, respectively. Mean values of the intrinsic frac￾ture toughness of monolith materials, measured by SEVNB, are approximately the same for both the Si3N4 and the Si3N4–30 wt% TiN compositions, about 4 ± 1 MPa m1/2. These measured values of CTEs, Ei and KðiÞ 1c were used in all calculations. 4. Calculation of the apparent fracture toughness A weight function analysis has been used to estimate the apparent fracture toughness in laminates with resid￾ual stresses [3,13,17,18]. A schematic presentation of a two-component multilayered sample is shown in Fig. 1(a), where ti is the thickness of a ith layer, w is the total thickness of the specimen, b is the width, and N is the total number of layers. The choice of coordinate system is of great importance to the apparent fracture toughness K1c calculations because of a significant sim￾plification of the procedure. The most appropriate coor￾dinate origin is on the tensile surface of the sample Fig. 1. Scheme of a two-component multilayer specimen: (a) numbers of layers and layer boundary coordinates; (b) an analyzed crack location in a layered sample. Table 1 Geometrical characteristics of Si3N4/Si3N4–30 wt% TiN layered materials Layer # Specimens of type 1 Specimens of type 2 Composition Layer thickness (lm) Residual stress (MPa)a Composition Layer thickness (lm) Residual stress (MPa)a 1 Si3N4 752 223.6 Si3N4–30 wt% TiN 852 42.3 2 Si3N4–30 wt% TiN 618 186 Si3N4 142 274 3 Si3N4 181 225.7 Si3N4–30 wt% TiN 611 44.1 4 Si3N4–30 wt% TiN 625 184.4 Si3N4 145 272.5 5 Si3N4 176 227.3 Si3N4–30 wt% TiN 611 45.6 6 Si3N4–30 wt% TiN 626 182.7 Si3N4 143 271.1 7 Si3N4 176 228.8 Si3N4–30 wt% TiN 616 47.1 8 Si3N4–30 wt% TiN 620 181.1 Si3N4 143 269.6 9 Si3N4 726 230.9 Si3N4–30 wt% TiN 733 48.7 Thickness (lm) 4500 4000 a The average residual stress in layer. M. Lugovy et al. / Acta Materialia 53 (2005) 289–296 291

M. Lugoty et al. Acta Materialia 53 (2005)289-296 where Ki is the intrinsic fracture toughness of the ith B106K1 layer material, ual stresses, h(xla, a) is the weight function for an edge cracked sample [13, 17, 18]. xi is the coordinate of an boundary of the ith lay E=El/(1-v), and E; and v; are the elastic modulus and Poisson ratio of the ith layer, respectively. A Pois- son ratio of 0. 25 for both compositions was used in all calculations. The expressions for IL (=0, 1, 2)and LG=0, 1)were obtained in [18] as follows EO (4) 1200 where Ei is the strain in the ith layer, which is not asso- TK ciated with any stress. The thermal expansion and/or volume change due to a crystallographic phase transfor Fig.2. Thermal expansion coefficients(CTEs)of the Si, N (I)and the mation might be the source of this strain. However, the Si3Ne-30 wt% TiN(2)in the 290-1200 K temperature range. Dashed case of a phase transformation is out of the scope of this les show experimental range of CTEs. paper. In the case of thermal expansio B,(T)dT, under bending. The geometry of the multilayered mate- rial analyzed here is such that the problem can be re- where B(T) is the thermal expansion coefficient of the ith duced to one dimension and that analytically tractable solutions can be used [18]. A schematic presentation of layer at the temperature T. To and T, are the actual and ¨ joining the analyzed crack location in the layered specimen cooling, the strain mismatch due to the different thermal presented in Fig. I(b), where a is the crack length and n is the number of layers crossed by the crack expansion coefficients is accommodated by creep as long as the temperature is high enough. Below a certain tem- An experimental value of the apparent fracture perature, called the"joining"temperature, the different toughness can be found using the xpression 191 Kapp=Y(a)/ 2, measure experimentally, and in general, T; is adopted to be somewhere below the sintering temperature. If 阝( T)is a linear function,=(B)△T, where 1(x)1.99-2(1-x)(2.15-393x+2.7x2 AT=T-To. (B)=E To*(2 is the average value of (1+2)(1-2)y2 the thermal expansion coefficient in the temperature 1.5P(s1-s2) range from Toto T. Other authors [20] used 1200K as the"joining "temperature for calculation of the resid ual stresses of Si3N4 based laminates. However, in our where P is the critical load (the applied bending load case, 948 and 765 K were assumed for AT with the lam corresponding to specimen failure)and si and s2 are outer inates with compressive and tensile residual stresses in and inner support spans of the four-point bending fix- the top layers, respectively, because these temperatures ture for SEVNb samples provided the best fit between the calculated and experi- The apparent fracture toughness of the layered co mentally measured values of fracture toughness for both posite can be calculated analytically by [18] types of laminate 6(2-1(k1-k) 人m一(E一+X与x-山于

under bending. The geometry of the multilayered mate￾rial analyzed here is such that the problem can be re￾duced to one dimension and that analytically tractable solutions can be used [18]. A schematic presentation of the analyzed crack location in the layered specimen is presented in Fig. 1(b), where a is the crack length and n is the number of layers crossed by the crack. An experimental value of the apparent fracture toughness can be found using the expression [19]: Kapp ¼ Y ðaÞrma1=2 ; ð1Þ where Y ðaÞ ¼ 1:99 að1 aÞð2:15 3:93a þ 2:7a2Þ ð1 þ 2aÞð1 aÞ 3=2 ; rm ¼ 1:5Pðs1 s2Þ bw2 and a ¼ a=w; where P is the critical load (the applied bending load corresponding to specimen failure) and s1 and s2 are outer and inner support spans of the four-point bending fix￾ture for SEVNB samples. The apparent fracture toughness of the layered com￾posite can be calculated analytically by [18]: where KðiÞ 1c is the intrinsic fracture toughness of the ith layer material, Kr is the stress intensity due to the resid￾ual stresses, h(x/a,a) is the weight function for an edge￾cracked sample [13,17,18], xi is the coordinate of an upper boundary of the ith layer (Fig. 1), E0 i ¼ Ei=ð1 miÞ, and Ei and mi are the elastic modulus and Poisson ratio of the ith layer, respectively. A Pois￾son ratio of 0.25 for both compositions was used in all calculations. The expressions for ILj (j = 0, 1, 2) and JLj (j = 0, 1) were obtained in [18] as follows: ILj ¼ 1 j þ 1 XN i¼1 E0 i ðx jþ1 i x jþ1 i1 Þ; ð3Þ J Lj ¼ 1 j þ 1 XN i¼1 ~eiE0 i ðx jþ1 i x jþ1 i1 Þ; ð4Þ where ~ei is the strain in the ith layer, which is not asso￾ciated with any stress. The thermal expansion and/or volume change due to a crystallographic phase transfor￾mation might be the source of this strain. However, the case of a phase transformation is out of the scope of this paper. In the case of thermal expansion: ~ei ¼ Z T j T 0 biðT ÞdT ; where bi(T) is the thermal expansion coefficient of the ith layer at the temperature T. T0 and Tj are the actual and ‘‘joining’’ temperatures, respectively. During sample cooling, the strain mismatch due to the different thermal expansion coefficients is accommodated by creep as long as the temperature is high enough. Below a certain tem￾perature, called the ‘‘joining’’ temperature, the different components become bonded together and residual stres￾ses appear. The ‘‘joining’’ temperature is difficult to measure experimentally, and in general, Tj is adopted to be somewhere below the sintering temperature. If bi(T) is a linear function, ~ei ¼ hbi iDT , where DT = Tj T0, hbii ¼ biðT 0ÞþbiðT jÞ 2 is the average value of the thermal expansion coefficient in the temperature range from T0 to Tj. Other authors [20] used
1200 K as the ‘‘joining’’ temperature for calculation of the resid￾ual stresses of Si3N4 based laminates. However, in our case, 948 and 765 K were assumed for DT with the lam￾inates with compressive and tensile residual stresses in the top layers, respectively, because these temperatures provided the best fit between the calculated and experi￾mentally measured values of fracture toughness for both types of laminate. 400 800 1200 1 2 3 4 5 6 T,K β•106 ,K-1 2 1 Fig. 2. Thermal expansion coefficients (CTEs) of the Si3N4 (1) and the Si3N4–30 wt% TiN (2) in the 290–1200 K temperature range. Dashed lines show experimental range of CTEs. Kapp ¼ 6Y ðaÞa1=2 I 2 L1 IL0IL2
KðiÞ 1c Kr  w2 E0 nþ1 R a xn h x a ; a
½ IL0x IL1 dx þ Pn i¼1E0 i R xi xi1 h x a ; a
½ IL0x IL1 dx n o ; ð2Þ 292 M. Lugovy et al. / Acta Materialia 53 (2005) 289–296

M. Lugory et al. Acta Materialia 53 (2005)289-296 293 The stress intensity due to the residual stresses is 5. Apparent fracture toughness of the layered composite []: with residual compressive or tensile stresses in the tor The calculated values of the apparent fracture X[LIJLI-I2JLo +(ILrJLo-ILoJu1)rdx toughness as a function of the crack length parameter +>E厂e)- d in the si3N4/Si3N4-30 wt% TiN laminate with com re shown in Fig. 4(a). The tough ness increases in the layers with compressive stress with +(uiLo-ILoJuixdu increasing crack length, and it decreases in the layers with tensile stress as the crack continues to grow The apparent fracture toughness Kapp in layered spec The layers with compressive and tensile stresses are mens can be analyzed as a function of the crack length parameter a, where d= Y(a)a".The crack length SiaNA Si N30%TIN Si, SiN-30STiN parameter d is the most appropriate to demonstrate crit ical conditions of a crack growth. One of the advantages of this parameter is that the stress intensity factor of an edge crack for a fixed value of the applied stress om is a raight line from the coordinate origin in the coordinate system Kapp-d Since K1=oma, the slope of the straight line is the applied stress om. The conditions for unstable crack growth in the internal stress field are as follows [12]: K,(om, a)=Kapp(a): dK,(om, a)/da> dKapp(a)/da. Using parameter a, these conditions become omd= Kapp(a) and om kApp(a)/da, which can be reduced to: Kap(a)/a≥dKap(a/da It follows from Eq(6) that unstable crack growth oc curs if the slope of the straight line corresponding to 0.020.040.060.080.10 the stress intensity factor at constant applied stress is greater than or equal to the slope of the tangent line to the fracture resistance curve at the same point (Fig. 3). Also the applied stress intensity factor be- comes higher than the fractur material K Stress intensity factor Tangent dK./da SiN-MIeTiN SiNrMMeTiN SiN 0.020.040.06 0.080.10 Fig. 4. The apparent fracture toughness as a function of the crack Ka la= omm length parameter a in the laminate with compressive (a)and tensile(b) outer layers. Filled circles correspond to the experimental data. Inserts are optical micrographs of the two parts of Si3N4/Si3N4-30 wt% TIN laminate samples with (a) Si3N4 surface layers with a residual Fig. 3. General criterion of stable/unstable crack growth in a brittle compressive stress and(b)Si3N/30 wt% TIN surface layers with a residual tensile stress after SEvNB test

The stress intensity due to the residual stresses is [18]: Kr ¼ 1 I 2 L1 IL0IL2  E0 nþ1 Z a xn h x a ; a   ½ IL1J L1 IL2J L0 þ ð Þ IL1J L0 IL0J L1 x dx þXn i¼1 E0 i Z xi xi1 h x a ; a ½IL1J L1 IL2J L0 þ ð Þ IL1J L0 IL0J L1 xdx  : ð5Þ The apparent fracture toughness Kapp in layered spec￾imens can be analyzed as a function of the crack length parameter a˜, where a˜ = Y(a)a1/2. The crack length parameter a˜ is the most appropriate to demonstrate crit￾ical conditions of a crack growth. One of the advantages of this parameter is that the stress intensity factor of an edge crack for a fixed value of the applied stress rm is a straight line from the coordinate origin in the coordinate system Kapp–a˜. Since K1 = rma˜, the slope of the straight line is the applied stress rm. The conditions for unstable crack growth in the internal stress field are as follows [12]: K1(rm,a) = Kapp(a); dK1(rm,a)/da P dKapp(a)/da. Using parameter a˜, these conditions become rma˜ = Kapp(a˜) and rm P dKapp(a˜)/da˜, which can be reduced to: Kappð~aÞ=~a P dKappð~aÞ=d~a: ð6Þ It follows from Eq. (6) that unstable crack growth oc￾curs if the slope of the straight line corresponding to the stress intensity factor at constant applied stress is greater than or equal to the slope of the tangent line to the fracture resistance curve at the same point (Fig. 3). Also the applied stress intensity factor be￾comes higher than the fracture resistance of the material. 5. Apparent fracture toughness of the layered composite with residual compressive or tensile stresses in the top layer The calculated values of the apparent fracture toughness as a function of the crack length parameter a˜ in the Si3N4/Si3N4–30 wt% TiN laminate with com￾pressive outer layers are shown in Fig. 4(a). The tough￾ness increases in the layers with compressive stress with increasing crack length, and it decreases in the layers with tensile stress as the crack continues to grow. The layers with compressive and tensile stresses are K app ã dKa /dã K a /ã = σ m Fracture resistance Stress intensity factor at constant applied stress 0 Tangent line Fig. 3. General criterion of stable/unstable crack growth in a brittle material. Fig. 4. The apparent fracture toughness as a function of the crack length parameter a˜ in the laminate with compressive (a) and tensile (b) outer layers. Filled circles correspond to the experimental data. Inserts are optical micrographs of the two parts of Si3N4/Si3N4–30 wt% TiN laminate samples with (a) Si3N4 surface layers with a residual compressive stress and (b) Si3N4–30 wt% TiN surface layers with a residual tensile stress after SEVNB test. M. Lugovy et al. / Acta Materialia 53 (2005) 289–296 293