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