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 testThe 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