Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 A3(a) t()(1E) eve For the purely fractional interface (i.e, Sd=0), Eq (27)is reduced to the ACK result and expressed as 6VftsEYEcsm When interfacial bonding exists between the fiber and the matrix interface (i.e., Sa >0). the matrix cracking stress, cr, can be obtained by using the root finding technique to solve Eq. (27)with the equirement of Eq (25) 5. Results and discussion e The ceramic composite systems of SiC(SCS-6)/borosilicate. SiC(Nicalon )/LAS and C/borosilicate are ed for the theoretical study and their material properties are listed in Table I The influences of interfacial bonding toughness and frictional shear stress on the matrix cracking stresses are illustrated in Fig. 4, in which the matrix cracking stresses of SiC/borosilicate composite are plotted as a function of frictional shear stress, ts, for different relative critical strain energy release rate, Sa/sm. The prediction for a perfectly bonded interface by the bhe model is also plotted in Fig. 4 for the purpose of comparison. He and Hutchinson [13] indicated that the critical strain energy release rate of the interface Sa should be less than one-fourth of the critical strain energy release rate of the matrix 4 otherwise the matrix crack propagates into the fiber rather than deflecting along the fiber/matrix interface. Therefore, the maximum relative critical strain energy release rate, Sa/5m, is chosen as 0.25 in the present analysis If the interface is resisted by frictional shear stress only (i. e, Sa/sm=0), the present analysis is consistent with the ACK result. For the case of maximum facial bonding toughness (i.e Table I Properties of composite systems SiC/borosilicate C/ borosilicate SiC/LAS Er 400 Gpa 380 Gpa 200 Gpa 63 Gpa 8.92J 892Jm2 47J 6-8 MPa 10-25 MPa 1-2 MPa 35-40×10-6°C-(ong)2.6×10-6°-( radia) 0.1×10-6°C-1 3.2×10-6°C-1 3.2×10-6C-1 500° 500°C -675°Cto-1200°C SCS-6AVcO Data from Ref [15] Nicalon Data from Refs. [15 and 18A2…s† ˆ ÿ 2ts aEf l 2 d …28b† A3…s† ˆ 4 3 ts a 2 VfEc VmEmEf  l 3 d ÿ 4Vfzd a ld ÿ Vmzm …28c† For the purely fractional interface (i.e., zd ˆ 0), Eq. (27) is reduced to the ACK result and expressed as scr ˆ 6V 2 f tsEfEczm aVmE 2 m !1=3 …29† When interfacial bonding exists between the ®ber and the matrix interface (i.e., zd > 0), the matrix cracking stress, scr, can be obtained by using the root ®nding technique to solve Eq. (27) with the requirement of Eq. (25). 5. Results and discussion The ceramic composite systems of SiC(SCS-6)/borosilicate, SiC(Nicalon)/LAS and C/borosilicate are used for the theoretical study and their material properties are listed in Table 1. The in¯uences of interfacial bonding toughness and frictional shear stress on the matrix cracking stresses are illustrated in Fig. 4, in which the matrix cracking stresses of SiC/borosilicate composite are plotted as a function of frictional shear stress, ts, for di€erent relative critical strain energy release rate, zd=zm: The prediction for a perfectly bonded interface by the BHE model is also plotted in Fig. 4 for the purpose of comparison. He and Hutchinson [13] indicated that the critical strain energy release rate of the interface zd should be less than one-fourth of the critical strain energy release rate of the matrix zm, otherwise the matrix crack propagates into the ®ber rather than de¯ecting along the ®ber/matrix interface. Therefore, the maximum relative critical strain energy release rate, zd=zm, is chosen as 0.25 in the present analysis. If the interface is resisted by frictional shear stress only (i.e., zd=zm ˆ 0), the present analysis is consistent with the ACK result. For the case of maximum interfacial bonding toughness (i.e., Table 1 Properties of composite systems SiCa /borosilicateb C/ borosilicateb SiCc /LASd Ef 400 Gpa 380 Gpa 200 Gpa Em 63 Gpa 63 Gpa 85 Gpa a 70 mm 4 mm 8 mm zm 8.92 J/m2 8.92 J/m2 47 J/m2 ts 6±8 MPa 10±25 MPa 1±2 MPa af 3:5 ÿ 4:0 10ÿ6 8Cÿ1 (long.) 2:6 10ÿ6 8Cÿ1 (radial) 0:1 10ÿ6 8Cÿ1 3:1 10ÿ6 8Cÿ1 am 3:2 10ÿ6 8Cÿ1 3:2 10ÿ6 8Cÿ1 1:5 10ÿ6 8Cÿ1 DT ÿ5008C ÿ5008C ÿ6758C to ÿ12008C a SCS-6 AVCO. b Data from Ref. [15]. c Nicalon. d Data from Refs. [15 and 18]. 22 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28