1048 Journal of the American Ceramic Sociery-Jacobsen and Brondsted Vol 84. No 5 2VAb2 +b3)Emri simulated stress-strain curve for SiC/SiC shows good agreement Or (1 -Va1) (18) with measured behavior(Fig 3(b)) (19) (2) CAiC The constant crack spacing of the 90 plies implies that The interfacial debond fracture energy is I and the thermal strain S-received C/SiC is already in stage Ill. Using the rule of is defined as mixtures,E,=0.5(EL Er) and setting ET =0 GPa, we find E 1 17 GPa, which is within the measured range of Er, supporting an)△T the stage Ill assumption. The large residual stress g(Table (20) between the plies in the C/SiC material demonstrates why this The analysis of the hysteresis behavior of unidirectional materials aterial is precracked in the 90 plies in the as-received condition s based on the results obtained in Refs. 10 and 36. Generalized Also, the fiber/matrix interface is in residual tension(or >0)for hysteresis behavior in the case of unloading to nonzero stresses can the CsiC material (Table I). When matrix cracking in the 0 plies be found in ref 4 occurs followed interface debonding. there should be no connection between fiber and matrix in C/Sic due to the tensile stress in the interface, and an interfacial gap is predicted. 7If V. Simulation of Tensile Behavior asperities are present, there may be interfacial contact during (I SiC/SiC sliding between fiber and matrix. The transition from stages 0 to I occurs at the first deviation for he osore, a simple one-dimensional shear-lag model is applied from linearity of the stress-strain curve, i.e., at 41 MPa (Table D) and it requires that the interface is in compr With Co=1.20 and h= 0.30 the fractur for sion. 35 Models for one-dimensional stress-strain behavior have tunneling cracking is roo=3.9 J/m2 at h/Loo =0. The simulated Deen proposed. 4, 5, 48 The approach used here is based on a can be compared with the experimentally measured changes. stiffness due to relief of residual strain and interfacial debonding, emulations of stiffness change with stress are plotted in Fig. 10 Too values. The stiffness changes indicate a I A one-dimensional model neglects the Poisson contraction,i.e ncreasing from 3.9 to 32.5 J/m. Such an increase is unlikely, because the tunneling crack openings in general are small. It Up = 0. The last simplifying assumption is that the change in peculated whether there is a contribution from tunneling cracks radial stress on the outer boundary of the cylindrical unit cell is rowing into the 0 plies, where fiber bridging increases the zero, which is a type-I boundary condition, equivalent to the apparent fracture energy, as shown by Ref. 46, and thus affecting implicit assumption for earlier-derived one-dimensional models The nondimensional constants(as and bs)become simple expres- The damage characterization shows that significant matrix sions. as shown below cracking in the oo bundles starts at o/S of 0.45-0.55. The crack spacing in the 90% plies follows the evolution of the crack spacing II. To obtain the smoothest transition, a=Ea≈(1-E stage II is set active at o/S>0.55. At hLo >0.9 the residual Em sses are largely released, and a maximum permanent strain of 0.015% because of tunneling cracking has been calculated for SiC/SiC using Eq (9) The relation o, =0.86o is found from analysis of the slope of h≈. REM 4EEVTLo the hysteresis curve. The same relation has been found for a similar material. This functional relation comes directly from analysis of The stress quantities become the composite stress; i.e., the stress on the 0 ply is not explicitly needed. The frictional shear stress t= 80 MPa is derived from the (23) maximum hysteresis loop width at the last unloading peak stress (stage III) before fracture, and it is assumed to have this constant The overall monotonic stress-strain behavior can be written a value at all stresses where stages l and ll are active. Th 2H(o +0)- E E (24) ar=26MPaSic/sic where the damaged elastic modulus Eo is given by 「e=39Jm2123047 11 +4H(+σ1) Simulated Keith and Kedward have used a constant TLo to simulate the 0.5EL stress-strain behavior of a SiC-fiber-reinforced AlPOa-matri composite and have found reasonable agreement between experi- 04 Stages 02回 (25). A good fit for the elastic modulus degradation has bee obtained with TLo =6.0 MPar'mm(Fig. 11). The initial residual contribution because of a constant TLo is subtracted from all the 0.0 equations before simulation begins, i.e., at o=0 MPa. The loss of 020.4 stiffness is sensitive to TLo (Fig. 11) Therefore, an experimentally observed maximum hysteresis Fig. 10. Stiffness change and onset of tunneling cracking as a function loop width eMax and the permanent strain at zero unloading stress nneling cracking toughness Tgo in SiC/SiC, To obtain the agreement Euo have been fitted to various TLo values. bEma can be fitted with experimental results and theory it is necessary to increase T9o Euo= 4.5 MPa'mm, whereas TLo can be fitted with values ranging accordingly, At stresses >0.55S, the compliance change is mainly due to from 4.5 to 6 MPa'mm. The tensile behavior of C/SiC(0, 90)is matrix cracking and debonding in the 0 plies. simulated using TLo = 4.5 MPa, mm(Fig 4(b)sD 5 2Vf~b2 1 b3! 1/ 2 ~1 2 Vfa1! S EmGi R D 1/ 2 (18) sT 5 Vfa2 1 2 Vfa1 EmεT (19) The interfacial debond fracture energy is Gi , and the thermal strain is defined as εT 5 ~afL 2 am!DT (20) The analysis of the hysteresis behavior of unidirectional materials is based on the results obtained in Refs. 10 and 36. Generalized hysteresis behavior in the case of unloading to nonzero stresses can be found in Ref. 45. IV. Simulation of Tensile Behavior (1) SiC/SiC The transition from stages 0 to I occurs at the first deviation from linearity of the stress-strain curve, i.e., at 41 MPa (Table I). With C1 0 5 1.20 and h 5 0.30 mm, the fracture energy for tunneling cracking is G90 5 3.9 J/m2 at h/L90 5 0. The simulated stiffness change for G90 5 3.9 J/m2 is shown in Fig. 10, where it can be compared with the experimentally measured changes. Simulations of stiffness change with stress are plotted in Fig. 10 for various G90 values. The stiffness changes indicate a G90 increasing from 3.9 to 32.5 J/m2 . Such an increase is unlikely, because the tunneling crack openings in general are small. It is speculated whether there is a contribution from tunneling cracks growing into the 0° plies, where fiber bridging increases the apparent fracture energy, as shown by Ref. 46, and thus affecting the composite compliance. The damage characterization shows that significant matrix cracking in the 0° bundles starts at s/S of ;0.45–0.55. The crack spacing in the 90° plies follows the evolution of the crack spacing in the 0° plies during stage II. To obtain the smoothest transition, stage II is set active at s/S . 0.55. At h/L90 . 0.9, the residual stresses are largely released, and a maximum permanent strain of 0.015% because of tunneling cracking has been calculated for SiC/SiC using Eq. (9). The relation si 5 0.86sp is found from analysis of the slope of the hysteresis curve. The same relation has been found for a similar material.12 This functional relation comes directly from analysis of the composite stress; i.e., the stress on the 0° ply is not explicitly needed. The frictional shear stress t 5 80 MPa is derived from the maximum hysteresis loop width at the last unloading peak stress (stage III) before fracture, and it is assumed to have this constant value at all stresses where stages II and III are active. The simulated stress–strain curve for SiC/SiC shows good agreement with measured behavior (Fig. 3(b)). (2) C/SiC The constant crack spacing of the 90° plies implies that as-received C/SiC is already in stage III. Using the rule of mixtures, Ey 5 0.5(EL 1 ET) and setting ET 5 0 GPa, we find Ey 5 117 GPa, which is within the measured range of Ey, supporting the stage III assumption. The large residual stress sR (Table I) between the plies in the C/SiC material demonstrates why this material is precracked in the 90° plies in the as-received condition. Also, the fiber/matrix interface is in residual tension (sr R . 0) for the C/SiC material (Table I). When matrix cracking in the 0° plies occurs followed by interface debonding, there should be no connection between fiber and matrix in C/SiC due to the tensile stress in the interface, and an interfacial gap is predicted.47 If asperities are present, there may be interfacial contact during sliding between fiber and matrix. Therefore, a simple one-dimensional shear–lag model is applied for the 0° plies, and it requires that the interface is in compres￾sion.35 Models for one-dimensional stress–strain behavior have been proposed.4,5,48 The approach used here is based on a simplified version of the present model and includes the loss of stiffness due to relief of residual strain and interfacial debonding, which are not included in the previous models. A one-dimensional model neglects the Poisson contraction, i.e., nf 5 nm 5 0. The debond fracture energy is neglected, leading to sD 5 0. The last simplifying assumption is that the change in radial stress on the outer boundary of the cylindrical unit cell is zero, which is a type-I boundary condition,35 equivalent to the implicit assumption for earlier-derived one-dimensional models. The nondimensional constants (as and bs) become simple expres￾sions, as shown below: a1 5 Ef EL a2 5 ~1 2 Vf!Ef EL b2 5 Em Ef b3 5 Vf 1 2 Vf (21) H 5 RVm 2 Em 2 4EfEL 2 Vf 2 tL0 (22) The stress quantities become si 5 2sT sT 5 2Vfsf R (23) The overall monotonic stress–strain behavior can be written as ε 5 s E0 1 S 1 E0 2 1 EL DsT 1 2H~s 1 sT! 2 (24) where the damaged elastic modulus E0 is given by 1 E0 5 1 EL 1 4H~s 1 sT! (25) Keith and Kedward48 have used a constant tL0 to simulate the stress–strain behavior of a SiC-fiber-reinforced AlPO4-matrix composite and have found reasonable agreement between experi￾ment and simulation, although they neglected the last term in Eq. (25). A good fit for the elastic modulus degradation has been obtained with tL0 5 6.0 MPazmm (Fig. 11). The initial residual contribution because of a constant tL0 is subtracted from all the equations before simulation begins, i.e., at s 5 0 MPa. The loss of stiffness is sensitive to tL0 (Fig. 11). Therefore, an experimentally observed maximum hysteresis loop width dεmax and the permanent strain at zero unloading stress εu0 have been fitted to various tL0 values. dεmax can be fitted with εu0 5 4.5 MPazmm, whereas tL0 can be fitted with values ranging from 4.5 to 6 MPazmm. The tensile behavior of C/SiC (0°,90°) is simulated using tL0 5 4.5 MPazmm (Fig. 4(b)). Fig. 10. Stiffness change and onset of tunneling cracking as a function of tunneling cracking toughness G90 in SiC/SiC. To obtain the agreement between experimental results and theory it is necessary to increase G90 accordingly. At stresses .0.55S, the compliance change is mainly due to matrix cracking and debonding in the 0° plies. 1048 Journal of the American Ceramic Society—Jacobsen and Brøndsted Vol. 84, No. 5