May 2001 Mechanical Properties of Two Plain-Woven Cvl SiC-Matrix Composites 047 This investigation revealed regions with a low crack spacing Lo of Tunneling cracking releases residual on the lamina 10-20 um)and regions with no matrix cracks in the 0 bundles level, leading to permanent deformation ons and tun unneling crack SEM investigations of the fracture surface also revealed a high openings at complete unloading Rewriting Its obtained in crack density(Fig. 8). The tunneling crack spacing Lgo in the 90 Ref 18, the permanent strain ep at complete unloading is given by bundles remained constant at an average of 124 um. Conse- quently, there were no experimental observations of evolution of matrix cracks in the bundles as a function of stress for this SoCE+E h material The above solution is unbounded, but an upper bound is the situation where the stress in the o' ply is zero, i.e., the permanent Ill. Theory of Tensile Behavior strain is the negative of the initial strain in the 0' plies. Therefore, (1 Damage-Free Properties the solution is bounded by the limit The stacked, plain-woven composite is treated as E +ero symmetric cross-ply laminate. The height of each ply equal to the maximum height of a bundle. The analytica Ey for calculating damage-free properties based on the properties is shown elsewhere" and reviewed briefly here. The interphase adheres preferably to the fiber after debonding and stage Il, multiple matrix cracking occurs simultaneo therefore, assumed to connect to the fiber(Fig. 9). The porosity is compliance of the 0 ply Eo can be written as in stage Il.The and 90. plies. We assume that lo =loo in assumed to alter the matrix properties. Such a procedure leads to a fiber/matrix system with an interface that subsequently can be used for interpretation of the inelastic behavior E-E+ D (10) (2) Tensile Behavior The nonlinear behavior is divided into five damage stages Stage 0 defines the damage-free composite exhibiting a linea D.=E(-en2+b)(= (11) elastic behavior. In Stage I, the cracking is confined to the 90o olies until stage Il is reached, where the 90 ply cracks penetrate where a and b are nondimensional constants defined in Ref. 35. Vr into the 0 plies. Stage Ill defines 90 ply crack saturatio is the volume fraction of fibers, Em the Young modulus of the delamination between the plies, and continuous cracking in the 0 matrix, and T a constant interfacial shear stress. The composite lies. Stage IV is fiber fracture and pullout. This idealized compliance is given by- composite behavior is consistent with experimental observations on brittle-matrix fiber-reinforced cross-ply materials. 2,13, 15-7 (A) Stage F90 Ply-Tunneling Cracks: The increase 1+C1+2D (12) composite compliance due to tunneling cracks can be written as Assuming that the strains in the 90 and 0 plies are equal, the 11 1+C1 effective stress(o) acting on the 0 ply is appr An approximation for CI based on finite-element calculations 1+ Cit+ 2DIEL (C) Stage I-0 Ply Cracking and Delamination: In stage C (4) the mechanical behavior is fully controlled by the o ply, and compliance becomes The nondimensional constant Ci depends weakly on strongly on the ply modulus ratio ELeT. A relationship EOR 1+D Loo and the applied composite stress o is derived in Ref. p between can be rewritten as and the stress on the 0 ply is given by uo= 2o E+E The analysis of the nonlinear monotonic tensile behavior of the 0 ply is based on the stress-displacement analysis by Hutchinson and Jensenof a fiber being pulled out of a brittle matrix against friction. The analysis can be rewritten in terms of stress and strain, 时时m finite-element solutions for small debond lengths with the fiber radius R(R< 1).The ove train E for a multiple-cracked unidirectional laminate onsists of three strain contributions that can be written ass Ee tEttE ET or E+(En-E1+2H2o0-o)n+-(5) From Eq(5), the critical stress for the onset of tunneling cracking where the strain subscipts e, T, and s refer to the elastic, thermal (onst) can be derived as loo→∞ and sliding contributions, respectively, and E t E Rb2(1-V4a1)2 4EmVTLo Using Eq. (5), the compliance change can be simulated as a In Eq (15), the stresses are given by function of composite stress. At crack densities h/Lgo 2, further cracking in the 90 plies has little effect on stiffnessThis investigation revealed regions with a low crack spacing (L0 of ;10–20 mm) and regions with no matrix cracks in the 0° bundles. SEM investigations of the fracture surface also revealed a high crack density (Fig. 8). The tunneling crack spacing L90 in the 90° bundles remained constant at an average of 124 mm. Conse￾quently, there were no experimental observations of evolution of matrix cracks in the bundles as a function of stress for this material. III. Theory of Tensile Behavior (1) Damage-Free Properties The stacked, plain-woven composite is treated as a classical symmetric cross-ply laminate. The height of each ply (2h) is set equal to the maximum height of a bundle. The analytical approach for calculating damage-free properties based on the constituent properties is shown elsewhere45 and reviewed briefly here. The interphase adheres preferably to the fiber after debonding and is, therefore, assumed to connect to the fiber (Fig. 9). The porosity is assumed to alter the matrix properties. Such a procedure leads to a fiber/matrix system with an interface that subsequently can be used for interpretation of the inelastic behavior. (2) Tensile Behavior The nonlinear behavior is divided into five damage stages. Stage 0 defines the damage-free composite exhibiting a linear elastic behavior. In Stage I, the cracking is confined to the 90° plies until stage II is reached, where the 90° ply cracks penetrate into the 0° plies. Stage III defines 90° ply crack saturation, delamination between the plies, and continuous cracking in the 0° plies. Stage IV is fiber fracture and pullout. This idealized composite behavior is consistent with experimental observations on brittle-matrix fiber-reinforced cross-ply materials.12,13,15–17 (A) Stage I—90° Ply-Tunneling Cracks: The increase in composite compliance due to tunneling cracks can be written as22 1 Ey 5 1 Ey 0 S 1 1 C1 h L90D (3) An approximation for C1 based on finite-element calculations is18,22 C1 5 C1 0 tanhS ET C1 0 EL L90 h D (4) The nondimensional constant C1 0 depends weakly on Vf , but strongly on the ply modulus ratio EL/ET. 22 A relationship between L90 and the applied composite stress s is derived in Ref. 18, and can be rewritten as s 5 3 G90E# y 0 hC1 0 tanhS ET C1 0 EL L90 h D4 1/ 2 2 EL 1 ET 2ET sR (5) where G90 is the toughness of the 90° ply in the tunneling cracking mode and E# y 0 the plane strain modulus of the cross ply, given as E# y 0 5 1 2 ELS1 1 EL ET D EL ET 2 nL 2 (6) From Eq. (5), the critical stress for the onset of tunneling cracking (sonset) can be derived as L90 3 `: sonset 5 S G90E# y 0 hC1 0 D 1/ 2 2 EL 1 ET 2ET sR (7) Using Eq. (5), the compliance change can be simulated as a function of composite stress. At crack densities h/L90 . 2, further cracking in the 90° plies has little effect on stiffness. Tunneling cracking releases residual stresses on the lamina level, leading to permanent deformations and tunneling crack openings at complete unloading. Rewriting the results obtained in Ref. 18, the permanent strain εp 90 at complete unloading is given by εp 90 5 C1 EL 1 ET 2ETE# y 0 h L90 sR (8) The above solution is unbounded, but an upper bound is the situation where the stress in the 0° ply is zero; i.e., the permanent strain is the negative of the initial strain in the 0° plies. Therefore, the solution is bounded by the limit εp,max 0 5 EL 1 ET 2EL sR E# y 0 (9) (B) Stage II—Simultaneous 0° and 90° Ply Cracking: In stage II, multiple matrix cracking occurs simultaneously in the 0° and 90° plies. We assume that L0 5 L90 in stage II. The compliance of the 0° ply E0 can be written as38 1 E0 5 1 EL S1 1 D1 R L0 D (10) where D1 5 EL Em ~1 2 Vfa1! 3 ~b2 1 b3! Vf 2 ~s 2 si ! t (11) where a and b are nondimensional constants defined in Ref. 35. Vf is the volume fraction of fibers, Em the Young modulus of the matrix, and t a constant interfacial shear stress. The composite compliance is given by22 1 Ey 5 1 Ey 0 S 1 1 C1 h L0 1 2D1 Ey 0 EL R L0 D (12) Assuming that the strains in the 90° and 0° plies are equal, the effective stress (s0) acting on the 0° ply is approximated by22 s0S 1 1 D1 R L0 D 5 EL Ey 0 S1 1 C1 h L0 1 2D1 Ey 0 EL R L0 D (13) (C) Stage III—0° Ply Cracking and Delamination: In stage III, the mechanical behavior is fully controlled by the 0° ply, and the analysis is similar to unidirectional laminates. The composite compliance becomes 1 Ey 5 1 Ey 0 S 1 1 D1 Ey 0 EL R L0 D (14) and the stress on the 0° ply is given by s0 5 2s. The analysis of the nonlinear monotonic tensile behavior of the 0° ply is based on the stress–displacement analysis by Hutchinson and Jensen35 of a fiber being pulled out of a brittle matrix against friction. The analysis can be rewritten in terms of stress and strain, including finite-element solutions for small debond lengths l compared with the fiber radius R (l/R , 1).38 The overall monotonic strain ε for a multiple-cracked, unidirectional laminate consists of three strain contributions that can be written as38 ε 5 εe 1 εT 1 εs 5 s0 E0 1 S 1 E0 2 1 EL DsT 1 2H@2~s0 2 si !sT 1 s0 2 2 si 2 # (15) where the strain subscipts e, T, and s refer to the elastic, thermal, and sliding contributions, respectively, and H 5 Rb2~1 2 Vfa1! 2 4EmVf 2 tL0 (16) In Eq. (15), the stresses are given by si 5 sD 2 sT (17) May 2001 Mechanical Properties of Two Plain-Woven CVI SiC-Matrix Composites 1047