September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under load 2407 micromechanical equilibrium of the composite using a shear- lag analysis gives access to interfacial sliding stress. The fol (e)+() (10) lowing assumptions are made: (i) isostrain conditions prevail in the cross-sectional area of the sample at the end of the sliding length, (ii) a constant interfacial friction stress exists within the sliding length, and (iii)the stress in the matrix is completely transferred to the fiber in the crack plane f the poisson ratio of the fiber and the matrix are assumed to be the same, the far-field stress of the composite(oo)is In the same manner, the stress in the matrix om() must de- related to the far-field fiber stress(o and the far-field matrix crease from the far-field stress o at the end of the cohesive stress(om)by the relation length to zero in the crack plane in : 12. Thus, σn+V,σ (12) where Ve and Vm represent the volume fractions of the fibers and the matrix, respectively Therefore, the strain variation in the matrix, 8(), is also The isostrain condition is expressed this way G Ooo of om (13) where the indices c, f, and m refer to the composite, fiber, and is given br acement of the free crack surface of the matrix, 8ms The elementary cell that was used in the model is depicted in [((-a (14) size on each side of the crack is given by in2. This distance is TH itself divided into two parts: the debonding length upon which the sliding occurs(a), and the length upon which the interface remains cohesive(/). R is the fiber radius () (15) Assuming that a constant sliding friction stress T opposes the between the fiber and matrix displacements in the crack plane 2+8 The fiber strain e=), the derivative of the fiber displacement along the tensile axis(8 )) increases the same way: By using Eq(7). d64(-) Using Eqs. (5)and(17), the measured inelastic strain(e inelastic) In the cohesive zone, only elasticity occurs. Within the can be related to the sliding friction stress lebon, the fiber displacement in the z-direction(see Fig. 2) vill be maximum in the crack plane; that is to say, 2= 2 Therefore we have (18) Is to say, aE8-21/+RE (19) Because east= oo/Ee, we finally obtain the relation Debonding crack inelastic elastic 22 2u- Crack opening Transverse he interfacial sliding stress is directly dependent upon Matrix Crack both the elastic and inelastic strains as well as the transverse crack density and the area upon which the sliding occurs, hrough a, Ie, and I IIL E accurately study the damage evolution in composites, experimental device that couples an ultrasonic immersion tanl Fig. 2. Schematic depiction of an elementary cell containing an open to a tensile machine and an extensometer has been developed. 29 This apparatus allows the study of the loss of stiffness of themicromechanical equilibrium of the composite using a shear￾lag analysis gives access to interfacial sliding stress. The fol￾lowing assumptions are made: (i) isostrain conditions prevail in the cross-sectional area of the sample at the end of the sliding length, (ii) a constant interfacial friction stress exists within the sliding length, and (iii) the stress in the matrix is completely transferred to the fiber in the crack plane. If the Poisson ratio of the fiber and the matrix are assumed to be the same, the far-field stress of the composite (s`) is related to the far-field fiber stress (sf ) and the far-field matrix stress (sm) by the relation s` = Vmsm + Vfsf (6) where Vf and Vm represent the volume fractions of the fibers and the matrix, respectively. The isostrain condition is expressed this way: s` Ec = sf Ef = sm Em (7) where the indices c, f, and m refer to the composite, fiber, and matrix, respectively, for the Young’s moduli (E) and the stresses (s). The elementary cell that was used in the model is depicted in Fig. 2. Because the intercrack distance is given as l, the cell size on each side of the crack is given by l/2. This distance is itself divided into two parts: the debonding length upon which the sliding occurs (ld), and the length upon which the interface remains cohesive (lc). R is the fiber radius. Assuming that a constant sliding friction stress t opposes the relative sliding motion of the fiber and the matrix and that the fiber stress (sf (z)) increases linearly from the far-field fiber stress sf at the end of the cohesive length, we can write sf~z! = sf + S 2t R Dz (8) The fiber strain «f (z), the derivative of the fiber displacement along the tensile axis (df (z)), increases the same way: «f~z! = ddf~z! dz = sf Ef + S 2t REf Dz (9) In the cohesive zone, only elasticity occurs. Within the debond, the fiber displacement in the z-direction (see Fig. 2) will be maximum in the crack plane; that is to say, z 4 l/2. Therefore, we have df = FS sf Ef DzGU 0 lc + FS sf Ef Dz + S t REf Dz 2 GU l c l/2 (10) Thus, df = S sf Ef D l 2 + t REf FS l 2D 2 − lc 2 G (11) In the same manner, the stress in the matrix sm(z) must de￾crease from the far-field stress sm at the end of the cohesive length to zero in the crack plane in z 4 l/2. Thus, sm~z! = smS1 − z l D (12) Therefore, the strain variation in the matrix, «m(z), is also linear: «m~z! = sm Em S1 − z l D (13) and the displacement of the free crack surface of the matrix, dm, is given by dm = FS sm Em DzGU 0 lc + F sm Em Sz − z 2 2l DGU l c l/2 (14) Thus, dm = sm Em S lc 2 2l + 3l 8 D (15) Because of the symmetry of the representative cell, one-half the crack-opening displacement, U, is simply the difference between the fiber and matrix displacements in the crack plane: U = df − dm = sf Ef S l 2D + t REf FS l 2D 2 − lc 2 G − sm Em S lc 2 2l + 3l 8 D (16) By using Eq. (7), U = s` Ec S l 8 − lc 2 2l D + t REf FS l 2D 2 − lc 2 G (17) Using Eqs. (5) and (17), the measured inelastic strain («inelastic) can be related to the sliding friction stress: «inelastic = bTS U a D (18) This is to say, «inelastic = bT a H s` Ec S l 8 − lc 2 2l D + t REf FS l 2D 2 − lc 2 GJ (19) Because «elastic 4 s`/Ec, we finally obtain the relation t = REf FS l 2D 2 − lc 2 G FS a bT D«inelastic − S l 8 − lc 2 2l D«elasticG (20) Finally, the interfacial sliding stress is directly dependent upon both the elastic and inelastic strains as well as the transverse crack density and the area upon which the sliding occurs, through a, lc, and l. III. Experimental Results To accurately study the damage evolution in composites, an experimental device that couples an ultrasonic immersion tank to a tensile machine and an extensometer has been developed.29 This apparatus allows the study of the loss of stiffness of the Fig. 2. Schematic depiction of an elementary cell containing an open crack. September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under Load 2407