S. Baste / Composites Science and Technology 61(2001)2285-2297 cm=∑c4 and Sem=∑SB (14) depends on the both the geometry of the crack and the elastic properties of material surrounded the crack here. the initial non cracked material The effective medium is considered as a two-phase medium [21]. Let phase I be the uncracked fibre rein- forced composite material. Phase 2 is a set of ellipsoids 4. Several crack arrays consisting of voids. The overall stifness tensor for the two-phase medium follows from Eq (14) 4..D Sic-Sic Cet=C-Cr C(C-Cr)Ar, Sem =S+crS- Sr)B The 3D representation of the cracks was applied to multiple arrays microcracking in a unidirectional Sic. (15) Micrographs(Fig 8)and experimental stiffness tensor where f index is for ellipsoidal voids. The crack locali- changes [23](Fig. 12)allow us to identified three arrays sation tensors are of microcracks: a transverse microcracks. which is topped and deviated in longitudinal cracks at the fibre Ar=(l-PC) and Br=-o (16) matrix interface. Those longitudinal arrays are growing cracks along the interface(Fig 9) and the effective elasticity tensor is given by A succ cessive process of prediction experimental data confrontation allows the optimal determination [24] of Ceff=C-cr C(I-PC) and Sefr=S+cr2-(17) the fixed sizes of the multiplication of the transverse cracks and of evolution laws of the cracks density, of with ce the volume concentration of the cracks the cracks thickness aspect ratio(Fig. 10) and of the semi-axes of the growing longitudinal cracks(Fig. 11 Fig. 12 plots the changes of the nine stiffnesses iden- 32x12x2x3 tified from the phase velocities as a function of tensile stress for the ID SiC-SiC. There is a good agreement where 2x are the average distances between two cracks in the i direction. They give the unit cell of the cracked material(Fig. 7) P and Q tensors appearing in Eq(9)depend upon the shape of the considered inclusion and the stiffness of the effective medium C. That leads to the self-consistent scheme. Here, we replace C with Co, the stiffness tensor of the uncracked material, in Eqs. (9). (16) and (17) This method, similar to the mori-Tanaka method [22] requires less calculations and gives a good approxima ion of the effective stiffness tensor for reasonable ig. 8. Micrograph of the transverse cracks in the ID Sic-Sic volume concentration of cracks [16] Eq.(17)is the equation used to evaluate the effe stiffness tensor for an anisotropic medium permeated by ellipsoidal cracks. It requires the determination of the tensor Q and P by a numerical evaluation of Eq. (10) [16]. P, the symmetrized derivative of the Green's tensor, fibre transverse crack Fig. 7. Unit cell of a cracked body Fig 9. Unit cell of the cracked ID SiC-SiC.Ceff ¼ X r crCrAr and Seff ¼ X r crSrBr: ð14Þ The effective medium is considered as a two-phase medium [21]. Let phase 1 be the uncracked fibre rein￾forced composite material. Phase 2 is a set of ellipsoids consisting of voids. The overall stiffness tensor for the two-phase medium follows from Eq. (14): Ceff ¼ C cfC Cð Þ Cf Af ; Seff ¼ S þ cf ð Þ S Sf Bf ð15Þ where f index is for ellipsoidal voids. The crack locali￾sation tensors are Af ¼ ð Þ I PC 1 and Bf ¼ Q1 ; ð16Þ and the effective elasticity tensor is given by Ceff ¼ C cfCðI PCÞ 1 and Seff ¼ S þ cfQ1 ð17Þ with cf the volume concentration of the cracks: cf ¼ 4 3 abc 2x12x22x3 ð18Þ where 2xi are the average distances between two cracks in the i direction. They give the unit cell of the cracked material (Fig. 7). P and Q tensors appearing in Eq. (9) depend upon the shape of the considered inclusion and the stiffness of the effective medium C. That leads to the self-consistent scheme. Here, we replace C with C0, the stiffness tensor of the uncracked material, in Eqs. (9), (16) and (17). This method, similar to the Mori–Tanaka method [22], requires less calculations and gives a good approxima￾tion of the effective stiffness tensor for reasonable volume concentration of cracks [16]. Eq. (17) is the equation used to evaluate the effective stiffness tensor for an anisotropic medium permeated by ellipsoidal cracks. It requires the determination of the tensor Q and P by a numerical evaluation of Eq. (10) [16]. P, the symmetrized derivative of the Green’s tensor, depends on the both the geometry of the crack and the elastic properties of material surrounded the crack: here, the initial non cracked material. 4. Several crack arrays 4.1. 1D SiC–SiC The 3D representation of the cracks was applied to a multiple arrays microcracking in a unidirectional SiC– SiC. Micrographs (Fig. 8) and experimental stiffness tensor changes [23] (Fig. 12) allow us to identified three arrays of microcracks: a transverse microcracks, which is stopped and deviated in longitudinal cracks at the fibre matrix interface. Those longitudinal arrays are growing cracks along the interface (Fig. 9). A successive process of prediction experimental data confrontation allows the optimal determination [24] of the fixed sizes of the multiplication of the transverse cracks and of evolution laws of the cracks density, of the cracks thickness aspect ratio (Fig. 10) and of the semi-axes of the growing longitudinal cracks (Fig. 11). Fig. 12 plots the changes of the nine stiffnesses iden￾tified from the phase velocities as a function of tensile stress for the 1D SiC–SiC. There is a good agreement Fig. 7. Unit cell of a cracked body. Fig. 9. Unit cell of the cracked 1D SiC–SiC. Fig. 8. Micrograph of the transverse cracks in the 1D SiC–SiC. S. 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