H Ismar, F Streicher/ Computational Materials Science 16(1999)17-24 o=/4o)1 where f is the surface area of the interface element (7) and f is a reference area. Examinations on in- terfacial properties can be found in the literature Fiber cracking usually begins at defects or flaws at [14-17]. The strength values referring to an inter the fiber surface and leads to brittle fiber damage face area F=5x 10-12 m2 are o.=300. Thereby the crack surface can be assumed to be douc 2400 and oos=300 MPa. They are results vertical to the fiber axis. To model this behavior of our parameter studies on interfacial behavior of the fiber is subdivided vertical to the fiber axis in unidirectional composites with a fiber volume pieces of the length L. If the stress in a fiber piece fraction uf= 45%. The Weibull parameter m and eaches the failure surface a crack is introduced the friction coefficient u are 8 and 0.6, respectively vertical to the fiber axis. The fiber strength values used [6, 10-12] for a regarded length of 5 x 10-m are oout= 3054, 0ouc=6980 and oos =2704 MPa3Substructure n The actual statistical fiber distribution in uni- 2.3. Fiber-matrix interface directionally reinforced ceramics is well described by a regular hexagonal fiber disposition. For this reason an elementary cell with a hexagonal fiber The fiber-matrix interfacial zone plays a key arrangement is used. The numerical simulations role in the mechanical behavior of composite ma- are carried out on substructures of different fiber terials. Its properties can be influenced by different volume fractions tr. Thereby the length of the fiber coatings such as C. bn or SiC. The most substructure . =100 10-6 m and the fiber di important task of the interface is the crack de ameter r=15x 10-6 m are the same for all fiber flection at the fiber surface after initiation of ma trix cracking. Thereby debonding length, matrix volume fractions. a different fiber content of the crack saturation and frictional sliding in the fiber substructure is realized by changing the edge lengths I, and Iy. In the case of uf= 45%, for ex- matrix interface after debonding are controlled by ample, I, and L, are 18.45 x 10-6 and 10.65x10-6 [13] to occur under the combined action of normal ress o and interfacial shear stress t In this case The calculations were performed with different q(2)can be rewritten as statistical strength distributions. At the beginning of the simulations each finite element randomly a1Gn+a1a2+2(a1-a2)2=1 obtains its strength values according to the Wei- bull distribution and the element specific fracture In the finite element mesh the interface is urface is defined. The values are distributed to the modelled as a region of negligible thickness, which elements in such a way that only the size of the forces a bonding of fiber and matrix. If the d fracture surface is varied according to statistics, bonding criterion( 8) is fulfilled for an interface but not the shape itself. element, debonding is introduced by reducing all as a result of the mismatch in coefficients of elastic constants of this element to zero. After thermal expansion between the constituer nts resid- debonding the interfacial stress caused by friction ual stresses are developed by the cooling down between the fiber and the matrix is calculated b ess after fabrication of the o take Ising Coulomb's law. Analogous to fiber and these stresses into account the substructure is matrix the strength of the interface is considered to cooled down from the stress free temperature 1025 be Weibull distributed. The constant oo is defined to 25 K [7. The side and front surfaces of the el- ementary cell are kept plane during thermal and mechanical loading. As a result of the cooling =(F (9) process there are compressive stresses in the 2-di- rection in the fibers while the matrix is loaded byr 0 ˆ L0 L  1=m r0: …7† Fiber cracking usually begins at defects or ¯aws at the ®ber surface and leads to brittle ®ber damage. Thereby the crack surface can be assumed to be vertical to the ®ber axis. To model this behavior the ®ber is subdivided vertical to the ®ber axis in pieces of the length L. If the stress in a ®ber piece reaches the failure surface a crack is introduced vertical to the ®ber axis. The ®ber strength values used [6,10±12] for a regarded length of 5 10ÿ6 m are r 0;ut ˆ 3054, r 0;uc ˆ 6980 and r 0;s ˆ 2704 MPa and mf ˆ 8. 2.3. Fiber±matrix interface The ®ber±matrix interfacial zone plays a key role in the mechanical behavior of composite ma￾terials. Its properties can be in¯uenced by di€erent ®ber coatings such as C, BN or SiC. The most important task of the interface is the crack de- ¯ection at the ®ber surface after initiation of ma￾trix cracking. Thereby debonding length, matrix crack saturation and frictional sliding in the ®ber± matrix interface after debonding are controlled by the interfacial properties. Debonding is postulated [13] to occur under the combined action of normal stress rn and interfacial shear stress s. In this case Eq. (2) can be rewritten as a1rn ‡ a11r2 n ‡ 2…a11 ÿ a12†s2 ˆ 1: …8† In the ®nite element mesh the interface is modelled as a region of negligible thickness, which forces a bonding of ®ber and matrix. If the de￾bonding criterion (8) is ful®lled for an interface element, debonding is introduced by reducing all elastic constants of this element to zero. After debonding the interfacial stress caused by friction between the ®ber and the matrix is calculated by using Coulomb's law. Analogous to ®ber and matrix the strength of the interface is considered to be Weibull distributed. The constant r 0 is de®ned as r 0 ˆ F0 F  1=m r0; …9† where F is the surface area of the interface element and F0 is a reference area. Examinations on in￾terfacial properties can be found in the literature [14±17]. The strength values referring to an inter￾face area F ˆ 5 10ÿ12 m2 are r 0;ut ˆ 300, r 0;uc ˆ 2400 and r 0;s ˆ 300 MPa. They are results of our parameter studies on interfacial behavior of unidirectional composites with a ®ber volume fraction vf ˆ 45%. The Weibull parameter mi and the friction coecient l are 8 and 0.6, respectively. 3. Substructure The actual statistical ®ber distribution in uni￾directionally reinforced ceramics is well described by a regular hexagonal ®ber disposition. For this reason an elementary cell with a hexagonal ®ber arrangement is used. The numerical simulations are carried out on substructures of di€erent ®ber volume fractions vf. Thereby the length of the substructure lz ˆ 100 10ÿ6 m and the ®ber di￾ameter rf ˆ 15 10ÿ6 m are the same for all ®ber volume fractions. A di€erent ®ber content of the substructure is realized by changing the edge lengths lx and ly . In the case of vf ˆ 45%, for ex￾ample, lx and ly are 18:45 10ÿ6 and 10:65 10ÿ6 m, respectively. The calculations were performed with di€erent statistical strength distributions. At the beginning of the simulations each ®nite element randomly obtains its strength values according to the Wei￾bull distribution and the element speci®c fracture surface is de®ned. The values are distributed to the elements in such a way that only the size of the fracture surface is varied according to statistics, but not the shape itself. As a result of the mismatch in coecients of thermal expansion between the constituents resid￾ual stresses are developed by the cooling down process after fabrication of the composite. To take these stresses into account the substructure is cooled down from the stress free temperature 1025 to 25 K [7]. The side and front surfaces of the el￾ementary cell are kept plane during thermal and mechanical loading. As a result of the cooling process there are compressive stresses in the z-di￾rection in the ®bers while the matrix is loaded by 20 H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24