Chapter 7:Multiscale Modeling of Composites 281 Application to a two-phase system When just one type of fiber is present,it follows from(7.28)and(7.29)that 吹公瓷 (7.30) The result(7.30)is now written in the form of the mixtures estimate plus a correction term as follows: h=4+G-匹-E (7.31) 以+Vn+吸 This result corresponds to one of the bounds derived using variational methods [1,2,10]and is identical to the result obtained when using the concentric cylinder model for a unidirectional,fiber-reinforced composite. 7.3.3 Transverse Shear It has been shown [8]that the effective transverse shear modulus for the multiphase composite is given by 24在引 (7.32) where 11 1 m (7.33) 121 Application to a two-phase system When just one type of fiber is present,it follows from(7.32)and (7.33)that 2+'k+1+')4k" A=听+2+0+0+2T (7.34) The result (7.34)is now written as a mixtures estimate plus a correction term so thatApplication to a two-phase system When just one type of fiber is present, it follows from (7.28) and (7.29) that f m eff m f A mA A A f m mA f A (1 ) . (1 ) V V V V µ µ µ µ µ µ + + = + + (7.30) The result (7.30) is now written in the form of the mixtures estimate plus a correction term as follows: f m2 eff f m A A fm A f A mA m fm f A mA A ( ) . V V V V V V µ µ µ µµ µ µ µ − =+ − + + (7.31) This result corresponds to one of the bounds derived using variational methods [1, 2, 10] and is identical to the result obtained when using the concentric cylinder model for a unidirectional, fiber-reinforced composite. 7.3.3 Transverse Shear It has been shown [8] that the effective transverse shear modulus for the multiphase composite is given by eff m m m T TT T 1 11 12 , µ µµ 1 k ⎡ ⎛ ⎞⎤ = −Λ + ⎢ ⎜ ⎟⎥ + Λ ⎣ ⎝ ⎠⎦ (7.32) where m m eff TT TT f 1 m m m m eff TT T TT T 11 1 1 . 121 12 1 N i i i i V k k µµ µµ µ µµ µ = − − Λ ≡ = ++ ++ ∑ (7.33) Application to a two-phase system When just one type of fiber is present, it follows from (7.32) and (7.33) that f m mm f m eff m T T mT T f TT T T m m f m m m2 m T T T f TT f T 2 (1 ) . ( 2 ) (1 ) 2 ( ) Vk V k V k Vk V µµ µ µ µ µ µµ µ µ + ++ = + ++ + (7.34) The result (7.34) is now written as a mixtures estimate plus a correction term so that Chapter 7: Multiscale Modeling of Composites 281