Chapter 7:Multiscale Modeling of Composites 279 A,-v:@itvak)-(aitvax) 11 i=1 (7.21) _(a+v"aj")-(ap+vaR) 11 Application to a two-phase system It follows from(7.16)to (7.21)that 1 L=21-)_4Y_停4k号陧 (7.22) E E 十 安 (7.23) a+VAOAT-aT+VRaR 1.1 where Vr=1-Vm=na2/b2 is the volume fraction of n fibers of radius a embedded in matrix within a cylinder of radius b. The relations (7.22)(7.24)are now written as the sum of a mixtures term plus a correction term so thatm mm T AA T A A 3 f 1 m T T eff eff eff m m m T A A T AA m eff T T ( )( ) 1 1 ( )( ). 1 1 N i ii i i i V k k α να α να µ α ν α α να µ = + −+ Λ = + + −+ = + ∑ (7.21) Application to a two-phase system It follows from (7.16) to (7.21) that m f eff eff 2 m m f m f m T A T T TT T T eff eff eff TT A m f mf m T TT 1 1 2(1 ) 4( ) , 1 V V k kk k kE E V V k k ν νµ µ µ + + − ≡ −= + + (7.22) f m eff m A A A Af m eff T T m f T T 1 1 , 1 1 V k k ν ν ν ν µ µ − ⎛ ⎞ = + ⎜ ⎟ + + ⎝ ⎠ (7.23) eff eff eff m m m T A A T AA f f f m mm T AA T A A f m eff T T m f T T ( )( ) 1 1 , 1 1 V k k α ν α α να α να α να µ µ + =+ + −+ ⎛ ⎞ + + ⎜ ⎟ + ⎝ ⎠ (7.24) where 2 2 f m V V na b =− = 1 / is the volume fraction of n fibers of radius a embedded in matrix within a cylinder of radius b. The relations (7.22)–(7.24) are now written as the sum of a mixtures term plus a correction term so that Chapter 7: Multiscale Modeling of Composites 279