Chapter 7:Multiscale Modeling of Composites 275 1=11-3 (7.2) kr1+△(km44m 1-(7-5vm)r 4m=m1+2(4-5ym)r (7.3) de=am (7.4) where 1 1 Λ= 1 p (7.5) 3 (4m-)Pg (7.6) 台2(4-5vm).+(7-5ym)4m (7.7) i=l Application to a two-phase system Consider a cluster of n spherical particles,having the same properties and the same radius a,embedded in an infinite matrix of different properties. The cluster is just enclosed by a sphere of radius b and the particle distribution is sufficiently homogeneous for it to lead to isotropic properties for the composite formed by the cluster and the matrix lying within this sphere.If the particulate volume fraction of the composite is denoted by Ip,then =h 6°=1-， (7.8)eff m m 1 1 13 , k k 1 4µ ⎛ ⎞ Λ = − ⎜ ⎟ + Λ ⎝ ⎠ (7.2) m eff m m 1 (7 5 ) , 1 2(4 5 ) ν µ µ ν − − Γ = + − Γ (7.3) eff m eff m 1 3 , k 4 α α µ ⎛ ⎞ = +Ω + ⎜ ⎟ ⎝ ⎠ (7.4) where m p p 1 p m 1 1 , 1 3 4 N i i i i k k V k µ = − Λ = + ∑ (7.5) m pp 1 mp mm ( ) , 2(4 5 ) (7 5 ) N i i i i µ µ V = ν µ νµ − Γ = − +− ∑ (7.6) p m p 1 p m 1 3 4 N i i i i k α α µ = − Ω = + ∑ Application to a two-phase system Consider a cluster of n spherical particles, having the same properties and the same radius a, embedded in an infinite matrix of different properties. The cluster is just enclosed by a sphere of radius b and the particle distribution is sufficiently homogeneous for it to lead to isotropic properties for the composite formed by the cluster and the matrix lying within this sphere. If the particulate volume fraction of the composite is denoted by Vp, then 3 p 3 m 1 , na V V b = =− (7.8) Chapter 7: Multiscale Modeling of Composites V . (7.7) 275