276 L.N.McCartney where Vm is the volume fraction of the matrix.The suffices p and m will be used to refer properties k,u,and a to the particles and matrix,respectively. It can be shown from (7.2)to (7.7)that the effective bulk modulus, shear modulus,and thermal expansion coefficient are given by 4 3亚+3 p 1 kgkm km (7.9) ke V p+ +3 k。44m 15(l-Vnm4。-4m)P, 2(4-5ym)Wn4。+,m)+(7-5aunJ (7.10) 13 de =an+he 4Mm V (d-ap). 1,3 (7.11) k。44m It has been shown [5]that it is possible to express these relations as the sum of mixtures estimates plus correction terms so that 1 m (7.12) kmk。kn V ++3 kmk。44m (4-4m)P Le =Vplp +Vmklm- 9ka+8m (7.13) Vom +Valp+6(km +24m) (7.14) ++3 kmk。44
where Vm is the volume fraction of the matrix. The suffices p and m will be used to refer properties k, µ, and α to the particles and matrix, respectively. It can be shown from (7.2) to (7.7) that the effective bulk modulus, shear modulus, and thermal expansion coefficient are given by m m p pm m p eff p m m mp m 4 3 3 1 , 3 4 4 V V kk k k k V V k k µ µ µ + + = ⎛ ⎞ ⎜ ⎟ + + ⎝ ⎠ (7.9) m p mp eff m m mp pm m m 15(1 )( ) 1 , 2(4 5 )( ) (7 5 ) V V V νµµ µ µ ν µ µ νµ ⎡ ⎤ − − = + ⎢ ⎥ − + +− ⎣ ⎦ eff m eff m p p m p m 1 3 4 ( ). 1 3 4 k V k µ α α α α µ + = + − + (7.11) 2 p m p m p m eff p m p m mp m 1 1 1 , 3 4 V V V V k k k kk V V k k µ ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ =+ − + + (7.12) 2 p m eff p p m m p m m m pm mp m m m ( ) , 9 8 6( 2 ) V V V V k V V k µ µ µ µµ µ µµ µ µ − =+ − + + + + (7.13) p m pm p m eff p p m m p m mp m 1 1 ( ) . 3 4 V V k k V V V V k k α α α αα µ ⎛ ⎞ ⎜ ⎟ − − ⎝ ⎠ =+ − + + (7.14) L.N. McCartney (7.10) It has been shown [5] that it is possible to express these relations as the sum of mixtures estimates plus correction terms so that 276
Chapter 7:Multiscale Modeling of Composites 277 The mixtures estimates are given by the first two terms on the right-hand side of (7.12)(7.14),while the third term is the correction term that must be applied to the mixtures rule.The results(7.12)and (7.14)are identical to those derived by applying the spherical shell model of the particulate composite to a representative volume element comprising just one particle and a matrix region that is consistent with the volume fraction of the composite.The values of the results (7.12)(7.14)are identical to one of the bounds obtained when using variational methods [1,2,10].These results suggest that the assumption by Maxwell [3]of low volume fractions is not necessary;an issue that will be discussed in [5]. The formulae(7.2)-(7.4)and (7.12)(7.14)completely characterize the properties of an isotropic particulate composite and are expressed in the form of a mixtures estimate and a correction term.These formulae can be used to estimate the effective properties of a microreinforced matrix where the reinforcing phase is particulate in nature. 7.3 Application of Maxwell's Method to Fiber Composites Because of the use of the far field in Maxwell's methodology for estimating the properties of composites,it is now possible to consider multiple fiber,rather than particulate reinforcements.Suppose in a cluster of fibers that there are N different types such that for i=1,...,N there are n fibers of radius a.The properties of the fibers of type i are denoted by a superscript i.Poisson's ratios are to be denoted by v and axial and transverse properties will be denoted by suffices A and T,respectively. The cluster is assumed to be homogeneous regarding the distribution of fibers and leads to transverse isotropic effective properties.The suffix or superscript m will be used to denote matrix properties. The cluster of all types of fiber is now considered to be enclosed in a cylinder of radius b such that the volume fraction of fibers of type i within the cylinder of radius b is given by V=n,a2/b2.The volume fractions must satisfy the relation (7.15)
The mixtures estimates are given by the first two terms on the right-hand side of (7.12)–(7.14), while the third term is the correction term that must be applied to the mixtures rule. The results (7.12) and (7.14) are identical to those derived by applying the spherical shell model of the particulate composite to a representative volume element comprising just one particle composite. The values of the results (7.12)–(7.14) are identical to one of the bounds obtained when using variational methods [1, 2, 10]. These The formulae (7.2)–(7.4) and (7.12)–(7.14) completely characterize the properties of an isotropic particulate composite and are expressed in the form of a mixtures estimate and a correction term. These formulae can be used to estimate the effective properties of a microreinforced matrix where the reinforcing phase is particulate in nature. 7.3 Application of Maxwell’s Method to Fiber Composites Because of the use of the far field in Maxwell’s methodology for estimating the properties of composites, it is now possible to consider multiple fiber, rather than particulate reinforcements. Suppose in a cluster i fibers of radius ai. The properties of the fibers of type i are denoted by a superscript i. Poisson’s ratios are to be denoted by ν, and axial and transverse properties will be denoted by suffices A and T, respectively. The cluster is assumed to be homogeneous regarding the distribution of The cluster of all types of fiber is now considered to be enclosed in a cylinder of radius b such that the volume fraction of fibers of type i within the cylinder of radius b is given by 2 2 f / i V na b = i i . The volume fractions must satisfy the relation m f 1 1. N i i V V= +∑ = (7.15) Chapter 7: Multiscale Modeling of Composites results suggest that the assumption by Maxwell [3] of low volume fractions and a matrix region that is consistent with the volume fraction of the is not necessary; an issue that will be discussed in [5]. fibers and leads to transverse isotropic effective properties. The suffix or of fibers that there are N different types such that for i = 1,…,N there are n superscript m will be used to denote matrix properties. 277
278 L.N.McCartney 7.3.1 Properties Derived from the Lame Solution By making use of Maxwell's methodology in conjunction with the Lame solution for two bonded concentric cylinders,several properties of a fiber- reinforced composite can be estimated.It has been shown [8]that the following effective properties for the multiphase fiber-reinforced com- posite Transverse bulk modulus: Axial Poisson's ratio: Axial thermalexpansion coefficient: Transverse thermal expansion coefficient:a may be estimated using the formulae 京》 (7.16) 吹吹-好 (7.17) a+听a欧)-a+a+(+) (7.18) where 11 1 1 11 (7.19) 1 i=l - 1 (7.20) 1 11
solution for two bonded concentric cylinders, several properties of a fiberreinforced composite can be estimated. It has been shown [8] that the T A A T may be estimated using the formulae 1 eff m m T 1T T 1 11 , k k 1 µ ⎛ ⎞ Λ = + ⎜ ⎟ − Λ ⎝ ⎠ (7.16) eff m A A2 m eff T T 1 1 , k ν ν µ ⎛ ⎞ = −Λ + ⎜ ⎟ ⎝ ⎠ (7.17) eff eff eff m m m T AA T A A 3 m eff T T 1 1 ( ) ( ) , k α να α ν α µ ⎛ ⎞ + = + + Λ + ⎜ ⎟ ⎝ ⎠ (7.18) where m eff m TT T T 1 f 1 m meff TT TT 11 1 1 , 11 1 1 N i i i i kk k k V µ µ k k = − − Λ = = + + ∑ (7.19) m meff AA AA 2 f 1 m meff TT TT , 11 1 1 N i i i i V k k νν νν µ µ = − − Λ = = + + ∑ (7.20) L.N. McCartney posite Transverse bulk modulus: k Axial thermalexpansion coefficient:α Transverse thermal expansion coefficient:α Axial Poisson’s ratio:ν eff eff eff eff following effective properties for the multiphase fiber-reinforced comBy making use of Maxwell’s methodology in conjunction with the Lamé 7.3.1 Properties Derived from the Lamé Solution 278
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 that
m 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
280 L.N.McCartney 1 ++ (7.25) m w- 711 vAn =VivA +VmVa- (7.26) ai+va=V(af+via)+V(a+vam) 11 @+a-安+ m十 (7.27) These results correspond to one of the bounds derived using variational methods [1,2,10]which are identical to those obtained using the concentric cylinder model for a unidirectional,fiber-reinforced composite 7.3.2 Axial Shear It has been shown [8]that the effective transverse shear modulus for the multiphase fiber-reinforced composite is given by =必会 (7.28) where A=之公-4-- (7.29) 台+吸+吸
2 f m f m T T eff f m f m T TT f m mf m TTT 1 1 1 , 1 V V k k V V k kk V V k k µ ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ =+ − + + (7.25) f m A A f m eff f m T T A f A mA f m f m mf m TTT 1 1 ( ) , 1 k k V V V V V V k k ν ν ν ν ν µ ⎛ ⎞ − − ⎜ ⎟ ⎝ ⎠ =+ − + + (7.26) eff eff eff f f f m m m T A A f T AA m T A A f m T T f f f m mm T AA T A A fm f m mf m TTT ( )( ) 1 1 [( ) ( )] . 1 V V k k V V V V k k α ν α α να α να α να α να µ + =++ + − − + − + + + (7.27) These results correspond to one of the bounds derived using variational methods [1, 2, 10] which are identical to those obtained using the concentric cylinder model for a unidirectional, fiber-reinforced composite. 7.3.2 Axial Shear It has been shown [8] that the effective transverse shear modulus for the multiphase fiber-reinforced composite is given by eff m A A 1 , 1 µ µ − Λ = + Λ (7.28) where m meff AA AA f m eff m 1 AA A A . N i i i i V µµ µµ = µ µµµ − − Λ ≡ = + + ∑ (7.29) 280 L.N. McCartney