Chapter 7:Multiscale Modeling of Composites Using Analytical Methods L.N.McCartney NPL Materials Centre National Physical Laboratory Middlesex TW11 OLW.UK 7.1 Introduction Both fiber and particulate composite materials provide applications in materials science where the multiscale microstructure leads to the need for multiscale modeling.The length scales encountered range from the fiber and particle sizes whose dimensions are measured in microns,to the individual plies in laminates whose thicknesses are measured in fractions of millimeters,to the laminates themselves whose thicknesses in the laboratory are measured in millimeters,e.g.,40-50 mm.The laminates then form parts of composite structures whose sizes are measured in meters,although modeling at this scale will not form part of this chapter. While conventional composites are based on essentially homogeneous matrices,which can be polymeric,metallic,or ceramic,advanced com- posites are also being considered to have matrices,which are themselves composites reinforced by submicron particles or whiskers,e.g.,carbon nanotubes.Such developments lead to the need to be able to estimate the properties of composite laminates that have multiscale reinforcements, e.g.,fibers in particulate/whisker-reinforced matrices.Also,there is a need to predict the onset of damage in the materials when they are operating in service conditions. This chapter will focus on two aspects of the problem that complement work already published dealing with multiscale analytical modeling for damaged composite laminates [7].The first aspect,considered in Sects. 7.2-7.4,is the development of new understanding with regard to the prediction of the properties of undamaged particulate and unidirectional
Chapter 7: Multiscale Modeling of Composites Using Analytical Methods L.N. McCartney NPL Materials Centre National Physical Laboratory Middlesex TW11 0LW, UK 7.1 Introduction Both fiber and particulate composite materials provide applications in materials science where the multiscale microstructure leads to the need for multiscale modeling. The length scales encountered range from the fiber and particle sizes whose dimensions are measured in microns, to the individual plies in laminates whose thicknesses are measured in fractions of millimeters, to the laminates themselves whose thicknesses in the laboratory are measured in millimeters, e.g., 40–50 mm. The laminates then form parts of composite structures whose sizes are measured in meters, although modeling at this scale will not form part of this chapter. While conventional composites are based on essentially homogeneous matrices, which can be polymeric, metallic, or ceramic, advanced composites are also being considered to have matrices, which are themselves composites reinforced by submicron particles or whiskers, e.g., carbon nanotubes. Such developments lead to the need to be able to estimate the properties of composite laminates that have multiscale reinforcements, e.g., fibers in particulate/whisker-reinforced matrices. Also, there is a need to predict the onset of damage in the materials when they are operating in service conditions. This chapter will focus on two aspects of the problem that complement work already published dealing with multiscale analytical modeling for prediction of the properties of undamaged particulate and unidirectional damaged composite laminates [7]. The first aspect, considered in Sects. 7.2–7.4, is the development of new understanding with regard to the
272 L.N.McCartney fiber-reinforced composites.Recent work [8]has involved the study of a methodology first published in 1873 that was developed by Maxwell [3] when considering the effective electrical conductivity of a conducting medium in which a dilute distribution of particles having a different conductivity was dispersed.It has now been shown [8]that this methodology can be applied much more widely with the result that many other effective properties of both particulate and fiber-reinforced composites can be estimated.A description of the key results of this investigation will first be presented in this chapter together with a discussion of the relationship of the new results to existing formulae that can be used to estimate effective composite properties.The results given will be very useful when estimating undamaged properties for matrices if reinforced by submicron particles,and the undamaged properties of the individual plies of laminates that will become damaged when loaded in service conditions.Recent work to be reported in [8]has reconsidered the bounds on properties that arise from the use of variational methods [1,2, 10];and in Sect.7.4,sets of conditions are given that identify whether the extreme values of properties are upper or lower bounds. The onset of microstructural damage in the form of fiber and interface fracture for unidirectional composites,and of ply cracking and delamination in laminated composites,leads to a deterioration of thermoelastic properties. For structural applications of composites,such as in plates with bolt holes, stress concentrations lead to localized damage and to localized changes in modulus,Poisson's ratios,and thermal expansion coefficients that cause load to be transferred to other parts of the structure.Damage development in structures is,thus,a gradual inhomogeneous process of material dete- rioration that eventually culminates in the catastrophic failure of the structure.The local damage-induced load transfer can lead to composite components out performing their expected performance on the basis of laboratory coupon data.Sections 7.5 and 7.6 of this chapter are concerned with laminated composites for which ply cracking is the only damage mode,although results are expected to be valid more generally.The damage model for laminates will require the properties of undamaged plies,and use can be made of the results presented in Sects.7.2-7.4 that relate to the fiber/matrix length scale rather than the ply/laminate length scales. While a methodology for the prediction of damage formation has been described in [7],it involves the use of various interrelationships between the effective properties of damaged laminates.These relationships were derived from the development of an accurate stress-transfer model [4-6] that estimated the values of the effective thermoelastic properties of the laminate.The objective here is to show how the interrelationships
fiber-reinforced composites. Recent work [8] has involved the study of a methodology first published in 1873 that was developed by Maxwell [3] when considering the effective electrical conductivity of a conducting medium in which a dilute distribution of particles having a different methodology can be applied much more widely with the result that many other effective properties of both particulate and fiber-reinforced composites can be estimated. A description of the key results of this investigation will first be presented in this chapter together with a discussion of the relationship of the new results to existing formulae that can be used to estimate effective composite properties. The results given will be very useful when estimating undamaged properties for matrices if individual plies of laminates that will become damaged when loaded in service conditions. Recent work to be reported in [8] has reconsidered the bounds on properties that arise from the use of variational methods [1, 2, 10]; and in Sect. 7.4, sets of conditions are given that identify whether the extreme values of properties are upper or lower bounds. The onset of microstructural damage in the form of fiber and interface fracture for unidirectional composites, and of ply cracking and delamination in laminated composites, leads to a deterioration of thermoelastic properties. For structural applications of composites, such as in plates with bolt holes, stress concentrations lead to localized damage and to localized changes in modulus, Poisson’s ratios, and thermal expansion coefficients that cause load to be transferred to other parts of the structure. Damage development in structures is, thus, a gradual inhomogeneous process of material deterioration that eventually culminates in the catastrophic failure of the structure. The local damage-induced load transfer can lead to composite components out performing their expected performance on the basis of laboratory coupon data. Sections 7.5 and 7.6 of this chapter are concerned with laminated composites for which ply cracking is the only damage mode, although results are expected to be valid more generally. The damage model for laminates will require the properties of undamaged plies, and use can be made of the results presented in Sects. 7.2–7.4 that relate to the fiber/matrix length scale rather than the ply/laminate length scales. While a methodology for the prediction of damage formation has been described in [7], it involves the use of various interrelationships between the effective properties of damaged laminates. These relationships were derived from the development of an accurate stress-transfer model [4–6] that estimated the values of the effective thermoelastic properties of the laminate. The objective here is to show how the interrelationships L.N. McCartney reinforced by submicron particles, and the undamaged properties of the conductivity was dispersed. It has now been shown [8] that this 272
Chapter 7:Multiscale Modeling of Composites 273 developed can be derived independently of the stress-transfer model.In addition,the analysis is extended to demonstrate that the interrelationships are valid also for laminates having orthogonal sets of ply cracks. Ply cracking damage is usually the first significant form of damage that occurs when a laminate is loaded.The occurrence of ply cracks leads to a degradation of laminate properties;and such property degradation needs to be taken into account when assessing the integrity of composite structures using numerical methods,such as finite element analysis (FEA) The use of FEA in a structural setting demands that the three-dimensional properties of laminates be available to the software.When considering the structural integrity of a component,account needs to be taken of the degradation of all local properties when damage occurs in the form of ply cracking.Such phenomena,leading to the redistribution of stress in the structure that will affect the onset of component failure during loading, need to be modeled realistically.The results given in Sects.7.5 and 7.6 provide most of the necessary property degradation relationships.The prediction of damage formation in laminates is,thus,set on a firm basis; and the results of these sections provide a rigorous framework of general validity that can be used with confidence in design methodologies. 7.2 Application of Maxwell's Method to Particulate Composites In the field of electricity and magnetism,Maxwell [3](as early as 1873) developed a method of estimating the electrical conductivity of an isotropic cluster of spherical particles of the same size embedded in a matrix having different conduction properties.The method was based on the exact solution for an isolated sphere embedded in an infinite matrix subject to a uniform gradient of electrostatic potential.This solution is applied both to the individual particles in the cluster(which were assumed to be noninteracting),and to the effective composite material that can be used to replace the particle cluster without affecting the potential distribution in the matrix at large distances from the cluster. The effective composite medium is taken to have a radius such that the matrix and particles enclosed have the same particle volume fraction as the composite for which properties are required.When observed at large distances from an isolated particle,the electrostatic potential has the form of the sum of the unperturbed potential distribution (that arises when the particle is not present)plus a term that is inversely proportional to the square of the radial distance from the center of the particle.The coefficient
developed can be derived independently of the stress-transfer model. In addition, the analysis is extended to demonstrate that the interrelationships are valid also for laminates having orthogonal sets of ply cracks. Ply cracking damage is usually the first significant form of damage that occurs when a laminate is loaded. The occurrence of ply cracks leads to a degradation of laminate properties; and such property degradation needs to be taken into account when assessing the integrity of composite structures using numerical methods, such as finite element analysis (FEA). The use of FEA in a structural setting demands that the three-dimensional properties of laminates be available to the software. When considering the structural integrity of a component, account needs to be taken of the degradation of all local properties when damage occurs in the form of ply cracking. Such phenomena, leading to the redistribution of stress in the structure that will affect the onset of component failure during loading, need to be modeled realistically. The results given in Sects. 7.5 and 7.6 provide most of the necessary property degradation relationships. The prediction of damage formation in laminates is, thus, set on a firm basis; and the results of these sections provide a rigorous framework of general validity that can be used with confidence in design methodologies. 7.2 Application of Maxwell’s Method to Particulate Composites In the field of electricity and magnetism, Maxwell [3] (as early as 1873) developed a method of estimating the electrical conductivity of an isotropic cluster of spherical particles of the same size embedded in a matrix having different conduction properties. The method was based on the exact solution for an isolated sphere embedded in an infinite matrix subject to a uniform gradient of electrostatic potential. This solution is applied both to the individual particles in the cluster (which were assumed to be noninteracting), and to the effective composite material that can be used to replace the particle cluster without affecting the potential distribution in the matrix at large distances from the cluster. The effective composite medium is taken to have a radius such that the matrix and particles enclosed have the same particle volume fraction as the composite for which properties are required. When observed at large distances from an isolated particle, the electrostatic potential has the form of the sum of the unperturbed potential distribution (that arises when the particle is not present) plus a term that is inversely proportional to the square of the radial distance from the center of the particle. The coefficient Chapter 7: Multiscale Modeling of Composites 273
274 L.N.McCartney of the perturbation term depends on the particle radius and the electrical conductivities of both the particle and the matrix. The isolated particle solution is applied both to the single sphere of effective composite material representing the cluster and to each particle in the cluster.Maxwell states that an assumption is made that the particles do not interact,in which case at large distances from a cluster of particles,the perturbing effect of the particles can be expressed as the sum of the perturbations that each particle would cause if isolated in the matrix at a given point.This assumption implies that results are likely to be valid only for low volume fractions of reinforcement,although evidence is presented suggesting much wider applicability. The purpose of this section is to report results that have recently been derived [8]where Maxwell's methodology has been applied to the estimation of many other properties for both isotropic particulate com- posites and anisotropic fiber-reinforced composites. 7.2.1 Applying Maxwell's Approach to Multiphase Particulate Composites Because of the use of the far field in Maxwell's methodology for estimating the properties of particulate composites,it is possible to consider multiple spherical reinforcements.Suppose,in a cluster of particulate reinforcements embedded in an infinite matrix that there are N different types such that for i=1,...N there are n spherical particles of radius a.The properties of the particles of type i are denoted by a super- script i and subscript p,where k will denote bulk moduli,u will denote shear moduli,and a will denote thermal expansion coefficients.The cluster is assumed to be homogeneous regarding the distribution of particles and leads to isotropic effective properties.A suffix m will be used to denote matrix properties. The cluster of all types of particle is now considered to be enclosed in a sphere of radius b such that the volume fraction of particles of type i within the sphere of radius b is given by V=na/b3.The volume fractions must satisfy the relation m+∑=, (7.1) where Vm is the volume fraction of matrix material.For the case of multiple phases,it has been shown that the effective bulk modulus kem, shear modulus uerr,and thermal expansion coefficient aerr are given by [8]
of the perturbation term depends on the particle radius and the electrical conductivities of both the particle and the matrix. effective composite material representing the cluster and to each particle in the cluster. Maxwell states that an assumption is made that the particles do not interact, in which case at large distances from a cluster of particles, the perturbing effect of the particles can be expressed as the sum of the perturbations that each particle would cause if isolated in the matrix at a given point. This assumption implies that results are likely to be valid only derived [8] where Maxwell’s methodology has been applied to the estimation of many other properties for both isotropic particulate composites and anisotropic fiber-reinforced composites. 7.2.1 Applying Maxwell’s Approach to Multiphase Particulate Composites different types such that for i = 1,…,N there are ni spherical particles of i The cluster of all types of particle is now considered to be enclosed in a sphere of radius b such that the volume fraction of particles of type i within the sphere of radius b is given by 3 3 p i V n = i i fractions must satisfy the relation m p 1 1, N i i V V= +∑ = (7.1) where Vm is the volume fraction of matrix material. For the case of multiple phases, it has been shown that the effective bulk modulus keff, shear modulus µeff, and thermal expansion coefficient αeff are given by [8] L.N. McCartney The isolated particle solution is applied both to the single sphere of suggesting much wider applicability. The purpose of this section is to report results that have recently been consider multiple spherical reinforcements. Suppose, in a cluster of Because of the use of the far field in Maxwell’s methodology for particulate reinforcements embedded in an infinite matrix that there are N a / b . The volume moduli, and α will denote thermal expansion coefficients. The cluster is assumed to be homogeneous regarding the distribution of particles and leads estimating the properties of particulate composites, it is possible to properties. radius a . The properties of the particles of type i are denoted by a superfor low volume fractions of reinforcement, although evidence is presented script i and subscript p, where k will denote bulk moduli, µ will denote shear to isotropic effective properties. A suffix m will be used to denote matrix 274
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
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