218 Mechanics of Composite Materials,Second Edition tm/2 t/2 FIGURE 3.4 A longitudinal stress applied to the representative volume element to calculate the longitudinal Young's modulus for a unidirectional lamina. The fibers and matrix follow Hooke's law (linearly elastic). The fibers possess uniform strength. The composite is free of voids. 3.3.1.1 Longitudinal Young's Modulus From Figure 3.4,under a uniaxial load F.on the composite RVE,the load is shared by the fiber F,and the matrix F so that F=F+Fu (3.29) The loads taken by the fiber,the matrix,and the composite can be written in terms of the stresses in these components and cross-sectional areas of these components as F=cAc, (3.30a) F=Afr (3.30b) Fm =GmAm (3.30c) where fm=stress in composite,fiber,and matrix,respectively A=area of composite,fiber,and matrix,respectively Assuming that the fibers,matrix,and composite follow Hooke's law and that the fibers and the matrix are isotropic,the stress-strain relationship for each component and the composite is 2006 by Taylor Francis Group,LLC
218 Mechanics of Composite Materials, Second Edition • The fibers and matrix follow Hooke’s law (linearly elastic). • The fibers possess uniform strength. • The composite is free of voids. 3.3.1.1 Longitudinal Young’s Modulus From Figure 3.4, under a uniaxial load Fc on the composite RVE, the load is shared by the fiber Ff and the matrix Fm so that (3.29) The loads taken by the fiber, the matrix, and the composite can be written in terms of the stresses in these components and cross-sectional areas of these components as (3.30a) (3.30b) (3.30c) where σc,f,m = stress in composite, fiber, and matrix, respectively Ac,f,m = area of composite, fiber, and matrix, respectively Assuming that the fibers, matrix, and composite follow Hooke’s law and that the fibers and the matrix are isotropic, the stress–strain relationship for each component and the composite is FIGURE 3.4 A longitudinal stress applied to the representative volume element to calculate the longitudinal Young’s modulus for a unidirectional lamina. t m/2 t m/2 t f t c σc σc h FFF c fm = + . F A c cc = σ , F A f ff = σ , F A m mm = σ , 1343_book.fm Page 218 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Micromechanical Analysis of a Lamina 219 c=EEc, (3.31a) Gf=EEfr (3.31b) and Gm=Enm (3.31c where em=strains in composite,fiber,and matrix,respectively E=elastic moduli of composite,fiber,and matrix,respectively Substituting Equation(3.30)and Equation(3.31)in Equation(3.29)yields EcAc=ErerAt+EmemAm. (3.32) The strains in the composite,fiber,and matrix are equal (=E=Em);then, from Equation (3.32), ALEn Ac E-EA (3.33) Using Equation (3.28),for definitions of volume fractions, E1=E/Vt+EnVm (3.34) Equation 3.34 gives the longitudinal Young's modulus as a weighted mean of the fiber and matrix modulus.It is also called the rule of mixtures. The ratio of the load taken by the fibers F,to the load taken by the composite F.is a measure of the load shared by the fibers.From Equation (3.30)and Equation (3.31), ELVS (3.35) In Figure 3.5,the ratio of the load carried by the fibers to the load taken by the composite is plotted as a function of fiber-to-matrix Young's moduli ratio EE for the constant fiber volume fraction V.It shows that as the fiber to matrix moduli ratio increases,the load taken by the fiber increases tre- mendously. 2006 by Taylor Francis Group,LLC
Micromechanical Analysis of a Lamina 219 (3.31a) (3.31b) and (3.31c) where εc,f,m = strains in composite, fiber, and matrix, respectively E1,f,m = elastic moduli of composite, fiber, and matrix, respectively Substituting Equation (3.30) and Equation (3.31) in Equation (3.29) yields (3.32) The strains in the composite, fiber, and matrix are equal (εc = εf = εm); then, from Equation (3.32), (3.33) Using Equation (3.28), for definitions of volume fractions, (3.34) Equation 3.34 gives the longitudinal Young’s modulus as a weighted mean of the fiber and matrix modulus. It is also called the rule of mixtures. The ratio of the load taken by the fibers Ff to the load taken by the composite Fc is a measure of the load shared by the fibers. From Equation (3.30) and Equation (3.31), (3.35) In Figure 3.5, the ratio of the load carried by the fibers to the load taken by the composite is plotted as a function of fiber-to-matrix Young’s moduli ratio Ef /Em for the constant fiber volume fraction Vf . It shows that as the fiber to matrix moduli ratio increases, the load taken by the fiber increases tremendously. σ ε c c = E1 , σ ε f ff = E , σ ε m mm = E , EA E A E A 1εε ε c c f f f mm m = + . E E A A E A A f f c m m c 1 = + . E EV EV 1 = + f f mm. F F E E V f c f = f 1 . 1343_book.fm Page 219 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
220 Mechanics of Composite Materials,Second Edition peol asodwo 0.6 V=0.2 -V=0.4 ······V=0.6 0.4 V=0.8 0.2L 0 20 40 60 80 100 Fiber to matrix moduli ratio,EE FIGURE 3.5 Fraction of load of composite carried by fibers as a function of fiber volume fraction for constant fiber to matrix moduli ratio. Example 3.3 Find the longitudinal elastic modulus of a unidirectional glass/epoxy lamina with a 70%fiber volume fraction.Use the properties of glass and epoxy from Table 3.1 and Table 3.2,respectively.Also,find the ratio of the load taken by the fibers to that of the composite. Solution From Table 3.1,the Young's modulus of the fiber is E=85 GPa. From Table 3.2,the Young's modulus of the matrix is E =3.4 GPa. Using Equation(3.34),the longitudinal elastic modulus of the unidirectional lamina is E1=(85)(0.7)+(3.4)(0.3) =60.52GPa. Using Equation(3.35),the ratio of the load taken by the fibers to that of the composite is 2006 by Taylor Francis Group,LLC
220 Mechanics of Composite Materials, Second Edition Example 3.3 Find the longitudinal elastic modulus of a unidirectional glass/epoxy lamina with a 70% fiber volume fraction. Use the properties of glass and epoxy from Table 3.1 and Table 3.2, respectively. Also, find the ratio of the load taken by the fibers to that of the composite. Solution From Table 3.1, the Young’s modulus of the fiber is Ef = 85 GPa. From Table 3.2, the Young’s modulus of the matrix is Em = 3.4 GPa. Using Equation (3.34), the longitudinal elastic modulus of the unidirectional lamina is Using Equation (3.35), the ratio of the load taken by the fibers to that of the composite is FIGURE 3.5 Fraction of load of composite carried by fibers as a function of fiber volume fraction for constant fiber to matrix moduli ratio. 1 0.8 0.6 0.4 0.2 0 20 40 60 80 100 Fiber to matrix moduli ratio, Ef /Em Vf = 0.2 Vf = 0.4 Vf = 0.6 Vf = 0.8 Fiber to composite load ratio, Ff/Fc E GPa 1 85 0 7 3 4 0 3 60 52 = + = ( )( . ) ( . )( . ) . . 1343_book.fm Page 220 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Micromechanical Analysis of a Lamina 221 0 60 50 人 40 Experimental data points 30 20 9 0 03 0.4 0.5 0.6 Fiber volume fraction,V FIGURE 3.6 Longitudinal Young's modulus as function of fiber volume fraction and comparison with experimental data points for a typical glass/polyester lamina.(Experimental data points repro- duced with permission of ASM International.) 85 60.520.7) =0.9831. Figure 3.6 shows the linear relationship between the longitudinal Young's modulus of a unidirectional lamina and fiber volume fraction for a typical graphite/epoxy composite per Equation(3.34).It also shows that Equation (3.34)predicts results that are close to the experimental data points.3 3.3.1.2 Transverse Young's Modulus Assume now that,as shown in Figure 3.7,the composite is stressed in the transverse direction.The fibers and matrix are again represented by rectan- gular blocks as shown.The fiber,the matrix,and composite stresses are equal.Thus, 0c=6f=0m' (3.36) where om=stress in composite,fiber,and matrix,respectively. Now,the transverse extension in the composite A,is the sum of the trans- verse extension in the fiber△，and that is the matrix,△m 2006 by Taylor Francis Group,LLC
Micromechanical Analysis of a Lamina 221 Figure 3.6 shows the linear relationship between the longitudinal Young’s modulus of a unidirectional lamina and fiber volume fraction for a typical graphite/epoxy composite per Equation (3.34). It also shows that Equation (3.34) predicts results that are close to the experimental data points.3 3.3.1.2 Transverse Young’s Modulus Assume now that, as shown in Figure 3.7, the composite is stressed in the transverse direction. The fibers and matrix are again represented by rectangular blocks as shown. The fiber, the matrix, and composite stresses are equal. Thus, (3.36) where σc,f,m = stress in composite, fiber, and matrix, respectively. Now, the transverse extension in the composite Δc is the sum of the transverse extension in the fiber Δf, and that is the matrix, Δm. FIGURE 3.6 Longitudinal Young’s modulus as function of fiber volume fraction and comparison with experimental data points for a typical glass/polyester lamina. (Experimental data points reproduced with permission of ASM International.) 70 60 50 40 30 20 10 0 0.3 0.4 Fiber volume fraction, Vf Experimental data points Longitudinal Young’s modulus, E1 (GPa) 0.5 0.6 F F f c = = 85 60 52 0 7 0 9831 . (.) . . σσ σ cfm = = , 1343_book.fm Page 221 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
222 Mechanics of Composite Materials,Second Edition t/2 FIGURE 3.7 A transverse stress applied to a representative volume element used to calculate transverse Young's modulus of a unidirectional lamina. △c=△y+△m (3.37) Now,by the definition of normal strain, △e=tec, (3.38a) △r=ffEf, (3.38b) and △m=tmew' (3.38c) where =thickness of the composite,fiber and matrix,respectively =normal transverse strain in the composite,fiber,and matrix, respectively Also,by using Hooke's law for the fiber,matrix,and composite,the normal strains in the composite,fiber,and matrix are Ec= (3.39a) E21 E= (3.39b) and Em (3.39c) 2006 by Taylor Francis Group,LLC
222 Mechanics of Composite Materials, Second Edition (3.37) Now, by the definition of normal strain, (3.38a) (3.38b) and (3.38c) where tc,f,m = thickness of the composite, fiber and matrix, respectively εc,f,m = normal transverse strain in the composite, fiber, and matrix, respectively Also, by using Hooke’s law for the fiber, matrix, and composite, the normal strains in the composite, fiber, and matrix are (3.39a) (3.39b) and (3.39c) FIGURE 3.7 A transverse stress applied to a representative volume element used to calculate transverse Young’s modulus of a unidirectional lamina. t m/2 t f t c σc σc h t m/2 ΔΔΔ cfm =+. Δc cc = ε t , Δ f ff = ε t , Δm mm = ε t , ε σ c c E = 2 , ε σ f f Ef = , ε σ m m Em = . 1343_book.fm Page 222 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC