《纺织复合材料》课程参考文献（Mechanics of Composite Materials）03 Micromechanical Analysis of a Lamina

3 Micromechanical Analysis of a Lamina Chapter Objectives Develop concepts of volume and weight fraction(mass fraction)of fiber and matrix,density,and void fraction in composites. Find the nine mechanical and four hygrothermal constants:four elastic moduli,five strength parameters,two coefficients of thermal expansion,and two coefficients of moisture expansion of a unidirec- tional lamina from the individual properties of the fiber and the matrix,fiber volume fraction,and fiber packing. Discuss the experimental characterization of the nine mechanical and four hygrothermal constants. 3.1 Introduction In Chapter 2,the stress-strain relationships,engineering constants,and fail- ure theories for an angle lamina were developed using four elastic moduli, five strength parameters,two coefficients of thermal expansion(CTE),and two coefficients of moisture expansion(CME)for a unidirectional lamina. These 13 parameters can be found experimentally by conducting several tension,compression,shear,and hygrothermal tests on unidirectional lamina (laminates).However,unlike in isotropic materials,experimental evaluation of these parameters is quite costly and time consuming because they are functions of several variables:the individual constituents of the composite material,fiber volume fraction,packing geometry,processing,etc.Thus,the need and motivation for developing analytical models to find these param- eters are very important.In this chapter,we will develop simple relationships for the these parameters in terms of the stiffnesses,strengths,coefficients of thermal and moisture expansion of the individual constituents of a compos- ite,fiber volume fraction,packing geometry,etc.An understanding of this 203 2006 by Taylor Francis Group,LLC

203 3 Micromechanical Analysis of a Lamina Chapter Objectives • Develop concepts of volume and weight fraction (mass fraction) of fiber and matrix, density, and void fraction in composites. • Find the nine mechanical and four hygrothermal constants: four elastic moduli, five strength parameters, two coefficients of thermal expansion, and two coefficients of moisture expansion of a unidirec￾tional lamina from the individual properties of the fiber and the matrix, fiber volume fraction, and fiber packing. • Discuss the experimental characterization of the nine mechanical and four hygrothermal constants. 3.1 Introduction In Chapter 2, the stress–strain relationships, engineering constants, and fail￾ure theories for an angle lamina were developed using four elastic moduli, five strength parameters, two coefficients of thermal expansion (CTE), and two coefficients of moisture expansion (CME) for a unidirectional lamina. These 13 parameters can be found experimentally by conducting several tension, compression, shear, and hygrothermal tests on unidirectional lamina (laminates). However, unlike in isotropic materials, experimental evaluation of these parameters is quite costly and time consuming because they are functions of several variables: the individual constituents of the composite material, fiber volume fraction, packing geometry, processing, etc. Thus, the need and motivation for developing analytical models to find these param￾eters are very important. In this chapter, we will develop simple relationships for the these parameters in terms of the stiffnesses, strengths, coefficients of thermal and moisture expansion of the individual constituents of a compos￾ite, fiber volume fraction, packing geometry, etc. An understanding of this 1343_book.fm Page 203 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

204 Mechanics of Composite Materials,Second Edition 10000 Nonhomogeneous lamina Homogeneous lamina FIGURE 3.1 A nonhomogeneous lamina with fibers and matrix approximated as a homogeneous lamina. relationship,called micromechanics of lamina,helps the designer to select the constituents of a composite material for use in a laminated structure. Because this text is for a first course in composite materials,details will be explained only for the simple models based on the mechanics of materials approach and the semi-empirical approach.Results from other methods based on advanced topics such as elasticity are also explained for completeness. As mentioned in Chapter 2,a unidirectional lamina is not homogeneous. However,one can assume the lamina to be homogeneous by focusing on the average response of the lamina to mechanical and hygrothermal loads(Figure 3.1).The lamina is simply looked at as a material whose properties are different in various directions,but not different from one location to another. Also,the chapter focuses on a unidirectional continuous fiber-reinforced lamina.This is because it forms the basic building block of a composite structure,which is generally made of several unidirectional laminae placed at various angles.The modeling in the evaluation of the parameters is dis- cussed first.This is followed by examples and experimental methods for finding these parameters. 3.2 Volume and Mass Fractions,Density,and Void Content Before modeling the 13 parameters of a unidirectional composite,we intro- duce the concept of relative fraction of fibers by volume.This concept is critical because theoretical formulas for finding the stiffness,strength,and hygrothermal properties of a unidirectional lamina are a function of fiber volume fraction.Measurements of the constituents are generally based on their mass,so fiber mass fractions must also be defined.Moreover,defining the density of a composite also becomes necessary because its value is used in the experimental determination of fiber volume and void fractions of a composite.Also,the value of density is used in the definition of specific modulus and specific strength in Chapter 1. 3.2.1 Volume Fractions Consider a composite consisting of fiber and matrix.Take the following symbol notations: 2006 by Taylor Francis Group,LLC

204 Mechanics of Composite Materials, Second Edition relationship, called micromechanics of lamina, helps the designer to select the constituents of a composite material for use in a laminated structure. Because this text is for a first course in composite materials, details will be explained only for the simple models based on the mechanics of materials approach and the semi-empirical approach. Results from other methods based on advanced topics such as elasticity are also explained for completeness. As mentioned in Chapter 2, a unidirectional lamina is not homogeneous. However, one can assume the lamina to be homogeneous by focusing on the average response of the lamina to mechanical and hygrothermal loads (Figure 3.1). The lamina is simply looked at as a material whose properties are different in various directions, but not different from one location to another. Also, the chapter focuses on a unidirectional continuous fiber-reinforced lamina. This is because it forms the basic building block of a composite structure, which is generally made of several unidirectional laminae placed at various angles. The modeling in the evaluation of the parameters is dis￾cussed first. This is followed by examples and experimental methods for finding these parameters. 3.2 Volume and Mass Fractions, Density, and Void Content Before modeling the 13 parameters of a unidirectional composite, we intro￾duce the concept of relative fraction of fibers by volume. This concept is critical because theoretical formulas for finding the stiffness, strength, and hygrothermal properties of a unidirectional lamina are a function of fiber volume fraction. Measurements of the constituents are generally based on their mass, so fiber mass fractions must also be defined. Moreover, defining the density of a composite also becomes necessary because its value is used in the experimental determination of fiber volume and void fractions of a composite. Also, the value of density is used in the definition of specific modulus and specific strength in Chapter 1. 3.2.1 Volume Fractions Consider a composite consisting of fiber and matrix. Take the following symbol notations: FIGURE 3.1 A nonhomogeneous lamina with fibers and matrix approximated as a homogeneous lamina. Nonhomogeneous lamina Homogeneous lamina 1343_book.fm Page 204 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

Micromechanical Analysis of a Lamina 205 volume of composite,fiber,and matrix,respectively Pm=density of composite,fiber,and matrix,respectively. Now define the fiber volume fraction V and the matrix volume fraction V as and Vin =Va Ve (3.1a,b) Note that the sum of volume fractions is V,+Vm=1, from Equation (3.1)as Uf+Um=Vc. 3.2.2 Mass Fractions Consider a composite consisting of fiber and matrix and take the following symbol notation:m=mass of composite,fiber,and matrix,respectively. The mass fraction (weight fraction)of the fibers (W)and the matrix(W) are defined as w,and W,= 0哑 W= (3.2a,b) Note that the sum of mass fractions is Wr+Wm =1, 2006 by Taylor Francis Group,LLC

Micromechanical Analysis of a Lamina 205 vc,f,m = volume of composite, fiber, and matrix, respectively ρc,f,m = density of composite, fiber, and matrix, respectively. Now define the fiber volume fraction Vf and the matrix volume fraction Vm as and (3.1a, b) Note that the sum of volume fractions is , from Equation (3.1) as 3.2.2 Mass Fractions Consider a composite consisting of fiber and matrix and take the following symbol notation: wc,f,m = mass of composite, fiber, and matrix, respectively. The mass fraction (weight fraction) of the fibers (Wf ) and the matrix (Wm) are defined as (3.2a, b) Note that the sum of mass fractions is , V v v f f c = , V v v m m c = . V V f m + = 1 v v f m + = vc . W w w f f c = , and W w w m m c = . W W f m + = 1 1343_book.fm Page 205 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

206 Mechanics of Composite Materials,Second Edition from Equation (3.2)as 0r+0m=0c· From the definition of the density of a single material, Wc=rVcr ws=ror,and (3.3a-c) Wm=YmUm Substituting Equation(3.3)in Equation(3.2),the mass fractions and vol- ume fractions are related as vand W= Wm= P碰Vw (3.4a,b) Pe in terms of the fiber and matrix volume fractions.In terms of individual constituent properties,the mass fractions and volume fractions are related by P WI= Pm一Vf' PLV+Vm Pm Wn= 1 Vm (3.5a,b) L(1-Vm)+Vm P One should always state the basis of calculating the fiber content of a composite.It is given in terms of mass or volume.Based on Equation(3.4), it is evident that volume and mass fractions are not equal and that the mismatch between the mass and volume fractions increases as the ratio between the density of fiber and matrix differs from one. 2006 by Taylor Francis Group,LLC

206 Mechanics of Composite Materials, Second Edition from Equation (3.2) as . From the definition of the density of a single material, (3.3a–c) Substituting Equation (3.3) in Equation (3.2), the mass fractions and vol￾ume fractions are related as (3.4a, b) in terms of the fiber and matrix volume fractions. In terms of individual constituent properties, the mass fractions and volume fractions are related by . (3.5a, b) One should always state the basis of calculating the fiber content of a composite. It is given in terms of mass or volume. Based on Equation (3.4), it is evident that volume and mass fractions are not equal and that the mismatch between the mass and volume fractions increases as the ratio between the density of fiber and matrix differs from one. w f m + w = wc w r v w r v w r v c c c f f f m m m = = = , , . and f f c W = V f , ρ ρ and m m c W = V m, ρ ρ f f m f m f m W = f V +V V , ρ ρ ρ ρ W V V m V f m m m = m − + 1 1 ρ ρ ( ) 1343_book.fm Page 206 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

Micromechanical Analysis of a Lamina 207 3.2.3 Density The derivation of the density of the composite in terms of volume fractions is found as follows.The mass of composite w.is the sum of the mass of the fibers w,and the mass of the matrix w as We=Wf+wm. (3.6) Substituting Equation(3.3)in Equation(3.6)yields Pevc=PrUr+PmUmr and UfPm vc Pe=PyVe (3.7) Using the definitions of fiber and matrix volume fractions from Equation (3.1), Pe=PVr+puVm (3.8) Now,consider that the volume of a composite v is the sum of the volumes of the fiber v and matrix () 0c=0r十0m· (3.9) The density of the composite in terms of mass fractions can be found as 工_W+W血 (3.10) Pe Pr Pm Example 3.1 A glass/epoxy lamina consists of a 70%fiber volume fraction.Use proper- ties of glass and epoxy from Table 3.1*and Table 3.2,respectively,to deter- mine the +Table 3.1 and Table 3.2 give the typical properties of common fibers and matrices in the SI sys- tem of units,respectively.Note that fibers such as graphite and aramids are transversely isotro- pic,but matrices are generally isotropic.The typical properties of common fibers and matrices are again given in Table 3.3 and Table 3.4,respectively,in the USCS system of units. 2006 by Taylor Francis Group,LLC

Micromechanical Analysis of a Lamina 207 3.2.3 Density The derivation of the density of the composite in terms of volume fractions is found as follows. The mass of composite wc is the sum of the mass of the fibers wf and the mass of the matrix wm as (3.6) Substituting Equation (3.3) in Equation (3.6) yields and . (3.7) Using the definitions of fiber and matrix volume fractions from Equation (3.1), (3.8) Now, consider that the volume of a composite vc is the sum of the volumes of the fiber vf and matrix (vm): . (3.9) The density of the composite in terms of mass fractions can be found as (3.10) Example 3.1 A glass/epoxy lamina consists of a 70% fiber volume fraction. Use proper￾ties of glass and epoxy from Table 3.1* and Table 3.2, respectively, to deter￾mine the * Table 3.1 and Table 3.2 give the typical properties of common fibers and matrices in the SI sys￾tem of units, respectively. Note that fibers such as graphite and aramids are transversely isotro￾pic, but matrices are generally isotropic. The typical properties of common fibers and matrices are again given in Table 3.3 and Table 3.4, respectively, in the USCS system of units. w w c f = + wm. ρ ρ c c v v = + f f ρmvm , ρ ρ c f ρ f c m m c v v v v = + c f ρ ρ = V f + ρmV m. v v c f = + vm 1 = W + W . c f f m m ρ ρ ρ 1343_book.fm Page 207 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

208 Mechanics of Composite Materials,Second Edition TABLE 3.1 Typical Properties of Fibers(SI System of Units) Property Units Graphite Glass Aramid Axial modulus GPa 230 85 124 Transverse modulus GPa 22 85 6 Axial Poisson's ratio 0.30 0.20 0.36 Transverse Poisson's ratio 0.35 0.20 0.37 Axial shear modulus GPa 22 35.42 3 Axial coefficient of thermal expansion um/m/C -1.3 5 -5.0 Transverse coefficient of thermal expansion um/m/C 7.0 5 4.1 Axial tensile strength MPa 2067 1550 1379 Axial compressive strength MPa 1999 1550 276 Transverse tensile strength MPa 77 1550 > Transverse compressive strength MPa 4 1550 > Shear strength MPa 36 35 21 Specific gravity 1.8 2.5 1.4 TABLE 3.2 Typical Properties of Matrices(SI System of Units) Property Units Epoxy Aluminum Polyamide Axial modulus GPa 3.4 71 3.5 Transverse modulus GPa 3.4 71 3.5 Axial Poisson's ratio 0.30 0.30 0.35 Transverse Poisson's ratio 一 0.30 0.30 0.35 Axial shear modulus GPa 1.308 27 1.3 Coefficient of thermal expansion um/m/C 63 23 90 Coefficient of moisture expansion m/m/kg/kg 0.33 0.00 0.33 Axial tensile strength MPa 72 276 54 Axial compressive strength MPa 102 276 108 Transverse tensile strength MPa 2 276 54 Transverse compressive strength MPa 102 276 108 Shear strength MPa 34 138 54 Specific gravity 1.2 2.7 1.2 1.Density of lamina 2.Mass fractions of the glass and epoxy 3.Volume of composite lamina if the mass of the lamina is 4 kg 4.Volume and mass of glass and epoxy in part(3) Solution 1.From Table 3.1,the density of the fiber is P=2500k8/m3. 2006 by Taylor Francis Group,LLC

208 Mechanics of Composite Materials, Second Edition 1. Density of lamina 2. Mass fractions of the glass and epoxy 3. Volume of composite lamina if the mass of the lamina is 4 kg 4. Volume and mass of glass and epoxy in part (3) Solution 1. From Table 3.1, the density of the fiber is TABLE 3.1 Typical Properties of Fibers (SI System of Units) Property Units Graphite Glass Aramid Axial modulus Transverse modulus Axial Poisson’s ratio Transverse Poisson’s ratio Axial shear modulus Axial coefficient of thermal expansion Transverse coefficient of thermal expansion Axial tensile strength Axial compressive strength Transverse tensile strength Transverse compressive strength Shear strength Specific gravity GPa GPa — — GPa μm/m/°C μm/m/°C MPa MPa MPa MPa MPa — 230 22 0.30 0.35 22 –1.3 7.0 2067 1999 77 42 36 1.8 85 85 0.20 0.20 35.42 5 5 1550 1550 1550 1550 35 2.5 124 8 0.36 0.37 3 –5.0 4.1 1379 276 7 7 21 1.4 TABLE 3.2 Typical Properties of Matrices (SI System of Units) Property Units Epoxy Aluminum Polyamide Axial modulus Transverse modulus Axial Poisson’s ratio Transverse Poisson’s ratio Axial shear modulus Coefficient of thermal expansion Coefficient of moisture expansion Axial tensile strength Axial compressive strength Transverse tensile strength Transverse compressive strength Shear strength Specific gravity GPa GPa — — GPa μm/m/°C m/m/kg/kg MPa MPa MPa MPa MPa — 3.4 3.4 0.30 0.30 1.308 63 0.33 72 102 72 102 34 1.2 71 71 0.30 0.30 27 23 0.00 276 276 276 276 138 2.7 3.5 3.5 0.35 0.35 1.3 90 0.33 54 108 54 108 54 1.2 f 3 ρ = 2500 kg / m . 1343_book.fm Page 208 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

Micromechanical Analysis of a Lamina 209 TABLE 3.3 Typical Properties of Fibers(USCS System of Units) Property Units Graphite Glass Aramid Axial modulus Msi 33.35 12.33 17.98 Transverse modulus Msi 3.19 12.33 1.16 Axial Poisson's ratio 0.30 0.20 0.36 Transverse Poisson's ratio 0.35 0.20 0.37 Axial shear modulus Msi 3.19 5.136 0.435 Axial coefficient of thermal expansion uin./in./F -0.7222 2.778 -2.778 Transverse coefficient of thermal expansion μuin./in./oF 3.889 2.778 2.278 Axial tensile strength ksi 299.7 224.8 200.0 Axial compressive strength ksi 289.8 224.8 40.02 Transverse tensile strength ksi 11.16 224.8 1.015 Transverse compressive strength ksi 6.09 224.8 1.015 Shear strength ksi 5.22 5.08 3.045 Specific gravity 1.8 2.5 1.4 TABLE 3.4 Typical Properties of Matrices(USCS System of Units) Property Units Epoxy Aluminum Polyamide Axial modulus Msi 0.493 10.30 0.5075 Transverse modulus Msi 0.493 10.30 0.5075 Axial Poisson's ratio 0.30 0.30 0.35 Transverse Poisson's ratio 0.30 0.30 0.35 Axial shear modulus Msi 0.1897 3.915 0.1885 Coefficient of thermal expansion uin./in./o℉ 35 12.78 50 Coefficient of moisture expansion in./in./Ib/Ib 0.33 0.00 0.33 Axial tensile strength ksi 10.44 40.02 7.83 Axial compressive strength ksi 14.79 40.02 15.66 Transverse tensile strength ksi 10.44 40.02 7.83 Transverse compressive strength ksi 14.79 40.02 15.66 Shear strength ksi 4.93 20.01 7.83 Specific gravity 1.2 2.7 1.2 From Table 3.2,the density of the matrix is Pm=1200kg/m3. Using Equation(3.8),the density of the composite is P.=(2500)(0.7)+(1200)0.3) =2110kg/m. 2.Using Equation(3.4),the fiber and matrix mass fractions are 2006 by Taylor Francis Group,LLC

Micromechanical Analysis of a Lamina 209 From Table 3.2, the density of the matrix is Using Equation (3.8), the density of the composite is 2. Using Equation (3.4), the fiber and matrix mass fractions are TABLE 3.3 Typical Properties of Fibers (USCS System of Units) Property Units Graphite Glass Aramid Axial modulus Transverse modulus Axial Poisson’s ratio Transverse Poisson’s ratio Axial shear modulus Axial coefficient of thermal expansion Transverse coefficient of thermal expansion Axial tensile strength Axial compressive strength Transverse tensile strength Transverse compressive strength Shear strength Specific gravity Msi Msi — — Msi μin./in./°F μin./in./°F ksi ksi ksi ksi ksi — 33.35 3.19 0.30 0.35 3.19 –0.7222 3.889 299.7 289.8 11.16 6.09 5.22 1.8 12.33 12.33 0.20 0.20 5.136 2.778 2.778 224.8 224.8 224.8 224.8 5.08 2.5 17.98 1.16 0.36 0.37 0.435 –2.778 2.278 200.0 40.02 1.015 1.015 3.045 1.4 TABLE 3.4 Typical Properties of Matrices (USCS System of Units) Property Units Epoxy Aluminum Polyamide Axial modulus Transverse modulus Axial Poisson’s ratio Transverse Poisson’s ratio Axial shear modulus Coefficient of thermal expansion Coefficient of moisture expansion Axial tensile strength Axial compressive strength Transverse tensile strength Transverse compressive strength Shear strength Specific gravity Msi Msi — — Msi μin./in./°F in./in./lb/lb ksi ksi ksi ksi ksi — 0.493 0.493 0.30 0.30 0.1897 35 0.33 10.44 14.79 10.44 14.79 4.93 1.2 10.30 10.30 0.30 0.30 3.915 12.78 0.00 40.02 40.02 40.02 40.02 20.01 2.7 0.5075 0.5075 0.35 0.35 0.1885 50 0.33 7.83 15.66 7.83 15.66 7.83 1.2 ρm = 1200 kg m3 / . ρc kg m = + = ( )( . ) ( )( . ) / . 2500 0 7 1200 0 3 2110 3 1343_book.fm Page 209 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

210 Mechanics of Composite Materials,Second Edition 2500 W22110 0.3 =0.8294 1200 Wm ×0.3 2110 =0.1706 Note that the sum of the mass fractions, W+Wnm=0.8294+0.1706 =1.000. 3.The volume of composite is 0e= We 4 2110 =1.896×10-3m3. 4.The volume of the fiber is Uf=VyUc =(0.7)1.896×10-3) =1.327×10-3m3. The volume of the matrix is Um=Vmve =0.3)0.1896×10-3) 2006 by Taylor Francis Group,LLC

210 Mechanics of Composite Materials, Second Edition . Note that the sum of the mass fractions, 3. The volume of composite is . 4. The volume of the fiber is . The volume of the matrix is Wf = × = 2500 2110 0 3 0 8294 . . Wm = × = 1200 2110 0 3 0 1706 . . W W f m + = + = 0 8294 0 1706 1 000 . . . . v w c c c = ρ = 4 2110 = × − 1 896 10 3 3 . m v V f f = vc = × − ( . 0 7)(1.896 10 ) 3 = × − 1 327 10 3 3 . m v V m m = vc =(0.3)(0.1896 × − 10 3 ) 1343_book.fm Page 210 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

Micromechanical Analysis of a Lamina 211 =0.5688×10-3m3. The mass of the fiber is Wf=Prof =(2500)1.327×10-3) =3.318k8. The mass of the matrix is Wm =PmUm =(1200)(0.5688×10-3) =0.6826kg. 3.2.4 Void Content During the manufacture of a composite,voids are introduced in the com- posite as shown in Figure 3.2.This causes the theoretical density of the composite to be higher than the actual density.Also,the void content of a FIGURE 3.2 Photomicrographs of cross-section of a lamina with voids. 2006 by Taylor Francis Group,LLC

Micromechanical Analysis of a Lamina 211 . The mass of the fiber is . The mass of the matrix is = 0.6826 kg . 3.2.4 Void Content During the manufacture of a composite, voids are introduced in the com￾posite as shown in Figure 3.2. This causes the theoretical density of the composite to be higher than the actual density. Also, the void content of a FIGURE 3.2 Photomicrographs of cross-section of a lamina with voids. = × − 0 5688 10 3 3 . m w v f f = ρ f = × − ( ) 2500 (1.327 10 ) 3 = 3.318 kg w v m m = ρ m = × − ( ) 1200 (0.5688 10 ) 3 1343_book.fm Page 211 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC

212 Mechanics of Composite Materials,Second Edition composite is detrimental to its mechanical properties.These detriments include lower Shear stiffness and strength Compressive strengths Transverse tensile strengths 。Fatigue resistance ·Moisture resistance A decrease of 2 to 10%in the preceding matrix-dominated properties gen- erally takes place with every 1%increase in the void content.! For composites with a certain volume of voids V,the volume fraction of voids V,is defined as (3.11) Then,the total volume of a composite (v)with voids is given by 0c=0r十0m十0 (3.12) By definition of the experimental density Pe of a composite,the actual volume of the composite is (3.13) Pce and,by the definition of the theoretical density Par of the composite,the theoretical volume of the composite is 巡 0f+0m= (3.14) Pa Then,substituting the preceding expressions(3.13)and (3.14)in Equation (3.12), 0=0+0, Pae Pa The volume of void is given by 2006 by Taylor Francis Group,LLC

212 Mechanics of Composite Materials, Second Edition composite is detrimental to its mechanical properties. These detriments include lower • Shear stiffness and strength • Compressive strengths • Transverse tensile strengths • Fatigue resistance • Moisture resistance A decrease of 2 to 10% in the preceding matrix-dominated properties gen￾erally takes place with every 1% increase in the void content.1 For composites with a certain volume of voids Vv the volume fraction of voids Vv is defined as (3.11) Then, the total volume of a composite (vc) with voids is given by (3.12) By definition of the experimental density ρce of a composite, the actual volume of the composite is (3.13) and, by the definition of the theoretical density ρct of the composite, the theoretical volume of the composite is (3.14) Then, substituting the preceding expressions (3.13) and (3.14) in Equation (3.12), . The volume of void is given by V v v v v c = . v v c f = + vm + vv. v w c c ce = ρ , v v w f m c ct + = ρ . w w v c ce c ct v ρ ρ = + 1343_book.fm Page 212 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC