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 composite 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 generally 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