6 Mechanics of Composite Materials,Second Edition TABLE 1.1 Specific Modulus and Specific Strength of Typical Fibers,Composites,and Bulk Metals Young's Specific Specific Material Specific modulus strength modulus strength Units gravity (Msi) (ksi) (Msi-in.3/Ib) (ksi-in.3/1b) System of Units:USCS Graphite fiber 1.8 33.35 299.8 512.9 4610 Aramid fiber 1.4 17.98 200.0 355.5 3959 Glass fiber 2.5 12.33 224.8 136.5 2489 Unidirectional graphite/epoxy 1.6 26.25 217.6 454.1 3764 Unidirectional glass/epoxy 1.8 5.598 154.0 86.09 2368 Cross-ply graphite/epoxy 1.6 13.92 54.10 240.8 935.9 Cross-ply glass/epoxy 1.8 3.420 12.80 52.59 196.8 Quasi-isotropic graphite/epoxy 1.6 10.10 40.10 174.7 693.7 Quasi-isotropic glass/epoxy 1.8 2.750 10.60 42.29 163.0 Steel 7.8 30.00 94.00 106.5 333.6 Aluminum 2.6 10.00 40.00 106.5 425.8 Young's Ultimate Specific Specific Material Specific modulus strength modulus strength Units gravity (GPa) (MPa) (GPa-m2/kg) (MPa-m2/kg) System of LInits:SI Graphite fiber 1.8 230.00 2067 0.1278 1.148 Aramid fiber 1.4 124.00 1379 0.08857 0.9850 Glass fiber 2.5 85.00 1550 0.0340 0.6200 Unidirectional graphite/epoxy 1.6 181.00 1500 0.1131 0.9377 Unidirectional glass/epoxy 1.8 38.60 1062 0.02144 0.5900 Cross-ply graphite/epoxy 1.6 95.98 373.0 0.06000 0.2331 Cross-ply glass/epoxy 1.8 23.58 8825 0.01310 0.0490 Quasi-isotropic graphite/epoxy 1.6 69.64 276.48 0.04353 0.1728 Quasi-isotropic glass/epoxy 1.8 18.96 73.08 0.01053 0.0406 Steel 7.8 206.84 648.1 0.02652 0.08309 Aluminum 2.6 68.95 275.8 0.02652 0.1061 Specific gravity of a material is the ratio between its density and the density of water. A comparison is now made between popular types of laminates such as cross-ply and quasi-isotropic laminates.Figure 1.2 shows the specific strength plotted as a function of specific modulus for various fibers,metals, and composites. Are specific modulus and specific strength the only mechanical parameters used for measuring the relative advantage of composites over metals? No,it depends on the application.+Consider compression of a column, where it may fail due to buckling.The Euler buckling formula gives the critical load at which a long column buckles ass 2006 by Taylor Francis Group,LLC
6 Mechanics of Composite Materials, Second Edition A comparison is now made between popular types of laminates such as cross-ply and quasi-isotropic laminates. Figure 1.2 shows the specific strength plotted as a function of specific modulus for various fibers, metals, and composites. Are specific modulus and specific strength the only mechanical parameters used for measuring the relative advantage of composites over metals? No, it depends on the application.4 Consider compression of a column, where it may fail due to buckling. The Euler buckling formula gives the critical load at which a long column buckles as5 TABLE 1.1 Specific Modulus and Specific Strength of Typical Fibers, Composites, and Bulk Metals Material Units Specific gravitya Young’s modulus (Msi) Ultimate strength (ksi) Specific modulus (Msi-in.3/lb) Specific strength (ksi-in.3/lb) System of Units: USCS Graphite fiber Aramid fiber Glass fiber Unidirectional graphite/epoxy Unidirectional glass/epoxy Cross-ply graphite/epoxy Cross-ply glass/epoxy Quasi-isotropic graphite/epoxy Quasi-isotropic glass/epoxy Steel Aluminum 1.8 1.4 2.5 1.6 1.8 1.6 1.8 1.6 1.8 7.8 2.6 33.35 17.98 12.33 26.25 5.598 13.92 3.420 10.10 2.750 30.00 10.00 299.8 200.0 224.8 217.6 154.0 54.10 12.80 40.10 10.60 94.00 40.00 512.9 355.5 136.5 454.1 86.09 240.8 52.59 174.7 42.29 106.5 106.5 4610 3959 2489 3764 2368 935.9 196.8 693.7 163.0 333.6 425.8 Material Units Specific gravity Young’s modulus (GPa) Ultimate strength (MPa) Specific modulus (GPa-m3/kg) Specific strength (MPa-m3/kg) System of Units: SI Graphite fiber Aramid fiber Glass fiber Unidirectional graphite/epoxy Unidirectional glass/epoxy Cross-ply graphite/epoxy Cross-ply glass/epoxy Quasi-isotropic graphite/epoxy Quasi-isotropic glass/epoxy Steel Aluminum 1.8 1.4 2.5 1.6 1.8 1.6 1.8 1.6 1.8 7.8 2.6 230.00 124.00 85.00 181.00 38.60 95.98 23.58 69.64 18.96 206.84 68.95 2067 1379 1550 1500 1062 373.0 88.25 276.48 73.08 648.1 275.8 0.1278 0.08857 0.0340 0.1131 0.02144 0.06000 0.01310 0.04353 0.01053 0.02652 0.02652 1.148 0.9850 0.6200 0.9377 0.5900 0.2331 0.0490 0.1728 0.0406 0.08309 0.1061 a Specific gravity of a material is the ratio between its density and the density of water. 1343_book.fm Page 6 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Introduction to Composite Materials 7 5000 0 Graphite fiber 4000 O Unidirectional graphite/epoxy 3000 2000 1000 Quasi-isotropic graphite/epoxy 0 OCross-ply graphite/epoxy Aluminum Steel 0 0 100 200 300 400 500 600 Specific modulus (Msi-in3/lb) FIGURE 1.2 Specific strength as a function of specific modulus for metals,fibers,and composites. π2E1 Pa= 2, (1.4) where P=critical buckling load (Ib or N) E=Young's modulus of column (lb/in.2 or N/m2) I second moment of area (in.or m) L=length of beam (in.or m) If the column has a circular cross section,the second moment of area is I= d (1.5) 64 and the mass of the rod is πdL M=P (1.6) 2006 by Taylor Francis Group,LLC
Introduction to Composite Materials 7 , (1.4) where Pcr = critical buckling load (lb or N) E = Young’s modulus of column (lb/in.2 or N/m2) I = second moment of area (in.4 or m4) L = length of beam (in. or m) If the column has a circular cross section, the second moment of area is (1.5) and the mass of the rod is , (1.6) FIGURE 1.2 Specific strength as a function of specific modulus for metals, fibers, and composites. 5000 4000 3000 2000 1000 0 0 100 200 Quasi-isotropic graphite/epoxy Aluminum Specific modulus (Msi-in3/lb) Cross-ply graphite/epoxy Unidirectional graphite/epoxy Graphite fiber Steel Specific strength (Ksi-in3/lb) 300 400 500 600 Pcr EI L = π2 2 I d = π 4 64 M = d L 4 ρ π 2 1343_book.fm Page 7 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
8 Mechanics of Composite Materials,Second Edition where M=mass of the beam (lb or kg) p density of beam (Ib/in.3 or kg/m3) d diameter of beam (in.or m) Because the length,L,and the load,p are constant,we find the mass of the beam by substituting Equation (1.5)and Equation (1.6)in Equation (1.4)as M 21Pa 1 (1.7) √元E2/p This means that the lightest beam for specified stiffness is one with the highest value of E12/p. Similarly,we can prove that,for achieving the minimum deflection in a beam under a load along its length,the lightest beam is one with the highest value of E1/3/p.Typical values of these two parameters,E12/p and E1/3/p for typical fibers,unidirectional composites,cross-ply and quasi-isotropic laminates,steel,and aluminum are given in Table 1.2.Comparing these numbers with metals shows composites drawing a better advantage for these two parameters.Other mechanical parameters for comparing the perfor- mance of composites to metals include resistance to fracture,fatigue,impact, and creep. Yes,composites have distinct advantages over metals.Are there any draw- backs or limitations in using them? Yes,drawbacks and limitations in use of composites include: High cost of fabrication of composites is a critical issue.For example, a part made of graphite/epoxy composite may cost up to 10 to 15 times the material costs.A finished graphite/epoxy composite part may cost as much as $300 to $400 per pound ($650 to $900 per kilogram).Improvements in processing and manufacturing tech- niques will lower these costs in the future.Already,manufacturing techniques such as SMC (sheet molding compound)and SRIM (structural reinforcement injection molding)are lowering the cost and production time in manufacturing automobile parts. Mechanical characterization of a composite structure is more com- plex than that of a metal structure.Unlike metals,composite mate- rials are not isotropic,that is,their properties are not the same in all directions.Therefore,they require more material parameters.For example,a single layer of a graphite/epoxy composite requires nine 2006 by Taylor Francis Group,LLC
8 Mechanics of Composite Materials, Second Edition where M = mass of the beam (lb or kg) ρ = density of beam (lb/in.3 or kg/m3) d = diameter of beam (in. or m) Because the length, L, and the load, P, are constant, we find the mass of the beam by substituting Equation (1.5) and Equation (1.6) in Equation (1.4) as . (1.7) This means that the lightest beam for specified stiffness is one with the highest value of E1/2/ρ. Similarly, we can prove that, for achieving the minimum deflection in a beam under a load along its length, the lightest beam is one with the highest value of E1/3/ρ. Typical values of these two parameters, E1/2/ρ and E1/3/ρ for typical fibers, unidirectional composites, cross-ply and quasi-isotropic laminates, steel, and aluminum are given in Table 1.2. Comparing these numbers with metals shows composites drawing a better advantage for these two parameters. Other mechanical parameters for comparing the performance of composites to metals include resistance to fracture, fatigue, impact, and creep. Yes, composites have distinct advantages over metals. Are there any drawbacks or limitations in using them? Yes, drawbacks and limitations in use of composites include: • High cost of fabrication of composites is a critical issue. For example, a part made of graphite/epoxy composite may cost up to 10 to 15 times the material costs. A finished graphite/epoxy composite part may cost as much as $300 to $400 per pound ($650 to $900 per kilogram). Improvements in processing and manufacturing techniques will lower these costs in the future. Already, manufacturing techniques such as SMC (sheet molding compound) and SRIM (structural reinforcement injection molding) are lowering the cost and production time in manufacturing automobile parts. • Mechanical characterization of a composite structure is more complex than that of a metal structure. Unlike metals, composite materials are not isotropic, that is, their properties are not the same in all directions. Therefore, they require more material parameters. For example, a single layer of a graphite/epoxy composite requires nine M L P E cr = 2 1 2 1 2 π ρ / / 1343_book.fm Page 8 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Introduction to Composite Materials 9 TABLE 1.2 Specific Modulus Parameters E/p,E12/p,and E//p for Typical Materials Young's Material Specific modulus Elp ER/p E/p Units gravity (Msi)(Msi-in.3/Ib)(psi-in.3/b)(psit-in./Ib) System of UInits:UISCS Graphite fiber 1.8 33.35 512.8 88,806 4,950 Kevlar fiber 1.4 17.98 355.5 83836 5,180 Glass fiber 2.5 12.33 136.5 38,878 2,558 Unidirectional graphite/epoxy 1.6 26.25 454.1 88,636 5,141 Unidirectional glass/epoxy 1.8 5.60 86.09 36384 2,730 Cross-ply graphite/epoxy 1.6 13.92 240.8 64545 4,162 Cross-ply glass/epoxy 1.8 3.42 52.59 28,438 2317 Quasi-isotropic graphite/epoxy 1.6 10.10 174.7 54,980 3,740 Quasi-isotropic glass/epoxy 1.8 2.75 42.29 25,501 2,154 Steel 7.8 30.00 106.5 19,437 1,103 Aluminum 2.6 10.00 106.5 33,666 2294 Young's Material Specific modulus Elp E/p Elp Units gravity (GPa) (GPa-m3/kg) (Pa-m3/kg) (Pais-m3/kg System of UInits:SI Graphite fiber 1.8 230.00 0.1278 266.4 3.404 Kevlar fiber 1.4 124.00 0.08857 251.5 3.562 Glass fiber 2.5 85.00 0.034 116.6 1.759 Unidirectional graphite/epoxy 1.6 181.00 0.1131 265.9 3.535 Unidirectional glass/epoxy 1.8 38.60 0.02144 109.1 1.878 Cross-ply graphite/epoxy 1.6 95.98 0.060 193.6 2.862 Cross-ply glass/epoxy 1.8 23.58 0.0131 8531 1.593 Quasi-isotropic graphite/epoxy 1.6 69.64 0.04353 164.9 2.571 Quasi-isotropic glass/epoxy 1.8 18.96 0.01053 76.50 1.481 Steel 7.8 206.84 0.02652 58.3 0.7582 Aluminum 2.6 68.95 0.02662 101.0 1.577 stiffness and strength constants for conducting mechanical analysis. In the case of a monolithic material such as steel,one requires only four stiffness and strength constants.Such complexity makes struc- tural analysis computationally and experimentally more compli- cated and intensive.In addition,evaluation and measurement techniques of some composite properties,such as compressive strengths,are still being debated. Repair of composites is not a simple process compared to that for metals.Sometimes critical flaws and cracks in composite structures may go undetected. 2006 by Taylor Francis Group,LLC
Introduction to Composite Materials 9 stiffness and strength constants for conducting mechanical analysis. In the case of a monolithic material such as steel, one requires only four stiffness and strength constants. Such complexity makes structural analysis computationally and experimentally more complicated and intensive. In addition, evaluation and measurement techniques of some composite properties, such as compressive strengths, are still being debated. • Repair of composites is not a simple process compared to that for metals. Sometimes critical flaws and cracks in composite structures may go undetected. TABLE 1.2 Specific Modulus Parameters E/ρ, E1/2/ρ, and E1/3/ρ for Typical Materials Material Units Specific gravity Young’s modulus (Msi) E/ρ (Msi-in.3/lb) E1/2/ρ (psi1/2-in.3/lb) E1/3/ρ (psi1/3-in.3/lb) System of Units: USCS Graphite fiber Kevlar fiber Glass fiber Unidirectional graphite/epoxy Unidirectional glass/epoxy Cross-ply graphite/epoxy Cross-ply glass/epoxy Quasi-isotropic graphite/epoxy Quasi-isotropic glass/epoxy Steel Aluminum 1.8 1.4 2.5 1.6 1.8 1.6 1.8 1.6 1.8 7.8 2.6 33.35 17.98 12.33 26.25 5.60 13.92 3.42 10.10 2.75 30.00 10.00 512.8 355.5 136.5 454.1 86.09 240.8 52.59 174.7 42.29 106.5 106.5 88,806 83,836 38,878 88,636 36,384 64,545 28,438 54,980 25,501 19,437 33,666 4,950 5,180 2,558 5,141 2,730 4,162 2,317 3,740 2,154 1,103 2,294 Material Units Specific gravity Young’s modulus (GPa) E/ρ (GPa-m3/kg) E1/2/ρ (Pa-m3/kg) E1/3/ρ (Pa1/3-m3/kg ) System of Units: SI Graphite fiber Kevlar fiber Glass fiber Unidirectional graphite/epoxy Unidirectional glass/epoxy Cross-ply graphite/epoxy Cross-ply glass/epoxy Quasi-isotropic graphite/epoxy Quasi-isotropic glass/epoxy Steel Aluminum 1.8 1.4 2.5 1.6 1.8 1.6 1.8 1.6 1.8 7.8 2.6 230.00 124.00 85.00 181.00 38.60 95.98 23.58 69.64 18.96 206.84 68.95 0.1278 0.08857 0.034 0.1131 0.02144 0.060 0.0131 0.04353 0.01053 0.02652 0.02662 266.4 251.5 116.6 265.9 109.1 193.6 85.31 164.9 76.50 58.3 101.0 3.404 3.562 1.759 3.535 1.878 2.862 1.593 2.571 1.481 0.7582 1.577 1343_book.fm Page 9 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
10 Mechanics of Composite Materials,Second Edition 一2a ↓↓↓↓↓↓↓ FIGURE 1.3 A uniformly loaded plate with a crack. Composites do not have a high combination of strength and fracture toughness*compared to metals.In Figure 1.4,a plot is shown for fracture toughness vs.yield strength for a 1-in.(25-mm)thick mate- rial.3 Metals show an excellent combination of strength and fracture toughness compared to composites.(Note:The transition areas in Figure 1.4 will change with change in the thickness of the specimen.) Composites do not necessarily give higher performance in all the properties used for material selection.In Figure 1.5,six primary material selection parameters-strength,toughness,formability, In a material with a crack,the value of the stress intensity factor gives the measure of stresses in the crack tip region.For example,for an infinite plate with a crack of length 2a under a uniaxial load o (Figure 1.3),the stress intensity factor is K=6yia. If the stress intensity factor at the crack tip is greater than the critical stress intensity factor of the material,the crack will grow.The greater the value of the critical stress intensity factor is,the tougher the material is.The critical stress intensity factor is called the fracture toughness of the material.Typical values of fracture toughness are 23.66 ksivin.(26 MPavm)for aluminum and 25.48 ksivin.(28 MPavm)for steel. 2006 by Taylor Francis Group,LLC
10 Mechanics of Composite Materials, Second Edition • Composites do not have a high combination of strength and fracture toughness* compared to metals. In Figure 1.4, a plot is shown for fracture toughness vs. yield strength for a 1-in. (25-mm) thick material.3 Metals show an excellent combination of strength and fracture toughness compared to composites. (Note: The transition areas in Figure 1.4 will change with change in the thickness of the specimen.) • Composites do not necessarily give higher performance in all the properties used for material selection. In Figure 1.5, six primary material selection parameters — strength, toughness, formability, FIGURE 1.3 A uniformly loaded plate with a crack. * In a material with a crack, the value of the stress intensity factor gives the measure of stresses in the crack tip region. For example, for an infinite plate with a crack of length 2a under a uniaxial load σ (Figure 1.3), the stress intensity factor is . If the stress intensity factor at the crack tip is greater than the critical stress intensity factor of the material, the crack will grow. The greater the value of the critical stress intensity factor is, the tougher the material is. The critical stress intensity factor is called the fracture toughness of the material. Typical values of fracture toughness are for aluminum and for steel. σ σ 2a K a =σ π 23.66 ksi in. (26 MPa m ) 25.48 ksi in. (28 MPa m ) 1343_book.fm Page 10 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC