Equation 3.17 now gives oLtu =1441.28 MPa. This example demonstrates that with the same fiber length,it is possible to achieve a high longitudinal tensile strength for the composite by increasing the interfacial shear stress.Physically,this means that the bonding between the fibers and the matrix must be improved. 3.1.1.3 Microfailure Modes in Longitudinal Tension In deriving Equations 3.9 and 3.17,it was assumed that all fibers have equal strength and the composite lamina fails immediately after fiber failure.In practice,fiber strength is not a unique value;instead it follows a statistical distribution.Therefore,it is expected that a few fibers will break at low stress levels.Although the remaining fibers will carry higher stresses,they may not fail simultaneously. When a fiber breaks(Figure 3.11),the normal stress at each of its broken ends becomes zero.However,over a distance of le/2 from each end,the stress builds back up to the average value by shear stress transfer at the fiber-matrix interface (Figure 3.11c).Additionally,the stress states in a region close to the broken ends contain 1.Stress concentrations at the void created by the broken fiber 2.High shear stress concentrations in the matrix near the fiber ends 3.An increase in the average normal stress in adjacent fibers(Figure 3.11b) Fiber breakage (a) 2 3 (b) (c) FIGURE 3.11 Longitudinal stress distributions (a)in unidirectional continuous fibers before the failure of fiber 3,(b)in fibers 2 and 4 after the failure of fiber 3,and(c)in fiber 3 after it fails. 2007 by Taylor Francis Group,LLC
Equation 3.17 now gives sLtu ¼ 1441.28 MPa. This example demonstrates that with the same fiber length, it is possible to achieve a high longitudinal tensile strength for the composite by increasing the interfacial shear stress. Physically, this means that the bonding between the fibers and the matrix must be improved. 3.1.1.3 Microfailure Modes in Longitudinal Tension In deriving Equations 3.9 and 3.17, it was assumed that all fibers have equal strength and the composite lamina fails immediately after fiber failure. In practice, fiber strength is not a unique value; instead it follows a statistical distribution. Therefore, it is expected that a few fibers will break at low stress levels. Although the remaining fibers will carry higher stresses, they may not fail simultaneously. When a fiber breaks (Figure 3.11), the normal stress at each of its broken ends becomes zero. However, over a distance of lc=2 from each end, the stress builds back up to the average value by shear stress transfer at the fiber–matrix interface (Figure 3.11c). Additionally, the stress states in a region close to the broken ends contain 1. Stress concentrations at the void created by the broken fiber 2. High shear stress concentrations in the matrix near the fiber ends 3. An increase in the average normal stress in adjacent fibers (Figure 3.11b) (a) (b) (c) 1234 P P Fiber breakage FIGURE 3.11 Longitudinal stress distributions (a) in unidirectional continuous fibers before the failure of fiber 3, (b) in fibers 2 and 4 after the failure of fiber 3, and (c) in fiber 3 after it fails. 2007 by Taylor & Francis Group, LLC
Owing to these local stress magnifications,possibilities for several microfailure modes exist: 1.Partial or total debonding of the broken fiber from the surrounding matrix due to high interfacial shear stresses at its ends.As a result,the fiber effectiveness is reduced either completely or over a substantial length(Figure 3.12a). 2.Initiation of a microcrack in the matrix due to high stress concentration at the ends of the void (Figure 3.12b). P P Debonding at the Matrix fiber-matrix cracking interface (a) (b) Fiber breakage (c) FIGURE 3.12 Possible microfailure modes following the breakage of fiber 3. 2007 by Taylor Francis Group.LLC
Owing to these local stress magnifications, possibilities for several microfailure modes exist: 1. Partial or total debonding of the broken fiber from the surrounding matrix due to high interfacial shear stresses at its ends. As a result, the fiber effectiveness is reduced either completely or over a substantial length (Figure 3.12a). 2. Initiation of a microcrack in the matrix due to high stress concentration at the ends of the void (Figure 3.12b). P Fiber breakage (c) P 1234 P Matrix cracking P (b) 1234 P P (a) 12 3 4 Debonding at the fiber–matrix interface FIGURE 3.12 Possible microfailure modes following the breakage of fiber 3. 2007 by Taylor & Francis Group, LLC
3.Plastic deformation (microyielding)in the matrix,particularly if the matrix is ductile. 4.Failure of other fibers in the vicinity of the first fiber break due to high average normal stresses and the local stress concentrations (Figure 3.12c).Each fiber break creates additional stress concentrations in the matrix as well as in other fibers.Eventually,many of these fiber breaks and the surrounding matrix microcracks may join to form a long micro- crack in the lamina. The presence of longitudinal stress (ov)concentration at the tip of an advan- cing crack is well known.Cook and Gordon [5]have shown that the stress components oxx and Tx may also reach high values slightly ahead of the crack tip (Figure 3.13a).Depending on the fiber-matrix interfacial strength,these stress components are capable of debonding the fibers from the surrounding matrix even before they fail in tension(Figure 3.13b).Fiber-matrix debonding ahead of the crack tip has the effect of blunting the crack front and reducing the notch sensitivity of the material.High fiber strength and low interfacial strength promote debonding over fiber tensile failure. With increasing load,fibers continue to break randomly at various loca- tions in the lamina.Because of the statistical distribution of surface flaws,the Debonding ahead of the crack tip (a) (b) FIGURE 3.13 Schematic representation of (a)normal stress distributions and (b)fiber- matrix debonding ahead of a crack tip. 2007 by Taylor Francis Group,LLC
3. Plastic deformation (microyielding) in the matrix, particularly if the matrix is ductile. 4. Failure of other fibers in the vicinity of the first fiber break due to high average normal stresses and the local stress concentrations (Figure 3.12c). Each fiber break creates additional stress concentrations in the matrix as well as in other fibers. Eventually, many of these fiber breaks and the surrounding matrix microcracks may join to form a long microcrack in the lamina. The presence of longitudinal stress (syy) concentration at the tip of an advancing crack is well known. Cook and Gordon [5] have shown that the stress components sxx and txy may also reach high values slightly ahead of the crack tip (Figure 3.13a). Depending on the fiber–matrix interfacial strength, these stress components are capable of debonding the fibers from the surrounding matrix even before they fail in tension (Figure 3.13b). Fiber–matrix debonding ahead of the crack tip has the effect of blunting the crack front and reducing the notch sensitivity of the material. High fiber strength and low interfacial strength promote debonding over fiber tensile failure. With increasing load, fibers continue to break randomly at various locations in the lamina. Because of the statistical distribution of surface flaws, the Debonding ahead of the crack tip s syy sxx s x y (a) (b) FIGURE 3.13 Schematic representation of (a) normal stress distributions and (b) fiber– matrix debonding ahead of a crack tip. 2007 by Taylor & Francis Group, LLC
Crack surface b☑ Crack a surface FIGURE 3.14 Schematic representation of fiber pullout and matrix bridging by broken fibers (a)fiber breakage;(b)fiber pullout;and (c)matrix bridging. fiber failure does not always occur in the crack plane(Figure 3.14).Therefore, the opening of the matrix crack may cause broken fibers to pull out from the surrounding matrix(Figure 3.15),which is resisted by the friction at the fiber- matrix interface.If the interfacial strength is high or the broken fiber lengths are greater than le/2,the fiber pullout is preceded by either debonding or fiber failure even behind the crack front.Thus,broken fibers act as a bridge between the two faces of the matrix crack.In some instances,multiple parallel cracks are formed in the matrix normal to the fiber direction.If these cracks are bridged by fibers,the volume of matrix between the cracks may deform significantly before rupture. Fracture toughness of a unidirectional 0 lamina is the sum of the energies consumed by various microfailure processes,namely,fiber fracture,matrix cracking or yielding,debonding,and fiber pullout.Theoretical models to calculate the energy contributions from some of these failure modes are given in Table 3.2.Although the true nature of the fracture process and stress fields are not known,these models can serve to recognize the variables that play 2007 by Taylor&Francis Group.LLC
fiber failure does not always occur in the crack plane (Figure 3.14). Therefore, the opening of the matrix crack may cause broken fibers to pull out from the surrounding matrix (Figure 3.15), which is resisted by the friction at the fiber– matrix interface. If the interfacial strength is high or the broken fiber lengths are greater than lc=2, the fiber pullout is preceded by either debonding or fiber failure even behind the crack front. Thus, broken fibers act as a bridge between the two faces of the matrix crack. In some instances, multiple parallel cracks are formed in the matrix normal to the fiber direction. If these cracks are bridged by fibers, the volume of matrix between the cracks may deform significantly before rupture. Fracture toughness of a unidirectional 08 lamina is the sum of the energies consumed by various microfailure processes, namely, fiber fracture, matrix cracking or yielding, debonding, and fiber pullout. Theoretical models to calculate the energy contributions from some of these failure modes are given in Table 3.2. Although the true nature of the fracture process and stress fields are not known, these models can serve to recognize the variables that play a a c b cc a a a Crack surface Crack surface FIGURE 3.14 Schematic representation of fiber pullout and matrix bridging by broken fibers (a) fiber breakage; (b) fiber pullout; and (c) matrix bridging. 2007 by Taylor & Francis Group, LLC
0 FIGURE 3.15 Fracture surface of a randomly oriented discontinuous fiber composite showing the evidence of fiber pullout. major roles in the development of high fracture toughness for a fiber-reinforced composite lamina.It should be noted that energy contributions from the fracturing of brittle fibers and a brittle matrix are negligible (<10%)compared with those listed in Table 3.2. 3.1.2 TRANSVERSE TENSILE LOADING When a transverse tensile load is applied to the lamina,the fibers act as hard inclusions in the matrix instead of the principal load-carrying members. Although the matrix modulus is increased by the presence of fibers,local stresses and strains in the surrounding matrix are higher than the applied stress. Figure 3.16b shows the variation of radial stress (and tangential stress (ee) in a lamina containing a single cylindrical fiber.Near the fiber-matrix interface, the radial stress is tensile and is nearly 50%higher than the applied stress. Because of this radial stress component,cracks normal to the loading direction 2007 by Taylor Francis Group,LLC
major roles in the development of high fracture toughness for a fiber-reinforced composite lamina. It should be noted that energy contributions from the fracturing of brittle fibers and a brittle matrix are negligible (<10%) compared with those listed in Table 3.2. 3.1.2 TRANSVERSE TENSILE LOADING When a transverse tensile load is applied to the lamina, the fibers act as hard inclusions in the matrix instead of the principal load-carrying members. Although the matrix modulus is increased by the presence of fibers, local stresses and strains in the surrounding matrix are higher than the applied stress. Figure 3.16b shows the variation of radial stress (srr) and tangential stress (suu) in a lamina containing a single cylindrical fiber. Near the fiber–matrix interface, the radial stress is tensile and is nearly 50% higher than the applied stress. Because of this radial stress component, cracks normal to the loading direction FIGURE 3.15 Fracture surface of a randomly oriented discontinuous fiber composite showing the evidence of fiber pullout. 2007 by Taylor & Francis Group, LLC