《纺织复合材料》课程参考文献（Fiber-Reinforced Composites，THIRD EDITION）03 Mechanics

where 1-v[1-(Em/E)] K,=1-4w/mPT1-(Ea/E】 Equation 3.27 assumes that the transverse tensile strength of the composite is limited by the ultimate tensile strength of the matrix.Note that Ko represents the maximum stress concentration in the matrix in which fibers are arranged in a square array.The transverse tensile strength values predicted by Equation 3.27 are found to be in reasonable agreement with those predicted by the finite difference method for fiber volume fractions <60%[2].Equation 3.27 predicts that for a given matrix,the transverse tensile strength decreases with increasing fiber modulus as well as increasing fiber volume fraction. 3.1.3 LONGITUDINAL COMPRESSIVE LOADING An important function of the matrix in a fiber-reinforced composite material is to provide lateral support and stability for fibers under longitudinal compres- sive loading.In polymer matrix composites in which the matrix modulus is relatively low compared with the fiber modulus,failure in longitudinal com- pression is often initiated by localized buckling of fibers.Depending on whether the matrix behaves in an elastic manner or shows plastic deformation,two different localized buckling modes are observed:elastic microbuckling and fiber kinking. Rosen [12]considered two possible elastic microbuckling modes of fibers in an elastic matrix as demonstrated in Figure 3.20.The extensional mode of (a) (b) FIGURE 3.20 Fiber microbuckling modes in a unidirectional continuous fiber compos- ite under longitudinal compressive loading:(a)extensional mode and (b)shear mode. 2007 by Taylor Francis Group,LLC

where Ks ¼ 1 vf[1 (Em=Ef)] 1 (4vf=p) 1=2[1 (Em=Ef)] : Equation 3.27 assumes that the transverse tensile strength of the composite is limited by the ultimate tensile strength of the matrix. Note that Ks represents the maximum stress concentration in the matrix in which fibers are arranged in a square array. The transverse tensile strength values predicted by Equation 3.27 are found to be in reasonable agreement with those predicted by the finite difference method for fiber volume fractions <60% [2]. Equation 3.27 predicts that for a given matrix, the transverse tensile strength decreases with increasing fiber modulus as well as increasing fiber volume fraction. 3.1.3 LONGITUDINAL COMPRESSIVE LOADING An important function of the matrix in a fiber-reinforced composite material is to provide lateral support and stability for fibers under longitudinal compres￾sive loading. In polymer matrix composites in which the matrix modulus is relatively low compared with the fiber modulus, failure in longitudinal com￾pression is often initiated by localized buckling of fibers. Depending on whether the matrix behaves in an elastic manner or shows plastic deformation, two different localized buckling modes are observed: elastic microbuckling and fiber kinking. Rosen [12] considered two possible elastic microbuckling modes of fibers in an elastic matrix as demonstrated in Figure 3.20. The extensional mode of (a) (b) FIGURE 3.20 Fiber microbuckling modes in a unidirectional continuous fiber compos￾ite under longitudinal compressive loading: (a) extensional mode and (b) shear mode.
2007 by Taylor & Francis Group, LLC.

microbuckling occurs at low fiber volume fractions (ve<0.2)and creates an extensional strain in the matrix because of out-of-phase buckling of fibers.The shear mode of microbuckling occurs at high fiber volume fractions and creates a shear strain in the matrix because of in-phase buckling of fibers.Using buckling theory for columns in an elastic foundation,Rosen [12]predicted the compressive strengths in extensional mode and shear mode as Extensional mode:OLeu 2vf VrEmEr12 31-v) (3.28a) Shear mode:oLcu= Gm (3.28b) (1-v)i where Gm is the matrix shear modulus Vr is the fiber volume fraction Since most fiber-reinforced composites contain fiber volume fraction >30%,the shear mode is more important than the extensional mode.As Equation 3.28b shows,the shear mode is controlled by the matrix shear modulus as well as fiber volume fraction.The measured longitudinal compres- sive strengths are generally found to be lower than the theoretical values calculated from Equation 3.28b.Some experimental data suggest that the longitudinal compressive strength follows a rule of mixtures prediction similar to Equation 3.9. The second important failure mode in longitudinal compressive loading is fiber kinking,which occurs in highly localized areas in which the fibers are initially slightly misaligned from the direction of the compressive loading.Fiber bundles in these areas rotate or tilt by an additional angle from their initial configuration to form kink bands and the surrounding matrix undergoes large shearing deformation(Figure 3.21).Experiments conducted on glass and car- bon fiber-reinforced composites show the presence of fiber breakage at the ends of kink bands [13];however,whether fiber breakage precedes or follows the kink band formation has not been experimentally verified.Assuming an elastic- perfectly plastic shear stress-shear strain relationship for the matrix,Budiansky and Fleck [14]have determined the stress at which kinking is initiated as Tmy Ock= (3.29) o+Ymy where Tmy=shear yield strength of the matrix Ymy=shear yield strain of the matrix =initial angle of fiber misalignment 2007 by Taylor Francis Group.LLC

microbuckling occurs at low fiber volume fractions (vf < 0.2) and creates an extensional strain in the matrix because of out-of-phase buckling of fibers. The shear mode of microbuckling occurs at high fiber volume fractions and creates a shear strain in the matrix because of in-phase buckling of fibers. Using buckling theory for columns in an elastic foundation, Rosen [12] predicted the compressive strengths in extensional mode and shear mode as Extensional mode: sLcu ¼ 2vf vfEmEf 3(1 vf)  1=2 , (3:28a) Shear mode: sLcu ¼ Gm (1 vf) , (3:28b) where Gm is the matrix shear modulus vf is the fiber volume fraction Since most fiber-reinforced composites contain fiber volume fraction >30%, the shear mode is more important than the extensional mode. As Equation 3.28b shows, the shear mode is controlled by the matrix shear modulus as well as fiber volume fraction. The measured longitudinal compres￾sive strengths are generally found to be lower than the theoretical values calculated from Equation 3.28b. Some experimental data suggest that the longitudinal compressive strength follows a rule of mixtures prediction similar to Equation 3.9. The second important failure mode in longitudinal compressive loading is fiber kinking, which occurs in highly localized areas in which the fibers are initially slightly misaligned from the direction of the compressive loading. Fiber bundles in these areas rotate or tilt by an additional angle from their initial configuration to form kink bands and the surrounding matrix undergoes large shearing deformation (Figure 3.21). Experiments conducted on glass and car￾bon fiber-reinforced composites show the presence of fiber breakage at the ends of kink bands [13]; however, whether fiber breakage precedes or follows the kink band formation has not been experimentally verified. Assuming an elastic￾perfectly plastic shear stress–shear strain relationship for the matrix, Budiansky and Fleck [14] have determined the stress at which kinking is initiated as sck ¼ tmy w þ gmy , (3:29) where tmy ¼ shear yield strength of the matrix gmy ¼ shear yield strain of the matrix w ¼ initial angle of fiber misalignment
2007 by Taylor & Francis Group, LLC.

FIGURE 3.21 Kink band geometry.a=Kink band angle,B=Fiber tilt angle,and @=Kink band width. Besides fiber microbuckling and fiber kinking,a number of other failure modes have also been observed in longitudinal compressive loading of unidirectional continuous fiber-reinforced composites.They include shear failure of the com- posite,compressive failure or yielding of the reinforcement,longitudinal split- ting in the matrix due to Poisson's ratio effect,matrix yielding,interfacial debonding,and fiber splitting or fibrillation(in Kevlar 49 composites).Factors that appear to improve the longitudinal compressive strength of unidirectional composites are increasing values of the matrix shear modulus,fiber tensile modulus,fiber diameter,matrix ultimate strain,and fiber-matrix interfacial strength.Fiber misalignment or bowing,on the other hand,tends to reduce the longitudinal compressive strength. 3.1.4 TRANSVERSE COMPRESSIVE LOADING In transverse compressive loading,the compressive load is applied normal to the fiber direction,and the most common failure mode observed is the matrix shear failure along planes that are parallel to the fiber direction,but inclined to the loading direction (Figure 3.22).The failure is initiated by fiber-matrix debonding.The transverse compressive modulus and strength are considerably lower than the longitudinal compressive modulus and strength.The transverse compressive modulus is higher than the matrix modulus and is close to the transverse tensile modulus.The transverse compressive strength is found to be nearly independent of fiber volume fraction [15]. 2007 by Taylor Francis Group,LLC

Besides fiber microbuckling and fiber kinking, a number of other failure modes have also been observed in longitudinal compressive loading of unidirectional continuous fiber-reinforced composites. They include shear failure of the com￾posite, compressive failure or yielding of the reinforcement, longitudinal split￾ting in the matrix due to Poisson’s ratio effect, matrix yielding, interfacial debonding, and fiber splitting or fibrillation (in Kevlar 49 composites). Factors that appear to improve the longitudinal compressive strength of unidirectional composites are increasing values of the matrix shear modulus, fiber tensile modulus, fiber diameter, matrix ultimate strain, and fiber–matrix interfacial strength. Fiber misalignment or bowing, on the other hand, tends to reduce the longitudinal compressive strength. 3.1.4 TRANSVERSE COMPRESSIVE LOADING In transverse compressive loading, the compressive load is applied normal to the fiber direction, and the most common failure mode observed is the matrix shear failure along planes that are parallel to the fiber direction, but inclined to the loading direction (Figure 3.22). The failure is initiated by fiber–matrix debonding. The transverse compressive modulus and strength are considerably lower than the longitudinal compressive modulus and strength. The transverse compressive modulus is higher than the matrix modulus and is close to the transverse tensile modulus. The transverse compressive strength is found to be nearly independent of fiber volume fraction [15]. b a w FIGURE 3.21 Kink band geometry. a ¼ Kink band angle, b ¼ Fiber tilt angle, and v ¼ Kink band width.
2007 by Taylor & Francis Group, LLC

Fiber direction Fiber direction (a) () FIGURE 3.22 Shear failure (a)in longitudinal compression(compressive load parallel to the fiber direction)and (b)in transverse compression (compressive load normal to the fiber direction). 3.2 CHARACTERISTICS OF A FIBER-REINFORCED LAMINA 3.2.1 FUNDAMENTALS 3.2.1.1 Coordinate Axes Consider a thin lamina in which fibers are positioned parallel to each other in a matrix,as shown in Figure 3.23.To describe its elastic properties,we first define two right-handed coordinate systems,namely,the 1-2-z system and the 2￥ FIGURE 3.23 Definition of principal material axes and loading axes for a lamina. 2007 by Taylor&Francis Group.LLC

3.2 CHARACTERISTICS OF A FIBER-REINFORCED LAMINA 3.2.1 FUNDAMENTALS 3.2.1.1 Coordinate Axes Consider a thin lamina in which fibers are positioned parallel to each other in a matrix, as shown in Figure 3.23. To describe its elastic properties, we first define two right-handed coordinate systems, namely, the 1-2-z system and the Fiber direction (a) Fiber direction (b) FIGURE 3.22 Shear failure (a) in longitudinal compression (compressive load parallel to the fiber direction) and (b) in transverse compression (compressive load normal to the fiber direction). z y 1 x q 2 FIGURE 3.23 Definition of principal material axes and loading axes for a lamina.
2007 by Taylor & Francis Group, LLC

(a) (b) FIGURE 3.24 Right-handed coordinate systems.Note the difference in fiber orientation in (a)and (b). x-y-z system.Both 1-2 and x-y axes are in the plane of the lamina,and the z axis is normal to this plane.In the 1-2-z system,axis 1 is along the fiber length and represents the longitudinal direction of the lamina,and axis 2 is normal to the fiber length and represents the transverse direction of the lamina.Together they constitute the principal material directions in the plane of the lamina.In the xyz system,x and y axes represent the loading directions. The angle between the positive x axis and the 1-axis is called the fiber orientation angle and is represented by 6.The sign of this angle depends on the right-handed coordinate system selected.If the z axis is vertically upward to the lamina plane,0 is positive when measured counterclockwise from the positive x axis(Figure 3.24a).On the other hand,if the z axis is vertically downward,6 is positive when measured clockwise from the positive x axis(Figure 3.24b).In a 0 lamina,the principal material axis I coincides with the loading axis x,but in a 90 lamina,the principal material axis I is at a 90 angle with the loading axis x. 3.2.1.2 Notations Fiber and matrix properties are denoted by subscripts f and m,respectively. Lamina properties,such as tensile modulus,Poisson's ratio,and shear modu- lus,are denoted by two subscripts.The first subscript represents the loading direction,and the second subscript represents the direction in which the par- ticular property is measured.For example,vi2 represents the ratio of strain in direction 2 to the applied strain in direction 1,and v2 represents the ratio of strain in direction I to the applied strain in direction 2. Stresses and strains are also denoted with double subscripts(Figure 3.25). The first of these subscripts represents the direction of the outward normal to the plane in which the stress component acts.The second subscript represents 2007 by Taylor Francis Group,LLC

x-y-z system. Both 1-2 and x-y axes are in the plane of the lamina, and the z axis is normal to this plane. In the 1-2-z system, axis 1 is along the fiber length and represents the longitudinal direction of the lamina, and axis 2 is normal to the fiber length and represents the transverse direction of the lamina. Together they constitute the principal material directions in the plane of the lamina. In the xyz system, x and y axes represent the loading directions. The angle between the positive x axis and the 1-axis is called the fiber orientation angle and is represented by u. The sign of this angle depends on the right-handed coordinate system selected. If the z axis is vertically upward to the lamina plane, u is positive when measured counterclockwise from the positive x axis (Figure 3.24a). On the other hand, if the z axis is vertically downward, u is positive when measured clockwise from the positive x axis (Figure 3.24b). In a 08 lamina, the principal material axis 1 coincides with the loading axis x, but in a 908 lamina, the principal material axis 1 is at a 908 angle with the loading axis x. 3.2.1.2 Notations Fiber and matrix properties are denoted by subscripts f and m, respectively. Lamina properties, such as tensile modulus, Poisson’s ratio, and shear modu￾lus, are denoted by two subscripts. The first subscript represents the loading direction, and the second subscript represents the direction in which the par￾ticular property is measured. For example, n12 represents the ratio of strain in direction 2 to the applied strain in direction 1, and n21 represents the ratio of strain in direction 1 to the applied strain in direction 2. Stresses and strains are also denoted with double subscripts (Figure 3.25). The first of these subscripts represents the direction of the outward normal to the plane in which the stress component acts. The second subscript represents (a) (b) z y x x 1 z y +q +q 1 FIGURE 3.24 Right-handed coordinate systems. Note the difference in fiber orientation in (a) and (b).
2007 by Taylor & Francis Group, LLC