M. Shioya, M. Nakatani/Composites Science and Technology 60(2000)219-229 alternately along the longitudinal direction of the com posite. By using a two-dimensional composite model, the axial compressive strength of the composite, oe, has been derived 1.0 Em v2(1+v)(1-v 0.8 for the shear mode and b06 yeE Oc 80.4 for the extension mode. In the above equations, Em, Gr and Vm are the tensile modulus, shear modulus and 0.2 Poisson ratio of the matrix and Er and v the tensile modulus and volume fraction of the fibres For the carbon fibre/epoxy resin systems used in this study, the critical stress which causes fracture in the Tensile modulus/GPa extension mode is higher than that in the shear mode Thus, the compressive strength of the composite strands Fig. Il. Compressive strength of composite strand versus tensile used in this study is given by Eq- (3). The dotted line in modulus of matrix for (O)XS and (o) Ta foe, relation of Eq(3) Fig. II shows the compressive strength of the composite using epoxy. A B and C matrices. Dotted line show strand predicted by Eq(3)for v=0.42, Vm=0.33 and strength of X5 and T4 fibre composite strands using b=tnF various values of Emm. In this figure, the compressive (5) different matrix resins are also shown. The experimental values are extremely lower than the predicted values. As where I is the moment of inertia of area, t the thickness qualitatively predicted by Rosen's model, however, the of the specimen measured in the bending direction, Fb compressive strength increases with increasing matrix the compression load at fracture and 8, the deflection at modulus for the T4 fibre composite strands. The matrix fracture. The deflection can be calculated from the axial modulus dependence of the compressive strength of the displacement, &a, using the equations, X5 fibre composite strands is much smaller than that of the T4 fibre composite strands δa E(P) This is because the compressive strength of the X5 ()」 fibre composite strand is almost limited by the fibre trength. That is, the reduced compressive strength of 8r p the X5 fibre at the lowest matrix modulus of 2. 4 GPa is L K(p) 0.49 GPa, which is close to the compressive strength of this fibre determined with the micro-compression test, where K(p) and E(p) are the complete elliptic integrals 0.51 GPa. Thus, the compressive strength of this com- of the first and the second kinds and L is the specimen posite strand cannot be increased even if the matrix length. If the composite strand shows a falling load modulus is increased above 2. 4 GPa. On the other hand, compression curve or the accurate determination of the the reduced compressive strength of the T4 fibre at the cross-section size is difficult, the bending strength can be matrix modulus of 2.4 GPa is only 0.59 GPa, which is practically estimated by using an effective cross-section lower than the compressive strength of this fibre deter- size of the composite strand at fracture, which can be mined with the micro-compression test, 2.0 GPa. Thus, obtained with the knowledge of the elastic modulus of the compressive strength of this composite strand can be the composite strand [ll increased by increasing the matrix modulus until the The bending strength of the composite strands using fibre strength is reached the matrix with a tensile modulus of 3.0 GPa and the reduced strength of the component fibres are shown in 3.5. Compressive strength determined with the axial Table 3. Among the composite strands shown in Table compression bending test of composite strand 3, only the H4 fibre composite strand finally fractured from compression. It is found that the reduced strength With the axial compression bending test of the com- is larger than the tensile or compressive strength of the posite strand, the bending strength, Ob, is obtained by fibre corresponding to the fracture mode of the comp he equationalternately along the longitudinal direction of the com￾posite. By using a two-dimensional composite model, the axial compressive strength of the composite, c, has been derived as c ˆ Gm 1 ÿ vf ˆ Em 2 1… † ‡ m 1 ÿ vf ÿ
…3† for the shear mode and c ˆ 4v3 fEfEm 3 1 ÿ vf ÿ
" #1=2 …4† for the extension mode. In the above equations, Em; Gm and m are the tensile modulus, shear modulus and Poisson ratio of the matrix and Ef and vf the tensile modulus and volume fraction of the ®bres. For the carbon ®bre/epoxy resin systems used in this study, the critical stress which causes fracture in the extension mode is higher than that in the shear mode. Thus, the compressive strength of the composite strands used in this study is given by Eq.(3). The dotted line in Fig. 11 shows the compressive strength of the composite strand predicted by Eq. (3) for vf ˆ 0:42, m ˆ 0:33 and various values of Em. In this ®gure, the compressive strength of X5 and T4 ®bre composite strands using di€erent matrix resins are also shown. The experimental values are extremely lower than the predicted values. As qualitatively predicted by Rosen's model, however, the compressive strength increases with increasing matrix modulus for the T4 ®bre composite strands. The matrix modulus dependence of the compressive strength of the X5 ®bre composite strands is much smaller than that of the T4 ®bre composite strands. This is because the compressive strength of the X5 ®bre composite strand is almost limited by the ®bre strength. That is, the reduced compressive strength of the X5 ®bre at the lowest matrix modulus of 2.4 GPa is 0.49 GPa, which is close to the compressive strength of this ®bre determined with the micro-compression test, 0.51 GPa. Thus, the compressive strength of this com￾posite strand cannot be increased even if the matrix modulus is increased above 2.4 GPa. On the other hand, the reduced compressive strength of the T4 ®bre at the matrix modulus of 2.4 GPa is only 0.59 GPa, which is lower than the compressive strength of this ®bre deter￾mined with the micro-compression test, 2.0 GPa. Thus, the compressive strength of this composite strand can be increased by increasing the matrix modulus until the ®bre strength is reached. 3.5. Compressive strength determined with the axial compression bending test of composite strand With the axial compression bending test of the com￾posite strand, the bending strength, b, is obtained by the equation, b ˆ ttFb 2I …5† where I is the moment of inertia of area, t the thickness of the specimen measured in the bending direction, Fb the compression load at fracture and t the de¯ection at fracture. The de¯ection can be calculated from the axial displacement, a, using the equations, a L ˆ 2 1 ÿ E p… † K p… †   …6† t L ˆ p K p… † …7† where K p… † and E p… † are the complete elliptic integrals of the ®rst and the second kinds and L is the specimen length. If the composite strand shows a falling load compression curve or the accurate determination of the cross-section size is dicult, the bending strength can be practically estimated by using an e€ective cross-section size of the composite strand at fracture, which can be obtained with the knowledge of the elastic modulus of the composite strand [11]. The bending strength of the composite strands using the matrix with a tensile modulus of 3.0 GPa and the reduced strength of the component ®bres are shown in Table 3. Among the composite strands shown in Table 3, only the H4 ®bre composite strand ®nally fractured from compression. It is found that the reduced strength is larger than the tensile or compressive strength of the ®bre corresponding to the fracture mode of the compo￾site strand. Fig. 11. Compressive strength of composite strand versus tensile modulus of matrix for (*) X5 and (*) T4 ®bre composite strands using epoxy-A, B and C matrices. Dotted line shows relation of Eq. (3) for vf ˆ 0:42 and m ˆ 0:33. M. Shioya, M. Nakatani / Composites Science and Technology 60 (2000) 219±229 227