Model composites were made of carbon fibers and epoxy resin. A polyacrylonitrile based carbon fiber (T800 S-24 K-10 E, Toray Industries) was used with the mechanical properties shown in Table 2 [25] . A carbon fiber was taken from a 24,000 tow supplied by the manufacturer. An epoxy resin (105/206, West system) was selected that cured at room temperature.

A unidirectional CFRP laminate was also fabricated using a unidirectional prepreg tape (T800SC/#2592, Toray Industries). The fiber volume fraction of the laminate was approximately 65%. The compression test was performed in accordance with ASTM D695 using an end-loading test fixture [26] . Average applied compressive strain and compressive strength were 0.92% ± 0.025% and 1445 MPa ± 55.9 MPa, respectively.

The compression tests of the model composites were performed by means of a four-point bending test. The

schematic of the four-point bending test is shown in Figure 5. Since the carbon fibers were embedded in an epoxy matrix on the epoxy beam, the four-point bending test of the epoxy beam was performed so as to develop a compressive strain in the carbon fibers. A loading span of 100 mm with a supporting span of 20 mm was used for the single-fiber model composite, as shown in Figure 5(a). A loading span of 130 mm and with a supporting span of 30 mm was used for the other model composites, as shown in Figure 5(b).

The electrical resistance of a carbon fiber was measured by means of a four-probe method during the compression

test to detect the onset of compressive failure. Electrical resistance of a carbon fiber R is expressed as:

where is specific electrical resistance, L is length, and A is cross-sectional area.

Electrical resistance of a carbon fiber decreases with an increase in compressive load due to the shortening of the fiber. If a carbon fiber fractures, electrical resistance increases. Electrical resistance of a carbon fiber was only measured in compression test of a single-fiber model composite to precisely detect the onset of fiber compressive failure and measure compressive failure strain. Electrical resistance of the carbon fiber was measured using a LCR HiTester, Model 3522 from Hioki.

The compression test of a single-fiber model composite was performed by means of a four-point bending test. Figure 6 shows an example of the electrical resistance change ratio due to compressive loading. The ordinate shows the electrical resistance change ratio. The abscissa shows the compressive strain of the carbon fiber measured using the strain gauge. The electrical resistance of the carbon fiber decreased as the compressive strain increased. Electrical resistance then increased due to the onset of compressive failure. The compressive failure strain was approximately 2.5% which included the pre-strain of approximately 0.17%. This strain value was much higher than that of the unidirectional CFRP fabricated using a unidirectional prepreg tape (see Section 2.1). This indicates that the compressive failure mechanism of a unidirectional CFRP is different from that of a single-fiber model composite. The difference between the single-fiber model composite and a unidirectional CFRP is explained by the difference in the number of carbon fibers as discussed in the following sections.

Compressive failure of a carbon fiber once the compressive load was completely removed is shown in Figure 7. The compressive loading was maintained after the increase in electrical resistance was observed to ensure the fiber fracture fully developed. Once the fracture, or crush, was fully developed, the load was removed. The fiber fractured in shear at an angle β, which was approximately 40˚. The angle was different from the kink-band angle for a unidirectional CFRP (see Table 1). Additionally, it has been reported that carbon fiber fails by buckling when the stiffness of the epoxy matrix is low [23] . However, the epoxy matrix was sufficiently stiff to prevent buckling of a carbon fiber.

The compression test of a two-fiber model composite was performed by means of a four point bending test. The compressive failure of the carbon fibers after the compression test is shown in Figure 8. The figure shows the fiber fracture after the compressive load was removed. The fracture point of the two carbon fibers was observed at almost same location. This would indicate mutual interaction between adjacent fibers during compressive failure when the distance between each fiber was less than 15 μm. From this result, multiple-fiber model composites were made with a distance between each fibers of less than 15 μm.

The compression test of a multiple-fiber model composite was performed by means of a four-point bending test. The distances between each fiber were less than 15 μm based on the results discussed in Section 3.2 where the fibers interacted mutually during compressive failure. The number of carbon fibers was increased from 3 to 11,000 to investigate the effect on the compressive failure. For the purposes of this research, the term “fiber volume fraction” is not used to describe model composites since the fraction is almost zero because the fiber embedded area is much smaller than the total cross-sectional area of the specimen as shown in Figure 9(a). Since the local fiber volume fraction in the fiber embedded area should be constant, as schematically shown in Figure 9(b); the number of carbon fibers was used as a parameter for discussing fracture modes.

The compressive failure of carbon fibers after the compression test is shown in Figure 10, which was taken when the compressive load was completely removed after fiber fracture. Fiber crush was observed when the number of carbon fibers was small as shown in Figure 10(a) to Figure 10(d). This is similar to compressive failure of a single-fiber model composite. Fractures propagated at an angle β, which was dependent of the number of aligned carbon fibers. The fracture angle β decreased slightly as the number of carbon fibers increased. By contrast, kink-band failure was observed when the number of carbon fiber was large as shown in Figure 10(e) to Figure 10(g). The kink-band angle (also expressed using β) decreased as the number of carbon fibers increased. The kink-band width ω increased as the number of carbon fibers increased. Kink-band parameters, i.e., kink- band angle and kink-band width were dependent on the number of adjacent carbon fibers.

The results are summarized in Figure 11. In the case of a unidirectional CFRP with approximately 60% fiber volume fraction, the kink-band angle was 5˚ - 30˚ and the kink-band width was 60 - 150 μm [3] [7] [11] [15] [17] . The kink-band parameters for the 11,000 fibers model composite were approaching that of the unidirectional CFRP. The number of carbon fibers was an influencing factor on compressive failure of a unidirectional CFRP when fibers were closely aligned within the mutually interacting distance.