Critical fibre volume

In order that the material may follow the curve in Fig. 6.23b, it is necessaiy for the critical fibre volume to be exceeded. The critical fibre volume is defined as the volume of fibres which, after matrix cracking will cany the load which the composite sustained before cracking. In such a way, a brittle material in which one cracking surface could be formed with low energy, transforms into the pseudo-plastic body, which can cany momentaiy the overioad with no visible damage. Therefore, the... 

The fibre volume fraction for which the strength ceases to decrease and begins to increase is called the critical fibre volume fraction, Below this value, the behaviour of the composite is only governed by the matrix ... 

For a given matrix, the critical fibre volume fraction decreases with the increasing strength of the fibres. This means that for fibres which are much stiffer than the matrix, is very low. 

Critical fibre volumes calculated from Eqs 4.29 and 4.30 are in the range of 0.3% to 0.8% for steel, glass and polypropylene reinforced cements [9]. Flowever, Eqs 4.29 and 4.30 were derived for continuous and aligned reinforcement. If the... 

Equations 4.30,4.34 and 4.35 can then be used to calculate the approximate critical fibre volumes by equating acu with... 

Equations 4.36-4.38 indicate that for short fibres, the critical fibre volume is a function of both the aspect ratio of the fibre and the fibre-matrix bond. Measured values of Tfu are in the range of 1-10 MPa. Figure 4.11 demonstrates the influence of bond (Figure 4.11 (a)) and orientation (Figure 4.11(b)). For typical aspect ratios of 50-100, I4(crit) is in the range of 1-3%. This is the same order of magnitude as the value derived from Eq. 4.30. 

Figure 4.11 Plots of calculated critical fibre volume, Vf(crit) vs. aspect ratio l/d for short fibres (a) composites with different fibre-matrix bond strength tfu, with Omu = 3 MPa and random 2D fibre array and (b) composites of different fibre orientations, with (Xmi = 5 MPa and tifu = 3 MPa.

By introducing the appropriate terms of the equations for i 2 and Ei, the critical fibre volume, Ff(crit) is ... 

For typical values of conventional and high performance systems (Table 4.1), Tjiptobroto and Hansen [68] calculated the values for 1-2 and 2, and determined the critical fibre volume, to be 3.3% and 15% for high performance and conventional composites, respectively. 

Substituting Eqs 12.1 and 12.2 into 12.3, and solving it for critical fibre volume for tensile strain hardening, yields the following relation ... 

Figure 12.3 Relations between critical fibre volume to achieve deflection hardening and fibre characteristics (bond and aspect ratio), based on Eq. 12.8 (after Naaman [I]).

Figure 12.3 provides a graphical presentation of this equation for characteristic composite values. It shows, as expected, that the critical fibre volume required for deflection hardening is smaller than for strain hardening (compare curves in Figures 12.3 and 12.2). 

The conditions set forth in Eqs 12.4 and 12.8 to obtain high performance FRC imply the need for the design of the fibres to achieve this purpose (fibre content, bond and aspect ratio). Li etaL [3] developed a complementary approach based on fracture mechanics in which the design of the matrix is also taken into consideration as one of the means to obtain cost-effective (e.g. low fibre content) strain hardening composites (for additional details see Section 4.4.3). The critical fibre volume, (I4)crit can be calculated in terms of the fibre properties and the composite fracture mechanics toughness and characteristic crack opening ... 

The dependency of critical fibre volume on fibre-matrix bond and the composite crack-tip toughness is demonstrated in Figure 12.4, for ultra-high density polyethylene (Spectra) fibre, with modulus of 117 GPa and 38 fim diameter. It clearly shows that to maintain a fibre content below 2% there is a need for a bond strength of about 1 MPa. I n order to accommodate fibres with bond of about 0.5 MPA, which is typical to many polymeric fibres, a fracture toughness below... 

The critical fibre volume according to this model is ... 

The input parameters for the normal and high density matrix composites are provided in Table 12.1. The calculated critical fibre volume for the dense matrix (DSP) is 3.8%, whereas for the conventional matrix composite it is 15%. This is the range of the reinforcement in dense FRC systems such as DSP and RPC, which provide strain and deflection hardening (see Section 12.3). 

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