Fibre fracture strain

Note that as shown below, this must be the matrix stress at the fibre fracture strain. 

As shown in Fig. 3.4 stress-strain tests on uniaxially aligned fibre composites show that their behaviour lies somewhere between that of the fibres and that of the matrix. In regard to the strength of the composite, Ocu, the rule of mixtures has to be modified to relate to the matrix stress, o at the fracture strain of the fibres rather than the ultimate tensile strength, o u for the matrix. 

This is because, with brittle fibres, failure of the composite will occur when the fibres reach their fracture strain. At this point the matrix is subjected to the full applied load, which it is unable to sustain. 

Fig. 15 Initiation of fibre fracture by a crack oriented parallel to the chain direction in a domain. It is proposed that a circular crack with a radius q releases the strain energy in a sphere around the crack with the same radius. Note that in this two-dimensional drawing only the circular crack is shown in perspective...

Impurities and flaws have a detrimental effect on the fibre strength. Due to shear stress concentrations at structural irregularities and impurities, the ultimate debonding stress r0 ( rm) or the critical fracture strain / may be exceeded locally far sooner than in perfectly ordered domains. Thus, during the fracture process of real fibres the debonding from neighbouring chains occurs preferably in the most disoriented domains and presumably near impurities. At the same time, however, the chains in the rest of the fibre are kept under strain but remain bonded together up to fracture. 

So far we have employed in this discussion a critical shear stress as a criterion for fibre fracture. In Sect. 4 it will be shown that a critical shear strain or a maximum rotation of the chain axis is a more appropriate criterion when the time dependence of the strength is considered. 

The strength of a fibre is not only a function of the test length, but also of the testing time and the temperature. It is shown that the introduction of a fracture criterion, which states that the total shear deformation in a creep experiment is bounded to a maximum value, explains the well-known Coleman relation as well as the relation between creep fracture stress and creep fracture strain. Moreover, it explains why highly oriented fibres have a longer lifetime than less oriented fibres of the same polymer, assuming that all other parameters stay the same. 

Some modes may dominate for example, for large bending strains in a flexible structure, fibre fracture in tension and fibre kinking in compression wdl dominate near both surfaces. Matrix cracks can cause delamination when they reach a ply interface. If the structure is stiff enough to resist with a significant force, then local indentation damage, and shear-driven delamination in the interior, wdl occur. Figure 9.2 shows schematically the different modes of fadure in three zones of a laminate. The peanut shape deformations (3) have this shape because the compression under the impact force suppresses the delaminations. 

Based on the assumption that the deformation governs the fracture and realising an analysis of the deformations to fracture of the two components, the matrix and fibre, three distinct cases are found. Figure 3.452. The most frequent case corresponds to the situation when the matrix fracture deformation is lower that than of fibre in this case, a multiple fracture of matrix occurs before of the sudden fibres fracture. The stress-strain curve presents a discontinuity, at the deformation level, in the moment of matrix failure. Figure 3.452a. The final fracture takes place at / > / ... 

If the failure strain in the matrix is larger than in the fibre, the fibres fracture before the matrix fails. This is frequently the case in composites with metallic or polymeric matrix. Figure 9.3 shows the resulting stress-strain diagram. It is assumed that the matrix yields plastically before the fibre breaks. The material deforms elastically until the matrix yields. On further increasing the strain, the strengthening fibres fracture, and the stress-strain curve drops to a small stress value that lies below that of the pure matrix material because of the reduced volume. Eventually, failure by fracturing of the matrix occurs. The fracture strain is smaller than in a pure matrix material. This is due... 

In composites with a brittle matrix i. e., mainly in ceramic matrix composites, the aim is not to increase the strength, but the fracture toughness. The fracture strain of the matrix is usually smaller than that of the fibre, leading to crack propagation in the matrix when the load increases. 

We assume that the fracture strain of the matrix is larger than that of the fibre. 

Table 9.1. Density and mechanical parameters (Young s modulus, tensile strength, fracture strain) of some important fibre materials [29,41,100,117,131,141]...

A further important point is that the fracture strain of high-strength carbon fibres is about 2% although they deform only elastically. Considering that the strains in the polymer matrix locally exceeds that of the fibre (see figure 9.4), we see that the fracture strain in the matrix has to be rather large. To avoid crack formation in the matrix, its fracture strain should be about twice that of the fibre i. e., 4% to 5%. Currently available duromers do not... 

It was already stressed that the properties of fibre and matrix have to be carefully adjusted to obtain optimal properties of the component under mechanical loads. Under tensile loads, the fracture strain of the matrix has to be... 

Thus, the fracture strain of the fibre is reached first. The failure stress can be estimated using equation (9.7),... 

Fig. 10 Variation of fibre strain with time for a polyDCHD fibre held at a constant stress corresponding to 50% of Its fracture strain (given by dashed line).

The shape of the resulting stress-strain curve is presented in Figure 4.2. The composite strength value of <7fu 14. and Eq. 4.69 are based on the assumption that failure by fibre fracture will precede failure by fibre pull-out. This is less likely to occur in short fibre-reinforced composites. 

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 ... 

Both types of fibres, PE and PVA, could provide strain hardening at modest fibre contents of 2-3% by volume, but the PE had some advantage in its ability to provide a greater extent of multiple cracking. This difference could be correlated with the mode of failure, in which the PVA fibre composite exhibited fibre fracture while the PE composite was characterized by a greater extent of fibre pull-out. The fraction of fibre fracture was 70% for the PVA and 35% for the PE [82]. 

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