Fibre interfacial shear stress

In the macrocomposite model it is assumed that the load transfer between the rod and the matrix is brought about by shear stresses in the matrix-fibre interface [35]. When the interfacial shear stress exceeds a critical value r0, the rod debonds from the matrix and the composite fails under tension. The important parameters in this model are the aspect ratio of the rod, the ratio between the shear modulus of the matrix and the tensile modulus of the rod, the volume fraction of rods, and the critical shear stress. As the chains are assumed to have an infinite tensile strength, the tensile fracture of the fibres is not caused by the breaking of the chains, but only by exceeding a critical shear stress. Furthermore, it should be realised that the theory is approximate, because the stress transfer across the chain ends and the stress concentrations are neglected. These effects will be unimportant for an aspect ratio of the rod Lld> 10 [35]. 

Figure 4.30 Stress transfer to a single fibre in a thermoset matrix, under tension in the x direction. The graph shows the variation of the fibre tensile stress and the interfacial shear stress, when the interface yields at both ends of the fibre.

Reynaud. P., Rouby, D. and Fantozzi, G., Effects of temperature and of oxidation on the interfacial shear stress between fibres and matrix in ceramic-matrix composites , Acta mater., vol 46, No. 7, pp 2461-2469, 1998... 

Distributions along a short fibre of tensile stress tr, and interfacial shear stress T. as predicted by eqns 6.33 and 6.35. 

The interfacial shear stress t is concentrated at the fibre ends (see Figure 6.23). With increase in strain, these are the sites where the interfrice first fails and debonding of fibres and matrix or shear failure of the matrix begins. This occurs when T here reaches the interfiicial shear strength. The magnitude of this strength is determined by three factors ... 

With further straining, the debonding regions of the fibre slip within their hole in the matrix. Slippage is resisted by a constant friction stress T, and the stress distributions switch from those shown in Figure 6.23 to the form shown in Figure 6.26. Since the fibre ends now carry a constant interfacial shear stress, the tensile stress here has a constant gradient (eqn 6.32) stress buUds up linear from each end. 

Schematic sketch of distributions of fibre tensile stress aiKl interfacial shear stress after fibre-matrix debonding has commenced. 

The fibre aspect ratio, which is its length to diameter ratio is a critical parameter in a composite. A relationship has been proposed by Cox to relate the critical fibre aspect ratio, IJd, to the interfacial shear stress, Zy, viz. ... 

Equation 1 assumes that the shear stress at the interface is constant as a result of complete interfacial debonding. With good adhesion, only partial debonding or other micro-mechanical events such as transverse matrix cracking are observed, which invalidate the assumption of a constant interfacial shear stress. As a result, alternative data reduction techniques have been developed. For example, Tripathi and Jones developed the cumulative stress-transfer function, which deals with the limitations given above. This has been further refined by Lopattananon et al into the stress-transfer efficiency from which an ineffective length of that fibre in that resin can be determined. In this model, the matrix properties and frictional adhesion at debonds can be included in the analysis. It is also possible to use the three-phase stress-transfer model of Wu et al to include the properties of an interphase. 

Many of the uncertainties associated with the above tests arise from the model used to calculate the stress or strain transferred between the matrix and fibre. (Laser Raman or Fluorescence Spectroscopy) LRS is a technique for determining directly the strain in the fibre in analogy to an embedded strain gauge. The strain is probed by recording a Raman (or fluorescence) spectrum of the reinforcing fibre at increments of applied strain. The shift in the frequency at which a stress-sensitive peak occurs can be used to estimate the strain profile in fhe reinforcing fibre. The rate of strain development in the fibre can be used to assess the quality of the interface. Quantification still requires an appropriate stress-transfer model although a simple force-balance calculation can be used to estimate the interfacial shear stress. 

Figure 8.3 Interfacial shear stress (a) and fibre tensile stress (b) as a function of fibre length (schematic)...

Average interfacial shear stress has no physical meaning and is used only to compare the pull-out behaviour of fibres of different dimensions. The question of whether the stress is in fact uniformly distributed cannot be answered by a simple pull-out test only, because the measurements of force and displacement are executed outside the embedment zone. 

The load transfer between fibre and matrix is mainly due to friction caused by interface roughness or to adhesion between fibre and matrix on the lateral surface of the fibre. Although the strains at the front and back side of the fibre can be large, only small loads are transferred there because these sides are much smaller. Crucial in determining the properties of the composite is the maximum interfacial shear stress t. ... 

To estimate the stress (7f within the fibre, we consider an infinitesimal fibre segment with a constant interfacial shear stress —t acting on its surface (figure 9.5). The forces within the fibre are —erf 7rd /4 at position x and... 

Fig. 9.6. Stress distribution in a fibre for two different values of the interfacial shear stress...

The stress distribution in the fibre is plotted in figure 9.6 for a constant interfacial shear stress T[. It has to be noted that in a sufficiently long fibre, there is no interfacial shear stress midway in the fibre. 

The crack resistance is the higher, the larger the critical fibre length is. In a ceramic matrix composite, it is thus useful to have a low value of the interfacial shear stress. If fibres are shorter than the critical length, they will not break but will be pulled out on one side of the crack, if they are longer, they will fracture first and be pulled out afterwards. 

If the load on the composite is increased, the weakest fibre will break and will thus not transfer any tensile stresses at the position of failure. This fracture, however, will not unload the whole fibre. If it is much longer that the critical length, the load will be transferred by interfacial shear stresses from the matrix to both fibre fragments. At some distance from this region, both fibre fragments bear the same load as before. Near to the fracture position, the material is weakened and the load is transferred to the surrounding material. 

The maximum interfacial shear stress in polymer matrix composites is determined by the adhesion between fibre and matrix, not by the yield strength of the matrix. 

It is possible to derive the variation of interfacial shear stress r with distance X along the fibre from the data in Figures 8.14(a) and 8.14(b) using the relationship [88] ... 

Figure 8.16 Dependence of the maximum interfacial shear stress upon matrix strain for the sized and de-sized Kevlar 49 fibres in an epoxy resin matrix (data taken from Figure 8.1S). The horizontal dashed line represents the shear yield stress of the epoxy resin, along with the scatter band of the measurements (after [38])...

At these ageing conditions, a lower interfacial shear stress for fibre-matrix debonding initiation was measured, which was explained also by the occurrence of matrix softening and void formation. 

Figure 3.5 Interfacial shear stress distribution along a fibre intersecting a crack immediately after cracking (a) debonding preceded cracking (b) no debonding prior to cracking.

In many simplified treatments of the stress-transfer problem, reference is made to an average interfacial shear stress value, t, assuming a uniform interfacial shear stress distribution aiong the whole fibre length for a pull-out load P,... 

Figure 3.6 Partially debonded fibre configuration and the interfacial shear stresses to calculate combined elastic and frictional resistance to pull-out.

Lin and Li [14] modelled the pull-out resistance of a slip hardening fibre by assuming a linear relation between the interfacial shear stress x and the slip A ... 

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