Prediction of Composite Strength and Rigidity

In multicomponent systems, a statistical approach is used for the prediction of a particular property. A statistical approach requires the knowledge of the distribution of the individual phases. In a composite it is very difficult to ascertain the distribution of individual phases. Hence a statistical approach cannot be used to predict the mechanical properties of a composite. This is why a two-phase model in which average stresses and strains are considered to exist in each of the phases is used. If we assume that fibre and matrix experience equal strain, then a parallel model can be used to 

This suggests that the longitudinal modulus of a unidirectional composite is practically dictated by the axial modulus of the fibres. If the load is applied in a transverse direction or in the case of shear loading, the parallel model mentioned previously cannot be applied. Rather the matrix and fibre can be considered to act in series. Assuming an iso-stress condition where the matrix and fibre carry the same load, the transverse modulus of the composite can be expressed by the following equations  

Similarly, under shear loading, the shear modulus (G ) of a composite can be expressed 

Hirsch [29] proposed a model considering the stress transfer between fibre and matrix. The model is a combination of parallel and series model and can be expressed as  

This is the well-known rule-of-mixture which describes a rather idealised situation and can predict the modulus only for continuous fibre-reinforced composites where there is sufficient stress transfer from the matrix to the fibre. However, short fibres are usually much shorter than the specimen length. For short fibres we must consider the matrix-fibre stress transfer. When the matrix is under stress, the maximum stress transferred to the fibre is described by the interfacial stress transfer (t). The stress transfer depends on the fibre length (/), so that at some critical length, /, the stress transferred is large enough to break the fibre. The stress transferred to the fibre builds up to its maximum value (o that which causes breakage) over a distance 1 from the end of the fibre. This means that the long fibres carry load more efficiently than short fibres. 

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