Statistics of composite failure

So far, we considered one fibre of the composite only, assuming it to be rep -resentative of all fibres. However, this implies that all fibres have the same properties. 

In reality, fibre properties are statistically distributed. This is true for their geometry (length and diameter), but also, especially in the case of ceramic fibres, for their mechanical properties that are distributed according to Weibull statistics (see section 7.3). Non-ceramic fibres are also usually not identical since they may contain surface defects, for instance. Because of this statistical distribution of their properties, not all fibres fail simultaneously even in a homogeneously loaded composite. Instead, the weakest fibre will fail first. Due to the volume effect (see section 7.3.1), the failure probability of a long fibre is greater than that of a short one. 

In the following, we consider the case of long fibres with a length several times larger than the critical length (see equation (9.9)). In this case, the fibre is loaded in tension over most of its length, for load transfer occurs only near its end points (see figure 9.6). The fibre will thus fail by fracture. 

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. 

If the fracture toughness of the matrix is low, this increase in stress can cause local failure of the matrix, initiating a crack that propagates from the site of fibre fracture. Because fibre properties are statistically distributed, the crack will usually not cause the next fibre it encounters to fracture and will be stopped there. The increased load is thus distributed to the surrounding fibres. 

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