Strength-based and fracture-based approaches in the analysis of fibre debonding

The debonding behaviour of fibres strongly affects the properties of fibre-reinforced composites. In the literature, two different approaches to the fibre debonding problem have been developed. In strength-based approaches [1-4], interracial debonding is assumed to occur once an interfacial strength is reached. In fracture-based approaches [5-9], the debonded interfacial zone is regarded as a tunnel crack which grows in size once an interfacial toughness is overcome at the crack tip. In this communication, the conditions for validity of these two different approaches will be discussed with respect to different possible stress distributions along the interface. A method to determine which approach is appropriate for a given composite system will be suggested. In most debonding analysis, it has been assumed that there is a sharp boundary between the debonded and undebonded regions of the interface. However, in reality, between the elastic undebonded zone and the frictional debonded zone, there may exist a transition zone (see Fig. 1) where breakdown of material takes place. If the transition zone is large (in comparison to fibre length), there is a smooth transition ofinterfacial stress from the undebonded zone to the debonded zone. On the other hand, if the transition zone is very small, there is an abrupt change between the two zones and a singular stress field will be present. In general, for large and small transition zone sizes and large and small T2/T i (Note: Here, zs is the interfacial strength while ri is the interfacial friction), four different cases can be distinguished. The various cases (I to IV) are shown schematically in Fig. 1. Cases I and II are for small z~/q. For case I, where the transition zone is large, there is no singularity in the stress field and a strength-based approach is appropriate. Since the difference between interfacial strength and interracial friction is not significant, a single parameter zi can be used to characterize both the transition zone and the frictional zone. For case II, where the transition zone is small, a stress singularity exists and a fracture-based approach should be more appropriate. However, if the interracial toughness (usually denoted by a critical interfacial energy release rate Go) is low, once the debonded zone has extended beyond several fibre diameters, the contribution of frictional shear stress becomes significant compared with the contribution of elastic stresses in the undebonded zone. If one is interested in global composite behaviour (such as the relation between applied stress and fibre displacement) which is insensitive to the inaccuracy of stresses at local points, the use of a strength-based theory with an approximate stress field (such as one obtained from the shear lag analysis) may provide a good approximation. However, if the interracial toughness is high, debonding is always dominated by the singular stress field and a fracture-based approach has to be used. Cases III and IV are for large r~/r~. For case III, where there is a large transition zone, the change of stress with slip in the transition zone (or the slip-weakening relation) can significantly affect interfacial behaviour. In this case, to study the debonding behaviour, linear elastic fracture mechanics will not be applicable because of the invalidity of the small scale yielding requirement. However, approaches similar to Barenblatt's [10] or Hillerborg's [11] for mode I "cohesive crack" or Li's [12] for mode II shear rupture, with the cohesive stresses given by the slip-weakening relation, can be employed. For case IV, where the transition zone is small, debonding behaviour is governed by the singular stress field and a fracture-based theory based on a single fracture parameter (such as the critical energy release rate) is appropriate. In general, a singularity may be maintained even when the transition zone is large such as in coarse grain alumina or certain fibre reinforced ceramics. For this case, a nonlinear fracture analysis similar to that for case III but which can consider both the crack tip singularity and the slip-weakening in the crack wake has to be carried out for the interfacial crack. Experimental observations in support of the above arguments are available. For a silicon carbide reinforced lithium alumino-silicate system where there is negligible chemical bond between the fibre and matrix (i.e., z~/~i = 1), Marshall and Oliver [13] have shown that a frictional sliding analysis (which is equivalent to a strength-based analysis with the effect of elastic shear stress transfer neglected) gave good agreement with experimental measurement of the applied load versus fibre displacement curve. This observation is in agreement with the above discussions which suggest that the strength-based approach is valid for low r~/zi (case I or case II). On the other hand, Piggott [7] shows that for a glass reinforced polyester resin system where ri is greatly reduced by Poisson's contraction of fibre near the loaded end (i.e., ~/z~ >> 1), a strength-based analysis cannot provide a