where (/> is a function of the volume fraction, f of the intermediate size particles, and C is a constant.

As far as the fracture of large particles is concerned, it must be expected that the stress or strain required for their separation depends upon their size [27, 29, 30]. This may account for the particle-size dependence of toughness as expressed by eq (11.18).

The final separation of the ligament between the crack tip and the nearest cracked large particle (figure 11.6) is dictated by void initiation at the smaller particles. This may occur at a lower strain if the yield stress is higher. Incorporation of this argument in the crack analysis would require the addition of a strain criterion for fracture of the ligament.

Since the large particles are of paramount importance for fracture toughness [9], it seems reasonable to assume that the size of the ligament is equal to the distance between the crack tip and the first large particle. Experiments have shown [29] that cracking of large particles in aluminium alloys occurs already at 0.8Klc. This observation supports the argument that some additional criterion is required, namely for void initiation at small particles in the fracture stage.

Although these arguments plea for a modified analysis of the problem one may be tempted to assume that a useful approximation is obtained by substituting eq (11.18) into (11.17). But it has to be noted that the strain at the crack tip is some function of K. For simplicity the strain may be assumed proportional to K. In that case Klc has to be proportional to the fracture strain in order to attain the fracture condition in the ligament. This would yield an equation of the type:

Such an equation may properly account for the effect of yield stress on toughness and for the role of the intermediate size particles. There is no experimental evidence to support the equation. Weis and Sengupta [31] have derived an equation showing Klc to be proportional to and they showed data for a number of steels confirming the equation. It should be emphasized that the equation merely serves as an illustration. In many materials it is not possible to distinguish between intermediate size particles and large particles, which suggests that an equation of the type (11.19) may never have a general applicability. Besides, commercial materials may contain particles with a variety of compositions and properties. This implies that the different alloying elements have largely different effects on fracture toughness. The presence of carbon in carbides has a significantly different influence than the occurrence of sulfur in sulfides. As an illustrative

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