9.9. Recapitulation

1. As an alternative to the standard form of the cohesive crack model characterized by a unique stress-displacement relation, a nonstandard form characterized by a fixed density profile of the stress-intensity factor (SIF) in the sense of the smeared-tip superposition method may be used. This new form facilitates analytical solutions.

2. Asymptotically for large enough structures, both forms are equivalent. The stress-displacement law can be obtained from a given profile of SIF density by solving an integral equation of the first kind with a weakly singular kernel, and vice versa.

3. It is shown that, for the nonstandard form of the cohesive crack model, the laws for the size effect, including their dependence on structure geometry, are essentially the same as for the standard form. This provides justification for the proposed nonstandard form.

4. The asymptotic large-size scaling of the nonstandard model for structures with a notch or preexisting stress-free crack is essentially the same as that established for the standard model by Planas and Elices (1992, 1993), but it is easier to establish.

5. Matching the large size asymptotic properties to the small size asymptotic properties derived from a compliance formulation leads to approximate size effect formulae for the entire size range.

6. In the nonstandard cohesive crack model, three cases of failure, leading to three different laws for the size effect, must be distinguished, depending on the location of the fracture process zone (FPZ) at maximum load:

   • Case 1. When the FPZ is attached to the tip of a notch or...