Stability of Cohesive Crack Model: Part II—Eigenvalue Analysis of Size Effect on Strength and Ductility of Structures

The preceding paper is extended to the analysis of size effect on strength and ductility of structures. For the case of geometrically similar structures of different sizes, the criterion of stability limit is transformed to an eigenvalue problem for a homogeneous Fredholm integral equation, with the structure size as the eigenvalue. Under the assumption of a linear softening stress-displacem ent relation for the cohesive crack, the eigenvalue problem is linear. The maximum load of structure under load control, as well as the maximum deflection under displacement control (which characterizes ductility of the structure), can be solved explicitly in terms of the eigenfunction of the aforementioned integral equation. crack model is a nonlinear theory of fracture me­ chanics in which the condition of stability limit is expressed in terms of the singularity condition of the second variation of the energy potential with respect to cohesive stresses or crack­ opening displacements. Although the criterion of stability limit can also be formulated in terms of energy variation with respect to the crack length, the resulting equation is not very useful, since the energy release rate in the cohesive crack model de­ pends on the cohesive stresses or crack-opening displacements. For a given structure, the criterion of stability limit leads to a highly nonlinear equation for crack length. However, when a class of geometrically similar structures of different sizes is considered and the relative crack length is given, the criterion of stability limit can be treated as an equation for the structure size at which the stability limit occurs at the given relative crack length. In this manner, the criterion of the stability limit is transformed into an eigenvalue problem, with the structure size as the eigenvalue. In the special case of linear softening, the eigenvalue problem is linear. It can be solved independently of the solution of the cohesive crack model. Furthermore, the corresponding maximum value of the load or loading parameter can be expressed explicitly in terms of the eigenfunction. In this way, the size effect curve can be obtained readily, without having to calculate the load-deflection curves for structures of various sizes.