Differential formulation and solution of a cohesive crack problem

Attempts at constructing a closed-form solution to the cohesive crack model have been unsuccessful. This paper develops a procedure to derive analytical solutions to a class of non-trivial cohesive crack problems within the theory of linear elastic fracture mechanics. The nonlinear integral equation describing the cohesive model for mode I crack growth is, under certain conditions, transformed into a boundary value problem for which an analytical treatment is possible. Closed-form solutions of the stress in the cohesive zone is derived for the case of a large compact tension geometry with constant and linear softening functions. The analytical expressions of load and crack mouth opening displacement are given and compared with finite element results. A closed-form expression of size-effect is also reported. In addition, a mathematical proof of Barenblatt’s third hypothesis of smooth crack face closure at the crack tip is provided. The existing solution for a built-in cantilever beam on softening foundation is recovered for a very slender geometry, and its use for $a_0/h<12$ should be avoided because transverse deformation does not occur in simple beam theory. Finally, the governing nonlinear differential equation corresponding to an exponential softening function is derived and solved numerically.