Relationship between incremental J integral and crack tip opening angle in elastic plastic materials

Crack tip opening angle (CTOA) was found to be nearly a constant during stable crack propagation, and thus it is suitable for modeling stable crack growth and instability. The purpose of this paper is to shed some further lights on the physical nature of CTOA. Starting from the power available for separating crack surfaces, a crack driving force parameter was defined. Two different incremental J integrals (one is based on stress work density, and the other is based on free energy density) were proved to be consistent with this crack driving force when the integral contour shrinks to the crack tip. Taking the fracture process zone (FPZ) into consideration, a relationship between critical CTOA and crack driving force was established under steady state condition. In particular, critical CTOA was associated with the incremental J integral directly when crack grows steadily. A rudimentary verification was also provided. By dividing the fracture process into two steps (Crack grows firstly while the load keeps invariant, and then a load increment is enforced under fixed crack length), a general relation between incremental J integral and critical CTOA was then obtained. In this relation, critical CTOA was related to both the incremental J integral and the slope of incremental J resistance curves. According to this relation, three main conclusions are obtained: 1) critical CTOA was consistent with crack driving force during steady crack growth; 2) crack tunneling is one reason of the initial non-constant CTOA phase; 3) critical CTOA was insensitive to thickness when the thickness is small or large enough. In the last part, the single-edge notched bend (SENB) specimen with fully yielded ligament was discussed. Critical CTOA was only dependent upon the slope of incremental J curves in this case. This special case has proved both terms of the general relation are necessary.