A novel Approach for Determining and Analyzing the Magnitude of the Plastic Zone around a Crack Tip: an Algorithm for an approximate analytical Solution

The straight crack of the length 2 a , in the thin, infinite plate is considered. A thin plate is assumed, in which a plane state of stress is occurred, which means that only the stress components, σ x , σ y and τ xy , [MPa], are different from zero, while all the other components of the stress tensor, σ ij , are equal to zero. Two pairs of the concentrated forces, F , act on the crack surface, they are monotonously increased and open the crack. The forces, F , are acting in-plane and are related to the unit of a plate thickness, [N/m]. The plate is made of a ductile metal material. When loading the plate, the plastic zones are formed around crack tips. It is assumed that the plate material is hardened at the plastic deformation. An isotropic and non-linear strain hardening is assumed, in accordance with Ramberg-Osgood´s hardening law . The investigation of the plastic zone magnitude is in the focus of this article, value is determined according to the premises of a well-known cohesive model (Dugdale´s model). The analytical methods were used in the analysis. The fully exact analytical solution was found while afterwards an assumption about small plastic zone around the crack tip was introduced and the approximate solution was found. The comparison of the exact and approximate analytical solution was in the focus of interest of this article. Also, the assessment of the accuracy of the approximate solution was of interest, for which much simpler mathematical apparatus is necessary.