Based on the linear theory the stresses at the crack tip are infinity but in reality there is always a plastic zone at the crack tip that limits the stresses to finite values. It is very difficult to model and calculate the actual stresses in the plastic zone and compare them to the maximum allowable stresses of the material to determine whether a crak is going to grow or not.
An engineering approach is to perform a series of experiments and reach at a critical stress intensity factor Kc for each material, called the fracture toughness of the material. One can then determine the crack stability by comparing K and Kc directly.
Kc's for a number of common engineering materials are listed in this page.