The paper is devoted to the investigation of the nucleation and propagation of cohesive cracks in 2D elastic structures. It is assumed that the body contains neither a notch nor any corner, which would induce elastic singularities, the elastic stress field is smooth and bounded, but non-uniform. In the framework of the Dugdale model, the crack propagation process is studied and the stabilizing effects of the stress gradients highlighted. It is shown that the first stage of the crack growth is controlled by the second derivatives of the stress field. In the second stage, when the loading reaches the value such that the crack opening at its center attains the critical values, then a non-cohesive zone appears in the center, leading to a brutal crack propagation. The crack size jumps instantaneously to a value, fixed by the stress gradient characteristic length. This second critical loading corresponds to macroscopic and non-cohesive cracks. The main goal of the paper consists in finding all results in closed form by using the methods of complex potentials and two-scale techniques. The analysis of the influence of the material length, stress gradient length and sensitivity of the response to the imperfections is considered. A short comparison with Griffith's theory is performed. In particular, since Dugdale model contains a critical stress, one can have a crack nucleation in a sound body at a finite loading in contrast with Griffith's model. However, only the first stage of the nucleation, at which the entire crack is submitted to cohesive forces, leads to a continuous crack length evolution with the loading. The evolution is necessarily discontinuous and leads to a crack length jump because of the presence of a snap-back in the equilibrium branch. In the Dugdale model, the entire solutions are obtained in closed form, allowing one to study the size effects. In particular, it is shown that in presence of stress gradient, the response is very sensitive to the ratio between the material length and the stress gradient characteristic length. The loading, at which the crack jump occurs, can be considered as the loading at which a ``macroscopic crack'' nucleates in the body. The snap-back in the partially non-cohesive branch plays an important role in presence of imperfections.