## 1 Introduction
It is recommended to use the linear elastic fracture mechanics or the traditional damage mechanics for hard adhesives (with high tensile modulus). In the case of soft adhesives, in which high energy dissipation and large deformation may occur, a cohesive zone model can be used [27]. Many scientists investigated the problem of the repairing and reinforcing of notched metal structures by composite overlays with the aim of increasing fatigue strength and life. One of the weakest parts of adhesive joints is the adhesive interface [28]. Many works consider the influence of the stiffness of the system core/overlays on the distribution of stresses in the adhesive interface [6]. However, only limited studies are focused on the analyses of the fracture mechanisms of adhesive joints subjected to fatigue loadings. The important conclusions, however, obtained only for static loading conditions, were provided by [PERSON] et al. [29] and [PERSON] et al. [30]. [PERSON] et al. [29] observed that the application of more ductile adhesives (Araldite 420) leads to more uniform and smaller shear stresses in the adhesive interface. On the other hand, the application of less ductile and stiffer and stronger adhesive (Sikadur 30) resulted only in the cohesive failure of the interface. The comparison of different adhesives under static loading conditions was carried out by [PERSON] et al. [30] and they observed that the failure mode depends on the mechanical properties of the adhesive. In contradiction to the results shown by [PERSON] et al. [29], they obtained cohesive failure for the adhesive with the largest stiffness (Sika 30--Elastic modulus 12 GPa). [PERSON] et al. [31] investigated cracked steel structures reinforced by CFRP laminates and observed that the largest increase in fatigue life was obtained for double-side reinforcement. The authors observed cohesive failure in the adhesive layer when normal modulus CFRP (165 GPa) was applied, and mixed-mode damage (CFRP rupture and delamination) in the case of the application of CFRP with an ultra-high modulus (460 GPa). A similar reinforcement technique was applied by [PERSON] et al. [32]. They observed two different failure mechanisms--composite failure (when patches with high a CFRP modulus of 640 GPa were applied; the increase in fatigue life was 4.7-7.9 times) and interfacial debonding (a normal CFRP modulus of 240 GPa overlays was used; the increase in fatigue life was 2.2-2.7 times). They also noted that bond width is a significant parameter in the context of fatigue life. The problems of interfacial debonding can be investigated with the use of finite element methods (FEM) and fracture mechanics (i.e., mixed-mode cohesive law) [21; 33]. Studies of steel specimens with cracks reinforced by CFRP with the use of epoxy resin adhesive were performed by [PERSON] et al. [34], in which debonding failure was spotted (the increase in fatigue life was 97-186%). The experimental tests for aluminum specimens with circular holes and reinforced by CFRP multilayered patches and with the use of Sikadur 330 CN adhesive were studied by [PERSON] et al. [18]. The typical observed failure form was the interfacial debonding and the increase of the fatigue life was in the range of 25-69%. Similar studies and results were also reported by [PERSON] et al. [15] for steel specimens with circular holes in which interfacial debonding occurred at the steel/adhesive interface. The studies concluded that insufficient composite thickness with respect to the core stiffness causes excessive loading of the adhesive interface and leads to adhesive and reinforcement failure. The fatigue tests with the use of different steels reinforced by CFRP overlays glued by E2500S resin were carried out by [PERSON] et al. [35] (the increase in fatigue life was 1.3-3.1 times). They observed different contributions of mixed-mode failure (delamination and debonding) dependent on the applied stress level and steel.