## 1 Introduction  
Recent technological developments have enabled the fabrication of electronic devices with high-density integration. Small size components, e.g., at the nanometer scale, can be fabricated with different shapes including features such as notches and may have defects such as cracks [1, 2]. These circumstances have brought problems commonly addressed by fracture mechanics and fatigue theory to a completely new scale level, raising several new questions, experimental challenges, but also attractive new scientific possibilities [3, 4, 5]. The demand for static and fatigue assessment of nanoscale components is increasing, on the one hand, and the validity of continuum-based approaches is questioned on the other. Indeed, at a very small scale, the simplification of a body as continuum and homogeneous may not hold, and the discrete nature of atoms should be considered [6, 7, 8]. Clarification of these aspects could not only bring enormous development in the field of nanotechnology, but macroscale could benefit as well, e.g., multi-scale modeling of fatigue with focus on short cracks and interaction with local micro-structure [9, 10], atomistic investigation of stresses, strains and grain boundaries [11], fracture properties of advanced materials [12, 13], experimental evaluation of fatigue curve of microscale samples [14]. Several studies have recently quantified the small scale breakdown of continuum-based linear elastic fracture mechanics (LEFM) and have demonstrated that continuum-based methods are still valid at small scale if that limit is not reached [15; 16; 17; 18]. The breakdown was expressed in terms of the ratio between the crack singular stress field length, which surprisingly is still characterizing the crack at a very small scale [19; 20; 21], and the fracture process zone. When this ratio is in the order of 4-5, continuum-based LEFM breaks down and the discrete nature of atoms cannot be ignored. Some researchers have gone at a further small size, and have tried to propose approaches that consider the atomic structures and would be valid at different scales. In this regard, it is worth mentioning those based on [PERSON]'s criterion applied to single crystal silicon [22; 23] and graphene [24], and more recently on averaged strain energy density concept [17]. On the other hand, as briefly mentioned earlier, when that limit is not reached, continuum-based approaches should be valid. This conclusion opens up a large number of possibilities in the field of nanotechnology since well-known tools could be extended to small scale for, e.g., nanoscale mechanical characterization, static/fatigue assessment of cracked or notched components [14; 25]. For example, the continuum formulation of averaged strain energy density concept has been demonstrated to be valid for nanoscale notched-samples [26; 27]. Similarly, by using the Theory of Critical Distances (TCD) [28; 29], a fast and extremely simple method was proposed to evaluate with high accuracy the fracture toughness of single crystal silicon by using notched samples [30]. At small scales, indeed, the fabrication of pre-cracks is extremely difficult, since an atomic sharp crack is ideally necessary to obtain a reliable fracture toughness when dealing with brittle materials. Even for small values of the tip radius, the defect would behave as a notch and would result in an apparent fracture toughness [19; 31]. In regard to the experimental challenges, it is also worth mentioning a recent study on experimental conditions affecting the measurement of the fracture toughness in elastic-plastic materials [32]. Among well-known fracture models, the Finite Fracture Mechanics (FFM) approach has several similarities with the methods mentioned before [33]. The FFM was originally proposed to deal with crack initiation in brittle material and to overcome the limitation of classical fracture mechanics, which assumes the pre-existence of a crack and deals only with its growth [34; 35].