# Energy Concepts and Critical Plane for Fatigue Assessment of Ti-6 Al-4V Notched Specimens
## 1 Introduction  
From the pioneering work by [PERSON] at the beginning of the 1920s [1], energy-based criteria have been widely used for estimating multiaxial fatigue lifetime of engineering components subjected to time-varying loading [2; 3; 4; 5; 6; 7; 8; 9; 10]. The fundamental idea on which such criteria are based is the assumption that energy (calculated in different ways) is always proportional to fatigue damage. Among the different energy-based criteria available in the literature [2], those proposed by [PERSON] [3] and [PERSON] et al. [4; 5; 6] are very interesting. The multiaxial fatigue assessment has to be performed through the cyclic plastic deformation according to [PERSON], whereas [PERSON] et al. argued that both cyclic plastic energy and elastic energy have to be properly taken into account in the fatigue lifetime estimation. It is worth noting that the fatigue life of notched structural components subjected to cyclic loading can be evaluated by means of energy-based concepts. In particular, the concept of strain energy density (SED) has originally been implemented in different criteria available in the literature to predict the fatigue behavior of notched components under uniaxial tensile loading [7; 8]. Subsequently, SED-based criteria have been formulated for multiaxial loading [9; 10]. The main drawback of the above criteria is that the fatigue behavior is assumed to depend only on the stresses at the notch tip. Therefore, any SED-based criterion cannot be applied at the tip of sharp notches since both the stress state and SED tend toward infinite. In order to overcome the above problem, [PERSON] and [PERSON] suggested considering a small but finite volume of material close to the notch tip (that is, the point of stress singularity), over which the SED has a finite value [11]. More precisely, the fatigue damage parameter for blunt and sharp notched structural components under tensile loading (Mode I) is the mean value of the SED, related to a control volume around the notch tip [11; 12]. The radius of the above volume depends on the unnotched specimen fatigue limit, the notch stress intensity factor (NSIF) range and the elastic Poisson'sratio. The [PERSON] and [PERSON] criterion has also been extended to notched structural components subjected to multiaxial loading [13; 14; 15; 16; 17; 18] as well as to welded joints [19; 20; 21]. Recently, an attempt to perform the fatigue lifetime assessment of notched specimens through energy concepts has been made by [PERSON] and co-workers [22; 23; 24], relatively to Ti-6 Al-4V titanium alloy specimens under uniaxial and multiaxial loading (biaxiality ratio 0.6 and 2.0), where the control volume concept has been implemented in the original formulation of the critical plane-based multiaxial fatigue criterion [25; 26]. In the present paper, the above criterion is proposed to be applied together with the control volume concept, and fatigue assessment is performed in a verification point at a distance related to energy concepts. It is validated by means of experimental data related to V-notched specimens made of titanium grade 5 alloy, subjected to mixed mode loading [27]. Such material and other titanium alloys have attracted significant interest being extensively used in leading industries, due to their low density and high specific strength at elevated temperature (aeronautics, nuclear energy) and their compatibility with human tissues (applications in the biomedical field) [28; 29; 30; 31; 32]. In the latter field, such materials can be used, for example, in the form of \(\beta\)-type titanium porous structures [30; 31], and functionally graded Ti-6 Al-4V alloy interconnected mesh structures [32]. In Section 2, the theoretical framework of the strain-based criterion by [PERSON] et al. is outlined, also implementing the concept of control volume [22; 23; 24].