Issue 35

G. Meneghetti et alii, Frattura ed Integrità Strutturale, 35 (2016) 172-181; DOI: 10.3221/IGF-ESIS.35.20 173 of fatigue cracks. To account for the notch support effect, Peterson postulated that the controlling factor is the stress at the distance of the structural size ahead the notch tip [12]; on the other hand, Neuber introduced the structural volume concept, inside which stresses are to be averaged [13]. In view of the extension of the heat energy-based approach to severely notched specimens, a theoretical frame and an experimental procedure have been established in the present paper, by considering a specimen containing a propagating fatigue crack. In particular, the specific heat loss Q has been averaged over a volume V surrounding the tip of the propagating crack, leading to the definition of the averaged energy parameter Q*. The volume V, even though on the order of the size of the structural volume for construction steels, has been chosen arbitrarily, since the focus of the present paper is the thermal problem and not yet the validation of a fatigue assessment method. Q* has been estimated starting from the temperature field measured close to the fatigue crack tip. Experimental temperature distributions have been compared with an analytical solution available in the literature. T HEORETICAL BACKGROUND n order to derive the energy per cycle dissipated in a volume V surrounding a crack tip, a previous theoretical model [8] has been adopted. Let us consider a material undergoing a fatigue test and consider a control volume V surrounding the crack tip, as shown in Fig. 1. The external surface S of the control volume V can be divided into three parts, namely S cv , S cd and S ir through which the heat Q is transferred to the surroundings by convection, conduction and radiation, respectively. The first law of thermodynamics states: ( ) V V W dV Q U dV        (1) where W is the input mechanical energy and  U the variation of the internal energy. All quantities are referred to a unit volume of material per cycle. Eq. (1) can be written in terms of mean power exchanged over one loading cycle as:   m ij ij L p V V V T d f dV H dV c E dV t                            (2) where f L is the frequency of the applied mechanical load, H=H cd +H cv +H ir is the thermal power dissipated by conduction, convention and radiation, respectively,  the material density, c the specific heat and p E  the rate of accumulation of damaging energy in a unit volume of material. Let us consider a plane problem and assume that the temperature of a material undergoing a constant amplitude, sinusoidal fatigue loading is given by:   ( , ; ) ( , ) 2 ( , ; ) a L m T r t T r sen f t T r t         (3) where T a is the amplitude of temperature oscillations due to the thermoelastic effect and T m is the mean temperature evolution. T a and T m depend on the position (r,  ) considered in the component. It is worth noting that the thermoelastic effect consists of a reversible exchange between mechanical and thermal energy, that does not produce a net energy dissipation or absorption over one loading cycle [14-17]. Since Eq. (2) considers the rate of energy contributions averaged over one cycle, then only the mean temperature evolution T m (t) appears on the right hand side of Eq. (2). The specific net heat generation H gen is given by:   gen ij ij L p H d f E          (4) Therefore Eq. (2) can be written in order to put into evidence only the thermal problem: m gen V V V T H dV H dV c dV t              (5) I