# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework
### Fatigue Fracture Entropy  
[PERSON], [PERSON], and [PERSON] proposed a fatigue fracture entropy (FEE) concept, which can serve as a measure of fatigue damage [16]. FEE is the cumulative entropy of material up to the fracture. They showed that FEE was nearly constant for the tested materials (Al 6060-T6 and stainless steel 304), independent of specimen geometry, load (bending, tension-compression, torsion, etc.), and frequency.  
[PERSON] and [PERSON] further elaborated the concept for low cycle fatigue (LCF) and high cycle fatigue (HCF) based on the infrared thermographic measurements under the cyclic loadings. They defined the entropy production rate as given by [16; 19]:  
\[\dot{S}=\frac{Q_{t}}{T} \tag{24}\]  
where \(Q_{t}=\sigma:\dot{\epsilon}_{p}\), which can be evaluated from the initial slope, \(R_{\theta}\), of the thermographic curve as:  
\[Q_{t}=\rho c_{p}R_{\theta}=\rho c_{p}\left.\dot{\theta}\right|_{t=0} \tag{25}\]  
Then, the total entropy for the fatigue process is obtained by integration of Equation (24) up to the time of failure, \(t_{f}\), as:  
\[FEE=\int_{0}^{t_{f}}\frac{Q_{t}}{T}dt\approx\frac{N_{f}}{fT_{s}}\oint\sigma de _{p} \tag{26}\]  
where \(T_{s}\) is the stabilized temperature, as the major part of self-heating is spent in Phase II, as shown in Figure 2. In the fatigue process, the plastic strain energy per cycle, \(\oint\sigma de_{p}\) (the area of the hysteresis loop), is almost constant [54; 55].  
[PERSON] and [PERSON] also applied the FEE concept to the HCF of SS304 and CS1018 steels [19]. As discussed previously, uniform heat generation and dissipation could come from two sources: (i) internal friction involving reversible movement of atoms and dislocations; and (ii) microplastic deformation and formation of microcracks. In LCF, the energy dissipation generated by internal friction could be negligible compared to the plastic hysteresis energy. However, in HCF, the energy dissipation generated by internal friction occurs at a level comparable to microplastic deformation, especially near the fatigue endurance limit. Therefore, for HCF, the initial temperature increase rate \(R_{\theta}\) could consist of two parts: (i) the part related to internal friction (\(R_{\theta}^{f}\))--which is non-damaging--and (ii) the part related to microplastic deformation and microcrack nucleation (\(R_{\theta}^{p}\)), which is damaging. Accordingly,  
\[Q_{t}=Q_{t}^{p}+Q_{t}^{f}=\rho c_{p}R_{\theta}^{p}+\rho c_{p}R_{\theta}^{f} \tag{27}\]  
The fatigue fracture entropy in HCF can be defined concerning the plastic energy generation (\(Q_{t}^{p}\)) as:  
\[FEE=\frac{N_{f}Q_{t}^{p}}{fT_{s}} \tag{28}\]Similar to the behavior observed by [PERSON] et al. (Figure 4), [PERSON] and [PERSON] observed for SS 304 and CS 1018 that linear dependence of \(R_{\theta}\) on stress below the fatigue endurance limit, and there was a sudden increase in the curve of \(R_{\theta}\) versus the stress amplitude above that. They attributed the low slope \(R_{\theta}^{f}\) to internal friction. As the stress increases, the portion of \(R_{\theta}^{f}\) in the total \(R_{\theta}\) (\(=R_{\theta}^{p}+R_{\theta}^{f}\)) becomes insignificant as compared to \(R_{\theta}^{p}\), which means intrinsic dissipation would dominate the self-heating process above the fatigue endurance limit. They used the Morrow energy approach to characterize the relationship between the stress amplitude and the fatigue life [20,22].