# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework
### The Energy Argument
Considering the formation of microcracks as part of the energy process, then the first law of thermodynamics of the deformation process should be:  
\[\Delta\Pi=\Delta W_{0}+\sum_{i}\frac{\partial\Delta W}{\partial a_{i}}\Delta a _{i}-\Delta Q_{h}=\Delta W_{0}-\sum_{i}G_{i}\Delta a_{i}-\Delta Q_{h} \tag{39}\]  
with  
\[G=-\frac{\partial\Delta W}{\partial a} \tag{40}\]  
where \(\Delta\Pi\) is the total energy, \(\Delta W_{0}\) is the stored strain energy without cracking, \(a_{i}\) is the newly formed crack area, \(G_{i}\) is the Griffith energy release rate [61], and, here, \(Q_{h}\) signifies the heat loss due to energy dissipation. Three instances can be inferred from Equation (39):  
1. In a deformation process without structural changes and cracking, \(\Delta\Pi=0\) and \(G=0\), the dissipated energy (heat) can be equal to the mechanical work input, \(\Delta Q_{h}=\Delta W_{0}\). 2. In a deformation process without structural changes, but cracking occurs due to fatigue, e.g., at low strain levels, \(\Delta\Pi=0\) and \(G\
eq 0\), the dissipated energy (heat) would be equal to the mechanical work less the crack-forming energy, \(\Delta Q_{h}=\Delta W_{0}-\sum\limits_{i}G_{i}\Delta a_{i}\). 3. In a deformation process with significant structural changes and cracking, e.g., at high strain levels, \(\Delta\Pi\
eq 0\) and \(G\
eq 0\), the dissipated energy (heat) would be equal to the mechanical work less the structural change and crack-forming energies, \(\Delta Q_{h}=\Delta W_{0}-\Delta W_{s}-\sum\limits_{i}G_{i}\Delta a_{i}= \Delta W_{h}\), i.e., \(\Delta Q_{h}<\Delta W\). Figure 8: A schematic of microstructure containing a dislocation structure with plastic deformation. Instances I and II can be utilized to interrogate fatigue endurance limits using thermographic methods. One of the difficulties in dealing with instance III is that it is hard to formulate the entropy change in a deformed body in terms of internal state variables at the continuum level. An internal state variable can be defined as the cross-sectional area damage parameter \(\omega\) (=0-1), as in the continuum damage mechanics [44]. As a result, the effective stress is increased as given by \(\sigma_{\mathit{eff}}=\sigma/(1-\omega)\), leading to increased strain energy stored within the material. This would partly account for the increased energy dissipation, or increased temperature, towards the end of the fatigue process. However, additional heat sources may be created along the process: once cracks have been nucleated, mechanical rubbing between the mating crack surfaces will also generate heat during fatigue. Therefore, physical links between the self-heating characteristics and the internal state variables still need to be established, which can be a significant area of future research. Another subject that needs to be considered along with thermodynamics is ultra-high cycle fatigue (UHCF) with failure cycles exceeding 10\({}^{7}\). The UHCF behavior typically has a duplex S-N curve behavior, where the location of fatigue crack nucleation has been found almost exclusively in the specimen's interior, usually at non-metallic inclusions [62; 63; 64]. In this case, local stiffness mismatch between the inclusion and the matrix could induce local stress concentration, as treated by [PERSON] [65], and thus cause crack nucleation and subsequent propagation. This article does not mean to give a comprehensive review of the UHCF studies but highlights the important aspects of self-heating. In UHCF testing (exceeding 10\({}^{7}\) cycles), to achieve the fatigue life results in a matter of days, the loading frequency would have to be in the kHz range [66]. It is then expected that the test results are frequency-dependent. For example, fatigue tests run on 316 LN stainless steel showed that the fatigue lives in air at 10 Hz were longer than those at 700 Hz [67]. It was observed that the specimen temperature reached a maximum of 270 \({}^{\circ}\)C during the 700 Hz test.