# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework
### Intrinsic Dissipation  
As deformation proceeds into the plastic regime, irreversible deformation occurs with microstructural changes in terms of dislocation structures. From the continuum point of view, attempts to track the development of irreversible processes in the energy balance (the first law of thermodynamics) are to find internal state variables admissible by the second law of thermodynamics, i.e., also satisfying the Clausius-Duhem inequality [22; 42; 43; 44]. However, due to the limitation of the continuum mechanics that assumes material homogeneity, it seems difficult to uniquely specify the irreversible changes in the microstructure concerning macroscopic strain in the framework of continuum thermodynamics. Nevertheless, a set of internal state variables \(\alpha_{\rm i}\) (i = 1, 2,...) are often introduced to represent the effect of irreversible microstructural changes at the continuum level. As far as dissipation at the macroscopic scale is concerned (with a measurable temperature variation), the heat source of internal energy dissipation is given by:  
\[d_{1}=\sigma:\dot{\varepsilon}_{p}-\rho\frac{\partial\psi}{\partial\dot{\alpha }}\dot{\alpha} \tag{17}\]  
Figure 4: Stress amplitude dependence of the intrinsic dissipation of the FV520B stainless steel. Data are after [22].  
The first term in Equation (17) represents the hysteresis energy in the cyclic process. The second represents the contributions from irreversible microstructural changes associated with inelastic deformation, i.e., changes in internal state variables, \(\dot{\alpha}\). With the involvement of internal state variables, it is even more difficult and often ambiguous to calculate the heat source, \(d_{1}\), straightly from its definition, using Equation (17). Most often, \(d_{1}\) is back-calculated from the thermographic measurements.  
[PERSON] et al. used a power-law function to describe intrinsic dissipation energy above the critical stress \(\sigma_{c2}\), as [22]:  
\[d_{1}=k_{d}\sigma^{p}H(\sigma-\sigma_{c2}) \tag{18}\]  
where \(k_{d}\) is the proportional constant and \(p\) (= 8.05 for FV520B) is the power exponent, \(H(x)\) is the Heaviside step function, which equals zero when \(x\leq 0\) and equals 1 when \(x>0\).  
They further demonstrated that the high cycle fatigue (HCF) life can be expressed by [22]:  
\[N_{f}=\frac{E_{C}f}{k_{d}\sigma^{p}}H^{-1}(\sigma-\sigma_{c2}) \tag{19}\]  
where \(E_{C}\) is the energy dissipation threshold for fatigue failure to be determined by fitting with experimental data. The inverse of the Heaviside function implies that fatigue life is infinite at \(\sigma_{c2}\), which is thus deemed to be fatigue endurance limit.  
Intrinsic dissipation occurs by irreversible dislocation motion, which is a damaging process resulting in fatigue under cyclic loading. Heat generation has also been observed in associated with localized slip zones (Luders band propagation) [47] and martensite transformation [48]. Even though both the internal friction and intrinsic dissipation processes are classified based on dislocation motion mechanisms, the exact stress-dependence functions for each stage are not derived from the underlying physical mechanisms yet, and the division between the damaging/non-damaging stages and the subsequent fatigue damage function (the S-N relation) is still largely phenomenological.