# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework
### The Tanaka-Mura-Wu Model
Here, the first case study is to compare the TMW model, Equation (33), with the S-N data (\(\sigma_{0}\) = 101 MPa) from [PERSON]'s study [9], as shown in Figure 6. The aluminum alloy properties used in the calculation are: \(\mu\) = 27 GPa, \(v\) = 0.3, \(w_{s}\) = 1.125 J/m\({}^{2}\), \(b\) = 2.86 \(\times\) 10\({}^{-10}\) m (same as 7075-T6 and 2024-T3 in [59]), and 2\(k\) = 101/\(\sqrt{3}\) MPa. It can be seen that the agreement between the experimental and predicted life is very well. The second case study takes the experimental data from [PERSON] and [PERSON] [12]. Uniaxial strain-controlled fully reversed (R = \(-\)1) fatigue tests were conducted on SS304L at frequencies between 0.1 and 5 Hz. The test frequency was selected to limit the stabilized temperature below 70 \({}^{\circ}\)C. In the medium and high cycle fatigue regimes, tests were run at the highest frequency compatible with the available experimental equipment. The cyclic stress-strain curve (connecting peak points of the stabilized hysteresis loops) was described by the Ramberg-Osgood equation:  
\[\epsilon=\epsilon_{e}+\epsilon_{p}=\frac{\sigma}{E}+\left(\frac{\sigma}{K^{ \prime}}\right)^{\frac{1}{n^{\prime}}} \tag{34}\]  
with \(K^{\prime}\) = 2250 MPa, \(n^{\prime}\) = 0.337 [12]. The experimental fatigue life was given by the Coffin-Manson equation:  
\[\epsilon=\epsilon_{e}+\epsilon_{p}=\frac{\sigma^{\prime}}{E}\Big{(}2N_{f} \Big{)}^{b}+\epsilon_{f}^{\prime}\Big{(}2N_{f}\Big{)}^{c} \tag{35}\]  
with \(\sigma^{\prime}\) = 3641 MPa, \(b\) = \(-\)0.238, \(\epsilon_{f}^{\prime}\) = \(-\)0.085, \(c\) = \(-\)0.375 [12]. Using Equation (34) to calculate the plastic strain at the given stress level, and using Equation (32) with \(E\) = 193 GPa, \(w_{s}\) = 2.4 J/m\({}^{2}\), \(b\) = 2.48 \(\times\) 10\({}^{-10}\) m (same as steels in [45; 59]), \(R_{s}\) = 1/3 [45] to calculate the fatigue life, the TMW model is compared with the Coffin-Manson relation, as shown in Figure 5:0 It is seen that the agreement of the TMW model with the experiment is very good. It should be emphasized that, in this calculation, Equation (32) provides class-A predictions without the need for the event data for calibration. Furthermore, using the stress-life Equation (33), the fatigue endurance limits can be evaluated as follows:  
Figure 5:1  
where \(N_{f}\) is the endurance cycle limit \(\sim\)10\({}^{7}\), and \(\sigma_{a}\) can either be calculated using the Ramberg-Osgood equations or taken from the fatigue test data. Figure 5:2 S-N data from fatigue tests including thermographic fatigue tests (data after [9]) and the curve by the TMW model for aluminum alloy AA6082 type E welded joint. [PERSON] and [PERSON], using the calorimetric data from their experiment, assumed that \(Q^{c}2N_{f}=const\). [12], where \(Q\) is the heat loss and \(c\) is the power exponent, apparently an empirical relation. Both Figures 6 and 7 demonstrate that the TMW model can predict fatigue life, independent of conventional and thermographic fatigue data, which offers huge savings in both time and cost for fatigue design.