# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework
### The Tanaka-Mura-Wu Model  
It has been well understood from the physical metallurgy point of view that metal fatigue is a process involving alternating slips under cyclic loading, leading to the formation of persistent slip bands, and eventually crack nucleation and propagation. Crystallographic slip is the mechanism of plastic deformation at both macro and microscopic scales. A tremendous amount of literature exists on this topic, but due to the limited space, it cannot all be cited in this article, except that readers are referred to see Deformation and Life Evolution in Crystalline Materials [46]. [PERSON] and [PERSON] [56] developed a theoretical model where fatigue crack nucleation is envisaged to occur via the accumulation of dislocation dipole pile-ups within the slip band. At the surface, these dislocation dipole pile-ups appear to be intrusion/extrusions, as shown in Figure 5. Dislocation pile-ups can also nucleate intergranular microcracks at their intersections with grain boundaries [57]. Scenarios of the heterogeneity of persistent slip band evolution are still of interest in current metallurgical studies [58]. Despite that, the Tanaka-Mura model does not cover all the complicated scenarios in real materials, but serves as a physical baseline from which mathematical derivations could proceed, as given in the following. Figure 5: Dislocations in (**a**) vacancy dipoles (forming an intrusion), (**b**) interstitial dipoles (forming an extrusion) and (**c**) tripoles (forming an intrusion-extrusion pair) at the surface. In the original Tanaka-Mura model, the plastic strain of dislocation pile-up was calculated by displacement integration, which resulted in a physical dimension of m\({}^{2}\) in the strain unit. Recently, [PERSON] [45] revisited the Tanaka-Mura model and revised the plastic strain expression using the true strain definition. In the revised model, hereafter called the Tanaka-Mura-Wu (TMW) model, the dislocation pile-up energy is given by [45; 56]:  
\[\Delta U_{c}=\frac{1}{2}(\Delta\tau-2k)\Delta\gamma \tag{29}\]  
where \(\Delta\tau\) is the applied shear stress range, \(k\) is the lattice friction resistance, and \(\Delta\gamma\) is the plastic strain range given by [45]:  
\[\Delta\gamma=\frac{2(1-v)(\Delta\tau-2k)}{\mu} \tag{30}\]  
where \(\mu\) is the shear modulus and \(v\) is Poisson's ratio. When the total accumulated energy is equal to the energy of forming new crack surfaces of 2a by the Griffith criterion [45; 56]:  
\[N\Delta U_{c}ba=2w_{s}a \tag{31}\]  
where \(a\) is the crack nucleation size (dislocation pile-up length), \(b\) the Burgers vector, and \(w_{s}\) the surface energy, J/m\({}^{2}\). Equation (31) can be turned into [45]  
\[N_{f}=\frac{8(1-v)w_{s}R_{s}}{\mu b}\frac{1}{\Delta\gamma^{2}} \tag{32}\]  
or  
\[N_{f}=\frac{2\mu w_{s}}{(1-v)b}\frac{1}{\left(\Delta\tau-2k\right)^{2}} \tag{33}\]  
It is noted that a surface roughness factor \(R_{s}\) (=1 for ideally smooth surface, e.g., electropolished; \(\approx\)1/3 for machined surface) is introduced into the strain-life equation, Equation (32), as it could affect the local intrusion/extrusions, while in the stress-life equation its effect is hidden in \(k\) to reduce the coupon fatigue limit. Under uniaxial loading conditions, the shear stress/strain ranges can be converted to the normal stress/strain by the Taylor factor: \(\Delta\tau=\Delta\sigma/\sqrt{3}\), \(\Delta\gamma=\sqrt{3}\Delta\varepsilon\). Equations (32) and (33) express the fatigue crack nucleation life explicitly in terms of the elastic modulus, the Burgers vector, and the surface energy of the material, which has been validated against LCF experiments on copper, titanium, tungsten, Type 316 stainless steel, Waspaloy, and MAR-M 509 [45], and recently extended to the full range fatigue life of many engineering alloys of Al, Ti, Fe, and Ni-base families, equiatomic high entropy alloy (HEA) CoCrFeMnNi [59], as well as additive manufactured materials [60].