# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework
### The Energy Argument  
As Equation (31) represents the Griffith energy criterion, the dislocation pile-up energy (which later will be converted to crack formation energy) contributed by each cycle in terms of plastic strain is given by combining Equations (29) and (30), as:  
\[\Delta U_{c}=\frac{3\mu}{4(1-v)}\big{(}\Delta e_{p}\big{)}^{2} \tag{37}\]  
The crack nucleation energy is calculated to be about 0.2 MJ/(m\({}^{3}\) cycle) at the strain amplitude of 0.16%, which is close to the heat energy \(Q\) of 0.3 MJ/(m\({}^{3}\) cycle) measured by [PERSON] and [PERSON] [12]. This means a fraction of the mechanical energy would be converted to crack nucleation by forming new crack surfaces. To explain the energy partition, a schematic of the simplified microstructure containing dislocation structures with plastic deformation is shown in Figure 8 (For understanding the dislocation structure in real materials, the readers are referred to look at transmission electron micrographs reported in the literature). A dislocation structure typically consists of slip lines and persistent slip bands, within which dislocation dipole pile-ups also exist (note that only vacancy dipoles are drawn here for the sake of fatigue crack nucleation). At low strain amplitudes, dislocation movement may lead to the formation of a ladder structure with dislocation walls and channels. At high strain amplitudes, it may evolve into labyrinth or cell structures. While the moving dislocations contribute to heat generation by lattice friction, dislocations in the (non-moving) walls cause structural changes, increasing the internal energy and the entropy by increasing the disorder of the lattice. When the dislocation dipole pile-up burst into a crack, the energy stored in the pile-up array converts to the energy of new surface formation, while the energy of those structural dislocations is stored in the material. Thus, the total irreversible work per cycle can be transformed into three parts:  
\[\Delta W=\int_{\mathcal{D}}\int\sigma\,d\varepsilon_{p}\;dV=\int_{\mathcal{D}} \Delta U_{s}\;dV+\int_{\mathcal{D}}\Delta U_{c}\;dV+\int_{\mathcal{D}}\Delta U _{h}\;dV \tag{38}\]  
Figure 7: The predicted fatigue life by the TMW model in comparison with the best-fit Coffin-Manson relation for the test data. where \(U_{s}\) is the energy stored in the material with irreversible structural changes, \(U_{c}\) is the work contributing to the energy of fatigue crack nucleation (forming new surfaces), and \(U_{h}\) is the part convertible to heat generation. It should be noted that a material coupon in a mechanical testing environment is not an isolated system. The heat generated by self-heating mechanisms in the material coupon will eventually dissipate into the environment, as outgoing heat flux. From Equation (38), it can be inferred that the energy stored for disorder (irreversible structural change), \(U_{s}\), should always be less than the total mechanical work input. As a material's entropy is almost impossible to be directly calculated by counting the order/disorder of the microstructural change involving dislocation arrangements for a real crystalline material, as expressed by [PERSON]'s equation: \(S=k_{B}\ln\Omega\), (\(k_{B}\) is the Boltzmann constant, \(\Omega\) is the number of microstates that the system will exist relative to all the possible microstates), Equation (38) serves as an indirect means to estimate that, once self-heating is precisely described.