# An Overview of Self-Heating Phenomena and Theory Related to Damping and Fatigue of Metals
## 3 The Thermodynamic Framework  
The thermodynamic framework for heat generation and energy dissipation at the macroscopic level is given by the continuum thermodynamics [44]. For a deformable body, the first principle of thermodynamics states that:  
\[\dot{E}=\dot{W}+\dot{Q}=\frac{d}{dt}\int_{\mathfrak{D}}\rho e\;dV=\int_{ \mathfrak{D}}\sigma:\dot{\varepsilon}dV+\int_{\mathfrak{D}}r\;dV-\int_{ \mathfrak{D}\mathfrak{D}}q\cdot n\;dS \tag{1}\]  
where \(\rho\) is the mass density, \(e\) is the specific internal energy, \(\sigma\) is the Cauchy stress tensor, \(\dot{\varepsilon}\) is the total strain rate tensor, the symbol \":\" signifies the scalar product of tensors, \(r\) is an external heat source, \(q\) is the heat flow vector through the material surface (with the normal vector \(n\)) of the material volume \(\mathfrak{D}\). By the divergence theorem (Gauss's theorem), Equation (1) turns into  
\[\dot{\rho}\dot{e}=\sigma:\dot{\varepsilon}+r-\
abla\cdot q \tag{2}\]  
The second law of thermodynamics requires that the local heat generation satisfies the Clausius-Duhem inequality:  
\[\dot{\rho}\dot{s}-\frac{r}{T}+\
abla\cdot\left(\frac{q}{T}\right)\geq 0 \tag{3}\]  
where \(s\) is the specific entropy, \(T\) is the absolute temperature. Figure 3: A schematic \(\Delta T_{AS}\) vs. \(\sigma\) curve to determine the fatigue endurance limit. Introducing the Helmholtz free energy, as:  
\[\psi=e-Ts \tag{4}\]  
Combining Equations (2)-(4) leads to:  
\[\sigma:\dot{\epsilon}-\rho\dot{\psi}-s\dot{T}-q.\frac{\
abla T}{T}\geq 0 \tag{5}\]  
The thermodynamic equilibrium of a homogenous control volume can be identified with local state variables: \(\dot{\epsilon}=\epsilon_{\epsilon}+\dot{\epsilon}_{p}\), \(T\), \(\alpha\), which represents respectively, the total strain (the sum of elastic and plastic strains), the absolute temperature, and internal state variables of the material element. The free energy and the entropy are state functions of these variables, i.e., \(\psi=\psi(\epsilon_{e},T,\alpha)\) and \(s=-\partial\dot{\psi}/\partial T\) (according to the second law of thermodynamics requirements) [44]. Then, the rate of change of these state functions are:  
\[\dot{\psi}=\frac{\partial\psi}{\partial\epsilon_{e}}\dot{\epsilon}_{e}+\frac{ \partial\psi}{\partial T}\dot{T}+\frac{\partial\psi}{\partial\alpha}\dot{\alpha} \tag{6}\]  
\[\dot{s}=\frac{\partial s}{\partial\epsilon_{e}}\dot{\epsilon}_{e}+\frac{ \partial s}{\partial T}\dot{T}+\frac{\partial s}{\partial\alpha}\dot{\alpha} =-\frac{\partial^{2}\psi}{\partial\epsilon_{e}\partial T}\dot{\epsilon}_{e} -\frac{\partial^{2}\psi}{\partial T^{2}}\dot{T}-\frac{\partial^{2}\psi}{ \partial\alpha\partial T}\dot{\alpha} \tag{7}\]  
\[\dot{e}=\dot{\psi}+T\dot{s}+s\dot{T} \tag{8}\]  
With the specific heat defined as \(c_{p}=-T.\partial^{2}\psi/\partial T^{2}\) and \(q=-k\
abla T\) (\(k\) is thermal conductivity), the first law of thermodynamics can be written as [44]:  
\[\rho c_{p}\dot{T}-k\
abla^{2}T=\sigma:\dot{\epsilon}_{p}+\rho T\frac{\partial ^{2}\psi}{\partial\epsilon\partial T}:\dot{\epsilon}_{e}-\rho\left(\frac{ \partial\psi}{\partial\alpha}-T\frac{\partial^{2}\psi}{\partial\alpha\partial T }\right)\dot{\alpha}+r \tag{9}\]  
This is the heat equation governing heat generation, dissipation, and transfer processes. Note that basically the same formulation has been adopted in the previous self-heating-thermographic studies [7; 8; 16; 22; 42; 43]. For coupons under uniform stress conditions, \(\
abla T=0\).