## 3 An Energy-Based Fatigue Life Prediction Method  
### Intrinsic Dissipation  
According to the literature [23; 24], the intrinsic dissipation based on a quantitative thermo-graphic method is considered as the fatigue indicator, instead of temperature produced on the material surface under cyclic loads. From the perspective of the fatigue damage mechanism, repeated work generated by the external loads facilitates micro grain movement and friction until the macro crack appears. During this damage evolution process, the mechanical energy transforms into heat energy, elastic and plastic strain energy, storage energy, and so on. Certainly, most of them are released in the form of heat energy, which results in thermal variations in the procedure of heat conduction, heat exchange, heat convection and heat radiation. However, it is not hard to see that the intrinsic dissipation determines thermal evolution, which is more suitable to describe fatigue failure behavior. It is assumed that there is an external heat source \(r\) during cyclic loading, while the total heat source \(s\) mainly includes three parts:  
\[s=s_{the}+d_{1}+r \tag{1}\]  
where \(s_{the}\) is the thermoelastic heat source generated by the reversible elastic strain, and \(d_{1}\) represents the intrinsic dissipation caused by the irreversible strains.  
In view of thermodynamic laws, the heat source could be expressed by temperature differential equation, presented in the previous references [20; 21; 22; 23; 24]:  
\[s=\rho C\frac{\partial\theta}{\partial t}-k\Delta\theta \tag{2}\]  
where \(\rho\) is density, \(C\) is specific heat, \(k\) is isotropic thermal conductivity, \(\theta\) is temperature increment with respect to the initial temperature, \(t\) is time, and \(\Delta\) is the Laplace operator. According to [PERSON] and co-authors' work [25], proposing 0D, 1D, 2D methods to solve the above differential equation, the 0D model is considered as the most appropriate approach for the heat solution in the high cycle fatigue test. Then, the term \(k\Delta\theta\) could be removed from Equation (2).  
When the stress ratio \(R\) is minus one, the heat source \(s_{the}\) tends to be zero because of the thermo-elastic coupling effect under tension-compression loading. If there is a mean stress, the heat contribution provided by the heat source \(s_{the}\) is extremely small, so that it could be neglected in the thermal differential equation. The external heat source \(r\) is defined to describe heat conversion  
Figure 8: The thermal evolution in whole life cycles under stress amplitude 260 MPa.  
between the specimen and the environment. However, since the thermal parameters and exchange conditions are not easy to accurately determine, it has to be simplified according to the heat loss behavior. Finally, a linear equation is utilized to establish the relationship between the heat loss and temperature variation:  
\[r=-\rho C\frac{\theta}{\tau} \tag{3}\]  
where \(\tau\) stands for a time constant characterizing the heat loss.  
After Equation (1), Equation (2) and Equation (3) are integrated and deduced, the intrinsic dissipation \(d_{1}\) could be expressed like this:  
\[d_{1}=\rho C(\frac{\partial\theta}{\partial t}+\frac{\theta}{\tau}) \tag{4}\]  
It is worth mentioning that the intrinsic dissipation \(d_{1}\) is defined as the average dissipated energy intensity of the affected region on the surface of the testing specimen. The heat diffusion along the thickness of specimen is assumed to be identical, while the heat loss between the specimen and the environment is still characterized by the term \(\rho C\theta/\tau\), as shown in Equation (4).