# Assessment of Fatigue Life and Failure Criteria in Ultrasonic Testing Through Thermal Analyses
## 2 Fundamentals for Fatigue and Temperature Rise  
Thermodynamic approaches related to mechanical fatigue have been extensively reported in the literature [12, 13, 14, 15, 16, 17, 18, 19] to predict fatigue life. Some methods have been tested using temperature parameters derived from the T-N curve. [PERSON] [15, 18, 19] specifically correlated fatigue life with heat dissipation. However, these methods have primarily been applied to low cycle fatigue (LCF) and high cycle fatigue (HCF) regimes.  
The temperature evolution during fatigue testing is represented by the T-N curve, as shown in Figure 1. This curve is divided into three distinct phases, each representing a specific stage of the fatigue process [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]:  
* Phase I: At the start of the test, the specimen's temperature rises rapidly from the initial room temperature (T\({}_{0}\)) to the stabilized or steady-state temperature (T\({}_{s}\)).  
Phase II tends to decrease and R\({}_{0}\) is more pronounced with the higher applied stress amplitude (\(\sigma_{a}\)) with constant operational frequency (Figure 2). This behavior is often described in low-frequency tests.  
As it is known, fatigue is an irreversible process and the energy dissipation during the cyclic loading is responsible for the temperature increase at the beginning of the test. When the temperature achieves the T\({}_{s}\), the thermal energy is totally dissipated [12; 16; 17].  
Table 1 exhibits the main models based on the T-N curve to estimate the fatigue life. These models were obtained from LCF and HCF tests [16].  
[PERSON] et al. [22] correlated fatigue life with the steady-state temperature T\({}_{s}\), measured from the initial room temperature T\({}_{0}\). This model can be applied if the test demonstrates a well-defined steady-state temperature and concludes with specimen fracture.  
Figure 1: Theoretical T–N curve (adapted by [16]).  
Figure 2: Illustration of the temperature evolution with the applied \(\sigma_{a}\) (adapted by [17]).  
[PERSON] et al. (2025) proposed that the area beneath the T-N curve remains constant and is unaffected by the stress amplitude \(\sigma_{\text{a}}\), provided that the steady-state temperature is achieved during the test.  
[PERSON] (2017); [PERSON] (2018); [PERSON] (2018); [PERSON] (2018) conducted extensive fatigue testing at low frequencies using plain and notched specimens under low- and high-cycle-fatigue regimes. [PERSON] found that the heat dissipation per cycle (Q\({}_{\text{cyc}}\)) is a material property independent of specimen geometry, test frequency, and operational conditions, making it a potential fatigue damage parameter. Q\({}_{\text{cyc}}\) is grounded in the first law of thermodynamics, with its analytical derivation detailed in [PERSON] (2018); [PERSON] (2018). Initially, Q\({}_{\text{cyc}}\) was reported as being proportional to the temperature slope at the beginning of the cooling stage [PERSON] (2018); [PERSON] (2018); [PERSON] (2018). However, [PERSON] and [PERSON] (2018) observed that the temperature slope at the start of the test (t = 0) matches the slope observed during the cooling stage. This approach enables the estimation of fatigue failure based on the heat generated and converted from mechanical energy during the fatigue process.  
The most recent model, developed by [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018), is notably straightforward. It relies solely on the temperature increase rate at the start of the test (R\({}_{\Theta}\)). Their findings indicate that fatigue life can be predicted using a universal curve, irrespective of stress conditions. They validated this model for two steel types under torsion, rotating bending, and bending tests, highlighting the temperature rise slope as a reliable predictor of fatigue life.