Estimating the intrinsic dissipation using the second harmonic of the temperature signal in tension-compression fatigue. Part II: Experiments

The theoretical approach developed in the companion paper is applied to the temperature signals acquired by using an infrared camera, during step-load fatigue tests carried out on AISI 304L cold-drawn bars. The intrinsic dissipation, calculated according to the analytical model and by means of the discrete Fourier transform of the material temperature, is compared with that measured by using a well-established experimental approach based on the measurement of the cooling gradient of the material temperature, after the fatigue test has been suddenly stopped. The results of this study indicate that the intrinsic dissipation calculated according to the two approaches become increasingly closer to one another for progressively higher applied stress amplitudes.


| INTRODUCTION
In Part I of this paper, 1 an analytical frame has been developed to evaluate the intrinsic dissipation of a metallic material under fatigue averaged over one loading cycle, Q p , by using the Fourier series of the material temperature signal. The theoretical model is based on the first law of thermodynamics, refers to zero-mean stress fatigue, and was developed under the hypothesis that the plastic strain energy density, W, is completely converted into heat and that uniform stress and temperature fields exist in the specimen. In this framework, two different material models were considered, namely, an elasticperfectly plastic and an elastic-plastic material obeying a Ramberg-Osgood law. Since in actual laboratory test conditions, the average material temperature stabilizes after achieving thermal equilibrium with the surroundings, a heat sink was introduced to simulate heat extraction from the material in a simplified and idealized fashion. The intrinsic dissipation has been linked to the second harmonic of the material temperature, ΔT 2 , as follows (eq. 2 of Part I) 1 : where β is a parameter greater than 1. In more detail, in the case of an elastic-perfectly plastic material, β depends on the ratio between the applied strain amplitude and the yield strain, while for an elastic-plastic material behavior, β depends on the cyclic strain hardening exponent, n 0 . Moreover, β was generally found to be dependent on the thermal boundary conditions (BCs). However, by assuming standard laboratory test conditions, the sensitivity of β to the thermal BCs is negligible (see the appendix of Part I 1 ). It has been also shown that when the material temperature stabilizes, the intrinsic dissipation Q p is equal to the heat energy exchanged with the surroundings by a unit volume of material per cycle, the Q parameter. 2 Previously, the heat energy per cycle (the Q parameter) was proposed as a fatigue damage index because it is independent of the mechanical and thermal BCs 2,3 for a given load cycle and stress state. 4 Q can be easily evaluated by stopping the fatigue test at t = t* after thermal equilibrium has been achieved and by measuring the cooling gradient immediately after t*, 2 according to Equation 2: where T(t) is the time-variant temperature of the material, ρ is the material density, c is the materialspecific heat, and f L is the load test frequency applied before the test interruption. In this Part II paper, the theoretical approach presented in Meneghetti and Ricotta 1 is applied to material temperature signals recorded during fully reversed, stress controlled, axial step-load fatigue tests performed on AISI 304L cold-drawn bars.
Q calculated according to Equation 1, hereafter called the second harmonic approach, has been compared with that evaluated by means of Equation 2, hereafter called the cooling gradient approach, as reported in Figure 1. Figure 1 shows also the experimental procedure adopted in the present paper, which will be presented in the relevant section.
The paper is organized as follows. First, the adopted experimental setup is described, and then the results are presented by comparing the Q values obtained using Equations 1 and 2. Afterwards, thermal finite element (FE) analyses are conducted to compare the second harmonic approach and the cooling gradient approach by considering virtual experiments to avoid the influence of noise. A discussion section, which compares the second harmonic and the cooling gradient approaches, concludes the paper.

| MATERIAL, SPECIMEN GEOMETRY, AND TEST METHODS
Based on the second harmonic approach, the Q parameter was evaluated using the fatigue data published by the authors previously, 5 dealing with different approaches for the rapid determination of the fatigue limit. In view of this, in previous study, 5 step-load, zero mean stress, axial fatigue tests were conducted on specimens made of cold-drawn AISI 304L stainless steel, having the elastic modulus E = 192,200 MPa, the tensile strength R m = 691 MPa, the proof strength R p02 = 468 MPa and the percent elongation after fracture A = 43%. 4 The analyzed material had density ρ = 7940 kg/m 3 , specific heat c = 507 J/(kg K) 6 and isotropic coefficient of thermal expansion α = 15.1 Â 10 À6 /K. 4 The fatigue limit of the material was evaluated to be equal to σ A,À1 = 330 MPa by F I G U R E 1 Schematic view of the second harmonic-and the cooling gradient-approach and adopted experimental procedure 2154 performing a short stair-case procedure at 10 million cycles and using eight specimens having their geometry machined according to Figure 2A, while Figure 2B shows the specimen geometry defined for the step-load fatigue tests. 5 Shorter specimens were adopted in the stair-case procedure to speed up long life tests thanks to their higher stiffness and the load test frequency was set in the range of 7 to 23 Hz, in order to keep the material temperature below 60 C. 5 The specimen surface was covered with a black paint having an emissivity κ = 0.93. The material temperature measurements were performed using an infrared camera, as described later. The fatigue tests were conducted by using a servo-hydraulic Schenck Hydropuls PSA 100 machine equipped with a 100 kN load cell and a TRIO Sistemi RT3 digital controller.
During step-load fatigue tests, the applied stress amplitude ranged from 270 to 405 MPa by imposing a step load equal to 15 MPa. After some preliminary fatigue tests, the number of cycles spent during each step was set equal to 10,000 cycles in order to guarantee the stabilization of the material temperature T s . The area of the hysteresis cycle was measured by using the signals acquired from the load cell and from an MTS extensometer having a gauge length of 25 mm. During these tests, f L was constant and equal to 1 Hz, in order to limit the stabilized material temperature T s (see Figure 1) below 60 C, for all the applied step-loads. During the step-load fatigue tests, the material temperature was measured using an FLIR SC7600 infrared camera with 50-mm focal lens, a 1.5-5.1-μm spectral response range, a noise equivalent temperature difference (NETD) < 25 mK, an overall accuracy of 0.05 C and operating with ALTAIR 5.90.002 commercial software. The spatial resolution was equal to 0.12 mm/px. The infrared camera was equipped with an analogue input interface, which was used to synchronize the force signal of the load cell with the temperature signal and eventually perform motion compensation. To compare the second harmonic and the cooling gradient approach, two consecutive temperature acquisitions were performed using a 10-s-long sampling window with f acq = 204.8 Hz (2048 frames). In particular, the first acquisition consisted of exactly 10 s of running test (n acq = 2048 frames between t s and t*, in Figure 1), followed by the second one capturing the machine stop at time t* to measure the cooling gradient. The temperature maps recorded during the first acquisition were first processed by using the FLIR MotionByInterpolation tool to allow for relative motion compensation between the fixed camera lens and the moving specimen subject to cyclic loads. Afterwards, the mean temperature signal taken from a small area, A, (12 Â 7 px 2 in size, i.e., 1.44 Â 0.84 mm 2 ) located in the center of the specimen net-section was processed to evaluate ΔT 2 (see Figure 1). It is worth noting that in this paper, the analytical approach presented in the companion paper 1 has been applied to the data acquired experimentally; therefore, the discrete Fourier transform (DFT) was applied to the temperature data points. In view of this approach, Eq. (20) and Eq. (21) of Meneghetti and Ricotta 1 must be modified as follows: F I G U R E 2 Geometry specimen for (A) stair case fatigue tests and (B) step-load fatigue tests where T i is the temperature data at a sampling rate of f acq , n acq is the number of picked-up samples and j* is the value assumed by the j-index relevant to the second harmonic. The DFT was performed by using the dedicated fast Fourier transform (FFT) routine available in the MATLAB 2016 commercial software. Concerning the cooling gradient approach (see the second line of Figure 1), Q was evaluated according to Equation 2, and the cooling gradient was evaluated considering the average temperature in the area A, as shown in Figure 1.

| EXPERIMENTAL RESULTS
The hysteresis loops measured during the step-load fatigue tests are shown in Figure 3. 5 Having the area of the hysteresis loops for each applied stress amplitude, the W hyst -σ a results are reported in Figure 4 and were fitted using Equation 5, 7 which assumes a Masing-material behavior 8 : where W L , W U , and W C were defined in Part I of the companion paper 1 and are recalled in Figure 3A for ease of reading. Figure 4 shows the resulting K 0 and n 0 parameters of the fitting procedure, which was conducted separately for the data, both clearly below and above the characteristic stress knee-point, σ A,À1, W hyst , respectively, as suggested in Ricotta et al. 5 In Ricotta et al., 5 the correlation of σ A,À1, W hyst was discussed with the material fatigue limit obtained by a short stair-case procedure. The n 0 and K 0 values fitted on the data reported in Figure 4 and the relevant β parameter, evaluated according to Equation 6 1 : are listed in Table 1.  Figure 4 shows that for applied stress amplitudes higher than σ A,À1, W hyst , the W hyst values were not stable for a given applied stress amplitude, but increased with the elapsed number of cycles, due to the material softening, as reported in the literature for the AISI 304L stainless steel. [9][10][11] Strain hardening usually takes place after the softening phase owing to the austeniticmartensitic plastic-induced transformation, and it is revealed by a reduction of the strain amplitude. 10,11 However, a limited number of cycles was spent at each stress amplitude (N = 10,000 cycles); therefore, strain hardening was never detected according to the monitoring of the strain amplitude which was performed for all applied stress levels. In light of this experimental outcome, the hypothesis formed in the Part I paper 1 was supported that plasticity-induced phase transformations did not take place.
To verify the validity of the Masing behavior assumption, first the origin of the measured hysteresis cycles was set at their lower tip and then the branches of the hysteresis loops were described by using the Ramberg-Osgood law, magnified by a factor of two: Step load-fatigue tests reanalyzed in terms of input mechanical W hyst for (A) P_9, (B) P_10, and (C) P_11 specimen T A B L E 1 Values of n 0 , K 0 , and β evaluated during the step-load fatigue tests  Figure 5 shows the comparison among the measured hysteresis cycles and Equation 7, which was evaluated with material parameters n 0 and K 0 , reported in Table 1. It is seen that, although the Masing behavior is not fulfilled in the entire range of applied stress Δσ; nevertheless, Equation 7 fits the apexes of the hysteresis loops with a good level of accuracy for both groups, that is, for σ a < σ A,À1, W hyst and for σ a > σ A,À1, W hyst . Figure 6 reports a characteristic example of the temperature versus time trends measured during the stepload fatigue tests of the P_11 specimen (Figure 6a), the details of the second harmonic of the temperature signal evaluated with Equation (4) (Figure 6b and d) and the cooling gradient evaluated according to Equation 2 ( Figure 6C,E), measured for two load steps. Figure 6A shows that T s achieved stabilization for all applied stress amplitudes and that the temperature increase with respect to the ambient temperature ranged from 10 C to 34 C, while the temperature oscillations due to the thermoelastic effect was on the order of tenth of a Celsius degree. Concerning the regularity of the distribution of both dissipation and the thermoelastic heat sources, careful analyses are reported in the works of Boulanger et al. 12 and Giancane et al. 13 Figure 6B,D shows that ΔT 2 is of the order of thousandths and hundredths of a Celsius degree for σ a = 300 MPa and σ a = 345 MPa, respectively, while the temperature drop measured during the cooling gradient is at least one order of magnitude higher in both cases (see Figure 6C,E). Therefore, the noise can affect the measurement of ΔT 2 to a greater extent than the cooling gradient, making in some cases its evaluation impossible. In light of the temperature signals reported in Figure 6, a discussion concerning the accuracy of the second harmonic and the cooling gradient approaches will be reported in the relevant section. Figure 7 shows the DFT results relevant to the temperature measured for the P_9 specimen, when the applied stress amplitude was equal to σ a = 270 MPa ( Figure 7A) and σ a = 330 MPa ( Figure 7B) and for the P_11 sample, in the case of σ a = 300 MPa ( Figure 7C) and σ a = 390 MPa ( Figure 7D). It is worth noting that in the x-axis of Figure 7, the j index was translated into discrete frequencies, f j , for ease of reading. The maximum value of the vertical axis was set to highlight the range of the second harmonic, ΔT 2 . As a consequence, the T s value is out of scale. For both specimens, it is seen that when the applied stress amplitude is lower than σ A,À1, W hyst , that is, σ a = 270 MPa ( Figure 7A) and σ a = 300 MPa ( Figure 7C), ΔT 2 does not arise clearly, while for σ a = 330 MPa ( Figure 7B) and σ a = 390 MPa ( Figure 7D), a sharp peak of ΔT 2 is evident. Moreover, Figure 7 shows that harmonics higher than the second are also present, as highlighted by the theoretical model presented in the companion paper 1 and according to the experimental results reported in Bar et al. 14,15 F I G U R E 5 Comparison among the experimental hysteresis cycles measured during the step-load fatigue tests and the cyclic stress-strain curves calculated by using the n 0 and the K 0 values evaluated according to Equation 5 (light blue hysteresis cycles:σ a < σ A,À1, Whyst ; red hysteresis cycles: σ a > σ A,À1, Whyst ) [Colour figure can be viewed at wileyonlinelibrary.com] Figure 8 compares the Q parameter evaluated according to the second harmonic approach of Equation 1 and to the cooling gradient approach of Equation 2 and shows that they are in closer agreement at progressively higher applied stress amplitudes, since the peak relevant to the second harmonic becomes increasingly higher, as shown in Figure 7.
To verify the hypothesis that the disagreement observed between the two approaches (particularly for the lowest applied stress amplitudes) is due to the presence of noise, idealized conditions have been simulated by carrying out dedicated thermal FE analyses, where the second harmonic and the cooling gradient approaches were applied to the numerical temperatures, as explained in the next section.

| FE ANALYSES
Three-dimensional thermal FE analyses of the P_11 specimen were performed with Ansys 2019 R3 commercial software using the eight-node SOLID70 element with an element size of 0.48 mm. The step-loads highlighted by the arrows in Figure 8C were considered, which are characterized by σ a = 300 MPa and σ a = 360 MPa. Taking advantage of the geometric and thermal loads symmetry, one-fourth of the specimen controlled by the gauge length of the extensometer was modeled, as shown in Figure 9. In the same figure, the adopted thermal BCs are reported as well. The BC were defined by averaging the 2048 frames captured by the infrared camera and the measured temperatures T s,1 and T s,2 were imposed to the transverse section, having z = 0 (corresponding to the bottom knife edge of the extensometer adopted in the experimental tests) and z = 24 mm (corresponding to the upper knife edge of the extensometer), respectively. A reasonable heat transfer coefficient by natural convection was assumed to be equal to α conv = 10 W/(m 2 ‧K) 16 and the material emissivity was set equal to κ = 0.93 to consider the heat transfer by radiation. Finally, λ = 16 W/(m K) was adopted, according to, 16 and the ambient temperature, T amb , was set in accordance to the experimental tests. The influence of the adopted α conv values on the FE results will be discussed later.
Equation 6 of Part 1 of the present paper states that (the dot symbol indicates the time derivative) 1 : where _ Q p and _ Q the are the equivalent plastic and thermoelastic power heat sources in a purely thermal problem. According to the hypothesis that the plastic strain power density is completely converted into heat, we have _ W t ð Þ ¼ _ Q p t ð Þ, while _ Q the appears in the first law of thermodynamics in 1 and can be expressed as: where T 0 ¼ T s , as calculated from the FE analyses. For each step-load, first a steady-state analysis was performed by only considering the rate of intrinsic dissipation per cycle _ Q p to calculate the temperature field distribution of T s at thermal equilibrium with surroundings. Afterwards, in order to simulate the experimental procedure shown in Figure 1, a 300-s-long transient analysis was performed by introducing time-dependent intrinsic dissipation and thermoelastic power heat sources into the FE model (see Equation 8), in order to evaluate the entire material temperature evolution from the beginning of the step-load fatigue test until temperature stabilization. Next, at time equal to 300 s, _ Q p t ð Þ, and _ Q the t ð Þ were suddenly set equal to zero to simulate the test machine stop and the material cooling, according to the second line of Figure 1. The second harmonic and the cooling gradient approaches were applied by considering the temperature signal taken from the reference node located at z = 12 mm (see Figure 9). which does not produce a net energy dissipation or absorption over one loading cycle, [17][18][19] therefore _ Q the = 0. Since _ Q p ¼ _ W , the plastic strain power density per cycle was calculated averaging _ W t ð Þ (see eq. 46 of Part I and Figure 3A) over one loading cycle, as follows:

| Steady-state FE analysis
According to Figure 3A, one can see that the plastic strain energy per cycle evaluated according to Equation 10 is equal to the area of the hysteresis loop W hyst evaluated according to Equation 5, when the material experiences a purely elastic behavior during the unloading phase, that is, when W C = 0 in Figure 3A. Considering the n 0 values listed in Table 1, the difference between the plastic strain energy per cycle from Equations 5 and 10 ranges from approximately 0.0% for n 0 = 0.0646 to 15.4% when n 0 = 0.354. The stabilized temperature obtained by the steadystate analyses was compared with the experimental results. Figure 10A,B report the comparison for σ a = 300 MPa and Figure 10C,D for σ a = 360 MPa. It is seen that there is a very good agreement between the numerical and experimental results. It is worth noting that the temperature of the lower grip of the fatigue test machine is higher than the upper one due to its proximity to the hydraulic power pack. This is more evident in the case of σ a = 300 MPa, due to the weaker effect of the material-self heating caused by plasticity. Figure 10B,D show the influence of the adopted coefficient by natural convection, α conv . By changing α conv from 10 to 6 W/(m 2 K), there is a maximum variation of the stabilized temperature profile equal to 0.10% and F I G U R E 1 0 Comparison between experimental and steady-state FE results, in the case of P_11 specimen for σ a = 300 MPa (A,B) and σ a = 360 MPa (C,B) [Colour figure can be viewed at wileyonlinelibrary.com] 0.39% for σ a = 300 MPa and σ a = 360 MPa, respectively. In light of this, it can be concluded that the α conv values taken in a range consistent with standard laboratory conditions have a negligible influence on the numerical results. It is interesting to note that the hypothesis that all plastic strain power density _ W is fully converted into specific heat energy rate _ Q p leads to a good agreement between experimental and numerical temperature distributions along the specimen's axis reported in Figure 10B,D.

| Transient FE analyses
In the case of transient FE analyses, all the terms in Equation 8 are different from zero. The thermoelastic heat source was calculated according to Equation 9. The plastic strain power density, _ W t ð Þ, recalling that _ Q p t ð Þ ¼ _ W t ð Þ, was evaluated according to eq. (46) of Part I. 1 However, in this paper, _ W t ð Þ was shifted by a factor ofT 4 to refer the plastic strain power density to the unstressed state σ = 0; in fact eq. (46) of Part I has as starting point t = 0 at the lower tip of the hysteresis loop. Figure 11 shows the input heat power adopted for the transient thermal analyses for σ a = 300 MPa and σ a = 360 MPa. The figure reports also the stress signal in a non-dimensional form to appreciate the timing between the stress cycle and the input heat power.
The results, obtained by adopting a time increment equivalent to a numerical sampling rate, f FE = 128 Hz, are plotted in Figure 12A,B, where it is seen that temperature stabilization is achieved for t > 100 s. The FE results are compared with the relevant experimental data in the close-up view reported inside Figure 12A,B, and a satisfactory agreement can be appreciated, in that a maximum difference equal to 0.1 C between the numerical and stabilized temperatures T s is seen. Next, the FFT algorithm was applied to the temperature data within the time window Δt = 32 s in Figure 12A,B. The results are reported in Figure 13, which highlights the range of the second harmonic ΔT 2 = 7.43 Â 10 À3 C and ΔT 2 = 3.53 Â 10 À2 C for σ a = 300 MPa and σ a = 360 MPa, respectively. Clearly, the sharp peaks corresponding to the harmonics are now much more visible compared to the experimental data reported in Figure 7, due to the absence of noise. From experiments, it was found ΔT 2 = 9.68 Â 10 À3 C and ΔT 2 = 2.22 Â 10 À2 C for σ a = 300 MPa and σ a = 360 MPa, respectively.  Table 2. Moreover, recalling that Q p ¼ W, the Q p values adopted as input in the FE analyses are also reported in Table 2. It is worth noting that Q 2 nd harmonic from FE analyses reported in Table 2 has been estimated by using β values found in the companion Part I paper, 1 where the material diffusivity has been neglected. The applicability of such β values is demonstrated by the excellent level of agreement between assigned ( Q p ) and calculated ( Q 2 nd harmonic ) heat generation per cycle. However, the applicability may not be guaranteed for materials having higher diffusivity than the stainless steel material adopted here.
Finally, the cooling gradient approach was applied to the numerical temperature trends reported in Figure 14. The cooling gradients were calculated at t* = 300 s when the heat generation was suddenly stopped, and it was found _ T t¼tÃ = À 3.33 Â 10 À2 C/s and _ T t¼tÃ = À 1.21 Â 10 À1 C/s for σ a = 300 MPa and for σ a = 360 MPa, respectively. The experimental cooling gradients were found equal to _ T t¼tÃ = À 3.43 Â 10 À2 C/s for σ a = 300 MPa and _ T t¼tÃ = À 1.12 Â 10 À2 C/s for σ a = 360 MPa. Then, the relevant Q values were calculated according to Equation 2 and reported in Table 2. The relevant experimental values are also reported in the table. Table 2 shows that (i) the Q values evaluated according to the second harmonic and the cooling gradient approaches are in excellent agreement with the input values of the FE analysis (consequently, the second harmonic and cooling gradient approaches are consistent with one another); (ii) the cooling gradient approach applied to the experimental data is in a better agreement with the hypothesis that the plastic strain energy is completely converted into heat.

| DISCUSSION
In this paper, the intrinsic dissipation per cycle Q p has been evaluated experimentally, starting from the second harmonic of the temperature signal. It has also been evaluated in parallel by using the well-established cooling gradient approach proposed by Meneghetti. 2 Some pros and cons of the two approaches are reported in the following.
The cooling gradient approach requires suddenly stopping the test machine. However, the machine takes some time to completely stop cycling (for example some tenths of a second); therefore, the material cooling starts slightly before the time t* when the cooling gradient can actually be measured. This aspect may restrict the use of this approach when the time window to be adopted for the cooling gradient evaluation is on the same order of the time required by the machine to completely stop cycling. This situation typically occurs in presence of severe stress raisers, as discussed in 3,20 and/or when the stabilized temperature T s is close to room temperature, that is, when the heat power dissipated by the material due to plasticity is limited. Conversely, the second harmonic approach does not suffer these drawbacks because it is based on the measurement of the material temperature during a running fatigue test. However, the present paper shows that evaluating the intrinsic dissipation by means of the DFT of the temperature signal is not trivial, since the magnitude of ΔT 2 ranged from thousandths to hundredths of a Celsius degree. Therefore, controlled laboratory test conditions and/or a testing chamber (see, 14 as an example) can be suggested to reduce the noise coming from the surroundings. The accuracy of the equipment adopted to measure the temperature is another issue.
Equation 1 requires the β parameter is evaluated, which depends on the assumed elastic-plastic material behavior. In this paper, the Ramberg-Osgood law and the Masing behavior have been assumed. Thanks to the reduced number of cycles spent at each step-load (N = 10,000 cycles), a limited cyclic evolution was observed for the tested material at each applied stress amplitude. Therefore, an average value of W hyst was considered for a given σ a , as described above. On the contrary, when material stabilization is not reached in terms of W along a fatigue test, the n 0 evolution is required to evaluate β. Conversely, the cooling gradient approach does not require any hypothesis regarding the elastic-plastic material behavior, because it relies solely on the density and the specific heat of the material. Moreover, strictly speaking the β parameter generally depends on thermal BCs, 1 while the cooling gradient approach does not require any control of the thermal boundary conditions. However, it must be highlighted that for practical applications, β proved to be quite independent of the thermal boundary conditions. In fact, dedicated thermal FE analyses were carried out, and reduced variations of β were found, despite the significant variations of the convective heat transfer coefficient α conv . More precisely, β was seen to vary by only 0.41% and 2.49% when α conv was increased from 10 W/(m 2 K) to 100 W/(m 2 K) and 1000 W/(m 2 K), respectively.
The cooling gradient approach (Equation 2) enables one to estimate the specific heat loss per cycle regardless the mean stress and stress state of the fatigue loading cycle, as it relies solely on the energy balance of the first principle of thermodynamics. As opposed to the cooling gradient approach, the second harmonic approach requires that the mean stress dependence of the relationship between ΔΤ 2 and the specific heat loss per cycle is investigated.

| CONCLUSIONS
The intrinsic dissipation per cycle (the Q p parameter) was evaluated by means of the range of the second harmonic, ΔT 2 , of the temperature signal measured during a fatigue test (the second harmonic approach). The relationship between ΔT 2 and Q p was derived analytically and discussed in the companion paper.
Step-load fatigue tests were performed on AISI 304L cold-drawn bars and the material temperature was acquired by using an infrared camera. Utilizing the DFT, the temperature signal was processed, and the Q p values were calculated according to the analytical model. In parallel, Q p was also measured during the same tests by using the well-established experimental approach proposed in the past by one of the authors, based on the measurement of the cooling gradient of the material temperature after having suddenly stopped the fatigue test (the cooling gradient approach).
Considering the material and test conditions analyzed in the present paper, it was determined that the results obtained by following the two approaches become increasingly closer for progressively higher applied stress amplitudes. In fact, the higher the applied stress is, the higher the ΔT 2 values are. As a consequence, the signal to noise ratio becomes increasingly favorable.
To verify the hypothesis that the disagreement observed between the two approaches is due to noise, virtual experiments were performed by means of dedicated thermal FE analyses, where the second harmonic and the cooling gradient approaches were applied to the numerical temperatures. In the noise-free virtual experiments, excellent agreement was observed between the second harmonic and the cooling gradient approaches.