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

An analytical framework was developed to correlate the intrinsic dissipation to the second harmonic of the temperature signal evaluated by means of the Fourier transform. The theoretical model assumes, without loss of generality, total conversion of the input mechanical energy into heat. Elastic-perfectly plastic or elastic – plastic materials obeying a Ramberg – Osgood law are considered. For the sake of simplicity, analytical solution of the thermoelastic and plastic temperature evolutions are obtained in the case of uniform stress and temperature fields. It was found that the intrinsic dissipation is correlated to the second harmonic of the temperature signal by means of a parameter (the β parameter) that, in the case of an elastic – plastic material, depends on the cyclic hardening exponent n 0 .


| INTRODUCTION
Fatigue is an irreversible process, accompanied by microstructural changes, localized plastic strains and energy dissipation. In the past, the input mechanical energy per cycle, W (i.e., the area of the hysteresis loop) has been proposed as a fatigue index to correlate the fatigue behavior of steels [1][2][3][4] and ductile cast irons. 5,6 The temperature increment of a metallic material undergoing a fatigue test is a manifestation of the thermal energy dissipation, and it has been experimentally observed that the higher the applied stress amplitude is, the more pronounced the temperature increase of the material is for a given set of mechanical and thermal boundary conditions (including load test frequency, stress ratio, room temperature, and specimen geometry). 7 The material temperature has been adopted as an index for the rapid estimation of the material fatigue limit or the high cycle fatigue limit of components, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] including the assessment of the fatigue curve scatter band on the basis of a probabilistic model, [22][23][24][25] for detecting fatigue damage and for monitoring crack propagation. [26][27][28][29][30][31] Moreover, temperature has recently been adopted as a fatigue damage indicator to analyze the fatigue life under constant amplitude, [32][33][34][35][36][37][38][39][40] multi-stage, [41][42][43] and multiaxial loading. 44,45 The dissipated heat energy density per cycle (the Q parameter), on which the material temperature distribution depends, was proposed as a promising fatigue index 46 because it is independent of mechanical and thermal boundary conditions 46,47 for a given load cycle and stress state. 48 Alternatively stated, Q has been recognized as a material property in the same way that W is considered to be a material property.
Q can be easily evaluated from temperature measurements, 46 as will be described later on. The Q parameter was initially adopted to rationalize in a single scatter band approximately 160 experimental results generated from constant amplitude, completely reversed, stress-or strain-controlled fatigue tests on plain or notched hot rolled stainless steel specimens, with notch radii ranging from 0.5 to 8 mm, 47,49,50 and from cold drawn un-notched bars of the same steel under fully reversed axial or torsional fatigue loading conditions. 51 Finally, the Q parameter was recently assumed to be a fatigue index that is capable of rapidly estimating the material fatigue limit by carrying out stepload fatigue tests, and it was successfully adopted to evaluate the fatigue limit of cold drawn AISI 304L bars. 52 Generally, during a fully reversed, elastic-plastic fatigue cycle, the material temperature oscillates due to the superposition of different phenomena: the thermoelastic effect, ΔT the , which is a reversible phenomenon, and the temperature change, ΔT p, which is related to the plastic energy dissipated as heat twice in one cycle, that is, during both the tensile and the compressive part of the fatigue cycle. An additional contribution to heat energy dissipation is due to anelastic phenomena, which are not damaging in a fatigue sense. 53,54 In more detail, when the dependence of material properties, such as E, ν, and α, on temperature is negligible (as is generally the case for metallic materials fatigued at ambient temperature), ΔT the is correlated with the applied stress range according to the following expression 55,56 : The above equation requires that the material is isotropic and undergoing adiabatic transformation, which is practically achieved by applying a cyclic loading above a suitable threshold frequency. If a cyclic sinusoidal loading is applied, the temperature change due to the thermoelastic effect is modulated at the same frequency of the applied load. Therefore, ΔT the can be evaluated experimentally by considering the first harmonic of the discrete Fourier transform (DFT) of the temperature signal (see Figure 1A). Regarding the temperature variation ΔT p related to plastic deformations, it has been shown that in fully reversed tension-compression fatigue tests, ΔT p is spread out into several harmonics of the temperature signal, starting from that modulated at the double frequency of the applied load, which however, retains the most significant contribution. 57,58 On this basis, some researchers have investigated the DFT of the temperature signal during fatigue tests 12,16,[19][20][21]27,59 and in some cases evaluated the intrinsic plastic dissipation per unit volume of material per cycle, Q p , starting from the range of the second harmonic ΔT 2. 57,58,60,61 Akai et al. 60 proposed a rapid technique to estimate the fatigue limit of AISI 304 and AISI 316L stainless steel by carrying out step-load fatigue tests and by monitoring the dissipated energy, which is evaluated from ΔT 2 . They pointed out that satisfactory results were found in the case of AISI 316L, whereas in the case of AISI 304, the fatigue limit estimated by dissipated energy measurements provided a conservative value compared to that evaluated by means of conventional staircase fatigue tests. Palumbo et al. 61 presented an accurate experimental technique to estimate the heat dissipated at the crack tip in the cyclic plastic zone. The proposed approach investigates the heat source operating at twice the load F I G U R E 1 (A) Material temperature oscillations during a fully-reversed fatigue test and its first and second harmonic of Fourier series and (B) energy balance for a material undergoing fatigue loadings and evaluation of the cooling gradient during a fatigue test frequency by assuming a bilinear elastic-plastic material behavior, adiabatic conditions and the plastic strain energy (i.e., the area of the hysteresis cycle) converted to heat. The authors highlight that in real applications, care must be paid to evaluate the intrinsic dissipation from the second harmonic of the temperature signal because adiabatic conditions cannot be easily guaranteed due to the thermal diffusivity close to the crack tip 61 . Bar et al. carried out fatigue tests on plane and U-notched specimens made of EN AW 7475-T761 clad sheet material and they pointed out that the experiments performed on flat specimens showed that the temperature change due to dissipative effects could not be described with a sine wave with the double loading frequency. 58 Similar conclusions were drawn by Bar et al. 57 in fatigue tests carried out on flat specimens of oxygen free copper (OF-Cu) and aluminum alloy AA7475 T761. The fatigue tests were performed with a test frequency f L = 1 Hz under pulsating tension (R = 0) and fully reversed axial loading conditions (R = À1).
Starting from the intuitive observation that, under completely reversed elastic-plastic fatigue, two temperature shots occur per cycle, then correlation between the intrinsic dissipation and the second harmonic of the temperature signal has been investigated by introducing the following general expression: where β is a coefficient to be determined, and the factor 2 takes into account that the intrinsic dissipation associated with plasticity is produced twice over one loading cycle. In particular, Part I of the paper presents an analytical framework for evaluating the intrinsic dissipation Q p , starting from the second harmonic of the material temperature, here called "second harmonic approach," evaluated by means of the Fourier Series. The theoretical model is based on the first principle of thermodynamics and the hypotheses that (i) the plastic strain energy is totally converted into heat, (ii) the stress state is uniform, and (iii) the temperature field is uniform; the model is developed for elastic-perfectly plastic and elastic-plastic materials obeying a Ramberg-Osgood law. In Part II of the paper, the theoretical approach is applied to experimental data obtained by carrying out step-load fatigue tests on AISI 304L cold drawn bars. The intrinsic dissipation has been evaluated starting from the DFT of the temperature signal and according to the procedure defined in Part I of this paper. Afterwards, the intrinsic dissipation has been successfully compared with the experimental intrinsic dissipation measured according to the experimental procedure proposed in a previous work, 46 hereafter named "cooling gradient approach."

| THEORETICAL BACKGROUND: ENERGY BALANCE
According to the classical continuum mechanics reported in, 62,63 the energy balance equation can be written in terms of power per unit volume by introducing the Helmholtz free energy as a thermodynamic potential [63][64][65][66] (the dot symbol indicates the time derivative) 67 : where T is the material temperature, _ W ¼ σ : _ ε p is the plastic strain power density, and _ E s the stored energy rate, including also the stored energy responsible for fatigue damage, 68-70 _ Q the ¼ Àα Á T Á Tr _ σ ð Þ is the heat rate per unit volume due to thermoelasticity and _ v e is the heat generation rate per unit volume supplied to or extracted from the material by external sources. The first two terms on the right-hand side of Equation 3 are the so-called intrinsic dissipation d 1 67 : To derive Equation 3 starting from the general energy balance equation, it has been assumed that the dependence of the material state on temperature is negligibly small because temperature variations are too small to induce phase transformations, which would imply additional coupling heat sources further to thermoelasticity; as an example, Chrysochoos et al. 66 tested dual phase steel specimens and noted that temperature variations were less than 15 K, therefore could not modify the internal state of the material. Second, it has been assumed that plasticity-induced phase transformations do not occur or can be considered negligible, as highlighted in previous work. 67 These assumptions will be commented on the Part II of this paper, where experimental results are reported. If the intrinsic dissipation and thermomechanical coupling in the form of thermoelasticity are null, Equation 3 simplifies to which is the general heat conduction equation.
Letting λr 2 T ¼ _ Q 67 and _ Q p ¼ d 1 , Equation 3 can be written as follows: Assuming that the material is homogeneous and that the temperature distribution is not affected by the nature of the internal heat source, 54 _ Q p and _ Q the can be thought of as equivalent plastic and thermoelastic power heat sources in a purely thermal problem, respectively. Concerning the regularity of the distribution of both dissipation and the thermoelastic heat sources, accurate analyses are reported in the works of Boulanger et al. 13 and Morabito et al. 27

| THE COOLING GRADIENT APPROACH TO ESTIMATE THE INTRINSIC DISSIPATION
If the power quantities in Equation 6 are averaged over one loading cycle, then the contribution of the thermoelastic heat source vanishes because it consists of a reversible exchange between mechanical and thermal energy, which does not produce a net energy dissipation or absorption over one loading cycle [71][72][73] ; moreover, in absence of external heat sources _ v e ¼ 0, Equation 6 can be written as follows: The energy balance Equation 7 is illustrated in Figure 1B, which shows the positive energy exchanges per cycle, that is, the mechanical energy W and the heat energy Q. When the temperature stabilizes during a fatigue test because equilibrium is achieved between heat production in the specimen and heat transfer to the surroundings, _ T becomes null, as shown in Figure 1B, and Equation 7 simplifies further to 46 : If the fatigue test is stopped suddenly at t = t* (see Figure 1B), then just after t* (i.e., at t = t* + ), _ Q p becomes zero (i.e., σ = 0 and _ E s =0 in Equation 4). By re-writing the energy balance equation in Equation 7, one can obtain: It is worth noting that the heat energy rate _ Q dissipated to the surroundings just before and just after t* is the same in Equation 8 and in Equation 9, respectively, because the temperature field does not change through t*. Finally, the thermal energy released in a unit volume of material per cycle can be calculated by simply accounting for the load test frequency, f L , 46 :

| THE SECOND HARMONIC APPROACH TO ESTIMATE THE INTRINSIC DISSIPATION
When equilibrium is achieved between heat production in the specimen and heat transfer to the surroundings, the stabilized material temperature can be assumed to be a periodic function and it can be expressed in terms of Fourier series, as follows: where The second harmonic of temperature and its range are defined according to Equations 13a and 13b, respectively, and both are plotted in Figure 1A: To evaluate the β parameter in Equation 2 and for the sake of simplicity of the analytical developments, the following hypotheses are considered: i. the plastic strain power density _ W is fully converted into specific heat energy rate _ ; this hypothesis, though idealized, is supported by different experimental outcomes 49,[74][75][76][77] ; however, it could also be assumed that _ Q p is a fraction of _ W , but the results in terms of β coefficient obtained later on would not change because Equation 2 takes into account the thermal problem; ii. a uniform stress state and a uniform temperature distribution exist in the material (i.e., internal adiabatic conditions exist, r 2 T = 0, then _ Q ¼ 0 in Equation 6 and temperature T becomes a function only of time); iii. to achieve the stabilized thermal situation illustrated in Figure 1B, heat extraction from the material volume is simulated by a heat sink v e such that _ v e ¼ À W Á f L , that is, heat production per cycle owing to plasticity effects is exactly removed by the heat sink.
The instantaneous energy balance Equation 6 can be written as follows: In Equation 14, T p and T the are the plastic and thermoelastic temperatures, respectively, which originate from the plastic and thermoelastic heat sources, according to Equation 6; T sink is the temperature associated to the heat sink _ v e . In what follows, the Q p parameter is evaluated theoretically by starting from the range of the second harmonic, ΔT 2 , of the temperature signal. The case of an elastic-perfectly plastic material is considered first, and then an elastic-plastic material obeying a Ramberg-Osgood law is analyzed.

| Elastic-perfectly plastic material
Let us consider a sinusoidal, fully reversed, straincontrolled fatigue test, according to With the aim of simplifying the analytical frame, the coordinate reference system has been set at the lower tip of the hysteresis cycle (see Figure 2) by introducing the auxiliary co-ordinate axes Δσ * and Δε * . Therefore, the applied strain becomes The strain is fully elastic until the timet, when the stress reaches the value 2Áσ y , where σ y is the material yield strength, as shown in Figure 2. Considering that for 0 ≤ t ≤t t can be calculated as follows: F I G U R E 2 Hysteresis cycle for an elasticperfectly plastic material For t >t , the resulting stress remains equal to 2σ y , whereas the plastic strain increases from zero and reaches its maximum value when the applied strain is at its maximum, that is, at t =T=2. Therefore, fort ≤ t ≤T 2 , the plastic strain is equal to At time t=T=2 , the plastic strain achieves its maximum value (2ε a À 2σ y E Þ and it remains constant as long as Δσ * returns again to zero, that is, for t =T=2 þt . ForT=2 þ t ≤ t ≤T , Δσ * and the elastic strain Δε Ã e equal zero, whereas the applied strain Δε * (t) decreases to zero; accordingly, in this time window, Δε Ã p (t) = Δε * (t), as shown in Figure 3.
To manage the plastic strain evolution during a single fatigue cycle, let us consider the rectangular wave reported in Figure 4A as having a period equal toT , defined according to Equation 19: where δ ¼ τ=T . Following Figure 4A, three rectangular wave functions are necessary, as reported in Figure 4B: where δ ε ¼T 2 Àt =T , δ ε y ¼t=T and the subscript L and U mean loading and unloading, respectively.
Therefore, the evolution of the plastic strain over one loading cycle can be defined as follows: Afterwards, the elastic component of the applied strain can be easily calculated according to Equation 24 Δε and the applied stress results Δσ Ã t ð Þ ¼ E Á Δε Ã e t ð Þ. Let us evaluate the material temperature related to the thermoelastic effect in Equation 14. From the definition of _ Q the , we have where T 0 is the material temperature when σ = Àσ y (i.e., Δσ * (t) = 0). Then, we have Figure 5A shows the trend of ΔT the versus time according to Equation 26. Let us consider now the material temperature related to plastic dissipation in Equation 14; we have or alternatively F I G U R E 3 Applied sinusoidal strain and its elastic and plastic component for an elastic-perfectly plastic material From Equation 28 the temperature trend due to plasticity effects, ΔT p (t), can be calculated as follows: Figure 5B shows the trend of ΔT p (t) vs. time according to Equation 29. From this figure, one can see that, in one loading cycle, the material temperature increases and achieves the maximum value when t ¼T , according to Equation 30: Under the hypothesis of a Masing material, 78 the plastic strain energy per cycle, W , can be calculated according to Halford 3 : where ε a,p is the amplitude of the plastic strain, and n 0 is the cyclic strain hardening exponent. In the case of an elastic-perfectly plastic material, n 0 = 0, and the plastic strain amplitude is equal to ε a À σ y E ; therefore, and Equation 30 can be expressed alternatively as Regarding the heat sink temperature Equation 14, we have Therefore, the solution of Equation 14 combines Equations 26, 29 and 34b: This idealized description of the heat extraction mechanism from the material is discussed in the Appendix A of this paper.
To plot the temperature function T(t) according to the strain function ε(t) of Equation 15, and not Δε * (t) of Equation 16, we have to shift ΔT(t) (Equation 35) by a factorT 4 and refer the thermoelastic temperature to the unstressed state σ = 0: Given T(t), then T s , A 1 , B 1 , A 2 and B 2 were analytically evaluated according to Equation (12a-c), as follows: It is worth noting that thermo-elasticity involves odd harmonics (k = 1, 3, 5 …), whereas plasticity involves even harmonics (k = 2, 4, 6, etc.). Moreover, the presence of the idealized heat sink (Equation 34a) influences only the even B k coefficients. Figure 6 shows the T(t) versus time graph evaluated according to Equation 36, the first, second, third, and fourth harmonic waves of the temperature signal T(t) and the plastic temperature, which was evaluated by subtracting from T(t) the thermoelastic harmonics, that is, the odd ones, according to Equation 38: Even though the heat extraction from the material has been idealized by means of a heat sink, Figure 6 highlights that the range of the second harmonic wave ΔT 2 is not equal to the range of plastic temperature ΔT p and the temperature signal obtained after cleaning the thermoelastic signal, Equation 38, is not a sine wave, differently from the applied strain, as pointed out by previous works. 57,58,61 To evaluate the intrinsic dissipation per cycle, Q p , by using ΔT 2 , the β parameter was evaluated according to Equation 2, and it was found to depend on the ratio ε a σ y =E ð Þ , that is, the ratio between the applied strain amplitude and the yield strain of the elastic-perfectly plastic material. Conversely, β is independent of ρ, c, and f L . β is plotted against the strain amplitude ratio in Figure 7 and it is seen that it ranges from π/2 and 3π/2 when ε a σ y =E ð Þ > 1.
The open and filled symbols reported in this figure will be discussed in Appendix A.
It is worth remembering that the parameter β reported in Figure 7 has been evaluated starting from the idealized temperature signal in Equations 35 and 36, where thermal equilibrium with the surroundings has been imposed by means of a heat sink, which linearly removes the heat generated by the intrinsic dissipation during one loading cycle (see Equation 34b). In Appendix A, more realistic hypotheses regarding the heat transfer from the specimen to the surroundings have been considered. Even though Appendix A illustrates that strictly speaking β depends on the thermal boundary conditions of the experimental test, the difference of the β parameter with respect to the values reported in Figure 7 is limited to a few percentage points, at least by considering usual laboratory test conditions. Therefore, the results reported in Figure 7 are substantially robust while varying the thermal boundary conditions of the tests.

| Elastic-plastic material
Let us consider an elastic-plastic material obeying a Ramberg-Osgood law in a fully reversed, forcecontrolled fatigue test with a sinusoidal waveform. By evaluating the macroscopic stress as σ(t) = σ a sin(ω L t), with a stress amplitude σ a , the elastic and plastic deformations can be written according to Equations 39a and 39b, respectively: where K 0 and n 0 are the cyclic strength coefficient and the cyclic strain hardening exponent of the material, respectively. Equation 39b can be defined only when the sinusoidal function is positive. To analytically evaluate the plastic strain power density over one load cycle, the coordinate reference system has been set once again at the lower tip of the hysteresis cycle (see Figure 8). As a consequence, in the time-domain, the stress-range can be written as F I G U R E 7 β parameter as a function of the strain amplitude ratio for an elastic-perfectly plastic material Accordingly, the time evolution of the elastic and plastic strain ranges is The rates of the plastic strain energy for 0 ≤ t ≤T=4 , , are equal to (see Figure 9): The rates of the plastic strain energy forT=4 ≤ t ≤T=2 , To manage a single equation defining the evolution of the plastic strain power density during a single fatigue cycle, proper rectangular waves were defined once again. According to Equation 19, a first rectangular wave was defined as equal to one in the 0 ≤ t ≤T=4 time window and equal to zero in the remaining part of the fatigue cycle. In this case, we have τ ¼T=4 and therefore δ C ¼T =4 T : F I G U R E 8 Plastic hysteresis cycle for an elastic-plastic material obeying a Ramberg-Osgood law F I G U R E 9 Plastic strain energy densities dissipated during one loading cycle The second rectangular wave was defined to be equal to one whenT=4 ≤ t ≤T=2 and equal to zero in the remaining part of the fatigue cycle. In this case, τ ¼T 4 and δ L ¼T =4 T : Then, the rate of plastic strain energy can be defined as follows: where R C t þT 2 and R L t þT 2 are the rectangular functions of Equations 44 and 45, respectively, which are translated ofT 2 , as shown in Figure 10. It is worth noting that _ W C = 0, when the material experiences a purely elastic behavior during the unloading phase. In this case, the plastic strain energy density averaged over one loading cycle, W , is equal to the area of the hysteresis cycle Figure 9).

Recalling Equation 14
, the thermoelastic temperature can be evaluated again by using previous Equation 26, where now Δσ * (t) is given by Equation 40. As a result, Figure 11A shows the elastic strain and the thermoelastic temperature evolution over one loading cycle.
Regarding the temperature due to plastic dissipation, ΔT p (t), it can be calculated according to Equation 27, by using _ W t ð Þ Equation 46, as follows: Figure 11B plots ΔT p versus time according to Equation 47. From this figure, one can see that, in one loading cycle, the plastic temperature increases and reaches its maximum value when t ¼T , as in Equation 48: Following the track adopted for the elastic-perfectly plastic material, the material temperature ΔT(t) was calculated according to Equation 35 and eventually T(t) was evaluated according to Equation 36 and reported in Figure 12. Once T(t) was defined by means of Equation 36, the parameters of the Fourier Transform T s , A 1 , B 1 , A 2 , and B 2 were calculated according to Equation (12). Differently from the elastic-perfectly plastic material, a closed form solution for the A k and B k coefficients was not found, therefore they were calculated numerically. Taking advantage of a sensitivity analysis, it was found that convergence is guaranteed for j ≥ 500 (see Equations 44 and 45). The first, second, and fourth harmonics of FT of T(t) are reported in Figure 12; in this case the odd FT terms, starting from the third one, are null because the thermoelastic temperature is a sine function. The plastic temperature was evaluated once again according to Equation 38, that is, by subtracting the thermoelastic harmonics from T(t). It is seen that also in the present case of an elastic-plastic material, the temperature evolution related to plasticity is not a sine wave, as reported by previous works. 57,58,61 Furthermore, Figure 12 highlights that the range of the second harmonic wave ΔT 2 is not equal to the plastic temperature range ΔT p .
Finally, the β parameter was evaluated according to Equation 2, and it was found that it depends only on the strain hardening coefficient, n 0 . Figure 13 reports β versus n 0 , where it should be kept in mind that usually, in the case of metals, n 0 ranges from 0.05 to 0.6. 4

| CONCLUSIONS
The evaluation of the intrinsic dissipation per cycle (the Q p parameter) by means of the Fourier Transform of the material temperature has been addressed. In view of this, an analytical model was developed based on the first principle of thermodynamics under the hypotheses that the rate of plastic strain energy is totally converted into the rate of specific heat energy, and uniform stress as well as temperature fields exist.
The analytical model was then applied in the case of an elastic-perfectly plastic material and an elastic-plastic  F I G U R E 1 3 β parameter as a function of the cyclic strain hardening exponent n 0 for an elastic-plastic material material. In both circumstances, it was found that the range of the second harmonic is correlated to the Q p parameter by means of a multiplicative parameter, β, which was found to be dependent on the assumed material behavior and the thermal boundary conditions. Nevertheless, assuming usual laboratory test conditions, the dependence of β on the thermal boundary conditions appeared very weak. Conversely, a strong dependence was found on the material stress-strain constitutive law. Specifically, in the case of the elastic-perfectly plastic material, β depends on the ratio between the applied strain amplitude and the yield strain, whereas in case of an elastic-plastic material obeying a Ramberg-Osgood type behavior, β depends on the cyclic strain hardening exponent.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request. In Equation 35, an idealized thermal boundary condition has been used, to simulate the achievement of thermal stabilization over time, which consists in a space-and time-independent heat sink (Equation 34a). Let us analyze the dependence of the β parameter on the thermal boundary conditions; two scenarios were considered referring to the elastic-perfectly plastic material:

NOMENCLATURE
a. a constant heat power equal to W =2t is extracted from the material by the heat sink only during the two linear elastic paths of the stress-strain loop reported in Figure 2; instead, during the perfectly plastic path adiabatic conditions are supposed to exist; b. to approach little more realistic thermal boundary conditions, the heat power is extracted steadily by convection during cyclic loading, by keeping the hypothesis of uniform temperature distribution valid; in this case the heat sink does not work any longer (_ v e ¼ 0).
In both cases, two different virtual tests were considered, as follows: 1. ε a = 2%, σ y = 300 MPa, E = 194,700 MPa, f L = 0.1 Hz; ρ = 7,940 kg/m 3 ; c = 507 J/(kg K); α = 16‧10 À6 K À1 ; T amb = 293.15 K 2. ε a = 0.15%, σ y = 250 MPa, E = 194,700 MPa, f L = 5 Hz; ρ = 7,940 kg/m 3 ; c = 507 J/(kg K); α = 16‧ 10 À6 K À1 ; T amb = 293.15 K Case (a) In this case, the material temperature evolution governed by Equation 35 can be re-written as where R 1 (t) is a rectangular function, with τ ¼t and δ 1 t=T (see Equation 19) and ΔT the (t) is  Figure A.1A reports the temperature T(t) derived from Equation A.1 taking into account Equation 36 and the plastic the β parameter depends on the thermal boundary conditions of the test, this dependence is very limited in the case of usual laboratory test conditions, which have been simulated here in the case (b). To investigate further the stability of β parameter to the thermal boundary conditions, Table A1 reports some scenarios where the load test frequency f L and the convection coefficient α conv were varied very significantly from the usual laboratory test conditions. Simulations were performed by using the heat transfer mode case (b). As an example, Table A1 demonstrates that by increasing the convection coefficient by two orders of magnitude (from the realistic value 10 W/(m 2 K) to the unrealistic value 1,000 W/(m 2 K) for usual laboratory testing conditions) β increases from 3.732 to 3.891, respectively, that is, only 4.3%.