PSI - Issue 77

Tomasz Rogala et al. / Procedia Structural Integrity 77 (2026) 11–17 Tomasz Rogala et al. / Structural Integrity Procedia 00 (2026) 000–000

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The loading phase in subsequent steps (shown by red colour line) is conducted until the stabilized temperature is achieved, which corresponds to no further accumulation of internal energy and the rate of heat generation approxi mately equals to heat dissipation rate ˙ q gen ≃ ˙ q diss . The unloading phase ˙ q gen = 0 (shown by blue colour line) remains until the material specimen cool down to the constant ambient temperature. The duration of the every step is usually determined based on the number of cycles m required to obtain the steady-state temperature response for the highest cyclic stress level T k ( σ k ) and very often is used for other lower stress levels. The set of gathered stabilized responses during cyclic loading [ T 1 ( σ 1 ) , T 2 ( σ 2 ) , ..., T k ( σ k )] enable to construct the T s − σ chart presented in Figure 1(b). Another approach, presented in the Figure 1(c) takes advantage of the unloading phase and estimation of the heat dissipation rate ˙ q diss based on the gradient of the stabilized temperature at the instant of time corresponding to the beginning of unloading phase. For subsequent steps i = 1 ... k the ˙ q diss is calculated using the expression: ˙ q diss , 1 = ρ c p ∂ T 1 ∂ t | t = ( N 1 + m ) f , where f is the loading frequency, ρ is the density and c p is the specific heat capacity of the material. Both of the above mentioned approaches enables to characterize the thermomechanical response of PMCs and shows the noticeable changes which corresponds to the fatigue strength σ TT . To minimize the influence of human error on the measurement outcome σ TT and due to the challenges in interpreting the location of the this feature, various estimation methods are used Luong (1998), Huang et al. (2017), Dolbachian et al. (2025), The thermomechanical characterization of PMCs can be done by using ∆ T s − σ or ˙ q − σ approaches. Although the a noticeable changes in this chart are not always clear to estimate the literature is proposing di ff erent estimation methods. Figure 2 shows concepts of di ff erent estimation methods based on ∆ T s − σ approach. In Figure 2(a) an estimation method based on bilinear model is presented. This method is based on the best approximation of the two linear models intersecting at σ TT point. This method refers to the Luong approach Luong (1998) applied firstly to metallic materials. The data is split into two subsets, each fitted with a separate linear model, based on the criterion of minimizing the root mean square error (RMSE) determined through a systematic search for the optimal split point J RMSE ( p ∗ ). The next approach presented in the Figure 2(b) refers to the normalized angle change which allows to divide the set of data at the stress value when maximum angle change is observed Huang et al. (2017). Figure 2(c) presents minimum curvature radius approach, which is based on the nonlinear exponential model approximation Amraei and Katunin (2025) that allows to calculate the radius curvature and find the stress which refers to the lowest value of the curvature radius Huang et al. (2017). The last one of the methods presented in Figure 2(d) is related to estimate the σ TT at the point of stress which corresponds to the maximum perpendicular distance spanned along the whole thermographic dataset Dolbachian et al. (2025). A comprehensive evaluation of fatigue strength in PMCs was carried out using thermographic techniques and classical HCF testing. The materials investigated included neat glass fiber-reinforced polymer (GFRP), GFRP with graphene nanoparticle reinforcement (GFRP-GNPs), and GFRP with hybrid nanoparticle reinforcement (GFRP HNPs). Fatigue tests were performed under fully reversed loading conditions ( R = − 1) at a frequency of 40 Hz. The details about the testing procedure can be find in Amraei and Katunin (2025) and Amraei et al. (2025). Two thermographic methods were applied: the temperature rise–stress approach ( ∆ T s – σ ) and the heat dissipation rate-stress approach ( ˙ q – σ ). Fatigue strength values at the HCF boundary range of 10 6 –10 7 cycles were estimated using both approaches and compared against classical fatigue testing results. The corresponding values are presented inTable 1. The thermographic estimations were further analyzed using four curve approximation methods: BL Luong (1998), AC Huang et al. (2017), MCR Huang et al. (2017), and MPD Dolbachian et al. (2025), Amraei et al. (2025). RMSE were calculated for all the methods, but for the each bilinear model the following expression was applied: RMSE = √ ((SSE 1 + SSE 2 ) / ( n 1 + n 2 )), where SSE is the sum of squared errors and n is the number of data points for each linear segment and subscripts refer to selected segment. The results are shown in Table 2. The values derived from existing literature are marked with an asterisk ( ∗ ). 3.2. Methods of fatigue strength estimation using thermographic approaches 4. Results and discussion