Identification of invisible fatigue damage of thermosetting epoxy resin by non-destructive thermal measurement using entropy generation

Abstract

To quantify carbon fiber-reinforced plastic (CFRP) fatigue, herein, we investigate the relationship between fatigue and an epoxy resin used in CFRPs. Generally, fatigue is related to the entropy, which comprises the mechanical entropy, calculated from the dissipated energy and temperature, and thermal entropy, calculated from the relationship between specific heat capacity and temperature. According to previous studies, mechanical entropy generation and thermal entropy generation are equal. Herein, 100 cyclic loading tests are conducted on epoxy resin specimens consisting of 4,4’-DDS and bisphenol a diglycidyl ether. The dissipated energy is determined based on stress – strain curves, and mechanical entropy generation is quantified. An equation for the relationship between the specific heat capacity and temperature is developed based on the Debye model, and the increase in specific heat capacity is calculated for equal mechanical and thermal entropy generations. Generally, differential scanning calorimetry is used for specific heat capacity measurements; however, because these measurements are performed by cutting the specimen, a nondestructive measurement method is required. In this study, the specific heat capacity is measured using lock-in thermography (LIT), and the measured and estimated values are comparable. Thus, fatigue can be estimated by quantifying the thermophysical properties, and the lock-in thermography method is a suitable thermophysical property measurement method for this application.

KEYWORDS:

• CFRP, Fatigue
• Specific heat capacity
• Nondestructive measurement
• Entropy

1 Introduction

Carbon fiber-reinforced plastics (CFRPs) are widely used in the aerospace industry and other applications [1,2]. However, the durability of CFRPs is a key issue [3–9]. Generally, CFRP durability depends on the durability of the resin used [9–14]. The most common CFRP resin is the epoxy resin. Generally, transverse cracking occurs in CFRPs owing to fatigue [15–18], which is caused by resin degradation [19–21]. However, the onset of transverse cracking in CFRPs with unknown load histories is difficult to determine.

Some studies have demonstrated that resin damage is associated with the entropy [22–33]. Notably, the entropy can be calculated by either dividing the dissipated energy with temperature, or by integrating heat capacity with temperature. Sato et al. [21] predicted the time-dependent fracture of thermoset CFRPs. This prediction was based on the entropy; however, no actual entropy measurements were involved. Several previous studies have proposed numerous entropy-based fracture models [7,12,21,33–39]. However, only a few actual entropy measurements have been performed [40–42].

Herein, we aim to predict the remaining life and residual strength of CFRPs subjected to prolonged use and with unknown loading histories. This prediction would be revolutionary if it were made using entropy measurements, which is why we make this prediction using nondestructive measurements [43]. Sakai et al. [42] compared the entropy generation between experiments and molecular dynamics (MD) simulations for the tensile loading of PA6. However, the study had several limitations. Because differential scanning calorimetry (DSC) requires samples to be broken down for testing, the measurements render the samples unusable. When DSC tests destroy the specimens, the heat capacity increases. Entropy generation was quantified by a linear and quadratic extrapolation of the relationship between heat capacity and temperature. Monotonic tensile load tests were conducted; however, cyclic loading tests were not involved. Furthermore, a thermoplastic resin was used, but the thermosetting resin used was not mentioned. Indeed, for commercial use, a study on thermoset resins is required.

In this study, entropy generation is quantified by performing cyclic loading tests on epoxy resins and by measuring the dissipated energy. We determine the relationship between the specific heat capacity and temperature using an equation based on the Debye model. From this equation, we construct a relationship between entropy generation and specific heat capacity increase. Based on the results, an increase in the specific heat capacity of the epoxy specimen after cyclic loading tests is predicted.

We have developed a thermal diffusivity, thermal conductivity, and volumetric heat capacity measurement method based on a periodic laser heating method using lock-in thermography (LIT) [42–45]. Using the LIT method, the fatigue damage evaluation through thermal diffusivity has been conducted for a CFRP laminate [46,47]. The LIT method has the advantage of obtaining a high accuracy 2-D image of thermophysical properties. It can provide the information about the location dependence of fatigue damage on composite material such as CFRP. The sensitivity of conventional pulse thermography [48], which uses pulse excitation to heat a sample and can measure 2-D thermal diffusivity, is restricted by nominal sensitivity of the infrared camera. Typical nominal sensitivity of cooled quantum detector infrared camera is 20 mK [49]. The LIT method can achieve a high signal-to-noise ratio and is suitable for very weak temperature signals due to its averaging nature of periodic signal [49]. The nominal sensitivity of LIT can reach less than 1 mK [49]. The LIT method is appropriate for thermal properties measurement, which needs high accurate measurement. In this study, we applied the LIT method to the fatigue sample to obtain volumetric heat capacity without cutting process compared to DSC analysis. We also conducted an Archimedes method to measure the density. Finally, we obtained the specific heat capacity to quantify the entropy generation due to fatigue damage.

2. Cyclic loading test

Fifty grams of bisphenol a diglycidyl ether (DGEBA) and 16.3 g of 4,4’-DDS were stirred at 100 ºC for 10 min and defoamed at 80 ºC. The mixture was both defoamed and stirred four times and to make the sample 3 mm thick, the mixture was poured between glass plates kept at 3 mm. It was baked at 180 °C for 2 h and cut with a diamond cutter to create four specimens with dimensions of 100 mm x 15 mm. A net was attached 30 mm from the top and bottom edges of each specimen, as shown in Figure 1. The net is attached to prevent destruction in the pinched area. A universal testing machine (AG-X plus, Shimadzu Co., Ltd., Kyoto, Japan) was used for cyclic loading testing, as shown in Figure 2. A strain gauge was mounted in the center of the specimen to measure strain. The specimen was maintained at 50 °C for 30 min in a thermostatic chamber. The stress – strain diagram corresponding to the cyclic loading tests is shown in Figure 3. The maximum stress in the cyclic loading test was 60 MPa, and the minimum stress was 10 MPa. Two of the four specimens were subjected to repeated loading tests, whereas the other two were not loaded. These two no-loaded specimens were used as undamaged reference samples.

Figure 1. Specimen used for the fatigue test.

Figure 2. Universal mechanical testing machine (AG-X plus, SHIMADZU Co., Ltd., Kyoto, Japan).

Figure 3. Stress–strain diagram of the epoxy resin obtained experimentally via the cyclic loading test, from which the dissipated energy is calculated.

3. Estimation of the specific heat capacity based on mechanical and thermal entropies

Next, the specific heat capacity of the specimen after cyclic loading is evaluated by combining two types of entropies: mechanical entropy and thermal entropy. In the estimations of these entropies, energy dissipation due to mechanical loading is determined using the stress – strain diagram and specific heat capacity, respectively. Thus, the two entropies are distinguished according to the method of energy dissipation measurement, but they present identical values, as supported by previous studies [42]. The specific heat capacity of the specimen after fatigue loading is estimated based on the entropies determined using these methods.

The mechanical entropy generation $\mathrm{\Delta }{\gamma }_M$ is expressed by following equation as [33]

(1) $\begin{array}{c}\mathrm{\Delta }{\gamma }_{\mathrm{M}}={\int }_0^{{\mathrm{t}}_{\mathrm{s}}}\left(\frac{{\mathrm{W}}_{\mathrm{P}}}{\mathrm{T}}\right)\mathrm{dt}\end{array}$(1)

(2) $\begin{array}{c}{W}_P=\mathit{W}-W_e\end{array}$(2)

where t is the time, T is the temperature, and $W_P$ is the dissipated energy, which is the total work ($\mathit{W})$ minus the internal energy ($W_e$) based on energy balance. These energies are obtained using the stress – strain diagram shown in Figure 3. The total work is obtained by integrating the product of the strain increment and stress with time. The internal energy, that is, the elastic strain energy, can be estimated from the elastic modulus and stress after cyclic loading, where the elastic modulus is determined by the slope of the stress around 0.01%–0.2% strain in the first cycle.

As a result, the dissipated energy is calculated to be 1.81 × 10⁴ J/m³K. Furthermore, given that the density of the epoxy resin is 1233 kg/m³, the entropy generated per unit mass is 1.468 × 10⁻² J/gK.

Next, thermal entropy generation $\mathrm{\Delta }{s}_T$ is represented by [42]

(3) $\begin{array}{c}\mathrm{\Delta }{\mathrm{s}}_{\mathrm{T}}={\int }_{{\mathrm{T}}_0}^{{\mathrm{T}}_1}\left(\frac{{\mathrm{c}}_{\mathrm{p}\mathrm{S}}}{\mathrm{T}}\right)\mathrm{d}\mathrm{T}\end{array}$(3)

where $c_{\mathrm{p}S}$ is the heat capacity of the epoxy resin. The temperature dependence of specific heat capacity over a wide temperature range is necessary to determine thermal entropy generation. It is well known that the Debye model can represent the temperature dependence of solid materials; that is, the specific heat capacity is proportional to the cube of the temperature in the low-temperature region and is constant in the high-temperature region [50]. However, previous experiments have demonstrated that thermoplastic polymers, such as PA6, have a specific heat capacity that increases with temperature at approximately room temperature [42]. One of the primary factors causing the difference in the temperature dependence of specific heat capacity is the arrangement of atoms constituting a molecule. The Debye model assumes that the atoms are arranged in a regular lattice. By contrast, the thermosetting polymer used in this study has an amorphous structure in which atoms are arranged randomly in a system similar to that of a thermoplastic polymer. To capture the temperature dependence resulting from the irregular arrangement of atoms, we propose the following equation:

(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4)

where $\mathit{A},\mathit{a},\mathit{b},\mathrm{and}\mathit{n}$ are fitting parameters. $T_0$ is the reference temperature used to switch the magnitudes of the contributions of the first and second terms to the total specific heat capacity. This equation appropriately interpolates the behavior of the specific heat capacity between the low- and high-temperature regions. The validity of this equation is not discussed in the present study. For this study, we need an equation such as Equation (4)(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) to estimate the increase in specific heat capacity due to entropy increase. For this purpose, Equation (4)(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) is not strictly valid. The validity of this equation will be discussed in a subsequent study, but we will adopt it in the meantime.

Figure 4 shows the specific heat capacities obtained using Equation (4)(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) and the DSC experiments using a small piece cut from the specimen after cyclic loading. Equation (4)(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) adequately expresses the experimentally obtained specific heat capacity: Although DSC measurements of the cut specimens are used to demonstrate the validity of this equation, the specific heat capacity generally increases before and after cutting. To avoid this artificial influence on the heat capacity, this study derives the heat capacity after fatigue loading without cutting, as described in the next section.

Figure 4. Specific heat capacity change of the specimen after cyclic loading obtained by the DSC measurement and relationship between temperature and specific heat capacity fitted with experimental values based on Equation (4)(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) . Note that the DSC measurement is destructive; hence, the specific heat capacity is inherently greater than the actual. The results are only used to prove the temperature dependence of specific heat capacity.

As described above, the generated mechanical entropy is 1.468 × 10⁻² J/gK, which is consistent with the thermal entropy obtained using Equations (3)(3) $\begin{array}{c}\mathrm{\Delta }{\mathrm{s}}_{\mathrm{T}}={\int }_{{\mathrm{T}}_0}^{{\mathrm{T}}_1}\left(\frac{{\mathrm{c}}_{\mathrm{p}\mathrm{S}}}{\mathrm{T}}\right)\mathrm{d}\mathrm{T}\end{array}$(3) and (4(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) ). Furthermore, as will be detailed in the next section, the experimentally measured heat capacity before fatigue loading is 1.06 J/gK. The measurement method is nondestructive and is an original contribution of the present study. These two conditions yield the fitting parameters in Equation (4)(4) $\begin{array}{c}{c}_{\mathit{pS}}=\mathit{A}\times T^{3}exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)+\left(\mathit{a}\sqrt{\mathit{T}}+\mathit{b}\right)\left(1-exp\left(-{\left(\frac{T}{T_0}\right)}^{\mathit{n}}\right)\right)\end{array}$(4) , resulting in the temperature dependence of the specific heat capacity shown in Figure 5. Consequently, the specific heat capacity after fatigue loading at 24 °C increases by 2.49% compared to that before the loading.

Figure 5. Relationship between temperature and specific heat capacity after (red line) and before (blue line) the cyclic loading test when mechanical and thermal entropy generation coincide; the difference in specific heat capacity increases by 2.49% at 24 °C.

4. Measurement of the specific heat capacity using LIT-based laser periodic heating method

4.1 Measurement principles and procedures

The specific heat capacity can be calculated using the thermal conductivity, thermal diffusivity, and density as follows:

(5) $\begin{array}{c}{C}_p=\mathit{\lambda }/\mathit{\rho D}\end{array}$(5)

These three values were measured separately to determine the specific heat capacity. Alasli et. al [45] have developed the mapping measurement method of the volumetric heat capacity using the LIT-based laser periodic heating method. First, a one-dimensional heat conduction model is introduced to obtain the thermal diffusivity in the out-of-plane direction. The rear surface of the sample is uniformly heated using a laser beam, which is modulated by a square-shaped signal. The other boundary conditions along the perimeter of the sample and camera-side surface are defined as adiabatic. By applying these boundary conditions, the quasi-steady-state temperature solution $\mathit{T}\left(\mathrm{K}\right)$ is expressed as follows [45]:

(6) $\begin{array}{c}\mathit{T}\left(\mathit{t},\mathit{z}\right)=T_{\mathit{DC}}\left(\mathit{z}\right)+\tilde{T}\left(\mathit{t},\mathit{z}\right)\end{array}$(6)

where $T_{\mathit{DC}}\left(\mathrm{K}\right)$ is the DC temperature and $\tilde{T}\left(\mathrm{K}\right)$ is the periodic temperature at angular frequency $\mathit{\omega }\left(=2\mathit{\pi f}\right)$, which is expressed by complex function as follows:

(7) $\begin{array}{c}\tilde{T}\left(\mathit{t},\mathit{z}\right)=\frac{q}{\mathit{\lambda \pi \sigma }sinh\left(\mathit{\sigma d}\right)}cosh\left[\left(\mathit{z}-\mathit{d}\right)\mathit{\sigma }\right]\cdot e^{i\left(\omega t-\frac{\pi }{2}\right)}\end{array}$(7)

where d(m) is the sample thickness, $\mathit{q}$(W/m²) is the input heat flux, $\mathit{\lambda }$(W/mK) is the thermal conductivity, and $\mathit{f}$(Hz) is the frequency of laser modulation. The complex number $\mathit{\sigma }$ is expressed as follows:

(8) $\begin{array}{c}\mathit{\sigma }=\sqrt{\mathit{i\omega }/\mathit{D}}\end{array}$(8)

where $\mathit{D}$(m²/s) is the thermal diffusivity. The amplitude of the temperature oscillation $\mathit{A}\left(\mathrm{K}\right)$ and the phase delay $\mathit{\varphi }$(rad) at $\mathit{z}=\mathit{d}$ are as follows:

(9) $\begin{array}{c}\mathit{A}=\left|\tilde{T}\left(\mathit{d}\right)\right|\end{array}$(9)

(10) $\begin{array}{c}\mathit{\varphi }=arg\left[\tilde{T}\left(\mathit{d}\right)\right]\end{array}$(10)

Figure 6 shows the analysis procedure based on the principle. At first, the infrared camera captured the periodic temperature response of the sample surface. At the same time, lock-in analysis was conducted to calculate the amplitude and phase delay. This process was applied to all pixels of the photodetector. Amplitude and phase were measured at multiple frequencies to obtain the frequency dependence. Experiment data was fitted to the theoretical curve of Equations (9)(9) $\begin{array}{c}\mathit{A}=\left|\tilde{T}\left(\mathit{d}\right)\right|\end{array}$(9) and (10(10) $\begin{array}{c}\mathit{\varphi }=arg\left[\tilde{T}\left(\mathit{d}\right)\right]\end{array}$(10) ) to calculate the thermal diffusivity, $\mathit{D}$. Thermal conductivity $\mathit{\lambda }$ can be calculated using the following equation by measuring the input heat flux dependence of the temperature amplitude $\mathrm{\partial A}/\mathrm{\partial q}$.

Figure 6. (a) Schematic diagram of lock-in analysis to obtain amplitude and phase delay. (b) Analysis procedure to measure thermal diffusivity, thermal conductivity, and volumetric heat capacity.

(11) $\begin{array}{c}\mathit{\lambda }=\left|\frac{1}{\mathit{\pi \sigma }sinh\left(\mathit{\sigma d}\right)}\right|\frac{1}{\mathrm{\partial A}/\mathrm{\partial }q}\end{array}$(11)

Because the input heat flux and temperature amplitude depend on the emissivity and absorptivity of the sample, the front and rear surfaces of the sample should be coated with a black-body spray. In the experiment, input heat flux dependence of amplitude was fitted to the theoretical solution to calculate the $\mathit{\lambda }$. Volumetric heat capacity $\mathit{\rho }{C}_{\mathit{p}}$was calculated based on Equation (5)(5) $\begin{array}{c}{C}_p=\mathit{\lambda }/\mathit{\rho D}\end{array}$(5) on each pixel. Finally, the spatial distribution of $\mathit{\rho }{C}_p$ was obtained. The density $\mathit{\rho }$(g/cm³) of the sample was measured using the Archimedes method, which was then used to calculate the average specific heat capacity of the sample.

4.2 Measurement apparatus

The measurement set-up is illustrated in Figure 7. The quasi-steady-state temperature response was recorded using a LIT system (ELITE System, Thermo Fisher Scientific). The detailed specification of the infrared camera is shown in Table 1. The acquired data were processed using digital lock-in processing with commercial software to extract the temperature amplitude and phase delay. Here, a square wave signal was required as the reference signal for the lock-in analysis. The diode laser beam intensity was directly modulated by the reference signal. The laser beam periodically heated the rear side of the sample through a square-core multimode fiber, a mode mixer, and an adjustable collimator. A square iris was used to shape the beam and match the sample geometry, thereby preventing laser incidence in the infrared camera. The intensity distribution of the laser beam output from this configuration was sufficiently uniform [47]. The resin sample was placed in a sample holder, and its rear surface was uniformly heated. The sample density was determined using the Archimedes method with a semi-micro analytical balance (GH-200, A&D) and a density determination kit (AD1653, A&D).

Figure 7. Schematic diagram of lock-in thermography system with optical laser heating set-up.

Table 1. Specifications of the infrared camera.

Parameter	Specification
IR sensor type	Cooled focal-plane array photon detector
Detector element	Indium antimonide (InSb)
Spectral range of detector element	3–5
Image format	640 512 pixels
Pixel pitch	15 15
Frame rate	357.89 Hz
Spatial resolution	/pixel
Noise equivalent temperature difference	<20 mK

4.3 Measurement sample

The specifications of each sample are listed in Table 2. The increase in specific heat capacity was evaluated for two sets of samples to verify the reproducibility: N0₁ and N0_2, N100₁ and N100₂. The center of each fatigue sample was cut to dimensions of 40 mm × 15 mm. This process is necessary for the density measurement. In principle, there is no limitation on the size in the in-plane direction. Figure 8 shows the surface preparation of the sample. To prevent laser penetration into the resin, platinum (Pt) thin films were deposited on the bare surfaces using the magnetron sputtering system (MSP-1S, Vacuum device). Additionally, a black-body paint (JSC 3, Japan Sensor Corp.) was applied on the Pt coating to enhance laser absorptivity and emissivity. If the black-body paint can effectively absorb the laser beam, the presence of Pt becomes unnecessary. Nevertheless, we observed laser transmission when Pt coating was absent.

Figure 8. Surface preparation of the sample for each step before the measurement. Pt layer was coated on the bare surface of the epoxy resin. Pt layer was applied only on the laser heating side to reduce the laser transmission. Black-body paint was applied on Pt layer to improve the absorptivity and emissivity. For camera-side sample surface, black-body paint directly on the bare surface.

Table 2. Specifications of the measurement samples.

Sample ID	N01	N02	N1001	N1002
Number of loading cycles	0	0	100	100
Size	40 mm $\times$ 15 mm
Thickness	3.130 mm	3.185 mm	3.184 mm	3.170 mm
Pt coating	Laser heating side only (Thickness of layer, 50nm)
Black-body paint	Both sides

4.4 Measurement conditions

The measurement conditions are presented in Table 3. The heating frequency for the thermal diffusivity measurements ranged from 0.02 Hz to 0.05 Hz, and four points were measured in 0.01 Hz increments. Once quasi-steady state temperature was achieved, the lock-in analysis was conducted for 30 periods. In the thermal conductivity measurement, the heating frequency was 0.02 Hz, and the laser intensity was measured every five points from 200 mW to 1000 mW. The density was measured after measuring these two properties. Because the sample position was fixed between both measurements, the volumetric heat capacity of each pixel was calculated from the thermal diffusivity and thermal conductivity distributions. Finally, the average specific heat capacity was calculated from the density using Equation (5)(5) $\begin{array}{c}{C}_p=\mathit{\lambda }/\mathit{\rho D}\end{array}$(5) .

Table 3. Measurement conditions.

Thermophysical property	Thermal diffusivity $\mathit{D}$	Thermal conductivity $\mathit{\lambda }$
Intensity of the laser beam	1000 mW	200 mW to 1000 mW ($\mathrm{\Delta }200\mathrm{mW}$)
Modulation frequency range	0.02 Hz to 0.05 Hz ($\mathrm{\Delta }0.01\mathrm{Hz}$)	0.02 Hz
Number of periods for lock-in analysis at each frequency	30
Environment	Atmosphere, 24${ }^{\circ }\mathit{C}$

4.5 Results

Figure 9(a) shows the spatial distribution of the amplitude and phase delay of the N100₁ sample. Figure 9(b) shows the frequency dependence of the amplitude and phase at the position indicated in Figure 9(a). In addition, the laser power dependence of the amplitude can be seen in Figure 9(b). The experimental plot was sufficiently fitted to the theoretical curve. Figure 10 shows the distribution of the thermophysical properties measured by the LIT method for the N100₁ sample. The broken lines in the distribution indicate areas sufficiently far from the sample boundary. One-dimensional heat conduction in the out-of-plane direction was ensured in this area. In addition, the effect of the cutting process was also negligible. The average thermophysical values within this region and the densities measured using the Archimedes method are listed in Table 4. The change in density before and after fatigue loading was sufficiently small. The average specific heat capacity of the unloaded sample was compared with the literature value [51] for DGEBA-4,4’-DDM was used to verify the validity of the measurement, which contains a different curing agent from that used in the experiment. $C_p$ was up to 19% lower than the literature value of 1.304 $\pm$ 0.023 J/gK. Furthermore, the literature values of density and thermal diffusivity were 1.160$\pm$0.004 g/cm³ and 0.136$\pm$0.001 mm²/s, respectively, which agreed with the experimental results within 6%. By contrast, the thermal conductivity was 0.21 W/mK, which was approximately 20% lower than the results. Molecular dynamics simulation for DGEBA-4,4’-DDS revealed that the thermal conductivity varies from 0.14 to 0.22 W/mK [52]. This difference was attributed to the crosslinking portion of the epoxy resin. Because the thermal conductivity obtained in this study was within this range, the specific heat capacity is also consistent because of the validity of the density and thermal diffusivity results. Figure 11 presents a histogram of the volumetric heat capacity. The peak of the histogram increased in both sets. As the density difference was less than 0.24%, this increase directly leads to an increase in the specific heat capacity. This also suggests that a cutting process for density measurements is not necessary for fatigue quantification. As listed in Table 4, the specific heat capacity increased by 4.9% on average. This value is comparable to an increase of 2.49% in the specific heat capacity predicted from the entropy-based $C_p$ estimation, which was specified in Section 3. These results suggest the possibility of estimating fatigue damage based on specific heat capacity measurements. In addition, the LIT method is suitable for this measurement because it can measure the volumetric heat capacity without requiring the cutting of the sample into tiny pieces, which is a mandatory process for the DSC method.

Figure 9. (a) Spatial distribution of the amplitude and phase delay of N100₁ sample at $\mathit{f}$=0.02Hz. (b) the first two plot shows the frequency dependence of the amplitude and phase delay. The third plot shows the laser power dependence of the amplitude. Each plot was obtained at the position indicated by the plus sign in Figure 9(a). The fitted curve and experimental result are shown in each plot.

Figure 10. Distribution of each thermophysical property of the N100₁ sample. Each broken line specifies the area for calculating the average thermal diffusivity, thermal conductivity, and volumetric heat capacity.

Figure 11. Histogram of the volumetric heat capacity of the samples. Upper histogram compares the results of N0₁ and N100₁. Lower histogram compares the results of N0₂ and N100₂.

Table 4. Results of each thermophysical property, which denotes the average value for the samples with the same loading cycles.

Sample ID	N01, N02	N1001, N1002
Number of loading cycles	0	100
Thermal diffusivity (mm2/s)	0.130	0.134
Thermal conductivity (W/mK)	0.168	0.183
Density (g/cm3)	1.234	1.231
Specific heat capacity (J/gK)	1.06	1.11
Relative difference in between an unloaded sample and fatigue loaded sample (%)	-	+4.9

5 Conclusion

Epoxy resins consisting of DGEBA and 4,4’-DDS were tested over 100 loading cycles. The specific heat capacity after the cyclic loading test was estimated using entropy generation. Based on previous studies, mechanical entropy generation was assumed to have the same value as thermal entropy generation. The mechanical entropy was calculated from the stress-strain diagram. To determine the temperature dependence of the specific heat capacity, the equation was devised with reference to the Debye model. The specific heat capacity after the cyclic loading test was estimated from the thermal entropy generation, that is, the mechanical entropy generation. The specific heat capacities of the specimens before and after the cyclic loading tests were measured nondestructively. The density was measured using the Archimedes method, and the thermal diffusivity, thermal conductivity, and volumetric heat capacity were measured via thermophysical property mapping using LIT. The values obtained from the thermophysical property measurements and those predicted from the entropy generation were comparable. The results suggest that sample fatigue can be estimated based on thermophysical property analysis, and the lock-in thermography method is a suitable method for this application.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the JST-Mirai Program [221036344]; Japan Society for the Promotion of Science [21KK0063].