Reliability Prediction of Acrylonitrile O-Ring for Nuclear Power Applications Based on Shore Hardness Measurements

The degradation of polymeric components is of considerable interest to the nuclear industry and its regulatory bodies. The objective of this work was the development of a methodology to determine the useful life—based on the storage temperature—of acrylonitrile O-rings used as mechanical sealing elements to prevent leakages in nuclear equipment. To this aim, a reliability-based approach that allows prediction of the use-suitability of different storage scenarios (that involve different storage times and temperatures) considering the further required in-service performance, is presented. Thus, experimental measurements of Shore A hardness have been correlated with storage variables (temperature and storage time). The storage (and its associated hardening) was proved to have a direct effect on in-service durability, reducing this by up to 60.40%. Based on this model, the in-service performance was predicted; after the first three years of operation the increase in probability of failure (POF) was practically insignificant. Nevertheless, from this point on, and especially, from 5 years of operation, the POF increased from 10% to 20% at approximately 6 years (for new and stored). From the study, it was verified that for any of the analysis scenarios, the limit established criterion was above that of the storage time premise considered in usual nuclear industry practices. The novelty of this work is that from a non-destructive test, like a Shore A hardness measurement, the useful life and reliability of O-rings can be estimated and be, accordingly, a decision tool that allows for improvement in the management of maintenance of safety-related equipment. Finally, it was proved that the storage strategies of our nuclear power plants are successful, perfectly meeting the expectations of suitability and functionality of the components when they are installed after storage.


Introduction
The mechanical characterization of materials provides the basis for the fundamental understanding of the behavior of components that can experience degradation in operation and/or during storage. A representative example is the thermal aging mechanism that severely affects materials that are ultimately intended to operate in the harsh service environment of a nuclear reactor. Materials based on organic polymers have many applications (sealings, insulations, etc.) in nuclear power plants (NPPs).
Nowadays, polymer materials hold an important role in the industry, thanks to their unique properties, such as a wide range of operating temperatures, high thermal/electrical insulation, corrosion-and light-resistance, and sufficient mechanical properties (high strength-to-weight ratio, stiffness, toughness, and ductility) [1,2]. that allows us to obtain the health condition at different supposed storage scenarios, considering the required in-service performance. For this study, NBR was selected as a gasket material, since a previous work [20] has shown that acrylonitrile is the best option to withstand moderate levels of radiation thresholds extracted from databases [29,30] as well as its recyclability, providing a sustainable life cycle. The evaluated parameter was the Shore A hardness in accordance with ISO 868 [31] during a period of five years. Measurements of Shore A hardness consisted of vertical immersion of the indenter into the composite surface [32]. The thermal hardening was quantified based on an adaptation of Arrhenius model-based correlation between hardness and temperature and storage time. This study incorporates a comparison between the results obtained for recently manufactured and existing O-rings in the warehouse, considering several statistical scenarios.
Using an adaptation of the Arrhenius model, predictions based on hardness results can be made over the 5-year period, including supplies stored for at least 18 years. Once the calculation model had been proposed, different storage limit conditions were obtained after validating the methodology comparing the predicted allowable storage periods and conditions with the real ones.

Methodology
The methodology ( Figure 1) is based on the analysis (Stage 1) of experimental data of Shore A hardness obtained during qualification processes (between 2014 and 2018) of recently manufactured (when they were measured) and previously stored NBR O-rings. Thus, by adapting the Arrhenius model for thermal aging, along with the activation energies indicated in the standard EPRI TR 1,009,748 [33], predictions (Stage 2) based on three scenarios are considered-very conservative, moderately conservative, and minimally conservative. Finally, a validation methodology is performed along with the estimation of in-service durability and the determination of critical storage conditions (Stage 3).
preventing industrial accidents while operating a machine. In general reactions, the linear Arrhenius equation is used to predict the aging characteristics [28].
The objective of this work is the development of a methodology to determine the useful life-based on the storage temperature-of NBR O-rings using a reliability-based approach that allows us to obtain the health condition at different supposed storage scenarios, considering the required in-service performance. For this study, NBR was selected as a gasket material, since a previous work [20] has shown that acrylonitrile is the best option to withstand moderate levels of radiation thresholds extracted from databases [29,30] as well as its recyclability, providing a sustainable life cycle. The evaluated parameter was the Shore A hardness in accordance with ISO 868 [31] during a period of five years. Measurements of Shore A hardness consisted of vertical immersion of the indenter into the composite surface [32]. The thermal hardening was quantified based on an adaptation of Arrhenius model-based correlation between hardness and temperature and storage time. This study incorporates a comparison between the results obtained for recently manufactured and existing O-rings in the warehouse, considering several statistical scenarios.
Using an adaptation of the Arrhenius model, predictions based on hardness results can be made over the 5-year period, including supplies stored for at least 18 years. Once the calculation model had been proposed, different storage limit conditions were obtained after validating the methodology comparing the predicted allowable storage periods and conditions with the real ones.

Methodology
The methodology ( Figure 1) is based on the analysis (Stage 1) of experimental data of Shore A hardness obtained during qualification processes (between 2014 and 2018) of recently manufactured (when they were measured) and previously stored NBR O-rings. Thus, by adapting the Arrhenius model for thermal aging, along with the activation energies indicated in the standard EPRI TR 1,009,748 [33], predictions (Stage 2) based on three scenarios are considered-very conservative, moderately conservative, and minimally conservative. Finally, a validation methodology is performed along with the estimation of in-service durability and the determination of critical storage conditions (Stage 3).  The experimental procedure consisted of a dimensional checking (a) and polymer composition characterization (b) before performing a Shore A hardness measurement (c),  The experimental procedure consisted of a dimensional checking (a) and polymer composition characterization (b) before performing a Shore A hardness measurement (c), with all testing performed at 22 ± 1 • C and 25 ± 5% of humidity, using a thermohygrometer Testo 608-H1 (Testo SE & Co. KGaA, Lenzkirch, Germany). Table 1 shows the expected Shore A hardness of NBR O-rings and the hardness acceptance criterion along with the homogenous dimensions of O-rings.  Table 1) were analyzed in the present study. The dimensional checking was performed using as an acceptance criterion of just ±1% for external and internal diameters, and therefore, for thickness. The thickness seems to be a critical aspect that could substantially influence the measurement as many studies have demonstrated [34]. In addition, the standardized procedure according to ASTM D2240 [35] and some authors [36] recommended that thickness should be at least equal to 6 mm. The measurements were performed on more than 140 O-rings from 14 different supplies and on an additional batch consisting of previously stored O-rings. The dimensional measurements were carried out using an equipment ScanMaker 9800XL PLUS TMA1600 III (Microtek, Hsinchu, Taiwan) as it is shown in Figure 2.
with all testing performed at 22 ± 1 °C and 25 ± 5% of humidity, using a thermo-hygrometer Testo 608-H1 (Testo SE & Co. KGaA, Lenzkirch, Germany). Table 1 shows the expected Shore A hardness of NBR O-rings and the hardness acceptance criterion along with the homogenous dimensions of O-rings.  Table 1) were analyzed in the present study. The dimensional checking was performed using as an acceptance criterion of just ±1% for external and internal diameters, and therefore, for thickness. The thickness seems to be a critical aspect that could substantially influence the measurement as many studies have demonstrated [34]. In addition, the standardized procedure according to ASTM D2240 [35] and some authors [36] recommended that thickness should be at least equal to 6 mm. The measurements were performed on more than 140 O-rings from 14 different supplies and on an additional batch consisting of previously stored O-rings. The dimensional measurements were carried out using an equipment ScanMaker 9800XL PLUS TMA1600 III (Microtek, Hsinchu, Taiwan) as it is shown in Figure 2.

(b) Polymer Composition Characterization
Before performing the hardness test, each O-ring was also analyzed to assess the composition of components. In this case, the expected (and the acceptance) criterion was acrylonitrile butadiene rubber (NBR). The technique used was the Fourier-transform infrared spectroscopy (FTIR) that is based on the concept of absorption of infrared radiation by sample. The resulting signal at the detector is a spectrum that characterizes the polymer analyzed and, therefore, it allows composition data to be obtained [37]; this technique is the method used to determine if the O-ring composition is the expected one and, therefore, if the component is ready to continue the characterization process (hardness test, in this case study). The equipment used was a Nicolet 5700 (Thermo Electron Corporation, Waltham, MA, USA). Thus, spectra of the NBR components were recorded over a wavenumber range of 4000-500 cm −1 , with 32 scanning times at a resolution of 4 cm −1 . Figure 3 exhibits the FTIR spectra along with the indicated characteristic peaks of NBR provided in Table 2.

(b) Polymer Composition Characterization
Before performing the hardness test, each O-ring was also analyzed to assess the composition of components. In this case, the expected (and the acceptance) criterion was acrylonitrile butadiene rubber (NBR). The technique used was the Fourier-transform infrared spectroscopy (FTIR) that is based on the concept of absorption of infrared radiation by sample. The resulting signal at the detector is a spectrum that characterizes the polymer analyzed and, therefore, it allows composition data to be obtained [37]; this technique is the method used to determine if the O-ring composition is the expected one and, therefore, if the component is ready to continue the characterization process (hardness test, in this case study). The equipment used was a Nicolet 5700 (Thermo Electron Corporation, Waltham, MA, USA). Thus, spectra of the NBR components were recorded over a wavenumber range of 4000-500 cm −1 , with 32 scanning times at a resolution of 4 cm −1 . Figure 3 exhibits the FTIR spectra along with the indicated characteristic peaks of NBR provided in Table 2.
Once the composition was checked, the Shore A hardness according to ISO 868 [31] was performed.

Polymer
Number of Peak in Figure 3 Wave Number (cm −1 ) Indication/Type of Bond Identified Once the composition was checked, the Shore A hardness according to ISO 868 [31] was performed.

(c) Hardness Test
The shore hardness was measured by the depth of indentation caused by a rigid ball under a spring load or dead load, the indentation being converted to hardness degrees on a scale ranging from 0 to 100. The reading from a dead-load hardness meter is called the international rubber hardness degree (IRHD). The spring-loaded meter gives Shore A values [39]. The energy absorbed by the sample material on impact is then related to the product of a "dynamic yield pressure" and the volume of the indent [40]. As indicated by Brown [41], the test results are affected by the operator, the time of application, and the deviations from perfectly elastic despite correct calibration and measurements according to the standard testing procedure. Spetz [42] examined the repeatability of hardness measurements on rubber materials and concluded that the operator was the main source of variability [43]. Thus, during the indentation experiments, hardness changes not only with the hold time but also with loading and unloading rate [44].    Table 2. Characteristic bonds of NBR as a function of wave number in the Fourier transform infrared spectroscopy test [38].

Polymer
Number of Peak in Figure 3 Wave (c) Hardness Test The shore hardness was measured by the depth of indentation caused by a rigid ball under a spring load or dead load, the indentation being converted to hardness degrees on a scale ranging from 0 to 100. The reading from a dead-load hardness meter is called the international rubber hardness degree (IRHD). The spring-loaded meter gives Shore A values [39]. The energy absorbed by the sample material on impact is then related to the product of a "dynamic yield pressure" and the volume of the indent [40]. As indicated by Brown [41], the test results are affected by the operator, the time of application, and the deviations from perfectly elastic despite correct calibration and measurements according to the standard testing procedure. Spetz [42] examined the repeatability of hardness measurements on rubber materials and concluded that the operator was the main source of variability [43]. Thus, during the indentation experiments, hardness changes not only with the hold time but also with loading and unloading rate [44].

Polymer
Number of Peak in Figure 3 Wave Number (cm −1 ) Indication/Type of Bond Identified Once the composition was checked, the Shore A hardness according to ISO 868 [31] was performed.

(c) Hardness Test
The shore hardness was measured by the depth of indentation caused by a rigid ball under a spring load or dead load, the indentation being converted to hardness degrees on a scale ranging from 0 to 100. The reading from a dead-load hardness meter is called the international rubber hardness degree (IRHD). The spring-loaded meter gives Shore A values [39]. The energy absorbed by the sample material on impact is then related to the product of a "dynamic yield pressure" and the volume of the indent [40]. As indicated by Brown [41], the test results are affected by the operator, the time of application, and the deviations from perfectly elastic despite correct calibration and measurements according to the standard testing procedure. Spetz [42] examined the repeatability of hardness measurements on rubber materials and concluded that the operator was the main source of variability [43]. Thus, during the indentation experiments, hardness changes not only with the hold time but also with loading and unloading rate [44].   Therefore, all measurements were performed by the same operator, using calibrated equipment and not repeating the hardness measurement at the same place because it provides permanent local changes in the material [45,46]. Figure 5 shows the testing measurement locations (TMLs) used in each characterized O-ring. The hardness testing was performed by using a durometer Zwick Roel Digi-Test Shore A/B/O (Zwick Roel, Ulm, Germany). Once all hardness data was collected, an analytical procedure was carried out.

Analytical Procedure
Hardness dispersion of rubber samples can be statistically well described by a normal distribution model [47]. Thus, Shore A hardness was fitted by a random normal distribution. Certainly, the Gaussian or normal distribution is the most-established model to characterize quantitative variation of original data. Accordingly, data are typically summarized using the arithmetic mean and the standard deviation, by µ ± σ [48]. Additionally, this type of representation allowed us to easily compare the mean and deviation among different supplies (from 2014 to 2018). The expression [49] for the one-dimensional normal density is often written according to Equation (1).
where HSA is Shore A hardness, µ is the mean, and σ is the standard deviation Figure 6 provides the normal distribution (density function versus measured hardness) for each supply. Table 3 shows mean values (µ) along with the standard deviation (σ) between measurements in each group of supplies (batches) and percentage variation in hardness of these measurements of the O-rings (as supplied) compared to stored O-rings.
The hardening experienced by the O-rings was between 11.80 and 15.98% with a difference in means (recently manufactured versus stored ones) of 13.81% (according to Table 4). Consequently, Table 4 shows the mean value of more than 140 Shore A hardness measurements made during the period between 2014 and 2018. Likewise, the study incorporated 12 hardness tests on stored O-rings without a defined date [50]. Nevertheless, it is known that they were entered into inventory in 2000 and that they could be dated form as early as 1994 (calculated on the test date in 2018).     A recent study stated that the mean hardening of some NBR samples after 18 years was of 11.66% [51]; therefore, there is coherence in the observed results, especially considering that the analyzed storage time was between 18 and 24 years. Consequently, this could be considered as a validated starting point to perform further methodological analysis. Using a normal representation, Figure 7 provides the mean hardening for recently manufactured and stored O-rings. The hardening experienced by the O-rings was between 11.80 and 15.98% with a difference in means (recently manufactured versus stored ones) of 13.81% (according to Table 4). Consequently, Table 4 shows the mean value of more than 140 Shore A hardness measurements made during the period between 2014 and 2018. Likewise, the study incorporated 12 hardness tests on stored O-rings without a defined date [50]. Nevertheless, it is known that they were entered into inventory in 2000 and that they could be dated form as early as 1994 (calculated on the test date in 2018). A recent study stated that the mean hardening of some NBR samples after 18 years was of 11.66% [51]; therefore, there is coherence in the observed results, especially considering that the analyzed storage time was between 18 and 24 years. Consequently, this could be considered as a validated starting point to perform further methodological analysis. Using a normal representation, Figure 7 provides the mean hardening for recently manufactured and stored O-rings.   Figure 7 shows that the mean value for recently manufactured (HSA mean = 61.33) O-rings was close to the expected value of 60 Shore A, whereas the mean value for stored (HSA mean = 69.78) O-rings was very close to the maximum allowable hardness (HSA max = 70 Shore A). Experimental findings demonstrated that O-rings with a Shore A hardness near to 70 are prone to failure [52]; subsequently, a three differentiated ranges are defined ( Figure 8)   There are a lot of characteristics that have to be considered when a polymer candidat is evaluated for an application in the harsh environment of a nuclear plant. Some of thes features are related mainly to thermal and radiation tolerance and its influence on me chanical properties [53]. Thus, defined normal conditions ( Figure 9) allow us to conside different scenarios depending on the parameters' variability inside the constructed range After defining the storage parameters' window and the ranges of hardness associated to degradation and risk of failure (ranges that do not fulfill the safety function; i.e., pre venting leakages), an Arrhenius-based model was developed [50], according to Equation (2), to correlate operation (or storage) time with operation (or storage) temperature: where ts is the estimated lifetime in service (hours), ta is the time considering acceleration in aging/degradation (hours), Ts is the normal operating temperature (K), Ta is the hard ening temperature (K), Ea is the activation energy (eV), and K is the Boltzmann constan (0.8617·10 4 eV/K). The activation energy used in the calculation was provided by EPRI TR 1,009,748 [26] that for NBR is equal to 0.88 eV.
As was mentioned before, 14 new supplies were compared with a large, stored batch There is, therefore, an uncertainty related to the manufacturing date of stored O-rings Considering this uncertainty about the date of manufacture of the previously stored O rings, three scenarios have been defined for the analysis-very conservative, moderately conservative, and minimally conservative. Subsequently, for the conservative interval, i was considered that the age of O-rings was 24 years, for the middle one (moderately con servative), 22.5 years and for the least conservative one, 18 years old (calculated on the test date in 2018). There are a lot of characteristics that have to be considered when a polymer candidate is evaluated for an application in the harsh environment of a nuclear plant. Some of these features are related mainly to thermal and radiation tolerance and its influence on mechanical properties [53]. Thus, defined normal conditions (Figure 9) allow us to consider different scenarios depending on the parameters' variability inside the constructed range.  [52]; subsequently, a three differentiated ranges are defined (Figure 8) as suitable, safe, and embrittled zones according to O-ring hardness. Thus, the risk associated with O-ring failure increases with Shore A hardness. There are a lot of characteristics that have to be considered when a polymer candidate is evaluated for an application in the harsh environment of a nuclear plant. Some of these features are related mainly to thermal and radiation tolerance and its influence on mechanical properties [53]. Thus, defined normal conditions (Figure 9) allow us to consider different scenarios depending on the parameters' variability inside the constructed range. After defining the storage parameters' window and the ranges of hardness associated to degradation and risk of failure (ranges that do not fulfill the safety function; i.e., preventing leakages), an Arrhenius-based model was developed [50], according to Equation (2), to correlate operation (or storage) time with operation (or storage) temperature: where ts is the estimated lifetime in service (hours), ta is the time considering acceleration in aging/degradation (hours), Ts is the normal operating temperature (K), Ta is the hardening temperature (K), Ea is the activation energy (eV), and K is the Boltzmann constant (0.8617·10 4 eV/K). The activation energy used in the calculation was provided by EPRI TR 1,009,748 [26], that for NBR is equal to 0.88 eV.
As was mentioned before, 14 new supplies were compared with a large, stored batch. There is, therefore, an uncertainty related to the manufacturing date of stored O-rings. Considering this uncertainty about the date of manufacture of the previously stored Orings, three scenarios have been defined for the analysis-very conservative, moderately conservative, and minimally conservative. Subsequently, for the conservative interval, it was considered that the age of O-rings was 24 years, for the middle one (moderately conservative), 22.5 years and for the least conservative one, 18 years old (calculated on the test date in 2018). After defining the storage parameters' window and the ranges of hardness associated to degradation and risk of failure (ranges that do not fulfill the safety function; i.e., preventing leakages), an Arrhenius-based model was developed [50], according to Equation (2), to correlate operation (or storage) time with operation (or storage) temperature: where t s is the estimated lifetime in service (hours), t a is the time considering acceleration in aging/degradation (hours), T s is the normal operating temperature (K), T a is the hardening temperature (K), E a is the activation energy (eV), and K is the Boltzmann constant (0.8617·10 4 eV/K). The activation energy used in the calculation was provided by EPRI TR 1,009,748 [26], that for NBR is equal to 0.88 eV.
As was mentioned before, 14 new supplies were compared with a large, stored batch. There is, therefore, an uncertainty related to the manufacturing date of stored O-rings. Considering this uncertainty about the date of manufacture of the previously stored Orings, three scenarios have been defined for the analysis-very conservative, moderately conservative, and minimally conservative. Subsequently, for the conservative interval, it was considered that the age of O-rings was 24 years, for the middle one (moderately conservative), 22.5 years and for the least conservative one, 18 years old (calculated on the test date in 2018).

Results and Discussion
Once the testing and the first statistical analysis (Stage 1) had been performed, a reliability estimation was carried out in order to develop a degradation model with respect to storage conditions, such as temperature or time (Stage 2).

Stage 2.: Reliability Estimation and Degradation Model Development
Considering a well-stablished correlation between hardening and temperature, the Arrhenius model can be rearranged [52], according to Equation (3), to obtain in-service durability (t s ): where HSA augm is the Shore A hardness augmented due to thermal aging with respect to HSA exp (expected HSA). Thus, with the measured hardness for recently manufactured and stored O-rings, in-service durability was calculated ( Figure 10).

Results and Discussion
Once the testing and the first statistical analysis (Stage 1) had been performe ability estimation was carried out in order to develop a degradation model with to storage conditions, such as temperature or time (Stage 2).

Stage 2.: Reliability Estimation and Degradation Model Development
Considering a well-stablished correlation between hardening and tempera Arrhenius model can be rearranged [52], according to Equation (3), to obtain in durability (ts): where HSAaugm is the Shore A hardness augmented due to thermal aging with re HSAexp (expected HSA). Thus, with the measured hardness for recently manufactured and stored O-r service durability was calculated ( Figure 10).  It can be concluded that the materials' response could be considered similar vious operation time of 6 years (52,560 h). If we considered the extreme case in w shore A is reached, TTIL would be equal to 4 years (35,040 h). On the other hand, reformulation of Arrhenius model is performed, Equation (5) provides the hard a function of the durability of recently manufactured O-rings (ts) and stored ones the measured hardness once stored (HSA).
(%) = 100• • • Subsequently, Figure 11 exhibits the maximum recommendable in-service t function of hardening (from hardness values). This representation was performed ing to Equation (5). The storage conditions (and their associated hardening) have a direct effect on inservice durability, reducing it by up to 60.40%. Thus, time to integrity loss (TTIL), considered as the time in operation where a Shore A hardness equals 65 (beginning of the embrittlement; according to Figure 9), can be calculated by using the Equation (4).
It can be concluded that the materials' response could be considered similar to a previous operation time of 6 years (52,560 h). If we considered the extreme case in which 70 shore A is reached, TTIL would be equal to 4 years (35,040 h). On the other hand, if a new reformulation of Arrhenius model is performed, Equation (5) provides the hardening as a function of the durability of recently manufactured O-rings (t s ) and stored ones (t a ) and the measured hardness once stored (HSA).
Subsequently, Figure 11 exhibits the maximum recommendable in-service time as a function of hardening (from hardness values). This representation was performed according to Equation (5).
Polymers 2021, 13, x FOR PEER REVIEW Figure 11. Maximum recommendable in-service time (ts) as a function of hardening (from h ness values).
As Figure 11 indicates, the measured hardening can be a useful parameter to e the maximum recommendable in-service time. Considering that no measurable h ing (i.e., a value of 60 HSA) implies the maximum in-service time (10 years, that i usual qualified lifetime for O-rings in the nuclear industry), a hardening of 5% ge a reduction of the recommended in-service time of 50% (i.e., a recommended time of 5 years), while a hardening of 10% implies a usability for only 3 years.
Using an exponential distribution for the degradation (according to the Arr model), the reliability function R(t) can be calculated [24] according to Equation (6 ( ) = −λt where λ is the failure rate, calculated as = 1 and t, the considered time.
The reliability of new (recently manufactured) and stored O-rings, represent function of the hardness, is shown in Figure 12.   As Figure 11 indicates, the measured hardening can be a useful parameter to estimate the maximum recommendable in-service time. Considering that no measurable hardening (i.e., a value of 60 HSA) implies the maximum in-service time (10 years, that it is the usual qualified lifetime for O-rings in the nuclear industry), a hardening of 5% generates a reduction of the recommended in-service time of 50% (i.e., a recommended time of use of 5 years), while a hardening of 10% implies a usability for only 3 years.
Using an exponential distribution for the degradation (according to the Arrhenius model), the reliability function R(t) can be calculated [24] according to Equation (6).
where λ is the failure rate, calculated as λ = 1 TTIL and t, the considered time. The reliability of new (recently manufactured) and stored O-rings, represented as a function of the hardness, is shown in Figure 12. As Figure 11 indicates, the measured hardening can be a useful parameter to estimate the maximum recommendable in-service time. Considering that no measurable hardening (i.e., a value of 60 HSA) implies the maximum in-service time (10 years, that it is the usual qualified lifetime for O-rings in the nuclear industry), a hardening of 5% generates a reduction of the recommended in-service time of 50% (i.e., a recommended time of use of 5 years), while a hardening of 10% implies a usability for only 3 years.
Using an exponential distribution for the degradation (according to the Arrhenius model), the reliability function R(t) can be calculated [24] according to Equation (6).
where λ is the failure rate, calculated as = 1 and t, the considered time.
The reliability of new (recently manufactured) and stored O-rings, represented as a function of the hardness, is shown in Figure 12.   As Figure 13 shows, the R(t) of stored manufactured O-rings was greater than the R(t) of the recent ones, independently of the hardness range. Nevertheless, the loss of reliability for recently manufactured O-rings was bigger when the hardening was greater. This is very reasonable because a hardening found in a recently manufactured O-rings probably implies a defective mechanical integrity or a degraded composition, while the same value for a stored O-ring just indicates that a hardening process took place. On the other hand, probability of failure distribution (POF (t)) can be calculated [24] according to Equation (7).
Thus, R(t) and POF (t) related to the performance fulfillment are represented ( Figure  14) as a function of the measured hardness. As a degradation and, therefore, a loss of integrity is expected when hardening takes place, in the case of stored O-rings (Figure 14a), a loss of more than the 20% of reliability is presented from a value equal to 65 Shore A hardness, being more than 50% from 70 shore A hardness. On the other hand, in the case of recently manufactured O-rings a hardness equal to 60 ± 5 is expected (as typically required by manufacturer; see Figure 9) showing a good reliability. Thus, a loss of 20% of reliability takes places when hardness is increased up to 65 shore A hardness, and from 68 HSA when the reliability is less than 50% (Figure 14b). In addition, POF as a function of the operation time (years), for both recently manufactured and stored O-rings (with a hardness close to 65 HSA) is shown in Figure  15. As Figure 13 shows, the R(t) of stored manufactured O-rings was greater than the R(t) of the recent ones, independently of the hardness range. Nevertheless, the loss of reliability for recently manufactured O-rings was bigger when the hardening was greater. This is very reasonable because a hardening found in a recently manufactured O-rings probably implies a defective mechanical integrity or a degraded composition, while the same value for a stored O-ring just indicates that a hardening process took place. On the other hand, probability of failure distribution (POF (t)) can be calculated [24] according to Equation (7).
Thus, R(t) and POF (t) related to the performance fulfillment are represented ( Figure 14) as a function of the measured hardness. As Figure 13 shows, the R(t) of stored manufactured O-rings was greater than the R(t) of the recent ones, independently of the hardness range. Nevertheless, the loss of reliability for recently manufactured O-rings was bigger when the hardening was greater. This is very reasonable because a hardening found in a recently manufactured O-rings probably implies a defective mechanical integrity or a degraded composition, while the same value for a stored O-ring just indicates that a hardening process took place. On the other hand, probability of failure distribution (POF (t)) can be calculated [24] according to Equation (7).
Thus, R(t) and POF (t) related to the performance fulfillment are represented ( Figure  14) as a function of the measured hardness. As a degradation and, therefore, a loss of integrity is expected when hardening takes place, in the case of stored O-rings (Figure 14a), a loss of more than the 20% of reliability is presented from a value equal to 65 Shore A hardness, being more than 50% from 70 shore A hardness. On the other hand, in the case of recently manufactured O-rings a hardness equal to 60 ± 5 is expected (as typically required by manufacturer; see Figure 9) showing a good reliability. Thus, a loss of 20% of reliability takes places when hardness is increased up to 65 shore A hardness, and from 68 HSA when the reliability is less than 50% (Figure 14b). In addition, POF as a function of the operation time (years), for both recently manufactured and stored O-rings (with a hardness close to 65 HSA) is shown in Figure  15. As a degradation and, therefore, a loss of integrity is expected when hardening takes place, in the case of stored O-rings (Figure 14a), a loss of more than the 20% of reliability is presented from a value equal to 65 Shore A hardness, being more than 50% from 70 shore A hardness. On the other hand, in the case of recently manufactured O-rings a hardness equal to 60 ± 5 is expected (as typically required by manufacturer; see Figure 9) showing a good reliability. Thus, a loss of 20% of reliability takes places when hardness is increased up to 65 shore A hardness, and from 68 HSA when the reliability is less than 50% (Figure 14b). In addition, POF as a function of the operation time (years), for both recently manufactured and stored O-rings (with a hardness close to 65 HSA) is shown in Figure 15.  Figure 15 indicates how the accumulated in-service time affects the risk of failure (losing reliability). During the three first years of operation the increase of POF is practically insignificant. Nevertheless, from this point, and especially, from 5 years of operation, the POF increased from 10% to 20% at approximately 6 years (for new and stored). From 6 years, the behavior of curves (for new and stored) are more different. In the case of stored ones, there is a linear progression up to reaching a POF equal to 0.78 at 10 years; while, in the case of the new ones, the POF is practically 100% when they reach an accumulated in-service time equivalent to 10 years. Seen from another point of view, the annualized loss of reliability can be quantified as a function of the hardening (or the measured value of hardness). As the last compared supply was dated in 2018, three comparative scenarios were established (very conservative, 24 years; medium, 22.5 years; and minimally conservative, 18). Thus, Figure 16 provides the annualized loss of reliability considering the three scenarios of analysis; since the loss of reliability is considered, in this case, to be due to the storage (and, therefore, the comparison of the three scenarios needs to be more precise). POF (t) represented in Figure 15 indicates how the accumulated in-service time affects the risk of failure (losing reliability). During the three first years of operation the increase of POF is practically insignificant. Nevertheless, from this point, and especially, from 5 years of operation, the POF increased from 10% to 20% at approximately 6 years (for new and stored). From 6 years, the behavior of curves (for new and stored) are more different. In the case of stored ones, there is a linear progression up to reaching a POF equal to 0.78 at 10 years; while, in the case of the new ones, the POF is practically 100% when they reach an accumulated in-service time equivalent to 10 years. Seen from another point of view, the annualized loss of reliability can be quantified as a function of the hardening (or the measured value of hardness). As the last compared supply was dated in 2018, three comparative scenarios were established (very conservative, 24 years; medium, 22.5 years; and minimally conservative, 18). Thus, Figure 16 provides the annualized loss of reliability considering the three scenarios of analysis; since the loss of reliability is considered, in this case, to be due to the storage (and, therefore, the comparison of the three scenarios needs to be more precise). POF (t) represented in Figure 15 indicates how the accumulated in-service time affects the risk of failure (losing reliability). During the three first years of operation the increase of POF is practically insignificant. Nevertheless, from this point, and especially, from 5 years of operation, the POF increased from 10% to 20% at approximately 6 years (for new and stored). From 6 years, the behavior of curves (for new and stored) are more different. In the case of stored ones, there is a linear progression up to reaching a POF equal to 0.78 at 10 years; while, in the case of the new ones, the POF is practically 100% when they reach an accumulated in-service time equivalent to 10 years. Seen from another point of view, the annualized loss of reliability can be quantified as a function of the hardening (or the measured value of hardness). As the last compared supply was dated in 2018, three comparative scenarios were established (very conservative, 24 years; medium, 22.5 years; and minimally conservative, 18). Thus, Figure 16 provides the annualized loss of reliability considering the three scenarios of analysis; since the loss of reliability is considered, in this case, to be due to the storage (and, therefore, the comparison of the three scenarios needs to be more precise). In the range of 55-65 HSA, the loss of reliability due to storage was small, with hardly any difference between 55 and 60 and 60 and 65 (zone of acceptable values). The loss of reliability accelerated in the range 65-70 (the slope is greater), being higher from 70 Shore A. Equation (8) provides the time to damage (TTD) that is calculated from TTIL obtained from Equation (4).

POF (t) represented in
On the other hand, using a defined safety factor (SF), a recommended replacement interval (RPI) can be calculated using the Equation (9).
According to Equations (8) and (9) and as function of different values of maximum allowable loss of reliability, TTD and RPI are obtained for recently manufactured O-rings (as the worst favorable scenario) with HSA > 65. This is presented in Table 5. Thus, applying this model, if O-rings are replaced in annual operation of maintenance, the reliability of O-rings with a hardness of 65 shore A is 0.85, whereas in the case of O-rings with a hardness of 68 shore A it is 0.78. Nevertheless, the recommendation is to use O-rings with a hardness lower than 60 HSA, to ensure a reliability above 0.90. Table 6 shows the maximum temperature obtained using Equation (1) and the calculation parameters indicated in Note * 3 (at the bottom of the Table) and considering the three scenarios (as defined in Section 3.1) Table 6. Prediction of the maximum allowable storage temperature according to the Arrhenius model.

Scenario of Analysis
Maximum Allowable Storage Temperature ( • C) * 3
In view of the results presented in Table 6, it can be concluded that the limit conditions for prolonged storage considering any of the three contemplated scenarios would be above the real conditions. That is, even in the case of the least conservative scenario, the maximum temperature predicted by the model is 25.17 • C, which is slightly higher than the maximum real temperature (according to Note * 1 in Table 4 On the other hand, a validation (Table 7) is performed to check if the analyzed assumptions stated in the analytical procedure (18, 22.5, and 24 years) and the maximum allowable hardness value according to the catalog would be reached for these NBR gaskets, that is, a value of 70 Shore A [42]. Adapting the model to predict in each of the three scenarios which maximum allowable hardness value (70 Shore A) (defined as the upper limit) would be reached, it was verified that for any of the scenarios the upper limit value is above the considered storage time premise (18.35 > 18 years for the least conservative scenario, 22.93 > 22.50 years for the medium scenario, and 24.46 > 24 years for the most conservative scenario). Therefore, it is possible to validate the model by ensuring that in the predictions (both for temperature ranges and for storage times) the allowable limit value of 70 Shore A is not reached in any case. Finally, an analysis to validate the methodology was performed ( Figure 17) on the basis of the representation of the expected maximum storage time (using the three considered scenarios) versus the time to reach the HSA max (70 HSA). In addition, the starting data (hardness values) showed coherence with another experimental work such as the one of Zhong et al., that provides similar embrittlement by storage for a period of 18 years [51].  Adapting the model to predict in each of the three scenarios which maximum allowable hardness value (70 Shore A) (defined as the upper limit) would be reached, it was verified that for any of the scenarios the upper limit value is above the considered storage time premise (18.35 > 18 years for the least conservative scenario, 22.93 > 22.50 years for the medium scenario, and 24.46 > 24 years for the most conservative scenario). Therefore, it is possible to validate the model by ensuring that in the predictions (both for temperature ranges and for storage times) the allowable limit value of 70 Shore A is not reached in any case. Finally, an analysis to validate the methodology was performed (Figure 17) on the basis of the representation of the expected maximum storage time (using the three considered scenarios) versus the time to reach the HSAmax (70 HSA). In addition, the starting data (hardness values) showed coherence with another experimental work such as the one of Zhong et al., that provides similar embrittlement by storage for a period of 18 years [51].

Conclusions and Future Work
The major conclusions resulting from this work can summarized as follows:

•
The measured hardening can be a useful parameter to estimate the maximum recommended in-service time. A hardening of 5% generates a reduction of the recommended in-service time of 50% (i.e., a recommended time of use of 5 years), while a hardening of 10% implies a usability of only 3 years.

•
The storage (and its associated hardening) of the NBR O-rings has a direct effect on the in-service durability, reducing this by up to 60.40%. Thus, the calculated time to integrity loss (TTIL), which is considered as the time in operation where a Shore A hardness equal to 65 is reached (beginning of the embrittlement), is 6 years (52,560 h).

•
During the first three years of operation the increase of POF is practically insignificant. Nevertheless, from this point, and especially, from 5 years of operation, the POF increases from 10% to 20% at approximately 6 years (for new and stored).

Conclusions and Future Work
The major conclusions resulting from this work can summarized as follows: • The measured hardening can be a useful parameter to estimate the maximum recommended in-service time. A hardening of 5% generates a reduction of the recommended in-service time of 50% (i.e., a recommended time of use of 5 years), while a hardening of 10% implies a usability of only 3 years.

•
The storage (and its associated hardening) of the NBR O-rings has a direct effect on the in-service durability, reducing this by up to 60.40%. Thus, the calculated time to integrity loss (TTIL), which is considered as the time in operation where a Shore A hardness equal to 65 is reached (beginning of the embrittlement), is 6 years (52,560 h).

•
During the first three years of operation the increase of POF is practically insignificant. Nevertheless, from this point, and especially, from 5 years of operation, the POF increases from 10% to 20% at approximately 6 years (for new and stored). • From 6 years of operation, the behavior of curves (for new and stored) are very different. In the case of stored ones, there is a linear progression up to reaching a POF equal to 0.78 at 10 years, while, in the case of the new ones, the POF is practically 100% when an accumulated in-service time equivalent to 10 years is reached. • A validation of the methodology was performed by comparing the predicted allowable storage periods and conditions with the real ones. Thus, applying this model, if O-rings are replaced in annual operation of maintenance, the reliability of O-rings with a hardness of 65 shore A is 0.85, whereas in the case of O-rings with a hardness of 68 shore A it is 0.78. • From the study, the general recommendation is using O-rings with a HSA less than 60 HSA, to ensure a reliability above 0.90. Finally, it was proved that the storage strategies of our nuclear power plants are successful, perfectly meeting the expectations of suitability and functionality of the components when they are installed after storage.
Finally, this methodology can be used in the future to analyze the suitability of other polymers after a long storage period.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.