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Service life prediction for aircraft coatings
Polymer Degradation and Stability 82 (2003) 1–13 www.elsevier.com/locate/polydegstab
Service life prediction for aircraft coatings Olga Guseva*, Samuel Brunner, Peter Richner Swiss Federal Laboratory for Materials Testing and Research, U¨berlandstrasse 129, CH-8600 Du¨bendorf, Switzerland Received 3 February 2003; received in revised form 14 March 2003; accepted 16 March 2003
Abstract A model with three stress types, the temperature, UV and aerosol, for estimating the service life of organic coatings under service conditions has been proposed. The model has been applied to the estimation of the service life for aircraft coatings concerning loss of gloss. The results obtained from a unique natural exposure programme for a reference polyurethane coating have been compared with the results from the specially designed accelerated ageing tests, which were performed with a new designed and constructed weathering device. As a special point of interest, the influence of sulphuric aerosol, which has simulated the stratosphere conditions after a big volcano eruption, on the service life of the reference coating has been investigated. # 2003 Elsevier Ltd. All rights reserved. Keywords: Coatings; Weathering; Accelerated tests; Service life prediction
1. Introduction The organic coating industry has been performing research in order to find new chemistry formulations with enhanced stability against environmental influences, and at the same time with possibly smaller contents of hazardous substances. The service life of these new chemistry formulations is often unknown. Usually, the field experiments (natural exposure) for modern coatings take prolonged time of several years, which is not acceptable for manufactures. Therefore a need exists for a methodology which is capable of making reliable service life predictions from accelerated experiments. Such Service Life Prediction (SLP) methodology was proposed by Martin et al. [1]. In this paper we applied this approach to the ageing of aircraft coatings. The degradation of aircraft coatings was considered as a loss of gloss provided that other requirements, such as the absence of cracking and delamination, were fulfilled. The work was supported by a special research programme, whereby in-service data from the exposure of products on B-747 escape hatches were compared to the data obtained using special designed accelerated tests,
* Corresponding author Tel.: +41-1-823-4361; fax: +41-1-8234015. E-mail address: [email protected] (O. Guseva). 0141-3910/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0141-3910(03)00124-1
focusing on the development of a service life prediction model. As the initial step, the environmental factors that are relevant for the ageing of aircraft coatings had been analysed [2]. It is known that the weathering of those coatings is not only determined by UV radiation, temperature and humidity but also by pollutants. This was clearly demonstrated during the eruption of volcano Pinatubo: in June 1991 the volcanic eruption brought more than 20 million tons of sulphur dioxide into the stratosphere raising the SO2-level from normal range of 0.01–0.05 ppbv to more than 10 ppbv in September 1991. Within the next four months the concentration of gaseous SO2 in the stratosphere had gradually decreased toward its usual value [3]. The aircrafts, however, showed a fast decrease of gloss in their coating during the following 4 years. It was found that SO2 actually formed a sulphuric acid aerosol, which remained dispersed in the stratosphere for a period of several years after eruption that acted destructively on aircraft coatings [2,3].
2. Natural exposure: Escape Hatch Programme In order to investigate the degradation of the aircraft coatings under service conditions, the so-called use level conditions, a research programme (Escape Hatch Programme) was carried out by AKZO Nobel together with
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KLM starting September 1994. According to this Programme, four different coatings were applied to each escape hatch of a B-747 aircraft for natural exposure (Fig. 1). Every 3 months gloss and colour of the hatches were measured. Altogether, 15 aircrafts were involved in the Programme. One of the pigmented polyurethane coatings with a light blue colour was applied to different escape hatches providing the possibility for statistical analysis. This coating is being referred to as a ‘‘reference coating’’ throughout the paper. The colour values for the unexposed panels in the CIE 1976 L*a*b* colour system were L*=55, a*=26, b*=41. Definitions and details of the measure procedure of colour can be found in Ref. [4]. The gloss degradation curves for gloss 60 for the reference coating are shown in Fig. 2. One of the curves in Fig. 2 (with black filled squares) shows the faster degradation than others. The reason for this fast degradation is that the aircraft with this coating at the
Fig. 1. Escape hatch of the B-747 (shown with the arrow).
escape hatch was in operation during the years 1994– 1996, when the sulphuric aerosol content in the stratosphere after the Pinatubo eruption was still noticeable (for details see Ref. [2]). This curve was omitted from the statistical analysis as an outlier. All other degradation curves in Fig. 2 were measured when the aerosol concentration at a flight height of about 10 km was normal. Definitions and details of the measure procedure of gloss can be found in Ref. [4]. The failure criterion for the gloss loss, which was defined by airline representatives, was 60 GU measured for gloss 60 . The interceptions of degradation curves and the failure-criterion-line (see Fig. 2) gave times-tofailure. It should be noted that the experimental points were measured every 3 months. Therefore, in order to obtain the exact times-to failure, the degradation curves were interpolated with polynomials. As shown in Fig. 2 there were 11 degradation curves altogether, whereby two of them (dashed curves) did not reach the failure criterion at the time of paper writing. These curves were considered as censored data in the further calculations. From this data it was possible to form the sample cumulative distribution function (cdf) and fit it to a particular theoretical cdf (for basic definitions of the reliability theory see Appendix A). There are many different lifetime distributions that can be used for the data analysis. The most commonly used are exponential, Weibull, normal and lognormal distributions. The currently available sample size of 10 times-to-failure, however, was too small for a reliable determination of the underlying distribution. The simplest suitable parametric distributions, which include the degradation mechanism (their hazard function can be increasing), are the Weibull, normal and lognormal. The comparison of the Likelihood Function values for the earliermentioned distributions showed that the Weibull dis-
Fig. 2. Gloss 60 -curves for the reference coating applied on escape hatches. The bold horizontal line represents a failure criterion of 60 GU.
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tribution suited slightly better to the experimental data. Therefore, the Weibull distribution was chosen as an underlying distribution throughout the work. The application of the basic definitions of the reliability theory to the Weibull distribution are presented in Appendix A. Linearized Weibull cdf for the Escape Hatch data is shown in Fig. 3, the reliability function and probability density function (pdf) in Fig. 4. The calculations were made with Weibull++ program [5]. With the Weibull distribution parameters, the shape parameter =9.4 and the scale parameter =52.0 months, available, one can calculate various reliability parameters—some of
Fig. 3. Linearized cdf (solid line) with 90% two-sided confidence bounds (dashed lines) of Escape Hatch Data for the reference coating for gloss 60 with a failure criterion of 60 GU.
them are presented in Table 1. From the Table 1 follows, that the expected life (mean life) for the reference coating was about 49 months, whereas the warranty time 90%, which means that only 10% of the coatings reached the failure criterion, was about 41 months.
3. SLP model for organic coatings with three stresses: temperature, UV and aerosol 3.1. Accelerated testing In typical life data analysis one determines a life distribution that describes the times-to-failure of a product at the usual operating conditions. Statistically speaking, one wishes to obtain ‘‘the use level probability density function’’ of the times-to-failure. Once this pdf is obtained, all other desired reliability parameters, such as for example, percentage failing under warranty (percentile), mean time to failure, mode life can be easily determined. However, coatings are supposed to have a rather prolonged life of several years. At the stage of development of a new formulation, researchers usually do not have enough time to carry out the natural ageing at the so-called use level conditions. To reduce the time of experiment one performs experiments at the enhanced stress levels to obtain pdfs at these high stress levels. The goal is to determine the use level pdf from these accelerated life test data rather than from data obtained under use conditions. To accomplish this, a relationship should be found that allows extrapolating from data collected at accelerated conditions to use level conditions, the so-called life-stress relationship. Some of the most commonly used life-stress relationships in accelerated life testing are Arrhenius (when the temperature is an accelerating factor), inverse power law (usually used for non-thermal accelerated stresses) and Eyring (for thermal or humidity stresses) (see Appendix B). Schematically the procedure of service life prediction described above is presented in Fig. 5. Here it is worth
Table 1 Reliability parameters together with 90% two-sided confidence bounds for gloss loss of reference coating obtained in the Escape Hatch Programme for gloss 60 with a failure criterion of 60 GU 90% Two-sided confidence bounds
Fig. 4. Reliability function (solid line) with 90% two-sided confidence bounds (dashed lines) (left) and pdf (dotted line) (right) of Escape Hatch Data for gloss 60 with a failure criterion of 60 GU.
b Characteristic life (Z) [month] Mean life [month] Median life [month] Mode life [month] Warranty time 90% [month]
Lower limit
Upper limit
9.4 52.0
6.5 49.5
13.7 54.5
49.3 50.0 51.3 40.9
46.7 47.4 48.7 36.8
52.0 52.6 54.1 45.5
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effect, was treated as an indicator variable, taking the discrete values of zero (0), when no aerosol was applied, and one (1), when aerosol was put into a chamber. To combine all these stress types together, we used the General Log–Linear relationship (see Appendix B). The General Log–Linear–Weibull model was derived by setting the scale parameter to the life–stress relationship for a model with three stress types; the shape parameter was assumed to remain constant across different stress levels (Appendix C). The general model for three stress factors is, therefore, ¼e Fig. 5. General scheme of SLP model for organic coatings.
to note that the accelerated test should be designed in such a way that it does not change the failure modes that would be encountered at use conditions. For example, a high temperature stress level for a coating should not exceed the glass transition temperature. Analysis of accelerated life test data, then, consists of an underlying life distribution that describes the product at different stress levels and a life–stress relationship that quantifies the manner in which the life distribution changes across different stress levels. 3.2. SLP model with three stresses
¼ e0 e
1 1 V V 1V 2e 3 3; 2
ð1Þ
where is the scale parameter of Weibull distribution (characteristic life), 0, 1, 2, 3 are the model parameters, V1 is the thermal stress, with the temperature expressed in absolute units (K), V2 is the UV irradiance at 340 nm in [W/(m2 nm)], V3 is the aerosol stress, assuming values of 0 and 1. Note, that the parameter B of the Arrhenius relationship from Table 6 is equal to the log-linear coefficient 1, and the parameter n of the inverse power relationship from Table 6 is equal to (2). The General Log–Linear–Weibull pdf for these three stresses is given by: 1 1 0 þ1 V1 þ2 lnðV2 Þþ3 V3 fðt; V1 ; V2 ; V3 Þ ¼ t e ð2Þ 1 et
As discussed earlier, aircraft coatings are sensitive to the following stress parameters: temperature, UV radiation, humidity and sulphuric aerosol [2]. Although the actual stresses vary with time both in the natural weathering and in the artificial ageing employed, in this paper we made the assumption of constant stress levels, that the stress loads applied to panels were constant with respect to time. The averaging procedures are described later. Humidity as a stress parameter was excluded from further considerations because the variation of this parameter across different aircrafts is not so distinguishable. Therefore the influence of this parameter was not investigated directly in this work. However, in the artificial weathering procedure employed the effect of humidity was not ignored: relative humidity changes from 10% to 100% during the weathering cycle were applied. The data obtained from the accelerated weathering were analysed assuming an Arrhenius life–stress relationship for temperature and the inverse power life– stress relationship for UV irradiance (Appendix B). The Arrhenius term is widely used for the dependence of coating photooxidation on temperature [6,7] and a power law expression for the influence of ultraviolet irradiance [6,7]. The third stress factor, the aerosol
1 0 þ1 V þ2 lnðV2 Þþ3 V3 1
e
0 þ1
V1 þ2 lnðV2 Þþ3 V3
:
Parameters , 0, 1, 2, 3 can be obtained by maximising the log-likelihood function, (see Appendix C), using ALTA PRO program [8].
4. Experimental apparatus In order to test coatings for aircraft applications, a device based on a modified commercial Accelerated Weathering Tester QUV was designed and constructed at EMPA [9]. The commercial device was equipped with eight separate thermally insulated chambers where eight weathering experiments can be run simultaneously. The damaging effects of sunlight were simulated by fluorescent UVA-340 lamps. The selection of the lamp type suitable for the weathering of aircraft coatings was discussed in details in Ref. [2]. The following weathering cycle was used: 8 h of UV with approximately 15% humidity followed by 4 h of condensation without irradiation with approximately 100% humidity. An additional water cooling system was installed in the chambers 1 and 2 that allowed us to achieve a maximal temperature difference of 20 K between the different chambers. For more details see Ref. [9].
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In the following calculations we considered the temperature and UV stresses as constant stresses with the values obtained by averaging over the cycle. For the temperature and UV the weighted averages over the cycle for each chamber were calculated using procedure described later [see Eq. (4)] which are presented in Table 2. In order to investigate the influence of the sulphuric aerosol on coatings, sulphuric acid with a concentration of 0.5 mol/l was injected using a nebulizer into five chambers. The duration of aerosol application was 40 min once a week per chamber. For the further calculations, this third stress, aerosol, was treated as an indicator variable, assuming discrete values of zero (no aerosol application) and one (aerosol application). Two test panels with the reference coating were placed in each chamber for exposure. It should be noted that those panels had been purposely manufactured using different batches and by different persons. In order to obtain the uniform exposition of all panels, a rotational procedure was employed: once a week the position of all panels was changed according to the schema described in Ref. [9]. Measurements of gloss were performed every 3 weeks. Five points from the top to the middle of the panel were measured. These points were treated independently in the following calculations, so that for each combination of stresses we dealt with a collection of 10 points. The averaged experimental curves for all chambers for gloss measured at 60 are shown in Fig. 6, where the filled black symbols were used for the chambers with aerosol application. From Fig. 6 follows that the sulphuric aerosol application considerably accelerated the degradation of the reference coating. The panels in the chamber 2 had not reached the failure criterion at the time of paper writing. In order to obtain times-to-failure the graphical extrapolation was used.
Table 2 Averaged stress levels used in each chamber for the accelerated ageing of reference coating Chamber
1 2 3 4 5 6 7 8 a
Stress 1
Stress 2
Stress 3
Temperature T [ C]
Temperature T [K]
UV radiation UV [W/m2 nm]
Aerosol
37.5 38.1 57.5 58.2 53.8 54.3 56.2 56.5
310.7 311.2 330.7 331.4 326.9 327.4 329.4 329.6
0.58 0.41 0.42 0.57 0.60 0.43 0.63 0.62
1a 0 1 0 1 1 0 1
1 Means sulphuric aerosol application 40 min once a week; 0—no aerosol application.
5
Fig. 6. Experimental gloss-60 curves for the reference coating for chambers 1–8. Filled black symbols denote the chambers with aerosol application.
5. Results and discussion 5.1. Shape parameter First, separate calculations for each chamber assuming the Weibull distribution were performed by means of the program Weibull++ [5]. For parameter estimation, the Least Squares method, or more precisely, the Rank Regression on X (RRX), was used. This method requires that a straight line be fitted to a set of data points such that the sum of the squares of horizontal deviations from the points to the line is minimised. The measure of how well the linear model fits the data is the correlation coefficient, denoted here by . The summary is presented in Fig. 7 and Table 3. Although the values of seem to be different (see Table 3), they do not much influence the shape of the Weibull distribution: in Fig. 7 all lines are almost parallel. The coefficient of skewness (for definition see Appendix A), which is a measure for the distribution asymmetry, ranges between 0.75 (for =13) and 1.1 (for =121). For the analysis of accelerated test data the assumption of the common shape parameter across the chambers (which supposes the parallelism of the lines in Fig. 7) should be made. Assuming the underlying Weibull distribution with the common shape parameter and the general log–linear life–stress relationship, the calculation of the service life for reference coating at a particular use level was performed using the program ALTA PRO 6 [8]. For the parameter estimation, the Maximum Likelihood Parameter estimation (MLE) method was used. The calculated value for the common shape parameter was 18.9. The use level was assumed to be 32 C (305 K) for the temperature parameter and 0.24 W/(m2 nm) for the UV parameter; the aerosol parameter was set to zero (see the later definition of the use level conditions). The results are shown in Fig. 8. In order to justify the assumption of the common for our data
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Fig. 7. Cumulative distribution functions for gloss loss for the reference coating with a failure criterion of 60 GU for gloss 60 , shown on Weibull paper for all chambers together. Filled black symbols denote the chambers with aerosol application. Table 3 Shape parameter () and scale parameter (characteristic life ) for the gloss loss with a failure criterion of 60 GU measured at gloss 60 for the reference coating calculated for each chamber separately Chamber
1 2 3 4 5 6 7 8
29.9 13.3 55.7 19.4 121.2 64.8 27.2 39.8
11.4 20.7 5.9 8.2 5.3 5.7 8.2 4.9
0.96 0.96 0.97 0.90 0.95 0.91 0.95 0.98
set, the likelihood ratio (LR) test was applied (Appendix D) using ALTA PRO 6 software [8], which gave the result that the shape parameter estimates of 18.9 did not differ statistically significantly at the 90% level. 5.2. Life–stress relationship Using the procedure described in the section ‘‘SLP model with three stresses’’ the following parameters were obtained for the life–stress relationship using notations of the Eq. (1): 0=10.7114, 1=4215 K, 2=0.4608, 3=-0.5249. While the parameter 0 is a
Fig. 8. Cumulative distribution functions for gloss loss for the reference coating with a failure criterion of 60 GU for gloss 60 , shown on Weibull paper assuming the common shape parameter across all chambers. The dashed line represents the extrapolation to the use level. Filled black symbols denote the chambers with aerosol application.
constant, the parameters 1, 2 and 3 correspond to the influence of the temperature, UV irradiance, and aerosol, respectively, on the service life. The calculation of the apparent activation energy (Ea=1.R, where the gas constant R=8.314 JK1 mol1) gave a value of 8.4 kcal/mol, which was within the interval 7.4–10.3 kcal/mol (30.8–43.1 kJ/mol) obtained by Allen et al. for photooxidative degradation of low-density polyethylene film materials containing nine types of titanium dioxide pigments [10]. A value for the apparent activation energy of approximately 7.2 kcal/mol (30 kJ/mol) was also utilised in the work of Sampers [11] for the estimation of the lifetime for LDPE films with and without stabilisers. These values for the apparent activation energy are close to the values of 5–8 kcal/mol used by Bauer [6] for photooxidation in automotive coatings. A similar approach to Bauer’s was employed by Jorgensen et al. [7] for the service life prediction for clear coat/coloured basecoat paint systems. The following generalised cumulative dosage model for the loss in performance, P, was used in Ref. [7]: ðt DPðtÞ ¼ A ½IUV ð Þn eE=kTðÞ d; ð3Þ 0
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where IUV is the cumulative UV light dosage integrated over a bandwidth of 290–385 nm; A, n, E are fitting parameters, where E denotes the activation energy; k is the Boltzmann constant. Depending on the system, the values 3.8–8.4 kcal/mol for the apparent activation energy and 0.64–0.71 for the power parameter n were obtained. The parameter 2=0.46, gained in our calculations, which characterises the influence of the UV-A irradiance at 340 nm on the reference coating, is close to the model parameter n of Jorgensen et al. [7]. In other models this parameter was usually assumed to be 1 [11,12]. For the first time the influence of the aerosols on the service life of coatings was considered quantitatively. It is known that the service life of aircraft coatings was reduced by approximately 50% in the years from 1992 to 1995 following the eruption of the volcano Pinatubo in Philippines in June 1991 [2]. Our first simplified approximation using indicator variable showed that the presence of the sulphuric aerosol shortened the service life of the reference coating by at least of 40%: keeping other parameters V1 (temperature) and V2 (UV) fixed at the use level, the ratio of the characteristic lives () with and without aerosol V3 ¼1 application is ¼ 0:59. More precise conclusions V3 ¼0 can be made using the quantitative value for the concentration of the sulphuric atoms absorbed by the coating during the ageing process. A suitable method for obtaining such values is X-ray photoelectron spectroscopy (XPS). Our first preliminary XPS measurements showed that the chosen concentration of the sulphuric aerosol combined with the application duration in our weathering experiments roughly corresponded to the Pinatubo influence on the reference coating. It follows, therefore, that our experimental conditions simulated well the influence of the big volcano eruption on the aircraft coatings. 5.3. Determination of the use level conditions Definition of the use level conditions for aircrafts is quite a complicated problem. Our goal here was to obtain two numbers for the temperature and the UV-irradiance at 340 nm averaged over time (including different seasons) and space (across different destinations throughout the world). The averaging proposed later was not perfect because too many factors had to be taken into account. Nevertheless, for the first estimation of the use level conditions we used the following procedure, where the Arrhenius term was used for the averaging of temperature and a power law for the averaging of UV radiation. For the first simplified estimation, the whole time interval was represented by a sum of subintervals ti for the temperature and the UV intensity, respectively,
T ¼
X X ðTi pi Þ= pi i
i
I ¼
X X ðIi pi Þ= pi ; i
ð4Þ
i
where the weighting function has the form n pi ¼ IUVðat340nmÞi e½Ea =RTi Þ Dti
ð5Þ
with Ea=8.4 kcal/mol and n=0.46 as it was obtained in the previous section as fitting coefficients for the life–stress relationship in the accelerated weathering test for the reference coating. Calculations were made using the above approach with the averaging over one year. We assumed that the temperature ageing of aircraft coatings occurs mainly on ground and during take-offs and landings. Mathematically, for the calculations of use level conditions, the Arrhenius term in the Eq. (5) is very small for the temperature of approximately 50 C at the flight height. In order to determine how much time an aircraft spend on the ground, the flying time data available from the Escape Hatch Programme were analysed, from which an assumption of flying time of approximately 62% was made. From the climatic databases [13,14], air temperatures and UV data for some destination sites in the USA, Europe, Far East and Africa were taken. In our weathering experiments, however, we were dealing with the panel temperatures, measured at the backside of panels. It is known, that the panel temperature is considerably higher than the air temperature: 5–7 C for light and 15–30 C for dark colours [6]. To find out the actual temperature difference between the panel and air temperatures for our particular reference coating with a light blue colour (L*=55, a*=26, b*=41) an additional natural exposure experiment was carried out. The panels were exposed during one summer month in the suburban National Air Pollution Monitoring Network (NABEL) site in Du¨bendorf, Switzerland [15]. In order to model the airport conditions, the panels were placed above the asphalt ground. The sensors were mounted to the backside of the panels. The experiment was conducted for the panels with isolated and non-isolated back sides. All measured temperatures were compared with the air temperature measured at the test site. The following results were obtained for the noon time: the panel temperature was about 3 K higher than the meteorologically measured air temperature for the nonisolated panel and about 16 K higher for the isolated one. As in the case of aircraft coatings we are dealing with the thermally isolated aircraft body, the difference of 16 K for panel temperature was added to the air temperatures for all destination sites as the input temperature in Eq. (4) before the averaging. The calculated use level conditions obtained with this procedure were 32 C (305 K) for the temperature and 0.24 W/(m2 nm) for the UV-parameter. The aerosol parameter for the use level for the current conditions (after 1996) was assumed to be zero: there
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were no big volcano eruptions in the past since the Pinatubo eruption in 1991, which could have a significant influence on aircraft coatings. 5.4. Reliability parameters at the use level The inclined dashed line on the right in Fig. 8 shows the extrapolation to the use level. It should be noted that the calculated reliability parameters at the use level were dependent much on the choice of the use level conditions, especially temperature. Calculations showed that a difference of two degrees led to a difference of about 3 months in the prediction times.
The calculated cdf and pdf at the use level are shown in Fig. 9. The calculated reliability parameters at a use level of temperature of 32 C, UV of 0.24 W/(m2 nm) and aerosol parameter of zero are presented in Table 4. The linearised life–stress relationships and the acceleration factors were obtained by keeping two of the three stresses at the use level and varying the other one. The plots are presented in Fig. 10, for temperature as the acceleration factor, and in Fig. 11 for UV as the acceleration factor. The aerosol parameter was set to zero in these calculations. For example, at 60 C (333 K), UV=0.24 W/(m2 nm) and aerosol parameter=0, the acceleration factor is 3.2; and at UV=0.8 W/(m2 nm), if the temperature and aerosol parameter are kept at a use level of 32 C and 0, respectively, the acceleration factor is 1.7. The influence of the aerosol application on the service life is presented in Fig. 12. As Table 4 Reliability parameters together with 90% two-sided confidence bounds for gloss loss of reference coating at the use level for gloss 60 with a failure criterion of 60 GU 90% Two-sided confidence bounds
Fig. 9. Reliability function with 90% two-sided confidence bounds (left) and pdf (right) at the use level for the reference coating for gloss 60 with a failure criterion of 60 GU.
Characteristic life [month] Mean life [month] Median life [month] Mode life [month] Warranty time 90% [month]
Lower limit
Upper limit
18.9 43.2
16.4 40.1
21.8 46.6
42.0 42.4 43.1 38.4
39.0 39.3 40.0 35.5
45.3 45.7 46.4 41.5
Fig. 10. Linearized life–stress plot (left) and the acceleration factor (right) for the temperature when UV and aerosol parameter are kept at a use level of 0.24 W/(m2 nm) and zero, respectively. The dashed lines denote the 90% two-sided confidence bounds.
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
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Fig. 11. Linearized life–stress plot (left) and the acceleration factor (right) for the UV when the temperature is kept at a use level of 32 C (305 K) and the aerosol parameter is set to zero. The dashed lines denote the 90% two-sided confidence bounds.
For comparison of the service lives for the reference coating obtained from the natural exposure and the
predicted service life from the artificial ageing using the EMPA weathering device the two pdfs are shown together in Fig. 13. One pdf (Fig. 13, dashed line) was taken from the Escape Hatch Programme, Fig. 4, and the other (Fig. 13, solid line) presents the calculated pdf for a use level of 32 C, 0.24 W/(m2 nm) and zero for the temperature, UV and aerosol parameter, respectively (see Fig. 9). The parameters to compare for both Weibull distributions are presented in Tables 1 and 4. The shape of both curves in Fig. 13 is similar, slightly nega-
Fig. 12. Linearized life–stress plot for the aerosol parameter when the temperature and UV are kept at a use level of 32 C (305 K) and 0.24 W/(m2 nm), respectively.
Fig. 13. Probability density functions for gloss loss with a failure criterion of 60 GU measured at gloss 60 for the reference coating obtained from the natural exposure experiment (dashed line) and the artificial ageing (solid line) calculated for a use level of 32 C (305 K), 0.24 W/(m2 nm) and zero for the temperature, UV and aerosol parameter, respectively.
discussed earlier, the aerosol application procedure employed led to the reduction in the service life of the reference coating by approximately 40%. 5.5. Comparison of service lives obtained from artificial ageing and natural exposure (Escape Hatch Programme) for reference coating
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tively skewed, both shape parameters are close: obtained from the natural exposure experiment was 9.4 and calculated from the artificial ageing was 18.9 with the coefficients of skeweness 0.6 and 0.9, respectively. From the Fig. 13 follows, that the distribution obtained from the natural exposure is much broader than that from the artificial weathering, their standard deviations are 6.3 and 2.7 months, respectively. This can be explained both by variations in the actual conditions between different planes and by other factors that were not included in the weathering procedure employed, such, for example, as variations in coating application conditions and aircraft washing. Both curves in Fig. 13 are closely located with a difference of about 8 months: the mode life, which corresponds to the maximum of pdf-curve, for natural exposure was 51.3 months, whereas that from the artificial ageing experiment was 43.1 months. However, for the warranty time 90% (the 10th percentile 0.1), which perhaps is a more useful parameter for the prediction of the service life for coatings, the difference is much smaller: only 2.5 months.
6. Conclusions A model with three stress types, the temperature, UV and aerosol, was proposed for calculating the service life for organic coatings under service conditions. The model was applied to the estimation of the service life for aircraft coatings concerning the gloss loss. The failure criterion, which was defined by airline representatives, was set to 60 GU for gloss 60 . The validation of the model was performed for reference polyurethane coating with a light blue colour. Firstly, gloss degradation curves for the reference coating were obtained in a unique natural exposure programme (Escape Hatch Programme) and the service life was calculated using the Weibull distribution. Secondly, the specially designed accelerated ageing tests were performed with the same coating using the new designed and constructed weathering device. In order to compare both results the life– stress relationship was found and the use level conditions were determined. For the temperature, the Arrhenius life–stress relationship with a value for the apparent activation energy of 8.4 kcal/mol was used and for the UV-A irradiance at 340 nm the inverse power life–stress relationship with a power coefficient of 0.46 was taken. The third stress, the aerosol, was treated as an indicator variable taking discrete values of zero (no aerosol application) and one (aerosol application). The estimation of the service conditions for aircraft coatings were performed by using an Arrhenius weighting term for temperature and a power weighting term for UV. Here we based on the climatic data for air temperature and UV irradiance at 340 nm at some ran-
domly chosen destination sites together with the specially performed experiment for determination of the difference between air and panel temperatures. The calculated service conditions (the use level) were 32 C for the panel temperature, 0.24 W/(m2 nm) for the UV and zero for the aerosol parameter. The calculated mean life for the gloss loss of reference coating at the assumed use level with a failure criterion of 60 GU measured at gloss 60 was 42.0 months and the 90% warranty time (time at which only 10% of the panels had reached the failure criterion—this reliability parameter is perhaps more useful for the prediction of the service life for coatings) was 38.4 months. The mean life and the 90% warranty time obtained from the natural weathering (Escape Hatch Programme) of the same coating were 49.3 and 40.9 months, respectively. Both experimental results were in good agreement, which validated the model. Humidity as a stress parameter was excluded from analysis because the variation of this parameter across different aircrafts is not so distinguishable. Therefore the influence of this parameter was not investigated directly in this work. However, in the artificial weathering procedure employed the effect of humidity was not ignored: relative humidity changes from 10 to 100% during the weathering cycle were applied. For analysis of degradation of coatings for another applications, for example, for automotive coatings, the influence of humidity has to be studied more carefully. Our method gives a unique opportunity to investigate the aerosol influence on coatings. Starting from the fact, that the service life of aircraft coatings was reduced by approximately 50% in the years from 1992 to 1995 following the eruption of the volcano Pinatubo in June 1991, the specially designed weathering experiment was conducted which simulated the post-eruption stratosphere conditions. The most simple mathematical model for the aerosol parameter using indicator variable showed that the sulphuric aerosol shortened the service life of the reference coating by at least of 40%, which suits perfectly with the earlier mentioned 50% reduction. Therefore, the experimental conditions were chosen correctly. The further development of the model can be made using X-ray photoelectron spectroscopy measurements, which could introduce a quantitative amount of sulphuric atoms cumulated in the coating during the ageing. Regarding the degradation of the reference coating, the conclusion can be made that the reference polyurethane coating under investigation was sensitive to the sulphuric aerosol. For the time being, the use level parameter for sulphuric aerosol was set to zero based on the fact that no big volcano eruptions have taken place which could have brought a considerable amount of aerosol into the stratosphere for the duration of the Escape Hatch Programme.
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
For new coatings now under development, it can be a question of interest whether a new product is resistant to sulphuric aerosol. Moreover, other pollutants, for example, acid rain simulation can be included into the investigation, which is important for automotive coatings.
Acknowledgements This research was supported by the Swiss Commission for Technology and Innovation, Akzo Nobel, SR Technics and KLM. We are grateful to Dr. Dick van Beelen, Dr. Hans Polak, Dr. Ruud van Overbeek and Pierre J.M. Moors of Akzo Nobel Aerospace Coatings; Leo G.J. van der Ven, S.M. Koeckhoven, Ritse E. Boomgaard, Dr. Klaus Zabel, J.J. Udema and Rob Lagendijk of Akzo Nobel; Hanspeter Roth, Peter Mu¨ller, Wilbert Meijer, Herbert Ackermann and Hans Gut of SR Technics; Jan van der Woning, Adrian Gerritsen and Carlo N.J. Bakker of KLM and Oliver von Trzebiatowski of EMPA for helpful discussions.
Appendix A. Basic concepts of Reliability theory applied to the Weibull distribution Basic definitions and their application to the Weibull distribution can be found, for example, in [5,16–21] and summarily are shown in Table 5.
Appendix B. Life–Stress relationships Some of most common used life–stress relationships [8,17,21] are presented in Table 6 When a test involves multiple accelerating stresses, a general log–linear relationship (GLL), which describes a life characteristic as a function of a vector of n stresses, or X ¼ ðX1 ; X2 ; . . . Xn Þ. Software ALTA 6 PRO [8] includes this model and allows up to eight stresses. Mathematically the model is given by, 0 þ
LðXÞ ¼ e
n P j¼1
j Xj
;
where j are model parameters, X is a vector of n stresses. This relationship can be further modified through the use of transformations and can be reduced to the models discussed previously (Table 6). If one applies an inverse transformation, such that V=1/X, the relationship would reduce to the Arrhenius relationship (with j=B from the Table 6), a logarithmic transformation, V=ln(X) leads to the inverse power relationship (with j=n from the Table 6). If no transformation is applied, the stress–life relationship has an exponential form. Furthermore, if more than one stress is present, one could choose to apply a different transformation to each stress to create combination models.
Table 5 Basic concepts of Reliability theory applied to the Weibull distribution Concept
Description
Cumulative distribution function F(t) (cdf)
Gives the probability that a panel will fail before time t: FðtÞ ¼ PrðT 4 tÞ
Reliability function R(t)
Gives the probability of surviving beyond time t RðtÞ ¼ PrðT > tÞ ¼ 1 FðtÞ
Probability density function (pdf)
Represents relative frequency of failure times as a function of time fðtÞ ¼ dFdtðtÞ
Hazard function (h.f.), or instantaneous failure rate function
Expresses the propensity to fail in the next small interval of time, given survival to time t: hðtÞ ¼ lim PrðttÞ ¼ RfððttÞÞ Dt !0
11
Weibull distribution " # t FðtÞ ¼ 1 exp
" # t RðtÞ ¼ exp
" # t 1 t fðtÞ ¼ exp t 1 b<1—decreasing h.f., describes products that improves with age (infant mortality) =1—constant h.f., describes products with random failures (exponential distribution) >1—increasing h.f., describes products ageing hðtÞ ¼
12
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
Table 5 (continued) Concept
Description
Weibull distribution 1 T ¼ G þ 1 , Ð1 where GðxÞ ¼ 0 et tx1 dt ðx > 0Þ
Ð1
Mean life T or expectation E(T)
T ¼ EðTÞ
Variance
Measure of the spread of the distribution. Var(T) hasÐ the units of time squared. 1 VarðTÞ 1 ½t EðTÞ2 fðtÞdt ¼ j Ð1 2 2 1 t fðtÞdt ½EðTÞ
1 tfðtÞdt
Standard deviation
Has the units of life, for example, months or hours ðTÞ ¼ ½VarðTÞ1=2
Coefficient of skewness
Measure for the asymmetry of the distribution Ð 1
s
1
½tEðTÞ3 fðtÞdt
½VarðTÞ3=2
2
VarðTÞ ¼
2 ! 2 1 G þ1 G þ1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 T ¼ G þ 1 G þ 1
s¼ 3 2 1 1 3 G 1þ 3G 1 þ G 1 þ þ2 G 1þ " #3=2 2 1 2 G 1þ G 1þ for 1<<3.6, s> 0 the pdf is skewed to the right (has a right tail) for > 3.6, s<0 the pdf is skewed to the left (has a left tail) for 3.0<<4.6, jsj <0.2 the distribution is almost symmetric (similar in shape to a normal distribution) 1 T ¼ ½lnð2Þ
Median life T
Value of the random variable that has exactly one-half of the area under the pdf to Ð T its left and one-half to its right 1 fðtÞdt ¼ 0:5
Mode T~
The maximum value of T that satisfies d½fðtÞ dt ¼ 0
100Pth Percentile of a distribution P
Age by which a proportion P the population fails. It is the solution of P ¼ FðP Þ:
P ¼ ½lnð1 PÞ1=
Warranty time 90%
is the 10th percentile, which means that 90% of all panels survived (have not reached the failure criterion)
0.1
1 1 T~ ¼ 1
Important examples: 0.632—scale parameter 0:5 ¼ ½lnð2Þ1= — median
Table 6 Life–Stress relationships
Appendix C. General Log-Linear-Weibull model Model parameters
Arrhenius
LðVÞ ¼ Ce
B V
C>0, B
1 K>0, n KV n A B Eyring LðVÞ ¼ eV A, B V L, is a quantifiable life measure, such as mean life, median life, characteristic life, percentile, etc.; V, represents the stress levels, [temperature values should be expressed in absolute units (K)]. Inverse Power Law (IPL)
LðVÞ ¼
The General Log–Linear–Weibull (GLL–Weibull) model can be derived by setting the scale parameter to the life–stress relationship LðXÞ (see Appendix B). Here we will follow the notations assumed in ReliaSoft [8]. ¼ LðXÞ; while the shape parameter is assumed to remain constant across different stress levels. This yield the following GLL-Weibull pdf,
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
fðt; XÞ ¼ t1 e
0 þ
n P
j Xj
et
j¼1
0 þ
n P
j X j
j¼1
e
:
The total number of unknowns to solve for in this model is n+2, i.e. , 0, 1, . . ., n. The maximum likelihood estimation method [16,17] can be used to determine the parameters for the GLL relationship and the selected life distribution. The loglikelihood function for the Weibull distribution for right censored data is given by Ref. [8], lnðLÞ ¼ L
2 F X 6 1 T e 6 i ¼ ln4Ti e
n P
0 þ
j¼1
0 Ti e
0 þ
n P j¼1
0 þ
e
i¼1
S X
j Xi;j
n P j¼1
3 j Xi;j
7 7 5
j Xi;j
;
i¼1
where F is the number of exact times-to-failure data points; is the Weibull shape parameter (unknown, to be estimated); 0, 1, . . ., n, are the (n+1) GLL parameters (unknown, to be estimated); X ¼ ðX1 ; X2 ; . . . Xn Þ is the vector of n stresses; Ti is the exact failure time; S is the number of suspension data points; and Ti0 ; is the running time of the suspension data points. The solution (parameter estimates) can be found by maximising L, i.e. by simultaneous solving of (n+2) equations such that @L ¼ 0; @ @L ¼ 0; i ¼ 0; :::; n @i Appendix D. Likelihood ratio test In order to justify the assumption of a common shape parameter among the data obtained at different stress levels, the likelihood ratio (LR) test can be performed [17]. The LR test statistic T, can be calculated from
13
experimental data: T ¼ 2ðL1 þ L2 þ . . . þ Ln L0 Þ, where L1, L2, . . ., Ln are the log–likelihood values obtained by fitting a separate distribution to the data from each of the n stress levels. The log–likelihood value L0 is obtained by fitting a model with a common shape parameter and a separate scale parameter for each of the n stress levels. If the shape parameters are equal, than the distribution of T is approximately w2 with (n1) degrees of freedom. References [1] Martin JW, Saunders SC, Floyd FL, Wineburg JP. Methodologies for predicting the service life of coatings systems. Blue Bell, PA, USA: Federation of societies for coatings technology; 1996. [2] Guseva O, Brunner S, Richner P. Macromol Symposia 2002; 187:883–93. [3] Read WG, Froidevaux L, Waters JW. Geophys Res Lett 1993; 20:1299–302. [4] Wicks ZWJr, Jones FN, Pappas SP. Organic coatings: science and technology. 2nd ed. New York: Wiley-Interscience; 1999. [5] ReliaSoft Corporation. Life data analysis reference, Weibull ++ Version 5.0. Tucson, AZ USA: ReliaSoft Publishing; 1997. [6] Bauer DR. Polym Degrad Stab 2000;69:297–306. [7] Jorgensen G, Bingham C, King D, Lewandowski A, Netter J, Terwilliger K. NREL/ CP-520-28579; 2000. Available from: http://www.doe.gov/bridge. [8] ReliaSoft Corporation. Accelerated life testing reference, ALTA version 6. Tucson, AZ, USA: ReliaSoft Publishing; 2001. [9] Brunner S, Richner P. Accelerated weathering device for service life prediction. [in preparation]. [10] Allen NS, Katami H. In: Clough RL, Billingham NC, Gillen KT, editors. Polymer durability. Advances in Chemistry Series249. New York: American Chemical Society; 1996. p. 537. [11] Sampers J. Polym Degrad Stab 2002;76:455–65. [12] Bauer DR. J Coat Technol 1997;69:85–96. [13] Available from: http://uvb.nrel.colostate.edu. [14] Available from: http://klimadiagramme.de. [15] Available from: http://www.empa.ch/nabel. [16] Nelson W. Applied life data analysis. New York: John Wiley & Sons; 1982. [17] Nelson W. Accelerated testing: statistical models, test plans, and data analyses. New York: John Wiley & Sons; 1990. [18] Lawless JF. Statistical models and methods for lifetime data. New York: John Wiley & Sons; 1982. [19] Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions, Vol. 1. New York: John Wiley & Sons; 1994. [20] Kececioglu D. Reliability engineering handbook, Vol. 1. Englewood Cliffs, New Jersey: Prentice Hall; 1991. [21] Meeker WQ, Escobar LA. Statistical methods for reliability data. New York: John Wiley & Sons; 1998.
Service life prediction for aircraft coatings Olga Guseva*, Samuel Brunner, Peter Richner Swiss Federal Laboratory for Materials Testing and Research, U¨berlandstrasse 129, CH-8600 Du¨bendorf, Switzerland Received 3 February 2003; received in revised form 14 March 2003; accepted 16 March 2003
Abstract A model with three stress types, the temperature, UV and aerosol, for estimating the service life of organic coatings under service conditions has been proposed. The model has been applied to the estimation of the service life for aircraft coatings concerning loss of gloss. The results obtained from a unique natural exposure programme for a reference polyurethane coating have been compared with the results from the specially designed accelerated ageing tests, which were performed with a new designed and constructed weathering device. As a special point of interest, the influence of sulphuric aerosol, which has simulated the stratosphere conditions after a big volcano eruption, on the service life of the reference coating has been investigated. # 2003 Elsevier Ltd. All rights reserved. Keywords: Coatings; Weathering; Accelerated tests; Service life prediction
1. Introduction The organic coating industry has been performing research in order to find new chemistry formulations with enhanced stability against environmental influences, and at the same time with possibly smaller contents of hazardous substances. The service life of these new chemistry formulations is often unknown. Usually, the field experiments (natural exposure) for modern coatings take prolonged time of several years, which is not acceptable for manufactures. Therefore a need exists for a methodology which is capable of making reliable service life predictions from accelerated experiments. Such Service Life Prediction (SLP) methodology was proposed by Martin et al. [1]. In this paper we applied this approach to the ageing of aircraft coatings. The degradation of aircraft coatings was considered as a loss of gloss provided that other requirements, such as the absence of cracking and delamination, were fulfilled. The work was supported by a special research programme, whereby in-service data from the exposure of products on B-747 escape hatches were compared to the data obtained using special designed accelerated tests,
* Corresponding author Tel.: +41-1-823-4361; fax: +41-1-8234015. E-mail address: [email protected] (O. Guseva). 0141-3910/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0141-3910(03)00124-1
focusing on the development of a service life prediction model. As the initial step, the environmental factors that are relevant for the ageing of aircraft coatings had been analysed [2]. It is known that the weathering of those coatings is not only determined by UV radiation, temperature and humidity but also by pollutants. This was clearly demonstrated during the eruption of volcano Pinatubo: in June 1991 the volcanic eruption brought more than 20 million tons of sulphur dioxide into the stratosphere raising the SO2-level from normal range of 0.01–0.05 ppbv to more than 10 ppbv in September 1991. Within the next four months the concentration of gaseous SO2 in the stratosphere had gradually decreased toward its usual value [3]. The aircrafts, however, showed a fast decrease of gloss in their coating during the following 4 years. It was found that SO2 actually formed a sulphuric acid aerosol, which remained dispersed in the stratosphere for a period of several years after eruption that acted destructively on aircraft coatings [2,3].
2. Natural exposure: Escape Hatch Programme In order to investigate the degradation of the aircraft coatings under service conditions, the so-called use level conditions, a research programme (Escape Hatch Programme) was carried out by AKZO Nobel together with
2
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
KLM starting September 1994. According to this Programme, four different coatings were applied to each escape hatch of a B-747 aircraft for natural exposure (Fig. 1). Every 3 months gloss and colour of the hatches were measured. Altogether, 15 aircrafts were involved in the Programme. One of the pigmented polyurethane coatings with a light blue colour was applied to different escape hatches providing the possibility for statistical analysis. This coating is being referred to as a ‘‘reference coating’’ throughout the paper. The colour values for the unexposed panels in the CIE 1976 L*a*b* colour system were L*=55, a*=26, b*=41. Definitions and details of the measure procedure of colour can be found in Ref. [4]. The gloss degradation curves for gloss 60 for the reference coating are shown in Fig. 2. One of the curves in Fig. 2 (with black filled squares) shows the faster degradation than others. The reason for this fast degradation is that the aircraft with this coating at the
Fig. 1. Escape hatch of the B-747 (shown with the arrow).
escape hatch was in operation during the years 1994– 1996, when the sulphuric aerosol content in the stratosphere after the Pinatubo eruption was still noticeable (for details see Ref. [2]). This curve was omitted from the statistical analysis as an outlier. All other degradation curves in Fig. 2 were measured when the aerosol concentration at a flight height of about 10 km was normal. Definitions and details of the measure procedure of gloss can be found in Ref. [4]. The failure criterion for the gloss loss, which was defined by airline representatives, was 60 GU measured for gloss 60 . The interceptions of degradation curves and the failure-criterion-line (see Fig. 2) gave times-tofailure. It should be noted that the experimental points were measured every 3 months. Therefore, in order to obtain the exact times-to failure, the degradation curves were interpolated with polynomials. As shown in Fig. 2 there were 11 degradation curves altogether, whereby two of them (dashed curves) did not reach the failure criterion at the time of paper writing. These curves were considered as censored data in the further calculations. From this data it was possible to form the sample cumulative distribution function (cdf) and fit it to a particular theoretical cdf (for basic definitions of the reliability theory see Appendix A). There are many different lifetime distributions that can be used for the data analysis. The most commonly used are exponential, Weibull, normal and lognormal distributions. The currently available sample size of 10 times-to-failure, however, was too small for a reliable determination of the underlying distribution. The simplest suitable parametric distributions, which include the degradation mechanism (their hazard function can be increasing), are the Weibull, normal and lognormal. The comparison of the Likelihood Function values for the earliermentioned distributions showed that the Weibull dis-
Fig. 2. Gloss 60 -curves for the reference coating applied on escape hatches. The bold horizontal line represents a failure criterion of 60 GU.
3
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
tribution suited slightly better to the experimental data. Therefore, the Weibull distribution was chosen as an underlying distribution throughout the work. The application of the basic definitions of the reliability theory to the Weibull distribution are presented in Appendix A. Linearized Weibull cdf for the Escape Hatch data is shown in Fig. 3, the reliability function and probability density function (pdf) in Fig. 4. The calculations were made with Weibull++ program [5]. With the Weibull distribution parameters, the shape parameter =9.4 and the scale parameter =52.0 months, available, one can calculate various reliability parameters—some of
Fig. 3. Linearized cdf (solid line) with 90% two-sided confidence bounds (dashed lines) of Escape Hatch Data for the reference coating for gloss 60 with a failure criterion of 60 GU.
them are presented in Table 1. From the Table 1 follows, that the expected life (mean life) for the reference coating was about 49 months, whereas the warranty time 90%, which means that only 10% of the coatings reached the failure criterion, was about 41 months.
3. SLP model for organic coatings with three stresses: temperature, UV and aerosol 3.1. Accelerated testing In typical life data analysis one determines a life distribution that describes the times-to-failure of a product at the usual operating conditions. Statistically speaking, one wishes to obtain ‘‘the use level probability density function’’ of the times-to-failure. Once this pdf is obtained, all other desired reliability parameters, such as for example, percentage failing under warranty (percentile), mean time to failure, mode life can be easily determined. However, coatings are supposed to have a rather prolonged life of several years. At the stage of development of a new formulation, researchers usually do not have enough time to carry out the natural ageing at the so-called use level conditions. To reduce the time of experiment one performs experiments at the enhanced stress levels to obtain pdfs at these high stress levels. The goal is to determine the use level pdf from these accelerated life test data rather than from data obtained under use conditions. To accomplish this, a relationship should be found that allows extrapolating from data collected at accelerated conditions to use level conditions, the so-called life-stress relationship. Some of the most commonly used life-stress relationships in accelerated life testing are Arrhenius (when the temperature is an accelerating factor), inverse power law (usually used for non-thermal accelerated stresses) and Eyring (for thermal or humidity stresses) (see Appendix B). Schematically the procedure of service life prediction described above is presented in Fig. 5. Here it is worth
Table 1 Reliability parameters together with 90% two-sided confidence bounds for gloss loss of reference coating obtained in the Escape Hatch Programme for gloss 60 with a failure criterion of 60 GU 90% Two-sided confidence bounds
Fig. 4. Reliability function (solid line) with 90% two-sided confidence bounds (dashed lines) (left) and pdf (dotted line) (right) of Escape Hatch Data for gloss 60 with a failure criterion of 60 GU.
b Characteristic life (Z) [month] Mean life [month] Median life [month] Mode life [month] Warranty time 90% [month]
Lower limit
Upper limit
9.4 52.0
6.5 49.5
13.7 54.5
49.3 50.0 51.3 40.9
46.7 47.4 48.7 36.8
52.0 52.6 54.1 45.5
4
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
effect, was treated as an indicator variable, taking the discrete values of zero (0), when no aerosol was applied, and one (1), when aerosol was put into a chamber. To combine all these stress types together, we used the General Log–Linear relationship (see Appendix B). The General Log–Linear–Weibull model was derived by setting the scale parameter to the life–stress relationship for a model with three stress types; the shape parameter was assumed to remain constant across different stress levels (Appendix C). The general model for three stress factors is, therefore, ¼e Fig. 5. General scheme of SLP model for organic coatings.
to note that the accelerated test should be designed in such a way that it does not change the failure modes that would be encountered at use conditions. For example, a high temperature stress level for a coating should not exceed the glass transition temperature. Analysis of accelerated life test data, then, consists of an underlying life distribution that describes the product at different stress levels and a life–stress relationship that quantifies the manner in which the life distribution changes across different stress levels. 3.2. SLP model with three stresses
¼ e0 e
1 1 V V 1V 2e 3 3; 2
ð1Þ
where is the scale parameter of Weibull distribution (characteristic life), 0, 1, 2, 3 are the model parameters, V1 is the thermal stress, with the temperature expressed in absolute units (K), V2 is the UV irradiance at 340 nm in [W/(m2 nm)], V3 is the aerosol stress, assuming values of 0 and 1. Note, that the parameter B of the Arrhenius relationship from Table 6 is equal to the log-linear coefficient 1, and the parameter n of the inverse power relationship from Table 6 is equal to (2). The General Log–Linear–Weibull pdf for these three stresses is given by: 1 1 0 þ1 V1 þ2 lnðV2 Þþ3 V3 fðt; V1 ; V2 ; V3 Þ ¼ t e ð2Þ 1 et
As discussed earlier, aircraft coatings are sensitive to the following stress parameters: temperature, UV radiation, humidity and sulphuric aerosol [2]. Although the actual stresses vary with time both in the natural weathering and in the artificial ageing employed, in this paper we made the assumption of constant stress levels, that the stress loads applied to panels were constant with respect to time. The averaging procedures are described later. Humidity as a stress parameter was excluded from further considerations because the variation of this parameter across different aircrafts is not so distinguishable. Therefore the influence of this parameter was not investigated directly in this work. However, in the artificial weathering procedure employed the effect of humidity was not ignored: relative humidity changes from 10% to 100% during the weathering cycle were applied. The data obtained from the accelerated weathering were analysed assuming an Arrhenius life–stress relationship for temperature and the inverse power life– stress relationship for UV irradiance (Appendix B). The Arrhenius term is widely used for the dependence of coating photooxidation on temperature [6,7] and a power law expression for the influence of ultraviolet irradiance [6,7]. The third stress factor, the aerosol
1 0 þ1 V þ2 lnðV2 Þþ3 V3 1
e
0 þ1
V1 þ2 lnðV2 Þþ3 V3
:
Parameters , 0, 1, 2, 3 can be obtained by maximising the log-likelihood function, (see Appendix C), using ALTA PRO program [8].
4. Experimental apparatus In order to test coatings for aircraft applications, a device based on a modified commercial Accelerated Weathering Tester QUV was designed and constructed at EMPA [9]. The commercial device was equipped with eight separate thermally insulated chambers where eight weathering experiments can be run simultaneously. The damaging effects of sunlight were simulated by fluorescent UVA-340 lamps. The selection of the lamp type suitable for the weathering of aircraft coatings was discussed in details in Ref. [2]. The following weathering cycle was used: 8 h of UV with approximately 15% humidity followed by 4 h of condensation without irradiation with approximately 100% humidity. An additional water cooling system was installed in the chambers 1 and 2 that allowed us to achieve a maximal temperature difference of 20 K between the different chambers. For more details see Ref. [9].
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
In the following calculations we considered the temperature and UV stresses as constant stresses with the values obtained by averaging over the cycle. For the temperature and UV the weighted averages over the cycle for each chamber were calculated using procedure described later [see Eq. (4)] which are presented in Table 2. In order to investigate the influence of the sulphuric aerosol on coatings, sulphuric acid with a concentration of 0.5 mol/l was injected using a nebulizer into five chambers. The duration of aerosol application was 40 min once a week per chamber. For the further calculations, this third stress, aerosol, was treated as an indicator variable, assuming discrete values of zero (no aerosol application) and one (aerosol application). Two test panels with the reference coating were placed in each chamber for exposure. It should be noted that those panels had been purposely manufactured using different batches and by different persons. In order to obtain the uniform exposition of all panels, a rotational procedure was employed: once a week the position of all panels was changed according to the schema described in Ref. [9]. Measurements of gloss were performed every 3 weeks. Five points from the top to the middle of the panel were measured. These points were treated independently in the following calculations, so that for each combination of stresses we dealt with a collection of 10 points. The averaged experimental curves for all chambers for gloss measured at 60 are shown in Fig. 6, where the filled black symbols were used for the chambers with aerosol application. From Fig. 6 follows that the sulphuric aerosol application considerably accelerated the degradation of the reference coating. The panels in the chamber 2 had not reached the failure criterion at the time of paper writing. In order to obtain times-to-failure the graphical extrapolation was used.
Table 2 Averaged stress levels used in each chamber for the accelerated ageing of reference coating Chamber
1 2 3 4 5 6 7 8 a
Stress 1
Stress 2
Stress 3
Temperature T [ C]
Temperature T [K]
UV radiation UV [W/m2 nm]
Aerosol
37.5 38.1 57.5 58.2 53.8 54.3 56.2 56.5
310.7 311.2 330.7 331.4 326.9 327.4 329.4 329.6
0.58 0.41 0.42 0.57 0.60 0.43 0.63 0.62
1a 0 1 0 1 1 0 1
1 Means sulphuric aerosol application 40 min once a week; 0—no aerosol application.
5
Fig. 6. Experimental gloss-60 curves for the reference coating for chambers 1–8. Filled black symbols denote the chambers with aerosol application.
5. Results and discussion 5.1. Shape parameter First, separate calculations for each chamber assuming the Weibull distribution were performed by means of the program Weibull++ [5]. For parameter estimation, the Least Squares method, or more precisely, the Rank Regression on X (RRX), was used. This method requires that a straight line be fitted to a set of data points such that the sum of the squares of horizontal deviations from the points to the line is minimised. The measure of how well the linear model fits the data is the correlation coefficient, denoted here by . The summary is presented in Fig. 7 and Table 3. Although the values of seem to be different (see Table 3), they do not much influence the shape of the Weibull distribution: in Fig. 7 all lines are almost parallel. The coefficient of skewness (for definition see Appendix A), which is a measure for the distribution asymmetry, ranges between 0.75 (for =13) and 1.1 (for =121). For the analysis of accelerated test data the assumption of the common shape parameter across the chambers (which supposes the parallelism of the lines in Fig. 7) should be made. Assuming the underlying Weibull distribution with the common shape parameter and the general log–linear life–stress relationship, the calculation of the service life for reference coating at a particular use level was performed using the program ALTA PRO 6 [8]. For the parameter estimation, the Maximum Likelihood Parameter estimation (MLE) method was used. The calculated value for the common shape parameter was 18.9. The use level was assumed to be 32 C (305 K) for the temperature parameter and 0.24 W/(m2 nm) for the UV parameter; the aerosol parameter was set to zero (see the later definition of the use level conditions). The results are shown in Fig. 8. In order to justify the assumption of the common for our data
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O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
Fig. 7. Cumulative distribution functions for gloss loss for the reference coating with a failure criterion of 60 GU for gloss 60 , shown on Weibull paper for all chambers together. Filled black symbols denote the chambers with aerosol application. Table 3 Shape parameter () and scale parameter (characteristic life ) for the gloss loss with a failure criterion of 60 GU measured at gloss 60 for the reference coating calculated for each chamber separately Chamber
1 2 3 4 5 6 7 8
29.9 13.3 55.7 19.4 121.2 64.8 27.2 39.8
11.4 20.7 5.9 8.2 5.3 5.7 8.2 4.9
0.96 0.96 0.97 0.90 0.95 0.91 0.95 0.98
set, the likelihood ratio (LR) test was applied (Appendix D) using ALTA PRO 6 software [8], which gave the result that the shape parameter estimates of 18.9 did not differ statistically significantly at the 90% level. 5.2. Life–stress relationship Using the procedure described in the section ‘‘SLP model with three stresses’’ the following parameters were obtained for the life–stress relationship using notations of the Eq. (1): 0=10.7114, 1=4215 K, 2=0.4608, 3=-0.5249. While the parameter 0 is a
Fig. 8. Cumulative distribution functions for gloss loss for the reference coating with a failure criterion of 60 GU for gloss 60 , shown on Weibull paper assuming the common shape parameter across all chambers. The dashed line represents the extrapolation to the use level. Filled black symbols denote the chambers with aerosol application.
constant, the parameters 1, 2 and 3 correspond to the influence of the temperature, UV irradiance, and aerosol, respectively, on the service life. The calculation of the apparent activation energy (Ea=1.R, where the gas constant R=8.314 JK1 mol1) gave a value of 8.4 kcal/mol, which was within the interval 7.4–10.3 kcal/mol (30.8–43.1 kJ/mol) obtained by Allen et al. for photooxidative degradation of low-density polyethylene film materials containing nine types of titanium dioxide pigments [10]. A value for the apparent activation energy of approximately 7.2 kcal/mol (30 kJ/mol) was also utilised in the work of Sampers [11] for the estimation of the lifetime for LDPE films with and without stabilisers. These values for the apparent activation energy are close to the values of 5–8 kcal/mol used by Bauer [6] for photooxidation in automotive coatings. A similar approach to Bauer’s was employed by Jorgensen et al. [7] for the service life prediction for clear coat/coloured basecoat paint systems. The following generalised cumulative dosage model for the loss in performance, P, was used in Ref. [7]: ðt DPðtÞ ¼ A ½IUV ð Þn eE=kTðÞ d; ð3Þ 0
7
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
where IUV is the cumulative UV light dosage integrated over a bandwidth of 290–385 nm; A, n, E are fitting parameters, where E denotes the activation energy; k is the Boltzmann constant. Depending on the system, the values 3.8–8.4 kcal/mol for the apparent activation energy and 0.64–0.71 for the power parameter n were obtained. The parameter 2=0.46, gained in our calculations, which characterises the influence of the UV-A irradiance at 340 nm on the reference coating, is close to the model parameter n of Jorgensen et al. [7]. In other models this parameter was usually assumed to be 1 [11,12]. For the first time the influence of the aerosols on the service life of coatings was considered quantitatively. It is known that the service life of aircraft coatings was reduced by approximately 50% in the years from 1992 to 1995 following the eruption of the volcano Pinatubo in Philippines in June 1991 [2]. Our first simplified approximation using indicator variable showed that the presence of the sulphuric aerosol shortened the service life of the reference coating by at least of 40%: keeping other parameters V1 (temperature) and V2 (UV) fixed at the use level, the ratio of the characteristic lives () with and without aerosol V3 ¼1 application is ¼ 0:59. More precise conclusions V3 ¼0 can be made using the quantitative value for the concentration of the sulphuric atoms absorbed by the coating during the ageing process. A suitable method for obtaining such values is X-ray photoelectron spectroscopy (XPS). Our first preliminary XPS measurements showed that the chosen concentration of the sulphuric aerosol combined with the application duration in our weathering experiments roughly corresponded to the Pinatubo influence on the reference coating. It follows, therefore, that our experimental conditions simulated well the influence of the big volcano eruption on the aircraft coatings. 5.3. Determination of the use level conditions Definition of the use level conditions for aircrafts is quite a complicated problem. Our goal here was to obtain two numbers for the temperature and the UV-irradiance at 340 nm averaged over time (including different seasons) and space (across different destinations throughout the world). The averaging proposed later was not perfect because too many factors had to be taken into account. Nevertheless, for the first estimation of the use level conditions we used the following procedure, where the Arrhenius term was used for the averaging of temperature and a power law for the averaging of UV radiation. For the first simplified estimation, the whole time interval was represented by a sum of subintervals ti for the temperature and the UV intensity, respectively,
T ¼
X X ðTi pi Þ= pi i
i
I ¼
X X ðIi pi Þ= pi ; i
ð4Þ
i
where the weighting function has the form n pi ¼ IUVðat340nmÞi e½Ea =RTi Þ Dti
ð5Þ
with Ea=8.4 kcal/mol and n=0.46 as it was obtained in the previous section as fitting coefficients for the life–stress relationship in the accelerated weathering test for the reference coating. Calculations were made using the above approach with the averaging over one year. We assumed that the temperature ageing of aircraft coatings occurs mainly on ground and during take-offs and landings. Mathematically, for the calculations of use level conditions, the Arrhenius term in the Eq. (5) is very small for the temperature of approximately 50 C at the flight height. In order to determine how much time an aircraft spend on the ground, the flying time data available from the Escape Hatch Programme were analysed, from which an assumption of flying time of approximately 62% was made. From the climatic databases [13,14], air temperatures and UV data for some destination sites in the USA, Europe, Far East and Africa were taken. In our weathering experiments, however, we were dealing with the panel temperatures, measured at the backside of panels. It is known, that the panel temperature is considerably higher than the air temperature: 5–7 C for light and 15–30 C for dark colours [6]. To find out the actual temperature difference between the panel and air temperatures for our particular reference coating with a light blue colour (L*=55, a*=26, b*=41) an additional natural exposure experiment was carried out. The panels were exposed during one summer month in the suburban National Air Pollution Monitoring Network (NABEL) site in Du¨bendorf, Switzerland [15]. In order to model the airport conditions, the panels were placed above the asphalt ground. The sensors were mounted to the backside of the panels. The experiment was conducted for the panels with isolated and non-isolated back sides. All measured temperatures were compared with the air temperature measured at the test site. The following results were obtained for the noon time: the panel temperature was about 3 K higher than the meteorologically measured air temperature for the nonisolated panel and about 16 K higher for the isolated one. As in the case of aircraft coatings we are dealing with the thermally isolated aircraft body, the difference of 16 K for panel temperature was added to the air temperatures for all destination sites as the input temperature in Eq. (4) before the averaging. The calculated use level conditions obtained with this procedure were 32 C (305 K) for the temperature and 0.24 W/(m2 nm) for the UV-parameter. The aerosol parameter for the use level for the current conditions (after 1996) was assumed to be zero: there
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O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
were no big volcano eruptions in the past since the Pinatubo eruption in 1991, which could have a significant influence on aircraft coatings. 5.4. Reliability parameters at the use level The inclined dashed line on the right in Fig. 8 shows the extrapolation to the use level. It should be noted that the calculated reliability parameters at the use level were dependent much on the choice of the use level conditions, especially temperature. Calculations showed that a difference of two degrees led to a difference of about 3 months in the prediction times.
The calculated cdf and pdf at the use level are shown in Fig. 9. The calculated reliability parameters at a use level of temperature of 32 C, UV of 0.24 W/(m2 nm) and aerosol parameter of zero are presented in Table 4. The linearised life–stress relationships and the acceleration factors were obtained by keeping two of the three stresses at the use level and varying the other one. The plots are presented in Fig. 10, for temperature as the acceleration factor, and in Fig. 11 for UV as the acceleration factor. The aerosol parameter was set to zero in these calculations. For example, at 60 C (333 K), UV=0.24 W/(m2 nm) and aerosol parameter=0, the acceleration factor is 3.2; and at UV=0.8 W/(m2 nm), if the temperature and aerosol parameter are kept at a use level of 32 C and 0, respectively, the acceleration factor is 1.7. The influence of the aerosol application on the service life is presented in Fig. 12. As Table 4 Reliability parameters together with 90% two-sided confidence bounds for gloss loss of reference coating at the use level for gloss 60 with a failure criterion of 60 GU 90% Two-sided confidence bounds
Fig. 9. Reliability function with 90% two-sided confidence bounds (left) and pdf (right) at the use level for the reference coating for gloss 60 with a failure criterion of 60 GU.
Characteristic life [month] Mean life [month] Median life [month] Mode life [month] Warranty time 90% [month]
Lower limit
Upper limit
18.9 43.2
16.4 40.1
21.8 46.6
42.0 42.4 43.1 38.4
39.0 39.3 40.0 35.5
45.3 45.7 46.4 41.5
Fig. 10. Linearized life–stress plot (left) and the acceleration factor (right) for the temperature when UV and aerosol parameter are kept at a use level of 0.24 W/(m2 nm) and zero, respectively. The dashed lines denote the 90% two-sided confidence bounds.
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
9
Fig. 11. Linearized life–stress plot (left) and the acceleration factor (right) for the UV when the temperature is kept at a use level of 32 C (305 K) and the aerosol parameter is set to zero. The dashed lines denote the 90% two-sided confidence bounds.
For comparison of the service lives for the reference coating obtained from the natural exposure and the
predicted service life from the artificial ageing using the EMPA weathering device the two pdfs are shown together in Fig. 13. One pdf (Fig. 13, dashed line) was taken from the Escape Hatch Programme, Fig. 4, and the other (Fig. 13, solid line) presents the calculated pdf for a use level of 32 C, 0.24 W/(m2 nm) and zero for the temperature, UV and aerosol parameter, respectively (see Fig. 9). The parameters to compare for both Weibull distributions are presented in Tables 1 and 4. The shape of both curves in Fig. 13 is similar, slightly nega-
Fig. 12. Linearized life–stress plot for the aerosol parameter when the temperature and UV are kept at a use level of 32 C (305 K) and 0.24 W/(m2 nm), respectively.
Fig. 13. Probability density functions for gloss loss with a failure criterion of 60 GU measured at gloss 60 for the reference coating obtained from the natural exposure experiment (dashed line) and the artificial ageing (solid line) calculated for a use level of 32 C (305 K), 0.24 W/(m2 nm) and zero for the temperature, UV and aerosol parameter, respectively.
discussed earlier, the aerosol application procedure employed led to the reduction in the service life of the reference coating by approximately 40%. 5.5. Comparison of service lives obtained from artificial ageing and natural exposure (Escape Hatch Programme) for reference coating
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O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
tively skewed, both shape parameters are close: obtained from the natural exposure experiment was 9.4 and calculated from the artificial ageing was 18.9 with the coefficients of skeweness 0.6 and 0.9, respectively. From the Fig. 13 follows, that the distribution obtained from the natural exposure is much broader than that from the artificial weathering, their standard deviations are 6.3 and 2.7 months, respectively. This can be explained both by variations in the actual conditions between different planes and by other factors that were not included in the weathering procedure employed, such, for example, as variations in coating application conditions and aircraft washing. Both curves in Fig. 13 are closely located with a difference of about 8 months: the mode life, which corresponds to the maximum of pdf-curve, for natural exposure was 51.3 months, whereas that from the artificial ageing experiment was 43.1 months. However, for the warranty time 90% (the 10th percentile 0.1), which perhaps is a more useful parameter for the prediction of the service life for coatings, the difference is much smaller: only 2.5 months.
6. Conclusions A model with three stress types, the temperature, UV and aerosol, was proposed for calculating the service life for organic coatings under service conditions. The model was applied to the estimation of the service life for aircraft coatings concerning the gloss loss. The failure criterion, which was defined by airline representatives, was set to 60 GU for gloss 60 . The validation of the model was performed for reference polyurethane coating with a light blue colour. Firstly, gloss degradation curves for the reference coating were obtained in a unique natural exposure programme (Escape Hatch Programme) and the service life was calculated using the Weibull distribution. Secondly, the specially designed accelerated ageing tests were performed with the same coating using the new designed and constructed weathering device. In order to compare both results the life– stress relationship was found and the use level conditions were determined. For the temperature, the Arrhenius life–stress relationship with a value for the apparent activation energy of 8.4 kcal/mol was used and for the UV-A irradiance at 340 nm the inverse power life–stress relationship with a power coefficient of 0.46 was taken. The third stress, the aerosol, was treated as an indicator variable taking discrete values of zero (no aerosol application) and one (aerosol application). The estimation of the service conditions for aircraft coatings were performed by using an Arrhenius weighting term for temperature and a power weighting term for UV. Here we based on the climatic data for air temperature and UV irradiance at 340 nm at some ran-
domly chosen destination sites together with the specially performed experiment for determination of the difference between air and panel temperatures. The calculated service conditions (the use level) were 32 C for the panel temperature, 0.24 W/(m2 nm) for the UV and zero for the aerosol parameter. The calculated mean life for the gloss loss of reference coating at the assumed use level with a failure criterion of 60 GU measured at gloss 60 was 42.0 months and the 90% warranty time (time at which only 10% of the panels had reached the failure criterion—this reliability parameter is perhaps more useful for the prediction of the service life for coatings) was 38.4 months. The mean life and the 90% warranty time obtained from the natural weathering (Escape Hatch Programme) of the same coating were 49.3 and 40.9 months, respectively. Both experimental results were in good agreement, which validated the model. Humidity as a stress parameter was excluded from analysis because the variation of this parameter across different aircrafts is not so distinguishable. Therefore the influence of this parameter was not investigated directly in this work. However, in the artificial weathering procedure employed the effect of humidity was not ignored: relative humidity changes from 10 to 100% during the weathering cycle were applied. For analysis of degradation of coatings for another applications, for example, for automotive coatings, the influence of humidity has to be studied more carefully. Our method gives a unique opportunity to investigate the aerosol influence on coatings. Starting from the fact, that the service life of aircraft coatings was reduced by approximately 50% in the years from 1992 to 1995 following the eruption of the volcano Pinatubo in June 1991, the specially designed weathering experiment was conducted which simulated the post-eruption stratosphere conditions. The most simple mathematical model for the aerosol parameter using indicator variable showed that the sulphuric aerosol shortened the service life of the reference coating by at least of 40%, which suits perfectly with the earlier mentioned 50% reduction. Therefore, the experimental conditions were chosen correctly. The further development of the model can be made using X-ray photoelectron spectroscopy measurements, which could introduce a quantitative amount of sulphuric atoms cumulated in the coating during the ageing. Regarding the degradation of the reference coating, the conclusion can be made that the reference polyurethane coating under investigation was sensitive to the sulphuric aerosol. For the time being, the use level parameter for sulphuric aerosol was set to zero based on the fact that no big volcano eruptions have taken place which could have brought a considerable amount of aerosol into the stratosphere for the duration of the Escape Hatch Programme.
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
For new coatings now under development, it can be a question of interest whether a new product is resistant to sulphuric aerosol. Moreover, other pollutants, for example, acid rain simulation can be included into the investigation, which is important for automotive coatings.
Acknowledgements This research was supported by the Swiss Commission for Technology and Innovation, Akzo Nobel, SR Technics and KLM. We are grateful to Dr. Dick van Beelen, Dr. Hans Polak, Dr. Ruud van Overbeek and Pierre J.M. Moors of Akzo Nobel Aerospace Coatings; Leo G.J. van der Ven, S.M. Koeckhoven, Ritse E. Boomgaard, Dr. Klaus Zabel, J.J. Udema and Rob Lagendijk of Akzo Nobel; Hanspeter Roth, Peter Mu¨ller, Wilbert Meijer, Herbert Ackermann and Hans Gut of SR Technics; Jan van der Woning, Adrian Gerritsen and Carlo N.J. Bakker of KLM and Oliver von Trzebiatowski of EMPA for helpful discussions.
Appendix A. Basic concepts of Reliability theory applied to the Weibull distribution Basic definitions and their application to the Weibull distribution can be found, for example, in [5,16–21] and summarily are shown in Table 5.
Appendix B. Life–Stress relationships Some of most common used life–stress relationships [8,17,21] are presented in Table 6 When a test involves multiple accelerating stresses, a general log–linear relationship (GLL), which describes a life characteristic as a function of a vector of n stresses, or X ¼ ðX1 ; X2 ; . . . Xn Þ. Software ALTA 6 PRO [8] includes this model and allows up to eight stresses. Mathematically the model is given by, 0 þ
LðXÞ ¼ e
n P j¼1
j Xj
;
where j are model parameters, X is a vector of n stresses. This relationship can be further modified through the use of transformations and can be reduced to the models discussed previously (Table 6). If one applies an inverse transformation, such that V=1/X, the relationship would reduce to the Arrhenius relationship (with j=B from the Table 6), a logarithmic transformation, V=ln(X) leads to the inverse power relationship (with j=n from the Table 6). If no transformation is applied, the stress–life relationship has an exponential form. Furthermore, if more than one stress is present, one could choose to apply a different transformation to each stress to create combination models.
Table 5 Basic concepts of Reliability theory applied to the Weibull distribution Concept
Description
Cumulative distribution function F(t) (cdf)
Gives the probability that a panel will fail before time t: FðtÞ ¼ PrðT 4 tÞ
Reliability function R(t)
Gives the probability of surviving beyond time t RðtÞ ¼ PrðT > tÞ ¼ 1 FðtÞ
Probability density function (pdf)
Represents relative frequency of failure times as a function of time fðtÞ ¼ dFdtðtÞ
Hazard function (h.f.), or instantaneous failure rate function
Expresses the propensity to fail in the next small interval of time, given survival to time t: hðtÞ ¼ lim Prðt
11
Weibull distribution " # t FðtÞ ¼ 1 exp
" # t RðtÞ ¼ exp
" # t 1 t fðtÞ ¼ exp t 1 b<1—decreasing h.f., describes products that improves with age (infant mortality) =1—constant h.f., describes products with random failures (exponential distribution) >1—increasing h.f., describes products ageing hðtÞ ¼
12
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
Table 5 (continued) Concept
Description
Weibull distribution 1 T ¼ G þ 1 , Ð1 where GðxÞ ¼ 0 et tx1 dt ðx > 0Þ
Ð1
Mean life T or expectation E(T)
T ¼ EðTÞ
Variance
Measure of the spread of the distribution. Var(T) hasÐ the units of time squared. 1 VarðTÞ 1 ½t EðTÞ2 fðtÞdt ¼ j Ð1 2 2 1 t fðtÞdt ½EðTÞ
1 tfðtÞdt
Standard deviation
Has the units of life, for example, months or hours ðTÞ ¼ ½VarðTÞ1=2
Coefficient of skewness
Measure for the asymmetry of the distribution Ð 1
s
1
½tEðTÞ3 fðtÞdt
½VarðTÞ3=2
2
VarðTÞ ¼
2 ! 2 1 G þ1 G þ1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 T ¼ G þ 1 G þ 1
s¼ 3 2 1 1 3 G 1þ 3G 1 þ G 1 þ þ2 G 1þ " #3=2 2 1 2 G 1þ G 1þ for 1<<3.6, s> 0 the pdf is skewed to the right (has a right tail) for > 3.6, s<0 the pdf is skewed to the left (has a left tail) for 3.0<<4.6, jsj <0.2 the distribution is almost symmetric (similar in shape to a normal distribution) 1 T ¼ ½lnð2Þ
Median life T
Value of the random variable that has exactly one-half of the area under the pdf to Ð T its left and one-half to its right 1 fðtÞdt ¼ 0:5
Mode T~
The maximum value of T that satisfies d½fðtÞ dt ¼ 0
100Pth Percentile of a distribution P
Age by which a proportion P the population fails. It is the solution of P ¼ FðP Þ:
P ¼ ½lnð1 PÞ1=
Warranty time 90%
is the 10th percentile, which means that 90% of all panels survived (have not reached the failure criterion)
0.1
1 1 T~ ¼ 1
Important examples: 0.632—scale parameter 0:5 ¼ ½lnð2Þ1= — median
Table 6 Life–Stress relationships
Appendix C. General Log-Linear-Weibull model Model parameters
Arrhenius
LðVÞ ¼ Ce
B V
C>0, B
1 K>0, n KV n A B Eyring LðVÞ ¼ eV A, B V L, is a quantifiable life measure, such as mean life, median life, characteristic life, percentile, etc.; V, represents the stress levels, [temperature values should be expressed in absolute units (K)]. Inverse Power Law (IPL)
LðVÞ ¼
The General Log–Linear–Weibull (GLL–Weibull) model can be derived by setting the scale parameter to the life–stress relationship LðXÞ (see Appendix B). Here we will follow the notations assumed in ReliaSoft [8]. ¼ LðXÞ; while the shape parameter is assumed to remain constant across different stress levels. This yield the following GLL-Weibull pdf,
O. Guseva et al. / Polymer Degradation and Stability 82 (2003) 1–13
fðt; XÞ ¼ t1 e
0 þ
n P
j Xj
et
j¼1
0 þ
n P
j X j
j¼1
e
:
The total number of unknowns to solve for in this model is n+2, i.e. , 0, 1, . . ., n. The maximum likelihood estimation method [16,17] can be used to determine the parameters for the GLL relationship and the selected life distribution. The loglikelihood function for the Weibull distribution for right censored data is given by Ref. [8], lnðLÞ ¼ L
2 F X 6 1 T e 6 i ¼ ln4Ti e
n P
0 þ
j¼1
0 Ti e
0 þ
n P j¼1
0 þ
e
i¼1
S X
j Xi;j
n P j¼1
3 j Xi;j
7 7 5
j Xi;j
;
i¼1
where F is the number of exact times-to-failure data points; is the Weibull shape parameter (unknown, to be estimated); 0, 1, . . ., n, are the (n+1) GLL parameters (unknown, to be estimated); X ¼ ðX1 ; X2 ; . . . Xn Þ is the vector of n stresses; Ti is the exact failure time; S is the number of suspension data points; and Ti0 ; is the running time of the suspension data points. The solution (parameter estimates) can be found by maximising L, i.e. by simultaneous solving of (n+2) equations such that @L ¼ 0; @ @L ¼ 0; i ¼ 0; :::; n @i Appendix D. Likelihood ratio test In order to justify the assumption of a common shape parameter among the data obtained at different stress levels, the likelihood ratio (LR) test can be performed [17]. The LR test statistic T, can be calculated from
13
experimental data: T ¼ 2ðL1 þ L2 þ . . . þ Ln L0 Þ, where L1, L2, . . ., Ln are the log–likelihood values obtained by fitting a separate distribution to the data from each of the n stress levels. The log–likelihood value L0 is obtained by fitting a model with a common shape parameter and a separate scale parameter for each of the n stress levels. If the shape parameters are equal, than the distribution of T is approximately w2 with (n1) degrees of freedom. References [1] Martin JW, Saunders SC, Floyd FL, Wineburg JP. Methodologies for predicting the service life of coatings systems. Blue Bell, PA, USA: Federation of societies for coatings technology; 1996. [2] Guseva O, Brunner S, Richner P. Macromol Symposia 2002; 187:883–93. [3] Read WG, Froidevaux L, Waters JW. Geophys Res Lett 1993; 20:1299–302. [4] Wicks ZWJr, Jones FN, Pappas SP. Organic coatings: science and technology. 2nd ed. New York: Wiley-Interscience; 1999. [5] ReliaSoft Corporation. Life data analysis reference, Weibull ++ Version 5.0. Tucson, AZ USA: ReliaSoft Publishing; 1997. [6] Bauer DR. Polym Degrad Stab 2000;69:297–306. [7] Jorgensen G, Bingham C, King D, Lewandowski A, Netter J, Terwilliger K. NREL/ CP-520-28579; 2000. Available from: http://www.doe.gov/bridge. [8] ReliaSoft Corporation. Accelerated life testing reference, ALTA version 6. Tucson, AZ, USA: ReliaSoft Publishing; 2001. [9] Brunner S, Richner P. Accelerated weathering device for service life prediction. [in preparation]. [10] Allen NS, Katami H. In: Clough RL, Billingham NC, Gillen KT, editors. Polymer durability. Advances in Chemistry Series249. New York: American Chemical Society; 1996. p. 537. [11] Sampers J. Polym Degrad Stab 2002;76:455–65. [12] Bauer DR. J Coat Technol 1997;69:85–96. [13] Available from: http://uvb.nrel.colostate.edu. [14] Available from: http://klimadiagramme.de. [15] Available from: http://www.empa.ch/nabel. [16] Nelson W. Applied life data analysis. New York: John Wiley & Sons; 1982. [17] Nelson W. Accelerated testing: statistical models, test plans, and data analyses. New York: John Wiley & Sons; 1990. [18] Lawless JF. Statistical models and methods for lifetime data. New York: John Wiley & Sons; 1982. [19] Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions, Vol. 1. New York: John Wiley & Sons; 1994. [20] Kececioglu D. Reliability engineering handbook, Vol. 1. Englewood Cliffs, New Jersey: Prentice Hall; 1991. [21] Meeker WQ, Escobar LA. Statistical methods for reliability data. New York: John Wiley & Sons; 1998.