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ScienceDirect Procedia Engineering 167 (2016) 10 – 17

Estimating fatigue life of structural components from accelerated data via a Birnbaum-Saunders model with shape and scale stress dependent parameters Giuseppe D’Annaa*, Massimiliano Giorgiob, Aniello Ricciob b

a CIRA Italian Aerospace Research Center, via Maiorise, 81043 Capua, Italy Second University of Naples, Departement of industrial and Information Engineering, via Roma, 81031 Aversa, Italy

Abstract The Birnbaum-Saunders model is widely applied to model fatigue failures caused by cyclic stresses both in the case of standard and accelerated life tests. This latter kind of tests are adopted when the product of interest is very reliable, in order to obtain failure data in a reasonably short amount of time. Estimates of the product’s reliability or the long-term performances at normal use condition are then obtained, from accelerated failure data, adopting functional relationships accounting for the effect of the accelerating variables on the product’s lifetime distribution. Customarily these models are formulated assuming that the accelerating variables affect the values of the lifetime distribution parameters, and not its form. In particular, in literature the Birnbaum-Saunders distribution is usually applied to accelerated data under the hypothesis that the scale parameter only depends on the stress conditions, while the shape parameter doesn’t depend on them. In this paper, an applicative example is presented in which this standard model does not work satisfactorily. In fact, it is shown as, in the case of the considered real set of accelerated fatigue failure data, the Birnbaum-Saunders distribution, in which both scale and shape parameters depend on the stress conditions, fits the data significantly better than the abovementioned standard option. Difference among lifetime distribution estimates provided by the two different considered models are highlighted and discussed.

©©2016 This is an open access article under the CC BY-NC-ND license 2016The TheAuthors. Authors.Published PublishedbybyElsevier ElsevierLtd. Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of DRaF2016. Peer-review under responsibility of the Organizing Committee of DRaF2016 Keywords: accelerated life testing, Birnbaum-Saunders Distribution, stress dependent shape parameter, maximum likelihood estimation.

Nomenclature ALT Accelerated life test BSR Birnbaum-Saunders based regression model IPL Inverse power law

BS PDF CDF

Birnbaum-Saunders distribution Probability density function Cumulative density function

1. INTRODUCTION Fatigue [1], [2] is one of the main causes of failure for structural components. Thus, demonstrating fatigue reliability of these components is of main concern. Nonetheless, it is a very challenging estimation problem because these components are very reliable, thus the collection of failure data, via standard life tests, would require many years [3], [4]. Thus, accelerated life tests (ALTs) are usually adopted in order obtaining failure data in a reasonably short amount of time. ALTs consist in testing the products of interest under conditions which are more severe than the standard ones. This practice allows forcing the products to fail more quickly. Estimates of the product’s reliability at normal use condition are then obtained (i.e., extrapolated), from accelerated failure data by adopting functional relations named link functions, that account for the effect of the accelerating variables on the product’s lifetime distribution. Customarily, extrapolation is operated assuming that the accelerating variables affect the

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of DRaF2016

doi:10.1016/j.proeng.2016.11.663

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Giuseppe D’Anna et al. / Procedia Engineering 167 (2016) 10 – 17

parameters of the lifetime distribution, and not its form. In fact, models adopted to analyze accelerated data are usually obtained by incorporating one or more link functions into the baseline distribution function (i.e., the distribution which describes the lifetime of the product of interest at standard condition). Resulting models are typically referred as regression models. The main feature of these models is that their use allows accounting, in an explicit way, for the effect of experimental conditions on the lifetime distribution of the products of interest (for a comprehensive review of ALT models see [5], [6], [7], [8], [9], and [10]). In particular, in literature, regression models, which adopt as baseline distribution the Birnbaum-Saunders (BS) model, are usually obtained by assuming that only the scale parameter depends on the stress conditions, whereas the shape parameter doesn’t depend on them (e.g., [11], [12]). This is a quite standard hypothesis, which is often made also in the case of other wide adopted baseline distributions (e.g., Weibull [13] and Lognormal). In this paper, an applicative example is presented in which this standard model (usually called Accelerate Failure Time model [10]) doesn’t work satisfactorily. In fact, it is shown as, in the case of the considered real set of accelerated fatigue failure time data, the Birnbaum-Saunders based regression (BSR) model proposed in this paper, in which both scale and shape parameters depend on the stress conditions, fits the data significantly better than the abovementioned standard option. The parameters of the considered models are estimated by means of maximum likelihood method, which doesn’t provide closed-form solutions. Thus estimates where obtained adopting an iterative numerical method. In the case of the BSR model, convergence problems can be encountered. Results reported in this paper have been obtained by using the Matlab Global Optimization tool box, which was found working satisfactorily with the considered data. Differences among the lifetime distribution estimates provided by the two considered regression models are highlighted and discussed. Formal comparisons among the considered models are performed using the likelihood ratio test and the Akaike information criterion. 2. THE MOTIVATING DATASET The dataset that stimulated the paper (see Table 1) was first presented in [14]. It is a very popular set of real data, that has been frequently used in the past to perform Birnbaum-Saunders (BS) based statistical analyses (see for example [11], [12] and [15]). It is constituted by fatigue failure times of N 304 6061-T6 aluminum coupons oscillated at 18 cycles per second under three different (constant) levels of maximum stress per cycle ( s1 31000 psi, s2 26000 psi and s3 21000 psi). In fact, a complete set of failure data is obtained for each stress level, with n1

101 , n2 102 and n3

101 failures observed at 31000 , 26000 and

21000 psi, respectively. Table 1. Failure data (cycles × 10-3) of 304 6061-T6 aluminum coupons tested for fatigue failure under cyclic stress at three different levels of maximum stress per cycle. Group 1: failure data collected at 31000 psi 70 90 96 97 99 112 113 114 114 114 124 128 128 129 139 134 134 134 134 134 142 142 142 142 144 157 157 157 158 159

100 116 130 136 144 162

103 119 130 136 145 163

104 120 130 137 146 163

104 120 131 138 148 164

105 120 131 138 148 166

107 121 131 138 149 166

108 121 131 139 151 168

108 123 131 139 151 170

108 124 132 141 152 174

109 124 132 141 155 196

109 124 132 142 156 212

112 124 133 142 157

Group 2: failure data collected at 26000 psi 233 258 268 276 290 342 342 344 349 350 367 370 370 372 372 400 400 403 404 406 428 432 432 433 433 466 468 470 470 473

310 350 374 408 437 474

312 351 375 408 438 476

315 351 376 410 439 476

318 352 379 412 439 486

321 352 379 414 443 488

321 356 380 416 445 489

329 358 382 416 445 490

335 358 389 416 452 491

336 360 389 420 456 503

338 362 395 422 456 517

338 363 396 423 460 540

342 366 400 426 464 560

Group 3: failure data collected at 21000 psi 370 706 716 746 785 1016 1018 1020 1055 1085 1235 1238 1252 1258 1262 1419 1420 1420 1450 1452 1567 1578 1594 1602 1604 1820 1868 1881 1890 1893

797 1102 1269 1475 1608 1895

844 1102 1270 1478 1630 1910

855 1108 1290 1481 1642 1923

858 1115 1293 1485 1674 1940

886 1120 1300 1502 1730 1945

886 1134 1310 1505 1750 2023

930 1140 1313 1513 1750 2100

960 1199 1315 1522 1763 2130

988 1200 1330 1522 1768 2215

990 1200 1355 1530 1781 2268

1000 1203 1390 1540 1782 2440

1010 1222 1416 1560 1792

3. BIRNBAUM-SAUNDERS DISTRIBUTION The Birnbaum-Saunders distribution was first proposed in [16] to model material lifetime under cyclic loads induced crack growth. An alternative derivation of the Birnbaum-Saunders model was introduced in [17], where the physical justification for the use of this distribution as a fatigue failure model has been reinforced, by relaxing the assumptions made in [16].

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By definition, the BS random variable, T a BS D , E , is linked to the standard normal random variable , Z

N 0,1 , by the

following (one to one) relationship:

1 ª T Eº « » , T t 0, D , E ! 0 T¼ where D and E are the shape and the scale parameters of the BS distribution, respectively. Z

D ¬ E

The probability density function (PDF), f t , the cumulative density function (CDF), F t , and the reliability function,

R t , of the BS random variable T can be expressed as:

f t

1

t 3 2 t E

2 2 S

D E

e

1 §t E · ¨ 2 ¸ 2D 2 © E t ¹

(1)

° 1 § t E ·½° F t ) ® ¨¨ ¸¾ t ¸¹ °¿ ¯°D © E R t

E ·°½ ° 1 § t P T ! t 1 F t 1 ) ® ¨¨ ¸¾ D E t ¸¹ °¿ ¯° ©

where ) • is the CDF of the standard normal random variable, Z . Other details about the model features are provided in literature (see for example, [12] and [18]). 4. BIRNBAUM-SAUNDERS REGRESSION MODELS The BS based regression (BSR) model proposed in this paper, is obtained combining BS distribution (i.e., the baseline distribution) and two life–stress relations, one for each model parameter. Indeed, it is assumed that, at any considered stress level, fatigue failure time follows a BS model in which both scale and shape parameters depend on the test conditions. Focusing on the experimental situation described in section 2, a unique accelerating variable is considered. In the following, this (possibly transformed) variable will be denoted as x . Consequently, the BS random variable, Ti , that denotes the lifetime at the stress condition x

xi , will be modelled as: BS D xi , E xi

Ti

(2)

where, D x and E x are the relations which account for the effect of the accelerating variable x on the parameters D and

E , respectively,. As previously mentioned, this model is more general than the BS based regression model (3), that is usually adopted in literature (see for example [11] and [12]) to analyze the data in table 1. In fact, this standard BS regression model is obtained assuming that the scale parameter E depends on the stress while the shape parameter D doesn’t depend on it.

Ti

BS D , E xi

(3)

Following [11] and [12], the very classical inverse power law function (4) (see for example [6] and [10]) is adopted to model the link function E x :

E x J E x

KE

(4)

where J E and KE are (positive valued) parameters that have to be estimated on the basis of the available data. In the following, it will be conventionally set x

s s3

s 21000 (see stress levels in section 2). Note that under this setting, given that

for x 1 it is E 1 J E , it results that J E in equation (4) can also be viewed as the scale parameter of the BS distribution at 21000 psi. The following inverse power law function, analogue to (4), is adopted to model D x in the model (2):

D x J D xKD where J D can be viewed as the value of the shape parameter of the BS distribution at 21000 psi.

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5. ESTIMATION OF THE PARAMETERS OF THE CONSIDERED BSR MODELS In this section the estimates of the parameters of the models (2) and (3) are reported. Following [11] and [12], these estimates Jˆ ,Kˆ , Jˆ ,Kˆ are obtained using the maximum likelihood method. The maximum likelihood estimates (MLEs), E E D D , of the J ,K , J ,K parameters of model (2) are the values of E E D D that maximize (over the parameter space) the log-likelihood function (5): 2J K ª tij J E xi D E 1 1 1 « 3 ni §¨ § · § · N ln ¨ ¸ N ¨ ln J D ln J E ¸ 2 ¦¦ 2 tij © ¹ 2 J D « i 1 j 1 ¨ J E xi 2J D KE © 2 2 S ¹ © ¬ 3 ni KE · 3 § 1 3 3 3 K ¦¦ ln tij J E xi E ¨KD ¸ ¦ ni ln xi 2 ¦ ni ln xi2 ¦ ln tij 2 ¹ i1 2 i1 JD i 1 i 1 j 1 ©

l1 J E ,KE , J D ,KD t

Similarly, the MLEs, likelihood function (6):

D , J E ,KE

1 § l2 D , J E ,KE t N ln ¨ © 2 2 S

t

, of the parameters of model (3) are the values of

N 1 · ¸ N ln D 2 D 2 D 2 ¹

^t

K ª 3 ni § tij J E xi E «¦¦ ¨ K tij «¬ i 1 j 1 ¨© J E xi E

; i 1, 2,3, j 1, 2,..., ni `

D , J E , and KE

.

that maximize the log-

·º 3 ni KE 3 K ¸» ¦¦ ln tij J E xi E ¦ ni ln xi ¸» i 1 j 1 2 i1 ¹¼ .

(5)

(6)

denotes the set of data in table 1, ti , j denotes the j-th data in i-th group n 101 n2 102 n 101 xi 268 ni ), is the number of failures which pertain to the i-th group (i.e., 1 , , and 3 ), is the

In equations (5) and (6), (e.g., t2,3

·º ¸» ¸» ¹¼

i, j

N ¦ i 1 ni 304 x 31000 21000 x2 26000 21000 x 21000 21000 associated stress level (i.e., 1 , , and 3 ), and . No closed form expression exists for the Maximum likelihood estimators of the parameters of the BS regression model, both in the case of log-likelihood functions (5) and (6). Thus, estimates have been found using numerical maximization techniques. In the case of the considered regression models, convergence problems can be encountered. Results reported in table 2 have been obtained using the Matlab Global Optimization tool box, which was found working satisfactorily with the considered data. 3

Table 2. MLEs of the parameters of models (2) and (3) obtained on the basis of data reported in Table 1. MLEs of parameters of the BSR model (2)

JˆD

0.288 , KˆD

1.582 , JÊ

1370.9 , KÊ

5.989

MLEs of parameters of the BSR model (3)

D

0.225 , J E

1354.5 , KE

5.989

6. ANALYSES OF RESULTS AND PRELIMINARY COMPARATIVE STUDY In order to check the ability of models (2) and (3) to fit the data in table 1, before employing formal statistical tools, we performed a preliminary quali-quantitative analysis. As a first step we estimated, for each i 1,2,3 , the parameters D i , E i of the variables Ti

BS D i , Ei which describes the fatigue life of the considered units at the i-th stress level, xi . The MLEs, D i , E i , of

the parameters D i , E i were obtained maximizing the following log-likelihood function:

l3,i Di , Ei t where, t i

^t

i, j

ni ni 3 ni 1 ln 2 S ni ln 2 Di Ei ¦ ln tij Ei ¦ ln tij 2 2 j1 2 Di2 j 1

ª ni § tij Ei «¦ ¨¨ «¬ j 1 © Ei tij

· º ni ¸¸ » 2 . ¹ »¼ Di

; j 1,2,..., ni ` is the set of data in table 1 pertaining to the group i. Hence, for example, estimates of parameters

D1 , E1 are obtained using only the data collected at 31000 psi. Results obtained for i 1,2,3 , are reported in the first row of Table 3 (see also [14]). These estimates show that the scale parameter E significantly depends on the stress level (it increases as the level of the stress decreases: i.e., E3 ! E 2 ! E1 ), whereas the dependence of D1 , D 2 , and D 3 on the stress level seems to be weaker. On the basis of these results, data in table 1 are typically analyzed using model (3). Nonetheless, differences exist among D1 , D 2 , and D 3 that don’t allow excluding that also this parameter depends on the stress level. In order to investigate this point, we estimated the same parameters using models (2) and (3). These estimates were computed (for each i 1,2,3 ) by means of formulas in equations (7) and (8), in the case of model (2) and (3) respectively:

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Dˆ xi JˆD xiKˆD , Eî

Dî

Di D , Ei where x1

36000 21000 , x2

Eˆ xi JÊ xi

KÊ

(7)

E xi J E x

KE i

(8)

21000 21000 1 , JˆD , KˆD , JÊ , and KÊ are the estimates in the first column of

26000 21000 , x3

table 2 and D , J E , and KE are those reported in the second column of the same table. Obtained results are reported in the second and in the third row of table 3, respectively. Table 3. Maximum likelihood estimates of the couples of parameters D1 , E1 , D 2 , E 2 , and D 3 , E 3 obtained using data in table 1. Estimates in the first row are achieved using only the data collected at the considered stress level,. Those reported in the other rows are obtained using models (2) and (3), on the basis of all the data. Stress level 36000 psi No link functions

D1 =0.170

Model (2) Model (3)

26000 psi

D 2 =0.161

Dˆ1 =0.155

E1 =131.81 Eˆ =133.05

D1 =0.226

E1 =131.46

1

21000 psi

D 3 =0.310

Dˆ 2 =0.205

E 2 =392.76 Eˆ =381.50

Dˆ3 =0.228

E3 =1336.4 Eˆ =1370.9

D 2 =0.226

E 2 =376.94

D 3 =0.226

E3 =1354.5

2

3

To check whether model (2) is able to describe the dependence of model parameters on the stress level better than model (3) or not, we compared these estimates to those reported in the first row of table 3. In fact, estimates provided by models (2) and (3) for D i and E i are obtained from all the data reported in table 1, using the link functions to extrapolate information about the life at stress level i from data collected at the other stress levels. Hence it seems quite natural to presume that the model who provides estimates that are closer to those obtained using only the data collected at the considered stress level (i.e., those in the first rows of table 3) is likely also the one who better describes the dependence of model parameters on the stress level. The difference between performances of models (2) and (3) can be better appreciated looking at the PDFs plotted in Figure 1.

Figure 1. Comparison PDFs of variables .. obtained computing equation (1) at the MLEs reported in table 3 The solid curve in figure 1a is the MLE estimate of the PDF (1) at 36000 psi, obtained using data pertaining to the group 1, only. This estimate, say f1 t , is achieved, as in equation (9), computing the PDF (1) at MLEs D1 and E1 reported in the first row of table 3:

f1 t

1 2 2 S

t 3 2 t E1

D1 E1

exp ª

·º 1 § t E1 2 ¸» « 2 ¨ t 2 D E «¬ 1 © 1 ¹ »¼

(9)

The solid curves in figure 1b and 1c, are the MLEs of the PDF (1), say f 2 t and f3 t , at 26000psi end 21000 psi, respectively. These estimates are obtained computing the PDF (1) at MLEs ( D 2 , E 2 ) and ( D 3 , E3 ), respectively. Similarly the dotted curves are the MLE of the same PDFs, say fˆ1 t , fˆ2 t and fˆ3 t , obtained under model (2) (i.e., these estimates are obtained computing the PDF (1) at the MLEs reported in the second row of table 3. Finally, the dashed curves, say f1 t , f 2 t and f3 t , are obtained using the estimates reported in the third row. Again, the idea is that a good model should provide estimates of the considered PDFs that are close to those one obtains directly using data collected at any specific stress level only (i.e., close to the solid curves). Unfortunately, although model (2) seems to provide results that are slightly better than those

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obtained using model (3), both estimates in table 3 and PDFs in figure 1 don’t give clear evidence about the fact that model (2) should be preferred to model (3). On the other side a clear difference exists between solutions provided by these two models, hence it is important to further investigate about this point by means of formal statistical model selection procedures. 7. FORMAL STATISTICAL ANALYSIS In this section we report the results obtained using two very commonly used model selection criteria: the likelihood ratio test and the Akaike information criterion. These criteria allow solving the trade-off between goodness of fit and complexity of the “best” model. In general, models with more parameters fit the data better than models with fewer parameters, hence in order to select the best model there is the need to balance against model simplicity. According to this criteria, given a set of candidate models, the model with more parameters will be preferred to the simpler ones only if it brings significant improvement in terms of fitting. The likelihood ratio test is a formal statistical test that can be used in the case models to compare, say M 0 and M 1 , are nested models. Under this circumstance, assumed that M 0 is nested in M 1 , for testing the (null) hypothesis that a certain model, M 0 , fits the considered data better than the alternative model M 1 , it is possible to use the following likelihood ratio statistic:

/

2 ˆ 0 ˆ 1

(10)

that is asymptotically distributed as a chi-square random variable with Q degrees of freedom. The null hypothesis is rejected for large value of / . In equation (10), ˆ 0 and ˆ 1 are the estimated log-likelihood functions formulated under models M 0 and

M 1 , respectively. These log-likelihood functions are obtained calculating

0

and

1

in correspondence of maximum likelihood

estimates of its parameters. The degrees of freedom, Q , of the likelihood ratio statistic are equal to the difference in the number of parameters between the two models. Note that, saying that the model M 0 , is nested within the model M 1 , is equivalent to say that M 0 can be obtained as a special case of the model M 1 , forcing a subset of its parameters to assume a (fixed) predetermined value. The Akaike Information Criterion [19] is a model selection criteria that can be adopted in the case candidate models are not nested. This criterion leads to prefer the model for which the value of the Akaike Information Criterion (AIC) index is lower. The AIC index for a given model is calculated as:

AIC 2h 2ˆ where h is the number of parameters in the model and ˆ is the estimated log-likelihood function. Main results obtained adopting the considered selection criteria are reported in table 4 and 5. Notation used in table 4 and 5 are introduced in table 6. Table 4. Results of the Likelihood ratio test (level of significance 0.05 ). Null hypothesis M1 M2 M3 M1 M1 M6

Alternative hypothesis M4 M4 M4 M5 M6 M5

Likelihood ratio statistics 925.20 259.2 624.8 908.8 864.6 44.2

p-value 5.8E-199 2.5E-57 2.1E-136 4.5E-198 4.9E-190 3.0E-11

Table 5. Estimated log-likelihood and AIC of considered models Model M1 M2 M3 M4 M5 M6

Estimated log-likelihood -2238.9 -1905.9 -2088.7 -1776.3 -1784.5 -1806.6

AIC 4481.8 3819.8 4185.5 3564.6 3576.9 3619.2

Decision M1 rejected M2 rejected M3 rejected M1 rejected M1 rejected M6 rejected

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Table 6. Details about notations used in table 4 and 5 Model

Assumptions

# parameters

Parameters

M1

Ti

BS D , E

Both parameters don’t depend on the stress

2

D , E

M2

Ti

BS D , Ei

4

D , E1 , E 2 , E 3

M3

Ti

BS D i , E

4

D1 , D 2 , D 3 , E

M4

Ti

BS D i , Ei

Only the scale parameter depends on the stress level. Doesn’t use link functions Only the shape parameter depends on the stress level. Doesn’t use link functions Both scale and shape parameters depend on the stress level. Doesn’t use link functions

6

D 1 , D 2 , D 3 , E1 , E 2 , E 3

M5 M6

description

Ti Ti

BS D xi , E xi

Model (2)

4

J D , KD , J E , KE

BS D , E xi

Model (3)

3

D , J E , KE

Note that all the models in the first column of table 4 are nested into the models in the second column of the corresponding row. Comparison between models that are not nested can be performed using the AIC index. The likelihood ratio test demonstrates that (at the level of significance 0.05), for the data in table 1, the best model among the considered candidates is the model (2) (i.e., M5). In particular, results reported in the first three rows of table 4, give evidence that the model M4 (which assumes that both scale and shape parameters depend on the stress) fits the data better than models M1, M2 and M3 (which assume that one or both the parameters are stress independent). Results in rows 4 and 5 show that models (2) and (3) fit the data better than model M1. Finally, the results in the last row shows that model (2) must be preferred to model (3). The Akaike information criterion leads to the same result. In fact the only model, among the considered candidates, which present a smaller AIC with respect to model (2) is the model M4, that is not suited for analyzing ALT data. In fact, this model, that doesn’t use any link function, doesn’t allow to estimate parameters that refers to stress conditions that differ from those at which data are collected. 8. CONCLUSIONS In this paper a Birnbaum-Saunders based regression model is presented in which both the scale and shape parameters depend on the stress conditions. The model is applied to a set of real accelerated fatigue life data. It is demonstrated that the proposed model allows fitting the available set of data better than the classical Birnbaum-Saunders based regression model, which assumes that only the scale parameter depends on the stress conditions. The parameters of the proposed model are estimated by using the maximum likelihood method. No closed form expression exists for the considered estimators. Thus, maximum likelihood estimates are obtained numerically. The ability of models to fit the considered data is checked using the likelihood ratio test and the Aikaike information criterion. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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