# A model for accelerated life testing

## A model for accelerated life testing

Structural Safety, 12 (1993) 129-136 129 Elsevier A model for accelerated life testing * H. Strelec Department of Statistics, University of Technol...

Structural Safety, 12 (1993) 129-136

129

Elsevier

A model for accelerated life testing * H. Strelec Department of Statistics, University of Technology Vienna, Wiedner Hauptstrasse 8-10, A- 1040 Wien, Austria

Abstract. After a short motivation for the problem of extremely high reliability a nonparametric model for accelerated life testing is described. Besides well introduced classical models for high reliabilities are mentioned. Finally a stress-dependent model for crack growth based on fatigue accumulation is discussed.

Key words: high reliability; accelerated life testing; acceleration model; acceleration function.

1. Introduction Many products in modern life are characterized by high reliability what means that the mean life time to failure (MTBF) is comparetivly h i g h . Examples for such products are given by satellites, cardiac pacemakers or articular implants where the failure probability within some few ten years have to be extremely small so that the MTBF will become very high. For instance the specification of a failure probability of only 1% within ten years makes an almost 1000 years (!) MTBF necessary if exponential life time distribution may be assumed. Another characteristic of technical products nowadays is the high complexity. Many technical systems like cars, production machines or power plants consist of several thousands of components. If the MTBF of such a system should be of a useful magnitude (some few thousand hours in operation mode), by a simplified approach (almost equal reliability for components, independent failing) you can derive that MTBFs for the components have to be about the required system M T B F multiplied by the number of components. In the case where a system consists of 1000 components, the requirement of a system MTBF of 1000 hours in operation mode makes a component MTBF of about 10 6 h necessary. That is more than 100 years! From an economical point of view life testing cannot be done on the whole system. Instead of that, reliability characteristics for the system are derived from those of the components the system consists of. Because of the above stated facts, analysis of items (components) of high reliability has to be considered here as well.

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In both cases it is obvious that the observation of life data within a reasonable time is impossible. Two main strategies are possible to overcome this problem: • extremely censoring A great number of items is under test and the life test will stop after the first failure--or perhaps after the second or third--has occured or if a time limit is reached which may seem to be absolutely long (one year or so), but which must actually be extremely short compared with the MTBF. Assuming some underlying model for the life time distribution, estimation and hypothesis testing of life time parameters may be performed on the base of those censored data. • accelerated life testing The key idea for accelerated life testing is the possibility of life time reduction when underlying stress will increase. For instance life time of a light bulb will decrease probably significantly if electrical voltage will increase for 10%. Therefore the observation of life tests becomes either possible at all or within a significantly shorter time. If it is possible to describe the relationship of life time distribution and underlying stress, information on life time distribution for high stress may be used for life time characteristics under usual stress. A short discussion about advantages and disadvantages of either methods is given in the next section and should be the base for the decision which strategy is to be preferred. An extensive description of concepts and methods in accelerated life testing can be found in Nelson [1].

2. Extremely censoring versus accelerated life testing The short description of both strategies to overcome the problem of high reliability, which is given in the preceding section, is sufficient to point out the advantages and disadvantages of both principles.

2.1. Extremely censoring Whatever kind of censoring is chosen, time censoring on the one hand a n d / o r failure censoring on the other hand, the main advantages of even extremely high censored sampling are: • Life tests are performed under usual conditions. • If a parametric life time model may be assumed, maximum likelihood estimation can be performed with its (asymptotically) optimal features. If for instance an exponentional life time model is given, the estimate of the MTBF is the reciprocal value of the total time on test (sum of observed operation times over all items on test). On the other hand the following disadvantages have to be considered which might be very severe: • The life time distribution type must be known otherwise any information about an unknown distribution will stop at the censoring limit. But unfortunately the life time distribution type can only be judged on the base of a very short starting phase of the life time of about one percent or less of the MTBF (see Fig. 1). The consequence is a great uncertainty on the assumption about the underlying life time distribution. • A great problem in connection with small portions of observed life time (see remark above) are early failures. Within this short starting phase of life time, early failures caused by atypical

131

F(t) 0.7

0.6 0.5 0.4 0.3

106 h

0.2 0.1

F(0.1) ~ 0.01 / j . , 1 0.0 -

a r e , ,

,

of

observation! ,

,

,

5

,

,

,

,

10

,

t (in 10 5 h)

0.1^=104h > 1year Fig. 1. Observation phase and life time distribution under usual stress with extremely high reliability.

material defects, carelessness at assembly, etc. are observered in an unproportional higher amount than usual. As a consequence, estimations of the life time parameter will be far too small and will therefore be useless for economical estimations of product reliability.

2.2. Accelerated life testing Naturally, advantages and disadvantages of acceleration methods are discussed within respect to extremely censoring. Of course some moderate form of censoring or trimming also makes sense in accelerated life testing because of cutting off early failures and unnecessary high values. The main advantages are: • Because of shorter MTBFs a greater part of a sample under test may be observed until failure. Trimming to the lower end becomes possible and therefore the influence of early failures may be reduced significantly. • Because of relatively short testing time more items may be tested within the time reserved for purpose. The consequences are larger samples, more information and therefore higher

precision. • Because of more information about a longer part of the life time distribution the type of distribution may be estimated much better. Of course there are also obvious disadvantages in accelerated life testing, the main facts being: • An extrapolation is necessary from high stress levels to usual stress. Every kind of extrapolation is problematical, but on the other hand when performing extremely censored life tests the type of life time distribution has to be extrapolated from a very short starting phase off. This is not less dangerous at least. • The relationship between life time distribution on high stress and under usual stress, which is the main key for accelerated life testing, may only rarely be given theoretically. In most of the cases only some more or less appropriate model must be used and estimated. When comparing advantages and disadvantages of the two discussed methods and when

132

F(t)

erh~hte Bel~tung S

E~(s)

1.0-

0.8

0.6

0.4

0.2

0.0 0

I0

t

20

a(s.,s;o

30

40

(in 103 h)

Fig. 2. Life time distribution u n d e r usual a n d high stress.

accepting the obvious neccessity of extremely high reliability, accelerated life testing seems to be the more reasonable and economically sensible possibility to overcome this problem.

3. The acceleration function Besides classical models to describe the relationship between stress and life time distribution which will shortly be discussed later, the concept of the acceleration function seems to be very general and efficient for the description of this relationship. It was first mentioned by Gonnot [2] and then developed by Viertl [3,4], Strelec and Viertl [5,6] and Strelec [7] among others. Figure 2 shows the situation where the life time T of some power transistor has exponential distribution with MTBF = 20.000 h. When raising the applied voltage for 10 percent the life t i m e distribution stays exponential but the MTBF will drop to 5.000 h. Now time t with probability p = P r ( T < t lS) = F ( t I S )

(1)

of T being below t under higher stress S is corresponding to time 4t under usual stress because the chance of failing before 4t under usual stress S. also equals p. Therefore the relationship between usual and high stress is given by

t ~a(Su, S; t ) = 4 t

(2)

that means, time under stress S is "running four times faster" than under usual stress. Here stress S describes voltage, but it can also consist of more stress components like temperature, pressure, humidity and so, so that it has to be considered as a stress vector. Generally the socalled acceleration function a(S1, \$2; t) (shortly af) describes the relationship of life time distribution under stress levels S 1 and S 2 by

F(tl \$2) = Pr(T ~< t l \$ 2 )

= F ( a ( S , , S2; t) I S~) = Pr(T ~ a(S,, 82; t) I \$1)

(3)

with F being the cumulative distribution function (cdf) of life time under the considered stress.

133 Two types of afs are of great importance, linear functions

a(S 1,S2; t ) = a ( S 1,\$2)t

for t > 0

(4)

on the one hand and powertype functions

a(S,, S2; t)=o/(Sl, S2)t ~8(s1"\$2) for t >0

(5)

on the other hand. The simple form of these functions is as important as the fact that for many classes of parametric life time models socalled type homogeneous acceleration functions are of linear or power type form, where type homogeneity means that the (parametric) family of life time distributions is not left when changing stress. In these cases the problem of accelerated life testing is solved by the estimation of the acceleration coefficients a(Sl, S 2) and /3(\$1, S2). The concept of accelerated life testing using acceleration functions is now given by the following three steps: (1) Many short life tests are performed under high stress levels and lead to an estimated acceleration function a(Sl, S2; t) between (high) stress levels S 1 and S 2. (2) The estimation of the life time distribution under some high stress level S may be done with high precision leading to an estimated cdf F ( t I S). (3) Retransformation F ( t ISu) = P ( a - ' ( S u, S; t)lS)

(6)

leads to the estimated cdf under usual stress which was the aim of the whole concept. Instead of steps 2 and 3 an alternate is given by (2a) "Simulation" of life time under usual stress by transforming observed life data under stress S, using the acceleration function determined above: tju = a(Su, S; ts)

(7)

(3a) Estimation of the cdf under usual stress using these simulated data leads t o / ~ ( t I Su) too.

4. Estimation of acceleration functions In the case of linear or power type acceleration functions the estimation of acceleration coefficients may be based on some loglinear form of the kind

a(Su, S ) = e x p

( /~__A l eftGu(S)+Go(S )

(8)

or /3(Su,S)=exp

~d,H,(S)+H0(S

u=l

) .

(9)

The functions G~, and H, are assumed to be known so that only the linear coefficients c~, and d, are to be estimated. In Refs. [5,6] methods are given for this purpose. The linear case is discussed below. The analysis starts with the determination of stress levels Su << So < Sl < S2 < " " • < Sr_ 1 < S r for which life test should be performed (r >1rA). For every stress level a sample of equal

134

sample size n has to be tested. C o m m o n trimming indices n 1 = [ne 1] + 1 and n 2 = [n(1 - e2)] + 1 for trimming portions e I and E2 with 0 < e 1 < 1 - ff2 ,( 1 define a moderate censoring scheme. The observation of life tests on these considered stress levels leads to ordered life data for every stress level St: tul ) ~< t/,(2) ~< " " " ~< tt,(n2). NOW

1 In o l ( S l _ l ,

Sl)=

n2 E

n 2 - - n I -~- 1 i=nl

In t'-"(/) tl,(i )

(10)

is an estimate for the logarithm of the acceleration coefficient between stress levels S t_ 1 and S t l = 1 . . . . , r) with asymptotic normality and stress independent variability. Therefore the solution for the coefficients c~ in the case of a linear acceleration function is given by the least squares solution of the following system rA ln---'-'~(S,_ 1, S t ) = Y'~ c~[G~(Sz)-Gu(SI_I) ] + [Go(Sl)-Go(Sz_a) ] (11) /z=l

of linear equations. In matrix form this solution c = (cl, c 2. . . . .

c = (X'£elX)-l(x'~,el)(y--g)

CrA)' is

given by (12)

with design matrix

X = (G~(S,)-G~(S,_I))I=I ...... ;~.=1 ...... a

(13)

and y : (1--n-~aol(S,_,, S , ) ) , : , ......

(14)

respectively

g=(Go(St)-Go(S,

,)),=1 ..... r"

(15)

The structure of the covariance matrix of y is given by £e = ( ~ v ) u ' = l ...... with ~ l = 2, o~u + 1 = - 1 and thr = 0 otherwise.

5. Classical acceleration models Since more than 40 years in the field of electronics parametric acceleration models have been used very extensively. Usually the exponential distribution is used as the life time distribution because almost no wear-out behaviour can be observed at classical electronic items like transistors, resistors, simple ICs and the like. Because of this fact the whole life time distribution is determined by the M T B F or by the (constant) failure rate A = M T B F - 1 in this case. Therefore the relationship between stress level and life time distribution may be expressed by a functional description of failure rate with respect to the underlying stress. With V being electrical voltage and T describing temperature, some of the best known models are

Power rule model: A ( V ) = c V p, c , p > 0 ;

(16)

135

Arrhenius model: a, b > 0 ;

) t ( T ) = e a-b/T,

(17)

Eyring model: ,~(T) = T e a-b/r,

a,b>0.

(18)

Generalized Eyring model: A(T, V) = aT e b/T+cV+dv/T,

a

> 0; b, c, d ~ ~.

(19)

Usually estimates ,~(St) for A(St) are determined from life tests on high stress levels S t and least squares method Y'~ [,~(S,) - A(St)] 2 ~ min!

(20)

l

leads to estimates for the unknown parameters of the considered model.

6. Acceleration model for crack growth In Sch~ibe [8] a stress-independent model for crack growth based on fatigue accumulation is described. A generalization with consideration of underlying stress is given by

x ( t ) = (b(S)t + g e ( t , h(S)) + o-(S)W(t)) a

(21)

with X(t) describing crack length at time t, P being a Poisson process with intensity parameter A and W defining a standard Wiener process. Without consideration of a the crack length is additively determined by a portion on the one hand which depends linearly on time and which is due to fatigue. On the other hand shocks occur following a Poisson process which contribute length b to crack length. Residual randomness of crack growth is described by a Wiener process with mean value 0 and variablity factor o-. Obviously fatigue, shock frequency and variabilty of random residual are depending on the underlying stress S. The parameter a characterizes crack growth in the following sense: a

{>} = 1 <

...

( progressive ~ ~linear ~crack growth ~ degressive )

and is assumed to be independent from underlying stress. Life time T of a device with crack propagation is determined by a critical crack length h 0 as T = min{t > 0 : X(t)>~h0}

(22)

the probability density function of which is given in [8]. Accelerated life testing using model (21) can now be performed in two steps: (1) For high stress levels S t the parameters a(St), b(St), A(S/) and ~r(S t) have to be estimated, for instance by maximum likelihood method. (2) Regression models for each of the parameters a(S), b(S), A(S) and ~r(S) should be assumed and the regression coefficients have to be estimated using the estimates of step 1.

136

References 1 W. Nelson, Accelerated Testing. Statistical Models, Test Plans and Data Analysis, Wiley, New York, 1990. 2 R. Gonnot, Recherche d'une fonction d'acceleration, pour le calcul de la fiabilite de systemes redondants--Application aux defaillances de mode commun, Reliability of Nuclear Power Plants, Proc. Symp. in Innsbruck 1975, IAEA, Vienna, 1975, pp. 199-220. 3 R. Viertl, Acceleration functions in reliability theory, Methods of Operations Research, 36 (1980) 321-326. 4 R. Viertl, Statistical Methods in Accelerated Life Testing, Vandenhoeck & Ruprecht, G6ttingen, 1988. 5 H. Strelec and R. Viertl, Estimation of acceleration functions in reliability theory under multicomponent stress, Prog. Cybernet. Syst. Res., 10 (1982) 455-459. 6 H. Strelec and R. Viertl, Estimation of power-type acceleration functions in life testing, Math. Comput. Modelling, 10 (1988) 711-717. 7 H. Strelec, Structure defects of acceleration models in reliability analysis, Osterr. Zeitschrift fiir Statistik und Informatik, 20 (1990) 185-207. 8 H. Sch~ibe, Modellieren von konkreten Sch~idigungsmechanismen, Qualitiit und Zuverliissigkeit, 36 (1991) 44-47.