An axiomatic approach to models of accelerated life testing

Engineering Fracture Mechanics Vol. 50. No. 2, pp. 203 217. 1995

Pergamon

Copyright L+ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0013-7944/95 $9.50 + 0.00

0073-7944(94)00202-9

AN AXIOMATIC

APPROACH

ACCELERATED

LIFE

TO MODELS

OF

TESTING

H E N D R I K SCHA, BEt and R E I N H A R D VIERTL~ ?An der Josefsh6he 21, D-53117 Bonn, F R . G . +Technische Universitat Wien, Institut ffir Statistik und Wahrscheinlichkeitstheorie, Wiedner HauptstraBe 8-10/107, A-1040 Wien, Austria Abstract In this paper an axiomatic approach to accelerated life testing is presented. It generalizes the transformation approach of Viertl. We are able to prove general results for the axiomatic approach under weak conditions. Several well-known models are covered by the axiomatic approach, e.g. the scale transformation model, Cox's model, step stress and damage accumulation theory. Using the axiomatic approach a theory of nonlinear damage accumulation and a model for accelerated life testing for crack propagation are developed.

1. INTRODUCTION ACCELERATED life testing has become an evident method in reliability. This is caused by the fact that life testing has to be carried out in continuously decreasing time intervals. To meet this requirement accelerated life tests have to be carried out. Several models for accelerated life tests exist in the literature, based on different assumptions. Sometimes the assumptions are rather mathematical and their physical sense is not quite clear. In other models implicit assumptions are present. The following classes of models are widely known and frequently used. (I) Cox's model q/" proportional hazards The failure rate under stress V is related to a basic failure rate 20(t) by 2(t) = 20(t) exp(flV),

(1)

where [t is some parameter. Note that V and fl can also be vector valued, for refs, see e.g. [I 3] and the refs cited therein. (2) The scale transformation model The lifetime distribution under stress V has form, see e.g. [4-6] ,~-(x I V) = F(x/g(V)),

(2)

hence changing load leads only to scale transformations keeping the shape of the distribution. The wide variety of models for accelerated life testing strengthens the question on possible classification and unification of different theories. A first step towards a unified theory of accelerated life testing was achieved by Viertl [7, 8]. (3) Viertl's general transformation model The lifetimes T~ and T~2 of an item under stresses Vt and V2, respectively are related by some transformation function a ( . , . ;.): Tv2 = a(V~, V2; Tv, ). (3) In the present paper we will introduce a model that is more general than the transformation model eq. (3) and study its properties. The model is based on an axiomatic approach. The axioms represent physical assumptions. The new, general model allows us to classify existing approaches of accelerated life testing and study their interdependence. In Section 2 the axiomatic approach is developed. In Section 3 we study several particular models, e.g. the proportional hazard, scale transformation, tampered failure rate, step stress, continuous changing stress, and cumulative damage models are discussed within the axiomatic approach. It is demonstrated in which manner these models are defined within the axiomatic approach and which assumptions coincide with them. In Section 4 we derive a particular model for fatigue crack growth. The model is able to take into account various loads. Section 5 presents a nonlinear damage accumulation theory. 203

204

H. SCH,~,BE and R. VIERTL

The paper demonstrates that (i) the axiomatic approach covers most of the well-known theories of accelerated life testing, see e.g. [9], (ii) the axiomatic approach provides insight into the basic assumptions particular theories of accelerated life testing are based on and (iii) the axiomatic approach is a valuable, easy tool to construct new approaches to accelerated life testing. 2. AXIOMATIC A P P R O A C H

In this section some basic assumptions have to be made. The number of assumptions, however, will be restricted to a sufficient minimum. The following basic assumptions will be made throughout the paper: (AI) Let the lifetime under load V, say Tj, be a real random variable defined on the probability space (~v, ~ , , Pu). The load V is an element of ¢~, the space of loads.

Remark : The space "/J~can be a space of real variables, vectors, matrices, functions or other mathematical objects and will be specified later. (A2) There exists a one-to-one measurable transformation function a ( . , . ;t), monotonous in t, having the property T~ ~ a ( V~, V2 ; T,, ), (4) for all V~, Vze ~ , where ~ denotes equality in distribution. Measurability of a ( . , . ; t ) is understood with respect to the a-algebras a(Tv, < t) and a(Tv~ < t).

Remark : Equation (4) defines a mapping of probability spaces for various V. Moreover, (4) relates the lifetime of a selected item under one load to the lifetime of the same items under another load. The following properties can be proved. Property 1: Convolution. We have T~ 3 = a(V2, V3; T~,) = a(V2, V~; a(V~, ~ ; Tr,)). =a(V~, K,; T4), for any V~, V2, V~e ~ , i.e.

a(V., V~; .)= a ( V 2, V3; a(Vl, V2; .))

V v2 e

~" .

(5)

The p r o o f is obvious. Property 2: Inverse transformation. The inverse transformation is given by

a I(V~, V 2 ; t ) = a ( V 2 , V l ; t ) . The p r o o f follows easily since a ( . , . ;.) is a one-to-one mapping. Property 3: Simplification. The model can be presented by a transformation a ( V ; . ) and a distribution function F(t): = ~vo(To <~ t). The p r o o f is obvious. F(t) can be found by the following procedure. Let us select a load V0 e ~/F. This can be the use load or any other fixed element of ~ . Then we define T0:=Tr,,a(V',.)=a(V0,

V;.),a

~ ( V ; . ) : = a ( V , V0;.)

F(t): = P4~(T0 ~< t). Consequently, we have

T, = a( Vo, V; To) = a( V; To).

An axiomatic approach to models

205

Remark: The meaning of To can be that o f an intrinsic time or a measure of damage. Proper O' 4: Standardization. Varying the scale and origin o f the time units o f To one can achieve a(V; O) = O. The assumptions (A1) and (A2) can be completed by additional assumptions. These assumptions are o f use to assure that the theory will be able to describe acceleration effects in life testing. (A3) Partial ordering relation. Assume a partial ordering relation, say ,~, on ~" such that there is V V~, V:, Vs e ~ : (i) V~ <~ V~ and V2'~ l/~ ~ V~ ,~ V3

(ii))'~ ,~ V~ (iii) V I ~ V, and V2 ~ V I ~ V 1= V~. Hence ~' is a partially ordered set. Example I Let ~' be the set o f positive reals denoting load amplitudes in fatigue life tests. Obviously, the set > ' i s ordered. £vample 2 Let :~ be the set o f pairs o f positive reals denoting temperature and voltage in an accelerated life test with integrated circuits. Then the relation ~ can be established if one o f the load characteristics differs and the other is held constant or if both load characteristics differ in the same direction. If one o f the load characteristics, e.g. the voltage, is smaller in the first case but temperature is smaller in the second case, no ordering between these two loads can be defined. Obviously, the set ' t i s partially ordered. (A4) Acceleration. Assume I/~ <~ V 2 then there is T~, > ~, Tv,, i.e. for all t

P(Tv, <~t) ~ P(Tv, ~< t). This assumption times. (A5) Continuity Assume a(V; t) derivative of a(.

assures that within our theory a greater load will lead to stochastic smaller life with respect to time. to be a continuous and differentiable function o f t for all V e ~ / . ,.) with respect to t is denoted by

The partial

:~(V; t): = ~a( V: t )/?t. This condition is reasonable because it assures that a lifetime distribution with no atoms on one load level is transformed to a distribution on another load level being also free of atoms. An a t o m in a lifetime distribution occurring only for some load levels would mean that the failure mechanism o f the item under study had changed. In these cases a theory o f accelerated life testing would not apply. (A6) Continuity with respect to load and existence of gradient. Let 1 " be a metrizable vector space, i.e. there exists a metric P(., .): ~ x: f--,[R. Assume that there exists a basis .{e,, i : I . . . . }, e, e * ' defined in such a manner that for <5 > 0 and for all i - 1, 2 . . . .

V + 6 e , > V. Furthermore, assume that, with respect to its first argument, a(. ; t) is continuous and differentiable for arbitrary, but fixed t. Hence

a(V,; O-a(V2; t) o(v,, v,) admits a limit as p(V~, V ) ~ O and p(V2, V)-)O. Define by Ca(V; t) dt

206

H. SCHABE and R. V I E R T L

the gradient of a ( . , . ; t ) with respect to the basis vectors ei, i = 1. . . . . i.e.

{'c~a(V; t)) a(V + 6ei; t) -- a ( V ; = lim "\ dt J~ ~o p ( V + fie, V)

t)

The gradient is assumed to exist uniquely. Due to assumption (A6), small changes of the load will yield only moderate changes of the lifetime. This assumption will hold in most practical applications since the failure mechanism will be the same for all loads. The assumptions (A1)-(A6) will now be used to prove some properties of the function a(. ;.). Note that (A1)-(A6) are still very general assumptions only assuring that the theory is really able to describe phenomena of accelerated life testing. Even the load V itself is not specified yet and can be described by rather arbitrary mathematical objects.

Lemma 1: Under the above assumptions, for any t, a(V; t) is a non-increasing function of V, i.e.

V1 ~ Vz~a(V,; t) >~a(V2; t). Proof: We have for Vl ~ V2

DZ(Tv, <~t) <~P( Tv2 <<-t ). This is equivalent to a-l(V,, t) <~a J(V2, t). Substituting t = a(V,, s) we get

s <~a l(V2, a(V,,s) ).

(6)

This yields

a(V2,s) <<.a(V2,a I(V2,a(Vi,s)))=a(Vi,s).

(7)

This is the assertion of the lemma. Lemma 2: There is (t~a(. ;t)/t3V)~ <<.0 for i = 1, 2 . . . .

Proof: If the limit

(c~a(. ; t)/SV)i of

a(V + 6ei; t ) - a(V; t) p (V + fie, V)

(8)

exists then its sign is the same as that of a(V + 6ei; t ) - a(V; t). Since V + 6ei ~ V, eq. (8) must also be non-positive which proves the lemma. Lemma 3: There exists a function a(V; t) such that

Tv= f f 0 et(V,t)dt. Since a(V; t) is differentiable with respect to t, the function a(V; t) can be expressed by means of 0(

a(V;t)=

~(V;s)ds "0

with properly chosen so. But the time unit can be chosen such that a(V; 0) (property 4) and consequently So = 0. This proves the lemma. Lemma 4: If the distribution function F(t) is absolutely continuous we can choose To such that it is uniformly distributed on the interval [0, 1].

An axiomatic approach to models

207

Proo] : Let us introduce a new transformation function a'(. ;.) defined by

a'(V', t): = a(V; F l(t)). Then To has to be replaced by To = F(To). Using To and a'(. ;.) we have the same theory of accelerated life testing as if we would use To and a(. ;.). One can observe that To has uniform distribution on [0, l] since To has absolutely continuous distribution F(t). Lemma 5: There exists a function /3(V; t) with property 7"o =

/3(v; t) at.

The p r o o f is similar to that of lemma 1: a(. ; .) is differentiable and hence the same holds for a-~(. : .). We have T0=a

'(V; T v ) = a - l ( V , Vo: rv),

using the chain rule we obtain

/3(V; t) = 1/e(V;

a I(v,

t)).

Lemma 6: Distribution of Tv If/3(V; t) is non-negative for all V e ~t~ and To has an absolutely continuous distribution, then Tv has distribution P(Tv ~< t) = F(a(V; t)).

ProoJ ] We have

To= ~0TV/3(V;s)ds,

(f0

f(t)=P(To<.t)=P

/3(V;s)ds~t

)

.

Expressing t in the form t =

/3(V; s) ds

) (f;

/3(V;s)ds<...

we have

P

/3(V;s)ds<...t \do

=P

io

/3(V;s)ds

)

=P(Tv~
Hence

P(Tv <<.u) = F(t) =

/3(V; s) ds

)

This result can be used to obtain the distribution of Tv for a given /3(. ;.) and F(t). Lemma 7: Assume that E{(~?~/0V) 2} is finite and assumptions (A1)-(A6) hold. Then ~ct/~V <<.O.

Proof: We have

~a(V,t)/Ol/ =

fo

(O~(V;s)/~V)ds <~0

for arbitrary but fixed t. This equation is valid for all t if and only if the assertion holds.

Remark : The first assertion of the lemma ensures the existence of the derivative of the integral and the interchangeability of integration and differentiation.

208

H. SCH)~BEand R. VIERTL

Lemma 8: We have (8/~/8 V),/> 0,

for i = 1 , 2 . . . .

Proof: Expressing fl(. ;.) by means of ~(. ;.) we have

(c~fl/g3V)i=(~[l/ot(V;a '( v; t))]), -1 (Sa ~(V; t)/Ot)i ~(V; a I(V; t)) 2 -1

1

~(V; a '(V; t)) 2 63a(Vi;a l(V; t))/~t

~>0.

Lemma 9: Let Tv have absolutely continuous distribution function F(tl V) with failure rate 2(t I V). Then

/~(v; t) = ~,(t[ v), when To is chosen to be a standard exponentially distributed random variable with cumulative distribution function F(t)= 1 - e x p { - t}.

Proof: Using the result of lemma 6, we obtain the distribution of Tv by means of F(t) and fl(. ;.):

, e p{ Now we have to show that

Subtracting 1, changing signs, taking logarithms, and taking derivatives we arrive at

~(tl v) =/~(v; t) which proves the assertion. 3. P O S S I B L E T Y P E S OF M O D E L S

In this section we will present particular models for accelerated life testing. These models are defined by the distribution function F(.) of To and the transformation function a(. ;.). The latter can be given using ~(. ;.) or equivalently fl(. ;.). Hence we will focus on the equation

TO= f O'Vfl(V; t)dt

(9)

and present several variants for the function fl(. ;.) leaving F(.) still arbitrary.

3.1. Theory without after effects Let us assume that the function fl(. ;.) does not depend explicitly on time and that load V is independent of time. Then there is

/~(v: t) =/~(v). Integrating eq. (9) we obtain

To = fl(V)Tv, hence Tv has distribution function given by

F(tfl(V)).

An axiomatic approach to models

209

Putting g(V) = 1//3(V) we see that we deal with the scale transformation model. Hence the assumption of absence of after effects, mathematically presented by the lack of explicit time dependence of/3(. ; .), leads to the scale transformation model. In turn, using a scale transformation model, we assume implicitly that there are no after effects. 3.2. Factorization of/3 Let us assume that load V does not depend on time and that fl(. ; .) can be expressed in the form /3(V; t) =/3~(V)/32(t), i.e. the dependence on V and t can be expressed in a product form. Then we easily obtain

To =/31 (V)B2(Tv), where B2(. ) is the integral of/32(.). Since/3(.) is non-negative for all t and V both of the functions/3~(.) and/32(.) shall have the same sign. Consequently,/32 (.) is non-negative or non-positive so that B 2(.) is a monotone function. Consequently the lifetime distribution under load V is

F(fl, ( V)B2(t )).

(lO)

Equation (lO) describes a model, where for the random variable not only a scale transformation is involved but also a change of the shape of the distribution function takes place. 3.3. Representation of/3(. ; .) as a sum Let us assume that V is again time independent and that/3(. ;.) can be expressed as a sum of two terms. /3(V; t) =/3, (V) +/32(0Then we obtain

To =/3~ (V)Tv + B2 (Tv). Since fl(. ;.) is non-negative and a(. ; 0 ) = 0 both functions /3,(.) and /32(.) must be nonnegative. Hence we obtain the distribution of Tv as

F(fl, (V)t + B2(t)), where B2(.) denotes the integral of f12(.). 3.4. Power law form of fl(. ;.) Let us assume that V is time independent and that/3(. ;.) is of the form

/3, (V)t ~'-~. This is the model discussed in [8, 10]. Since a(. ; 0) = 0 we conclude/32(V ) >~ 0 for all V ~ ~ and /3(. ; 3 / > 0 yields/3~(.) ~> 0. Integrating eq. (9) we arrive at

To = ~,,( V) T;~ (V~, where we have introduced new functions

72(v) =/3:(v) + 1, 7,(v) =/3,(v)/(1 +/3:(v)). These functions fulfil ~':(V)~>I

for all

VeU,

7~(V)~>0

Finally, we obtain the distribution function of Tv as F(7, (V)t;':(~). Details for the model can be found in [10].

for all

V e ' t ~.

210 3.5.

H. SCHABE and R. VIERTL

Cox's model of proportional hazards

Let us assume that V(t) is a time dependent non-random load with values in Nk. Moreover, let //(. ;.) be of the special product form fl(V(t); t) = 20(0

exp{flV(t)},

where 20(t) denotes a non-random function and /~ is a parameter taking values in N~. Then we obtain

To =

2o(t) exp{/~V(t)} dt.

The distribution of Tv is given by P(Tv ~< t) =

20(s) exp{/~V(s)} ds .

If we consider a special case with T0 being a standard exponential variable we find that Tv has cumulated hazard function A (t; V) =

)~0(s)exp{/~V(s)}

ds,

i.e. we have hazard rate 2(t; V) = 20(t) exp{/3V(t)}. This is Cox's model of proportional hazards with )~0(t) playing the role of the baseline hazard rate. This consideration demonstrates that a generalization of Cox's model of proportional hazards is given by P(Tv~
and arbitrary functions F(t). If T0 does not follow an exponential distribution 20(t ) has not the sense of a baseline hazard rate. In the next lemma we will prove that this model generally cannot be reduced to Cox's model. Lemma 10: Let

To =

~(T'v

20(s) exp{flV(s)} ds

(1 1)

)

with To having arbitrary absolutely continuous distribution function F(t). Then there does not exist another baseline hazard rate 20(t) such that eq. (1 1) has equivalent expression To =

IllIv

2o(s) exp{flV(s)} ds'

but with 17"o being standard exponential.

Proof: Let F(t) have cumulative hazard function A ( t ) = - l n ( 1 - F ( t ) ) . Since F(.) is absolutely continuous, A(.) is differentiable and the hazard rate denoted by 2(t). If a transformation to the Cox-model would be possible, there should hold A (fo2o(s)exp{flV(s)}_ ' ds) =

f ' 2o(s) exp{flV(s)} ds.

This equation is equivalent to

).

(i

)

2o(S) exp{/~V(s)} ds )~o(t) = io(t).

We see that ~0(t) is depending on load V(t) whenever 2(t) ¢ const. This contradicts the fact that in model (1 1) 2o(t) is load independent. Moreover, for the case 2(t) being constant, To has an exponential distribution in the original model, i.e. we are transforming a Cox-model of proportional hazards.

An axiomatic approach to models 3.6.

2ll

Step stress and the scale transformation model Let us now assume that V(t) is a step function defined by f V~, for t ~<

V(t) =~ V,, for t > 3" The function /3(. ;.) is assumed to be as in Section 3.1, i.e. will have

T0=

i{ TI~B(V(t)~dt }

=/3(V,)(~ A

/3(V, t) =/3(V). Integrating eq. (9) we

Tv)+~(V2)I(Tv>~)(Tv-~),

where(. A .) denotes the minimum of two variables and I(.) is the indicator function. Because/3(.) is non-negative we obtain the distribution function of Tv as

fF(/3(V1)t) for t ~< T P(Tv
for t > ~ .

This is the usual step stress model, see e.g. [11, 12]. 3.7.

Continuous stress change

Let us now assume /3(V; t) =/3(V) and V being a non-random continuous function of time V(t). From eq. (9) we immediately obtain To=

f0TV/3(V(t))dt.

The distribution of T~, is F (fo/3(V(s)) ' ds).

(12)

This is the model of continuous stress as discussed e.g. by refs [11-14]. Note that the result, eq. (12), is a direct consequence of eq. (9) and is derived without further proof, whereas [11] and [I 2] had to carry out a p r o o f in order to derive eq. (12). This fact shows the superiority of the axiomatic approach. 3.8.

Generalization of Arbutiski's long-term survivorship model

In this subsection we derive a generalization of a model presented by [15]. Let g(t; V) be a probability density with cumulative distribution function G (t; V) and V be a vector valued constant non-random load. Let us choose /3(. ;.) in the form /3(V; t) = g ( t ; V) exp{/3V}, where /3 represents a parameter having the same dimension as V. Then, by lemma 6 there is P(T,~
)

g(s; V)exp{/3V} ds = F(G(t; V)exp{/3V}).

Obviously this distribution has an atom at infinity, i.e. with probability F(exp{/3V}) an item will not fail. This is the general form of the model. Choosing the standard exponential distribution for To, i.e.

F(t) = 1 - e x p { - t } and G(.) to be a Weibull distribution

G(t)= 1 -exp{-(t/Z)"} with load dependent parameters

= ~0exp{~V}, we arrive at P ( T v > t ) = c(V) °~': ~, where 3.9.

q =~0exp{'/V}

c(V)= e x p { - e / ~ } , the model described by [15].

A generalization of the tampered failure rate model Let us choose /3(. ;.) in the form

/3(v; t) = ,~(t)/3, (v),

212

H. SCH~,BE and R. VIERTL

with/~(.) and 2(.) being non-negative functions, 2(.) being integrable with cumulative function A (.). The stress V is a step-function defined by fV, I

V(t)=

V2

for l ~ r "

Putting ),(t) = 1 the given model reduces to the step-stress model. After some algebra, the lifetime distribution is P(Tv<~t)=F(fl(V))A(t AT) + l(t > r ) ( A ( t ) - A ( r ) ) ) . This is a generalization of the tampered failure rate model of[16]. Let us now consider a special case with F(.) being the standard exponential distribution. If we introduce G(t): = I - e x p { - A ( z ) } ,

G(t) = 1 -

G(t)

the survival function becomes

P(Tv

t)

f G ( t ) t~<)') )~(z)/~(~)~(~~)O(t) ~ ~ )

for t ~< z for t > ~"

This survival function is the same as in [16].

3.10. Damage accumulation theories In this subsection we will show that eq. (9) is also capable of describing a d a m a g e accumulation hypothesis. First let us briefly give some basic principles of d a m a g e accumulation, see e.g. [17 26]. In d a m a g e accumulation theory, the d a m a g e S(t) is introduced as a main characteristic. This function is non-decreasing, zero at time zero and failure occurs at the time of S(t) approaching level one. In most cases the lifetime distribution under load with constant amplitude V is assumed to follow a scale t r a n s f o r m a t i o n model with T~ having distribution function

F(t/g(V)), where g(V) is non-increasing in V and F(t) is a lifetime distribution having unit mean. The load process V(t) under use conditions is assumed to be a stationary stochastic process being two times differentiable in quadratic mean. The increase of d a m a g e d S ( t ) is the quotient of the time the item was under load V(t) divided by the lifetime of this item under load with constant amplitude V - - V(t). Then we have the increase d S ( t ) of d a m a g e at time t in the form

JdlCV'(t) > 0)1 , (13) Tog( V(t )) where To is a variable distributed according to F(t) and hence Tug(V(t)) is the lifetime of the unit under load with constant amplitude V = V(t). Moreover, IdI(V'(t) > 0)] is the n u m b e r of load dS(t) =

cycles in the interval (t, t + dt), see also [27]. Consequently we have that the lifetime T) under stochastic load V(t) is defined by ~)

as(t) =

f~,,dl(V'(t)>O),> Tog(V(t))

This eqhation can be rewritten as

f(r)'Idl(V'(t)>O)] T0=

)

g(V(t))

fr' 6(V'(t))]V"(t)] dt" = , g(V(t))

Putting now /3(v(t), t) =

6( v'(t))l v"(t)l

g(v(t))

we see that eq. (9) is also able to describe a d a m a g e accumulation theory. Speaking strictly, [t(. ; .) is a function not of V(t) but of its first two derivatives and we should indicate this by ~(V(t), t) = ~(V'(t), V"(t)). We see that a linear d a m a g e accumulation theory is described by a function/~(. ;.) depending on time only implicitly through the load function V(t). M o r e o v e r the

An axiomatic approach to models

213

load function and its derivative are involved in dS(t) as given by eq. (13) only with their values at time' t, the whole history of the process V(.) from time zero up to time t is not included. This is the main reason that this theory is a linear damage accumulation theory. The above considerations also put light on the models discussed in Sections 3.1, 3.6 and 3.7. We see that the scale transformation model, the step-stress-model and the model with continuous varying stress are based on a linear damage accumulation theory. If the assumptions for such a theory are violated, e.g. hysteresis effect occurs in the material, these theories will not hold true in their form and have to be modified.

3.11. General damage accumulation theory In this subsection we will discuss the form of a general, nonlinear damage accumulation theory. A nonlinear damage accumulation theory should be capable of describing relaxation processes and hysteresis effects, compare [28]. Hence the increase of damage dS(t) as given by eq. (13) will depend not only on the value of the load function at time t but on the whole history. Consequently. we arrive at

~( wl

To=

~q({V(u):u<. r},r)dr. I

4. A M O D E L FOR L O A D D E P E N D E N T CRACK P R O P A G A T I O N

Let us first develop a crack propagation model as discussed by [29] generalizing a model of [30]. Let X(.) denote the crack length, W(.) a standard Wiener process and H(.) a Poisson process. Moreover, let h(.) be a given, continuous increasing function with h(0) = 0. Then crack propagation is described by the equation

h(X(s)) = bs + aW(s) + kt fI(,;os). The crack length X(s) might not be monotonous caused by W(s) which is also not monotonous. This deficiency, however, is minor and is present in many crack propagation models based on the Wiener process, see e.g. [30]. The function h(.) can be chosen, for example in one of the following forms (i) polynomial form: h(x) = a~x + a2x 2 + . . . (ii) power law form: h(x)= x" (iii) logistic form: h(x) --- a - b ln((c - x)/x). The item fails when crack length X(.) for the first time exceeds a critical crack length, say h<~ Hence lifetime will be T = inf{t : X(t) >~ho}. The density of the lifetime distribution can be evaluated as

./~,(s/h,,)= ~ hoh'(h°)

{

k=0 N,2~O'S 32 exp

[h(ho)-l~k-hs]2 } ()~s)k 2a~'s -- )-* k!

{ 14)

Now we have to introduce dependence on load into the model. Let us assume the load to be a process 1/(t). Now we will use a model without explicit time dependence

To=

f0TI/~(V(s)) ds

(15/

with T,, having probability density given by eq. (14) and [1(.)>~ O. Equation (15) is equivalent to P(Tv~< t ) =

)

fl(V(s))ds

)

(16)

From eq. (16) we obtain for the densities

ft~ (t) = froI;o'~(V(s)) ds l[t( V(t )).

(17)

214

H. SCH,~BE and R. VIERTL

Substituting eq. (15) into eq. (17) we obtain ./~. ( t ) =

[h(h0) - t~k - bu] 2 _ ).u ~ ()m) k

y, hoh'(ho) k=o ~,2/~0"/432 e x p

o fl(V(t)),

with u given by

f

(18)

' fl(V(s)) ds/t. )

Introducing load dependent parameters a(V), )~(V) and b ( V ) in the form b ( V ) = bu/t = b

fl( V(s))ds/t, O

~t

h ( V ) = bu/t = h l) fl(V(s)) ds/t we have " hoh'(ho) D, ( t ) = ~ , , j 2 ~ ( v ) t 3 ~ 2

exp{

[h(ho)-I~k-b(V)t]2 2a2(v) t

- 2(v)t

} (2(V)t) k o l<~fl(v).

Consequently, the model can be rewritten in the form h ( X ( t ) ) = b( V)t + ~ ( V ) W ( t ) + l~FI().(V)t).

(19)

The function h(t) and parameter It do not become load dependent. Note that, in the general case the load depending parameters are still depending on time. Two examples will show how time dependence of b(V), 2(V) and a ( V ) can be excluded. 4.1. Load does not depend on time Let us consider the special case V(t) = V being constant. Then we immediately obtain from (18) u = [~(V). Hence we have a simple scale transformation model similar to that discussed in Section 3.1 and u and hence also the parameters will not depend on time. The function fl(V) can be chosen in the form f i ( V ) = 1/g(V), where g ( V ) is one of the acceleration functions frequently used in scale transformation models. Especially. one can use the power law g ( V ) = A V "' 4.2. Stochastic load process Let us assume V(t) to be a stationary stochastic process satisfying ~E{I~(V(f)) 2} < ,~, ~{fl(V(t))} is absolutely integrable.

(20)

Then " l

u =

t

fl(V(s))ds/t

(21)

0

shall be interpreted as an lto-Integral with condition (20) assuring existence of eq. (21) as a random variable. Due to condition (20), eq. (21) is well-defined and exists. Moreover we have the following lemma. L e m m a 11 Under the above conditions, as t ~ 3c, u

=

,~,sp(v)}.

An axiomatic approach to models

;.!5

Here ~ v denotes convergence in probability. The p r o o f follows from the theory of stochas!.ic processes, see e.g. [31]. Let us denote L = l/[I:{fl(V)}. Then, for h0--> ~ we have T ~ e.~ and weak convergence lifetime is large c o m p a r e d with times t that are sufficient [25]. Hence ['or practical purposes we can use the model 2(V) given by h(V) = h/L
of u holds. In most practical applications that u is close enough to its limit, see eg. eq. (19) with parameters b(V), rr(t') and ,;.(V) = 2 ' L .

This result coincides with a result derived using the linear d a m a g e accumulation hypothesis and lifetime distribution density

/':,

~ hoh'(ho)

(t) = ~ , , ~ 7

e exp

{ [h(ho)-Iak

hu]Z 2u}(2u)"

2~reu

~

fl(V(t)),

with u = fl(V) with constant load. Once more. this fact demonstrates that eq. (9) can be gi\en ).he sense of a d a m a g e accumulation theory. 5. C O N S T R U C T I N G N O N L I N E A R D A M A G E A C C U M U L A T I O N THEORIES

In this section we will focus on theories as presented in Sections 3.10 and 3.11. A general d a m a g e accumulation theory is given by T() =

f

T, fl({ I/(U): l,~ ~ "r ], g)dz.

(122)

)

Choosing function fi(. . ) as

fi(u(t); t ) =

(~ ( v'(~ ))1 v"(t )i

!23)

g(V(t))

yields the linear d a m a g e accumulation theory. A simple generalization of eq. (23) is the scaled non-stationary d a m a g e accumulation theory [22] which can be expressed as

fl( V(t ), t) = A (t /g(V(t )))

a(v'(t))lV"(t)l ,~(v(t))

We have the following result Lemma 12: Assume to hold (i) V(t) is a stationary and ergodic stochastic process (ii) <5(V'(t))] V"(t)]/g(V(t)) has covariance process satisfying

f (iii) A(.) is bounded. Then, as t ~ + :

Y=

C ( t ) dt < r~.

'

we have convergence

I ~" 6(v'(s))l v"(s)i j A(s/g(V(s))) ds=> t <> g(V(s)1 P

1 i'{E,,{A(sig(V(s)))6(V'(s))!V"(s)ilds. t

)

'

<~(v(s))

J

Using this result we obtain the lifetime distribution under stochastic load F ( f a (' s

) d s )/ + O ( M ) ,

where

a( S) = E,. {A (s /g( V (s))) O( V'(s))l V"(s )I/,~( V(s))}, M ( t ) = ~,.{Y}.

(;!4)

216

H. SCHABE and R. VIERTL

T h e f o l l o w i n g e x a m p l e s h a l l s e r v e as a n i l l u s t r a t i o n . L e t us specify t h e m a i n c h a r a c t e r i s t i c s as follows: (i) V ( t ) is n o r m a l l y d i s t r i b u t e d h a v i n g a u t o c o v a r i a n c e f u n c t i o n R ( t ) a n d z e r o m e a n . (ii) g ( v ) = Cc ~, w h e r e C a n d 4~ a r e p o s i t i v e c o n s t a n t s . T h e n we c a n e v a l u a t e a ( s ) u s i n g t h e d e c o m p o s i t i o n of' t h e m e a n [F),{a(t)} = ~:,, {,5( V ( t ) ) } [I:rr {A ( t / g ( V ( t ) ) ) t V " ( t ) / g ( V(t))}. T h e r e s u l t is a g e n e r a l i z a t i o n o f M i l e s ' f o r m u l a , see e.g. [22, 25], 2,,,e a(t } =

,,/R,,(0)R(0)(~,

,)2

~:

J

e

z '~eA(t[2R(O)z]*=/C)dz

0

T h i s e x p r e s s i o n will n o w b e e v a l u a t e d for A(t) = 1 - a exp[-bt},

,=2.

We obtain a (t) = M e ( I -- a / ( 1 + 2 b t R ( 0 ) / C ) : ) , M~=>, ~

I

R"(O)R~e

1~2/(:zC).

The lifetime distribution under stochastic load becomes F{mot(1

_a/(1 + 2htR(O)/C))}.

REFERENCES [1] [2] [3] [4] [5]

D. R. Cox and D. Oakes, Analysis qf Survival Data. Chapman and Hall, London (1984). R. Gill, Understanding Cox's regression model: A martingale approach. J. Amer. Statist. Assoc. 79, 441 447 (1984). J. D. Kalbfleisch and R. L. Prentice, Estimation of the average hazard ratio. Biometrika 68, 105 112 (1981). J. I. McCool, Confidence limits for Weibull regression with censored data. 1EEE Trans. Reliab. 29, 145 149 (1980). J. Sethuraman and N. D. Singpurwalla, Testing of hypotheses for distributions in accelerated life tests. J. Amer. Statist. Assoc. 77, 204 208 (1982). [6] M. Shaked, W. J. Zimmer and C. A. Ball, A nonpararnetric approach to accelerated life testing. J. Amer. Statist. Assoc. 74, 694 699 (1979). [7] R. Viertl, Acceleration functions in reliability theory. Math. Oper. Res. 36, 321-326 (1980). [8] R. Viertl, Statistical Methods" in Accelerated Li[~, Testing. Vandenhoeck & Ruprecht, G6ttingen (1988). [9] H. Sch/-ibe, Parameter Estimation in a Model q/" Accelerated Lil'e Testing. Institut ffir Mechanik der Akademie der Wissenschaften der DDR, Preprint P-MECH-01/85 (1985) [in German]. [10] R. Viertl, Nonlinear acceleration functions in life testing. Math. Oper. Res. 47, 115 122 (1983). [I 1] K. A. Doksum and A. Hoyland, Model jbr Variable-Stress Accelerated Life Testing Experiments Based on Wiener Processes" and the Inverse Gaussian Distribution. Report No. 3, University of Ume~ (1989). [12] L. Nilsson, Estimation .[rom accelerated lile tests with increasing stress, Diss., University of Umedt (1985). [13] D. S. Bai, M. S. Cha and S. W. Chung, Optimum simple ramp tests for the Weibull distribution and type-I censoring. IEEE Trans. Rel. 41, 407 413 (1992). [14] X.-K. Yin and B.-Z. Sheng, Some aspects of accelerated life testing by progressive stress. IEEE Trans. Reliab. 36, 150 155 (1987). [15] T. Arbutiski, A family o1" multiplicative survival models incorporating a long term survivorship parameter c as a function of covariates. Commun. Statist. Theor. Meth. 14, 1627 1642 (1985). [16] G. K. Bhattacharya and Z. Soejoeti, A tampered failure rate model for step stress accelerated life test. Commun. Statist. -Theor. Meth. 18, 1627 1643 (1989). [17] A. Desmond, Stochastic models of failures in random environments. Can. J. Statist. 13, 171 183 (1985). [18] W. Gnilke, L!'/~, Tmw Computation lbr Machine Components, VEB Verlag Technik, Berlin (1980) [in German]. [19] W. Heinrich and K. Hennig, Random Vibrations o/ Mechanical Systems, Akademie-Verlag, Berlin (1977) [in German]. [20] S. Hohn and J. Mare, A simple model for fatigue life. IEEE Trans. Reliability 37, 314 322 (1988). [21] D. Joensson, Spectral analytic life time computation for components under stochastic load. Technische Mechanik 7, 65 7I (1986) [in German]. [22] C. Lange and H. Friedrich, Failure Prohabililies' /or Stochastically Loaded Mechanical Systems Failing Due to Level Crossings as well as due to Damage. AdW der DDR, lnstitut for Mechanik, FMC-Series No. 12, Berlin und Karl-Marx-Stadt (1984) [in German]. [23] H. Sch/ibe, Lil~, Length Calculation With Linear Damage Accumulation. Report d. Inst. f. Mech., AdW d. DDR, S-Reihe, No. 8. 221 226 (1986). [24] H. Sch/ibe, Life time computation using linear damage accumulation. 1/l-Mitteilungen 26, 181 187 (1987) [in German].

An axiomatic approach to models [25] H. Sch~ibe, Life length calculation using linear damage accumulation. F.~lC-~'erie~ N o 46, 43 ~;6 A~ld~t~il; { ~lis: d. DDR, Inst. f. Mechanik, Karl-Marx-Stadt (1990) [in German]. [26] H. Schfibe, On the construction of a nonlinear damage accumulation. Eng. Fracturc Mechamcs 45, 027 0~,:~ (1'~93) [27] W. !. Tichonov and W. I. Chimenko, Peaks o/" Realizations qf Stochastic Processes. Nauka, Moskwa i!');~ ": i~v

Russian]. [28] J. Kallenberg, A Mathematical Model for Nonlinear Damage Accumulation. Diss. (A), Bergakademie Freibe~g (1')89i [in German]. [29] H. Sch~ibe, A new stochastic model for crack propagation. Qual. Reliab. E,g. b , . 6, 341 344 i19901. [30] K. Sobczyk, Stochastic models for fatigue damage of materials. Adv. Appl. Prob. 19. 652 673 ~1987) [31] A. D. Wentzel, A Course on the Theory o[ Stochastic Processes. Nauka-Moskwa 11975) [in Russian]. (Reeeived 14 No~'ember 1993)