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A reliability model for a nonlinear damage process
Reliability Engineering 18 (1987) 73-99
A Reliability Model for a Nonlinear Damage Process
Anwar Khalil Sheikh and Munir Ahmad Departments of Mechanical Engineering and Mathematical Sciences, University of Petroleum and Minerals, Dhahran, Saudi Arabia (Received: 16 May 1986)
A BS TRA C T Several types of damage processes lead to the failure of mechanical devices. Some examples of these damage processes are: wear, fatigue, creep and corrosion. In this paper we consider a generalized damage process defined by the random function D(t); D(t)= Atn; t~ T, where A, B and T are random variables, and D(t)= damage incurred at time t. If the life of the device is terminated when the damage exceeds a limiting value D t (i.e. D(t) > D~), then it will be shown that, under certain assumptions regarding random variables A and B, the reliability model of the device is
O[
log~t--Ct
l
where
• (z) = ~
~ exp [-½¢23 d~
and Ct, ¢tt,fit are the functions of D t and the mean and variance of the random variables A and B. In this paper various statistical characteristics of the above-mentioned model are discussed. The generalized nature of the proposed model is emphasized, and it is shown that the lognormal model is a special case of the proposed model The method of estimation of the parameters of the reliability model from a set of realizations (sample functions) of the damage process is also illustrated. 73 Reliability Engineering 0143-8174/87/$03-50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
74
Anwar Khalil Sheikh, Munir Ahmad
1 CAUSES OF F A I L U R E Failure of a machine part* (or a system) is defined as the event associated with a shift in the operating characteristics of that part (or system) from its permissible limit. This shift in operating characteristics occurs due to accumulation of damage with the passage of time. The failure may be caused by the combination of the external, internal and initial factors. 1.1 External factors These factors are extrinsic or extraneous in character; that is, they originate from causes external to the system. They represent the environmental conditions under which the equipment is operating. For example, failure may occur due to unpredictable stresses arising from such environmental factors as sudden shocks. The rate of occurrence of this type of failure is dependent upon the severity of environmental conditions; the more severe the environments, the more frequently the occurrence of failure. The chance failure or a random failure is normally caused by the external factors) The set of external factors will be designated by the symbol aE. 1.2 Internal factors Internal factors are those factors which are due to the internal causes such as physical processes that take place inside the system. They are usually associated with certain inherent characteristics of the system, and are dominant in the initial and final stages of operation. During the initial phase of operation of a system, a type of failure known as initial failure may occur from the inherent defects in the system attributed to faulty design, manufacture or assembly (initial quality factor). As the system operates it wears out; hence during its final phase of operation a second type of failure, known as wear-out failure, may be experienced by the system. Wear-out failure is the outcome of accumulated wear and tear; a depletion process occurs through abrasion, fatigue, creep and like phenomena, which results in the loss of specified physical, mechanical, chemical or other properties characterizing the initial design features of the system. Ultimately, an operating system will fail as a result of wear and tear. The set of internal factors will be designated by the symbol at. 1.3 Initial quality factors The dimensional inaccuracies in manufactured products, inherent variability of material characteristics combined with faulty production * The terms m a c h i n e part, device or system will be used interchangeably.
Nonlinear damage process
75
techniques, improper quality control and defects in assembly could lead to a considerable variability in the degree of damage at the beginning of operation (i.e. initial damage at time t = 0). The above-mentioned factors are called as initial quality factors and will be denoted by %.
2 M O D E L S OF D A M A G E PROCESSES The life of a machine component is terminated by two types of failure: gradual failure (such as wear-related failures) and sudden failures. Most component failures are caused by various damage processes occurring in the machine which lead to a gradual decline in the initial efficiency of the component, which in turn leads to component failure when accumulated damage reaches a critical level. Various irreversible damage processes which lead to the system's failures are wear, corrosion, internal stress redistribution and plastic distortion of components, creep, fatigue, fracture, diffusion, evaporation, crystallization, burn-out, biological attack, deposition and seasonal fluctuations in temperatures. Wear and fatigue are two major causes of failures of machine parts. In this paper, for illustrative purposes, several examples from wear and fatigue damage processes will be given. However, the proposed mathematical model is not restricted only to these two types of damage processes; it can well be applied in other cases such as creep, corrosion, etc., where damage as a function of time conforms with the hypothesized nonlinear law. The life of a machine or its elements depends on the rate of the damage process and the limiting conditions of the device. The rate of damage process (for example, wear), dD(t)/dt, is a function of several independent variables:
dD(t)/dt = ~O(P, V, O; ~, %, %; t)
(1)
where P = V= 0 = ~ = ~E = So = t=
pressure sliding velocity set of other operating conditions, including service conditions set of internal factors causing damage to the system set of external factors causing damage to the system set of initial quality factors at time t = 0 the operating time.
F r o m eqn (1) we can write a general form of damage function as D(t) = g[P, V, O; %, al, aE; t] where g[¢] = S~ ~0[~] de.
(2)
76
Anwar Khalil Sheikh, Munir Ahmad
D(t)~
~-t
(a)
D(t)I
I Dlt) ~t ~e- to - . . ~ (b)
L••t (f)
D(t)~
~t (¢)
D(t)l J ~
D(t) ~t (d)
D(t)~
f
Ig)
~t (e)
Fig. I.
Various plausible models of damage propagation: D(t) represents the degree of total damage at time t.
Considering the occurrence of irreversible damage processes resulting in a given type of damage (wear, deformation, corrosion, fracture, etc.) it is possible to distinguish a certain type of relationship as plausible models for the damage function D(t). Typical examples of damage functions (i.e. the models of damage development) are given in Fig. 1. 1.
2.
3.
4.
All factors affecting the rate of a given damage process are stabilized and there are no factors which cause the rate to change. The relationship D(t) is linear, as for example in an established wear period (Fig. l(a)). The process of deterioration may not become evident for some period t o, after which it may rapidly cause failure of the component. In this case there is an accumulation of internal factors, for example fatigue fractures of the components (Fig. l(b)). As the degradation process proceeds those factors which affect intensity gradually decline. This is typical, for example, of wear during the initial period known as running-in period. This happens due to the fact that microirregularities on the surfaces of rubbing parts change (Fig. l(c)). This type of behavior is also observed in the initial zone of creep rate versus time curves. Factors causing damage gradually intensify and the damage process rate rises continuously. For example, if there is corrosion or fatigue
Nonlinear damage process
5.
77
damage of the surfaces, after initiation the process develops at an increasing rate (Fig. l(d)). Finally, the process causing the damage may be of an unstable nature; this happens when the component operates with sharp variations in working rates and in fluctuating operating conditions (Fig. l(e)).
In fact, damage of most components usually proceeds in several stages representing the typical cases listed above. The wear processes often consist of three stages (Fig. l(f)): (a) decreasing wear rate (running-in) (b) constant wear rate (normal wear) (c) increasing wear rate (catastrophic wear) i.e. the entire process of damage can be expressed by curves c, a and d. Creep rate curves show a similar behavior to Fig. l(f). However, there are several instances when initial wear period (running-in) can be made negligibly small and the process consists of approximately constant damage rate plus an increasing damage rate. Some types of tool wear and fatigue crack growth after its initiation exhibit this type of behavior. Deformation of a component as a result of redistribution of internal stresses first occurs at a high rate and then gradually slows down (Fig. l(g)). The rate of the damage process d D ( t ) / d t and the nature of the D(t) curve are greatly affected by the machine operating rates. As a rule an increase in load, speed, temperature or other operating conditions leads to an intensification of the failure process, i.e. reduces the reliable life of the product. In this paper we will discuss the damage process which exhibits a continuously increasing rate of damage, as shown in Fig. l(d). Two cases, i.e. one describing the wear of a cutting tool and the second describing the crack growth after initiation, are shown in Figs 2 and 3. These cases correspond to Fig. l(d), and the general model for the damage process applicable in these cases is D(t) = A t ~
(3)
where D ( t ) = a m o u n t of wear, or crack length or any other observable parameter of the damage process A, B = empirically determined constants. The rate of the damage process is d D ( t ) / d t = B A t B- ~
(4)
Anwar Khalil Sheikh, Munir Ahmad
78
6.5 6.0
AtB=0"3778 t 0615
W(t)=
5-5 5.0
4.5 E E 4.O S 0
× 3.5
• :3.0 "6
S 2.5 2 2.0
-~ 1.o ID
E 0.5
t~ a v
O
Fig. 2.
I
I
I
I
I
I
I
I
I
I
I
I
5
10
15
20
25
30
35
40
45
50
55
60
Average cutting time t. Tool wear as a function of cutting time. 2
thus to have dD(t)/dt as an increasing function of time B > 1. C o m p a r i n g eqn (4) with (1), and eqn (3) with (2), one notes that the cumulative effect of variables P, V, O; s0, s~, SE is incorporated in A and B.
3 STOCHASTIC
NATURE
OF
DAMAGE
PROCESSES
During the operation of a machine there are changes and fluctuations in loads (P), velocities (V), methods of lubrication, and the degree of contamination of the friction surfaces, and temperatures, etc. There is a manufacturing or assembling variability in the initial quality of the product (So); similarly there are inherent variabilities in internal and external factors (cq and SE) of the system. Sometimes part of these variables, such as pressure, velocity and initial quality, could be maintained at a fixed level without fluctuations (i.e. as deterministic quantities); however, most of the other factors are extremely difficult to control at a desired level in a deterministic sense. Since various input variables, P, V, O; s~, sF., So, are in general r a n d o m in nature, the damage function D(t) in real life represents a stochastic
Nonlinear damage process
Gmax-- (Train = 55--2 kglmm 2
79
Crmax-Gmin : 62--2 kglmm 2
+o,.'+.7
+.°' +T w
u
u
-"
0
" °
~u
20 30 40 Number of cycles N Cycle x 104
0 10 20 30 Number of cycles N Cycle x 104
Omax- ¢rmin : 44.0-1.Skg/mm 2 o
o crack length, 2a crack depth, 2b e&
initial
O-max --
flaw O'mi n = 3 9 , O - - 1 . 6
2a
A 2b e& (nitial flaw kglmm 2
0-2
)
~
2,20
~
:b:a
oJ o+ ~a E ~lE0.2
0.1
0.1~
r~ g
)u
) 20 40 60 N Cycle x 10 4
80
0
I
I
5 10 N Cycle x 104
Fig. 3. Crack growth behavior after initiation from the initial flow (a-N relation). For the initial flaw, a pit-like small surface notch 50/~m deep and 50/~m in diameter is used. 2a: Crack length on specimen surface; b: crack depth, a and/or b represent the level of damage in these cases and a model of the type a = A N B can be fitted to these crack propagation curves, a
80
Anwar Khalil Sheikh, Munir Ahmad c
i E,
I
I..tt. tit)
:
,
. . . . .
T
Fig. 4.
A set of sample functions of stochastic damage processes: D(t), and life distribution corresponding to a critical damage level D~.
phenomenon. Thus various types of damage functions sketched in Figs 1-3 represent merely a single realization of each specific random process. When several machine parts are tested and damage is monitored as a function of time, a set of several realizations of the random phenomena, O(t) = g[P, V, O; 0%,~, ~E; t]
te T
(5)
are observed, as shown in Fig. 4. Each curve in this set of realizations of damage process has a given probability of occurrence. Some experimentally obtained random functions of wear of cutting tools are shown in Figs 5-7. When damage reaches a limiting value DI the system fails and life at that limiting level shows scatter, as illustrated in Fig. 5. In laboratory conditions the various factors affecting the wear process can be strictly controlled, and the experimental curves of wear are therefore close m
M a t e r i a l : carbon steel m
ol r4
~
iI
k:
o
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1
2
3
4
5
6
7
8
9
10
11
Unit length number
Fig. 5.
Wear diagram for a carbon steel. Scatter of wear index: _+5%/
Nonlinear damage process 0'012 --
81
Parameters Speed 1000 sfprrl
0.010 - -
0'008
"2 o . o o e
0.004 ,'7 0.002
I
0
I
5
10 Cutting t i m e (rain)
I
15 t
Fig. 6. Real-life sample function of w(t), with weak mixing, which can be represented approximately by a linear wear model. Data supplied by J. Seidel, Scientific Lab, Ford Motor Co. 5
1.2]--
Drill w e a r history Flank w e a r individual tools
1.0
~, 0.8--
i,
0.6
0.4
0.2
0
50
Fig. 7,
I 100
150
I I I 200 250 300 Pieces produced (x 101)
I 350
Sample function of wear, with strong mixing. 6
400
I 450
82
Anwar Khalil Sheikh, Munir Ahmad
together, as shown in Figs 5 and 6. The scatter in the wear measurements is m u c h higher in normal operating conditions, as evident in Fig. 7. The stochastic representation of the damage process whose deterministic representation was given by eqns (3) and (4) is D(t) = At B
t~T
(6)
where A = D(1) -~ initial value of damage
(7)
B
(8)
dD(t)/D(t) dt/t
The results of repetitive tests on identical components, for example wear on cutting tool and crack growth in fatigue, do confirm the stochastic model of eqn (6). In the subsequent section a reliability model for this nonlinear d a m a g e process is developed.
4 RELIABILITY MODEL FOR A CRITICAL DAMAGE LEVEL D t If the life of the machine part is terminated when damage D(t) exceeds or reaches a critical level D t, then the corresponding life distribution (i.e. the conditional distribution of the r a n d o m variable T when D(t) = D~) can be obtained by using the standard techniques of probability theory. F o r this purpose we can rewrite the damage growth model D ( t ) = A t ~ as follows: log e D(t) = log e A + B log~ t
t~T
(9)
and the following assumptions are m a d e about the r a n d o m variables log A and B. (1) Both logeA and B are statistically independent, and distributed according to a normal law with mean #LA = E [ I o g e A] = E[B]
and variance a~A = V [log e A]
(2)
P [ l o g eA < 0] = 0 P [ B < 0] = 0
The second assumption implies that the coefficient of variations (COV) of the r a n d o m variables log e A and B are very small; for example a value of the
Nonlinear damage process
83
COV of loge A, or the COV of B < 0.3 will satisfy the condition expressed in the assumption (2) above. The joint probability function of the two independent random variables A and B in eqn (9) is given by
,
r
f(A'B)=2rurLAaaAexp[--2~.k- -aL--AA/
\-~B ,]]J
d>0, --~
(10)
Considering the transformation In A = In Dt - B In t B=B Then the Jacobian of the transformation is IAB/tl. Thus B
exp
--
!nD'-
B l n t - #L,. + (B-,,.B2~I
f(t'B)=2naLAant
\ aB
aLA
/
)J
--oo < B < + o o Integrating f(t, B) over B we have f(t) =
1 ~ 2naLBAant exp --2
lnDz-Blnt-#LA~ + B-#~ aLA / \ an ] ) J dB
(11) Let ~, = a~/#~
(12)
Al ~___GLA/~.,IB 2 2
(13)
and C~ -
In D t - ~LA
(14)
The probability density function f(t) can be rewritten after some simplification in the following form: otsCllnt+fl, f(t)=v/-~(t)[o~,(lnt)2 + flt]3n =f(t; off,Ct, fit)
F
1["
exp L-5
lnt-C~
/ 2]
t)2 + &/...I
t> 1
(15)
Equation (15) represents the probability density function of the random variable T, which defines the life of the product corresponding to the damage criteria DI, and is mathematically expressed in the following form: log~ T = l°ge Dt -- log~ A B
Dl ~ D(t)
(16)
Anwar Khalil Sheikh, Munir Ahmad
84
The functionf(t) is the probability density function of the time to first failure [failure criteria D ( t ) > D~] of the products degrading according to a nonlinear d a m a g e process of the type D(t) = At B. This is a three-parameter model with the following significance of the various parameters. Parameter Ct is the location parameter (eqn (14)) and depends u p o n the limiting value of the d a m a g e D t and mean values of the r a n d o m variables log e A and B. It is shown in the next section that Ct = 1Oge T~t, where TM is the median life. Parameter ~l is the shape parameter and is the ratio of the variances of the r a n d o m variable B and the m e a n value of the B (eqn (12)). F r o m eqn (9) it is obvious that B = d[loge D(t)]/d(log~ t) = slopes of log e D(t) versus log~ t curves, thus ct~is equivalent to the coefficient of variation of these slopes and characterizes the dispersion in the life of the product. By empirically fitting the three-parameter probability model f(t; C, at, fit) to several data sets on the fatigue life observations it is found by the authors that the parameter x/~/~ usually lies between 0.01 and 0.2. The third parameter, fit, which gave this distribution an added feature is the ratio of the variance of the r a n d o m variable, log~ A, to the square of the mean value of B (eqn (14)). Note that A is the a m o u n t of damage incurred at 1 = 1, D(1) = A. Since T~t >> 1, thus the m i n i m u m value of t, tmin, is tar away from zero, which means that for all practical purposes logeA represents the initial level of damage, i.e. D ( 1 ) = A - ~ D ( 0 ) . Therefore fl~ is a parameter which incorporates the variability in the initial quality of the product. The mathematical expressions of the reliability function,
fl 1~5
R(t) = P [ T > t] =
f(t) dt
the cumulative distribution function, F(t)= 1 -R(t), and the hazard rate function, 2(t) = f(t)/g(t)
= - [dg(t)/dt]/R(t)
are given below. (It is i m p o r t a n t to emphasize that limits of r a n d o m variable t are replaced from - v o < t < ~ to 1 < t_< 0% since the d o m a i n of t is tmin < t < ~ where t.,tn >> 1. If this condition is violated then the probability model o f e q n (15) will not be a proper probability model. In several damage processes, for example wear, fatigue, etc., where rate of damage propagation is slow and the median life is very high, for example in fatigue failures, the median life m a y be as high as 103 to 107 cycles, and the above-mentioned condition is always satisfied.) Reliability function:
R(t)
=
1
-
.~
log~t-C,
/ ,tlOge t): +
}
t> 1
(17)
85
Nonlinear damage process CUmulative distribution function: (I)~ log, t - Ct F(t) = [x/at(log ~ t) 2 + flit}
t> 1
(18)
and the hazard function is
[atCtlog ~ t + ill] exp [[_-gl~',:x/(log, ( / a t ( l o g t~-C)t) 2 d- flit}1 2(0 =
t> 1
x//-}-nn(t)[at(loget) 2 + R I"td-13/2(I)r-
Ct ~l_oget 2-
(19)
]
Lx/a,(log, t) + iltJ
For illustrative purposes plots of three-parameter modelf(t, C,, at, flit) and its associated probability functions R(t, Ct, flit, at) and 2(t, C t, flit, at) are given in Figs 8-10. For at = 0.003 and T~ = exp [C~] = 1000, for values of fit varying from 0 to 0.9 with an interval of 0.01. These figures illustrate the role of parameter flit. In Fig. 11,f(t; Ct, a~, f/) is plotted for a t = 0.003, fit = 0, and C t = l o g e T~t= logel000 to log¢ 10000 to illustrate the increase in guaranteed life (trot,) with an increase in Ct (or median life TM). The main statistical characteristics of the three-parameter reliability model R(t; Ct, at, flit) are: 1. 2. 3. 4.
Median: Mean: Variance: Mode:
T~t = exp [Ct] T = S~ tf(t)dt: need numerical evaluation V(T) = S~ ( t - T)2f(t)dt: need numerical evaluation is the solution ofdf(t)/dt = 0, which can be resolved by iterative methods by searching the value of tmode in the neighborhood of TM.
Two special cases of this model are discussed in the following sections. 4.1 Two-parameter model
f(t, C, ~1):
the variability in B = 0
In this case the variability in the parameter B is assumed zero, and therefore the parameter a t = 0, and the probability density functionf(t) (eqn (15)) will reduce to _
1
= f(t, Cl, fit)
(20)
In a similar way two-parameter functions R(t, Ct, flt), F(t,C~,ft) and 2(t, Ct, ft) can be obtained by letting at = 0 in eqns (17) to (19). at = 0 means that the damage process is modelled as D ( t ) - At n, t > 1, where A is the
Curve 1 J~l 0.1
2 02
3 4 .5 6 7 8 9 0"3 0.4 0.5 0'6 0"7 0-8 0-9
T@ : 1 0 0 0 , ~tt : 0 - 0 0 3
"L g
~s
Fig. 8. Three-parameter probability density model: f(t; G, ~ ) ,
Nonlinear damage process
87
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Anwar Khalil Sheikh, Munir Ahmad
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r a n d o m variable but B is a deterministic quantity. Equation (20) is a wellk n o w n lognormal model where m a x i m u m likelihood estimates of the parameters Ct and fJl are C"l - ~ l°ge li N and . / ~ (1Oge ti -- Ct) 2 Thus the lognormal model which is frequently used in reliability analysis is a special case of the three-parameter model f ( t ; Ct, fit, at) proposed in this paper. 4.2 Two-parameter model
f(t, C~ ~1): variability
in A is zero
This case arises when the variability in the initial quality is zero, i.e. V[log e A] = 0. For mass-produced items with proper quality control at manufacturing and assembly stages, this situation is easily realized. By letting fit = V[loge A]/E2 [B] = 0 in eqns (15), (17), (18) and (19) we obtain the following (two-parameter) probability models: f ( t ) = f ( t ; Cl, al) = ~
t(loge t) 2
R(t) = R(t; Cz, at) = 1 - q~
F(t) = F(t; C t, aj) = t9
1
loge t log~ i
t> 1
(21)
t> 1
(22)
t> 1
(24)
1 log e t
and C, exp - ~ 2(t) = 2(t; C l, a~) - - -
lo~t_] J
The graphical representation of these functions is given in Figs 12-15. The following features are observed in this model. (a) The distribution is a skewed distribution; this skewness is one of the major features of this model. (b) As C~ increases (i.e. when TM = exp C~ increases) the m i n i m u m guaranteed life of the product increases (see Fig. 11). This feature
Nonlinear damage process
91
4
Curve "
"
A 0.05 TM:
•
.~
B 0075
C 0.1
D E 0.125 0.150
F 0.175
G 0.2
I 0 4 = exp ~C,']
a
t . °
--
.
=
==========================
°
,,,,,...,.:::-------,-.-.-...-.
.
~= t ~ r ~ = . e . p [c,]
Fig. 12. Two-parameter probability density model:f (t; C~,e~).
"i!::.:..-.. Curve v~l
:i iii:i:L
A 0"05
B 0075
C 01
7 . = lO" = e,p
0 0'125
E F 0"1.50 0175
G 02
[c,]
11:1~111
<:
:,ir~=,i
e×. [c,]
Fig. 13. Two-parameter reliability model: R(t;Ct,~q).
Anwar Khalil Sheikh, Munir Ahmad
92
,.,.,., •
•
1
•
~
.
,.,..-r ,,1..¢ ,,,,,,,., ~.1pi ° r
~
)
t
,n,~*tn...~
"1
,~....
q
I
1
g
I
9
6 ,,,.,, .-
}
11
i
11
~J
.i
o
,,,~ •
,Ul
I
w~.. • •
6
d '1
vnpr .~ vn."~
Q *.o e'
•q
v v
,,
% : ~, :
o
...... , ~ , ~
v v
fl
:I
'
I
•
*q
•
lm
1 In
" ' 1 I%
,~.1 *!
'I
'I
%
0
.
•
II
• v
~A >
~."'°1
~n1~ ~ 7n1.1 v .~n~ w~
~I::~ ,
'I @
m
•
m
•
,,~-0
• ,H, "n n~
08"0
OgO
OL'O
"~I
(5
1l,~
% %%':: v
00~
it
*'
n a,*,~
q
,
6
og'o
OgO
Ol"O
0£'0
0~0
~.)
Nonlinear damage process Curve
f6~
A
B
C
D
E
F
93 G
o.o~ o.o~o o.o7~ o., o-~o O.~TS 0.2 r.-- ~0"= e~p [C,]
-T'-
4~44~4~9~4444 ~4444 ~
°=°.=1o.°======= J~
"=.=j=.
A
X
,~t
.! •
Fig. 15. Hazard function of two-parameter probability model: 2(t; C~, ~).
(c)
makes this two-parameter model superior to three-parameter life models such as Weibull or lognormal where a third parameter (minimum life parameter to) has to be incorporated to show guaranteed life period. Hazard function first increases for some period of time and then slowly decreases. The behavior of hazard function 2(t, C~,al) is somewhat similar to the behavior of lognormal and inverse Gaussian hazard function. The first increasing and then decreasing hazard function gives a rather thick tail to the distribution on the right-hand side. (This becomes more significant when shape parameter a t increases.) This significant thick tail on the right-hand side indicates that a large portion of the given type of devices or components will fail, say, before t _<4TM; however, a small fraction of the component will survive for a very long period of time which can be as high as 10-15 times the median value. This type of longevity behavior is commonly observed when the damage process is fatigue or wear.
94
A n w a r Khalil S h e i k h , M u n i r A h m a d
4.2.1 M a x i m u m likelihood estimate o f parameters The m a x i m u m likelihood estimates of the parameters C~ and a t are developed in Appendix I. The resulting expressions to obtain t2~ and 0it from a set of observations of life, t 1, t2,..., ti, i = 1, N, are as follows: N
{12 }-1
(25)
i=1 N
(26) i=1
Using these maximum-likelihood estimates of the parameters Ct and ~, the data from refs 8 and 9, as shown in Fig. 16, have been used to show the validity of the proposed model, [f(t; C~, at)I, and it turns out to be a better model than the other suggested models. In fatigue data analysis this model is particularly very appropriate and compatible to the physical p h e n o m e n a of crack propagation (see refs 7 and 8). 4.2.2 Parameter estimation from a sample o f damage curves An alternative way of estimating the parameters of the model from the sample functions of damage process is given below. Consider a set of M realizations of the damage propagating process 1)(t) = At B. Suppose the ith realization is given by the curve D i = Ai tni, where i = 1, 2 .... , M (M > 10), then the parameters C~ and a I are c o m p u t e d as follows: 1
M
~ - i=~l log e A i -- log~ D t 6't = M E Bi/M
(27)
i=1
and M
Z (B,~t =
l
i=1
(28)
log e A i
1/M i
where M
li~ = Z Bi/M i=1
(29)
Nonlinear damage process
95
~t (a)
o-
.......
o -0' {t 0uo
u o
6
[,
0.1
o1
¢,
I
,
<,
i
,,
.,
~u . o
c ~
, -
>
:
~J
~
1 = 0.01 2 = 0'02 3= 0'03 4 = 5 = 6 = 7 = 8 = 9=
0.04 0.05 006 0.07 0.08 0.09
'~" ' ~
id 16,
% .q
~,
,c,
,a
~r
r-
r,
L,
i.
6
u.
,t.
c,
,X
ul
p f-
,'
tl
c .,
r.-
I.-
L~,
,,
c c
dr"
o
• ,
.
c,;
6
,.Q .c, -,c
r_ r
r
R~
-I" t-t-
~c
r- r-
•
•,
r
r-
.
r-,r'-t-
r-:-
t
o~r._, rJ. . (1:,
c,.
( . , c : ? ,J,
o.,
',
:..
L 0, o .~ c. e .
i' r... . . . . . . . .
0/:
o..J
I,-,
0
0
(o .
.',r
r,
r~
c ~¢1 ¢
•
~r
r
~r, '
8
°
cl r'
'
¢ c;
~° 8
°
8 o
8
°
o
8
°
8
°
8
~
8
o ,~ o oo o,,
o
C# = I o g e TM
Fig. 16(a).
Dependence of coefficient of variation (COV) on median life TM.
Anwar Khalil Sheikh, Munir Ahmad
96
i,
,]
i
¢druC~l'~[J[JCdc;
1 = 0"10 2 = 0"11 3 = 0.12 4 = 0'13 5 = 0.14 ~r~
1'i
~.
~'¢
O-
0..
C:,~-
~
{'~
C',t
~
~"
r--
8 = 0.17 9 = 0.18
,0 ',0
03
r"
",O
O-
r'
6 = 0-15 7 = 0"16
r,,.
O,
u~
IZl
CO
'~
~.)
-,
o
r'-
,,u
O
0
~0
~3
r-
e,,.
(~
8:
*1
> 0 u
rl
1
8 o 8 o
8 .o, 8 . ,o o o o o o. o . .o. °. o . . o. o o o o o 8 log e TM =
Fig. 16(b).
Cl
Dependence of coefficient of variation (COV) on median life TM.
Nonlinear damage process
97
4.2.3 Main statistical characteristics of the rnodel f(t; Cl, ~t) The median life is Tu = exp [Ct]. The mode can be found by iterative method as a solution of (df(t)/dt)=O. The quantile life tq is obtained from P I T < tq] = F(t~) = q, and for small values of x/~t (<0-2) we have
c, t~ = exp {1 _ x/~tzq }
(30)
where z~ is the qth quantile of standardized and normal variate found from the condition ~[zq] = q. The exact values of mean and variance of T are difficult to obtain mathematically. However, fairly good approximations of the mean and variance of life T for small value of ~ (i.e. x/~t < 0.2) are given by the following expressions: MEAN: VARIANCE:
{
T = exp Ct 1 + Ctctt ~ + 1 tr 2 = C/2~qexp {2Ct}
(31) (32)
The coefficient of variation of life is Ct KT = 1 + Ct~t(1 + CJ2)
(33)
We note that coefficient of variation is dependent upon not only the shape parameter w/~t but also on the location parameter Ct = loge TM. Thus K r will vary with respect to the median life. This behavior of COV of life makes our proposed model distinctly different from several other models such as lognormal Weibull, etc., where COV is the function of shape parameter alone and is independent of median life. Plots of Kr versus Ct for various values of x/~t are given in Figs 16a and 16b.
5 CONCLUSIONS (1) A new three-parameter probability model, f(t;Ct, at, flt), is proposed which describes the probability density of time to first failure of the product degrading according to nonlinear scheme of damage expressed by a stochastic equation of the type D(t) = At B, and the life criterion is a critical level of damage D v (2) Two special cases of the proposed model f(t; Ct, o~l,fit) are explored. In the first case, when at -- 0, the model reduces tof(t; Ct, fl,), which is shown to be a lognormal model. In the second case, when fit = 0, the model reduces to a two-parameter model, f(t; Ct, at), which is found a very
98
Anwar Khalil Sheikh, Munir Ahrnad
useful model in several damage processes such as fatigue and wear processes. (3) F o r the two-parameter model f(t;C~,~t) various statistical characteristics, such as mean, variance, median and coefficient of variation, are developed. It is pointed out that a distinct feature of this life model is the dependency of the coefficient of variation on the median life of the product. The increasing-decreasing rate of the hazard function and its appropriateness in fatigue and wear-related life modelling is pointed out. (4) Methods of estimating the parameters ~t, C~, etc., from the life data, as well as from the sample of damage functions, are also developed.
6 ACKNOWLEDGEMENT The support provided by the University of Petroleum and Minerals ( U P M grant No. ME/Modelling/46) is gratefully acknowledged. The authors wish to thank M r Jawad Qureshi for careful preparation of the manuscript.
REFERENCES 1. Gertsbakh, I. B. and Kordonsky, Kh. B. Models of Failure, Springer-Verlag, New York, 1969. 2. Hitomi, K., Nakamura, N. and Inoue, S. Reliability analysis of cutting tools, ASME Paper 78- WA/PROD-9, 1978. 3. Kitagawa, H., Takahashi, S., Suh, C. M. and Miyashita, S. Quantitative analysis of fatigue process-microcracks and slip lines under cyclic strains, Fatigue Mechanisms, ASTM STP 675 (1979), pp. 420-49. 4. Field, M. Relation of microstructure to the machinability of wrought steels and cast iron, International Research in Production Engineering, 1963. 5. Duncon, J. Mathematical and numerical methods for scheduling cutting tool replacements and applications. PhD thesis, The University of Michigan, 1969. 6. Rhodes, J. S. Transfer machine operational analysis and cutting tool change policies, The Charles Draper Laboratory Report, November 1973. 7. Sheikh, A. K., Ahmad, M. and Younas, M. Stochastic nature of crack propagation and its implications in fatigue life prediction, ISTFA/82 Eighth Annual International Symposium for Testing and Failure Analysis, San Jos6, California, 25-27 October 1982. 8. Weibull, W. The Statistical Aspect of Fatigue Failures and its Consequences in Fatigue and Fracture of Metals, W. M. Murray (ed.), John Wiley & Sons, New York, 1952. 9. Shuaib, A. N., Mohiuddin, A. and ;Sheikh, A. K. Designer's approach to the selection of reliability models in fatigue life investigations, ASME paper No. 84- WA/DE-12.
99
Nonlinear damage process
A P P E N D I X : M A X I M U M L I K E L I H O O D OF T H E P A R A M E T E R S Consider the probability density function at (21)--when fl~ = 0. The loglikelihood function of a random sample of size n from the population is lnti-
ln L = n ln C, - 21n et - 2 i=1
1
2 2( ti - ~-~l
i=1
i=1
The estimating equations t~lnL ----0
and
~lnL ---0
give P1
°¢1= n
n
1-~/] i=1
and
C'l = n
Cl "~2
1-1-~i
(ln ti)-I i=1
~'t is the harmonic mean of logarithms of the observations.
}
A Reliability Model for a Nonlinear Damage Process
Anwar Khalil Sheikh and Munir Ahmad Departments of Mechanical Engineering and Mathematical Sciences, University of Petroleum and Minerals, Dhahran, Saudi Arabia (Received: 16 May 1986)
A BS TRA C T Several types of damage processes lead to the failure of mechanical devices. Some examples of these damage processes are: wear, fatigue, creep and corrosion. In this paper we consider a generalized damage process defined by the random function D(t); D(t)= Atn; t~ T, where A, B and T are random variables, and D(t)= damage incurred at time t. If the life of the device is terminated when the damage exceeds a limiting value D t (i.e. D(t) > D~), then it will be shown that, under certain assumptions regarding random variables A and B, the reliability model of the device is
O[
log~t--Ct
l
where
• (z) = ~
~ exp [-½¢23 d~
and Ct, ¢tt,fit are the functions of D t and the mean and variance of the random variables A and B. In this paper various statistical characteristics of the above-mentioned model are discussed. The generalized nature of the proposed model is emphasized, and it is shown that the lognormal model is a special case of the proposed model The method of estimation of the parameters of the reliability model from a set of realizations (sample functions) of the damage process is also illustrated. 73 Reliability Engineering 0143-8174/87/$03-50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
74
Anwar Khalil Sheikh, Munir Ahmad
1 CAUSES OF F A I L U R E Failure of a machine part* (or a system) is defined as the event associated with a shift in the operating characteristics of that part (or system) from its permissible limit. This shift in operating characteristics occurs due to accumulation of damage with the passage of time. The failure may be caused by the combination of the external, internal and initial factors. 1.1 External factors These factors are extrinsic or extraneous in character; that is, they originate from causes external to the system. They represent the environmental conditions under which the equipment is operating. For example, failure may occur due to unpredictable stresses arising from such environmental factors as sudden shocks. The rate of occurrence of this type of failure is dependent upon the severity of environmental conditions; the more severe the environments, the more frequently the occurrence of failure. The chance failure or a random failure is normally caused by the external factors) The set of external factors will be designated by the symbol aE. 1.2 Internal factors Internal factors are those factors which are due to the internal causes such as physical processes that take place inside the system. They are usually associated with certain inherent characteristics of the system, and are dominant in the initial and final stages of operation. During the initial phase of operation of a system, a type of failure known as initial failure may occur from the inherent defects in the system attributed to faulty design, manufacture or assembly (initial quality factor). As the system operates it wears out; hence during its final phase of operation a second type of failure, known as wear-out failure, may be experienced by the system. Wear-out failure is the outcome of accumulated wear and tear; a depletion process occurs through abrasion, fatigue, creep and like phenomena, which results in the loss of specified physical, mechanical, chemical or other properties characterizing the initial design features of the system. Ultimately, an operating system will fail as a result of wear and tear. The set of internal factors will be designated by the symbol at. 1.3 Initial quality factors The dimensional inaccuracies in manufactured products, inherent variability of material characteristics combined with faulty production * The terms m a c h i n e part, device or system will be used interchangeably.
Nonlinear damage process
75
techniques, improper quality control and defects in assembly could lead to a considerable variability in the degree of damage at the beginning of operation (i.e. initial damage at time t = 0). The above-mentioned factors are called as initial quality factors and will be denoted by %.
2 M O D E L S OF D A M A G E PROCESSES The life of a machine component is terminated by two types of failure: gradual failure (such as wear-related failures) and sudden failures. Most component failures are caused by various damage processes occurring in the machine which lead to a gradual decline in the initial efficiency of the component, which in turn leads to component failure when accumulated damage reaches a critical level. Various irreversible damage processes which lead to the system's failures are wear, corrosion, internal stress redistribution and plastic distortion of components, creep, fatigue, fracture, diffusion, evaporation, crystallization, burn-out, biological attack, deposition and seasonal fluctuations in temperatures. Wear and fatigue are two major causes of failures of machine parts. In this paper, for illustrative purposes, several examples from wear and fatigue damage processes will be given. However, the proposed mathematical model is not restricted only to these two types of damage processes; it can well be applied in other cases such as creep, corrosion, etc., where damage as a function of time conforms with the hypothesized nonlinear law. The life of a machine or its elements depends on the rate of the damage process and the limiting conditions of the device. The rate of damage process (for example, wear), dD(t)/dt, is a function of several independent variables:
dD(t)/dt = ~O(P, V, O; ~, %, %; t)
(1)
where P = V= 0 = ~ = ~E = So = t=
pressure sliding velocity set of other operating conditions, including service conditions set of internal factors causing damage to the system set of external factors causing damage to the system set of initial quality factors at time t = 0 the operating time.
F r o m eqn (1) we can write a general form of damage function as D(t) = g[P, V, O; %, al, aE; t] where g[¢] = S~ ~0[~] de.
(2)
76
Anwar Khalil Sheikh, Munir Ahmad
D(t)~
~-t
(a)
D(t)I
I Dlt) ~t ~e- to - . . ~ (b)
L••t (f)
D(t)~
~t (¢)
D(t)l J ~
D(t) ~t (d)
D(t)~
f
Ig)
~t (e)
Fig. I.
Various plausible models of damage propagation: D(t) represents the degree of total damage at time t.
Considering the occurrence of irreversible damage processes resulting in a given type of damage (wear, deformation, corrosion, fracture, etc.) it is possible to distinguish a certain type of relationship as plausible models for the damage function D(t). Typical examples of damage functions (i.e. the models of damage development) are given in Fig. 1. 1.
2.
3.
4.
All factors affecting the rate of a given damage process are stabilized and there are no factors which cause the rate to change. The relationship D(t) is linear, as for example in an established wear period (Fig. l(a)). The process of deterioration may not become evident for some period t o, after which it may rapidly cause failure of the component. In this case there is an accumulation of internal factors, for example fatigue fractures of the components (Fig. l(b)). As the degradation process proceeds those factors which affect intensity gradually decline. This is typical, for example, of wear during the initial period known as running-in period. This happens due to the fact that microirregularities on the surfaces of rubbing parts change (Fig. l(c)). This type of behavior is also observed in the initial zone of creep rate versus time curves. Factors causing damage gradually intensify and the damage process rate rises continuously. For example, if there is corrosion or fatigue
Nonlinear damage process
5.
77
damage of the surfaces, after initiation the process develops at an increasing rate (Fig. l(d)). Finally, the process causing the damage may be of an unstable nature; this happens when the component operates with sharp variations in working rates and in fluctuating operating conditions (Fig. l(e)).
In fact, damage of most components usually proceeds in several stages representing the typical cases listed above. The wear processes often consist of three stages (Fig. l(f)): (a) decreasing wear rate (running-in) (b) constant wear rate (normal wear) (c) increasing wear rate (catastrophic wear) i.e. the entire process of damage can be expressed by curves c, a and d. Creep rate curves show a similar behavior to Fig. l(f). However, there are several instances when initial wear period (running-in) can be made negligibly small and the process consists of approximately constant damage rate plus an increasing damage rate. Some types of tool wear and fatigue crack growth after its initiation exhibit this type of behavior. Deformation of a component as a result of redistribution of internal stresses first occurs at a high rate and then gradually slows down (Fig. l(g)). The rate of the damage process d D ( t ) / d t and the nature of the D(t) curve are greatly affected by the machine operating rates. As a rule an increase in load, speed, temperature or other operating conditions leads to an intensification of the failure process, i.e. reduces the reliable life of the product. In this paper we will discuss the damage process which exhibits a continuously increasing rate of damage, as shown in Fig. l(d). Two cases, i.e. one describing the wear of a cutting tool and the second describing the crack growth after initiation, are shown in Figs 2 and 3. These cases correspond to Fig. l(d), and the general model for the damage process applicable in these cases is D(t) = A t ~
(3)
where D ( t ) = a m o u n t of wear, or crack length or any other observable parameter of the damage process A, B = empirically determined constants. The rate of the damage process is d D ( t ) / d t = B A t B- ~
(4)
Anwar Khalil Sheikh, Munir Ahmad
78
6.5 6.0
AtB=0"3778 t 0615
W(t)=
5-5 5.0
4.5 E E 4.O S 0
× 3.5
• :3.0 "6
S 2.5 2 2.0
-~ 1.o ID
E 0.5
t~ a v
O
Fig. 2.
I
I
I
I
I
I
I
I
I
I
I
I
5
10
15
20
25
30
35
40
45
50
55
60
Average cutting time t. Tool wear as a function of cutting time. 2
thus to have dD(t)/dt as an increasing function of time B > 1. C o m p a r i n g eqn (4) with (1), and eqn (3) with (2), one notes that the cumulative effect of variables P, V, O; s0, s~, SE is incorporated in A and B.
3 STOCHASTIC
NATURE
OF
DAMAGE
PROCESSES
During the operation of a machine there are changes and fluctuations in loads (P), velocities (V), methods of lubrication, and the degree of contamination of the friction surfaces, and temperatures, etc. There is a manufacturing or assembling variability in the initial quality of the product (So); similarly there are inherent variabilities in internal and external factors (cq and SE) of the system. Sometimes part of these variables, such as pressure, velocity and initial quality, could be maintained at a fixed level without fluctuations (i.e. as deterministic quantities); however, most of the other factors are extremely difficult to control at a desired level in a deterministic sense. Since various input variables, P, V, O; s~, sF., So, are in general r a n d o m in nature, the damage function D(t) in real life represents a stochastic
Nonlinear damage process
Gmax-- (Train = 55--2 kglmm 2
79
Crmax-Gmin : 62--2 kglmm 2
+o,.'+.7
+.°' +T w
u
u
-"
0
" °
~u
20 30 40 Number of cycles N Cycle x 104
0 10 20 30 Number of cycles N Cycle x 104
Omax- ¢rmin : 44.0-1.Skg/mm 2 o
o crack length, 2a crack depth, 2b e&
initial
O-max --
flaw O'mi n = 3 9 , O - - 1 . 6
2a
A 2b e& (nitial flaw kglmm 2
0-2
)
~
2,20
~
:b:a
oJ o+ ~a E ~lE0.2
0.1
0.1~
r~ g
)u
) 20 40 60 N Cycle x 10 4
80
0
I
I
5 10 N Cycle x 104
Fig. 3. Crack growth behavior after initiation from the initial flow (a-N relation). For the initial flaw, a pit-like small surface notch 50/~m deep and 50/~m in diameter is used. 2a: Crack length on specimen surface; b: crack depth, a and/or b represent the level of damage in these cases and a model of the type a = A N B can be fitted to these crack propagation curves, a
80
Anwar Khalil Sheikh, Munir Ahmad c
i E,
I
I..tt. tit)
:
,
. . . . .
T
Fig. 4.
A set of sample functions of stochastic damage processes: D(t), and life distribution corresponding to a critical damage level D~.
phenomenon. Thus various types of damage functions sketched in Figs 1-3 represent merely a single realization of each specific random process. When several machine parts are tested and damage is monitored as a function of time, a set of several realizations of the random phenomena, O(t) = g[P, V, O; 0%,~, ~E; t]
te T
(5)
are observed, as shown in Fig. 4. Each curve in this set of realizations of damage process has a given probability of occurrence. Some experimentally obtained random functions of wear of cutting tools are shown in Figs 5-7. When damage reaches a limiting value DI the system fails and life at that limiting level shows scatter, as illustrated in Fig. 5. In laboratory conditions the various factors affecting the wear process can be strictly controlled, and the experimental curves of wear are therefore close m
M a t e r i a l : carbon steel m
ol r4
~
iI
k:
o
I
I
I
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
9
10
11
Unit length number
Fig. 5.
Wear diagram for a carbon steel. Scatter of wear index: _+5%/
Nonlinear damage process 0'012 --
81
Parameters Speed 1000 sfprrl
0.010 - -
0'008
"2 o . o o e
0.004 ,'7 0.002
I
0
I
5
10 Cutting t i m e (rain)
I
15 t
Fig. 6. Real-life sample function of w(t), with weak mixing, which can be represented approximately by a linear wear model. Data supplied by J. Seidel, Scientific Lab, Ford Motor Co. 5
1.2]--
Drill w e a r history Flank w e a r individual tools
1.0
~, 0.8--
i,
0.6
0.4
0.2
0
50
Fig. 7,
I 100
150
I I I 200 250 300 Pieces produced (x 101)
I 350
Sample function of wear, with strong mixing. 6
400
I 450
82
Anwar Khalil Sheikh, Munir Ahmad
together, as shown in Figs 5 and 6. The scatter in the wear measurements is m u c h higher in normal operating conditions, as evident in Fig. 7. The stochastic representation of the damage process whose deterministic representation was given by eqns (3) and (4) is D(t) = At B
t~T
(6)
where A = D(1) -~ initial value of damage
(7)
B
(8)
dD(t)/D(t) dt/t
The results of repetitive tests on identical components, for example wear on cutting tool and crack growth in fatigue, do confirm the stochastic model of eqn (6). In the subsequent section a reliability model for this nonlinear d a m a g e process is developed.
4 RELIABILITY MODEL FOR A CRITICAL DAMAGE LEVEL D t If the life of the machine part is terminated when damage D(t) exceeds or reaches a critical level D t, then the corresponding life distribution (i.e. the conditional distribution of the r a n d o m variable T when D(t) = D~) can be obtained by using the standard techniques of probability theory. F o r this purpose we can rewrite the damage growth model D ( t ) = A t ~ as follows: log e D(t) = log e A + B log~ t
t~T
(9)
and the following assumptions are m a d e about the r a n d o m variables log A and B. (1) Both logeA and B are statistically independent, and distributed according to a normal law with mean #LA = E [ I o g e A] = E[B]
and variance a~A = V [log e A]
(2)
P [ l o g eA < 0] = 0 P [ B < 0] = 0
The second assumption implies that the coefficient of variations (COV) of the r a n d o m variables log e A and B are very small; for example a value of the
Nonlinear damage process
83
COV of loge A, or the COV of B < 0.3 will satisfy the condition expressed in the assumption (2) above. The joint probability function of the two independent random variables A and B in eqn (9) is given by
,
r
f(A'B)=2rurLAaaAexp[--2~.k- -aL--AA/
\-~B ,]]J
d>0, --~
(10)
Considering the transformation In A = In Dt - B In t B=B Then the Jacobian of the transformation is IAB/tl. Thus B
exp
--
!nD'-
B l n t - #L,. + (B-,,.B2~I
f(t'B)=2naLAant
\ aB
aLA
/
)J
--oo < B < + o o Integrating f(t, B) over B we have f(t) =
1 ~ 2naLBAant exp --2
lnDz-Blnt-#LA~ + B-#~ aLA / \ an ] ) J dB
(11) Let ~, = a~/#~
(12)
Al ~___GLA/~.,IB 2 2
(13)
and C~ -
In D t - ~LA
(14)
The probability density function f(t) can be rewritten after some simplification in the following form: otsCllnt+fl, f(t)=v/-~(t)[o~,(lnt)2 + flt]3n =f(t; off,Ct, fit)
F
1["
exp L-5
lnt-C~
/ 2]
t)2 + &/...I
t> 1
(15)
Equation (15) represents the probability density function of the random variable T, which defines the life of the product corresponding to the damage criteria DI, and is mathematically expressed in the following form: log~ T = l°ge Dt -- log~ A B
Dl ~ D(t)
(16)
Anwar Khalil Sheikh, Munir Ahmad
84
The functionf(t) is the probability density function of the time to first failure [failure criteria D ( t ) > D~] of the products degrading according to a nonlinear d a m a g e process of the type D(t) = At B. This is a three-parameter model with the following significance of the various parameters. Parameter Ct is the location parameter (eqn (14)) and depends u p o n the limiting value of the d a m a g e D t and mean values of the r a n d o m variables log e A and B. It is shown in the next section that Ct = 1Oge T~t, where TM is the median life. Parameter ~l is the shape parameter and is the ratio of the variances of the r a n d o m variable B and the m e a n value of the B (eqn (12)). F r o m eqn (9) it is obvious that B = d[loge D(t)]/d(log~ t) = slopes of log e D(t) versus log~ t curves, thus ct~is equivalent to the coefficient of variation of these slopes and characterizes the dispersion in the life of the product. By empirically fitting the three-parameter probability model f(t; C, at, fit) to several data sets on the fatigue life observations it is found by the authors that the parameter x/~/~ usually lies between 0.01 and 0.2. The third parameter, fit, which gave this distribution an added feature is the ratio of the variance of the r a n d o m variable, log~ A, to the square of the mean value of B (eqn (14)). Note that A is the a m o u n t of damage incurred at 1 = 1, D(1) = A. Since T~t >> 1, thus the m i n i m u m value of t, tmin, is tar away from zero, which means that for all practical purposes logeA represents the initial level of damage, i.e. D ( 1 ) = A - ~ D ( 0 ) . Therefore fl~ is a parameter which incorporates the variability in the initial quality of the product. The mathematical expressions of the reliability function,
fl 1~5
R(t) = P [ T > t] =
f(t) dt
the cumulative distribution function, F(t)= 1 -R(t), and the hazard rate function, 2(t) = f(t)/g(t)
= - [dg(t)/dt]/R(t)
are given below. (It is i m p o r t a n t to emphasize that limits of r a n d o m variable t are replaced from - v o < t < ~ to 1 < t_< 0% since the d o m a i n of t is tmin < t < ~ where t.,tn >> 1. If this condition is violated then the probability model o f e q n (15) will not be a proper probability model. In several damage processes, for example wear, fatigue, etc., where rate of damage propagation is slow and the median life is very high, for example in fatigue failures, the median life m a y be as high as 103 to 107 cycles, and the above-mentioned condition is always satisfied.) Reliability function:
R(t)
=
1
-
.~
log~t-C,
/ ,tlOge t): +
}
t> 1
(17)
85
Nonlinear damage process CUmulative distribution function: (I)~ log, t - Ct F(t) = [x/at(log ~ t) 2 + flit}
t> 1
(18)
and the hazard function is
[atCtlog ~ t + ill] exp [[_-gl~',:x/(log, ( / a t ( l o g t~-C)t) 2 d- flit}1 2(0 =
t> 1
x//-}-nn(t)[at(loget) 2 + R I"td-13/2(I)r-
Ct ~l_oget 2-
(19)
]
Lx/a,(log, t) + iltJ
For illustrative purposes plots of three-parameter modelf(t, C,, at, flit) and its associated probability functions R(t, Ct, flit, at) and 2(t, C t, flit, at) are given in Figs 8-10. For at = 0.003 and T~ = exp [C~] = 1000, for values of fit varying from 0 to 0.9 with an interval of 0.01. These figures illustrate the role of parameter flit. In Fig. 11,f(t; Ct, a~, f/) is plotted for a t = 0.003, fit = 0, and C t = l o g e T~t= logel000 to log¢ 10000 to illustrate the increase in guaranteed life (trot,) with an increase in Ct (or median life TM). The main statistical characteristics of the three-parameter reliability model R(t; Ct, at, flit) are: 1. 2. 3. 4.
Median: Mean: Variance: Mode:
T~t = exp [Ct] T = S~ tf(t)dt: need numerical evaluation V(T) = S~ ( t - T)2f(t)dt: need numerical evaluation is the solution ofdf(t)/dt = 0, which can be resolved by iterative methods by searching the value of tmode in the neighborhood of TM.
Two special cases of this model are discussed in the following sections. 4.1 Two-parameter model
f(t, C, ~1):
the variability in B = 0
In this case the variability in the parameter B is assumed zero, and therefore the parameter a t = 0, and the probability density functionf(t) (eqn (15)) will reduce to _
1
= f(t, Cl, fit)
(20)
In a similar way two-parameter functions R(t, Ct, flt), F(t,C~,ft) and 2(t, Ct, ft) can be obtained by letting at = 0 in eqns (17) to (19). at = 0 means that the damage process is modelled as D ( t ) - At n, t > 1, where A is the
Curve 1 J~l 0.1
2 02
3 4 .5 6 7 8 9 0"3 0.4 0.5 0'6 0"7 0-8 0-9
T@ : 1 0 0 0 , ~tt : 0 - 0 0 3
"L g
~s
Fig. 8. Three-parameter probability density model: f(t; G, ~ ) ,
Nonlinear damage process
87
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88
Anwar Khalil Sheikh, Munir Ahmad
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Nonlinear damage process
89
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Anwar Khalil Sheikh, Munir Ahmad
90
r a n d o m variable but B is a deterministic quantity. Equation (20) is a wellk n o w n lognormal model where m a x i m u m likelihood estimates of the parameters Ct and fJl are C"l - ~ l°ge li N and . / ~ (1Oge ti -- Ct) 2 Thus the lognormal model which is frequently used in reliability analysis is a special case of the three-parameter model f ( t ; Ct, fit, at) proposed in this paper. 4.2 Two-parameter model
f(t, C~ ~1): variability
in A is zero
This case arises when the variability in the initial quality is zero, i.e. V[log e A] = 0. For mass-produced items with proper quality control at manufacturing and assembly stages, this situation is easily realized. By letting fit = V[loge A]/E2 [B] = 0 in eqns (15), (17), (18) and (19) we obtain the following (two-parameter) probability models: f ( t ) = f ( t ; Cl, al) = ~
t(loge t) 2
R(t) = R(t; Cz, at) = 1 - q~
F(t) = F(t; C t, aj) = t9
1
loge t log~ i
t> 1
(21)
t> 1
(22)
t> 1
(24)
1 log e t
and C, exp - ~ 2(t) = 2(t; C l, a~) - - -
lo~t_] J
The graphical representation of these functions is given in Figs 12-15. The following features are observed in this model. (a) The distribution is a skewed distribution; this skewness is one of the major features of this model. (b) As C~ increases (i.e. when TM = exp C~ increases) the m i n i m u m guaranteed life of the product increases (see Fig. 11). This feature
Nonlinear damage process
91
4
Curve "
"
A 0.05 TM:
•
.~
B 0075
C 0.1
D E 0.125 0.150
F 0.175
G 0.2
I 0 4 = exp ~C,']
a
t . °
--
.
=
==========================
°
,,,,,...,.:::-------,-.-.-...-.
.
~= t ~ r ~ = . e . p [c,]
Fig. 12. Two-parameter probability density model:f (t; C~,e~).
"i!::.:..-.. Curve v~l
:i iii:i:L
A 0"05
B 0075
C 01
7 . = lO" = e,p
0 0'125
E F 0"1.50 0175
G 02
[c,]
11:1~111
<:
:,ir~=,i
e×. [c,]
Fig. 13. Two-parameter reliability model: R(t;Ct,~q).
Anwar Khalil Sheikh, Munir Ahmad
92
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Nonlinear damage process Curve
f6~
A
B
C
D
E
F
93 G
o.o~ o.o~o o.o7~ o., o-~o O.~TS 0.2 r.-- ~0"= e~p [C,]
-T'-
4~44~4~9~4444 ~4444 ~
°=°.=1o.°======= J~
"=.=j=.
A
X
,~t
.! •
Fig. 15. Hazard function of two-parameter probability model: 2(t; C~, ~).
(c)
makes this two-parameter model superior to three-parameter life models such as Weibull or lognormal where a third parameter (minimum life parameter to) has to be incorporated to show guaranteed life period. Hazard function first increases for some period of time and then slowly decreases. The behavior of hazard function 2(t, C~,al) is somewhat similar to the behavior of lognormal and inverse Gaussian hazard function. The first increasing and then decreasing hazard function gives a rather thick tail to the distribution on the right-hand side. (This becomes more significant when shape parameter a t increases.) This significant thick tail on the right-hand side indicates that a large portion of the given type of devices or components will fail, say, before t _<4TM; however, a small fraction of the component will survive for a very long period of time which can be as high as 10-15 times the median value. This type of longevity behavior is commonly observed when the damage process is fatigue or wear.
94
A n w a r Khalil S h e i k h , M u n i r A h m a d
4.2.1 M a x i m u m likelihood estimate o f parameters The m a x i m u m likelihood estimates of the parameters C~ and a t are developed in Appendix I. The resulting expressions to obtain t2~ and 0it from a set of observations of life, t 1, t2,..., ti, i = 1, N, are as follows: N
{12 }-1
(25)
i=1 N
(26) i=1
Using these maximum-likelihood estimates of the parameters Ct and ~, the data from refs 8 and 9, as shown in Fig. 16, have been used to show the validity of the proposed model, [f(t; C~, at)I, and it turns out to be a better model than the other suggested models. In fatigue data analysis this model is particularly very appropriate and compatible to the physical p h e n o m e n a of crack propagation (see refs 7 and 8). 4.2.2 Parameter estimation from a sample o f damage curves An alternative way of estimating the parameters of the model from the sample functions of damage process is given below. Consider a set of M realizations of the damage propagating process 1)(t) = At B. Suppose the ith realization is given by the curve D i = Ai tni, where i = 1, 2 .... , M (M > 10), then the parameters C~ and a I are c o m p u t e d as follows: 1
M
~ - i=~l log e A i -- log~ D t 6't = M E Bi/M
(27)
i=1
and M
Z (B,~t =
l
i=1
(28)
log e A i
1/M i
where M
li~ = Z Bi/M i=1
(29)
Nonlinear damage process
95
~t (a)
o-
.......
o -0' {t 0uo
u o
6
[,
0.1
o1
¢,
I
,
<,
i
,,
.,
~u . o
c ~
, -
>
:
~J
~
1 = 0.01 2 = 0'02 3= 0'03 4 = 5 = 6 = 7 = 8 = 9=
0.04 0.05 006 0.07 0.08 0.09
'~" ' ~
id 16,
% .q
~,
,c,
,a
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r-
r,
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(o .
.',r
r,
r~
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•
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8
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°
8 o
8
°
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8
°
8
°
8
~
8
o ,~ o oo o,,
o
C# = I o g e TM
Fig. 16(a).
Dependence of coefficient of variation (COV) on median life TM.
Anwar Khalil Sheikh, Munir Ahmad
96
i,
,]
i
¢druC~l'~[J[JCdc;
1 = 0"10 2 = 0"11 3 = 0.12 4 = 0'13 5 = 0.14 ~r~
1'i
~.
~'¢
O-
0..
C:,~-
~
{'~
C',t
~
~"
r--
8 = 0.17 9 = 0.18
,0 ',0
03
r"
",O
O-
r'
6 = 0-15 7 = 0"16
r,,.
O,
u~
IZl
CO
'~
~.)
-,
o
r'-
,,u
O
0
~0
~3
r-
e,,.
(~
8:
*1
> 0 u
rl
1
8 o 8 o
8 .o, 8 . ,o o o o o o. o . .o. °. o . . o. o o o o o 8 log e TM =
Fig. 16(b).
Cl
Dependence of coefficient of variation (COV) on median life TM.
Nonlinear damage process
97
4.2.3 Main statistical characteristics of the rnodel f(t; Cl, ~t) The median life is Tu = exp [Ct]. The mode can be found by iterative method as a solution of (df(t)/dt)=O. The quantile life tq is obtained from P I T < tq] = F(t~) = q, and for small values of x/~t (<0-2) we have
c, t~ = exp {1 _ x/~tzq }
(30)
where z~ is the qth quantile of standardized and normal variate found from the condition ~[zq] = q. The exact values of mean and variance of T are difficult to obtain mathematically. However, fairly good approximations of the mean and variance of life T for small value of ~ (i.e. x/~t < 0.2) are given by the following expressions: MEAN: VARIANCE:
{
T = exp Ct 1 + Ctctt ~ + 1 tr 2 = C/2~qexp {2Ct}
(31) (32)
The coefficient of variation of life is Ct KT = 1 + Ct~t(1 + CJ2)
(33)
We note that coefficient of variation is dependent upon not only the shape parameter w/~t but also on the location parameter Ct = loge TM. Thus K r will vary with respect to the median life. This behavior of COV of life makes our proposed model distinctly different from several other models such as lognormal Weibull, etc., where COV is the function of shape parameter alone and is independent of median life. Plots of Kr versus Ct for various values of x/~t are given in Figs 16a and 16b.
5 CONCLUSIONS (1) A new three-parameter probability model, f(t;Ct, at, flt), is proposed which describes the probability density of time to first failure of the product degrading according to nonlinear scheme of damage expressed by a stochastic equation of the type D(t) = At B, and the life criterion is a critical level of damage D v (2) Two special cases of the proposed model f(t; Ct, o~l,fit) are explored. In the first case, when at -- 0, the model reduces tof(t; Ct, fl,), which is shown to be a lognormal model. In the second case, when fit = 0, the model reduces to a two-parameter model, f(t; Ct, at), which is found a very
98
Anwar Khalil Sheikh, Munir Ahrnad
useful model in several damage processes such as fatigue and wear processes. (3) F o r the two-parameter model f(t;C~,~t) various statistical characteristics, such as mean, variance, median and coefficient of variation, are developed. It is pointed out that a distinct feature of this life model is the dependency of the coefficient of variation on the median life of the product. The increasing-decreasing rate of the hazard function and its appropriateness in fatigue and wear-related life modelling is pointed out. (4) Methods of estimating the parameters ~t, C~, etc., from the life data, as well as from the sample of damage functions, are also developed.
6 ACKNOWLEDGEMENT The support provided by the University of Petroleum and Minerals ( U P M grant No. ME/Modelling/46) is gratefully acknowledged. The authors wish to thank M r Jawad Qureshi for careful preparation of the manuscript.
REFERENCES 1. Gertsbakh, I. B. and Kordonsky, Kh. B. Models of Failure, Springer-Verlag, New York, 1969. 2. Hitomi, K., Nakamura, N. and Inoue, S. Reliability analysis of cutting tools, ASME Paper 78- WA/PROD-9, 1978. 3. Kitagawa, H., Takahashi, S., Suh, C. M. and Miyashita, S. Quantitative analysis of fatigue process-microcracks and slip lines under cyclic strains, Fatigue Mechanisms, ASTM STP 675 (1979), pp. 420-49. 4. Field, M. Relation of microstructure to the machinability of wrought steels and cast iron, International Research in Production Engineering, 1963. 5. Duncon, J. Mathematical and numerical methods for scheduling cutting tool replacements and applications. PhD thesis, The University of Michigan, 1969. 6. Rhodes, J. S. Transfer machine operational analysis and cutting tool change policies, The Charles Draper Laboratory Report, November 1973. 7. Sheikh, A. K., Ahmad, M. and Younas, M. Stochastic nature of crack propagation and its implications in fatigue life prediction, ISTFA/82 Eighth Annual International Symposium for Testing and Failure Analysis, San Jos6, California, 25-27 October 1982. 8. Weibull, W. The Statistical Aspect of Fatigue Failures and its Consequences in Fatigue and Fracture of Metals, W. M. Murray (ed.), John Wiley & Sons, New York, 1952. 9. Shuaib, A. N., Mohiuddin, A. and ;Sheikh, A. K. Designer's approach to the selection of reliability models in fatigue life investigations, ASME paper No. 84- WA/DE-12.
99
Nonlinear damage process
A P P E N D I X : M A X I M U M L I K E L I H O O D OF T H E P A R A M E T E R S Consider the probability density function at (21)--when fl~ = 0. The loglikelihood function of a random sample of size n from the population is lnti-
ln L = n ln C, - 21n et - 2 i=1
1
2 2( ti - ~-~l
i=1
i=1
The estimating equations t~lnL ----0
and
~lnL ---0
give P1
°¢1= n
n
1-~/] i=1
and
C'l = n
Cl "~2
1-1-~i
(ln ti)-I i=1
~'t is the harmonic mean of logarithms of the observations.
}