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Evaluation of structural creep reliability using entropy principles
101
Materials Science and Engineering, 12 (1973) 101-109 ,© Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Evaluation of Structural Creep Reliability using Entropy Principles A. B. O. SOBOYEJO Faculty of Engineering, University of Lagos, Lagos (Nigeria) (Received October 10, 1972)
Summary* The principle of maximum entropy is used to obtain the prior probability distribution functions for critical structural creep strain and structural creep rupture at known high temperatures and uniaxial stresses. From the prior distribution function obtained, reliability function which is simply the probability of successful operation, can be derived for specified critical creep-strain and creep-rupture modes of failure. An attempt is made to derive the reliability functions from prior considerations of the mechanics of failure, and the mechanical and physical characteristics of engineering materials. This work assumes that mechanical creep design reliability functions for creep-rupture and critical creep-strain modes of structural elements can have values such that the failure of the elements can occur either by any of the modes of failure or by the assumed combined modes of failure. It is also pointed out that the prior probability distribution functions from which the reliability functions are derived can be improved by the use of Bayes' theorem in order to obtain a posterior probability distribution function, whenever more creep data are made available. The posterior probability distribution functions can then be used to derive more accurate reliability functions. Finally, these considerations and procedures which yield the reliability design criteria are illustrated by an application to the stress analysis of a structural member, with given mechanical, physical, environmental and creep properties.
for structural parts, components and systems from prior considerations of the physical characteristics of the materials and the anticipated loads and environmental conditions. The numerical value of the reliability function is the probability of successful operation as a function of time, as reliability is usually defined. For the proposed method, test data will be used primarily to verify the analysis and to obtain the values of the pertinent constants in the analysis, thus making more efficient use of the testing efforts. Maximum entropy principle is utilized to show that the distribution functions obtained for the structural creep failure at high temperatures and uniaxial stresses give the minimally biased probability distribution functions, consistent with the given information about the material characteristics concerning the creep rupture modes of failure of the materials. The probability distribution functions obtained for the mode of failure can be regarded as the prior distribution functions ; and Bayes' theorem can be used to improve the above prior results, in order to obtain better posterior distribution functions, whenever more data are made available from further experimental results or testing. From the probabilistic model developed, the reliability functions for the creep rupture modes of failure are also developed. It will be seen that material physical characteristics have a profound influence on the reliability design criteria for any engineering material.
II. USE OF M A X I M U M ENTROPY PRINCIPLE I. I N T R O D U C T I O N
This paper represents an attempt to establish a formal procedure for deriving reliability functions
The method proposed by Shannon 1 for measuring the informational content in a message is as follows : H --- - K ~ Pi log Pi
* R6sum6 en frangais fi la fin de l'article. Deutsche Zusammenfassung am SchluB des Artikels.
where H = entropy or uncertainty
(1)
102
A . B. O . S O B O Y E J O
K = arbitrary constant
Differentiating eqns. (4) and (5), we get
Pi Ai =i-th possible outcome of an event x = knowledge about the occurrence o fan event x. Equation (1) has been used for a long time in statistical mechanics. Boltzmann made the statement that "the entropy of a system is a measure of our ignoranc e as to its true state". This concept was never seriously used until Shannon showed that the expression used for the entropy has the mathematical properties needed to make it a reasonable measure of uncertainty. Jaynes 2 has formulated the principle of maximum entropy. According to this principle, the minimally prejudiced assignment of probabilities is that which maximizes the entropy subject to the given information 3. In many problems, the prior knowledge or the information is in the form of averages. If this is the case, the mathematical procedure takes a very simple form. Let us consider an example in which, say, ~ is the mean rupture time of a structural component operating under known temperature and uniaxial stress• If the information available is the mean value t above, then -- E [t] = expected value of creep rupture time. (2) Consistent with this information, we wish to determine the prior probability distribution of t. Let ti be the creep rupture time such that if we take a small increment in t up to any time, then
ti=iAt.
dP~ = 0
(9)
E ti dPi = 0.
(10)
i
and i
Multiplying eqns. (9) and (10) by Lagrangian multipliers (a o - 1) and al, we get (a o - 1) ~ d P i = 0 (11) i
and al~tidP i = 0 . •Adding eqns. (8), (1 I) and (12), we get (log -Pl+ ao + al ti) dPi = 0 i
whence
Pi=exp[-ao-alti]. (13) Equation (13) gives the minimally biased probability distribution of t, consistent with the information that the mean value is known. The values of a o and a 1 can be found as follows: From eqn. (13), we have the result: ~,Pi = ~ e x p [ - a o - a l i
Z P [tiff] = 1.0
this gives a o = log E e x p ( - a 1 ti).
(14)
i
Now - E ti e x p ( - a , ti) 0do
~_-
aal
i
~ e x p ( - al ti) i
= - exp ( - do)~
(4)
ti
exp(-
a1
ti)
i
i
and Z P [ti/[] ti = 1"
= --EtiPi i
(5)
= - t.
i
According to Jaynes' principle, we wish to maximize H under the two constraints given by the eqns. (4) and (5). Note that if
P, = P [tfl]
(15)
From eqn. (14) ao = log ~ exp ( - al ti) i
(6)
= log
e x p ( - a l t ) d t = log
1
3o
then -H K
ti] = 1.0 ;
i
(3)
Then we can write our information in the following form :
(12)
i
•. -
Z
i
Pi log Pi
d(~-H-) = ~ ( e + l o g P i ) d P i = 0 . i
(7)
(8)
--ao=logaa
gao Oai I .. a, = ~-
- 1 al (16)
EVALUATION
OF STRUCTURAL
and ao = log f. Substituting eqns. (16) and (15) in eqn. (13), we get
Pi=
f,,,,_- oxp{_ expl-" exp (-
-
/ for' 0
= 0,
As can be seen from eqn. (17), it is an exponential
type of distribution. Thus the maximum entropy principle says that the exponential distribution is the minimally biased distribution when only one parameter of the distribution is known.
Ill. S T R U C T U R A L
CREEP RUPTURE 4
for t < 0 .(19b)
The mean and variance of the distribution shown in either eqn. (19a) or eqn. (19b) are given by ~ and ~2 respectively. The reliability function for the structural creep rupture time is given by (see Fig. 1)
1.0
It can be shown that the scatter s'6 observed for creep rupture time, t, of structural elements operating at known temperature and uniaxial stress appears to be an inherent characteristic of the physical nature of the material itself, associated with a Markov process, or dynamical statistical process, as a rate process. From the results of this study, it appears that the mean rupture time ~ may be approximately related by 7 #=f
103
CREEP RELIABILITY
1 coxp
I 0.8 O.B
0.4
0.2
-~ C exp -
(18)
0
I
0
1.0
I
2.0
t___ 3.0
I 5.0
4.0
t
where R is gas constant; AU can be regarded as the activation energy for self diffusion; T is absolute temperature; C is a constant which depends on other variables such as the uniaxial sustained stress a applied to the material; # is the rate function for the process, which is assumed to be independent of time; and e is a constant for the problem. If the mean rupture time, ~, is the only prior information available, then the minimally biased distribution function which is consistent with this information is the exponential distribution. Therefore, using eqns. (17) and (18), the distribution function for creep rupture time under the conditions of known temperature and known uniaxial stress can be expressed as fl(t)=}-exp =0
-
,
for
t>10
,
for
t<0
which is the same expression as
Fig. I. Reliability function, R 1 (t) (see eqn. (20)).
R 1(t) = 1 - probability of failure =1-
f
oteXp
= ~f?exp {-~/dt
exp/ i1 =exp{-t'Cexp
IV. C R I T I C A L S T R U C T U R A L
(19a)
~ dt
(-~T1},
CREEP
t> 0. (2o/
STRAIN
It will be assumed here that over certain ranges of stress and temperature, the critical creep strain of an engineering material can be expressed as 8"
104
A . B . O. SOBOYEJO
=
+ f(rr,, r r 2 ) / ; /
\~/
(21)
where the characteristic time t~ can be identified with the reciprocal of the minimum creep rate Jmi., i.e., to -
1
(22)
= emin
and r~1 and rca are pi-parameters such that 7 : f(n~, re2) = { 1 - 3 ( ~ )
~}.
(23)
For a relatively simple material like pure aluminum, the minimum creep rate 7mi. can be related to the uniaxial stress, a, and the absolute temperature, T, by an Eyring type of reaction rate formula 9 as follows : 7mi n :
~ :
~--~,g ~ J
exp
(24)
where k/h is the ratio of Boltzmann's constant to Planck's constant; AU, R and T are as previously defined; and g (a/E) represents a usually rapidly increasing function of the ratio of uniaxial stress to Young's modulus. The quantity h/kT has the physical dimension of time and is of the order of magnitude of atomic vibration periods. The quantities A U/R T and ale in eqn. (24) are pi-parameters. Unless a and T vary during the course of creep, eqn. (24) indicates that a large variability of t¢ is ordinarily not to be expected. This is because AU and E vary little from one specimen to the next for relatively simple structural materials. Over the practical range of elongations, for some available creep data 10 on Hastelloy R-235 bar stock, eqn. (21) with f(nl, 7~2) computed from eqn. (23) can be fitted to the data aa°'11 with n1=0.08 and rt 2 = 24, that is 1
,(t)=0.08(~1+0.66
-
t
t 3,
+ 2 4
--
t >/0
(25a)
and t~(t) =
0.08 --
3t)
0.66
t-~ +--
t¢
+
24 × 3
t2 ,
t~3 t 20.
(25b)
Equations (25a) and (25b) can be assumed for all
practical purposes as the prior mean values of the critical creep strain rate, since there are no statistical data in ref. 10. Moreover, in the absence of statistical data, a distribution of time to reach a critical creep strain is assumed directly instead of a variability in nl and/or 7~2 of eqn. (21). Since eqns. (25a) and (25b) give the only set of prior information available, then the minimally biased distribution function consistent with this information, for the critical creep strain mode of failure, can be expressed as (see eqn. (17)) f2 (t) = 7(r) exp [ - 7 ( z ) . t ] , = 0
t >/0 t < 0,
(26)
where t/to = z is the time ratio corresponding to the critical creep strain rate 7(0 in eqn. (25b). Furthermore, the reliability function for the structural material's critical creep strain mode of failure can be expressed as R2 (t) = 1 - probability of failure = -
J(z) e x p [ - ~ ( z ) t ] d t o
'
= •
l
/
~(z) exp [ - ~ ( z ) t]dt t
=exp{-~(z)t},
t >0.
(27)
V. USE OF BAYES" T H E O R E M 12
Maximum entropy principle yields the prior probability density function; and when more data become available, the results of data can be combined with the prior results to yield the posterior probability distribution which is a more reliable distribution than the prior distribution. In the modem formulation of the decision rule 13, we normally have four sets. (1) Experiment E, (2) Result of Experiment Z, (3) Action A, and (4) State of Nature 0. Moves (2) and (4) are situations which are determined by' chance, while moves (1) and (3) are determined by the decision maker. After all the four moves, the decision maker receives utility u(E, Z A, 0). The optimum decision is that which maximizes his utility. Hence in looking at this problena, we can introduce the Bayes' theorem to give the posterior distribution function. P [0/Z, E] =
P' (0) P (Z/0, E) f ~i10(0)P(Z/0, E)d0
(28)
EVALUATION OF STRUCTURAL CREEP RELIABILITY w h e r e p l ( 0 ) is the prior distribution function and
P (Z/0, i~) is the sampling probability distribution, which can be obtained from more data. Equation (28) can now be used as a means of obtaining more accurate reliability functions for the structural creep rupture mode of failure and the critical creep strain mode of failure for the structural material. For example, Bayes' theorem can be applied to the prior probability distribution functions for the structural creep rupture time and critical structural creep strain respectively. For structural creep rupture time, the posterior probability distribution function is fl(t)
=
~
fl (t) gl (t)
t >~0
(29a)
f fl(t)gl(t)dt •
0
and for critical structural creep strain, the posterior probability distribution function is
105
reliability functions can be derived by the use of Bayes' theorem, as outlined above.
VII. PHYSICAL INTERPRETATION
The phenomenon of structural creep rupture appears that it can be interpreted as a Markov process, which is a dynamical stochastic process. Let us denote the probability that structural creep rupture will initiate in the next interval of time dr, provided that it has not initiated already up to time t, as/~ (t)dt, where # (t) is the rate function of the process. If the probability of successful operation, or the reliability, of the structural element is R1 (t), that is the probability that structural creep has not initiated is R1 (t); then the probability that it will initiate between t and t + dt is given by R, (t) # (t)dt = - dR 1 (t)
(3 la)
and from which ~2(t) =
f2(t)g2(t)
t >/0
(29b)
)o f2(t)g2(t)dt The functions g~ (t) and gz (t) (t ~0), are estimates of the probability distribution functions obtained from data, for structural creep rupture time and critical structural creep strain respectively. Equations (29a) and (29b) obtained above can now be used to derive posterior reliability functions for structural creep rupture time and critical structural creep strain modes of failure.
VI. COMBINED MODES OF STRUCTURAL FAILURE
The combined reliability function R12 (t), is given as
R , a ( t ) = R l ( t ) R 2 ( t ),
t~0
(30)
if there is no significant correlation between strain rate and rupture time, that is if the two individual modes of failure can be considered to be independent of each other. It is now possible with the use of the above developments to incorporate a certain reliability into a device or to assess the reliability of an intended design in which a critical creep-strain value or creeprupture is the predominant failure mode. The results given by eqn. (30) can be regarded as the 'prior' reliability function Rx2(t), for the combined strain-rupture modes of failure; and posterior
dRl (t) d ~ + # ( t ) ' R l ( t ) = 0.
(31b)
The reliability can be obtained approximately by the relation
R 1(t)= 1
V(t) U(t) + 1"
(32)
Equation (32) is the mean frequency of the V(t) th of N(t)th observed structural creep rupture time data. Figures 2a, 2b and 2c show the experimental results obtained for structural creep rupture time for identical copper wires, under known constant uniaxial stress conditions and known environmental temperatures. Assuming that the rate function #(t) is time independent, the integration of eqn. (31)gives R, (t) = exp { - # t } ,
t >/0,
(33a)
and considering eqn. (18), it can be shown that Rl(t)=exp (-t),
t>~0,
(33b)
which is the same physical result as that in eqn. (20), and from which In'Rl(t)=-#t=-
tt,
t>~0.
(34)
From the graph of In" R 1(t) against t, the negative of the slope enables # = 1/t to be determined (see eqn. (18)). This enables a good approximation to be obtained for the mean creep rupture time, ~, under the
106
A. B. O. SOBOYEJO
known condition of uniaxial stress and environmental temperature, T. From expression (18) it can be shown that AU I n - # = In- C - R-T'
(35)
and using the experimental data of the actual environmental temperatures T and the corresponding values of # obtained from experimental data, a plot of In .# against 1/T can be made for different stress levels, see Fig. 3 ; and since R is a constant which is the gas constant, the values of AU which is the actime,
Rupture
Rupture
2 O'
4
I
time, t (min) 6 8
I
I
I
0
~-
200
0
400
t (rain) 600
800
1000
10 I - 0.5
-
0.5
I
-1.0
-1.0
-1.s -1.: -2.0
-2.c -2.5
-2.5 Q
-3.0
Fig. 2a. Reliability function, R~ (t) (see eqn. {20)). Initial uniaxial stress, e,=24.5 kg/mm 2. Environmental temperature, 40~C or 104°F.
c
Fig. 2c. Reliability function, R 1(t) (see eqn. (20)). Initial uniaxial stress, ¢,=24.5 kg/mm 2. Environmental temperature, 8°C or 46.4°F. of oDsolute temperature, .1.¢T)xlO 4
Reciprocal time,
Rupture
0
50 i
[
i
t(mln) 100 I
15
.
17
18
19
I
I
I
150
initial
I
°I -0
16
unlaxial
I
stress,
,
5 °
~ 0"= 24.00 kg/mm z
1-2 ::k
-1.O
~In~tiat uniax[al stress, v\ ~S"g/mm~
-3
-4
-2.O -5
-E
-3.¢
20
b
Fig. 2b. Reliability function, R 1(t) (see eqn. (20)). Initial uniaxial stress, ~,=24.5 kg/mm 2. Environmental temperature, 20~C or 68° F.
Fig. 3. In. # = In' C -
AU
(eqn. (35)).
EVALUATION OF STRUCTURAL CREEP RELIABILITY
tivation energy for self diffusion can be obtained from the tangents to the lines in Fig. 3. These results for AU are finally plotted against the corresponding uniaxial stresses, as shown in Fig. 4. It appears that there is a linearity between activation energy, AU, and uniaxial stress, ~. That is, it appears that AU = ~a
(36)
107
t
.a c t~
- 1.5
g E
1.0
where ~ is a constant for the problem. Therefore, eqn. (18) can be expressed as =
~- C e x p
-
(37)
2.0
u
0.5
g z;
for this problem. 0
t
0.1
0.2
t__
0.3
I
0.4
0.5
Fig. 5. Mean creep ~- vs. dimensionless time (fit°) for Hastelloy R 235 bar stock, 4.(3 100
~0 3.0
80 .= 6 0
2.0 •
o
,-4
g "~ >
P
1.(3
40 30 1502 20
<
x
+J
o
24
I I I I 24.2 24.4 24.6 24.8 l : n i t i o t uniaxial s t r e s s , d ( k g / m m z)
I 25
Fig. 4. Relationship between activation energy AU and uniaxial stress, a.
le5O° :~, 101 8
61
Villi. N U M E R I C A L E X A M P L E
It is required to make a prior estimation of the creep rupture time and the initial allowable uniaxial tensile stress in a structural member made of Hastelloy R-235 bar material, for a given maximum creep elongation of 1.0 % while operating at a constant temperature of 1500° F, for a combined structural reliability of at least 90 %. Assume that the material is kept in operation for at least 200 h.
1
1 2
I
5 10 100 1000 Mean r u p t u r e t i m e ( { ) h
Fig. 6. Curves of initial stress vs. mean rupture time for Hastelloy R 235 bar material at various temperatures 6.
(2) Assuming that t = 200 h, therefore tc = 625 h, since t/tc=0.32 at 1.0% elongation from Fig. 3. (3) The specified combined structural reliability is R12 (t) = R, (t)" R 2 (t)
+
IX. S O L U T I O N
(1) The mean creep elongation of 1.0 % is reached when t/t~ = 0.32 and at this instant ~(t) = 0.011/0.250 t¢ per hour, see Fig. 5 and eqns. (25a) and (25b).
10000
(see eqn. (30)) hence -
---- + 0.230 x
-~ 0.10
'/--090
108
whence the creep rupture time [ can be obtained as [ = 2360 h. (4) From Fig. 6, since the mean structural creep rupture time [=2360 h, and the temperature is 1500° F, therefore the initial allowable uniaxial stress is given as 20,000 psi.
X. CONCLUSION
It is imporant to summarize the essential steps developed in this paper, so as to make the method more meaningful and easy to handle for practical engineering problems. These steps are as follows: (1) Estimate the necessary value for the mean rupture [ for the structural material, using expression (18) for the rate function of the process, or any other suitable expression for the rate function. (2) Obtain the prior probability distribution function ft (t) for the creep rupture mode of failure of the structural material, from eqn. (19); and in the absence of data obtain the prior reliability function Rx (t), for the creep rupture mode of failure of the structural material, from eqn. (20). It should be pointed out again that the prior distribution function fl (t) given by eqn. (19) is the minimally biased distribution function, consistent with the initial information about the value of the estimated mean rupture time L Expression (19) is derived from the maximization of the entropy expression. (3) Estimate by any suitable method the expression for the critical creep strain rate of the material, at the desired temperature and uniaxial stress conditions. Such an expression, e.g. eqn. (25b), can be regarded as the prior mean value expression, for most practical purposes. (4) Obtain the prior probability distribution function f2(t) for the critical creep strain mode failure of the structural material, from eqn. (26); and in the absence of data, obtain the prior reliability function R2(t), for the critical creep strain mode of failure of the structural material, from eqn. (27). In this case, as shown in step (2), the prior distribution function f2 (t) given by eqn. (26) is the minimally biased distribution consistent With the initial information about the value of the estimated critical creep strain rate. Expression (26) is again derived from the maximization of the entropy expression. (5) Obtain prior estimate of the reliability function R t2 (t)=Rt (t). R2 (t), for the combined strain-
A. B. O. SOBOYEJO
rupture modes of failure of the structural material, under the given conditions of temperature and uniaxial stress. (6) When more data become available, the above prior distribution functions fl (t) and t"2(t) carl be improved by means of Bayes' theorem, to yield the posterior probability distribution functions, fl (t) and f2 (t), given by eqns. (29a) and (29b), for creep rupture and critical creep strain modes of failure respectively. The posterior distribution functions now obtained can be used to derive posterior reliability functions for creep rupture, critical creep strain and combined creep strain-rupture modes of failure.
ACKNOWLEDGEMENTS
The facilities provided by Towne School of Civil and Mechanical Engineering of the University of Pennsylvania in Philadelphia, Ford Motor Company in Michigan, and the Faculty of Engineering of the University of Lagos, Nigeria, in order to carry out this work, are all thankfully acknowledged.
REFERENCES 1 C. E. Shannon, Bell. System Tech. J., July and Oct. (1948). 2 E. T. Jaynes, Probability Theory in Science and Engineering, McGraw-Hill, New York, 1961. 3 H. C. Shah, Principle of maximum entropy and its applications in reliability estimation of aircraft structures. Final Rept. to NAEC under Contract No. N156-45588 and to be presented at the AIAA/ASME Structures, Structural Dynamics, and Mater., Conf., Palm Springs, Calif., March, 1967. 4 J. Hult, Creep in Engineering Structures, Blaisdell Publishing Co., 1966. 5 S. Taira and R. Koterawaza, Japan Soc. Mech. Eng., 14 (1961) 238. 6 C. W. Phillips and J. J. Sinnott, Trans. Am, Soc. Metals, 46 (1954) 63. 7 I. Finnie and W. R. Heller, Creep of Engineering Materials, McGraw-Hill, New York, 1959. 8 R. Simon, J. Basic Eng., March (1966) 87. 9 H. Eyring, J. Chem. Phys., 4 (1936) 283. 10 J. G. McBride, B. Mulhern and R. Widmer, New England Mater. Lab., Medfor, Mass., Tech. Doc. Rept. No. WADDTR-61-199, prepared under Contract No. AF 33(616)-6200, August, 1962. 11 R. Mesloh, Ann. Reliability and Maintainability, 5 (1966) 590. 12 E. Parzen, Modern Probability Theory and Its Applications, Wiley, New York, 1960 p. 119. 13 H. Raiffa and R. Schlaifer, Lecture Notes, Harvard Graduate School of Business, Harvard Univ., Cambridge, Mass.
EVALUATION OF STRUCTURAL CREEP RELIABILITY
Utilisation de la notion d'entropie pour lYvaluation des risques de rupture par fluage dans les structures
L'auteur se sert du principe de l'entropie maximale pour d&erminer de premirres fonctions de distribution de la drformation structurale critique de fluage et du temps de rupture, dans le cas de structures soumises ~t des contraintes uniaxiales connues, /~ des temprratures 61ev~es connues. A partir de la premirre fonction de distribution on peut drduire la fonction de confiance, qui est simplement la probabilit6 d'un fonctionnement sans incident. Cette fonction est calculre pour des drformations critiques ~t rupture et des conditions de contrainte et de durre de vie bien sp~cifires. L'auteur cherche h 6tablir les fonctions de confiance/l l'aide de considrrations antrrieures sur la mrcanique de la rupture et des proprirtrs physiques et mrcaniques des matrriaux industriels. Dans son travail il suppose que les fonctions de confiance, destinres au calcul des structures devant rrsister au fluage et drfinies aussi bien ~ l'aide de la dfformation que du temps de rupture, peuvent prendre des valeurs telles que la rupture des pirces peut survenir par l'un ou l'autre des mrcanismes envisagrs ou par toute combinaison de ces mrcanismes. I1 signale 6galement que les premirres fonctions de distribution, it partir desquelles les fonctions de confiance sont dbduites, peuvent 6tre amrliorres grace au throrrme de Bayes. On peut ainsi chercher /l obtenir une deuxirme fonction de distribution qui tient compte des nouvelles donnres de fluage rendues disponibles. Les fonctions de distribution secondaires permettront de d~terminer les fonctions de confiance avec plus de prrcision. En fin de compte, l'auteur illustre ses consid+rations et ses mrthodes relatives aux critrres de dessin des pirces, en les appliquant ~ l'analyse des contraintes d'un 616ment de structure ayant des proprirt~s physiques et mrcaniques, des caractrristiques de fluage et un environment donnrs.
109
Eine Bestimmung der Verlgifllichkeit yon Proben in Kriechversuchen mit Hilfe yon Entropiebetrachtungen
Mit Hilfe des Prinzips maximaler Entropie werden die friiheren Wahrscheinlichkeitsverteilungen der kritischen Kriechdehnung und des Kriechbruchs bei bekannten hohen Temperaturen und einachsigen Spannungen gewonnen. Aus diesen Verteilungsfunktionen werden VerlaBlichkeitsfunktionen (die Wahrscheinlichkeit fiir einen erfolgreichen ProzeB) fiir spezielle kritische Kriechdehnungs- und Kriechbruchmoden abgeleitet. Es wird der Versuch untemommen, die VerlaBlichkeitsfunktionen aus vorhergehenden Betrachtungen der Bruchmechanik und der mechanischen und physikalischen Eigenschaften des Materials abzuleiten. In der vorliegenden Arbeit wird angenommen, dab VerlaBlichkeitsfunktionen fiir Kriechbruch und kritische Kriechdehnungsmoden der Strukturelemente Werte annehmen k6nnen, die es erlauben, dab der Bruch tier Elemente entweder durch jeden der Bruchmoden oder durch die Bruchmoden gemeinsam erfolgen kann. Es wird auBerdem darauf hingewiesen, dab friihere Wahrscheinlichkeitsverteilungen, aus denen VerlaBlichkeitsfunktionen abgeleitet wurden, mit Hilfe des Bayes-Theorems verbessert werden k6nnen, um immer dann neuere Wahrscheinlichkeitsverteilungen abzuleiten, wenn neue Kriechdaten vorliegen. Aus den neuen Verteilungsfunktionen k6nnen dann genauere Verl~iBlichkeitsfunktionen abgeleitet werden. SchlieBlich werden diese Betrachtungen und Verfahren, die VerlaBlichkeitskriterien liefern, anhand einer Anwendung auf die Spannungsanalyse einer Struktur mit gegebenen mechanischen, physikalischen und Kriecheigenschaften bei gegebenen ~iuBeren Einfltissen illustriert.
Materials Science and Engineering, 12 (1973) 101-109 ,© Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Evaluation of Structural Creep Reliability using Entropy Principles A. B. O. SOBOYEJO Faculty of Engineering, University of Lagos, Lagos (Nigeria) (Received October 10, 1972)
Summary* The principle of maximum entropy is used to obtain the prior probability distribution functions for critical structural creep strain and structural creep rupture at known high temperatures and uniaxial stresses. From the prior distribution function obtained, reliability function which is simply the probability of successful operation, can be derived for specified critical creep-strain and creep-rupture modes of failure. An attempt is made to derive the reliability functions from prior considerations of the mechanics of failure, and the mechanical and physical characteristics of engineering materials. This work assumes that mechanical creep design reliability functions for creep-rupture and critical creep-strain modes of structural elements can have values such that the failure of the elements can occur either by any of the modes of failure or by the assumed combined modes of failure. It is also pointed out that the prior probability distribution functions from which the reliability functions are derived can be improved by the use of Bayes' theorem in order to obtain a posterior probability distribution function, whenever more creep data are made available. The posterior probability distribution functions can then be used to derive more accurate reliability functions. Finally, these considerations and procedures which yield the reliability design criteria are illustrated by an application to the stress analysis of a structural member, with given mechanical, physical, environmental and creep properties.
for structural parts, components and systems from prior considerations of the physical characteristics of the materials and the anticipated loads and environmental conditions. The numerical value of the reliability function is the probability of successful operation as a function of time, as reliability is usually defined. For the proposed method, test data will be used primarily to verify the analysis and to obtain the values of the pertinent constants in the analysis, thus making more efficient use of the testing efforts. Maximum entropy principle is utilized to show that the distribution functions obtained for the structural creep failure at high temperatures and uniaxial stresses give the minimally biased probability distribution functions, consistent with the given information about the material characteristics concerning the creep rupture modes of failure of the materials. The probability distribution functions obtained for the mode of failure can be regarded as the prior distribution functions ; and Bayes' theorem can be used to improve the above prior results, in order to obtain better posterior distribution functions, whenever more data are made available from further experimental results or testing. From the probabilistic model developed, the reliability functions for the creep rupture modes of failure are also developed. It will be seen that material physical characteristics have a profound influence on the reliability design criteria for any engineering material.
II. USE OF M A X I M U M ENTROPY PRINCIPLE I. I N T R O D U C T I O N
This paper represents an attempt to establish a formal procedure for deriving reliability functions
The method proposed by Shannon 1 for measuring the informational content in a message is as follows : H --- - K ~ Pi log Pi
* R6sum6 en frangais fi la fin de l'article. Deutsche Zusammenfassung am SchluB des Artikels.
where H = entropy or uncertainty
(1)
102
A . B. O . S O B O Y E J O
K = arbitrary constant
Differentiating eqns. (4) and (5), we get
Pi Ai =i-th possible outcome of an event x = knowledge about the occurrence o fan event x. Equation (1) has been used for a long time in statistical mechanics. Boltzmann made the statement that "the entropy of a system is a measure of our ignoranc e as to its true state". This concept was never seriously used until Shannon showed that the expression used for the entropy has the mathematical properties needed to make it a reasonable measure of uncertainty. Jaynes 2 has formulated the principle of maximum entropy. According to this principle, the minimally prejudiced assignment of probabilities is that which maximizes the entropy subject to the given information 3. In many problems, the prior knowledge or the information is in the form of averages. If this is the case, the mathematical procedure takes a very simple form. Let us consider an example in which, say, ~ is the mean rupture time of a structural component operating under known temperature and uniaxial stress• If the information available is the mean value t above, then -- E [t] = expected value of creep rupture time. (2) Consistent with this information, we wish to determine the prior probability distribution of t. Let ti be the creep rupture time such that if we take a small increment in t up to any time, then
ti=iAt.
dP~ = 0
(9)
E ti dPi = 0.
(10)
i
and i
Multiplying eqns. (9) and (10) by Lagrangian multipliers (a o - 1) and al, we get (a o - 1) ~ d P i = 0 (11) i
and al~tidP i = 0 . •Adding eqns. (8), (1 I) and (12), we get (log -Pl+ ao + al ti) dPi = 0 i
whence
Pi=exp[-ao-alti]. (13) Equation (13) gives the minimally biased probability distribution of t, consistent with the information that the mean value is known. The values of a o and a 1 can be found as follows: From eqn. (13), we have the result: ~,Pi = ~ e x p [ - a o - a l i
Z P [tiff] = 1.0
this gives a o = log E e x p ( - a 1 ti).
(14)
i
Now - E ti e x p ( - a , ti) 0do
~_-
aal
i
~ e x p ( - al ti) i
= - exp ( - do)~
(4)
ti
exp(-
a1
ti)
i
i
and Z P [ti/[] ti = 1"
= --EtiPi i
(5)
= - t.
i
According to Jaynes' principle, we wish to maximize H under the two constraints given by the eqns. (4) and (5). Note that if
P, = P [tfl]
(15)
From eqn. (14) ao = log ~ exp ( - al ti) i
(6)
= log
e x p ( - a l t ) d t = log
1
3o
then -H K
ti] = 1.0 ;
i
(3)
Then we can write our information in the following form :
(12)
i
•. -
Z
i
Pi log Pi
d(~-H-) = ~ ( e + l o g P i ) d P i = 0 . i
(7)
(8)
--ao=logaa
gao Oai I .. a, = ~-
- 1 al (16)
EVALUATION
OF STRUCTURAL
and ao = log f. Substituting eqns. (16) and (15) in eqn. (13), we get
Pi=
f,,,,_- oxp{_ expl-" exp (-
-
/ for' 0
= 0,
As can be seen from eqn. (17), it is an exponential
type of distribution. Thus the maximum entropy principle says that the exponential distribution is the minimally biased distribution when only one parameter of the distribution is known.
Ill. S T R U C T U R A L
CREEP RUPTURE 4
for t < 0 .(19b)
The mean and variance of the distribution shown in either eqn. (19a) or eqn. (19b) are given by ~ and ~2 respectively. The reliability function for the structural creep rupture time is given by (see Fig. 1)
1.0
It can be shown that the scatter s'6 observed for creep rupture time, t, of structural elements operating at known temperature and uniaxial stress appears to be an inherent characteristic of the physical nature of the material itself, associated with a Markov process, or dynamical statistical process, as a rate process. From the results of this study, it appears that the mean rupture time ~ may be approximately related by 7 #=f
103
CREEP RELIABILITY
1 coxp
I 0.8 O.B
0.4
0.2
-~ C exp -
(18)
0
I
0
1.0
I
2.0
t___ 3.0
I 5.0
4.0
t
where R is gas constant; AU can be regarded as the activation energy for self diffusion; T is absolute temperature; C is a constant which depends on other variables such as the uniaxial sustained stress a applied to the material; # is the rate function for the process, which is assumed to be independent of time; and e is a constant for the problem. If the mean rupture time, ~, is the only prior information available, then the minimally biased distribution function which is consistent with this information is the exponential distribution. Therefore, using eqns. (17) and (18), the distribution function for creep rupture time under the conditions of known temperature and known uniaxial stress can be expressed as fl(t)=}-exp =0
-
,
for
t>10
,
for
t<0
which is the same expression as
Fig. I. Reliability function, R 1 (t) (see eqn. (20)).
R 1(t) = 1 - probability of failure =1-
f
oteXp
= ~f?exp {-~/dt
exp/ i1 =exp{-t'Cexp
IV. C R I T I C A L S T R U C T U R A L
(19a)
~ dt
(-~T1},
CREEP
t> 0. (2o/
STRAIN
It will be assumed here that over certain ranges of stress and temperature, the critical creep strain of an engineering material can be expressed as 8"
104
A . B . O. SOBOYEJO
=
+ f(rr,, r r 2 ) / ; /
\~/
(21)
where the characteristic time t~ can be identified with the reciprocal of the minimum creep rate Jmi., i.e., to -
1
(22)
= emin
and r~1 and rca are pi-parameters such that 7 : f(n~, re2) = { 1 - 3 ( ~ )
~}.
(23)
For a relatively simple material like pure aluminum, the minimum creep rate 7mi. can be related to the uniaxial stress, a, and the absolute temperature, T, by an Eyring type of reaction rate formula 9 as follows : 7mi n :
~ :
~--~,g ~ J
exp
(24)
where k/h is the ratio of Boltzmann's constant to Planck's constant; AU, R and T are as previously defined; and g (a/E) represents a usually rapidly increasing function of the ratio of uniaxial stress to Young's modulus. The quantity h/kT has the physical dimension of time and is of the order of magnitude of atomic vibration periods. The quantities A U/R T and ale in eqn. (24) are pi-parameters. Unless a and T vary during the course of creep, eqn. (24) indicates that a large variability of t¢ is ordinarily not to be expected. This is because AU and E vary little from one specimen to the next for relatively simple structural materials. Over the practical range of elongations, for some available creep data 10 on Hastelloy R-235 bar stock, eqn. (21) with f(nl, 7~2) computed from eqn. (23) can be fitted to the data aa°'11 with n1=0.08 and rt 2 = 24, that is 1
,(t)=0.08(~1+0.66
-
t
t 3,
+ 2 4
--
t >/0
(25a)
and t~(t) =
0.08 --
3t)
0.66
t-~ +--
t¢
+
24 × 3
t2 ,
t~3 t 20.
(25b)
Equations (25a) and (25b) can be assumed for all
practical purposes as the prior mean values of the critical creep strain rate, since there are no statistical data in ref. 10. Moreover, in the absence of statistical data, a distribution of time to reach a critical creep strain is assumed directly instead of a variability in nl and/or 7~2 of eqn. (21). Since eqns. (25a) and (25b) give the only set of prior information available, then the minimally biased distribution function consistent with this information, for the critical creep strain mode of failure, can be expressed as (see eqn. (17)) f2 (t) = 7(r) exp [ - 7 ( z ) . t ] , = 0
t >/0 t < 0,
(26)
where t/to = z is the time ratio corresponding to the critical creep strain rate 7(0 in eqn. (25b). Furthermore, the reliability function for the structural material's critical creep strain mode of failure can be expressed as R2 (t) = 1 - probability of failure = -
J(z) e x p [ - ~ ( z ) t ] d t o
'
= •
l
/
~(z) exp [ - ~ ( z ) t]dt t
=exp{-~(z)t},
t >0.
(27)
V. USE OF BAYES" T H E O R E M 12
Maximum entropy principle yields the prior probability density function; and when more data become available, the results of data can be combined with the prior results to yield the posterior probability distribution which is a more reliable distribution than the prior distribution. In the modem formulation of the decision rule 13, we normally have four sets. (1) Experiment E, (2) Result of Experiment Z, (3) Action A, and (4) State of Nature 0. Moves (2) and (4) are situations which are determined by' chance, while moves (1) and (3) are determined by the decision maker. After all the four moves, the decision maker receives utility u(E, Z A, 0). The optimum decision is that which maximizes his utility. Hence in looking at this problena, we can introduce the Bayes' theorem to give the posterior distribution function. P [0/Z, E] =
P' (0) P (Z/0, E) f ~i10(0)P(Z/0, E)d0
(28)
EVALUATION OF STRUCTURAL CREEP RELIABILITY w h e r e p l ( 0 ) is the prior distribution function and
P (Z/0, i~) is the sampling probability distribution, which can be obtained from more data. Equation (28) can now be used as a means of obtaining more accurate reliability functions for the structural creep rupture mode of failure and the critical creep strain mode of failure for the structural material. For example, Bayes' theorem can be applied to the prior probability distribution functions for the structural creep rupture time and critical structural creep strain respectively. For structural creep rupture time, the posterior probability distribution function is fl(t)
=
~
fl (t) gl (t)
t >~0
(29a)
f fl(t)gl(t)dt •
0
and for critical structural creep strain, the posterior probability distribution function is
105
reliability functions can be derived by the use of Bayes' theorem, as outlined above.
VII. PHYSICAL INTERPRETATION
The phenomenon of structural creep rupture appears that it can be interpreted as a Markov process, which is a dynamical stochastic process. Let us denote the probability that structural creep rupture will initiate in the next interval of time dr, provided that it has not initiated already up to time t, as/~ (t)dt, where # (t) is the rate function of the process. If the probability of successful operation, or the reliability, of the structural element is R1 (t), that is the probability that structural creep has not initiated is R1 (t); then the probability that it will initiate between t and t + dt is given by R, (t) # (t)dt = - dR 1 (t)
(3 la)
and from which ~2(t) =
f2(t)g2(t)
t >/0
(29b)
)o f2(t)g2(t)dt The functions g~ (t) and gz (t) (t ~0), are estimates of the probability distribution functions obtained from data, for structural creep rupture time and critical structural creep strain respectively. Equations (29a) and (29b) obtained above can now be used to derive posterior reliability functions for structural creep rupture time and critical structural creep strain modes of failure.
VI. COMBINED MODES OF STRUCTURAL FAILURE
The combined reliability function R12 (t), is given as
R , a ( t ) = R l ( t ) R 2 ( t ),
t~0
(30)
if there is no significant correlation between strain rate and rupture time, that is if the two individual modes of failure can be considered to be independent of each other. It is now possible with the use of the above developments to incorporate a certain reliability into a device or to assess the reliability of an intended design in which a critical creep-strain value or creeprupture is the predominant failure mode. The results given by eqn. (30) can be regarded as the 'prior' reliability function Rx2(t), for the combined strain-rupture modes of failure; and posterior
dRl (t) d ~ + # ( t ) ' R l ( t ) = 0.
(31b)
The reliability can be obtained approximately by the relation
R 1(t)= 1
V(t) U(t) + 1"
(32)
Equation (32) is the mean frequency of the V(t) th of N(t)th observed structural creep rupture time data. Figures 2a, 2b and 2c show the experimental results obtained for structural creep rupture time for identical copper wires, under known constant uniaxial stress conditions and known environmental temperatures. Assuming that the rate function #(t) is time independent, the integration of eqn. (31)gives R, (t) = exp { - # t } ,
t >/0,
(33a)
and considering eqn. (18), it can be shown that Rl(t)=exp (-t),
t>~0,
(33b)
which is the same physical result as that in eqn. (20), and from which In'Rl(t)=-#t=-
tt,
t>~0.
(34)
From the graph of In" R 1(t) against t, the negative of the slope enables # = 1/t to be determined (see eqn. (18)). This enables a good approximation to be obtained for the mean creep rupture time, ~, under the
106
A. B. O. SOBOYEJO
known condition of uniaxial stress and environmental temperature, T. From expression (18) it can be shown that AU I n - # = In- C - R-T'
(35)
and using the experimental data of the actual environmental temperatures T and the corresponding values of # obtained from experimental data, a plot of In .# against 1/T can be made for different stress levels, see Fig. 3 ; and since R is a constant which is the gas constant, the values of AU which is the actime,
Rupture
Rupture
2 O'
4
I
time, t (min) 6 8
I
I
I
0
~-
200
0
400
t (rain) 600
800
1000
10 I - 0.5
-
0.5
I
-1.0
-1.0
-1.s -1.: -2.0
-2.c -2.5
-2.5 Q
-3.0
Fig. 2a. Reliability function, R~ (t) (see eqn. {20)). Initial uniaxial stress, e,=24.5 kg/mm 2. Environmental temperature, 40~C or 104°F.
c
Fig. 2c. Reliability function, R 1(t) (see eqn. (20)). Initial uniaxial stress, ¢,=24.5 kg/mm 2. Environmental temperature, 8°C or 46.4°F. of oDsolute temperature, .1.¢T)xlO 4
Reciprocal time,
Rupture
0
50 i
[
i
t(mln) 100 I
15
.
17
18
19
I
I
I
150
initial
I
°I -0
16
unlaxial
I
stress,
,
5 °
~ 0"= 24.00 kg/mm z
1-2 ::k
-1.O
~In~tiat uniax[al stress, v\ ~S"g/mm~
-3
-4
-2.O -5
-E
-3.¢
20
b
Fig. 2b. Reliability function, R 1(t) (see eqn. (20)). Initial uniaxial stress, ~,=24.5 kg/mm 2. Environmental temperature, 20~C or 68° F.
Fig. 3. In. # = In' C -
AU
(eqn. (35)).
EVALUATION OF STRUCTURAL CREEP RELIABILITY
tivation energy for self diffusion can be obtained from the tangents to the lines in Fig. 3. These results for AU are finally plotted against the corresponding uniaxial stresses, as shown in Fig. 4. It appears that there is a linearity between activation energy, AU, and uniaxial stress, ~. That is, it appears that AU = ~a
(36)
107
t
.a c t~
- 1.5
g E
1.0
where ~ is a constant for the problem. Therefore, eqn. (18) can be expressed as =
~- C e x p
-
(37)
2.0
u
0.5
g z;
for this problem. 0
t
0.1
0.2
t__
0.3
I
0.4
0.5
Fig. 5. Mean creep ~- vs. dimensionless time (fit°) for Hastelloy R 235 bar stock, 4.(3 100
~0 3.0
80 .= 6 0
2.0 •
o
,-4
g "~ >
P
1.(3
40 30 1502 20
<
x
+J
o
24
I I I I 24.2 24.4 24.6 24.8 l : n i t i o t uniaxial s t r e s s , d ( k g / m m z)
I 25
Fig. 4. Relationship between activation energy AU and uniaxial stress, a.
le5O° :~, 101 8
61
Villi. N U M E R I C A L E X A M P L E
It is required to make a prior estimation of the creep rupture time and the initial allowable uniaxial tensile stress in a structural member made of Hastelloy R-235 bar material, for a given maximum creep elongation of 1.0 % while operating at a constant temperature of 1500° F, for a combined structural reliability of at least 90 %. Assume that the material is kept in operation for at least 200 h.
1
1 2
I
5 10 100 1000 Mean r u p t u r e t i m e ( { ) h
Fig. 6. Curves of initial stress vs. mean rupture time for Hastelloy R 235 bar material at various temperatures 6.
(2) Assuming that t = 200 h, therefore tc = 625 h, since t/tc=0.32 at 1.0% elongation from Fig. 3. (3) The specified combined structural reliability is R12 (t) = R, (t)" R 2 (t)
+
IX. S O L U T I O N
(1) The mean creep elongation of 1.0 % is reached when t/t~ = 0.32 and at this instant ~(t) = 0.011/0.250 t¢ per hour, see Fig. 5 and eqns. (25a) and (25b).
10000
(see eqn. (30)) hence -
---- + 0.230 x
-~ 0.10
'/--090
108
whence the creep rupture time [ can be obtained as [ = 2360 h. (4) From Fig. 6, since the mean structural creep rupture time [=2360 h, and the temperature is 1500° F, therefore the initial allowable uniaxial stress is given as 20,000 psi.
X. CONCLUSION
It is imporant to summarize the essential steps developed in this paper, so as to make the method more meaningful and easy to handle for practical engineering problems. These steps are as follows: (1) Estimate the necessary value for the mean rupture [ for the structural material, using expression (18) for the rate function of the process, or any other suitable expression for the rate function. (2) Obtain the prior probability distribution function ft (t) for the creep rupture mode of failure of the structural material, from eqn. (19); and in the absence of data obtain the prior reliability function Rx (t), for the creep rupture mode of failure of the structural material, from eqn. (20). It should be pointed out again that the prior distribution function fl (t) given by eqn. (19) is the minimally biased distribution function, consistent with the initial information about the value of the estimated mean rupture time L Expression (19) is derived from the maximization of the entropy expression. (3) Estimate by any suitable method the expression for the critical creep strain rate of the material, at the desired temperature and uniaxial stress conditions. Such an expression, e.g. eqn. (25b), can be regarded as the prior mean value expression, for most practical purposes. (4) Obtain the prior probability distribution function f2(t) for the critical creep strain mode failure of the structural material, from eqn. (26); and in the absence of data, obtain the prior reliability function R2(t), for the critical creep strain mode of failure of the structural material, from eqn. (27). In this case, as shown in step (2), the prior distribution function f2 (t) given by eqn. (26) is the minimally biased distribution consistent With the initial information about the value of the estimated critical creep strain rate. Expression (26) is again derived from the maximization of the entropy expression. (5) Obtain prior estimate of the reliability function R t2 (t)=Rt (t). R2 (t), for the combined strain-
A. B. O. SOBOYEJO
rupture modes of failure of the structural material, under the given conditions of temperature and uniaxial stress. (6) When more data become available, the above prior distribution functions fl (t) and t"2(t) carl be improved by means of Bayes' theorem, to yield the posterior probability distribution functions, fl (t) and f2 (t), given by eqns. (29a) and (29b), for creep rupture and critical creep strain modes of failure respectively. The posterior distribution functions now obtained can be used to derive posterior reliability functions for creep rupture, critical creep strain and combined creep strain-rupture modes of failure.
ACKNOWLEDGEMENTS
The facilities provided by Towne School of Civil and Mechanical Engineering of the University of Pennsylvania in Philadelphia, Ford Motor Company in Michigan, and the Faculty of Engineering of the University of Lagos, Nigeria, in order to carry out this work, are all thankfully acknowledged.
REFERENCES 1 C. E. Shannon, Bell. System Tech. J., July and Oct. (1948). 2 E. T. Jaynes, Probability Theory in Science and Engineering, McGraw-Hill, New York, 1961. 3 H. C. Shah, Principle of maximum entropy and its applications in reliability estimation of aircraft structures. Final Rept. to NAEC under Contract No. N156-45588 and to be presented at the AIAA/ASME Structures, Structural Dynamics, and Mater., Conf., Palm Springs, Calif., March, 1967. 4 J. Hult, Creep in Engineering Structures, Blaisdell Publishing Co., 1966. 5 S. Taira and R. Koterawaza, Japan Soc. Mech. Eng., 14 (1961) 238. 6 C. W. Phillips and J. J. Sinnott, Trans. Am, Soc. Metals, 46 (1954) 63. 7 I. Finnie and W. R. Heller, Creep of Engineering Materials, McGraw-Hill, New York, 1959. 8 R. Simon, J. Basic Eng., March (1966) 87. 9 H. Eyring, J. Chem. Phys., 4 (1936) 283. 10 J. G. McBride, B. Mulhern and R. Widmer, New England Mater. Lab., Medfor, Mass., Tech. Doc. Rept. No. WADDTR-61-199, prepared under Contract No. AF 33(616)-6200, August, 1962. 11 R. Mesloh, Ann. Reliability and Maintainability, 5 (1966) 590. 12 E. Parzen, Modern Probability Theory and Its Applications, Wiley, New York, 1960 p. 119. 13 H. Raiffa and R. Schlaifer, Lecture Notes, Harvard Graduate School of Business, Harvard Univ., Cambridge, Mass.
EVALUATION OF STRUCTURAL CREEP RELIABILITY
Utilisation de la notion d'entropie pour lYvaluation des risques de rupture par fluage dans les structures
L'auteur se sert du principe de l'entropie maximale pour d&erminer de premirres fonctions de distribution de la drformation structurale critique de fluage et du temps de rupture, dans le cas de structures soumises ~t des contraintes uniaxiales connues, /~ des temprratures 61ev~es connues. A partir de la premirre fonction de distribution on peut drduire la fonction de confiance, qui est simplement la probabilit6 d'un fonctionnement sans incident. Cette fonction est calculre pour des drformations critiques ~t rupture et des conditions de contrainte et de durre de vie bien sp~cifires. L'auteur cherche h 6tablir les fonctions de confiance/l l'aide de considrrations antrrieures sur la mrcanique de la rupture et des proprirtrs physiques et mrcaniques des matrriaux industriels. Dans son travail il suppose que les fonctions de confiance, destinres au calcul des structures devant rrsister au fluage et drfinies aussi bien ~ l'aide de la dfformation que du temps de rupture, peuvent prendre des valeurs telles que la rupture des pirces peut survenir par l'un ou l'autre des mrcanismes envisagrs ou par toute combinaison de ces mrcanismes. I1 signale 6galement que les premirres fonctions de distribution, it partir desquelles les fonctions de confiance sont dbduites, peuvent 6tre amrliorres grace au throrrme de Bayes. On peut ainsi chercher /l obtenir une deuxirme fonction de distribution qui tient compte des nouvelles donnres de fluage rendues disponibles. Les fonctions de distribution secondaires permettront de d~terminer les fonctions de confiance avec plus de prrcision. En fin de compte, l'auteur illustre ses consid+rations et ses mrthodes relatives aux critrres de dessin des pirces, en les appliquant ~ l'analyse des contraintes d'un 616ment de structure ayant des proprirt~s physiques et mrcaniques, des caractrristiques de fluage et un environment donnrs.
109
Eine Bestimmung der Verlgifllichkeit yon Proben in Kriechversuchen mit Hilfe yon Entropiebetrachtungen
Mit Hilfe des Prinzips maximaler Entropie werden die friiheren Wahrscheinlichkeitsverteilungen der kritischen Kriechdehnung und des Kriechbruchs bei bekannten hohen Temperaturen und einachsigen Spannungen gewonnen. Aus diesen Verteilungsfunktionen werden VerlaBlichkeitsfunktionen (die Wahrscheinlichkeit fiir einen erfolgreichen ProzeB) fiir spezielle kritische Kriechdehnungs- und Kriechbruchmoden abgeleitet. Es wird der Versuch untemommen, die VerlaBlichkeitsfunktionen aus vorhergehenden Betrachtungen der Bruchmechanik und der mechanischen und physikalischen Eigenschaften des Materials abzuleiten. In der vorliegenden Arbeit wird angenommen, dab VerlaBlichkeitsfunktionen fiir Kriechbruch und kritische Kriechdehnungsmoden der Strukturelemente Werte annehmen k6nnen, die es erlauben, dab der Bruch tier Elemente entweder durch jeden der Bruchmoden oder durch die Bruchmoden gemeinsam erfolgen kann. Es wird auBerdem darauf hingewiesen, dab friihere Wahrscheinlichkeitsverteilungen, aus denen VerlaBlichkeitsfunktionen abgeleitet wurden, mit Hilfe des Bayes-Theorems verbessert werden k6nnen, um immer dann neuere Wahrscheinlichkeitsverteilungen abzuleiten, wenn neue Kriechdaten vorliegen. Aus den neuen Verteilungsfunktionen k6nnen dann genauere Verl~iBlichkeitsfunktionen abgeleitet werden. SchlieBlich werden diese Betrachtungen und Verfahren, die VerlaBlichkeitskriterien liefern, anhand einer Anwendung auf die Spannungsanalyse einer Struktur mit gegebenen mechanischen, physikalischen und Kriecheigenschaften bei gegebenen ~iuBeren Einfltissen illustriert.