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Modelling of curvilinear random fatigue crack growth
~
Pergamon
Engineering Fracture Mechanics Vol. 52, No. 4, pp. 703 715, 1995 Copyright © 1995 Elsevier Science Ltd 0013--7944(95)00042-9 Printed in Great Britain. All rights reserved 0013-7944/95 $9.50 + 0.00
MODELLING OF CURVILINEAR R A N D O M FATIGUE CRACK GROWTH K. SOBCZYK and J. TREBICKI Center of Mechanics, Institute of Fundamental TechnologicalResearch, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland B. F. SPENCER, JR Department of Civil Engineering and GeologicalSciences, University of Notre Dame, Notre Dame, IN 465564)767, U.S.A. Abstract--In the paper, stochastic description of curvilinearfatigue crack growth is provided. The crack growth process is regarded as a cumulative random jump process consisting of a random number of random incrementswith random angles. The probabilitydistribution of the crack size representedby the model is determined and the relation of the basic model parameters to fracture mechanics is analyzed. The description presented seems to be capacious enough to include various realistic situations resulting in departure of crack growth from the straight-line direction.
1. I N T R O D U C T I O N FATIGUE crack growth in engineering materials is a complicated process which is sensitive to many factors, e.g. various loading conditions, material inhomogeneity, environment, etc. Because of the difficulties in accurately describing the fatigue process and the significant random scatter in empirical data, it is now commonly accepted that the prediction of fatigue fracture can rationally be performed by use of a stochastic approach [1]. The existing stochastic descriptions are, however, almost entirely concentrated on uniaxial loading (and pure mode I fatigue) and on growth along a straight path. Cracks rarely grow exactly along a straight line. Even when the applied stress is uniaxial, randomness in the crack geometry, material properties and metallurgical structure may result in unpredictable departures of the crack path from a straight trajectory. For example, examination of three-dimensional replicates of fatigue cracks in pure aluminium specimens by scanning electron microscopy shows the crack front to be highly non-uniform; the crack advances on many "minifronts", and the local direction of growth may deviate several tens of degrees from the average direction [2]. Very irregular crack growth has been observed (cf. Fig. 1), for example, under torsion of steel elements [3]. When the loading process is multiaxial, an interesting aspect of the fatigue process is that the crack may grow in a mixed mode manner. An adequate description of multiaxial fatigue should include both the crack growth direction and the rate; in general, a curvilinear fatigue crack path has to be considered [445]. Figure 2 shows an experimentally obtained irregular crack trajectory given in [6]. Also, as it was underlined by Kitagawa et al. [7], "in all efforts of combining micro and macro fracture processes, the crack morphology seems to have significant meaning, because a microscopic crack usually has a complicated geometry, for example, the concave and convex crack front, the zig-zag crack path or branching". In ref. [8], Suresh studied various types of crack deflections and conjectured that crack deflections, and in general the nonlinear crack path, play a major role in influencing the crack growth of short fatigue cracks. In engineering fracture mechanics, several criteria have been proposed to determine the initial crack growth direction in a mixed mode situation. For example, one of the best known is the criterion of Sih [9], according to which the crack is assumed to grow in the direction of the maximum energy release rate, or in the direction of the minimum of the strain energy density function. Although these criteria highlight the mechanical aspects of mixed mode fatigue cracking, existing studies indicate that difficulties remain in prediction of the crack growth trajectories. Any 703
704
K. SOBCZYK et al.
Fig. 1. Experimental crack growth trajectories under torsion (from Kocanda and Kur [3]).
material inhomogeneity or inaccuracy in determination of the stress intensity factors, as well as in the validity of the criterion chosen for initial crack direction, contribute to errors and an increase in unpredictability. In this situation, a stochastic description of the curvilinear crack growth seems to be both rational and useful. Of course, the lack of reliable experimental data associated with non-straight crack growth makes it difficult to perform fully satisfactory estimation of the parameters and the subsequent model verification. Nevertheless, it is justified to construct a mathematical scheme (a possible "pre-model") into which the random curvilinear crack growth process can be imbedded. In this paper, we propose a stochastic description for curvilinear crack growth in terms of the cumulative random jump processes. We regard the crack growth process as a discontinuous random process consisting of a random number of random increments with random angles. The model presented herein constitutes an extension of the modelling ideas reported in papers by Sobczyk [10] and Sobczyk and Trebicki [11, 12] (see also Sobczyk and Spencer [1]). 2. DESCRIPTION OF THE MODEL According to many experimental studies, the fatigue crack growth process is discrete and random in time. At the same time, it is a cumulative process. These basic observations led the authors of papers [11, 12] to construct a model for straight-line fatigue crack growth of the form N(t)
A(t,?)=Ao+ ~,A,(?), A,(7)=AA,
(I)
i=l
~0
i
i
l
i s¢
l,O
.~
E 2o.~,,
~
,
t
,
l~axiat stress ratio =05
o
#
#
,~....
o
-40
' ~
(----)predicted
'~
(~) -60
-8o 40
-;o-2o
I
o
experimentot
I
2o
I
to
8'0
80
Crock size (ram)
Fig. 2. Experimental crack growth trajectories under biaxial stress (from Lam [6]).
Modelling of curvilinear random fatigue crack growth
705
a)
.[ x-axis
c)
b)
Xi
Xi
x-axis
x-axis
Fig. 3. Illustration of the geometry of the model.
where A (t, 7) is the length of a dominant crack at time t. The letter y symbolizes the elementary event belonging to F, where F is a space (sample space) on which the probability measure is defined; for each y ~ F, A (t, y) represents a possible realization of the crack length process. The value A0 is the initial crack length, which is of sufficient size to propagate (assumed to be known from empirical data), Aj(7) are random variables characterizing the elementary crack increments; N(t) is an integer-valued stochastic process characterizing the number of crack increments in the interval
(0, t]. Let us assume now that (because of material heterogeneity, uncertainty in initial crack direction and position, departure from uniaxiality, etc.) the crack grows not along a straight line, but along a curvilinear path, fluctuating around the straight line (x-axis). In order to characterize such a process let us introduce the angle O i between the actual crack increments and x-axis (cfi Fig. 3a); in general Oi~ ( - ~ / 2 , n/2), but it seems that in situations of engineering interest, the "path fluctuation angles" Oi should not depart strongly from zero. We propose the general description of a random curvilinear crack growth process in the form of a two-dimensional process [A (t, 7), Z(t, ),)] N(0
A(t, 7)=Ao+ ~ Y~(7)
(2)
i=l
N(O
Z(t, 7 ) = Y'. Z,(7)
(3)
i=l
where A(t, 7) is now the stochastic process characterizing the crack length along the curved path at time t, Y~(~) are random variables representing the increments in the curvilinear crack length, Z(t, 7) is a stochastic process characterizing the departure of the crack tip (on the curved path) from the x-axis (that is, from a straight line) and Zi(7) are random variables describing elementary departures of the crack tip in the ith increment from the local straight line. In general, the random variables Y~(7) and Z~(7) can be related to the random variables X~(y)--characterizing the elementary increments in certain auxiliary process--as follows:
r,(y) = w[O,(7)lX,(y)
(4)
z~(~) = u[e,(~,)]x,(~)
(s)
where w(. ) and u(. ) are appropriate functions suggested by empirical information concerning elementary crack increments in the curvilinear crack growth, and O~(7) are the random variables which are assumed to be independent of Xi(~).
706
K. SOBCZYK et al. It seems that the most natural realization of the relationship in (4) and (5) takes the form:
0,(7)
(6)
2",.(7) = X~(7) tan 0,(7)
(7)
Y,(7) = x , ( 7 ) sec
where s e c ( . ) = l/cos(. ). The above formulae indicate that )(,.(7) (elementary increment along x-axis) is an orthogonal projection of Y~(7) on the x-axis and Z~(~) is shown in Fig. 3(b). Other possible examples of the functions w(. ) and u(. ) may result from specific relationships between the true curvilinear crack increments and an observable counterpart in the experiment; for instance, if Yg(7) ~< X~(y), one may assume that: Y~(7) = X~(~,)cos 0,(~,)
(8)
Z~(?) = X~(7) sin O~(7)
(9)
which indicates that the Y~(7) are taken as the orthogonal projections of the observables X~(7) with the actual value of O~(7) (cf. Fig. 3c). It is clear that formulae (4) and (5) are general enough to reflect the true empirical properties of the elementary (curvilinear) crack increments I1,.and their relation to X~(7). It should be noticed that formulae 4 and 5----expressing the basic quantities of the curvilinear crack growth in terms of increments of X~ (and simple functions of O,)--make the model represented by formulae (2) and (3) highly versatile; for example, one can use the results concerning X~(7) from the analysis of straight-line growth models.
3. RELATION TO FRACTURE M E C H A N I C S The mathematical scheme for the curvilinear fatigue crack growth presented above in the form of the formulae (2) and (3), and (4) and (5) includes as the model parameters: the initial crack length (A0), the intensity (2o) of the random counting process N(t), the parameters characterizing the probability distribution of X~(7), and the characteristics of the random angles O~(7). The model, like its predecessor described in papers [11, 12], is general and does not include the loading process explicitly; the characteristics of the applied stress (sinusoidal, or random) are incorporated into the properties of the process N(t), as well as in the characteristics of random variables X~(7) describing the elementary increments in the observable process; in the case of random loading, they can be identified in a manner analogous to that used in papers [11, 12]. Here, the basic quantity is the random fluctuation angle @~(7) characterizing the direction (with respect to the x-axis) of each elementary curvilinear crack increment. In general, the probability distribution re, (~) of random O~(7) should be estimated from the empirical data, or inferred from the physics of the phenomenon. In practice, it is easiest to hypothesize the probability distribution re, (,9) and then to estimate its parameters. As a first step in this direction, it seems to be useful to look at the possible relationship between the randomness in Or and the Sih criteria for initial crack growth direction (cf. [9, 18]). If the initial crack position in an infinite plate is perpendicular to the applied stress and the crack grows in mode I(Kl = a 2 x / / ~ , Ku = 0), the strain energy density function for the initial direction of the crack propagation is ¢r2A S = ~ (3 - 4v - cos @)(q + cos O)
(10)
where A is the crack length, a is the applied deterministic stress, p is the shear modulus, v is Poisson's ratio, and @ is the polar angle for the cylindrical coordinates of a point with respect to the crack tip. According to the Sih criterion, the crack will initially propagate in a radial direction along which the strain energy density function has a minimum, that is when aS 0-~ = 0,
3zS ~ - 7 > O.
(ll)
Modelling of curvilinear random fatigue crack growth
707
2.30
(xC) 2.25
2.20 2.15 2.10
'°~ - ..... .......:j ~ ' d....................
~
2.05
2.00 1.95 1.90 , , , , , , , , , ~,
, , , , , , , ,' L
SO'
, , , , , 2' ' ' ' I
' 0' '
' ' ' ' 4' 0I
' ' ' ' ' ' 't
50
' I ' ' ' '
'
a 60
Fig. 4. The strain energy density function S(~) (bold line) and (S[O(7)]) as function of a for uniform and Gaussian distribution of initial direction O(~), respectively; v = 0.25, G = 16#/a2A, ot in degrees.
This condition gives O = O0 = 0 and (1 -- 2v)o'2A Smin =
(12)
4g
Let us randomize formula (10) by assuming that the initial angle O has some random uncertainty, that is we treat O as a random variable O(V) with zero mean and a one-parameter probability density)Co (g; ~). The parameter ~ can be estimated (for example) from the following criterion max
< S(7) > - &.i°
~< E
(13)
Smin
where ( • ) denotes the mean value, ( S ( 7 ) ) = S S(0)fe (~; ~) dO, and c is a given number. Therefore, the parameter • quantifying the scatter in the angle 0 ( 7 ) (e.g. • could be the parameter in the uniform distribution on the interval [ - ~ , ~] or the standard deviation in the Gaussian distribution) is determined as a maximal distribution parameter for which the relative error between the deterministic Stain given by (12) and mean value of the random variable S(y) (being a function (10) 8.0
[~]
(1) -
/,1=0.15
(2)
u=o.25
-
Gatusmiaa
....
(3) - b'=0.35
.2 6.0
~ ~-
/ ""
.o~/"/"
¢s
(a!,
(3)/
'~4.0
0
~'~ 2 . 0 0
0.0
rlllll
0
i
i l l l l
10
ill
i
i
i l l [ i t
20
i
iii
ii
l i l t
30
ii
T ? I , I ]
i11
40
ii
i
r ii
jSl
50
i
lIT
III
Or. 60
Fig. 5. Variation coefficient q of S[O(7)] vs ct for different Poisson ratio v; initial direction 0(7) has uniform and Gaussian distribution with parameter a respectively; ~t in degrees. EFM 52/4~H
708
K. SOBCZYK et al.
4C10~
(1) - ~'=0.001 (2)
3o~ \
10 2
O~
~--~
........
O.15
-
~=o.oo~
(3) - ~=o.ol
~ .........
0.20
~
~ .........
0.25
~ .........
0.30
; .........
0.35
; .........
0.40
V
0.45
Fig. 6. The scatter parameter • vs the Poisson coefficient v for different errors E; initial direction O(y)
has uniform and Gaussian distribution with parameter a, respectively;~ in degrees. of 0(7)) is not greater than the given value of E. Figure 4 shows for ~ e [0, 60 °) the plots of the strain energy density function (10) for O = ~ and its mean values when O is treated as random variable with uniform and Gaussian distributions, respectively, with parameters ~. The percentage coefficient of variation q = ~ S / ( S ) , (var S = ( S 2) - (S)2), of the random variable S(7), as a function of 0t is presented in Fig. 5. Simple calculations show that even for small E (of order 0.005 = 0.5%) and for v e [0.15, 0.3] we obtain quite large values of ~ satisfying (13). This fact is illustrated in Fig. 6 and can be interpreted that even when a deterministic mode I loading condition is present, some random deviations from ~9 -- O0 = 0 may occur. This implies that the initial crack direction can be considered as a random variable O (7)- As long as the departure of the crack tip from the x-axis is small during the crack propagation, the hypothesis that KH = 0 and that the crack is perpendicular to the stress direction seems to be justified, and the reasoning presented above can be adopted for predicting the direction of such a curvilinear crack growth. If the departure of the crack tip from x-axis is significant and the actual ith crack position is inclined at some random angle fli with the uniaxial stress axis (y-axis), then K~i) ~ 0 and K~ ) ¢ 0. In this situation a possible crack growth direction ~;+~ can be calculated (for example) from the condition of the maximum circumferential stress (cf. Sih [18]), that is from equation KIi) sin ~ + l + KI~)(3 cos ~ + ,
--
1) = 0.
(14)
When this condition is randomized by introducing random variables into K~ and K, (e.g. analogously to Grigoriu et al. [19]), we obtain the probability distribution of the random angles ~ + i. This type of distribution can be adopted to characterize the sequence of random angles of the elementary crack increments. Of course, completely satisfactory linking of the model presented to fracture mechanics requires general solutions for the stress intensity factors for curvilinear (e.g. zig-zagging) cracks. This, however, is a task which seems to exceed the present possibilities of contemporary fracture mechanics [7, 8]. But, this difficulty may be seen as the primary impetus for constructing a stochastic model (such a suggestion was similarly made by Kitagawa et al. [7]). In further analysis and in the illustrative examples, the distributions of the angles ~9~ are assumed to be given. 4. P R O B A B I L I T Y
DISTRIBUTIONS
OF THE MODEL
PROCESSES
4.1. General formulae Since the mathematical structure of the processes A(t, 7) and Z(t, 7) with formulae (2) and (3) is the same, let us introduce, for mathematical convenience, the process N(t)
s(t, 7)= ~ ~,(7) i=1
05)
Modelling of curvilinear random fatigue crack growth
709
which can symbolize either the crack length along curved path or the departure of the crack tip from the x-axis. In the first case, the random variables ~i(?) are given by formulae (4), whereas in the second situation by (5). Let us assume that random variables ~i(?) with the probability densities f~e(s) are statistically independent of each other and are independent of the process N(t). The distribution function Fs(s; t) for the process in (15) is
Fs(s; t) = P[S(t, 7) ~
(16)
k=l
where Pk(t) = P[N(t) = k]. The probability densityfs(s; t) for S(t, ?) is expressed by the formula
fs(s; t) = O~2Fs(s; t) = ~ fs(s IN(t) = k )Pk(t) U~
(17)
k=|
where fs(slN(t)= k) is the density of the random variable k
Sk(~) = 2
~i(~)"
(18)
i=1
Since the ~(7) are independent
fs(s IN(t) = k) =f¢, (s)*f¢ 2(s)* • • • *f~k(s) = G(~)(s)
(19)
where * denotes convolution, and
k(~)G(k-°(s-~)dz,
G~k)(s) =
k=2,3 ....
(20)
Finally, the one-dimensional probability density for the process S(t, 7) takes the general form
fs(s; t) =
G(g)(S)Pk(t)= kffil
Pk(t) k=l
f0
f~ (z)G (k-|)(s -- z) dr.
(21)
To determiners(s; t) explicitly, one has to assume a specific random counting process N(t) with given Pk(t) and specific distribution f¢~(x). If, for example, the random variables ~i(?) are exponentially distributed, that is
f~,(x)=~gexp(-~ix), = 0,
x >10
x <0
(22)
then (cf. Feller [13]) G(k)(s) = (i=I~, ~ ) ( ~
~.k exp(-- ~is))
Oi.k = [(~, -- ~,) " "" ( ~ - , -- ~,)(~+, -- ~,)" "" (~k -- ~,)1-'.
(23)
If the ~(7) are identically exponentially distributed (i.e. ~ = ~), and N(t) is the Poisson, or pure birth process, the distribution density fs(s; t) is given in [11]. 4.2. Moments of the model processes Analytical determination of the probability distribution of the process S(t, 7) according to the formulae given above may not be easy, mainly because of difficulties in analytical calculations of the convolution integrals in (20) for large k. Also numerical calculations of these convolutions can be computationally cumbersome. Therefore, it is of interest to determine the moments of the process S(t, 7)- Under the assumptions regarding N(t) and ~(7) made in Section 4.1, the interesting moments are as follows
= ~ Pk(t) = ~ E'~Pk(t) where
(24)
k=O
k=O
E7 =
~,(?) i=1
.
(25)
710
K. SOBCZYK e t a l .
Making use of Newton's formula, we have the following expression for E~: m!
E~'=
~
hi + ' " + h k
=
mh, !"
• • hk!
<¢1,">.
(26)
If the ~ (7) are mutually independent, then m!
x"
L k ~_ h , ' hl+...+h
k
l-'l
h~,,L~ . . .=
(27)
xh~, (x) dx.
(28)
•. .
where (~f'(Y))
=
For instance, the first conditional moments (27) occurring in the general formula (24) are
,)k
i=1
2
k
= Y', ( ¢ ~ ) + 2 ~ (¢,)(¢j)
~,(7)
i= I
i
k
~i(7)3 = E <~> ''1-3 E <¢2><~j> i= 1
i
+ 3 ~ (~i)(~) +6 i
~
(¢i)(~j)(~k).
(29)
i
To make further analysis more efficient, we need to assume a specific form of the probability distribution of the elementary crack increments. The empirical observations support the hypothesis of an exponential distribution for X~(y) (cf. Kogaiev and Liebiedinskij [14]), i.e.
f~,(x) = ~iexp(-~ix),
x ~>0.
(30)
Therefore
= ~hi!.
(31)
Assuming that the random elementary increments Xi(Y) and the elementary angles Oi(7) are independent, the processes A (t, 7) and Z(t, 7) defining the curvilinear crack growth and given by (2), (3) and (4), (5), have moments determined by formulae (24) and (27), where for the process ~, - -~--N(,) t=, ~(7) A ~(t, "~
<~,~'(7)> =
(x~'(7)>
= ¢Theh~!
L
Wh' (9 if'o, (oa)dO
(32)
and fe, (~) is the probability density of O~(7), ~ ~ [a, b] = ( - x / 2 , z/2). For the process Z(t, 7) = Zi= N(t)t Z~(y) we have
<~,~'0)> =
(x~'o)F
f
= ~?h'h~!
uh'(~)fo, (~) d&
(33)
The moments calculated according to the formulae presented above provide important information about the model processes. However, they can also be used for determining the approximate probability via the maximum entropy principle (cf. Spencer and Bergman [15], Sobczyk and Trebicki [16]).
Modelling of curvilinear random fatigue crack growth 5. S I N U S O I D A L
711
LOADING
5.1. Estimation of mean elementary increments The analysis presented above, as has been previously indicated, does not explicitly consider the loading process. The applied stress enters the model through the properties of the counting process N(t) and by the characteristics of the random elementary crack increments. In general, the loading can be sinusoidal as well as random. In the case of random loading, the intensity of the counting process N ( t ) can be related to the peaks of the random applied stress (cf. [11]). Let us assume here, that the loading is sinusoidal with a given frequency and that the crack advances take place in each cycle. In this case, the process (15) reduces to the random variable N
SN(7) = ~ ~i(7)
(34)
i=l
where N denotes the number of cycles (number of increments of the crack growth). The probability density of SN(7) can be easily expressed by a convolution integral. If the ¢~(y) are independent, exponentially distributed random variables then formula (23) holds. After a crack has initiated, the number of cycles until the crack reaches its critical (or control) size A * is usually in the order of a few hundred thousand. In such situations one can use the central limit theorem to assert that for large N, SN(V) can be approximated by a Gaussian random variable with the mean value mSN and variance var SN, where N
N
nSN = ~ <¢i(7)>,
var SN = ~ var ~i
i=1
i=1
var ~, = <¢2(7)> - <~(V)> 2.
(35)
The parameter ~gin the exponential distribution of the crack increments Xi(7) can be estimated (using conventional data) from the following condition = Ap(N)
(36)
where the left-hand side denotes the mean value of AN(t, V) = A0 + E~=~ X~(7), and the right-hand side is the crack size predicted by the Paris equation for the same number of cycles. The mean of the random variables X~(7) characterizing the ith crack increment is taken as an increment in the ith cycle predicted by the Paris equation, that is = Ae(i) - Ae(i - 1) = AAp(i)
(37)
where Ae(O) = Ao, 1 <<.i <<.N*, and N* is the number of cycles to reach the critical (or control) size A *. Therefore. 1 AAp(i) where (cf. [17]) 2 --
Ae(N)=
K
A~o2 ~"2 + - ~ - - C , N
)2,'(2 - h') ,
C,=C(Aax/~) 2
(38)
and C, ~c are the material parameters occurring in the Paris equation; Aa is the difference between the maximum stress areax and the minimum stress ami, for the sinusoidal loading. Making use of the above values of the parameters ~t in the exponential distribution of the crack increments and taking the same material and experimental conditions as in Kocanda and Szala [17], (A0=0.01 m, C = 1.03 x 10 -12, m =3.89, area,= 120MPa, amin=40MPa, A* = 0 . 0 6 m , N* = 277329 cycles), one can obtain an effective characterization of the curvilinear crack growth according to the method described. The results of numerical calculations performed for the case of sinusoidal loading are shown in Figs 7-10. A uniform distribution on the interval [-~t, 7] ()co, (,9; ~) = 1/2a, ,9 ~ [ - ~, ~ ]) is taken for the random variables Oi (V). Figure 7 shows a comparison between the probability density of the crack length according to the proposed curvilinear modelling and the probability density of the crack length predicted by straight-line growth theory. Figure 8 represents the corresponding mean values of the crack length vs the number of cycles N e [1, N*]
712
K. SOBCZYK et al. 1.2
7/ o
~0.8
t
I / /
7/
i/~
it i
~ increments ~ a s I n Pig. 3 b
,
,
,'
',
~
I
I
,'
',
~0.4
mm (+A0)
0.0 41.5
42.5
43.5
44.5
45.5
46.5
Fig. 7. Probability density of the crack length, straight-line growth ( - - ) , (. . . . ) follows eqs (34) and (6). The directions O~(7) have uniform distribution on [ - ~ , ~]; ~ = 25 °, number of cycles N = 270 x 103.
for different values of ~. It is seen that modelling based on the curvilinear growth gives larger mean values as compared to the theory of straight-line crack growth, after the same number of cycles N. As far as the departure process ZN(~,) of the crack tip from the straight-line direction is concerned, the model analyzed above has the ability to predict the mean behaviour of the crack growth trajectories; however, calculations give a very small variance of ZN(y), and AN(y). This result implies that the model is inadequate to portray the macroscopic curvilinear crack growth. This is however, caused mainly by the assumption of independence of random elementary increments, which can be removed (cf. [12]). 5.2. Generalization It has been assumed in the previous sections that each load cycle generates a crack increment and each elementary increment has a different random angle ~9i. However, such a hypothesis may not be sufficiently realistic; the crack may change its direction only after some number of elementary (micro) increments. To incorporate this phenomenon into the model the following formula for the
60.0 /
£
(1)
- a=e5
(2)
- ~ =450
°
/
/
/~ (2)...///
40.0 o (1)~/
// /,7
"~ 20.0 2 "-(+~) 0.0 0
"~~llljl,tljl~llll 50
100
150
200
250
I
(xzooo)
300
Fig. 8. Mean value of the crack length dependent on the values of parameter ~t; directions O~(7) have uniform distribution on [-ct, ~t]; ( ) mean value of the straight-line growth, (. . . . ) follows eqs (34) and (6), N - - n u m b e r of cycles.
Modelling of curvilinear random fatigue crack growth
4.80
713
\ \ \
.0
\
3.80
03
\ \
ID
\a
=20
=
,.~ 2.80 "0 to
"~
0~=15
o
~
~
~
~'1 1.80
0180
. . . . . . . . .
tO
[ . . . . . . . . .
30
'
. . . . . . . . .
[ . . . . . . . . .
50
70
[ . . . . . . . . .
90
llO
Fig. 9. Standard deviation of the crack tip departure for different values of M ; ( ) follows eqs (34) and (7), (. . . . ) follows eqs (34), (9), directions Oi(y) have uniform distribution on [-~t, ~,]; number of cycles N = 270 x 10~.
model process SN(7) can be used M
Su(?) = ~ ~k(?)
(39)
k=l
where for the crack length ~'k(?)
= w [Ok (?)]
~ X~(?)
(40)
i=1
or for the departure of crack tip from x-axis Nk
~',(?)
= u[Ok(?)] ~ X,(?).
(41)
i=l
The total number o f increments N = Ni + N2 + " • • +
NM, where Nk (k
= 1. . . . . M ) are the number
of elementary increments along the direction specified by the random angle Ok(?), and M is the number of possible different directions occurring during crack propagation. In general, the numbers M, N x can be either deterministic or random, and the random variables ~'k(?) can be treated as I0.00
,",
5.00
•N o.oo ¢.)
tO r~ -5,00
-lO.OO
.........
0.00
~ .........
10.00
~ .........
20.00
~ .........
30.00
~ .........
40.00
50.00
Fig. 10. Simulated crack growth trajectories for sinusoidal loading using model (39); directions Oi(7) have uniform distribution on [ - e , e]; e = 25°; Nk---deterministic.
714
K. SOBCZYK et al.
(a)
10.00
5.00
•~
o.oo
-5.00
-10.00 0.0
(b)
I I l + l l l l l l l l l l l , l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
m.ln
10.00
20.00
30.00
40.00
50.00
10.00
P-O.O0
30.00
40.00
50.00
5.00
3.00
1.00 N
~-
¢S
1.00
-3.00
-5.00
0.00
Fig. 11. Simulated crack growth trajectories for sinusoidal loading using model (39); directions ei(7) have uniform distribution on [-ct, ct]; ~ = 25 °, Nk--random from [1, 3.3 x 103].
mutually correlated. It is seen that in the particular case when M, Ark are deterministic and Nk = ! for each k (hence M = N), then formula (39) reduces to (34). In the case of the deterministic values of M, Nk, the moments (S~v(7)), m = 1, 2 . . . . . are calculated [for independent ~k(7)] using appropriate formulae from Section 4.2. Figure 9 shows a dependence of the standard deviation of the crack tip departure from the x-axis (formulae 40, 41 and 7) on the number of changes M in the direction of crack propagation. An illustrative plot of crack growth trajectories simulated (for deterministic M > 1, Nk > 1) is shown in Fig. 10. The increments in the curvilinear crack growth are constructed in the manner shown in Fig. 3(b). The total number of increments is N = 277 x 103, with each number of local direction increments being Nk= N/M (M = 100). Figure l l(a and b) shows the simulated trajectories of the curvilinear crack growth when Nk--the number of increments along direction Ok(7)--is a uniformly distributed random variable on the intervals [1, 3.3 x 103] and [1, 2.0 x 103], respectively. 6. C O N C L U S I O N S
The method of modelling fatigue crack growth presented herein provides a mathematical scheme for characterization of specific features of curvilinear crack growth. The presented
Modelling of curvilinear random fatigue crack growth
715
description seems to be capacious enough to include various realistic situations resulting in departure of crack growth from the straight-line direction. The descriptive capability of the method presented is grounded in the flexibility of the choice of various counting processes (or random variables) characterizing the random number of crack increments, including various intensities of the crack growth process, and in various (depending on the specific situation) possible ways of characterization of the "angle fluctuation factors" w [O,], u[O~]. The representation of the model in the form of a cumulative jump process makes it probabilistically and computationally effective. As has been shown, one can efficiently calculate the exact moments and approximated probability distribution of the basic quantities. The basic restrictive hypothesis of the analysis presented in the paper is the independence of elementary crack increments; as it was shown in the paper[12], the correlation of the random increments significantly affects the variance of the crack size, especially for large values of time. The analysis presented in [12] can be extended directly to the curvilinear crack growth analysis discussed herein. In addition, it seems that in the situation when curvilinear growth is mainly due to biaxiality of the loading (and mixed mode fracture), there is the possibility of relating the proposed description with the analysis presented in refs [5], [9] (cf. also [12]). An effort toward such a relationship will be a future concern of the authors. Acknowledgement--The support of this research under the Maria Sklodowska-Curie Joint U.S./Polish Fund is gratefully acknowledged.
REFERENCES [1] K. Sobczyk and B. F. Spencer, Random Fatigue: From Data to Theory. Academic Press, New York (1992). [2] J. C. Grosskrentz, The mechanisms of metal fatigue. Phys. Stat. Sol. 47(6), 359-375 (1971). [3] S. Kocanda and J. Kur, Investigation of velocity and crack surfaces in steel 45 under varying torsion, Biuletyn WAT, No. 12, pp. 17-28, 1981 (in Polish). [4] G. H. Besterfield, W. K. Liu and M. A. Lawrence, Fatigue crack growth reliability by probabilistic finite elements, Comput. Math. appl. Mech. Engng 86, 297-320 (1991). [5] V. N. Shlyannikov and N. Z. Braude, A model for prediction crack growth rate for mixed mode fracture under biaxial loads. Fatigue Fract. Mater. Struct. 15(9), 825-844 (1992). [6] Y. C. Lain, Fatigue crack growth under biaxial loading, Fatigue Fract. Engng Mater. Struct. 16(4), 825-844 (1993). [7] H. Kitagawa, R. Yuuki and T. Ohira, Crack-morphological aspects in fracture mechanics, Engng Fracture Mech. 7, 515-529 (1975). [8] S. Suresh, Crack deflection: implications for the growth of long and short fatigue cracks, Metall. Trans. 14A, 2375-2385 (1983). [9] G. C. Sih, Strain energy density factor applied to mixed mode crack problems. Int. J. Fract. Mech. 10, 305-321 (1974). [10] K. Sobczyk, Stochastic models for fatigue damage of materials, Adv. Appl. Probability 19, 6524~73 (1987). [11] K. Sobczyk and J. Trebicki, Modelling of random fatigue by cumulative jump processes, Engng Fracture Mech. 34, 477--492 (1989). [12] K. Sobczyk and J. Trebicki, Cumulative jump-correlated model for random fatigue. Engng Fracture Mech. 40(1), 201-210 (1991). [13] W. Feller, Introduction to Probability Theory and Its Applications, Wiley, New York (1968). [14] V. H. Kogaiev and S. G. Liebiedinskij, Probabilistic model of fatigue crack growth. Mashinov&dien!je 4, 78 83, 1983 (in Russian). [15] B. F. Spencer and L. A. Bergman, On the estimation of failure probability having prescribed statistical moments of first passage time. Probabilistic Engng Mech. 1(3), (1986). [16] K. Sobczyk and J. Trebicki, Maximum entropy principle in stochastic dynamics, Probabilistic Engng Mech. 5(3), (1990). [17] S. Kocanda and J. Szala, Principles of Fatigue Calculations. PWN, Warsaw, 1985 (in Polish). [18] G. C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer, Utrecht (1991). [19] M. Grigoriu, S. E1 Borgi, M. Saif and A. R. Ingraffea, Probabilistic prediction of mixed mode fracture initiation and trajectory under random stress, Probabilistic Methods in Civil Engng (Edited P. D. Spanos), Proc. 5th ASCE Specialty Conf., Blacksburg, Virginia (1988). [20] J. Eftis and N. Subramonian, The inclined crack under biaxial load, Engng Fracture Mech. 10, 43~7 (1978). (Received 28 April 1994)
Pergamon
Engineering Fracture Mechanics Vol. 52, No. 4, pp. 703 715, 1995 Copyright © 1995 Elsevier Science Ltd 0013--7944(95)00042-9 Printed in Great Britain. All rights reserved 0013-7944/95 $9.50 + 0.00
MODELLING OF CURVILINEAR R A N D O M FATIGUE CRACK GROWTH K. SOBCZYK and J. TREBICKI Center of Mechanics, Institute of Fundamental TechnologicalResearch, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland B. F. SPENCER, JR Department of Civil Engineering and GeologicalSciences, University of Notre Dame, Notre Dame, IN 465564)767, U.S.A. Abstract--In the paper, stochastic description of curvilinearfatigue crack growth is provided. The crack growth process is regarded as a cumulative random jump process consisting of a random number of random incrementswith random angles. The probabilitydistribution of the crack size representedby the model is determined and the relation of the basic model parameters to fracture mechanics is analyzed. The description presented seems to be capacious enough to include various realistic situations resulting in departure of crack growth from the straight-line direction.
1. I N T R O D U C T I O N FATIGUE crack growth in engineering materials is a complicated process which is sensitive to many factors, e.g. various loading conditions, material inhomogeneity, environment, etc. Because of the difficulties in accurately describing the fatigue process and the significant random scatter in empirical data, it is now commonly accepted that the prediction of fatigue fracture can rationally be performed by use of a stochastic approach [1]. The existing stochastic descriptions are, however, almost entirely concentrated on uniaxial loading (and pure mode I fatigue) and on growth along a straight path. Cracks rarely grow exactly along a straight line. Even when the applied stress is uniaxial, randomness in the crack geometry, material properties and metallurgical structure may result in unpredictable departures of the crack path from a straight trajectory. For example, examination of three-dimensional replicates of fatigue cracks in pure aluminium specimens by scanning electron microscopy shows the crack front to be highly non-uniform; the crack advances on many "minifronts", and the local direction of growth may deviate several tens of degrees from the average direction [2]. Very irregular crack growth has been observed (cf. Fig. 1), for example, under torsion of steel elements [3]. When the loading process is multiaxial, an interesting aspect of the fatigue process is that the crack may grow in a mixed mode manner. An adequate description of multiaxial fatigue should include both the crack growth direction and the rate; in general, a curvilinear fatigue crack path has to be considered [445]. Figure 2 shows an experimentally obtained irregular crack trajectory given in [6]. Also, as it was underlined by Kitagawa et al. [7], "in all efforts of combining micro and macro fracture processes, the crack morphology seems to have significant meaning, because a microscopic crack usually has a complicated geometry, for example, the concave and convex crack front, the zig-zag crack path or branching". In ref. [8], Suresh studied various types of crack deflections and conjectured that crack deflections, and in general the nonlinear crack path, play a major role in influencing the crack growth of short fatigue cracks. In engineering fracture mechanics, several criteria have been proposed to determine the initial crack growth direction in a mixed mode situation. For example, one of the best known is the criterion of Sih [9], according to which the crack is assumed to grow in the direction of the maximum energy release rate, or in the direction of the minimum of the strain energy density function. Although these criteria highlight the mechanical aspects of mixed mode fatigue cracking, existing studies indicate that difficulties remain in prediction of the crack growth trajectories. Any 703
704
K. SOBCZYK et al.
Fig. 1. Experimental crack growth trajectories under torsion (from Kocanda and Kur [3]).
material inhomogeneity or inaccuracy in determination of the stress intensity factors, as well as in the validity of the criterion chosen for initial crack direction, contribute to errors and an increase in unpredictability. In this situation, a stochastic description of the curvilinear crack growth seems to be both rational and useful. Of course, the lack of reliable experimental data associated with non-straight crack growth makes it difficult to perform fully satisfactory estimation of the parameters and the subsequent model verification. Nevertheless, it is justified to construct a mathematical scheme (a possible "pre-model") into which the random curvilinear crack growth process can be imbedded. In this paper, we propose a stochastic description for curvilinear crack growth in terms of the cumulative random jump processes. We regard the crack growth process as a discontinuous random process consisting of a random number of random increments with random angles. The model presented herein constitutes an extension of the modelling ideas reported in papers by Sobczyk [10] and Sobczyk and Trebicki [11, 12] (see also Sobczyk and Spencer [1]). 2. DESCRIPTION OF THE MODEL According to many experimental studies, the fatigue crack growth process is discrete and random in time. At the same time, it is a cumulative process. These basic observations led the authors of papers [11, 12] to construct a model for straight-line fatigue crack growth of the form N(t)
A(t,?)=Ao+ ~,A,(?), A,(7)=AA,
(I)
i=l
~0
i
i
l
i s¢
l,O
.~
E 2o.~,,
~
,
t
,
l~axiat stress ratio =05
o
#
#
,~....
o
-40
' ~
(----)predicted
'~
(~) -60
-8o 40
-;o-2o
I
o
experimentot
I
2o
I
to
8'0
80
Crock size (ram)
Fig. 2. Experimental crack growth trajectories under biaxial stress (from Lam [6]).
Modelling of curvilinear random fatigue crack growth
705
a)
.[ x-axis
c)
b)
Xi
Xi
x-axis
x-axis
Fig. 3. Illustration of the geometry of the model.
where A (t, 7) is the length of a dominant crack at time t. The letter y symbolizes the elementary event belonging to F, where F is a space (sample space) on which the probability measure is defined; for each y ~ F, A (t, y) represents a possible realization of the crack length process. The value A0 is the initial crack length, which is of sufficient size to propagate (assumed to be known from empirical data), Aj(7) are random variables characterizing the elementary crack increments; N(t) is an integer-valued stochastic process characterizing the number of crack increments in the interval
(0, t]. Let us assume now that (because of material heterogeneity, uncertainty in initial crack direction and position, departure from uniaxiality, etc.) the crack grows not along a straight line, but along a curvilinear path, fluctuating around the straight line (x-axis). In order to characterize such a process let us introduce the angle O i between the actual crack increments and x-axis (cfi Fig. 3a); in general Oi~ ( - ~ / 2 , n/2), but it seems that in situations of engineering interest, the "path fluctuation angles" Oi should not depart strongly from zero. We propose the general description of a random curvilinear crack growth process in the form of a two-dimensional process [A (t, 7), Z(t, ),)] N(0
A(t, 7)=Ao+ ~ Y~(7)
(2)
i=l
N(O
Z(t, 7 ) = Y'. Z,(7)
(3)
i=l
where A(t, 7) is now the stochastic process characterizing the crack length along the curved path at time t, Y~(~) are random variables representing the increments in the curvilinear crack length, Z(t, 7) is a stochastic process characterizing the departure of the crack tip (on the curved path) from the x-axis (that is, from a straight line) and Zi(7) are random variables describing elementary departures of the crack tip in the ith increment from the local straight line. In general, the random variables Y~(7) and Z~(7) can be related to the random variables X~(y)--characterizing the elementary increments in certain auxiliary process--as follows:
r,(y) = w[O,(7)lX,(y)
(4)
z~(~) = u[e,(~,)]x,(~)
(s)
where w(. ) and u(. ) are appropriate functions suggested by empirical information concerning elementary crack increments in the curvilinear crack growth, and O~(7) are the random variables which are assumed to be independent of Xi(~).
706
K. SOBCZYK et al. It seems that the most natural realization of the relationship in (4) and (5) takes the form:
0,(7)
(6)
2",.(7) = X~(7) tan 0,(7)
(7)
Y,(7) = x , ( 7 ) sec
where s e c ( . ) = l/cos(. ). The above formulae indicate that )(,.(7) (elementary increment along x-axis) is an orthogonal projection of Y~(7) on the x-axis and Z~(~) is shown in Fig. 3(b). Other possible examples of the functions w(. ) and u(. ) may result from specific relationships between the true curvilinear crack increments and an observable counterpart in the experiment; for instance, if Yg(7) ~< X~(y), one may assume that: Y~(7) = X~(~,)cos 0,(~,)
(8)
Z~(?) = X~(7) sin O~(7)
(9)
which indicates that the Y~(7) are taken as the orthogonal projections of the observables X~(7) with the actual value of O~(7) (cf. Fig. 3c). It is clear that formulae (4) and (5) are general enough to reflect the true empirical properties of the elementary (curvilinear) crack increments I1,.and their relation to X~(7). It should be noticed that formulae 4 and 5----expressing the basic quantities of the curvilinear crack growth in terms of increments of X~ (and simple functions of O,)--make the model represented by formulae (2) and (3) highly versatile; for example, one can use the results concerning X~(7) from the analysis of straight-line growth models.
3. RELATION TO FRACTURE M E C H A N I C S The mathematical scheme for the curvilinear fatigue crack growth presented above in the form of the formulae (2) and (3), and (4) and (5) includes as the model parameters: the initial crack length (A0), the intensity (2o) of the random counting process N(t), the parameters characterizing the probability distribution of X~(7), and the characteristics of the random angles O~(7). The model, like its predecessor described in papers [11, 12], is general and does not include the loading process explicitly; the characteristics of the applied stress (sinusoidal, or random) are incorporated into the properties of the process N(t), as well as in the characteristics of random variables X~(7) describing the elementary increments in the observable process; in the case of random loading, they can be identified in a manner analogous to that used in papers [11, 12]. Here, the basic quantity is the random fluctuation angle @~(7) characterizing the direction (with respect to the x-axis) of each elementary curvilinear crack increment. In general, the probability distribution re, (~) of random O~(7) should be estimated from the empirical data, or inferred from the physics of the phenomenon. In practice, it is easiest to hypothesize the probability distribution re, (,9) and then to estimate its parameters. As a first step in this direction, it seems to be useful to look at the possible relationship between the randomness in Or and the Sih criteria for initial crack growth direction (cf. [9, 18]). If the initial crack position in an infinite plate is perpendicular to the applied stress and the crack grows in mode I(Kl = a 2 x / / ~ , Ku = 0), the strain energy density function for the initial direction of the crack propagation is ¢r2A S = ~ (3 - 4v - cos @)(q + cos O)
(10)
where A is the crack length, a is the applied deterministic stress, p is the shear modulus, v is Poisson's ratio, and @ is the polar angle for the cylindrical coordinates of a point with respect to the crack tip. According to the Sih criterion, the crack will initially propagate in a radial direction along which the strain energy density function has a minimum, that is when aS 0-~ = 0,
3zS ~ - 7 > O.
(ll)
Modelling of curvilinear random fatigue crack growth
707
2.30
(xC) 2.25
2.20 2.15 2.10
'°~ - ..... .......:j ~ ' d....................
~
2.05
2.00 1.95 1.90 , , , , , , , , , ~,
, , , , , , , ,' L
SO'
, , , , , 2' ' ' ' I
' 0' '
' ' ' ' 4' 0I
' ' ' ' ' ' 't
50
' I ' ' ' '
'
a 60
Fig. 4. The strain energy density function S(~) (bold line) and (S[O(7)]) as function of a for uniform and Gaussian distribution of initial direction O(~), respectively; v = 0.25, G = 16#/a2A, ot in degrees.
This condition gives O = O0 = 0 and (1 -- 2v)o'2A Smin =
(12)
4g
Let us randomize formula (10) by assuming that the initial angle O has some random uncertainty, that is we treat O as a random variable O(V) with zero mean and a one-parameter probability density)Co (g; ~). The parameter ~ can be estimated (for example) from the following criterion max
< S(7) > - &.i°
~< E
(13)
Smin
where ( • ) denotes the mean value, ( S ( 7 ) ) = S S(0)fe (~; ~) dO, and c is a given number. Therefore, the parameter • quantifying the scatter in the angle 0 ( 7 ) (e.g. • could be the parameter in the uniform distribution on the interval [ - ~ , ~] or the standard deviation in the Gaussian distribution) is determined as a maximal distribution parameter for which the relative error between the deterministic Stain given by (12) and mean value of the random variable S(y) (being a function (10) 8.0
[~]
(1) -
/,1=0.15
(2)
u=o.25
-
Gatusmiaa
....
(3) - b'=0.35
.2 6.0
~ ~-
/ ""
.o~/"/"
¢s
(a!,
(3)/
'~4.0
0
~'~ 2 . 0 0
0.0
rlllll
0
i
i l l l l
10
ill
i
i
i l l [ i t
20
i
iii
ii
l i l t
30
ii
T ? I , I ]
i11
40
ii
i
r ii
jSl
50
i
lIT
III
Or. 60
Fig. 5. Variation coefficient q of S[O(7)] vs ct for different Poisson ratio v; initial direction 0(7) has uniform and Gaussian distribution with parameter a respectively; ~t in degrees. EFM 52/4~H
708
K. SOBCZYK et al.
4C10~
(1) - ~'=0.001 (2)
3o~ \
10 2
O~
~--~
........
O.15
-
~=o.oo~
(3) - ~=o.ol
~ .........
0.20
~
~ .........
0.25
~ .........
0.30
; .........
0.35
; .........
0.40
V
0.45
Fig. 6. The scatter parameter • vs the Poisson coefficient v for different errors E; initial direction O(y)
has uniform and Gaussian distribution with parameter a, respectively;~ in degrees. of 0(7)) is not greater than the given value of E. Figure 4 shows for ~ e [0, 60 °) the plots of the strain energy density function (10) for O = ~ and its mean values when O is treated as random variable with uniform and Gaussian distributions, respectively, with parameters ~. The percentage coefficient of variation q = ~ S / ( S ) , (var S = ( S 2) - (S)2), of the random variable S(7), as a function of 0t is presented in Fig. 5. Simple calculations show that even for small E (of order 0.005 = 0.5%) and for v e [0.15, 0.3] we obtain quite large values of ~ satisfying (13). This fact is illustrated in Fig. 6 and can be interpreted that even when a deterministic mode I loading condition is present, some random deviations from ~9 -- O0 = 0 may occur. This implies that the initial crack direction can be considered as a random variable O (7)- As long as the departure of the crack tip from the x-axis is small during the crack propagation, the hypothesis that KH = 0 and that the crack is perpendicular to the stress direction seems to be justified, and the reasoning presented above can be adopted for predicting the direction of such a curvilinear crack growth. If the departure of the crack tip from x-axis is significant and the actual ith crack position is inclined at some random angle fli with the uniaxial stress axis (y-axis), then K~i) ~ 0 and K~ ) ¢ 0. In this situation a possible crack growth direction ~;+~ can be calculated (for example) from the condition of the maximum circumferential stress (cf. Sih [18]), that is from equation KIi) sin ~ + l + KI~)(3 cos ~ + ,
--
1) = 0.
(14)
When this condition is randomized by introducing random variables into K~ and K, (e.g. analogously to Grigoriu et al. [19]), we obtain the probability distribution of the random angles ~ + i. This type of distribution can be adopted to characterize the sequence of random angles of the elementary crack increments. Of course, completely satisfactory linking of the model presented to fracture mechanics requires general solutions for the stress intensity factors for curvilinear (e.g. zig-zagging) cracks. This, however, is a task which seems to exceed the present possibilities of contemporary fracture mechanics [7, 8]. But, this difficulty may be seen as the primary impetus for constructing a stochastic model (such a suggestion was similarly made by Kitagawa et al. [7]). In further analysis and in the illustrative examples, the distributions of the angles ~9~ are assumed to be given. 4. P R O B A B I L I T Y
DISTRIBUTIONS
OF THE MODEL
PROCESSES
4.1. General formulae Since the mathematical structure of the processes A(t, 7) and Z(t, 7) with formulae (2) and (3) is the same, let us introduce, for mathematical convenience, the process N(t)
s(t, 7)= ~ ~,(7) i=1
05)
Modelling of curvilinear random fatigue crack growth
709
which can symbolize either the crack length along curved path or the departure of the crack tip from the x-axis. In the first case, the random variables ~i(?) are given by formulae (4), whereas in the second situation by (5). Let us assume that random variables ~i(?) with the probability densities f~e(s) are statistically independent of each other and are independent of the process N(t). The distribution function Fs(s; t) for the process in (15) is
Fs(s; t) = P[S(t, 7) ~
(16)
k=l
where Pk(t) = P[N(t) = k]. The probability densityfs(s; t) for S(t, ?) is expressed by the formula
fs(s; t) = O~2Fs(s; t) = ~ fs(s IN(t) = k )Pk(t) U~
(17)
k=|
where fs(slN(t)= k) is the density of the random variable k
Sk(~) = 2
~i(~)"
(18)
i=1
Since the ~(7) are independent
fs(s IN(t) = k) =f¢, (s)*f¢ 2(s)* • • • *f~k(s) = G(~)(s)
(19)
where * denotes convolution, and
k(~)G(k-°(s-~)dz,
G~k)(s) =
k=2,3 ....
(20)
Finally, the one-dimensional probability density for the process S(t, 7) takes the general form
fs(s; t) =
G(g)(S)Pk(t)= kffil
Pk(t) k=l
f0
f~ (z)G (k-|)(s -- z) dr.
(21)
To determiners(s; t) explicitly, one has to assume a specific random counting process N(t) with given Pk(t) and specific distribution f¢~(x). If, for example, the random variables ~i(?) are exponentially distributed, that is
f~,(x)=~gexp(-~ix), = 0,
x >10
x <0
(22)
then (cf. Feller [13]) G(k)(s) = (i=I~, ~ ) ( ~
~.k exp(-- ~is))
Oi.k = [(~, -- ~,) " "" ( ~ - , -- ~,)(~+, -- ~,)" "" (~k -- ~,)1-'.
(23)
If the ~(7) are identically exponentially distributed (i.e. ~ = ~), and N(t) is the Poisson, or pure birth process, the distribution density fs(s; t) is given in [11]. 4.2. Moments of the model processes Analytical determination of the probability distribution of the process S(t, 7) according to the formulae given above may not be easy, mainly because of difficulties in analytical calculations of the convolution integrals in (20) for large k. Also numerical calculations of these convolutions can be computationally cumbersome. Therefore, it is of interest to determine the moments of the process S(t, 7)- Under the assumptions regarding N(t) and ~(7) made in Section 4.1, the interesting moments are as follows
(24)
k=O
k=O
E7 =
~,(?) i=1
.
(25)
710
K. SOBCZYK e t a l .
Making use of Newton's formula, we have the following expression for E~: m!
E~'=
~
hi + ' " + h k
=
mh, !"
• • hk!
<¢1,">.
(26)
If the ~ (7) are mutually independent, then m!
x"
L k ~_ h , ' hl+...+h
k
l-'l
h~,,L~ . . .=
(27)
xh~, (x) dx.
(28)
•. .
where (~f'(Y))
=
For instance, the first conditional moments (27) occurring in the general formula (24) are
,)k
i=1
2
k
= Y', ( ¢ ~ ) + 2 ~ (¢,)(¢j)
~,(7)
i= I
i
k
~i(7)3 = E <~> ''1-3 E <¢2><~j> i= 1
i
+ 3 ~ (~i)(~) +6 i
~
(¢i)(~j)(~k).
(29)
i
To make further analysis more efficient, we need to assume a specific form of the probability distribution of the elementary crack increments. The empirical observations support the hypothesis of an exponential distribution for X~(y) (cf. Kogaiev and Liebiedinskij [14]), i.e.
f~,(x) = ~iexp(-~ix),
x ~>0.
(30)
Therefore
(31)
Assuming that the random elementary increments Xi(Y) and the elementary angles Oi(7) are independent, the processes A (t, 7) and Z(t, 7) defining the curvilinear crack growth and given by (2), (3) and (4), (5), have moments determined by formulae (24) and (27), where for the process ~, - -~--N(,) t=, ~(7) A ~(t, "~
<~,~'(7)> =
(x~'(7)>
= ¢Theh~!
L
Wh' (9 if'o, (oa)dO
(32)
and fe, (~) is the probability density of O~(7), ~ ~ [a, b] = ( - x / 2 , z/2). For the process Z(t, 7) = Zi= N(t)t Z~(y) we have
<~,~'0)> =
(x~'o)F
f
= ~?h'h~!
uh'(~)fo, (~) d&
(33)
The moments calculated according to the formulae presented above provide important information about the model processes. However, they can also be used for determining the approximate probability via the maximum entropy principle (cf. Spencer and Bergman [15], Sobczyk and Trebicki [16]).
Modelling of curvilinear random fatigue crack growth 5. S I N U S O I D A L
711
LOADING
5.1. Estimation of mean elementary increments The analysis presented above, as has been previously indicated, does not explicitly consider the loading process. The applied stress enters the model through the properties of the counting process N(t) and by the characteristics of the random elementary crack increments. In general, the loading can be sinusoidal as well as random. In the case of random loading, the intensity of the counting process N ( t ) can be related to the peaks of the random applied stress (cf. [11]). Let us assume here, that the loading is sinusoidal with a given frequency and that the crack advances take place in each cycle. In this case, the process (15) reduces to the random variable N
SN(7) = ~ ~i(7)
(34)
i=l
where N denotes the number of cycles (number of increments of the crack growth). The probability density of SN(7) can be easily expressed by a convolution integral. If the ¢~(y) are independent, exponentially distributed random variables then formula (23) holds. After a crack has initiated, the number of cycles until the crack reaches its critical (or control) size A * is usually in the order of a few hundred thousand. In such situations one can use the central limit theorem to assert that for large N, SN(V) can be approximated by a Gaussian random variable with the mean value mSN and variance var SN, where N
N
nSN = ~ <¢i(7)>,
var SN = ~ var ~i
i=1
i=1
var ~, = <¢2(7)> - <~(V)> 2.
(35)
The parameter ~gin the exponential distribution of the crack increments Xi(7) can be estimated (using conventional data) from the following condition
(36)
where the left-hand side denotes the mean value of AN(t, V) = A0 + E~=~ X~(7), and the right-hand side is the crack size predicted by the Paris equation for the same number of cycles. The mean of the random variables X~(7) characterizing the ith crack increment is taken as an increment in the ith cycle predicted by the Paris equation, that is
(37)
where Ae(O) = Ao, 1 <<.i <<.N*, and N* is the number of cycles to reach the critical (or control) size A *. Therefore. 1 AAp(i) where (cf. [17]) 2 --
Ae(N)=
K
A~o2 ~"2 + - ~ - - C , N
)2,'(2 - h') ,
C,=C(Aax/~) 2
(38)
and C, ~c are the material parameters occurring in the Paris equation; Aa is the difference between the maximum stress areax and the minimum stress ami, for the sinusoidal loading. Making use of the above values of the parameters ~t in the exponential distribution of the crack increments and taking the same material and experimental conditions as in Kocanda and Szala [17], (A0=0.01 m, C = 1.03 x 10 -12, m =3.89, area,= 120MPa, amin=40MPa, A* = 0 . 0 6 m , N* = 277329 cycles), one can obtain an effective characterization of the curvilinear crack growth according to the method described. The results of numerical calculations performed for the case of sinusoidal loading are shown in Figs 7-10. A uniform distribution on the interval [-~t, 7] ()co, (,9; ~) = 1/2a, ,9 ~ [ - ~, ~ ]) is taken for the random variables Oi (V). Figure 7 shows a comparison between the probability density of the crack length according to the proposed curvilinear modelling and the probability density of the crack length predicted by straight-line growth theory. Figure 8 represents the corresponding mean values of the crack length vs the number of cycles N e [1, N*]
712
K. SOBCZYK et al. 1.2
7/ o
~0.8
t
I / /
7/
i/~
it i
~ increments ~ a s I n Pig. 3 b
,
,
,'
',
~
I
I
,'
',
~0.4
mm (+A0)
0.0 41.5
42.5
43.5
44.5
45.5
46.5
Fig. 7. Probability density of the crack length, straight-line growth ( - - ) , (. . . . ) follows eqs (34) and (6). The directions O~(7) have uniform distribution on [ - ~ , ~]; ~ = 25 °, number of cycles N = 270 x 103.
for different values of ~. It is seen that modelling based on the curvilinear growth gives larger mean values as compared to the theory of straight-line crack growth, after the same number of cycles N. As far as the departure process ZN(~,) of the crack tip from the straight-line direction is concerned, the model analyzed above has the ability to predict the mean behaviour of the crack growth trajectories; however, calculations give a very small variance of ZN(y), and AN(y). This result implies that the model is inadequate to portray the macroscopic curvilinear crack growth. This is however, caused mainly by the assumption of independence of random elementary increments, which can be removed (cf. [12]). 5.2. Generalization It has been assumed in the previous sections that each load cycle generates a crack increment and each elementary increment has a different random angle ~9i. However, such a hypothesis may not be sufficiently realistic; the crack may change its direction only after some number of elementary (micro) increments. To incorporate this phenomenon into the model the following formula for the
60.0 /
£
(1)
- a=e5
(2)
- ~ =450
°
/
/
/~ (2)...///
40.0 o (1)~/
// /,7
"~ 20.0 2 "-(+~) 0.0 0
"~~llljl,tljl~llll 50
100
150
200
250
I
(xzooo)
300
Fig. 8. Mean value of the crack length dependent on the values of parameter ~t; directions O~(7) have uniform distribution on [-ct, ~t]; ( ) mean value of the straight-line growth, (. . . . ) follows eqs (34) and (6), N - - n u m b e r of cycles.
Modelling of curvilinear random fatigue crack growth
4.80
713
\ \ \
.0
\
3.80
03
\ \
ID
\a
=20
=
,.~ 2.80 "0 to
"~
0~=15
o
~
~
~
~'1 1.80
0180
. . . . . . . . .
tO
[ . . . . . . . . .
30
'
. . . . . . . . .
[ . . . . . . . . .
50
70
[ . . . . . . . . .
90
llO
Fig. 9. Standard deviation of the crack tip departure for different values of M ; ( ) follows eqs (34) and (7), (. . . . ) follows eqs (34), (9), directions Oi(y) have uniform distribution on [-~t, ~,]; number of cycles N = 270 x 10~.
model process SN(7) can be used M
Su(?) = ~ ~k(?)
(39)
k=l
where for the crack length ~'k(?)
= w [Ok (?)]
~ X~(?)
(40)
i=1
or for the departure of crack tip from x-axis Nk
~',(?)
= u[Ok(?)] ~ X,(?).
(41)
i=l
The total number o f increments N = Ni + N2 + " • • +
NM, where Nk (k
= 1. . . . . M ) are the number
of elementary increments along the direction specified by the random angle Ok(?), and M is the number of possible different directions occurring during crack propagation. In general, the numbers M, N x can be either deterministic or random, and the random variables ~'k(?) can be treated as I0.00
,",
5.00
•N o.oo ¢.)
tO r~ -5,00
-lO.OO
.........
0.00
~ .........
10.00
~ .........
20.00
~ .........
30.00
~ .........
40.00
50.00
Fig. 10. Simulated crack growth trajectories for sinusoidal loading using model (39); directions Oi(7) have uniform distribution on [ - e , e]; e = 25°; Nk---deterministic.
714
K. SOBCZYK et al.
(a)
10.00
5.00
•~
o.oo
-5.00
-10.00 0.0
(b)
I I l + l l l l l l l l l l l , l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
m.ln
10.00
20.00
30.00
40.00
50.00
10.00
P-O.O0
30.00
40.00
50.00
5.00
3.00
1.00 N
~-
¢S
1.00
-3.00
-5.00
0.00
Fig. 11. Simulated crack growth trajectories for sinusoidal loading using model (39); directions ei(7) have uniform distribution on [-ct, ct]; ~ = 25 °, Nk--random from [1, 3.3 x 103].
mutually correlated. It is seen that in the particular case when M, Ark are deterministic and Nk = ! for each k (hence M = N), then formula (39) reduces to (34). In the case of the deterministic values of M, Nk, the moments (S~v(7)), m = 1, 2 . . . . . are calculated [for independent ~k(7)] using appropriate formulae from Section 4.2. Figure 9 shows a dependence of the standard deviation of the crack tip departure from the x-axis (formulae 40, 41 and 7) on the number of changes M in the direction of crack propagation. An illustrative plot of crack growth trajectories simulated (for deterministic M > 1, Nk > 1) is shown in Fig. 10. The increments in the curvilinear crack growth are constructed in the manner shown in Fig. 3(b). The total number of increments is N = 277 x 103, with each number of local direction increments being Nk= N/M (M = 100). Figure l l(a and b) shows the simulated trajectories of the curvilinear crack growth when Nk--the number of increments along direction Ok(7)--is a uniformly distributed random variable on the intervals [1, 3.3 x 103] and [1, 2.0 x 103], respectively. 6. C O N C L U S I O N S
The method of modelling fatigue crack growth presented herein provides a mathematical scheme for characterization of specific features of curvilinear crack growth. The presented
Modelling of curvilinear random fatigue crack growth
715
description seems to be capacious enough to include various realistic situations resulting in departure of crack growth from the straight-line direction. The descriptive capability of the method presented is grounded in the flexibility of the choice of various counting processes (or random variables) characterizing the random number of crack increments, including various intensities of the crack growth process, and in various (depending on the specific situation) possible ways of characterization of the "angle fluctuation factors" w [O,], u[O~]. The representation of the model in the form of a cumulative jump process makes it probabilistically and computationally effective. As has been shown, one can efficiently calculate the exact moments and approximated probability distribution of the basic quantities. The basic restrictive hypothesis of the analysis presented in the paper is the independence of elementary crack increments; as it was shown in the paper[12], the correlation of the random increments significantly affects the variance of the crack size, especially for large values of time. The analysis presented in [12] can be extended directly to the curvilinear crack growth analysis discussed herein. In addition, it seems that in the situation when curvilinear growth is mainly due to biaxiality of the loading (and mixed mode fracture), there is the possibility of relating the proposed description with the analysis presented in refs [5], [9] (cf. also [12]). An effort toward such a relationship will be a future concern of the authors. Acknowledgement--The support of this research under the Maria Sklodowska-Curie Joint U.S./Polish Fund is gratefully acknowledged.
REFERENCES [1] K. Sobczyk and B. F. Spencer, Random Fatigue: From Data to Theory. Academic Press, New York (1992). [2] J. C. Grosskrentz, The mechanisms of metal fatigue. Phys. Stat. Sol. 47(6), 359-375 (1971). [3] S. Kocanda and J. Kur, Investigation of velocity and crack surfaces in steel 45 under varying torsion, Biuletyn WAT, No. 12, pp. 17-28, 1981 (in Polish). [4] G. H. Besterfield, W. K. Liu and M. A. Lawrence, Fatigue crack growth reliability by probabilistic finite elements, Comput. Math. appl. Mech. Engng 86, 297-320 (1991). [5] V. N. Shlyannikov and N. Z. Braude, A model for prediction crack growth rate for mixed mode fracture under biaxial loads. Fatigue Fract. Mater. Struct. 15(9), 825-844 (1992). [6] Y. C. Lain, Fatigue crack growth under biaxial loading, Fatigue Fract. Engng Mater. Struct. 16(4), 825-844 (1993). [7] H. Kitagawa, R. Yuuki and T. Ohira, Crack-morphological aspects in fracture mechanics, Engng Fracture Mech. 7, 515-529 (1975). [8] S. Suresh, Crack deflection: implications for the growth of long and short fatigue cracks, Metall. Trans. 14A, 2375-2385 (1983). [9] G. C. Sih, Strain energy density factor applied to mixed mode crack problems. Int. J. Fract. Mech. 10, 305-321 (1974). [10] K. Sobczyk, Stochastic models for fatigue damage of materials, Adv. Appl. Probability 19, 6524~73 (1987). [11] K. Sobczyk and J. Trebicki, Modelling of random fatigue by cumulative jump processes, Engng Fracture Mech. 34, 477--492 (1989). [12] K. Sobczyk and J. Trebicki, Cumulative jump-correlated model for random fatigue. Engng Fracture Mech. 40(1), 201-210 (1991). [13] W. Feller, Introduction to Probability Theory and Its Applications, Wiley, New York (1968). [14] V. H. Kogaiev and S. G. Liebiedinskij, Probabilistic model of fatigue crack growth. Mashinov&dien!je 4, 78 83, 1983 (in Russian). [15] B. F. Spencer and L. A. Bergman, On the estimation of failure probability having prescribed statistical moments of first passage time. Probabilistic Engng Mech. 1(3), (1986). [16] K. Sobczyk and J. Trebicki, Maximum entropy principle in stochastic dynamics, Probabilistic Engng Mech. 5(3), (1990). [17] S. Kocanda and J. Szala, Principles of Fatigue Calculations. PWN, Warsaw, 1985 (in Polish). [18] G. C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer, Utrecht (1991). [19] M. Grigoriu, S. E1 Borgi, M. Saif and A. R. Ingraffea, Probabilistic prediction of mixed mode fracture initiation and trajectory under random stress, Probabilistic Methods in Civil Engng (Edited P. D. Spanos), Proc. 5th ASCE Specialty Conf., Blacksburg, Virginia (1988). [20] J. Eftis and N. Subramonian, The inclined crack under biaxial load, Engng Fracture Mech. 10, 43~7 (1978). (Received 28 April 1994)