Probabilistic model of early fatigue crack growth

Prob. Engng. Mech. Vol. PH: S 0 2 6 6 - 8 9 2 0 ( 9 7 ) 0 0 0 2 9 - 5

ELSEVIER

13, No. 3, pp. 227-232, 1998 © 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved. 0266-8920/98 $19.00 + 0.00

Probabilistic model of early fatigue crack growth V. V. Bolotin a, A. A. Babldn b & I. L. Belousov b alnstitute of Mechanical Engineering Research, Russian Academy of Sciences, 101830 Moscow, Russia

bMoscow Power Engineering Institute~Technical University, 111250 Moscow, Russia A model is developed in this paper to describe the nucleation and early growth of fatigue cracks. Polycrystalline materials are modelled as a set of elements (grains) with random properties. It is assumed that the resistance to damage of neighboring elements is mutually independent and follows the same probability distribution, except for the elements situated near the surface whose resistance is lower and which are subjected to higher scattering. Damage accumulation in each element due to cyclic loading is considered, and an element is treated as ruptured when a critical damage level is attained; then the ruptured element is included in the cracked domain. The finite element technique is applied to realize the modelling. Numerical results exhibit all the principal features of early fatigue crack growth such as nonmonotonous change of crack growth rates, statistical scatter of crack dimensions and growth rates, and stabilization of the process when a considerable number of grains enter the cracked domain. © 1998 Elsevier Science Limited. mesocracks in the meaning given above. However, even within this class of cracks, one needs a more detailed classification. Fatigue cracks usually originate from the surface. A typical mesoscopic fatigue crack is an elongated ensemble of damaged grains occupying the first rows of grains or sets of grains situated closely to the surface. Shallow cracks may be short along the surface, and then they are close to microscopic cracks. When a shallow crack is long enough, it may be considered as an almost sure candidate to be transformed into a regular macroscopic fatigue crack. One of the problems of fracture and fatigue mechanics is to describe, in terms of mathematical models, the transformation of a single microcrack or a bounded set of such microcracks in a regular macroscopic crack. There are two principal formation patterns of macroscopic cracks. First, it is the coalescence of microdefects into a connected damaged domain. This is possible as a result of clustering of defects randomly distributed in a body or at its surface. Damaged grains may be separated with a number of non-damaged ones, but the bridges between them are to be ruptured due to stress concentration around the interacting flaws. The second mechanism consists of the formation of short shallow cracks, their growth along the surface and the beginning of their in-depth penetration. The first pattern is met in composite materials such as unidirectional fiber compositesJ The second pattern is more typical for polycrystalline materials. This case is discussed in the present paper. It is well known that there is a significant difference between the behavior of mesoscopic and macroscopic cracks. This difference is illustrated in Fig. 1 where the

1 INTRODUCTION Fatigue crack initiation and growth are controlled by a number of random factors. Among them are: random microstructure of real materials; specimen-to-specimen and batch-to-batch scatter of mechanical properties; defects and imperfections of structural components; random character of loading and actions. Most of the listed factors act jointly, and this makes the prediction of fatigue life a rather complicated problem. Some aspects of this prediction have been widely discussed in literature. 1-3 This paper deals with one of the sides of the problem, with the influence of stochastic microstructure on early growth of fatigue cracks when crack sizes are comparable with dimensions of grains and other elements of material microstructure. One distinguishs microcracks as essential damage located in a single grain or a set of few neighboring grains, and macroscopic cracks, whose size is large compared with grain sizes. However, the macroscopic cracks occupy a wide scale interval beginning from visible cracks observed with ordinary equipment up to the cracks which cover almost all the cross-section of a specimen or structural component. There frequently exists a gap between micro- and macrocracks. For example, the grain size may be of the order of 10 #m, meantime a crack becomes visible when the order approaches 1 mm. The crack-like defects entering into such a gap need a special name, and we will call them mesoscopic cracks or, briefly, mesocracks. In literature, one often talks of 'small' or 'short' cracks. As it is seen from the context, most of these cracks in particular, 'physically short cracks', 4 may be considered as 227

228

V.V. Bolotin et al.

yl

3

L_t_L_ h (b)

? AK Fig. 1. Typical crack growth rate diagrams (lines 1 and 2 for microcracks, line 3 for mesocracks). (a) crack growth rate da/dN is plotted vs the range AK of the stress intensity factor K (hereafter we discuss model I fatigue cracks omitting the subscripts such as in ~rft). The branches 1 and 2 correspond to mesocracks, the branch 3 to macrocracks. When a crack is small enough covering, say one or two grains in the body's depth, the diminishment of the rate da/dN is frequently observed. Most micro- and mesocracks, being the candidates to develop in macroscopic cracks, cease to grow at all, while some of them begin to penetrate into depth; then we come to branch 3. Another feature of fatigue cracks is a significant scatter of their dimensions and, respectively, of their growth rates. This behavior is the result of the random structure of real materials, polycrystalline ones in particular. There is a scatter of mechanical properties of grains (elastic moduli, yield limits, ultimate stresses) as well as of the grains' orientation with respect to the directions of applied stresses and crack growth. When the number of ruptured grains entering the cracked domain is small, the effect of local material randomness is essential. It is in fact the only consistent way to explain all peculiarities of the behavior of mesoscopic fatigue cracks. When the number of grains crossed by the crack front becomes large, the role of local properties becomes secondary, and the process of fatigue crack growth is mostly controlled by average material properties. Let the surface of a body contain a number of microcracks. The aim is to follow the process of damage accumulation in the vicinity of a crack consisting of a moderate number of ruptured grains, and the further transformation of this mesocrack into a regular macroscopic fatigue crack. It is obvious that such a problem, stated in the whole scale, is very complicated. A number of simplifications are to be used, concerning both material properties and the crack shape, to obtain numerical results which can be discussed in terms of available experimental data.

2 M O D E L L I N G OF C R A C K G R O W T H Consider a mode I crack initiated from the surface of a body subjected to cyclic tension with remotely applied stresses cr~ (Fig. 2(a)). The initial surface notch has the characteristic size h (Fig. 2(b)). Due to damage accumulation in the

(c)

Fig. 2. Model of early crack growth: (a) scheme of loading; (b) crack nucleus; (c) mesoscopic crack.

neighboring grains, some of them begin to rupture. This means the propagation of the cracked area and the formation of a mesocrack. For simplification, we assume that only the grains situated in the cross-section plane are subjected to rupture. Hence, the crack is modelled as a slit of constant thickness h and of arbitrary shape in the plane (Fig. 2(c)). This means that we neglect kinking, branching, crack tip blunting, and other effects accompanying the fatigue crack propagation in real polycrystalline materials. As to material properties, we must also use far-going assumptions. With the intention to apply the finite element technique and eight-knot cubical mesh, we present grains as elementary cubes with dimensions equal to h. In real materials, it may be h = 10 or 100/xm, etc. Mechanical properties are randomly distributed among the grains. The first-row grains are assumed to be weaker than those situated deeper. This takes into account the presence of surface flaws and their significance for fatigue cracks initiation. However, the properties are assumed to be distributed independently among the neighboring grains. Among material parameters that are of stochastic nature, there are a lot of variables such as elastic moduli, yield stresses, ultimate stresses. When we start to consider grains as crystals, we must take into account their anisotropy and random distribution of their orientation with respect to applied stresses and crack growth directions. Involvement of these factors makes the problem too cumbersome. To minimize the number of parameters subjected to randomization, we assume that all the complex of grain properties is given by random value s interpreted as the resistance stress against damage accumulation. Compared with this resistance stress, the scatter of compliance characteristics, such as elastic moduli, seems to be less significant. In principle, following the idea of the stochastic finite element technique, one can take into account this kind of randomness, too. However, it will take a large amount of processor time because an extremely fine mesh is required. The damage of grains is discussed later in terms of continuum damage mechanics using a scalar damage measure

Early fatigue crack growth

229

80 70 O

60 50

ASthl

Scl ASth2

Sc2 30

Fig. 3. Cumulative distribution functions of resistance stresses: (1) for first-row grains; (2) for second-row and following grains.

0]

20 60.5,6 Here 0 --< w -< 1, and the limits correspond to virgin and ruptured grains, respectively. For each grain a separate equation of damage accumulation is used. In this study we assume this equation is of the form dw dN

Air

Atrth

(1 -- w) -

n

Here Ao is the range of the tensile stress in the considered grain; s, Aath, m, and n are material parameters. We assume that the resistance stress s is a random value, the power exponent m and n are given deterministically. As to the threshold resistance stress Atrth, it may be either random or deterministic. In the first case, not to multiply the number of random variables, we may assume Aath proportional to s. Certainly, when Aa < Aath, we must make the right-hand side in eqn (1) equal to zero. In the study of fatigue, both theoretical and experimental, the Weibull distribution is the most appropriate. 1,7 This distribution is valid both for fatigue life and characteristic stresses, such as the cyclic stress range at failure under a fixed cycle number. It is natural to use the Weibull distribution for the resistance stress s from eqn (1). For the first row grains, let F(s) = 1 - exp [ -

\(S'~-sclA---S'h' ]~/ ~' J

0 0

(1)

I

I

I

[

40

80

120

160

200

N - 10-3, cycles Fig. 4. Smoothed samples of in-depth early crack growth. entering into the fine mesh domain; generation of samples of resistance stresses sl . . . . . sn in these grains; solution of eqn (1) for these grains and comparison of the evaluated damage measures w L. . . . . w2 with the critical level; changing of the crack shape when some of the grains are ruptured or prolongation of the cyclic loading process Aa=(N). In addition, statistical treatment of results is included such as the estimation of means and standard deviations for crack dimensions and their rates. The fine mesh domain in the half-space counts 50 × 50 × 25 = 62500 elements; outside this domain a coarser mesh is used. The results of numerical simulation are presented in Figs 4 - 9 . The following numerical data for parameters 250

(2)

and for other grains 200

Here Scl, Sc2, ASthl, Astha, ~l, and oL2 are material parameters, and s >-- ASth2 in eqn (2), s >-- Asth2 in eqn (3). To take into account comparable weaknesses of the first-row grains, we assume s¢1 < Sc2. As higher scattering of resistance is expected for the first-row grains, we assume ~ < ~2. When the threshold stress Aoth in eqn (1) is deterministic, it is expedient to put A S t h : AtTth I for the first-row grains and ASth = Aoth2 for other grains. The distribution given in eqn (1) and eqn (2) are illustrated in Fig. 3. 3 N U M E R I C A L S I M U L A T I O N AND D I S C U S S I O N OF RESULTS

150

G 100

50

0

0

I 40

80

120

160

200

N-10 3, cycles The procedure of numerical simulation includes: computations of the stress ranges Aal(N) . . . . . Aan(N) in each grain

Fig. 5. Smoothed samples of along-the-surface early crack

growth.

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V . V . Bolotin et al.

107i

2.5

10~

2.0

1.5

1.0

10-9

0.5

0.0

0

10

20

30

40

50

~

I

70

80

10-,0/ 0

I

50

100

a, ~ m

I

2~

150

250

C, ~ m

Fig. 6. Smoothed samples of the crack dimensions ratio as a

Fig. 8. Smoothed samples of along-the-surface crack growth rate.

function of crack depth. entering eqns (1)-(3) were used: set = 2400 MPa, Sc2 = 3200 MPa, A s t h l = Asth2 = 320 MPa, o~1 = 2, a2 = 4, ml = m2 = 4, n = 1. The initial conditions were stated corresponding to a single completely ruptured grain which becomes the nucleus of a crack. Other grains were assumed to be initially non-damaged. Computations were performed rain max for Ao~ = 150 MPa, R = o~ /o~o = 0. Several samples a(N) of the early in-depth growth are shown in Fig. 4. The scatter of samples is significant: the cycle number at crack tip advancement in several grains varies from 70.103 to 180.103 . The crack propagation along the body surface exposes the same order of magnitude. To propagate a crack up to the size containing 40

grains, the cycle number varies from 80.103 to 180.103 (Fig. 5). The processes a(N) and c(N) are certainly stepwise. To make the pictures in Fig. 4 and Fig. 5 clearer, these step-wise processes are replaced with continuous ones by connecting the points at the middle of each step with straight lines. It is of interest to follow the change of the ratio c/a at the early stage of crack growth. Several samples of this ratio in function of cycle number are presented in Fig. 6. In the beginning of crack propagation, the ratio varies on a large scale, from 0.5 to 2.0 and even wider. But then this variation becomes more moderate. To the end of the numerical simulation the distribution of c/a becomes comparatively 10-7

10 7

104

~ 1 0 -9

10- I°

10"

0

I

I0

I

20

I

30

I

40

1

50

L

60

I

70

0 80

20

40

60

80

a, p.m

a, I.tm Fig. 7. Smoothed samples of in-depth crack growth rate.

Fig. 9. Estimates of the mean in-depth crack growth rates and their confidence margins.

Early fatigue crack growth compact. The coarse estimation gives c/a = 1.5, which is in agreement with experimental data. s'9 To estimate the crack growth rates da/dN and dc/dN, we use the samples a(N) and c(n) which have been already subjected to continualization. However, the resulting samples of rates are step-wise again. For lucidity, we apply the continualization procedure again: the middle points of each step of samples are connected with straight lines. This approach results in the samples presented in Fig. 7 and Fig. 8. The scatter of da/dN covers one order of magnitude (Fig. 7). It is a pleasant surprise to observe a fair agreement with experimental results on Short crack behavior. 4,s At the initial stage of crack growth, the rates are subjected to decrease, and the minima of da/dN are of the order of 10-1°m cycle -1. Then the steady increase of da/dN is observed. When a crack has penetrated in 6-8 grains, one may say that the stage of stable crack growth begins. A similar approach was applied to assess the rate dc/dN of crack propagation along the surface. Several samples are shown in Fig. 8. Two items ought to be mentioned. Compared with Fig. 7, the scatter of rates is much higher. One might explain this phenomenon by the initial assumption: the scatter of the properties of near-surface grains was assumed larger than that of internal grains. The second peculiarity is that, compared with da/dN samples, the samples of dc/dN do not expose the initial stage of decreasing that precedes the initiation of the steady growth stage. Because of significant scattering of numerical results, statistical treatment seems to be relevant. The standard approaches were applied to estimate the mean value of the in-depth crack growth rate as well as its standard deviation. The mean crack growth rate diagram is plotted in Fig. 9, where 31 samples are used. The middle line is the estimate of the mean value, and two other lines correspond to the 10-7 _

231

1.6

4f

1.2

1.0

0.8--

0.6

0.4 0

I

I

I

I

I

I

I

10

20

30

40

50

60

70

80

a, ~rtl

Fig. 11. Coefficient of variation of in-depth crack growth rate. confidence level equal to 0.9. The standard deviation of the crack growth rate is shown in Fig. 10, where three lines are also drawn: the estimate of o(da/dN) and the borders of 0.9 confidence margins. The character of both diagrams looks similar. In particular, the variance of the crack growth rate increases with N. The coefficient of variation of da/dN (Fig. 1 l) behaves 'reasonably'. The initial branch of the diagram is not very reliable; but the distinct decrease of the coefficient of variation to the magnitudes close to 0.5 is in agreement with experiments. 4 FORMATION OF MACROSCOPIC CRACKS When the number of ruptured grains entering the cracked area becomes sufficiently large, the further process of crack growth is controlled by averaged material characteristics such as mean resistance stress, besides loading parameters. For the case when distributions (2) and (3) are valid, the mean resistance stresses are

10-8

Sm I = ASth I "~- Sc I I~ ( 1 + 1/~ 1), Sm 2 : ASth2 "~- Sc2 ~ ( 1 + 1/a 2)

(4)

z where F(.) is gamma-function. 10-9

10

X

io 0

I

I

I

I

I

I

I

10

20

30

40

50

60

70

a,

80

~tm

B

2c Fig. 10. Estimates of the standard deviations of in-depth crack growth rates and their confidence margins.

Fig. 12. Schematic presentation of macroscopic crack.

V.V. Bolotin et al.

232

To estimate the rates da/dN and db/dN, let us assume that a crack advances in a step h when a corresponding grain on the crack boundary is ruptured. Then at ml = m2 = m, n = 0

da

h(KAAO~_=z~Sth2)m de

dN

k

Sm2

h (KBA~I~s~--ASthl)m

' dN

(5) Here XA and KB are stress concentration factors at points A and B (Fig. 12), Sml and Sin2 are mean stresses given in eqn (4). Let the crack have a semi-elliptical shape with semi-axes a and c. To find an approximate value of stress concentration factors, let us use the analogy between stress concentration and stress intensity factors. The stress intensity factor is mode I for near-surface semi-elliptical cracks is defined as 9:°

g(fl) =

Y°~(ra)'/2[sin23+(a)2cos2fl] TM E(K)

(6)

Here fl is an elliptical angle (Fig. 12), Y is a correction factor of the order of unity, E(K) is the elliptical integral of second kind, i.e.

f ~12

a2

(sin 2 3 + X2 COS2 fl)l/2df3, x 2 = 1 -- c--~

E(x) =

An heuristic formula or stress concentration factors based on the analogy with eqn (6), has the form

2Y(:)1/2[

K(3) = E--~

sin 2 13+

(a)2

cos 2 3

]1/4

( c) ml2 (

Sc2~ m

Despite a number of simplifications, the suggested model covers all principal features of early fatigue crack growth: high scattering of crack growth rates on the early stage when crack is to be considered as mesoscopic one; temporary retardation of crack growth on this stage; transition to a more regular growth when the depth of a crack becomes significant. In addition, the model predicts quite a reasonable ratio of the crack in-depth and along-the-surface dimensions. This ratio stabilizes when a crack approaches that which may be interpreted as a mesoscopic one. Several ways of generalization of the model are obvious such as including the scatter of elastic moduli and/or the anisotropy and disorientation of individual grains with respect to the direction of crack growth; however, these generalizations require a drastic increase of computational work.

ACKNOWLEDGEMENTS This paper is dedicated to the 60th anniversary of Professor Franz Ziegler, Technical University of Vienna, Austria. The research was partially supported by the Russian Foundation for Basic Research (grant 96-01-01488).

REFERENCES (7)

where 0(3) is the crack tip radius of curvature which actually characterizes stress concentration near the crack tip with complex fractography. 11 Its value, generally, varies along the crack border. Eqn (7) is similar to the formula suggested by Mura ~2 for internal ellipsoidal hollows. The finite element analysis shows satisfactory agreement with results given by formula (7) when a hollow is sufficiently flattened. However, for our purposes, rougher estimates are enough. As Y--I,E(K)--I, o--h, one may state that KA"~-" (a/o)II2, KB-~-a/(pc) 112. This means that KA/KB"~-~ (c/a) 1/2. At KAa~ ~> > Aath2 , KBa= > >Aathl, eqn (5) give the estimates for the ratio of crack tip advancements

Ao .~

5 CONCLUSIONS

(8)

1. Bolotin, V. V., Prediction of Service Life for Machines and Structures, ASME Press, New York, 1989 (Russian Edition 1984). 2. Bolotin, V. V., Fatigue life prediction of structures. In

3.

4.

5. 6. 7.

The ratio of semi-axis tends to become stationary for macroscopic (however shallow compared with the body thickness) cracks:

8.

C (Sc2~ 2m/(m+2) -d "~ \ s~i/

(9)

As sc~ <- s~2, w e have c -- a at all m > 0. In particular, at m = 2 , eqn (7) gives c/a = s~2/s~l, and at m = 4 w e obtain c/a = (ScJS~O 4/3.W h e n m > > I, the ratio c/a approaches (ScJS~O 2. W e see that most parameters of the micromechanical model enter eqns (5)-(7) which are valid, at least approximately, at the macroscopic level. This means that these parameters m a y be estimated from c o m m o n fatigue tests.

9. 10.

11. 12.

Probabilistic Structural Mechanics: Advances in Structural Reliability Methods, ed. P. D. Spanos and Y. T. Wu. Springer-Verlag, Berlin, 1994. Sobczyk, K. and Spencer, B. F., Random Fatigue: From Data to Theory, Academic Press, Boston, MA, 1992. Miller, K. J. and de los Rios, E. R. (ed.), Short Fatigue Cracks, ESIS Publ. 13. Institute of Mechanical Engineers, London, 1992. Rabotnov, Yu. N., Mechanics of Deformable Solids. Nauka, Moscow, 1979 (in Russian). Kachanov, L. M., Introduction in Continuum Damage Mechanics. Martinus Nijhoff, Dordrecht, 1986. Schijve, J. A normal distribution or a Weibull distribution for fatigue life. Fatigue Fract. Engng. Mater Struct., 1993, 16(8), 851-859. Goto, M. Scatter in small crack propagation and fatigue behavior in carbon steels. Fatigue Fract. Engng. Mater. Struct., 1993, 16(8), 795-809. Chang, J. (ed.), Part-through Crack Fatigue Life Prediction, ASTM STP 687. ASTM, Philadelphia, 1979. Carpinteri, A., Propagation of surface cracks under cyclic loading. In Handbook of Fatigue Crack Propagation in Metallic Structures, ed. A. Carpinteri. Elsevier, Amsterdam, 1994, pp. 653-705. Bolotin, V. V., Stability Problems in Fracture Mechanics. John Wiley, New York, 1996. Mura, T., Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht, 1987.