Stochastic modelling of crack growth based on damage accumulation

Theoretical and Applied Fracture Mechanics 6 (1986) 95-101 North-Holland

95

STOCHASTIC MODELLING OF CRACK GROWTH BASED ON DAMAGE ACCUMULATION F. E L L Y I N Department of Mechanical Engineering, The University of Alberta, Edmonton, Alberta, Canada T6G 2G8

The growth rate of a fatigue crack is modelled from a damage accumulation standpoint. The material ahead of the crack-tip is considered to be composedof assembly of uniaxial fatigue elements which accumulate damage per load cycle. Each element is subjected to increased levels of stress and strain ranges as the crack propagates. A linearly accumulated damage criterion is assumed, and failure of an element indicates a void initiation at its position. Both deterministic and stochastic analyses are included. The historical damage of the material before it reaches the crack tip vicinity is quantified and is shown to be significant for the first few elements. The predicted results agree fairly well with the experimental data.

1. Introduction When a material is subjected to varying loads or deformations, damage accumulation leads to formations of cracks and subsequent propagation of them. The second phase, i.e., the fatigue crack growth sometimes is viewed as incremental growth due to a critical amount of damage pile-up at the crack-tip. The success of such a theory depends upon the specification of an appropriate damage criterion. One of the earliest criterion is due to Miner [1], in which he observed that when the uniaxial fatigue specimens were subjected to different levels of cyclic loading, the sum of the cyclic ratio approached unity for a large number of loading ranges. If the material ahead of the crack is modelled as an assemblage of uniaxial material elements, then the crack growth can be seen as the successive failure of these elements. The cyclic ratio for the given loading level can be determined from the stress/strain range prevailing on each element. The breadth of the element is the extent of the crack extension when the element reaches its critical damage stage [2]. Some other researchers have proposed damage criterion based on the fatigue ductility [3], or a measure of strain energy density at the crack-tip [4,5]. There is a need to extend the deterministic models using probabilistic simulation so as to incorporate statistical aspects of the crack growth process. Such a need arises when one is required to make risk assessment of critical components in a system, or to plan inspection strategies [6]. This recognition has led to using some probabilistic

distributions on the deterministically calculated growth path. This type of approach may require extensive fracture data for the estimation of the distribution parameters. A proper probabilistic approach could reduce, to a large extent, the dependency on the experimental data. In this paper, a model of fatigue crack growth is considered based on damage accumulation and probabilistic considerations.

2. Crack-tip modelling The material ahead of the crack-tip is considered to be made of uniaxial fatigue elements which accumulate damage per cycle. The damage zone, DZ, considered here is the extent of the reversed plastic zone, RPZ, ahead of the crack-tip. Beyond this zone, the accumulated fatigue damage is insignificant. The fatigue damage resistance, FDR, of each element is either constant or is a random observation from the same probability distribution. Damage is accumulated linearly on each element due to the stress/strain level at its position. When the total damage accumulated on an element exceeds its FDR, the element failure takes place by initiation of a void at its position. As each void coalesces with the crack, the sequence of initiation is perceived as a stable crack growth. The element dimensions are finite so that the macroscopic laws and properties apply to each element. For each element in the DZ, the fatigue damage begins to accumulate when it enters the zone. This prior damage effect is termed 'histori-

0167-8442/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

F Ellyin / Stochastic modelling of crack growth based on damage accumulation

96 .....

historical d a m a g e

of stress [8] in the form of

......

d a m a g e a c c u m u l a t e d p e r c y c l e of loading

' i j _ _(1 _+ v) s,,,, + ~1 - 2 v

= f a t i g u e d a m a g e resisance

E0

3o[ o] i + 2

crack

i i

i .......

o,Jns,,

[o0l

07'

(2)

where s~j is the deviotoric stress, o~ is the equivalent stress, ~ ; v is the Poisson's ratio; n and a are material constants from eq. (1). Adopting Hutchinson's plane stress solution [8], the stress and strain components normal to the plane of the crack are given by

~ .......

_ °kk6ij

O0

4 3 2 1

Element No. Fig. 1. Schematic of fatigue elements, cumulative damage, and fatigue damage resistance.

O o [ J ] "/(l+') e = --~ aZ,,oo%X ( 0 0 - POr)

,[

, ]lj,+n

-

otlnOoEo X j

]n/(l+n)

so,

o = Oo ~ I . o o , oX

cal damage" and is subtracted from the initial strength of the element to indicate the instantaneous (residual) strength at a given life. Figure 1 shows a schematic of the elements, their stochastic fatigue damage resistance, the mean historic damage, and damage accumulation per loading cycle. The historic damage, in general, is a decreasing function of distance from the crack-tip, and attains a statistical equilibrium state with a stable mean [7]. The deterministic version of this model can be obtained by considering that the elements in Fig. 1 have identical damage resistance.

3. S t r e s s - s t r a i n f i e l d a h e a d of t h e c r a c k

where 50 and 6r are nondimensional functions of the strain hardening exponent n (and the polar coordinate 0 in the general case). The parameter I , is a nondimensional function of n. Note that

400 I--

o

ooff, .~

¢/¢ o = o / o o + a( O/Oo) 1/" ,

100

(1)

where o0 is an effective yield stress; % = oo/E; E is the Young's modulus and a and n are parameters chosen to fit the data. Hutchinson, using the -/2 deformation theory of plasticity, generalized eq. (1) to multiaxial states

M o n o t o n i c Stress - Strain Relationship

A~ _

2o0

A uniaxial stress-strain relation generally used to fit the experimental data is the R a m b e r g Osgood one, which may be written as,

(3)

2

l

t

o Tensile Mean Strain Data --Best Fit Line for Tensile M e a n Strain Data .... Best Fit Line for No Mean Strain

_

0

A~

2(2o~.s×1o~) + \ ~ J

r

I 04

I

I 0.8

I

I 1.2

J

I 1.6

A~/2 (%) Fig. 2. Cyclic and monotonic stress-chain curve for strain controlled tests, from [11].

97

F. Ellyin / Stochastic modelling of crack growth based on damage accumulation

eqs. (3) exhibit a singularity as x ~ 0. It has been widely recognized that a small region denoted "fracture process zone" exists immediately ahead of the crack tip where the stress and strain fields given by eqs. (3) may not provide a good approximation [5,9] (see Fig. 3). A measure of the extent of the fracture process zone is given in [5]. The above stress/strain field equations are for the monotonically increasing load. To extend the response to unloading, reloading and cyclic loading, we may use Rice's plastic superposition method [10]. The fundamental assumption is that the components of the plastic strain tensor remain in constant proportion to one another at each point in the plastic region. It is to be noted that close to the crack-tip, the stringent conditions of proportionality may not be satisfied, however, the method may be taken as a first approximation in the absence of a more accurate theory. When a material is subjected to cyclic loading, its stable cyclic stress-strain curve is different from that of the monotonic one. The uniaxial cyclic stress-strain curve is generally represented by

Ac

6¢,

2

2E +

( Ao ]~/"' ~K--7]

,

o--* A o , n ---~n ' ,

c--* Ac, a ---~ct',

=

E

].'/(] +n')(

AK 2 ,- . t x 2 a~.~Oy) x

+ -°~-a'

O0

itOr )

--

AK 2 ,_ .

,.2

a~.(oy)

oo - ½Or), x

] n ' / ( l + n')

AO

Oy'

=

AK 2 "1 t 2 O/"In(Oy ) X J

(7)

~

08.

The stress and stress range acting on each fatigue element, Fig. 1, can be obtained from eqs. (7), once the element size is determined. We note that for a uniaxial fatigue element, the maximum possible normal stress or strain amplitude cannot exceed the fatigue strength, of' or fatigue ductility, c~, respectively. The attainment of these critical values can be identified with the critical stress intensity range, A Kc, where unstable crack growth is observed. The minimum distance from the crack tip, x*, where eqs. (7) can be applied, is shown in Fig. 4. In this figure, the product of stress and strain amplitudes (½Ao. ½A~) is set equal to o[c~.

Plastic(Fatigue) Zone

Process Z o n e - -

,/••Reversed

Crack 8"--

Rp I

2 (Aa-Ae)

RY

M°n°t°niCe Plastic

Iu

[~'~--~HRRSolution /

(5)

where oy is the cyclic yield stress of the material, and a' = ( 2 E / o o ) ( O o / 2 K ' ) 1/"'. For the small scale yielding and plane stress condition, we have AJ = AK2/E.

Oy AE

(4)

where Ac and Ao are the strain and stress range, respectively; K ' is a coefficient with the dimensions of stress, and n' is the cyclic hardening coefficient. The parameters K ' and n' are chosen to fit the experimental data. Figure 2 shows a representative monotonic and cyclic stress-strain curve for a low alloy carbon steel (A516 Gr.70). For this type of steel, the effect of the tensilemean-strain is negligible [11]. Using Rice's superposition method, eqs. (3) can be modified for a cyclically loaded crack by the following transformation: J--* A J , Oo --* oy,

normal to the plane of the crack are given by

Crack x

(6)

~

Fly

Fig. 3. Schematic of the near tip zones, and J-dominance Therefore, the cyclic stress/strain components

region.

F. Ellyin / Stochastic modelfing of crack growth based on damage accumulation

98

sidering the contribution of the elastic strain component.

4. Damage criterion The stress and strain ranges on each element can be determined from eqs. (7) by substituting for x, the appropriate element location, see Fig. 4. The number of cycles to failure of the ith element subject to stress (strain) amplitude ½Aoi (½Ac i ) can be derived from a Manson-Coffin type of relation [11].

\

\

/x tr. A._ge 2 2

HRR Fields

s

t

of t N ]b ½A,,= ~ - t f,, + ' ~ ( N f , ) ~,

(9)

½Ao, = o[( Nf, ) b, where b and c are material constants. Figures 5 and 6 show an example of the best fit curves to determine material properties for a low alloy carbon steel, A516 Gr.70. This steel is now extensively used in the fabrication of m o d e m pressure vessels [11]. The stress and strain range product, Aoi Aci, which is a measure of the strain energy density, is given by

X*

Distance X Fig. 4. Definition of the element width, x *, and distribution of the stress and strain amplitude product along the crack-line.

For example, in the case of small scale yielding and neglecting the elastic strain in eq. (7), the magnitude of x* is given by 6K )

x* = D - 4EI, o~c'f '

p2

4of' t ~ ~:b AosAe,= E t , ' r , , + 4 o f " i ( N r , ) h+C-

(8)

For a given value of the stress and strain ranges, the number of cycles to failure can be calculated from a numerical inversion of eq. (10).

where D = ~0(#0- ½#r). However, in the analysis to follow the value of x* was calculated by con-

0.030

u

i

i

~

n

i

it]

r

J

i

r

i

i

i

]

I

o.olo

E

Ae 858 8 --= - .... ~ 1 ~ ~ 2 0 9 , 5 0 0

<

A~e

OOOllo2

e

I

i

i

r

i

r

o T e n s i l e M e a n Strain Data - - B e s t Fit Line for T e n s i l e M e a n Strain Data ---- B e s t Fit Line for No M e a n Strain Elastic Strain Plastic Strain

"0 o.

i

I

i

i

i

A-~P - - - ~

i

i

i

i1

i

i

i

i

I

-108 -50o Nf + 0 2 1 9 N f

-~6~0

v

103

(10)

i

i

I

I

104

I

i

~'~1

i

i

i

i

I

105

Life (cycles) Fig. 5. Elastic, plastic and total strain amplitude versus number of cycles to failure, Nr, from [11].

F. Ellyin / Stochastic modelling of crack growth based on damage accumulation 103

'

'

'

'

''

' ' I

'

'

'

'

''

' '1

~

V

o~

-....

b

.

°~D

Best Best

Fit Fit

Line Line

for for

Tensile Mean Strain No Mean Strain

.

.....

.

~

.

.

99

' '

..............

r

Data A¢

_ 858

8 Nf -0'108

2

I0]02

~

,

i ~ , i111103

,

,

,

,~J,,1104

. . . . . . .

J 501

Life (cycles) Fig. 6. Stress amplitude versus n u m b e r of cycles to failure for A516 Gr.70 low alloy carbon steel.

Let the number of cycles spent at a position i be denoted by n,, then the cyclic ratio for the element is

to the crack tip, is

d~i = n i / N f i .

x= x*

(11)

If k is the number of crack advances from the first instance this particular element enters the damage zone until it fractures, then the total damage to failure by linear damage law is k

E

Oljllj=dPiq-*2q- "'" q-dpjq- ' ' '

"}-~k,

(12)

j=l

where % = 1/Nfj. When the fatigue resistance is a r a n d o m variable, then it can be shown [7] that the r a n d o m sum Eq~j to be asymptotically normal, i.e., k

E doj=N(Ix, 0"2)

(13)

j=l

If/~ = 1, eq. (13) becomes a form of the statistical Miner's law [12].

t )

For a give load characterized by A K (or A J ) , the number of elements is determined by dividing R P Z by the parameter x*. The stress and strain range on the first element is obtained from eq. (7) by substituting for x = x*. For all other elements, the midpoint values are used, see Fig. 4. In the deterministic case, the damage accumulated by an element i, from the moment it enters the damage zone until it becomes adjacent

p,

(14)

where 0 < p < 1, (da/dN)i is the mean growth rate at the interval, and Rp is the extent of the reversed plastic zone, see Fig. 3. Equation (14) is an expression of the fact that the fatigue element does not fail in the region outside of x > x*. However, as to be seen later on, this is not the case with the probabilistic model. The cyclic ratio for the element adjacent to the crack until the failure is da

X*/ dN Nf(x*)

1 -p.

(15)

Combining eqs. (14) and (15), and replacing the sum by the integral, the crack growth rate becomes [2]

da

5. Rate of crack growth

i

Nfi

dN

x* Nf(x*) +

fxRP 1 * Ne(x----~ dx.

(16)

In the case of probabilistic model, the fatigue resistance of each element is generated from a standard normal random variate generator, truncated at the origin to avoid negative values, Fig. 1. Once the fatigue resistance of an element is exhausted, the crack advances to that position, and the random crack growth rate is recorded,

da ix*
(17)

F Ellyin / Stochastic modelling of crack growth based on damage accumulation

100 101

I0-3 [] o ,, * v

10°

E

o•e

10-~

121 .~

0

~

AK AK AK AK

= = = = AK =

20MPa~/m 30MPa~/m 50MPa~/m 80MPav/m 120MPa~m

C) >, 0

E

¥

i0-2

0

vv

,,i-,

A 5 1 6 - Gr 70- Steel X = 7.432 × 10-5m

i0 -4

V

A K c = 200 MPa 10-s

"~ c 0

10-6

o)

10-7

"l-

t- i0-3

e ,4

i0-4

10-a

0 10-5

i 4

0

i 8

i 12

i 16

I0 "9

20

Dimensionless Distance f r o m C r a c k - t i p .

Fig. 7. Historic damage versus fatigue element position from the crack-tip, A516 Gr.70 carbon steel.

10-10

I

1

I I IIIlJ

10

I

I

I [llll]

10

I

I

i I IIII

10

10

Stress Intensity Factor, MPa~/m Fig. 8. Predicted mean crack growth rate versus stress intensity range and comparisons with experimental data [13].

where i is the element distance from the crack tip, and n is the number of cycles spent in the configuration. field ahead of the crack, Fig. 4. The amount of predamage is a function of distance and AK. Similar results are observed for other types of metals [7]. The median crack growth rate, da/dN, is plotted against stress intensity range factor, AK, in Fig. 8, and is compared to the experimental data reported by Ellyin and Li [13]. It is seen that the agreement is quite good. Similar trends were observed for other materials [7]. As for the probability distribution of the crack growth simulation, the probability plots show that the lognormal distribution is a good approximation of the simulated data. Experimental data reported in [14] appear to confirm this prediction.

6. Results and discussion

To quantify the historic damage, its mean value is plotted against dimensionless distance from the crack tip in Fig. 7, for various A K values. The abscissa is the distance from the crack tip divided by x*, i.e., number of fatigue elements, and the material used for this figure is A516 Gr.70 a low alloy carbon steel. The experimental data used for the numerical calculations are summarized in Table 1. It is noted that the historic damage is significant for the first few elements. This is to be expected, given the nature of the stress/strain Table 1 Material properties Material A516 Gr.70

b -0.108

c -0.506

E (GPa)

n'

200

0.19

a; (MPa)

a t'

310

900

c~

(MPa)

AK c

x*

(MPa~/m) 0.26

200

7 . 4 3 × 1 0 -5

F. Ellyin / Stochastic modelling of crack growth based on damage accumulation

7. Conclusions T h e m o d e l p r e s e n t e d here is c a p a b l e of p r e d i c ting the m e a n or m e d i a n crack g r o w t h rate, a n d the a g r e e m e n t with the e x p e r i m e n t a l d a t a is fairly good. The s i m u l a t i o n process also leads to p r e d i c t ion of the t h r e s h o l d stress intensity range, b e l o w which there is no growth. T h e m o d e l considers the historic d a m a g e d u e to the p o s i t i o n of an e l e m e n t within the d a m a g e zone, p r i o r to its failure. I n c o r p o r a t i o n of this d a m a g e results in the residual s t r e n g t h of the few elements n e a r the c r a c k - t i p b e i n g less t h a n their virgin strength, thus affecting c r a c k g r o w t h rate. Acknowledgments T h e w o r k p r e s e n t e d here is p a r t of a general i n v e s t i g a t i o n into m a t e r i a l b e h a v i o u r u n d e r multiaxial states of stress a n d adverse e n v i r o n m e n t s . T h e research is s u p p o r t e d , in part, b y the N a t u r a l Sciences a n d E n g i n e e r i n g C o u n c i l of C a n a d a ( N S E R C G r a n t N o . A-3808). This p a p e r is a synthesis of the w o r k s b y the a u t h o r a n d his c o l l a b o r a t i o n whose n a m e s a p p e a r in the list of references. In p a r t i c u l a r , the a u t h o r wishes to ack n o w l e d g e the c o n t r i b u t i o n s o f Drs. F a k i n l e d e a n d K u j a w s k i to this work.

References [1] M.A. Miner, "Cumulative damage in fatigue", J. Appl. Mech. 12A, 159 (1945).

101

[2] F. Ellyin and C.O.A. Fakinlede, "Crack-tip growth rate model for cyclic loading", in: J.T. Pindera, ed., Modelling Problems in Crack-Tip Mechanics, Martinus Nijhoff, Dordrecht, The Netherlands, p. 224, (1984). [3] K.P. Oh, "A weakestqink model for the prediction of fatigue crack growth rate", J. Engrg. Mat. Tech. 100, 170 (1978). [4] G.C. Sih and E. Madenci, "Fracture initiation under gross yielding: a strain energy density criterion", Engrg. Fracture Mech. 18, 667 (1983). [5] D. Kujawski and F. Ellyin, "A fatigue crack propagation model", Engrg. Fracture Mech. 20, 694 (1984). [6] F. Ellyin, "A strategy for periodic inspection based on defect growth", Theoret. Appl. Fracture Mech. 4, 83 (1985). [7] F. Ellyin and C.O.A. Fakinlede, "Probabilistic simulation of fatigue crack growth by damage accumulation", Engrg. Fracture Mech. 22, 697 (1985). [8] J.W. Hutchinson, "Singular behaviour at the end of a tensile crack in a hardening material", J. Mech. Phys. Solids 16, 13 (1968). [9] J.W. Hutchinson, "Fundamentals of the phenomenological theory of nonlinear fracture mechanics", J. Appl. Mech. 50, 1042 (1983). [10] J.R. Rice, "Mechanics of crack-tip deformation and extension by fatigue", in: Fatigue Crack Propagation, ASTM-STP 415, p. 247 (1967). [11] F. Ellyin, "Effect of tensile-mean-strain on plastic strain energy and cyclic response", J. Engrg. Mater. Tech. 104, 119 (1985). [12] T. Shimokawa and S. Tanaka, "A statistical consideration of Miner's law", Internat. J. Fatigue 2, 165 (1980). [13] F. Ellyin and H.P. Li, "Fatigue crack growth in large specimens with various stress ratios", J. Pressure Vessel Tech. 106, 255 (1984). [14] M. Bertrand, D. Lefebvre and F. Ellyin, "Statistical analysis of crack initiation and fatigue fracture of thin-walled tubes using the Weibull law", J. de Mbcanique Theorique et Appliqube 1,493 (1983).