Temperature dependence of stage II fatigue crack growth rate

Engineering Frmrure Merhankr Printed in Great Britain.

45, Vol.

No. 6, pp. 741-750,

TEMPERATURE

$6.00 + 0.00 0013-7944/93 %: 1993 Pergamon Press Ltd.

1993

DEPENDENCE OF STAGE II FATIGUE CRACK GROWTH RATE A. IOST

Laboratoire de Metallurgic Physique, CNRS URA 234, Bltiment C6, Universite des Sciences et Technologies de Lille, F59655 Villeneuve d’Ascq, France Ecole Nationale Superieure d’Arts et Metiers, C.E.R. de Lille, 8 boulevard Louis XIV, 59046 Lille Cedex, France Abstract-A critical assessment has been performed by compiling experimental data concerning the temperature dependence of stage If fatigue crack growth. For aluminium alloys, high strength steels, austenitic stainless steels and superalloys, the power coefficient, M, for the Paris reIationship is temperature dependent, and all ln(du/d~) vs ht(AK) curves cross at one point, designated as the pivot point (PP), which is a material-dependent parameter. The assumption is made that PP corresponds to a transition point for the fatigue crack growth mechanism.

INTRODUCTION IN THE intermediate growth rate range for fatigue crack growth, the data can generally be fitted to the Paris-Erdogan law [I]:

da/dN = C(AK)“,

(1)

where C and m are material and test condition parameters. In the early 197Os, these parameters C and m were first observed to be interrelated by Kitagawa [2], according to the equation: m =a+blnC,

(2)

where the coefficients a and b are typically determined by regression analysis. As pointed out by Tanaka and Matsuoka [3], eq. (2) implies that the In(da/dN) vs ln(AK) curves have a pivot point at da/dN = (da/dN), and AK = (AK),, where all curves intersect. An equivalent representation of (2) can be given as: da[dN = (da/dN)~(AKfAK~)m with (da/dN), = exp( - a/b);

A& = exp( - l/b).

(3)

These relationships were extensively studied and values of PP were found for brittle steels, ductile steels, aluminium and titanium alloys [4]. In the present paper, experimental data were first compiled to account for the temperature dependence of stage II fatigue crack growth rate (FCGR), and this effect was analysed employing eq. (3). The physical meaning of eq. (3) was also studied. This paper is a sequel to earlier works on the PP [5] and on the effect of load ratio on the m-ln C relationship [6]. The units are mm/cycle (da/dN), MPaJm (AK) and Cal/mole (Q) unless otherwise stated,

DISCUSSION

ON SOME LITERATURE

REPORTS

From Jeglic ‘s empirical analysis

As typical of the temperature dependence of stage II fatigue crack growth rate, we shall consider first the work of Jeglic et al. [7] on the Al-2.6 Mg alloy between room temperature and 573 K. Thirty tests were performed for 12 levels of temperature to provide the empirical relationship (4), which shows that the intercept, C, exp( - Qo/RT), and the exponent, m, are both EFM WC-C

741

742

A. IOST

dependent on the temperature. temperature is increased:

The intercept

increases and the exponent

da JdN = C, exp( - Q,,/RT)AK”‘z

decreases as the

R7’ ’

or da/dN = C, AK ’ exp - [(Q. - C, In AK)/RT]. Equation (4) is an Arrhenius-type and Q = Q0 - C, In AK as the (inch/cycle - psiy/inch):

!4!

law, in which Q0 was interpreted as the activation energy apparent activation energy. With the authors’ units

C, = 21,870;

Cz = 3989;

Q. = 39,700 Cal/mole.

This value is comparable to the values of activation energy for volume diffusion in aluminium alloys containing 5-8% magnesium. Therefore Jeglic et al. concluded that the fatigue crack propagation rate is controlled by a diffusive process. However, if we take da/dN in mm/cycle and AK in MPaJm as units, new coefficients can be obtained for eq. (4): C; =0.672

and

Q;l= 12,525.

Now, Q;t is of the same order of magnitude as the activation energy for core di~usion in aluminiunl alloys. The foregoing Jeglic analysis makes no allowance for the dependence of Q;l on the unities. As a result Q; cannot correspond to an activation energy in the usual sense [7,9], and it must be emphasized that such an error is often found in the literature. From experimental data, the values of m for specimens tested at different temperatures were plotted against In C (Fig. 1). This figure shows that eqs (2) and (3) are verified with: a=

-2.27;

b = -0.332;

(AK), = 20.33 MPaJm,

(da/dN), = 1.08 x 10~-3mm/cycle;

with

r =

-0.994.

From empirical eq. (4), the m-ln C relation is perfectly verified (r = -0.999995) a= -2.13;

b = -0.319;

(dff~d~r)~= I.28 x lo--“mm/cycle;

and (AK),, = 23.24 MPaJm.

It is clear from Fig. 2 that all propagation curves intersect one another at the PP. If both eqs (2) and (4) are verified, that means:

if a and b are temperature

m = (Cz/RT)

- 2 = a + b(ln C; - Q;,/RT);

independent,

then:

b= -Cz/Q;t;

-22

-20

Fig. 1. m-111 C relationship

-18

a=

-16

-2-blnC;.

-14

-12

for Al-Z.6 Mg aluminium

-10

alloy. (From

Jeglic et ul. 17j.j

with:

743

Temperature dependence of fatigue crack growth rate

I.

T (“K)

I

T

Al-Z.6 Mg alloy

l” * 573 ’ 423 * 333

I! z O.OOl-2 4

0.000~ 14

AK (MPalfm)-> s

1

8

L

‘

16

18

20

22

24

1

_

0 A fi ' m +

373 353 523 473 413 443

26

Fig. 2. Plot of da/dN against AK for Al-2.6 Mg aluminium alloy. All curves obtained for different temperatures intersect the same point.

From

(5),

the coordinates

of PP are: (AK), = w(QX’d

(da/dN), = C; exp(-2&&G);

and the Jeglic equation (4) can be written in the equivalent form: da/dN = (da/dN),(AK/AK~)(C2i”~‘2.

(6)

Jeglic’s equation was applied to Wei’s data on the 7075 T 651 aluminium alloy [IO] in the temperature range 273-363 K. The m-In C relationship was again found: da/dN = (da/dN),(AK/AK)~545’RV-2 with (da/dN), = 1.96 x low4 mm/cycle;

AI$ = 16.59 MPa,/m

and a = -3.039;

b = -0.356.

A difference can already be noted between the PP values of these two aluminium alloys. From Yokobori et al.‘s kinetic theory of fatigue crack propagation

Yokobori et al. [8, 1l-l 71have proposed a kinetic theory based on dislocation dynamics. From the empirical Johnston-Gillman relationship between the velocity of moving dislocations and the effective stress r* applied on them: _ ZJ= Ug(T*/Toy

with

o0 = 10e2 m/s.

(7)

1o-2 2 m IO“ 2 w lo-' 10.' 10-610"

AK 1O-8!

-32

Fig.

-30

-28

-26

-24

-22

m-In C relationship for 25Mn-Xr-1Ni (From Yokobori et al. [I I].)

-20

steel.

10

100

1OlM

Fig. 4. Plot of da/dN against AK for GH 36 superalioy in log-log coordinates. All straight lines obtained for different temperatures intersect the same point.

744

A. IOST Table

1. Pivot point

Temperature (K)

for aluminium alloys (the values marked with an asterisk from experimental data found in the literature)

No. of data

RT 300-573 300-573 295 -380 RT RIRT 4 76 RT 4- 295 RT

148 30 theory 30 16 8 10% 6 13 9

r 0.494 I 0.99Y 0.996 0.99 0.93 0.90

I

(da:@+‘), X 10” 3.9 10.8 12.87 t .96 25.4 7.lh 10.8 3.5 1.1 5.73 0.4

are calculated

AK”

a

b

13.5 20.33 23.24 16.59 23.49

-3.02 -- 2.27 -2.12 - 3.04 - I .89 -2.67 -- I .98 -2.84 - 4.46 -2.89 - 5.45

-0.384 -0.33’ -0.318 .-.0.356 -0.317 --0.369 -0.290 ---0.357 0.490 ---0.3X8 --0.539

IS.04 31.45 16. 7.7 13.2 6.4

by us

Ref. ‘I 7*

7” lo* 24 29 30 31 II ?I 32*

According to thermal activation theory, it can be shown that: tT= Sv* exp(-tiH/RT),

(8)

where S is the product of the area swept out per successful thermal activation by the number of sites per unit dislocation length where thermal activation can occur and V* is the frequency of vibration of the dislocation segment involved in the thermal activation process. If this process is the thermally activated overcoming of the Peierls-Nabarro hills by the double kink nucleation model proposed by Seeger [I 81, AH is a decreasing function of the effective shear stress (the thermal component of the stress has been substituted for the total stress z in Seeger’s equation): AH = H,[l +0.25ln(l6r~/nt*)], where -F;r,is the energy of a single kink and 5: the Peierls-Nabarro into (8) and rearranging we get:

(9’)

stress at 0 K. Substituting (9)

2’= S~*(n!162:~)‘~~,~~~exp(-H,/4kT) where z?= MT*, with M = ~~*(~~/~~~, n = H,,/4U (19,201, tW = el16~E/x and k is the Boltzmann constant. According to the dynamics of crack propagation: da/dN = n *h, Table

2. Pivot

Material Steels 25Mn-SCr--lNi 300 series AS steels ASS welded 300 series 316 SS 316 SS Cr-Mo-V-S with bainit. structure Superalloy Superalloy 718 Ni base HSS Nickel base superalloy 35 NCD 16

point

(IO)

for steels or Ni-base superalloys (the values noted with an asterisk experimental data found in the literature) Experimental variable

R,f, r, struct.

state

T. R,f, T, R.f, T, R,.L T

r, environment thick., orient. T, R =O.l T, R = 0.3 T. R =O.S T T

T T, R,;, T, R,f, K R,f, . Tempering

Temperature (W RT 99%RI99-RT 4-RT 4- RT 4RT 300-866 RT-357 RT RT--.533 RT-533 RT--533 448-873 9733 1123 RT-922 173RT 4-RT RT 4-76 RT

No. of data 77 IO theory 50 31 29 12 16 ‘3 3 3 4 4 10 3 28 8 20 18

(da/dN),, X 104 2.35 3.43 1129 0.18 0.73 5.50 2.30 5.19 5.25 0.14 0.16 52.5 4.41 469 50 1.19 0.90 I.iO 1.80

are calculated

by us from

(AK!,

-a

--b

36.0 41.9 37.2 31.4 21.5 29.9 40.0 33.0 39.9 20.6 13.3 13.7 74.7 48.4 202 107 34.3 28.0 37.7 31.4

2.33 2.14

0.279 0.267 0.276 0.290 0.326 0.294 0.271 0.286 0.271 0.331 0.387 0.382 0.182 0.258 0.188 0.210 0.278 0.300 0.276 0.290

2.60 3.56 2.80 2.03 2.39 2.05 2.50 4.32 4.22 0.53 1.99 0.56 1.13 2.52 2.80 2.51 2.51

Ref‘. ??

II* i 1* 31 it 31 24 35* 36* ?I’ 27+ 27* 22* 23* 26* Ii* 31 31 31 34*

145

Temperature dependence of fatigue crack growth rate

Table 3. a vs b, deduced from striation spacing

a vs b

m

2 2 2 2.12 2.L

-6 -&.6

-0.2 -0.3 -0.4 -0.5 Fig. 5. a YSb variation for some aluminium alloys. Straight lines are related to striation spacing models (Table 3).

2+ 2-k 2+ 2.12 + 2.1 +

12.91 b 13.72b 12.82 b 13.52 b 14.74 b

Ref. 3 48 48 49 46

where n* is the number of dislocations emitted from a crack tip until the time concerned and b is the Burgers vector. By this method, Yokobori et al. [8, 1l-171 obtained the variation of da/dN vs AK on the rate process theory: da/dN = A AKb’ exp - [(U, - a, In AK)/kT].

(11)

A, b,, U, and a, are material constants: A =: 1.39& - ‘.45b(A ‘/4~,~~~ +j# b, = (n + l)/(n + 2);

+‘)Kn +‘1;

a, = H,(n -t 1)/4(n + 2);

u2 = EH& + U/4@ + 2)MhJE:), where 6 is the distance over which the applied stress is averaged and f is the frequency and /J the modulus of rigidity. On the one hand, the theoretical eq. (11) has been found to be in fairly good agreement with experimental data, i.e.: high strength steel from room temperature to 173 K (HSS) [I 31; 25Mn-5Cr-1Ni austenitic steel between 95 and 293 K (AS) ]l l]; 780 MFa high strength steel weldments from 223 K to RT [17]. On the other hand, for HSS and AS, the authors’ results are available and are in agreement with our phenomenological study about the temperature influence on PP. As is shown in ref. [13, Fig. 7, p. 4691 for HSS (low tempered martensitic structure), the In(da/dN)-ln(AK) straight lines corresponding to RT, 223 and 173 K intersected each other for: (AK), = 107 MPaJm; corresponding

(da/dN), = 5 x 10.-‘mm/cycle

to a = - 1.13; b= -0.21.

C and m values related to AS are available in ref. [I I] for different structural states (as rolled, solution treated and welded) and different temperatures (RT, 153, 113 and 99 K). Equations (2) and (3) can be applied to these results to give: m= -2.136-0.267lnC or (da/dN),, = 3.43 x 10e4 mm/cycle;

(AK), = 41.9 MPa,/m.

746

A. IOST

AS illustrated in Fig. 3, all data lie on a straight line with a very good regression coefficient in m--In C coordinates: r = -0.9995. If for this last material eqs (2) and (11) are both verified, we can deduce: m =h, +u,/kT=a+b(inA

- C:,/kT)

OK (n +

l)/(n -t 2) + n(n + l)/(n f 2) = a + b In A - bn(H + I>/@ + 2)ln(z,,jc);

if a,, 6,, a, b, A, tJ, (n + I)/@ + 2) are temperature = - l,‘ln(r,,,/‘c)

b = -a/U,

Substituting the approximate c = 1.5 x IO-’ m, we obtain:

independent [1 1, 12, 19,201 then: or

(AK), = ~~~~6.

(12)

values for iron used in ref. [ll], i.e. zoo= 9.62 x lo4 MPa\/m (AK,) = 37.2 MPaJm:

and

b = - 0.276.

This value matches very well with the 0.267 found directly by m-In C regression. Thus, the experimental results agree fairly well with dislocation dynamic theory and with our PP phenomenological study; meanwhile the uncertainty about the 12,L, zooand H, values limits the use of Yokobori’s equation (12), and the double kink nucleation model does not seem appropriate for FCC alloys. Therefore, macroscopic equations derived from an elementary microscopic model may lead to results disagreeing with the experiment and a phenomenological equation for the activation enthalpy such as AH = I& In@,*iz,) proposed by Dotsenko and Landau [2 I ] can be used without hypothesis on the mechanisms involved in the rate process. Using this method, the relation n = H,,/RT can also be obtained. From Jizhou and Shaolun? results The recent experimental work of Jizhou and Shaolun [22] discusses the resutts of a series of experiments conducted on GH36 superalloys used in turbine disks or aero-engines (R = 0.1 -f = 0.5 Hz). C and m were found to vary with respect to the temperature as follows: C(T) = [63,174.37+ - 63,754.9 x 10’ T’ f 1962.21” - 17,665.1 x IO’]10 ” m(T)=-9x10

xT”+1.028x10

“T’--4.0337xlO~‘T+8.744

16O’C Q T < 6OO’C. Experimental m and In C values have a linear relationship, as in the previous cases: m = - 1.21 -0.2318

In C;

r = -0.9999

or

da/dN = 5.25 x 10 ‘(AKi74.7)”

For theoretical values corresponding to Jizhou’s polynomial relation. the correlation m = a + b In C is perfect, and Fig. 4 (with authors’ units) shows that the FCGR curves focus to the PP

-8

-10

-12 1

Fig. 6. (da/dN),

vs (AK),, variation

2

3

for some aluminiurn alloys. Straight spacing models (Table 3).

4

lines arc related

to striation

Temperature dependence of fatigue crack growth rate

747

For other nickel-base superalloys (K5) tested by the same authors [23], we found: m = -

r =

1.9912 - 0.2577 In C;

-0.9997

or

da/dN = 4.412 x 10-4(AK/48.43)m.

This last result is obtained for two structural states named conventional casting and directional solidification. If all curves have the same PP, the relation between m and T differs with the structural state. From some other authors ’ results

Radhakrishnan [24] noted earlier that variations of both C and m with temperature lead to the possibility that the PP may remain unchanged as the temperature is increased. From the data of Shahinian et al. [25] for 316, 321 and 348 stainless steel, a temperature independent PP in the range 300866 K of 40 MPa,/ m and 5.5 x 10-4mm/cycle was found. Alloy 718 (a precipitation hardenable nickel-base superalloy) has been studied by James 1261 for different heat treatments, specimen geometries and product forms, at 297, 589, 700, 811 and 922 K. By computing these results, we found that the m-ln C relationship held very well (r = -0.997) with: (AK), = 201.7 MPaJm;

(da/dN), = 4.69 x IO-* mm/cycle.

It is to be noted that Jizhou and Shaolun’s interpretation of James’ results is erroneous: they take and C values for stage I and stage II FCGR. For the data of Liaw et al. [27] for Cr-Mo-V steel (bainitic structure), we found for T = 297, 422 and 533 K a variation of PP coordinates with the R ratio: m

R =

0.1:

(da/dN), = 5.25 x lo-‘mm/cycle;

(AK), = 20.58 MPa,/m

R =

0.3:

(da/dN), = 1.41 x 10m5mm/cycle;

(AK), = 13.26 MPa,/m

R =

0.5:

(da/dN), = 1.58 x lo-‘mm/cycle;

(AK), = 13.73 MPaJm

with r >

0.9996.

Beyond R = 0.3, no further shift in FCGR curves occurs with increasing R factor. This can be interpreted as the R cut-off effect [28], and agrees well with our previous work about the R ratio’s influence on the m-ln C relationship [6]. DISCUSSION Different alloy systems (aluminium alloys, bainitic steels, austenitic stainless steel and superalloys) have been studied. All results, and others taken from the literature are reported in Tables 1 and 2 and interpreted by means of the PP. The aim of this section is to discuss the temperature dependence of the coefficients C and m of the Paris law, and the influence of the temperature on the m-ln C correlation; to comment on some criticism of the PP and on the physical significance of PP. Temperature

dependence

of C and

m

C and m have been found to be temperature dependent. When the temperature increases, the power exponent, m, decreases. To analyse the relationship between the exponent, m, and the temperature T, linear and inverse plots of temperature were tried as follows. vs T

For stainless steels, Radhakrishnan [24] found that the best relation was obtained with an m plot; m = (GIRT) - 2 was found by Jeglic et al. for Al-2.6 Mg and 7075 T 651 aluminium alloy; m = b, + a, /RT is predicted by the kinetic theory of Yokobori.

No general relationship can be obtained in view of the literature reports. The general trend is that the exponent, m, decreases and the intercept, C, increases according to an Arrhenius-type relationship as the temperature is increased. These variations differ from the kinetic model of

748

A. IOST

MacGowan and Liu [37], in which m is a constant. in a vacuum.

This last model seems to be applied

principally

The m-ln C correlation is satisfied for FCGR corresponding to different temperatures, i.e. the PP remains unchanged as the temperature increases. All correlation coefficients are better than - 0.99 (- 1 implies a perfect correlation). For HS steel [13] and AS [l I], agreement is obtained with both Yokobori kinetic theory of fatigue crack propagation and the PP temperature invariance. In recent work, Tobler and Cheng (3 1] have found for nickel-base superalloys a slightly lower PP at 76 and 4 K than at 295 K; and for aluminium alloys a dearly lower PP at 76 and 4 K than at room temperature. The first set of results should be viewed with some degree of reservation because of the small number of specimens tested and especially the fact that the testing conditions are different. The second set can be explained by a transition in the Arrhenius law associated with a transition in the failure mechanisms. Such a transition can be seen in Yokobori et al.‘s [38] work on 304 SS. This austenitic stainless steel obeys one Arrhenius law between 300 and 763 K, and another between 773 and 973 K. m-In C relationship:

a defence

Recently, severai authors [39,40] claimed that the m-In C relationship does not possess any fundamental significance, but is the result of the method conventionally employed to plot the data. This criticism can be applied to the results of Niccols [29] or Tanaka [33], for instance, By collecting m and C values from different researchers and experimental conditions with a wide variety of steels or aluminium alloys, the correlations have no physical meaning; even so, the regression coefficient appears to fit very well. Previously, Tanaka [33] found one PP for brittle steels and another for ductile steels; further. Ishi et al. f41] questioned the presumption of a single relationship between m and C simply by classifying a material as ductife or brittle. fn fact, we assume that a material has its own PP. which depends on the test conditions (especially R ratio and loading frequency). The results presented in Tables 1 and 2 back up this hypothesis. On the other hand, Table 2 shows that for similar materials (i.e. austenitic stainless steels), the values of PP are quite similar. Figure 5, in which a was plotted versus b for aluminium alloys, invalidates the unicity of the PP for this class of materials, but it is clear from Figs 2 and 4 that a PP can be found for a particlll~~r material.

The physicul meaning of’ the PP From fractographic studies [38,4247], the striation mechanism is known stage 11 crack growth with a AK power dependence close to 2 (Table 3). s = C’ AK”” :

to operate

~1’ = 2.

Tanaka et al. [4] noted that C’ also satisfies the Paris law when m = m ’ = 2. This means curve in eq. (IO) passes the PP when A.K = (AK),,: ,P = (dn/dN),,

= <-“(AK)“”

during (Ii) that the

i 131

and with (2) and (3): a = m’ - b In C’.

(l5!

A variation of the PP coordinates implies a variation of a vs b according to (I 5). We found a good correlation between the a and b values deduced from the PP coordinates and eq. (I 2). in which we took experimental m’ and C’ values from Table 3. In the same way, Fig. 6 shows a variatioI1 of striation spacing vs AK ciose to the variation of (da,/dN), vs AK,,. This means that PP corresponds to a transition in crack growth behaviour. At low AK. s > da/dN and s < da,!dN at high AK. Pineau and Pelloux [50] noted that the transition in which the s and da/dN curves cross over corresponds to the appearance of voids on the fracture surface of austenitic stainless steels.

Temperature

dependence

of fatigue

crack

growth

rate

749

Equation (15) also implies that all m--In C curves cross over at the same point, the coordinates of which are, for aluminium alloys: m’=2

InC’=

-13.2.

For other alloys, C’ values can be found by using the rationalized form: s = C”(AK/‘E)*. On the other hand, other authors [5 l] found good agreement with FCGR and striation spacing for a wide range of AK. Thus, the physical meaning of eq. (13) is questionable. The PP may be a transition point in the FCGR curve. Ritchie and Knott [52] observed, in a low alloy steel which is subject to temper emb~ttlement, a sudden acceleration in growth rate for critical values of stress intensity factor. This acceleration was associated with a burst of cracking on the fracture surfaces as a consequence of the mode of static fracture occurring in addition to the striation mechanism. We find that this acceleration corresponds to these critical values of AK: (AK), [6]. CONCLUDING

REPARKS

From analysis of the fatigue crack propagation data, the following conclusions can be drawn. The PP depends on the material. For some ranges of temperatures, the PP remains unchanged as the temperature is increased. m and C are interrelated and the knowledge of m gives C. This is in agreement with expe~mental observation and is consistent with Yokobori’s theory. The PP may be associated with the intersection of macroscopic and microscopic crack growth rates or transition mechanisms in the FCGR curve. At the present stage it appears that further work, especially microstructure and fracture morphology observations, is necessary to more accurately identify the physical meaning of the PP. ~c~~o~~e~g~~e~rs-~ helpful discussions,

would like to thank Professor J. Fact for encouraging me to do this research and Professor T. Magnin for his reading of the manuscript.

work, as well as for

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;:2 [3] [41 [51

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