Fatigue Crack Growth Under Biaxial Stress at Room- 123docz.net

Elevated Temperature

REFERENCE: Brown, M. W. and Miller, K. J., "Mode I Fatigue Crack Growth Under Biaxial Stress at Room and Elevated Temperature," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 135-152.

ABSTRACT: Fatigue crack growth rates have been measured in different biaxial stress fields for a variety of stress ranges at two temperatures. It is shown that a negative T- stress accelerates the crack propagation, the increase in growth rate being greater for high stresses. Different methods of determining plastic zone size are compared, and a crack growth correlation with crack-tip plasticity is proposed for remote loads not greater than the yield stress.

KEY WORDS: biaxial stresses, crack propagation, fatigue (materials), plastic deforma- tion, stress intensity, stainless steels

a c Cm E K N r R T W*

Ymax

ACT

e„

A e

Crack length

Crack plus plastic zone size Constants

Young's modulus Stress intensity factor Number of cycles Distance from crack tip K IK

T-stress

Specimen width for /^-calibration Maximum shear strain amplitude Range of stress

Normal strain amplitude to maximum shear plane Polar coordinate

Biaxial stress ratio

'Senior research fellow and professor, respectively. Department of Mechanical Engineering, Uni- versity of Sheffield, Sheffield, U.K.

135

V Poisson's ratio (T Stress Subscripts

f,o Final and initial values p Plastic

u Ultimate tensile strength x,y,z Cartesian coordinates

ys Yield stress / Value for A = 1

Since the introduction of linear elastic fracture mechanics (LEFM) many studies of fatigue crack propagation have invariably demonstrated that the rate of growth of cracks is controlled by the crack-tip stress intensity factor. Although there may be differences between materials in their sensitivity to mean stress, which is characterized by the ratio R = K^iJK^^, the use of fracture mechanics equations has led to a wide acceptance of the LEPTVI approach in all situations involving nominally elastic loading, for example, the power law relationship due to Paris

da/dN = C{AK)" (1) relating crack growth rate to range of stress intensity factor.

The Paris law has been developed from empirical considerations, together with the knowledge that the stress intensity factors describe adequately the stress conditions at the tip of a crack. However, the experimental results supporting this approach have been obtained almost exclusively from uniaxial loading situations, and to ensure that true LEFM conditions exist, stresses have been well below yield stress. An important prediction of the Paris equation is that only the stresses which contribute to the crack tip singularity can affect fatigue crack growth. This has been upheld in later biaxial studies, but only where long cracks and low stress levels have been used to ensure that the LEFM conditions are strictly maintained.

However, when stresses are increased, particularly for the short crack growth regime, marked deviations from LEFM predictions have been found. A number of studies of Mode I fatigue crack propagation have been conducted in recent years for biaxial stress conditions, and these have been reviewed elsewhere [1,2], Unfortunately the results are in conflict, a fact which does not appear to be related solely to the different properties of the various materials studied. A compressive stress parallel to the plane of the crack has been shown to increase, to decrease, and to leave unaffected the rate of growth of cracks in carbon steels [3]. Reasons for such discrepancies are difficult to ascertain when comparing results from different laboratories; nevertheless, the foregoing effect of stress

range on the magnitude of the biaxial influence is apparent. The importance of plasticity is also underlined by low-cycle fatigue studies, which show a far greater dependence of strength on multiaxial strain state compared to crack propagation tests.

An investigation of biaxial Mode I fatigue crack growth, therefore, has been conducted over a wide range of cyclic loads to examine the dependence on stress range, at both room and elevated temperature. The results emphasize the importance of plasticity to fatigue crack growth, showing a distinct correlation between crack-tip plastic zone size and propagation rate.

Biaxial Test System

A servohydraulic biaxial test facility has been developed for fatigue and creep crack growth studies. Tensile or compressive loads may be applied to each pair of arms of a cruciform specimen (Fig. 1), developing a biaxial stress field in the working section. The loads are controlled such that equal forces are produced on opposing arms. Crack propagation may be observed in the uniform thickness portion of the plate, with growth initiated at a central slot formed by spark erosion. A five zone heater enables elevated temperature testing, giving a uniform temperature distribution over the specimen and avoiding thermally induced stresses [41

The specimen depicted in Fig. 1 is complex and expensive to produce, but it is designed to ensure that a constant, controlled biaxial stress field can be maintained over a wide range of crack lengths. The array of slots machined along each edge of the working area serves two purposes. First, they allow a uniform distribution of applied stress along the edge, as each "finger" experiences the same extension (or compression) under tensile (or compressive) load. Second, they eliminate cross sensitivity between the two axes, because, as the load is

FIG. 1—Cruciform specimen geometry.

applied along one axis deforming the gage area, the individual fingers on the other loading axis are able to flex freely, thereby allowing the gage section to deform without restraint along the edges. Since the edge constraint is small, a uniform distribution of strain is readily obtained over the working section of uncracked specimens from the separate loads applied to each axis, irrespective of their sense. The elastic strain distribution was observed in a photoelastic study, verifying both the uniformity of strain and the bending behavior of the fingers. This was subsequently checked by a finite-element analysis, revealing a stress variation of 2% over the central 70% of the working section for equibiaxial loading, and 5% deviation at the base of the fillet radius.

The applied stress was determined from applied load for each axis, divided by the cross-section area of the working region (421 mm^), assuming that a negligible proportion of the load was required to bend the fingers of the second axis. This procedure was verified by the finite-element analysis, indicating better than 1 % accuracy for the stress on the central axis of the specimen.

Strains were measured with a biaxial extensometer enabling one strain controlled test to be conducted. All other tests were under load control with a sinusoidal waveform. Mode I crack extension was monitored using a d-c potential drop technique, although the room-temperature tests were checked using a trav- eling microscope. Crack growth rates were determined by a least squares fit of a parabola to groups of five crack length readings, differentiated to give the extension per cycle according to ASTM Test Method for Constant-Load-Am- plitude Fatigue Crack Growth Rates Above 10'" m/Cycle (E 647-83).

Specimen Material and Test Conditions

All tests were conducted on AISI 316 austenitic stainless steel, taken from two different heats. The composition and tensile properties are given in Table 1. Both heats were solution annealed, although Material A showed a higher yield stress probably due to a number of small size grains remaining in the plate. The

TABLE 1 —Chemical Composition and Mechanical Properties.

Material A B

C 0.06 0.049

Mn 1.88 1.36

P 0.023 0.023

S 0.020 0.018

Si 0.62 0.54

Cr 17.30 17.26

Ni 13,40 11.20

Mo 2.34 2.15

Material A B

Temperature,

°C 20 550 20 550

0.2% Proof Stress, MPa

395 268 243 133

Tensile Strength,

MPa 611 489 597 474

Elongation,

55 39 68 44

Reduction in area,

71 54 71 55

tensile tests at 550°C for both materials showed marked serrated flow at a strain rate of IQ-'^s-'.

Three stress states were examined, equibiaxial (A = + 1), uniaxial (A = 0) and pure shear (A = - 1 ) , where A = a;t/o-^ is a measure of biaxiality, CT3, being normal to the crack plane. Loading was proportional, the value of A being held constant throughout the load controlled cycle. Stresses were chosen to cover the full range from threshold (corresponding to the 2 mm initial crack) up to the yield stress. Cyclic hardening was observed at the start of the tests above the yield point. An R ratio of - 1 was chosen to avoid ratchetting, except at the lowest stress level where zero to tension cycling was employed to reduce crack closure effects in measuring threshold. Frequencies were 1 Hz at high-stress levels and 20 Hz for threshold tests.

Results

The crack growth results are presented in Figs. 2 to 6 in terms of crack growth rate plotted against AK, where

AK = ACT \/(TTa sec (-aa/W*)) (2) Here Aa is the stress range normal to the crack, including the compressive

portion. The term W* (=101.9 mm) is the equivalent width of the specimen (nominally 100 mm) which allows for the stiffening of the edges, see Appendix.

Equation 2 is used only as a convenient parameter in plotting the results at the highest stresses, even though LEFM is strictly not applicable, since it provides a suitable correction factor for the effects of finite plate width. As each figure corresponds to a fixed stress range ACT normal to the crack plane, the results show relative crack growth rates for different biaxialities corresponding to each crack length a, given by Eq 2.

Figure 2 shows crack growth from threshold, defined here as a growth rate of 0.1 nm/cycle. Although there is a small variation in the threshold values, this is probably within experimental error, disguising any influence of stress state on the threshold. These thresholds are for growth from an initial crack of root radius 0.08 mm, giving higher values than might be expected from a sharper crack [5]. A slightly higher crack growth rate is observed in shear loading, the results obeying the Paris law with m = 2.

Figure 3 presents similar data to Fig. 2, but for /? = — 1 and with stress range increased by a factor of 3. The shear cracks clearly propagate more rapidly than in the equibiaxial case. The anomalous uniaxial points for low AK correspond to nonsymmetrical growth early in the test where only one fatigue crack was growing from the notch. Below 10 nm/cycle the shear and equibiaxial tests give m = 2, rising to 3.5 above 10 nm/cycle.

Figure 4 shows the effect of doubling the stress range, giving a very clear effect of biaxial stress state on fatigue crack growth. Here the Paris law exponent

iO'

158-

- -

•

; - -

- -

•

. . , 1 , , 1 r • ,

A0"= 6 4 MPa R = 0 2 0 '

Key 5 SHEAR 0 UNIAXIAL

• EOUIBIAXIAL 0 0 V 0"

V D

. ^

V D

^ 8 0 V D a

^ 0 oo 0 0

=^ 1

! i i t 1 1 i . ,

•

- - -

•

- - -- -

3 10 30 iK (MPa fm)

FIG. t~—Fatigue crack growth close to threshold (R = O)for Material A at 2(fC.

is approximately 2.8, increasing sharply when AA" exceeds 100 MPa Vm. Also included in this figure is a shear test conducted under strain control, giving very much lower propagation rates. The strain range Ae of 0.25% was chosen to give the same initial stress range as the load controlled tests. The shear tests are of note because both at this stress level and in the elevated temperature tests two pairs of cracks were produced. The first pair formed at the initial slot in the expected manner, following the Mode I direction closely along the horizontal plane. However, a pair of vertical cracks also nucleated at the initial notch, growing at a similar rate to the horizontal cracks since they experienced a similar stress field with A = - 1.

Figures 5 and 6 show elevated temperature results, both at 550°C, but for the two different materials. The clear biaxial effect of faster growth when A is negative is again observed. However, the slope of the power law equation, m, has reduced to about 1, increasing sharply to wi = 6 at about 33 MPaVm for

16'

I 0 ô

- 9

1 1 1 1

- i O ' = 193 MPa R = -1 20" C

Key

y. V SHEAR ° 0 UNIAXIAL ^ . n EOUIBIAXIAL 9

•in

0 ^ ° e 0 ô 9 V 0

V a 0,°

" •

V D

w 0 ' D ' ' n * v ^

'iff 'J?

<ô(ằ

" /

• 0

>ằ

. . --

. , -

- . .

! - - -

AK ( MPa/m)

FIG, 3—Biaxial fatigue crack growth for ACT = 193 MPa for Material A at 20°C.

Material B, 48 MPaVm for Material A. This change in slope corresponds with a transition to slant mode growth, introducing a Mode III component at the crack tip. Similar changes in slope have been observed particularly in Type 316 stainless steel [6,7], related to the influence of thermal aging. The low value for m of unity reflects the steadily increasing resistance of the material to fatigue crack growth as cyclic aging proceeds at this temperature, strengthening the micro- structure ahead of the crack tip [8,9]. As Mode I growth becomes difficult.

Mode III is favored, and the cracks turns to a slant mode, leading rapidly to failure.

Crack-Tip Plasticity and Crack Growth Rate

At low stresses with very small-scale yielding, crack-tip plasticity is very limited, and plastic zone size is governed by the stress intensity factor, irrespective of biaxiality. But at higher stresses, above the very small-scale yield

oiz

T3l*D 10

- 6 10

- 7 10

10 - 8

•

- -. -

•

r A a = 3 8 6

V o o

—

R ằ -1 Key SHEAR

' 1

MPa , °

2 0°C

o V UNIAXIAL

EOUIBIAXIAL , o •

SHEAR o

(strain control)

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0 / 0 o / D

/ ° / °

a o o

< . . 1 . 1 < 1

- - ,

• --

•

— - -

•

10 100 AK (MPa/m)

FIG. 4—Biaxial fatigue crack growth for ACT = 386 MPa for Material A at 2(TC.

regime, larger plastic zone sizes are found for shear loading compared to uniaxial [70,77]. For small-scale yielding, the plastic zone size may be estimated from the elastic stress distribution, but it is important to include the second term in the series expansion of CT^„ the stress parallel to the plane of crack, as follows

^ằ = 7 7 ^ = cos :: 1 + sm - sm — ) + 0(r"2) VZirr 2 \ I Z

K fl / fi IB^

a „ = f-^ cos - 1 - sin - sin — ) + (A - I)o- + 0(r"^) vl-nr 2 \ 2 2

K, ( . % e 36^, ^^ ,„^

T„ = ; sm - cos - cos —- + 0(r"^)

" V 2 ^ \ 2 2 2 '

(3)

MATERIAL A ACT • 199 MPo

R • -I 550*C Key

V SHEAR O UNIAXIAL 0 EQUIBIAXIAL

10'

„<?°

OP D

oo'

,..t,l.„l.„...l

10 100

AK (MPa/m)

FIG. 5—Elevated temperature crack growth for Material A at S50°C.

where K, = oVmi and a is the stress normal to the crack of length a. The term (A - l)o- is frequently called the T-stress, and it is clearly determined in biaxial stress tests by the load applied parallel to the crack. Since the stress (ACT) is parallel to the crack, it contributes nothing to the crack tip singularity, defined by the initial terms in Eq 3; therefore, K, takes the usual form of a V t r a , for a wide center cracked panel.

By using the von Mises yield criterion with the stresses in Eq 3, and neglecting terms of order r"^ and above, a quadratic equation for plastic zone radius r,, in terms of 6 is derived for the plane stress case. For plane strain an additional stress

v(0-.ô + CTvv) (4)

€11

10*

10^

MATERIAL - ACT . 2 0 4

R = -1

•

' ' 1 i 1 ' 1 1 . B

M P a 5 5 0 ' C

o X B V

X ~

X V

V n

J7~

xiS^

^ 0

J rf^

•

- -

0 D D

Key V SHEAR X A =-0.5 o UNIAXIAL

• EQUIBIAXIAL

. I

10 100 A K ( M P a / m )

FIG. 6—Elevated temperature crack growth for Material B al 550°C.

is required for the yield criterion. The quadratic may be solved for r^, choosing 9 to give the maximum value for plastic zone size. For plane stress 0 is always close to 71° for the /T-values covered by the tests, but for plane strain 6 reduces from a maximum value of 87 to 71 ° as A" is increased. Values for reversed plastic zone sizes are given in Table 2, showing that as A decreases, plasticity is more widespread. To obtain the reversed plastic zone size, observed in cyclic loading, AK was used in Eq 3 for K, and twice the monotonic yield stress was used in the yield criterion, following the suggestion of Rice [72], assuming that a ki- nematic hardening rule is applicable. In view of the high degree of cyclic hardening found in austenitic stainless steels, the monotonic yield stress may appear to be too low for cyclic conditions, giving an overestimate of plastic zone size.

However, the error will offset to some extent by the use of the elastic stress distribution to define Kp, which always gives a lower bound value because equi- librium will not be strictly satisfied with an elastic-plastic stress-strain law.

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} / / /

\ / 1 ^ 7 / / / "

/ -^ / "

1 / / / • 1^ / ''

/ / / / / / / /

y K?y

/ p MATERIAL PLANE STRAIN DBCS

A 0

•

a B

•

da/dN da/dNi

FIG. 7—Normalized biaxial crack growth rate as a function of plastic zone size. da/dN, and r^, refer to the equibiaxial test.

a — — a T

Fatigue Crack Growth Under Biaxial Stress at Room and