Plastic work during fatigue crack propagation in a high strength low alloy steel and in 7050 Al-Alloy

E@mring

Fmctvrr

Mechanics, 1977. Vol. 9, pp. ELM

Peq.~ooPress.

Printed in Great Britain

PLASTIC WORK DURING FATIGUE CRACK PROPAGATION IN A HIGH STRENGTH LOW ALLOY STEEL AND IN 7050 Al-ALLOYt SHOZO IKEDA$ YOSHITO IZUMI and MORRIS E. FINE Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60201,U.S.A. Ab@rae-Theoretically and empirically a fatigue crack propagation rate of the form dc (AK)4 dNajG% is indicated where ,L is the shear modulus, o is an appropriate measure of the ahoy’s strength, and u is the energy to de a unit area of fatigue crack. The local stress-strain curvesin the plastic zone around a propagating fatigue crack were determined using tiny foil strain gages. The areas in the hysteresis 100~swere integrated over the plastic zone for a unit area of crack advance to give an approximate value for u. The non-hysteretic plastic work was neglected in this calculation hut its contribution to the total plastic work in the plastic zone near the crack tip is small.

1.INTRODUCTION THE ROLEof

the plastic deformation which occurs at a crack tip in determining the rate of fatigue crack propagation is not well understood. Donahue et al. [l] and McClintock[21 assumed that the growth per cycle is some fraction of the crack tip opening and obtained for the rate of fatigue crack propagation & = (A AK’)/(uE)

(1)

A is a constant, AK is the stress intensity amplitude, u is an appropriate measure of the strength of the alloy, and E is Young’s modulus. Donahue et al. empirically established that in many materials A - a/E so that the dependence on strength is removed. They suggested that this arises because A is a function of the strain hardening rate and “in general it is to be expected that the higher the ratio of yield stress to modulus, the lower the work hardening rate”. Weertman[3] and Mura and Lin[4] derived theoretical equations for the fatigue crack propagation rate of the form where

dc aa-7

(AIQ4 CLULJ

(2)

where CLis the shear modulus and U may be considered the effective surface energy for fatigue crack propagation, i.e. the energy required to extend a fatigue crack by a unit area. Weertman’s approach was to use the formalism of continuous dislocations with the assumption that an element in the plastic zone at the tip of the crack breaks when the cumulative displacement reaches a critical value. Mura and Lin derived their equation from the plastic strip model with the failure criterion that an element at the crack tip fractures when the integrated plastic work reaches a critical value. The plastic work around the tip of a fatigue crack may be separated into two parts, non-hysteretic plastic work or work attributed to permanent strain and hysteretic plastic work which is obtained from the summation of the areas of hysteresis loops from repeated strgining[5]. The non-hysteretic plastic work required to extend the fatigue crack by a unit area depends on the maximum applied stress and AK but does not depend uniquely on the number of cycles of plastic deformation prior to fracture of the individual elements. Thus the non-hysteretic plastic work must be very small compared to the hysteretic plastic work for U to be approximately The research on steels was supported by the National Science Foundation through the Northwestern University Materials Research Center, Grant No. DMR-7203019.The research on aluminum alloys was supported by the Air Force Office of scientific Research, Oflice of Aerospace Research, Grant No. AF-AFOSR-73-2431. Wresent address: National Research Institute for Metals, 2-3-12Nakameguro, Meguro-ku, Tokyo, Japan. 123

124

SHOZO IKEDA etaf

constant in the Mura-Lin formulation. The non-hysteretic plastic work, of course, is of prime importance for static failure. Assuming that the crack renucleates ahead of the crack front, McClintock]2] derived an expression for the rate of fatigue crack propagation with dc /dN - AK 4. In this equation dc !d N was predicted to be inversely related to (e,‘)’ where ljp was considered to be the true plastic strain at fracture in a monotonic test. It would seem preferable to introduce a cyclic strain terin characteristic of a fatigue test. Cherepanov and Halmanov[61 derived an equation similar to eqn (2) on the basis of a balance between the local energy expended at the crack tip and the energy supplied. Experimentally, the dependence of dc/dN on AK4 has been observed in a aumber of materials[7], although lower and higher values have also been observed. The functional dependence of dc/dN on l/p (or l/E) is well accepted@] and the functional dependence on i/tr,’ has been observed in a number of systemsIP_111. If dcidN -x (AK)‘iva,‘, then from a dimensional point of view the proportionality constant must have the units of reciprocal energy per unit area, i.e. surface energy. In the present research, the plastic work associated with advancing fatigue cracks were measured in a Nb containing high strength low alloy steel and an aluminum alloy ~7050) by cementing foil strain gages on the sides of panel specimens. The objective was to determine LJ’in eqn (2) and clarify the role the material’s cyclic ductility plays in determining the rate of fatigue crack propagation.

2. EXPERIMENTAL PROCEDURE A hot rolled 2.5 mm thick low carbon niobium doped high strength low alloy steel plate was supplied by Inland Steel Research Laboratory. The chemical analyses in weight percent are as follows: C, 0.06; Mn, 0.33; P, 0.010; S, 0.030; Si, 0.003; Cu, 0.03; Ni, 0.02; MO,0.01; Nb, 0.027; AI, 0.039; N, 0.007. A steel of 7050 T76 Al ahoy was received from Alcoa Research Laboratory. Its chemical analysis in weight percent is: Cu, 2.25; Fe, 0.01; Si, 0.01; Mg, 2.36; Zn, 6.20; Ti, 0.02: Zr. 0.11. Sheet specimens 20 mm wide, 2.3-2.4 mm thick, and 100mm long were cut parallel to the rolling direction. The side surfaces of the specimens of Nb-steel and 70.50Al were polished using 600 grit SiC and 1 p diamond paste, respectivety. A 3 mm long 0.2 mm wide center notch was introduced by electro-discharge machining. Flat specimens 2.3 mm thick with gage sections 9 mm Iong and 4 mm wide were also prepared for determination of the cyclic stress-strain properties. Specimens of the 7050 alloy were tested in the T4 condition (solution treated 1 hr at 500°C followed by aging 7 days at room temperature) as well as the T76 condition (solution treated l/2 hr at 482°C and aged 4 days at room temperature, 3 hr at 121”C, and 9 hr at 163°C). Small strain gages (Micro Measurements MA-06-008CL-120 for Nb-steel and MA-13-008CL.. 120 for the Al alloy), 210 ym wide and 200 pm long in the gage section were cemented on the notched specimens astride and above the line where the fatigue crack was expected to run (Fig. 1). The distance in the X direction from the tip of the notch to the center of the strain gage was about 2 mm. The strain gage measured strain only along the Y direction. The plastic work from the strain component along the X direction is relatively small, especially when the strain along the Y direction is large. Cyclic tension-tension loads of 30 and 3 Hz were applied to the notched specimens of the steel and Al alloy, respectively, by a closed loop electrohydraulic MTS fatigue machine. The R ratio (minimum load divided by maximum load) was 0.05. All fatigue crack propagation tests were done in a dry argon atmosphere. The crack lengths were measured by a 30X telemicroscope. During the test, the frequency was decreased to 0.3 Hz to measure nominal stress vs strain gage output, i.e. a hysteresis loop, using an X-Y recorder. Hysteresis loops were also recorded under load control. To keep stress intensity amplitude (AK) essentially constant for the Nb-steel specimens, the cyclic load and minimim load were decreased in steps every 0.1-0.2 mm advance of the total crack length. Constant tension-tension cyclic load was applied to the 7050 Al alloy specimens because the plastic zone was so small that AK did not change much during crack propagation of the length of a plastic zone. The Isida correction factor was used to calculate AK. The crack propagation rate vs c at constant AK was nearly constant in the Nb-steel at the same value determined by constant load amplitude tests [ 121.A number of different values of AK were investigated in the range ~20MN~m3”.

Plastic work during fatigue crack pro~gation in a high strength low alloy steet and in 70.50Al-alloy

125

Fig. 1. Schematic drawing of fatigue crack propagation specimen showing location of the strain gages and definitions of X and Y.

The cyclic stress-strain curves were determined by the incremental strain-step method using the flat unnotched specimens and a clip-on strain gage. 3. EXPERIMENTAL RESULTS 3.1 Nominal stress-strain curves Nominal stress-strain curves are shown in Fig. 2 for the Nbsteel; (a) is a typical set of curves when the strain gage was cemented so that Y was small and (b) is a typical set of curves for moderately large Y. The curves have been arbitrarily shifted to the right on the strain axis to set

STRAIN

(%)

0 0

0.1 a2 0.3 STRAIN (%I

Fig. 2. Typical nominal stress-strain relations for the Nb-steel. (a) is a set of curves for Y P 38 pm and (b) is a set of curves for Y = 640pm. The number of cycles, iV,and X in mm corresponding to the curves are as follows: N A 3 C D E F

I 4.m 130,000 182,000 192,000 l!n,ooa

X 2.0 1.5 0.8 0.37 0.19 0.012

N G H I J K

1 20,000 44,000 86,000 116,OCQ

X 2.0 1.6 1.1 0.44 0.02

126

SHOZO IKEDA ef nl.

them apart. The letters on the figure designate the location of the crack front relative to the htri:m gage. The non-closures at the bottom of curves A and G are due to permanent slram. I‘he rncremeni of permanent strain in each cycle is very small except for the first cycle. When the strain gage is far from the crack tip, the hysteresis loop widths are too small to show on the scale of Fig. 2 (curvr:s B. C and H). Curves D and I show slight hysteresis, and the loop widths increase further Niith advance of the crack (curves E, F and J). The strain gage was broken just heforr the i:rack rip arrived at the gage (several hundred cycles after curve F). When the crack passed under the strain gage, the hysteresis loop width decreased with advance of the crack (K). and when the crack had propagated far beyond the gage, no plastic strain was observed in the gage during cyclic loading. Several series of nominal stress-strain curves were obtained with different gages cemented ;it different distances from the crack line for each AK value investigated.

3.2 Estimation

of the local stress and plastic work per cycle

The local stress in the volume represented by the strain gage is, of course, different from the nominal stress so that the local stress must be estimated. As discussed below, the local stress-strain curves in the material under the foil strain gages were estimated by comparison with the cyclic stress-strain curves of unnotched specimens under load control. In testing unnotched specimens under load control, when the applied load amplitude was less than 300 MN/m’ and the strain amplitude was 0.15%, the stress-strain loop for the Nb-steel was nearly stable even though the R ratio was zero (tension-tension). When the load amplitude was more than 350 MN/m* and the strain amplitude was more than 0.184%, the stress-strain curve did not form a closed loop when the R ratio was zero, but the strain at maximum stress increased a small amount for each cycle. As shown in Fig. 2, the stress-strain loop around a crack in a notched specimen is closed even when the strain amplitude was more than 0.184%. This means a residual compressive stress must have been present even though the nominal stress is always tensile since the loop would not close without the presence of a compressive stress when there has been plastic deformation. When the amplitudes of load control tests on unnotched flat specimens were adjusted to give a negative R ratio, i.e. the load was compressive during part of the cycle, an R ratio of -0.25 was required for a 350 MN/m’ load amplitude to essentially close the hysteresis loop. Moreover, the loop width did not change much on decreasing the R ratio to -1.

L.-i_ 0.4

!

-02’

02

0 STRAIN

(%)

Fig. 3. Estimation of local stress from comparison between cyclic stress-strain curve determined with strain control and clip-on strain gage, upper curve and hysteresis loop near crack tip using foil strain gage, lower curve. Data is for Nb-steel.

Plastic work during fatigue crack propagation in a high strength low alloy steel and in 7050Al-alloy

STRAIN

STRAIN

AMPLITUDE

AMPLITUDE

127

(%I

(%)

Fig. 4. Relation between loop width and strain amplitude. Points were taken with foil gages in notched specimens. Curves were taken usii clip-on strain gage and unnotched specimens. (a) NL+steel,@) 7050Al alloy. The sign&axe of the symbols on (a) are

% 2 A,

@G$) 12.4 913 12.4

(L) 57 El,

(M$& 15.5

138 26 0, A 279 A,

19.5 15.5 19.5

(i) 684 14 138

For the 7050 Al alloy specimens also, no open loops were observed with the foil gages after the first cycle. Therefore, compressive stresses must have been present with these as well. The upper curve in Fig. 3 shows cyclic stress-strain loops under strain control for the Nb-steel determined on an untracked flat specimen using a clip-on strain gage. The lower curve is a hysteresis loop obtained from a cracked specimen using a foil gage. The mean value of strain amplitude in the lower curve was shifted to coincide with zero strain of the upper curve to take account of the residual compressive stresses near the crack tip. The exact amount of the shift in the origins of the hysteresis loops determined with the foil strain gages has only a small effect on the amount of hysteretic work done per cycle determined from the areas in the loops since the slopes of the lines are parallel except at the large strain tips of the loops. For a local strain of value A on the center line of the hysteresis loop of the cracked specimen, the local stress was assumed to be the value S obtained from the cyclic stress-strain curve for an unnotched specimen at the same strain. The loop width BC was assumed to be the same as that of the upper curve, PQ. This method of determining local stress is, of course, only approximate. The loop widths of the hysteresis curves obtained from the notched specimens are plotted against strain amplitude along with the data using a clip-on gage and an unnotched specimen in Fig. 4(a) for the Nb-steel. For the most part, the two sets of data are in reasonably good agreement. Deviations from the solid curve indicated by the arrows in this figure were obtained when the strain gage was just above the crack tip. Figure 4(b) shows similar results for the 7050 Al alloy and again the agreement is good, although foil strain gage data was taken only at small amplitudes.

SHOZO KEDA it nl.

STRAIN

STRAIN

AMRITLDE

(~4

AMPLITUDE

Fig. S. Plastic work per cycle per unit volume vs strain ampiitude. Points represent data taken u&g f&f strain gages with local stress estimated as described in text and shown on Fig. 3. Line represents data taken with strain control on an unnotched specimen using strain control. (a) Nb-steel, (b) 7050 Al alloy. For significance of symbols on (a) see Fig. 4 caption.

The plastic work per cycle per unit volume for the Nb-steel, that is the area of a hysteresis loop determined with a foil strain gage on a cracked specimen estimated by the method described above, is plotted as a function of strain amplitude in Fig. 5(a) for various values of Y and AK. The solid line in Fig. 5(a) shows the plastic work vs strain amplitude observed in a strain controlled tension-compression test using an unnotched flat specimen and a clip-on strain gage. Except at low AK, the agreement between the two is reasonably good giving further support to the validity of the method for estimating plastic work as a first approximation. The lack of agreement at low AK may occur because the strain gage is too large compared to the local plastic zone. Some of the data points when Y is large and the crack had propagated beyond the maximum strain amplitude position also deviate from the curve, as indicated by the arrow in Fig. S(a). The contribution of these regions to the total plastic work are not very large. Similar plots for the 7050 Al alloy are shown in Fig. S(b). The data for the notched specimens is only for small amplitudes where the agreement is quite good. 3.3 Total hysteretic plastic work per unit area of fatigue crack The hysteretic plastic work expended for extension of a fatigue crack by a unit area U is given bv the eauation, (3) where 0; and a, are the upper and lower lines of the hysteresis Since

loops

and E is the local strain.

(4)

Plasticworkduringfatiguecrackpropagationin a highstrengthlowaUoysteeland in 7050fi-dk)Y

129

where the di&xen~e between the inte~ations is the area of a given hysteresis loop co~espondin~ to the bation X, Y with respect to the crack, eqn (3) can be rewritten simply as u=

Uxr dX dY.

If

(5)

some typical @CYdata for various X and Y values are shown in Fig. 6(a) for the Nb-steel where a

x ( DISTANCE AND

BETWEEN

STRAIN

CRACK

GAGE CENTER

TIP (mm>

f 6

s

4

5

B

5

2

2 0L 0

I

2

X, DISTANCE BETWEEN TIP AND STRAIN GAGE

0 X,

I DISTANCE TIP

AND

2

BETWEEN STRAIN

CRACK (mm)

GAGE

CRACK (mm)

Fig. 6. Typical local plastic work profdes along Xdimction for several different Y values (Ux, vs X). (a)Nb-steef,~)7050A1alloyT76,~c)7050A1alioyT4. EFM Vol. 9. No. l--t

130

SBOZO IKEDA et al.

&Y at constant Y vs X curves are shown. The strain amplitude has two maxima when the plastic zone is cut by a vertical plane normal to the crack plane, as shown schematically by the inserted figure in Fig. 6(a). A plastic zone of this kind is formed for an applied tensile stress when plane strain conditions are satisfied around the crack tip[13-181. As a result, the plastic work done at Y = 325 ,um is larger than that at Y = 26 pm in the range X > 400 pm. as shown in Fig. 6(a). The shape of the plastic zone will be discussed again. Similar results for the 7050-T4 and T76 specimens are shown in Fig. 6(b) and (c), respectively. Here the plastic zone is very small and it was not possible to observe the drop in Ii,,- as the value of X becomes small when Y is not small. In performing the integration of eqn (5), one problem is determining the large value of plastic work in the volume closer than 1OO~mto the crack tip where strain gage measurements were not possible with the present set-up. The strain amplitudes determined by the strain gages were extrapolated to I pm assuming a linear relation on a log-log plot as suggested by the data shown in Fig. 7. Hahn et al.[15] examined strain amplitudes around crack tips in Si-Fe by the etch pit method. Their values are of the same order as those shown in Fig. 7. According to Rice [ 131.for the case of a Mode III crackf 171,the equi-strain amplitude lines about a crack are circles as shown by the insert in Fig. 7. The plastic zone shape for a Mode III crack was used to estimate the strain amplitude vs X and Y in the region very close to the crack tip. As discussed subsequently, the data close to the crack tip confirms this shape for the plastic strain field. Values of U for the volume around a crack tip closer than 100 pm, designated UC,were estimated in this manner and are shown in Table 1. Another concern in experimentally determi~ng the total hysteretic plastic work is the “background” plastic work. The plastic work away from the crack tip is not zero for the Nb-steel. as shown by the tails in the curves for Fig. 6(a). The “background” pbstic work per unit volume depends on the nominal cyclic stress amplitude. Thus, the tail of the curves in Fig. 6(a) depend on experimental conditions. If the crack is long and the stress amplitude is small enough, the background level of U,, is almost zero. Because the plastic work far away from the crack can have little effect on processes near the crack tip, a “background” plastic work was subtracted

I

10 X, DISTANCE

u?*

IDO

FROM

CRACK

TIP

&m)

3

IO

r

x ,

DISTANCE

loco

loo FROM

CRACK

TIP

-

(pm)

Fig. 7. Strain amplitude vs distance from crack tip along X direction showing extrapolated curves to X = I pm for (a) Nb-steel, (b) 7050 Al alloy. These curves are for Y = 0. For a, c1= 9.3 MN/m3’2, x = 12.5MN/m”*, 0 = 15.5MN/m”*, () = 19.9MN/d’*.

Plastic work during fatigue crack propagation in a high strength low alloy steel and in 7050Al-alloy

131

Table I. Hysteretic plastic work for extension of fatigue crack by a unit area

m

m/4 Nb-steel

9.3

0.6

X lo=

0.37

x 106

0.25

x 106

0.3

x 10-8

12.4

1.2

x 106

0.77

x 10S

0.44

x 106

0.8

x 10-a

15.5

0.8

x 106

0.65

X IO6

0.12

X IO6

1.7

x 10.8

19.5

1.2

x Id

0.75

x 10”

0.40

x 10”

3.7

x lo-”

7050 T76

15.5

6.3

X 10’

4.2

x lo*

2.1

x 10”

3.0

*

7050 T4

12.4

5.4

Y 104

2.8

I

2.6

x Id

3.0

x 107

” uf

cycle

J/m2

J/8

J/n?

= total

hysteretic

= hysteretic

plastic

plastic

work

per

lti

unit

area

= “f

work

per

unit

area

in

region

work

per

unit

area

estimated

1oc

+ UC more distant

than

100 *ol from crack UC = hysteretic than de/a

plastic

for

region

Closer

100 pm to crack

= fatigue

crack

propagation

rate

before the integration for the Nb-steel. For the 7050 aluminum alloy, the observed “background” plastic work was very small and was neglected, Fig. 6(b). The background level, which is a function of X and AK, was estimated by comparison with the stress-strain loop of a flat specimen for the same stress amplitude. The amount of the subtraction at X = 0 mm was about one-third of that at X = 2 mm. For simplicity, the subtraction line was drawn as a straight line between the two points, UXUat X = 2 mm and one-third of Uxy (at X=2mm) at X=Omm. The areas under the lines in Fig. 6 for X and Y greater than 100 pm (U,) are plotted as functions of Y in Fig. 8. Twice the areas under the curves in Fig. 8 give the plastic work values for the different AK’s neglecting the plastic work near the crack tip (X and Y less than 100pm). These are designated Ur. The “background” plastic work was subtracted from the total in determining U,. The total cyclic plastic work U is a summation of UC and U,. Values of UC,U, and U are listed in Table 1. Values of U for various values of AK from 9 to 20 MN/m3’2are shown in Fig. 9

DISTANCE

FROM

CRACK

LINE

(Y) pm

‘p-c:~, 500 DISTANCE

FROM

1 lcim

CRACK

LINE

(Y),m

Fig. 8. Local plastic work in a belt of unit width along X plotted vs Y (U, vs Y). Plastic work when X or Y < lOOam is neglected in U,. (a) Nb-steel, (b) 7050 Al alloy. For a, 0 = 9.3 f&r/m”*, x = 12.5MN/m”2, 0 = 15.5MN/m”*, 0 = 19.9MN/m’“. For (b), 0 =T76 treatment, AK = 15.5MN/m”$ x = T4 heat treatment, AK = 12.4MN/m3’*.

132

SHOZO IKEDA

IO STRESS

12 INTENSfTY

14

cf

crf.

I6 FACTOR

18 20 AK (~N/m~‘*)

22

Fig. 9. Hysteretic plastic work far’ extension of a fatigue crack by a unit area. U, vs AK for Nb-steel.

for the Nb-steel. The average value is approx. 1 x 10”J/m*. The observed U for the 7050 Al alloy was about 6 x 10’J/m’ for both the T76 and T4 heat treatments. The U value of the 7050 Al alloy is 16times smaller than that of the Nb-steel in keeping with the high value of (dc/dN),, in the alloy. A fracture toughness test of the Nb-steel on a deep single edge notch specimen with a long notch gave KC equal to 56.0 MN/m’“, which corresponds to a plastic energy (E = KCZf~)[20]of 1.5 x 1O’Jfm’. This value is much smaller than U. The nonhysteretic plastic work cannot make an important contribution to U for the range of AK studied. The strain is measured on the surface of the specimen. Since the surface region of the specimen is in plane stress while the inside of specimen may be partially or completely in plane strain, the strain at the surface is not strictly the mean strain of the specimen. This is one source of error in the measurement. Extrapolation of strain to the small X range also gives an error. The resulting values of U for the dotted extrapolated lines shown in Fig. 7(a) for AK = 19.5 MNfm3n result in the upper and lower limits shown in Fig. 9. Estimation of the local stress by the method of Fig. 3 also leads to an error which probably does not exceed 1.5 times. For comparison, the accuracy of the values of dc/dN of the Nb-steel was about a factor 1.51121. identifying U with the total hysteretic plasti~plast~~ work neglects the non-hysteretic plastic work. The hysteresis loops appeared closed after the first cycle within the precision of the X-Y recorder indicating that the non-hysteretic plastic strain per cycle is very small. The minimum strain continually increased with cycling indicating that the loops were not actually closed; however, the non-hysteretic plastic work is quite small compared to the hysteretic plastic work.

3.4 PI&c zone and dispzucement The Ioop width is plotted in Fig. 10 vs distance from the crack line in the Y direction for various values of X. The areas under the curves correspond to the integrated plastic displacements near the edge of the plastic zone. These are shown in the table in the figure caption. The equi-hysteretic plastic work density lines around a fatigue crack tip can be determined from the curves of U,, vs X for various Y’s for each of the AK values investigated (Fig. 6). These are shown in Fig. 11 for the Nb-steel. The shapes and sizes are roughly the same for all the AK values, another indication that U is roughly constant in the AK range investigated. Note that near the crack center, the equi-hysteretic plastic work density lines are roughly circdar in three of the four cases lending support to the method for estimating UC.Only partial results for the 7050 Al alloy were obtained but the results are quaiitatively similar.

3.5 Monotonic and cyclic mechanical properties The monotonic and cyclic mechanical properties determined with unnotched tensile specimens are given in Table 2. The monotonic and cycfic 0.2% off set yield stress values a, and a: indicate a small amount of cyclic softening for the Nb-steel but substantial cyclic hardening for the 7050 Al base alloy especially for the T4 condition. For the Nb-steel, a stable set of incremental strain hysteresis loops were obtained after a few cycles at the maximum strain amplitude. This was not the case for the 7050 Al alloy but an essentially stable set of hysteresis loops were obtained after a large number of cycles.

Plastic work during fatigue crack propagation in a high strength low

0

alloy

moo

500

steel and in 7050M-tiOy

1500

Y tarn) Fig. 10. Relation between loop width and distance from the crack tine for Nb-steel. The area under the curves give the plastic displacements and are shown in table below. Plastic Displacements X = lOt$m X=200&m QO.113 flM l,0.074 pm A, 0.101 A, 0.065 x, 0.118 t, 0.042

AK 19.5MN/m’” 15.5 12.4

x

’ 1400

’

x=sOO&Un

0,0.095pm & 0.021 *, 0.027

’ 1000

c

600

200 I

c

Fig. 11. pi-hysteretic plastic work density lines for Nb-steel. Numbers are located at experimental points, In sequence toward origin numbers are 1, 2, . . , 9 x 10” J/m’ or 1, 2, 3, 4 x lOI J/m’.

Table 2. Mechanical properties determined on unnotched specimens

133

134

SHOZO IKEDA cl al.

4. DISCUSSION 4.1 Comparison of results for Nb-steel and 7050 Al alloy If one assumes that dc/dN obeys approxjmately an equation of the form of eqn (2) and that Lj may be identified with or is proportional to the cyclic hysteretic plastic work, then A = Idc/dN)aK(cr:)2p?I should be a constant for all metals and alloys. This may be tested for the Nb-steel and the 7050 Al base alloy of the present research at AK = 15.5MN/m3” using the data given in Tables I and 2. A computes to 130 and 120~N4~rn6~~yclefor 7050-T76 and the Nb-steel respectively. The agreement is very encouraging. The much larger value of Idc/dNlax for the 7050 Al-base alloy is compensated by its much smaller value of U and the smaller F. Note that the difference in Idc/dN[,, for the two alloys cannot be understood either from the differences in p or CL*or from the differences in a: or (a:)“. The effective surface energy term seems to be obviously playing the determination role. Note, this result is independent of the exponent on AK in the rate equation. 4.2 Campa~iso~ of measured U with t~earies of Mura and Lin and Weertman The crack propagation rate (dc/dN) equation derived by Mura and Lin[4] is

where

UJ+K %K cl

2

DJ+K

tan /3 @Lh l-tanpltanff’

=(l+K)

v

n

2

I+ 1 -t{l +4B(K/cr;)2}“2 [

1

K = (?rc)“ZS

(AK)’

16~~; 1 +{1 + B(AK/v;)~)“~’

(9)

and K = 3 - 4~ for plane strain. v = Poisson’s ratio, h = plastic zone width, K = stress intensity factor, tan a = Young’s modulus, tan p = work hardening rate obtained by fitting the hysteresis loops with parallelograms, rr: = cyclic yield stress, p = shear modulus, U, = effective surface energy, r; is the plastic zone radius in the X direction, and V, is the displacement on each side of the plastic zone. Using the data in Table 2, the observed values of Idc/djVIarc,and taking K = AK and v = 0.3, values of II,,,, and V, were calculated from eqns (6) to (9). The results are shown in Table 3. In the calculation of U,,,, V, was neglected since it is negligible in the range of AK studied. Comparison of Tables 1 and 3 shows that U,,, is approx. 10-1.5times larger than U measured from the integrated hysteretic plastic work. Table 3. Cumulativeplastic work (U,,,),plastic zone size (2r:), and displacementbetweenupper and lowerregions of plastic zone (V.) calculatedfrom the theory of Mura and Lin[4] 2rp flK “” % k m/ms LIn .Iit2 ii= --

Plastic work during fatigue crack propagation in a high strength 10~ doY

135

steel and in 7050A-dloY

The Weertman equation dc _ ?r(l- v)(AKY iii?-- sunup

(10)

gives an even larger value of U. Here 0; is the critical stress for fracture of the material at the crack tip with work hardening assumed to occur. While CGmay be somewhat larger than o:, a factor of three is required to give values of U essentially the same as from the Mura-Lin theories. In view of the approximations in the theories of Mura and Lin and Weertman, and in view of the sources of error in the measurements of U, the present results cannot on their face values be taken as either confirming or refuting these theories and the failure criteria on which they are based. However, the reconciling of the large differences between Idc /dNlaK in a steel and in an aluminum base alloy by the compensating large differences in the measured values of the integrated hysteretic plastic work gives credence to an equation for dc /dN of the form of eqn (2). The role of accumulation of damage in fatigue crack propagation requires discussion. Most of the measured integrated hysteretic plastic work goes into heating the specimen with only a small percentage being stored as damage in the material. The role of the stored energy is not clear since strain cycling is known to result in cyclic softening in cold worked metals and in martensitic steels where a reduction in dislocation density has been observed[l9]. However, the local dislocation density is very large in certain regions and this appears to be a common feature of the structure of metals subjected to extensive cyclic plastic deformation[20]. Pile-ups of dislocations sit second phase particles or grain boundaries may also play a role in the fatigue crack propagation process. The stored energy in the regions of high dislocation density thus appears to be more important than the total stored energy. The total hysteretic energy which derives primarily from the to and fro motion of dislocations, however, must be supplied to the specimen for the fatigue crack to grow. The present results suggest that U in eqn (2) is closely related to the total hysteretic energy. 4.3 Cyclic plastic zone size and displacement The data from the foil strain gages in the present study give a determination of the plastic zone sizes. These are best seen from Fig. 6 where Urn, the area of a single hysteresis loop for a given value of X and Y, are plotted vs X. The cyclic plastic zone size along the X axis for the Nb-steel at AK = 12.4MN/m3’*is approx. 1 mm. It is approx. 0.5 mm for 7050-T4 aluminum base alloy at the same value of AK. The cyclic plastic zone radius along the X axis, r:, is given by 0.0075 (AK/o;)* according to Hahn et al. 1151or (lr/64)(AK/ui)‘according to Rice[l3]. The value of 2r; from the first formula is 30-50 times smaller than the measured cyclic plastic zone size. The formulae derived by Mura and Lin, eqn (8) and Rice, predict a plastic zone size 5-10 times smaller than the measured values. The values of 2r6 calculated from eqn (8) are also given in Table 3. The computed static plastic zone sizes using the formulas of Rice [ 131or Hahn et al. [ 151are roughly four times the cyclic plastic zone size. The displacements calculated from eqn (9) are also given in Table 3. The calculated displacements 0.3-1.5 pm are roughly 10 times the measured values shown in Fig. 10, 0.02-O.12 pm. Acknowledgements-The authors are pleased to acknowledge many helpful discussions with Profs. T. Mura and J. Weertman. They especially appreciate the assistance of the Inland Steel and Alcoa Research Laboratories for supplying the materials.

[l] R. J. Donahue, H. M. Clark, P. Atamno, R. Kmnble and A. J. McEvily, Int. J. Fmcture Me&. 8, 209 (1972). [2] F. A. McClintock, ASTM STP 415 (1%7). Discussion to article by C. Laird, p. 170. [3] J. Weertman, J. Fracture 9, 125 (1973). [41 T. Mura and C. T. Lin, Int. J. Fracture Mech. 10, 284 (1974);private communication. [5] K. N. Raju, Int. J. Fracture Me&. 8, 1 (1972). [6] G. P. Cherepanov and H. Hahnanov, Engng Fracture Mech. 4, 219 (1972). Pl N. E. Frost, K. J. Marsh and L. P. Pook, MercrlFatigue, p. 212. Clarendon Press, Oxford (1974). [S] M. 0. Speidel, High Temperature Materials in Gas Turbines, p. 250. Elsevier, Amsterdam (1974). [9] L. H. Burck and J. Weertman, Met. Trans. 7A, 257 (1976).

136

SHOZO IKEDA et al.

[IO] J. S. Santner and M. E. Fine, Met. Trans. 7A, 583 (1976). [ll] A. Saxena and S. D. Antolovich, Met. Trans. 6A, 1809(1975). [12] S. Ikeda, J. J. Perout and M. E. Fine, unpublished data. [13] J. R. Rice, J. Appl. Mech. 34, 287 (1%7); ASTMSTP 415, 247 (1%7). [14] G. C. Sib, Methods of Analysis and Solutions of Crack Problems, p. 470. Noordhoff, Leyden (1973). [15] G. T. Hahn, R. G. Hoagland and A. R. Rosenfield, Met. Trans. 3, 1189(1972). [la] hf. A. Schroedl, J. J. McGowan and C. W. Smith, Engng Fracture Mech. 4. 801 (1972). [17] R. M. Schneiderovitch and 0. A. Levin, Strain fields investigation in connection with failure criteria analysis. 3rd Int. Co& Fracture, Vol. II, p. I-435, 8-13 April, Munich, Germany (1973). [18] 0. Vosikovsky, ht. J. Fracture 10, 141 (1974). 1191P. N. Thielen, R. A. Fournelle and M. E. Fine, Acta Met. 24, 1 (1976). [20] C. Laird, Cyclic deformation of metals and alloys. Treatise in Materials Science & Technology, Vol. 6, p. 101.Academic Press, New York (1975). (ReceivedDecember

1975)