Fatigue crack propagation rate and the crack tip plastic strain amplitude in polycrystalline copper

Materials Science and Engineering, 70 (1985) 91-100

91

Fatigue Crack Propagation Rate and the Crack Tip Plastic Strain Amplitude in Polycrystalline Copper P. LUK.~, L. KUNZ and Z. KNESL Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, Brno (Czechoslovakia) B. WEISS and R. STICKLER Institute of Material Sciences, University of Vienna, Vienna (Austria) (Received March 5, 1984)

ABSTRACT The size o f the dislocation cells produced by deformation at the fatigue crack tip in copper specimens was used as a measure o f the crack tip stress amplitude. Using the cyclic stress-strain curve, the values o f the crack tip plastic strain amplitude and their dependence on the macroscopic stress intensity factor were obtained. While the fatigue crack propagation rate expressed as a function o f the stress intensity factor is strongly influenced by the testing temperature and frequency, it is almost independent o f these parameters when expressed in terms o f the crack tip plastic strain amplitude. The results o f a finite element computation led to the same conclusion.

1. INTRODUCTION The fatigue crack propagation rate can be described by means of fracture mechanics parameters. This phenomenological description is certainly advantageous, as it is independent of the specimen geometry. In contrast, the fracture mechanics (i.e. continuum) description of the crack tip stress and deformation fields is not adequate for the physical modelling of crack growth and consequently for the optimization of the microstructural and metallurgical parameters. Several methods have been employed for characterizing the extremely localized cyclic plasticity around the fatigue crack tip. Perhaps the oldest is the m e t h o d of selected area channelling patterns in the scanning electron microscope (see for example refs. 1 and 2), 0025-5416/85/$3.30

which can be used for the mapping of the equivalent strain distribution to within approximately 20 pm of the crack tip or fracture surface. The equivalent plastic strain distribution can be measured also by the microhardness [3] and quantitative metallographic techniques applied to twins [3] or to deformation bands [4] determined for low cycle fatigue specimens. The minimum distance from the crack tip (or the fracture surface) at which these methods yield reliable results is again 2 0 - 3 0 pm. The recent stereoimaging strain analysis technique [ 5] can characterize the strain up to 1/am from the crack tip. The principal result obtained by the above-mentioned methods is that the local plastic strain at and near the tip is very high, exceeding 10% for loading in the Paris range and decreasing sharply with distance from the crack tip. Especially valuable are the results showing the dependence of the crack tip plastic strain on the macroscopic stress intensity factor, as these data relate the microscopic parameters to the macroscopic parameters. The crack tip plastic strain decreases rapidly with decreasing stress intensity factor [4, 5] down to values of the order of 10 -4 in the near-threshold range [4]. There are ample experimental data showing that for wavy slip materials the fatigue crack tip is surrounded by a dislocation cell structure. The majority of these observations were obtained by means of transmission electron microscopy (TEM). Recently also the backscattered electron imaging technique in the scanning electron microscope has been successfully applied to the detection of dislocation cells [6]. As the cell size is known to decrease with increasing stress amplitude, the experi© Elsevier Sequoia/Printed in The Netherlands

92 mentally determined distribution of cell sizes can be used as a measure of the distribution of stress amplitudes. Up to now this has been done only in a scanning electron microscopy study of the cell structure around the tip of a fatigue crack on the surface of a low carbon specimen [6]. In a previous investigation [7], it was found by means of TEM that the size of cells adjacent to the fracture surface in polycrystalline copper is a unique function of the macroscopic stress intensity factor amplitude. This offers a basis for a further study of the distribution of the crack tip stress and consequently the crack tip plastic strain. In particular, the relationship between these microscopic parameters and the macroscopic fatigue crack propagation rate was determined for various test temperatures and cyclic frequencies.

2. EXPERIMENTAL PROCEDURES Polycrystalline copper of commercial purity (99.98%) was used in the form of bars or plates. After the gauge section of the specimens had been machined and mechanically polished, the specimens were annealed at 600 °C for 1 h in vacuum. The grain size was 70 /zm. The basic mechanical properties were as follows: yield stress R p o . 2 = 37 MPa; tensile strength Rm = 220 MPa; dynamic elasticity modulus Edyn = 1.42 × 105 MPa for rodshaped specimens; dynamic elasticity modulus E d y n : 1.27 × 105 MPa for plate-shaped specimens. The following experiments were performed: (i) measurement of the fatigue crack propagation rate and the threshold stress intensity factor; (ii) measurement of the cyclic stress-strain curves (CSSCs); (iii) measurement of the size of the dislocation cells adjacent to the fracture surface. To vary the temperature and the frequency, the tests were run (a) at a frequency of 100 Hz at room temperature, (b) at a frequency of 100 Hz at --100 °C and (c) at a frequency of 20 kHz at room temperature (20 °C). All the tests were performed at zero mean stress (R = - - 1 ) . The geometry of the specimens for both the 100 Hz tests and the 20 kHz tests were essentially identical. The CSSCs were determined on cylindrical rod-shaped specimens

with a gauge section of 4 mm diameter. The crack growth rate was measured on centrenotched plate specimens (thickness, 5 mm; width, 20 mm). The 100 Hz tests were run in a modified Amsler pulsator. The CSSCs were determined by the companion test procedure using cylindrical specimens cycled at constant total strain amplitudes eat. The hysteresis loop was continuously recorded. The values of the stress amplitude oa and the plastic strain amplitude eap used for the construction of the CSSCs were obtained at one-half of the number of cycles to failure. The crack propagation rate da/dN was measured on the centrenotched sheet specimens cycled under controlled stress amplitudes. The crack length was monitored by means of a travelling microscope with an accuracy of +10 pm. For the low temperature tests the specimens were placed in an environmental chamber with a controlled a m o u n t of nitrogen gas;the temperature was kept constant within + 2 °C. The 20 kHz tests were carried out with a commercial test unit in an essentially totalstrain-controlled manner. In order to maintain the specimen temperature near ambient an efficient cooling system (deionized water as coolant) was employed. The total strain amplitude was measured by means of miniature strain gauges. The plastic strain amplitude was deduced and the cyclic saturation stress (for approximately N = 108 cycles) determined as described in ref. 8. The crack length was determined by means of an optical microscope-closed-circuit television system with an accuracy of + 2 pm. Forced-air cooling was found sufficient to assure a specimen temperature below 60 °C. The computation of the K values was carried out using a relationship published recently [9]. The TEM technique has been described in detail in ref. 7. The sheet specimens after complete fracture were electroplated with copper and cut into slices. Then thin foils were prepared by jet polishing from the interface between the electroplated deposit and the specimen. The crack length and the corresponding stress intensity factor were known for each foil. The sizes of the cells up to a distance of about 10 pm from the fracture surface (at least up to this distance the cell size is reasonably homogeneous) were determined as the mean intercept length.

93 3. RESULTS

Kat h. The experimental data can be described

by the equation

3.1. Fatigue crack propagation curves The results of the da/dN m eas ur em e nt are presented in Fig. 1. The stress intensity factor amplitude Ka is defined as Ka = (Kmax -Kmm)/2. It can be clearly seen that bot h the test f r e q u e n c y and the t e m p e r a t u r e influence the crack propagation rate da/dN and the threshold stress intensity factor amplitude

da dN

= A(Ka ~

-

-

Kath ~)

(1)

The constants for this equation for the presented near-threshold region are listed in Table 1. The full lines in Fig. 1 represent eqn. (1) for the constants listed in Table 1.

3.2. Cyclic stress-strain curves The CSSCs are presented in Fig. 2. It can be seen that the straight line approxi m at i on is acceptable in all three cases. Therefore, the equation

o

o a ~- k•ap n

168

I[)9

,

,

i

2

4

6

K o (MPa mI12)

Fig. 1. Fatigue crack propagation rate da/dN as a function of stress intensity factor amplitude Ka (R =--1): o, 20°C, 100 Hz; +, 20 kHz;A,--100°C, 100 Hz.

,

(2)

was applied for the description of the experimental data. The constants of eqn. (2) are listed in Table 2. The data presented were obtained under high cycle, low amplitude test conditions. Since bot h the low amplitude and the high amplitude data are relevant to the description of the stress and strain state at the crack tip, we assume that the experimental CSSCs can be extrapolated. The justification for this extrapolation is based on the published data showing a unique CSSC over the whole range of the plastic strain amplitudes. F o r example, the data of Klesnil and Pol~k [10] on polycrystalline copper (grain size, 30 pm) in the range of cap from 8 X 10 -5 t o 10 -e can be described by a single power law dependence. The same holds true for the data of Mughrabi and Wang [11] for copper (grain size, 25 pm) in the Gap range from 5 X 10 .5 to 10 -1. Very strong support for this assumption is offered by the paper of R o t h et al. [12] who compiled almost 100 individual CSSCs measured by a large num ber of investigators on c o p p e r at room temperature. T h e y f o u n d t hat all the

TABLE 1 Constants for eqn. (1) Test

A (ram cycle-1) x x (MPa m1/2)-~

~

Kath (MPa m 1/2)

20°C; 10() Hz --100°C; 100 Hz 20 kHz

1.1 × 10-l° 2.0 × 10-11 3.7 X 10-11

7.0 7.0 7.0

2.15 3.10 2.55

94

/

160

140

(/A,d

120

o

100 A

c~ 13.. v

b°

8O

/

¢o

o

!

60

!

10 .4

I0s

10 `3

Eap Fig. 2. CSSCs: o, 20°C, 100 Hz;,

,,20 kHz;~,--100°C, 100 Hz.

TABLE 2 Constants of the cyclic stress-strain curve (eqn. (2))

Test

k (MPa)

n

20°C, 100 Hz --100°C, 100 Hz 20 kHz

560 750 620

0.205 0.205 0.205

d a t a in the eap range f r o m 5 X 10 -5 t o 5 X 10 -1 are c o v e r e d b y a relatively n a r r o w s c a t t e r band, the m e a n o f w h i c h can be described b y a u n i q u e p o w e r law curve with c o n s t a n t s k = 550 MPa and n = 0.209.

Fig. 3. Dislocation cell structure adjacent to fracture surface in a specimen cycled at 20 kHz (Ka = 5 MPa

ml/2): D, deposited layer; S, specimen. (Magnification, 6800×.)

3.3. Transmission electron microscopy A t y p i c a l TEM m i c r o g r a p h is s h o w n in Fig. 3. I t reveals a d i s l o c a t i o n cell s t r u c t u r e adjac e n t t o the f r a c t u r e surface. F o r Ka values e x c e e d i n g 1.2 times the t h r e s h o l d value, the

cell size u p t o the distance o f at least 10 p m f r o m the f r a c t u r e surface is c o n s t a n t ; for higher stress intensities this range in w h i c h the cell size is c o n s t a n t is c o n s i d e r a b l y greater.

95

1.5 1.0

0.5 0.4 A

E

0.3 02

b

.1

2

i

i

I

I

3

5

10

15

K (MPa m 1/2) {3

Fig. 4. Ceil size d us. stress intensity factor amplitude Ka: l i n e a, d = 3.28/Ea; line b, d = 2.90/Ka; O, 20 °C, 100 Hz; +, 20 k H z ; ~ , - - 1 0 0 ° C , 100 Hz.

In the present investigation the mean intercept lengths for the cells lying less than 10 pm from the fracture surface were recorded. The dependence of cell size on stress intensity is shown in Fig. 4. The data for the low frequency cycling at room temperature are partly taken from ref. 7. Each point is based on at least eight micrographs of the type shown in Fig. 3. The experimental scatter was determined as the standard deviation. The accuracy of the cell size measurement is typically + 10%. It can be seen that the results for the 20 kHz tests do n o t differ from those for 20 °C, 100 Hz tests. In contrast, there is a small but statistically detectable difference between the low frequency tests at 20 °C and at --100 °C. The dependence of the cell size d on the stress intensity factor amplitude Ka can be described by the simple relation d = c/ga

(3)

The values of the constant c (least-squares fit) are listed in Table 3. Kayali and Plumtree [13] summarized the relevant results on the relation between cell size (measured at various temperatures in the bulk of low cycle fatigue specimens by means of TEM) and the saturation stress amplitude

TABLE 3 Constant c in eqn. (3) (pm MPa m 1/2)

Test

c

20°C, 100 Hz - 1 0 0 ° C , 100 Hz 20 kHz

3.28 2.90 3.28

for iron, copper and aluminium. An inverse relationship was found. For copper this relationship is d -- 1 1 5 / o a

(4)

where d (the mean intercept length) is expressed in micrometres and (:/a in megapascals. It should be pointed out that the cell size is a unique function of the stress amplitude and not of the strain amplitude. This is in agreement with the model of equilibrium between applied stress and internal stresses. The higher the applied stress, the shorter are the free dislocation segments and consequently the smaller is the cell size. Here we shall use the size of the cells just adjacent to the fracture surface as a measure of the local stress amplitude, i.e. as a measure of the crack tip

96

stress amplitude. Combining eqns. (3) and (4) we may deduce a dependence of the crack tip stress amplitude on the macroscopic stress intensity amplitude. When eqn. (2) is also em(2 300

250

p l o y e d , the relationship between crack tip plastic strain amplitude and stress intensity is obtained. This is shown in Figs. 5 and 6 for the range of stress intensities applied. The values of the stress amplitude at the crack tip (Fig. 5) are obviously too small to cause a decohesion process. Further it can be seen (Fig. 6) that the crack tip plastic strain amplitude very near the threshold is about 10 -4 and exceeds the value of 10 -2 for a crack propagation rate of the order of 10 -5 mm cycle-1.

D.. Z

4. FINITE ELEMENT ELASTIC-PLASTIC ANALYSIS

200 2

o

b° 150

100

~0

2

I

I

I

4

6

8

K (MPo m11z) O

Fig. 5. Crack tip stress amplitude as determined from cell size measurement v s . stress intensity factor amplitude Ka: line a, - 1 0 0 ° C , 100 Hz; line b, 20°C, 100 Hz and 20 kHz; ~, t, thresholds.

Q

b C

ld 2

~. 10-3

0 L u

j

10.4 I

10-5

I

2

I

4

I

I

6

I

I

8

K a (MPO m 1/2)

Fig. 6. Crack tip plastic strain amplitude as determined from cell size measurement v s . stress intensity factor amplitude Ka: curve a, 20 °C, 100 Hz; curve b, 20 kHz; curve c, --100°C, 100 Hz.

The m e t h o d of elastic-plastic analysis used here is essentially the same as that described in ref. 14. The geometry of the analysed specimen is identical with that of the specimen for the fatigue crack propagation rate measurement. Because of the symmetry, only one quadrant of the specimen was analysed. The computer program assumes plane stress conditions together with the yon Mises yield criterion and the Prandtl-Reuss stress-strain relation according to incremental plasticity. The material was characterized by the CSSC according to eqn. (2) with the constants listed in Table 2. The usual triangular finite element grid was used with constant-strain elements. The total number of elements and the total number of nodal points adopted for onequarter of the specimen were 689 and 384 respectively. The elements are essentially focused around the crack tip. Only a nonpropagating crack was considered and crack tip blunting was neglected. The characteristic size of the smallest element in the vicinity of the crack tip was 20 pm. As the region of constant cell size (see Section 3} covers at least 10 pm, this element size m a y be considered to be sufficiently small. The boundary conditions simulate the grips in the 100 Hz fatigue machine which are assumed to be rigid and the prescribed symmetry of loading. The change in the boundary condition in the 20 kHz resonance specimen was found to have no significant influence on the computed stress and strain distribution. The load is applied in increments at points remote from the crack in the form of vertical forces consistent with the experimental set-up.

97

siderably higher in the near-threshold region b u t are comparable at higher stress intensities. The same holds true for the crack tip plastic strain amplitudes (Fig. 8). The values of the plastic strain amplitude (component perpendicular to the crack) are larger in the threshold region by a factor of 10 (see Fig. 6), but they are almost identical with the experimentally found values in the region of crack rates in the range 10-6-10 -5 mm cycle -1. This difference in the threshold region may be attributed to the assumption of a zero crack tip radius in the finite element m e t h o d computations. The assumption of a non-zero crack tip radius would obviously lead to much lower stress and strain values just in the threshold region.

350

300 Q.. £-

250

o

b~

200

150

I00

'

'

'

'

2

4

6

8

5. D I S C U S S I O N

K (MPa m 1/2) (3

Fig. 7. Crack tip stress amplitude as c o m p u t e d by the finite e l e m e n t m e t h o d v s . stress intensity factor amplitude Ka: curve a, --100 °C, 100 Hz; curve b, 20 kHz; curve c, 20 °C, 100 Hz.

10,2

L o

10-3

Tr

W

_4

I

10

2

I

4

I

I

6 K a (MPa rn112)

I

I

8

Fig. 8. Crack tip plastic strain amplitude as computed by the finite element method v s . stress intensity

factor amplitude Ka: curve a, 20 °C, 100 Hz; curve b, 20 kHz;curve c,--100°C, 100 Hz.

The results of the computer analysis are presented in Figs. 7 and 8. The crack tip values of the stress amplitude ( c o m p o n e n t perpendicular to the crack) as a function of the macroscopic stress intensity factor are shown in Fig. 7. A comparison with the analogous experimental results (Fig. 5) shows that the c o m p u t e d crack tip stress values are con-

Both the experimental data and the results of the finite element analysis made it possible to correlate the crack tip plastic strain amplitude with the stress intensity factor amplitude (see Figs. 6 and 8). The stress intensity is in turn related to the crack propagation rate (Fig. 1), so that it is possible to express the measured macroscopic crack propagation rate as a function of the crack tip plastic strain amplitude. This is shown in Fig. 9 for the crack tip strain values deduced from the experimental data and in Fig. 10 for the computed crack tip plastic strains. In both cases the relation between crack propagation rate and crack tip plastic strain is not strongly affected by the test temperature and frequency (in contrast with the observation when the crack propagation rate is plotted as function of the stress intensity, see Fig. 1). The difference between Figs. 9 and 10 (higher plastic strains in the finite element m e t h o d case) may be attributed mainly to the abovementioned assumption of zero crack tip radius in the finite element m e t h o d computation. Moreover, the experimental determination of the crack tip stresses and strains is based on the uniaxial low cycle fatigue data; in reality, the stress-strain state at the crack tip is triaxial. This could also contribute to the difference. The principal result of this paper, i.e. the finding that the fatigue crack propagation rate is dependent primarily on the crack tip plastic

98

.5

10

.5

10

10- 6 .6

"T _,m

10

O ,O

E

T 10 "z

0

~Z

0

i

10 .8

E

_7

10

OZ

~ol-o O.

\

10-8

o, b c 10-9 10-5

i

i

10.4

10,3

10. 2

°g 10

Cap, crack

tip

Fig. 9. Fatigue crack propagation rate d a / d N vs. experimentally determined crack tip plastic strain amplitude: curve a, 20 °C, 100 Hz; curve b, 20 kHz; curve c, --100°C, 100 Hz.

strain a m p l i t u d e and is a l m o s t i n d e p e n d e n t o f t e s t t e m p e r a t u r e and f r e q u e n c y , favours c r a c k p r o p a g a t i o n m o d e l s based o n the d o m i n a n t role o f the plastic d e f o r m a t i o n at the c r a c k tip. T w o main representatives o f this c a t e g o r y o f the crack p r o p a g a t i o n m o d e l s are t h e models o f Laird and S m i t h [ 15] and N e u m a n n [16]. The Laird and S m i t h m o d e l o f blunting and resharpening is the m o s t general descriptive m o d e l of crack tip plastic d e f o r m a t i o n ; processes on the scale o f dislocations or slip bands are n o t specified. N e u m a n n ' s m o d e l specifies the geometrical changes o f the c r a c k tip b y the alternating a c t i o n o f coarse slip o n t w o slip systems. T h e TEM investigation r e p o r t e d in this and a previous p a p e r [7] can be s c h e m a t i c a l l y s u m m a r i z e d as s h o w n in Fig. 11. T h e z o n e s u r r o u n d i n g the crack tip contains dislocation cells. T h e slip bands observed outside this z o n e are persistent slip bands (PSBs) with a typical ladder-like structure. T h e dislocation s t r u c t u r e in the s u r r o u n d i n g m a t r i x consists o f veins. As the PSBs are k n o w n t o e x h i b i t a m u c h higher plastic strain a m p l i t u d e ( b y a

'

10"s

10 .4

'

10.3

' 10- 2

E~p, crack tip

Fig. 10. Fatigue crack propagation rate d a / d N vs. computed crack tip plastic strain amplitude: curve a, 20 °C, 100 Hz; curve b, 20 kHz; curve c , - 1 0 0 ° C , 100 Hz.

PSBs cell

structure

l a d d e r - !.ike structure

vein structure

Fig. 11. Schematic drawing of the zone of high plastic strain amplitude ([~) ahead of the crack tip and the corresponding types of dislocation structure.

f a c t o r of 100) t h a n the matrix, n o t o n l y the z o n e a r o u n d the crack tip b u t also this z o n e plus the region c o n t a i n i n g the PSBs m u s t be c o n s i d e r e d as the actual z o n e o f high plastic strain ahead o f the crack tip. T h e length o f the PSBs is limited b y the grain boundaries. This is m o s t p r o b a b l y also the reason w h y the crack p r o p a g a t i o n rate and the t h r e s h o l d d e p e n d on the grain size.

99 The increment of macroscopic crack length per cycle for low crack propagation rates is always smaller than the cell size. Moreover, the cell size decreases with increasing stress intensity, while the crack length increment increases. For example, for a macroscopic crack length increment of the order of 0.01 /lm and less, the cell size is about 1 pm (see Figs. I and 4). Thus the crack front sees an inhomogeneous material composed of dense cell walls and relatively dislocation-free cell interiors. Therefore the local microscopic crack propagation rate can be very different at different spots of the crack front; there can even be places at which the local strains are too small to produce any crack advance for several cycles at all. However, there m a y be places at which the crack front sees a higher local strain and consequently the crack length increments are higher than the average values. Thus the experimentally found dependence of the crack propagation rate on the crack tip plastic strain amplitude must be understood as the relation between the average macroscopic propagation rate and the average plastic strain. For high crack propagation rates the crack length increment becomes comparable with the cell size. In this case the differences in local plastic strains do not play such an important role. The crack growth is a continuous process with only negligible local deviations. The growing fatigue crack always tends to follow the path of highest local deformation. For small stress intensities where the PSBs start almost right from the crack tip (see Fig. 11), the crack will follow the PSBs as the zones of highest strain. This explains the wellknown crystallographic propagation in the near-threshold region. At higher stress intensities the crack tip is surrounded by an extensive zone of cells (produced by the slip activity on at least two slip systems) which does not contain preferred paths and, therefore, the crack propagates non-crystallographically. The fatigue crack growth is given by the micromorphological changes in the crack tip in a material exhibiting a dislocation cell structure. The cells are already formed into their final shape and size before the crack front reaches them. The average crack propagation rate is proportional to the average degree of the crack tip micromorphological change. The question is what this micro-

morphological change is. As the crack tip can be rather complex with respect to its geometry, it is not automatically justified to relate the growth rate to the crack-tip-opening displacement (defined as the distance between the crack flanges in the direction perpendicular to the crack plane) only. Whatever the micromorphological change in the crack tip, it is logical to assume that its magnitude is related to the magnitude and the distribution of the plastic deformation ahead of the crack tip. Thus the plastic strain amplitude at the crack tip must be (for the given shape of the zone shown in Fig. 11, which is believed to be influenced mainly by the grain size) the decisive controlling factor in fatigue crack growth. It is this point which has been verified in the present investigation.

6. CONCLUSIONS (1) The fatigue crack propagation rate in polycrystalline copper depends (for the same stress intensities) on the test temperature (20 °C and --100 °C) and on the test frequency (100 Hz and 20 kHz). (2) The size of dislocation cells adjacent to the fracture surface was used as a measure of the local crack tip stress amplitude. Values of the local crack tip plastic strain were deduced from CSSCs. Both the stress and the plastic strain crack tip values were expressed as functions of the macroscopic stress intensity. (3) The crack tip stress and plastic strain amplitudes were also computed by the finite element method. (4) The dependence of the fatigue crack propagation rate on the crack tip plastic strain amplitude is almost independent of the temperature and frequency. This confirms the dominant role of cyclic plastic deformation in the process of fatigue crack growth.

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100

Fatigue Eng. Mater. Struct., 3 (1980) 289-303. 3 G. Chalant and L. R~my, Plastic strain distribution at the tip of a fatigue crack, Eng. Fract. Mech., 16 (1982) 707-720. 4 M. Clavel and A. Pineau, Fatigue behaviour of two nickel alloys, Mater. Sci. Eng., 55 (1982) 157-180. 5 J. Lankford, Materials aspects of crack tip yielding and subcritical crack growth in engineering alloys. In J. Carlson and N, G. Ohlson (eds.), Proc. 4th Int. Conf. on Mechanical Behaviour o f Materials, Vol. 1, Pergamon, Oxford, 1983, pp. 3 29. 6 D. L. Davidson and J. Lankford, The effect of water vapor on fatigue crack tip stress and strain range distribution and the energy required for propagation in low-carbon steel, Int. J. Fract., 1 7 (1981) 257-272. 7 P. Luk£~ and L. Kunz, Threshold stress intensity and dislocation structures surrounding fatigue cracks in polycrystalline copper, Mater. Sci. Eng., 62 (1983) 149-157. 8 B. Weiss, H. Milliner, R. Stickler, P. Lukfi~ and L. Kunz, Influence of frequency on fatigue limit and fatigue crack growth behavior of polycrystalline Cu, Proc. 6th Int. Conf. on Fracture, New Delhi, December 4-10, 1984, to be published. 9 A. F. Blom, A. Hadrboletz and B. Weiss, Effect of crack closure on near-threshold crack growth behavior in a high strength A1 alloy up to ultrasonic frequencies. In J. Carlson and N. G. Ohlson (eds.),

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Proc. 4th Int. Conf. on Mechanical Behaviour o f Materials, Vol. 2, Pergamon, Oxford, 1983, pp. 755-762. M. Klesnil and J. Polfik, Cyclic plasticity of polycrystalline copper, Publ. Tech. Univ. Miskolc, Set. C, Mach., 38 (1983) 7 1 - 8 7 , 1 3 9 - 1 5 6 . H. Mughrabi and R. Wang, Cyclic deformation of face-centred cubic polycrystals: a comparison with observations on single crystals. In N. Hansen, A. Horsewell, T. Leffers and H. Lilholt (eds.), Proc. 2nd RiscJ Int. Symp. on Metallurgy and Materials Science, September 14-18, 1981, Ris6 National Laboratory, Ris6, 1981, pp. 87-98. L. D. Roth, L. E. Willertz and T. R. Leax, On the fatigue of copper up to ultrasonic frequencies. In J. M. Wells, O. Buch, L. D. Roth and J. K. Tien (eds.), Ultrasonic Fatigue, AIME, New York, 1982. E. S. Kayali and A. Plumtree, Stress-substructure relationship in cyclically and monotonically deformed wavy slip mode metals, MetaU. Trans. A, 13 (1982) 1033-1041. M. Kuna, Z. Bilek, Z. Kn~sl and V. Schmidt, The study of crack tip stress and strain field in elasticplastic materials, Czech. J. Phys. B, 28 (1978) 88-107. C. Laird and G. C. Smith, Crack propagation in high stress fatigue, Philos. Mag., 7 (1962) 847857. P. Neumann, Coarse slip model of fatigue, Acta Metall., 17 (1969) 1219-1225.