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Microcrack propagation in fatigued f.c.c. monocrystals II: crack faces and crack front shapes
Materials Science and Engineering, A 141 ( 1991 ) 49- 54
49
Microcrack propagation in fatigued f.c.c, monocrystals II: crack faces and crack front shapes C. Blochwitz Piidagogische Hochschule Dresden, lnstitut fiir Physik, Wigardstr. 17, 0-8060 Dresden (F.R.G.)
(Received October 1, 1990; in revised form December 19, 1990)
Abstract Nickel monocrystals were cyclically deformed at constant plastic strain amplitudes up to different fatigue stages (cf. part I). The crack faces and crack front shapes of microcracks which propagate along persistent slip bands were estimated by removing material layers parallel-to the specimen surface. The crack faces are similar to a "folded curtain" with deflections from the primary into the cross-slip plane which increase with the crack depth. The ratio c/l of the crack depth c and crack length l increases with increasing number of cycles; this means that the cracks propagate faster in depth than in length. The shapes of the crack fronts in the interior are trochoidal in most cases and not simple semielliptic. From the changes in frequency of crack lengths in different specimen depths, crack depth distribution curves were calculated which correspond well to the depth distribution curves that were determined from cut internal planes in part I.
1. Introduction In part I [ 1] of our work the frequency distributions of microcrack depths were estimated from longitudinal cuts of monocrystalline specimens at different fatigue stages. From the dependence of the distribution curves on the number of cycles, mean propagation rates and corresponding propagation laws could be determined. In this paper we report investigations of the shape of microcrack faces and of the ratio between crack depth c in the specimen interior and crack length I on the specimen surface. These studies were performed on pieces of the same nickel monocrystals as in part I. First results concerning this problem have been reported elsewhere [2]. An excellent overview of microcrack development in fatigued copper monocrystals, from the intrusion stage at the specimen corner up to the fatal crack determining the lifetime, has been given by Bao-Tong and Laird [3]. In this context, the present part of our paper comments on a special question of the fundamental problem of metal fatigue--the initia0921-5093/91/$3.50
tion and propagation of small cracks in smooth specimens.
2. Experiments Nickel single crystals of rectangular crosssection were fatigued at a constant plastic strain amplitude epa of 8 x 10 -4 with zero mean strain. Details of the specimen preparation and of the deformation technique can be found in part I [1]. The tests were interrupted at different cycle numbers (30 000, 60 000, 90 000 and 120 000). After measuring microcrack depths in cut faces (see part I [1]), the same specimen pieces were used for the investigation of microcrack lengths on the specimen surface. On the original surface the microcracks are hidden within the rough surface relief of persistent slip band (PSB) macrobands. The cracks or intrusions, respectively, were visible after removing a defined surface layer by mechanical polishing with an AI-H20 suspension using a vibrating tube or by hand. The thickness of the removed layer was © Elsevier Sequoia/Printed in The Netherlands
50
(a)
(b)
aO
2
j~ x : O,%×aD
N\
/Z// / t/
D
(c) Fig. 1. (a) Original surface with PSB macrobands and (b) the same place after removing a layer of about 1 /~rn showing intrusions within the PSB regions (specimen 17•2; N=30000); (c) illustration showing the estimation of the removed layer thickness from hardness indentations (from ref. 2).
estimated from the decrease in the diagonals of microhardness indentations (see Fig. 1 ). 3. Results and discussion
3.1. Crack face After repolishing the PSB surface relief, the remaining cracks became clearly visible. Therefore the serrated shape of the crack shores cannot be overlooked. The irregularity increases with increasing number of removed layers, i.e. with increasing crack depth. Figure 2 shows an example of the shape
changes of the crack shores during successive removal of the surface layer. It can be shown by variation of the polishing direction that the deflections of the crack shore from a straight line are not an artefact of the polishing method. Estimation of the surface traces of {111} primary slip planes from the stereographic projection after evaluation of the Laue diagrams shows that the direction of the trace of the crossslip plane seems to be connected with the mean deflection of the crack shores. Obviously, the spatial shape of the crack face (in any case at cracks with lengths greater than 100 /~m) is indeed similar to an irregular "folded curtain" as mentioned by Basinski and Basinski in 1985 [4]. Therefore the cross-slip plane has to pla~, an important role during microcrack propagation. Consequently the formation of the free surface during crack propagation along PSBs is influenced by the movement of screw dislocations. In Fig. 3 the possible shape of a microcrack in fatigued f.c.c, monocrystals is shown schematically. The increasing roughness of the crack shores can be understood when we take into account that the local stresses at the crack tip increase with increasing crack depth. This is why the cross-slip plane can be activated more easily than near the original specimen surface when the cracks are still short.
3.2. Depth-length relation of microcracks An attempt has been made to reconstruct the contours (these are the shapes of the crack fronts) of individual cracks by the following procedure. After removal of a surface layer of about 3/~m, as described in Section 2, the contours of microcracks following the traces of the primary {111} slip planes became clearly visible. On each specimen the population of cracks with different lengths was numbered and their starting length 1 could be estimated. The crack length l is the projected length, estimated by the shortest distance between the two crack tips. After that, layers of thickness Ax were removed step by step, and the new lengths l (decreasing, in general) of the numbered cracks were measured. Taking into account the inclination of the slip planes according to the crystallographic orientation, giving a factor of about 21/2, the symmetrized contours of the cracks could be drawn as shown schematically in Fig. 4. In Fig. 4 the crack vanishes after the fifth
51
Fig. 2. Example of the shape changes of the crack shores during surface layer removal for total removed layer thicknesses of (a) 2/am, (b) 11.5 /am and (c) 25.5/am. The traces of the primary slip plane (PP) and of the cross-slip plane (CP) are shown (specimen 9/1; N = 120 000).
removal step and the real crack depth c as well as the surface length l can be estimated roughly. Figure 5 shows an example of such crack contours for the specimen 36/2 ( N = 90 000). The relations between depth c and length l were calculated from the scattered distributions of c - l pairs (Fig. 6) of about 50 individual cracks on each specimen. Assuming a potential law of
the form c = k ( l - l*)"
(1)
with l* = 5/~m, the values of k and m were calculated by a regression programme for the plot of log c = l o g k + m log ( l - l*)
(2)
Although the scatter of the c - l points is
52
crack Front crmckface. lt~
~
cr~ck$t~ore
View ~t the polishedsurface
Fig. 3. Scheme of the possible spatial crack shape in fatigued f.c.c, monocrystals: PP, primary slip plane; CP, cross-slip plane; bp, primary Burgers vector.
l
r
surmce q
,
rl
\I/i
obviously high, there exists a correlation between c and l values with a correlation coefficient r between 0.61 and 0.86. This corresponds to the simple fact that the cracks which are the longest at the specimen surface are also in most cases the deepest. More important is a comparison of the calculated c-I curves of different specimens with different numbers of cycles (Fig. 7, Table 1 ). The c/l quotient increases markedly with increasing number of cycles. A comparison of the averaged values of surface crack length and crack depth (from part I [1]) also shows that microcrack propagation in fatigued monocrystals is faster in depth than along the surface. At higher numbers of cycles (N~> 90000) the increase in l results from the linking up of elementary parallel cracks within PSB macrobands. This also explains the existence of shallow cracks (low c value) with relatively high l values (see Fig.
8(c)).
Fig. 4. Principle of the estimation of crack front contours from surface removal experiments, x is the thickness of the removed layer, x' is the removed depth measured within the inclined crack plane. From the crystallographic orientation it follows that x'= 2 ~/:x.
N/IO 3 I 120
K
m
36:29
0675
60 2. 3S/+ I /+55
0586 0t~886
C'K({'I') m
ctwck 2
/
~o~
so ooo
//oooo
C Q~n)
Fig. 5. Examples of individual crack front contours (specimen 36/2; N = 90000).
i
i
10
i
i
~
i
i
i
0 t(/~m)~
i
i
100
Fig. 7. Mean c ( I ) curves for various numbers of cycles (ep. = 8 x 10-"). c = K ( l - l * ) m.
•
.
TABLE 1
N " 9 0 000
3C
•
o•
(life•
1o
%• •
•
Averaged crack depth c and crack length I after various numbers N of cycles ~lum~d
cracks
C." I
Fig. 6. Depth-length c - I pairs of individual cracks (specimen 36/2; N=90000). M gives the mean values of c and 1 for the evaluated cracks, c = 2.527 ( 1-/,)(i.659; r = 0.86.
Specimen
N ( x 10 3)
c(~m) l(~m)
K
m
28/1 28/2 36/2 9/1
30 60 90 120
8.6 16.6 26.2 50.0
1.455 2.345 2.527 3.629
0.4886 0.586 0.659 0.675
43 41 45-77 68
53 F~__. ¸ l'
-~ ~
-_-
A
_
x
_____ L1
Ix'
~zx/
b)
a~
c)
Fig. 8. Some possible contours of individual cracks.
3.3. Crack front shape The information about the crack contours, following from the change in length of individual cracks during the removal of surface layers, can be completed by the evaluation of crack length distribution curves. These curves change on stepby-step removal because the shortest cracks vanish first whereas longer cracks can be observed again at the next investigation step, but with reduced length. The reduction A l of the actual length l' after a removal step depends on the crack contour. At semielliptic (concave curvature) crack front contours, for example, the ratio A I/Ax' of a crack (Fig. 8) should increase with increasing removal depth, whereas at cracks with a "trochoidal" shape (convex curvature) the opposite behaviour
should be expected. Figure 9 and Table 2 show an example of absolute frequencies h of crack lengths after different removal steps. Given the result in the previous section, that longer cracks are also deeper in general, it can be assumed that the same group of cracks is considered after different removal steps if the h value is held constant. The intersection of a horizontal line (corresponding to h =constant) with the h(l) field therefore gives the length of a representative crack of the group considered for different depths. In Fig. 9, for example, 20 cracks exist within the length interval (75 + 12.5) ffm at step 0. The ratio of length reduction A l and removal depth Ax' decreases markedly at first with the removal depth. It should be noted further that the jump at constant h value from the first to the next step (step 0 to step 2 in our example) is higher at longer cracks than at shorter ones. Both observations, which can also be confirmed for other specimens, are in agreement with a trochoidal shape of cracks as shown in Fig. 8(b).
3.4. Surface removal and crack depth distribution
Fig. 9. Changes in the absolute frequency distribution of the crack length 1 after removing surface layers of depth x' (specimen 28/2; N=60000). The curves were drawn by averaging histograms with interval widths of 25/~m.
In part I, crack depth distributions have been estimated from cuts which have been made perpendicular to the PSB traces on the specimen surface. The frequency distributions may be false because the cracks are not generally cut at the highest depths. Therefore the depth distribution has been determined by another method. After removing a surface layer of thickness Ax the area density of cracks that are visible at the actual specimen surface by their crack length l' is reduced by an amount Az of cracks having depths between zero and Ax. From the actual area density of cracks after removal of the layer thickness x, the number of cracks (per square millimetre) having depths between x',, and x', + ~ has been calculated step by step (Table 3). The probability density ~(c) has been calculated by the formula
TABLE 2
~(c)
~ep 130
I
i.°2.0
too
50
lb
eb
~
b
x'~.,)
-
10
lb
s'o
Ib
lb
i
/ (pro)
-
Reductions A/in crack length after various removal steps
Step
x' (pm)
Ax' (pm)
AI/Ax'
0 2 4 6
0 9.0 19.0 30.9
0 9.0 10.0 11.9
2.9 1.4 1.2
Az 1 a x ' z,,
(3)
with x' is the interval width and z0 is the area density of cracks before layer removal. In Fig. 1'0, the ~ values determined from the crack depths in a cut internal face are compared with the ~ values determined from the crack densities in the removed areas, using the example
54 TABLE 3
Estimation of crack depth distributions from the reductions of the area crack density z during surface layer removal by x' Step
x' (ktm)
Ax' (/~m)
z (mm 2)
Az (mm -2)
¢
0 1 2 3 4 5 6
0 3.8 9.0 13.8 19.0 25.2 30.9
0 3.8 5.2 4.8 5.2 6.2 5.7
776 a 600 440 321 214 145 129
176 160 119 107 69 16
6.0 x 4.0 × 3.2 x 2.7 × 1.4 × 0.4X
10 -2 10 -2 10 -2 10- 2 10 -2 10 -2
azo.
t
6" 0
~ I "x :" :'.
5"
-o.-¢lO-
18 " ¢01~ N 160000 from c u t f a c e fr~
111
area den~
tn
dtfferent ~lOecimendepth5 ~,.
3-
x~ ko ~%
Epa=2.5~lO'* i Al=68500 :. : : from cut face asa~scriboc/in
111
similar depth distribution curves, as mentioned also in ref. 3 for copper monocrystals. This behaviour is reasonable because the microcracks propagate within PSBs which have local mean plastic strain amplitudes that do not depend on the external amplitudes. Only at the interaction of cracks in later fatigue stages [5] can strain amplitude effects on the crack distribution curves be expected because the PSB and crack densities, respectively, increase with increasing external plastic strain amplitude.
2" 1"
Acknowledgments Fig. 10. Crack depth distributions (probability densities): x , from the cut face [1] !eva = 8 x 10-4; N = 60000); o, from the area density in various specimen depths (epa=8× 10-4; N = 6 0 0 0 0 ) ; e, from the cut face as described in ref. 1 (epa = 2,5 × 10-4; N = 6 8 500).
of specimen 28/2 ( N = 60 000). The curves show a good correspondence within the error of the experimental values and consequently confirm the results of part I [ 1].
3.5. Influence of strain amplitude on crack depth distributions After finishing part I, crack depth distributions were estimated at amplitudes other than the value 8 × 1 0 - 4 used previously. In Fig. 10, for example, the values for a specimen fatigued at epa=2.5 × 1 0 - 4 to 68500 cycles are shown. At comparable numbers of cycles (corresponding to half of the lifetime at epa = 8 x 10 -4) we found
The author is very grateful to Professor H. Mughrabi, Universitfit Erlangen-Nfirnberg, and to Dr. K. Mecke and Dr. S. Steidten, P~idagogische Hochschule Dresden, for critical reading of the manuscript. He would like to thank Mrs. C. Herrmann and Mrs. J. Schmidt for technical help. References 1 C. Blochwitz, D. Heinrich and R. Frenzel, Mater. Sci. Eng. A, 118 (1989) 71-81. 2 C. Blochwitz and D. Heinrich, Microcrack propagation studies in fatigued nickel single crystals, in P. Lukfi~ and J. Pol~ik (eds.), Bas& Mechanisms in Fatigue o f Metals, Elsevier, Amsterdam, 1988, pp. 315-322. 3 M. Bao-Tong and C. Laird, Overview No. 77 parts I-V, Acta Metall., 37 (1989) 325-379. 4 Z. S. Basinski and J. S. Basinski, Acta Metall., 33 (1985) 1307-1317. 5 J. PoNk, K. Obrtlik and P. Ligkut/n, Mechanisms of fatigue crack initiation, in P. Lukfig and J. PoNk (eds.), Basic Mechanisms in Fatigue o f Metals, Elsevier, Amsterdam,
1988, pp. 101-109.
49
Microcrack propagation in fatigued f.c.c, monocrystals II: crack faces and crack front shapes C. Blochwitz Piidagogische Hochschule Dresden, lnstitut fiir Physik, Wigardstr. 17, 0-8060 Dresden (F.R.G.)
(Received October 1, 1990; in revised form December 19, 1990)
Abstract Nickel monocrystals were cyclically deformed at constant plastic strain amplitudes up to different fatigue stages (cf. part I). The crack faces and crack front shapes of microcracks which propagate along persistent slip bands were estimated by removing material layers parallel-to the specimen surface. The crack faces are similar to a "folded curtain" with deflections from the primary into the cross-slip plane which increase with the crack depth. The ratio c/l of the crack depth c and crack length l increases with increasing number of cycles; this means that the cracks propagate faster in depth than in length. The shapes of the crack fronts in the interior are trochoidal in most cases and not simple semielliptic. From the changes in frequency of crack lengths in different specimen depths, crack depth distribution curves were calculated which correspond well to the depth distribution curves that were determined from cut internal planes in part I.
1. Introduction In part I [ 1] of our work the frequency distributions of microcrack depths were estimated from longitudinal cuts of monocrystalline specimens at different fatigue stages. From the dependence of the distribution curves on the number of cycles, mean propagation rates and corresponding propagation laws could be determined. In this paper we report investigations of the shape of microcrack faces and of the ratio between crack depth c in the specimen interior and crack length I on the specimen surface. These studies were performed on pieces of the same nickel monocrystals as in part I. First results concerning this problem have been reported elsewhere [2]. An excellent overview of microcrack development in fatigued copper monocrystals, from the intrusion stage at the specimen corner up to the fatal crack determining the lifetime, has been given by Bao-Tong and Laird [3]. In this context, the present part of our paper comments on a special question of the fundamental problem of metal fatigue--the initia0921-5093/91/$3.50
tion and propagation of small cracks in smooth specimens.
2. Experiments Nickel single crystals of rectangular crosssection were fatigued at a constant plastic strain amplitude epa of 8 x 10 -4 with zero mean strain. Details of the specimen preparation and of the deformation technique can be found in part I [1]. The tests were interrupted at different cycle numbers (30 000, 60 000, 90 000 and 120 000). After measuring microcrack depths in cut faces (see part I [1]), the same specimen pieces were used for the investigation of microcrack lengths on the specimen surface. On the original surface the microcracks are hidden within the rough surface relief of persistent slip band (PSB) macrobands. The cracks or intrusions, respectively, were visible after removing a defined surface layer by mechanical polishing with an AI-H20 suspension using a vibrating tube or by hand. The thickness of the removed layer was © Elsevier Sequoia/Printed in The Netherlands
50
(a)
(b)
aO
2
j~ x : O,%×aD
N\
/Z// / t/
D
(c) Fig. 1. (a) Original surface with PSB macrobands and (b) the same place after removing a layer of about 1 /~rn showing intrusions within the PSB regions (specimen 17•2; N=30000); (c) illustration showing the estimation of the removed layer thickness from hardness indentations (from ref. 2).
estimated from the decrease in the diagonals of microhardness indentations (see Fig. 1 ). 3. Results and discussion
3.1. Crack face After repolishing the PSB surface relief, the remaining cracks became clearly visible. Therefore the serrated shape of the crack shores cannot be overlooked. The irregularity increases with increasing number of removed layers, i.e. with increasing crack depth. Figure 2 shows an example of the shape
changes of the crack shores during successive removal of the surface layer. It can be shown by variation of the polishing direction that the deflections of the crack shore from a straight line are not an artefact of the polishing method. Estimation of the surface traces of {111} primary slip planes from the stereographic projection after evaluation of the Laue diagrams shows that the direction of the trace of the crossslip plane seems to be connected with the mean deflection of the crack shores. Obviously, the spatial shape of the crack face (in any case at cracks with lengths greater than 100 /~m) is indeed similar to an irregular "folded curtain" as mentioned by Basinski and Basinski in 1985 [4]. Therefore the cross-slip plane has to pla~, an important role during microcrack propagation. Consequently the formation of the free surface during crack propagation along PSBs is influenced by the movement of screw dislocations. In Fig. 3 the possible shape of a microcrack in fatigued f.c.c, monocrystals is shown schematically. The increasing roughness of the crack shores can be understood when we take into account that the local stresses at the crack tip increase with increasing crack depth. This is why the cross-slip plane can be activated more easily than near the original specimen surface when the cracks are still short.
3.2. Depth-length relation of microcracks An attempt has been made to reconstruct the contours (these are the shapes of the crack fronts) of individual cracks by the following procedure. After removal of a surface layer of about 3/~m, as described in Section 2, the contours of microcracks following the traces of the primary {111} slip planes became clearly visible. On each specimen the population of cracks with different lengths was numbered and their starting length 1 could be estimated. The crack length l is the projected length, estimated by the shortest distance between the two crack tips. After that, layers of thickness Ax were removed step by step, and the new lengths l (decreasing, in general) of the numbered cracks were measured. Taking into account the inclination of the slip planes according to the crystallographic orientation, giving a factor of about 21/2, the symmetrized contours of the cracks could be drawn as shown schematically in Fig. 4. In Fig. 4 the crack vanishes after the fifth
51
Fig. 2. Example of the shape changes of the crack shores during surface layer removal for total removed layer thicknesses of (a) 2/am, (b) 11.5 /am and (c) 25.5/am. The traces of the primary slip plane (PP) and of the cross-slip plane (CP) are shown (specimen 9/1; N = 120 000).
removal step and the real crack depth c as well as the surface length l can be estimated roughly. Figure 5 shows an example of such crack contours for the specimen 36/2 ( N = 90 000). The relations between depth c and length l were calculated from the scattered distributions of c - l pairs (Fig. 6) of about 50 individual cracks on each specimen. Assuming a potential law of
the form c = k ( l - l*)"
(1)
with l* = 5/~m, the values of k and m were calculated by a regression programme for the plot of log c = l o g k + m log ( l - l*)
(2)
Although the scatter of the c - l points is
52
crack Front crmckface. lt~
~
cr~ck$t~ore
View ~t the polishedsurface
Fig. 3. Scheme of the possible spatial crack shape in fatigued f.c.c, monocrystals: PP, primary slip plane; CP, cross-slip plane; bp, primary Burgers vector.
l
r
surmce q
,
rl
\I/i
obviously high, there exists a correlation between c and l values with a correlation coefficient r between 0.61 and 0.86. This corresponds to the simple fact that the cracks which are the longest at the specimen surface are also in most cases the deepest. More important is a comparison of the calculated c-I curves of different specimens with different numbers of cycles (Fig. 7, Table 1 ). The c/l quotient increases markedly with increasing number of cycles. A comparison of the averaged values of surface crack length and crack depth (from part I [1]) also shows that microcrack propagation in fatigued monocrystals is faster in depth than along the surface. At higher numbers of cycles (N~> 90000) the increase in l results from the linking up of elementary parallel cracks within PSB macrobands. This also explains the existence of shallow cracks (low c value) with relatively high l values (see Fig.
8(c)).
Fig. 4. Principle of the estimation of crack front contours from surface removal experiments, x is the thickness of the removed layer, x' is the removed depth measured within the inclined crack plane. From the crystallographic orientation it follows that x'= 2 ~/:x.
N/IO 3 I 120
K
m
36:29
0675
60 2. 3S/+ I /+55
0586 0t~886
C'K({'I') m
ctwck 2
/
~o~
so ooo
//oooo
C Q~n)
Fig. 5. Examples of individual crack front contours (specimen 36/2; N = 90000).
i
i
10
i
i
~
i
i
i
0 t(/~m)~
i
i
100
Fig. 7. Mean c ( I ) curves for various numbers of cycles (ep. = 8 x 10-"). c = K ( l - l * ) m.
•
.
TABLE 1
N " 9 0 000
3C
•
o•
(life•
1o
%• •
•
Averaged crack depth c and crack length I after various numbers N of cycles ~lum~d
cracks
C." I
Fig. 6. Depth-length c - I pairs of individual cracks (specimen 36/2; N=90000). M gives the mean values of c and 1 for the evaluated cracks, c = 2.527 ( 1-/,)(i.659; r = 0.86.
Specimen
N ( x 10 3)
c(~m) l(~m)
K
m
28/1 28/2 36/2 9/1
30 60 90 120
8.6 16.6 26.2 50.0
1.455 2.345 2.527 3.629
0.4886 0.586 0.659 0.675
43 41 45-77 68
53 F~__. ¸ l'
-~ ~
-_-
A
_
x
_____ L1
Ix'
~zx/
b)
a~
c)
Fig. 8. Some possible contours of individual cracks.
3.3. Crack front shape The information about the crack contours, following from the change in length of individual cracks during the removal of surface layers, can be completed by the evaluation of crack length distribution curves. These curves change on stepby-step removal because the shortest cracks vanish first whereas longer cracks can be observed again at the next investigation step, but with reduced length. The reduction A l of the actual length l' after a removal step depends on the crack contour. At semielliptic (concave curvature) crack front contours, for example, the ratio A I/Ax' of a crack (Fig. 8) should increase with increasing removal depth, whereas at cracks with a "trochoidal" shape (convex curvature) the opposite behaviour
should be expected. Figure 9 and Table 2 show an example of absolute frequencies h of crack lengths after different removal steps. Given the result in the previous section, that longer cracks are also deeper in general, it can be assumed that the same group of cracks is considered after different removal steps if the h value is held constant. The intersection of a horizontal line (corresponding to h =constant) with the h(l) field therefore gives the length of a representative crack of the group considered for different depths. In Fig. 9, for example, 20 cracks exist within the length interval (75 + 12.5) ffm at step 0. The ratio of length reduction A l and removal depth Ax' decreases markedly at first with the removal depth. It should be noted further that the jump at constant h value from the first to the next step (step 0 to step 2 in our example) is higher at longer cracks than at shorter ones. Both observations, which can also be confirmed for other specimens, are in agreement with a trochoidal shape of cracks as shown in Fig. 8(b).
3.4. Surface removal and crack depth distribution
Fig. 9. Changes in the absolute frequency distribution of the crack length 1 after removing surface layers of depth x' (specimen 28/2; N=60000). The curves were drawn by averaging histograms with interval widths of 25/~m.
In part I, crack depth distributions have been estimated from cuts which have been made perpendicular to the PSB traces on the specimen surface. The frequency distributions may be false because the cracks are not generally cut at the highest depths. Therefore the depth distribution has been determined by another method. After removing a surface layer of thickness Ax the area density of cracks that are visible at the actual specimen surface by their crack length l' is reduced by an amount Az of cracks having depths between zero and Ax. From the actual area density of cracks after removal of the layer thickness x, the number of cracks (per square millimetre) having depths between x',, and x', + ~ has been calculated step by step (Table 3). The probability density ~(c) has been calculated by the formula
TABLE 2
~(c)
~ep 130
I
i.°2.0
too
50
lb
eb
~
b
x'~.,)
-
10
lb
s'o
Ib
lb
i
/ (pro)
-
Reductions A/in crack length after various removal steps
Step
x' (pm)
Ax' (pm)
AI/Ax'
0 2 4 6
0 9.0 19.0 30.9
0 9.0 10.0 11.9
2.9 1.4 1.2
Az 1 a x ' z,,
(3)
with x' is the interval width and z0 is the area density of cracks before layer removal. In Fig. 1'0, the ~ values determined from the crack depths in a cut internal face are compared with the ~ values determined from the crack densities in the removed areas, using the example
54 TABLE 3
Estimation of crack depth distributions from the reductions of the area crack density z during surface layer removal by x' Step
x' (ktm)
Ax' (/~m)
z (mm 2)
Az (mm -2)
¢
0 1 2 3 4 5 6
0 3.8 9.0 13.8 19.0 25.2 30.9
0 3.8 5.2 4.8 5.2 6.2 5.7
776 a 600 440 321 214 145 129
176 160 119 107 69 16
6.0 x 4.0 × 3.2 x 2.7 × 1.4 × 0.4X
10 -2 10 -2 10 -2 10- 2 10 -2 10 -2
azo.
t
6" 0
~ I "x :" :'.
5"
-o.-¢lO-
18 " ¢01~ N 160000 from c u t f a c e fr~
111
area den~
tn
dtfferent ~lOecimendepth5 ~,.
3-
x~ ko ~%
Epa=2.5~lO'* i Al=68500 :. : : from cut face asa~scriboc/in
111
similar depth distribution curves, as mentioned also in ref. 3 for copper monocrystals. This behaviour is reasonable because the microcracks propagate within PSBs which have local mean plastic strain amplitudes that do not depend on the external amplitudes. Only at the interaction of cracks in later fatigue stages [5] can strain amplitude effects on the crack distribution curves be expected because the PSB and crack densities, respectively, increase with increasing external plastic strain amplitude.
2" 1"
Acknowledgments Fig. 10. Crack depth distributions (probability densities): x , from the cut face [1] !eva = 8 x 10-4; N = 60000); o, from the area density in various specimen depths (epa=8× 10-4; N = 6 0 0 0 0 ) ; e, from the cut face as described in ref. 1 (epa = 2,5 × 10-4; N = 6 8 500).
of specimen 28/2 ( N = 60 000). The curves show a good correspondence within the error of the experimental values and consequently confirm the results of part I [ 1].
3.5. Influence of strain amplitude on crack depth distributions After finishing part I, crack depth distributions were estimated at amplitudes other than the value 8 × 1 0 - 4 used previously. In Fig. 10, for example, the values for a specimen fatigued at epa=2.5 × 1 0 - 4 to 68500 cycles are shown. At comparable numbers of cycles (corresponding to half of the lifetime at epa = 8 x 10 -4) we found
The author is very grateful to Professor H. Mughrabi, Universitfit Erlangen-Nfirnberg, and to Dr. K. Mecke and Dr. S. Steidten, P~idagogische Hochschule Dresden, for critical reading of the manuscript. He would like to thank Mrs. C. Herrmann and Mrs. J. Schmidt for technical help. References 1 C. Blochwitz, D. Heinrich and R. Frenzel, Mater. Sci. Eng. A, 118 (1989) 71-81. 2 C. Blochwitz and D. Heinrich, Microcrack propagation studies in fatigued nickel single crystals, in P. Lukfi~ and J. Pol~ik (eds.), Bas& Mechanisms in Fatigue o f Metals, Elsevier, Amsterdam, 1988, pp. 315-322. 3 M. Bao-Tong and C. Laird, Overview No. 77 parts I-V, Acta Metall., 37 (1989) 325-379. 4 Z. S. Basinski and J. S. Basinski, Acta Metall., 33 (1985) 1307-1317. 5 J. PoNk, K. Obrtlik and P. Ligkut/n, Mechanisms of fatigue crack initiation, in P. Lukfig and J. PoNk (eds.), Basic Mechanisms in Fatigue o f Metals, Elsevier, Amsterdam,
1988, pp. 101-109.