- Email: [email protected]
Crack initiation and growth behaviour of Al 2024-T4 under fretting fatigue
Int. J. Fatigue Vol. 19, No. 3, pp. 243-251, 1997 © 1997 Elsevier Science Limited. All rights reserved Printed in Great Britain 0142-1123/97/$17.00+.00
ELSEVIER
PIh 0142-1123(96)00055-2
Crack initiation and growth behaviour of AI 2024-T4 under fretting fatigue Hyung-Kyu Kim* and Soon-Bok Lee Korea Advanced Institute of Science and Technology, Science Town, Taejon, 305701, Korea (Received 17 April 1996; revised 15 July 1996; accepted 22 July 1996) A fretting fatigue experiment is conducted with AI 2024-T4 pad and specimen. A pad has a rectangular foot for simulating punch pressure on the contact surface of the specimen. Fretting fatigue cracks are monitored and analysed by image processing. The crack initiation sites and obliquities are examined. The behaviour of the crack growth rate is also shown. An analysis is carried out along the mid plane of specimen to deal with a plane strain condition. The elastic solution of square punch is utilized to estimate the crack initiation site with assuming the elastic-perfectly plastic behaviour of material. The predictions are in good agreement with the experimental data. The surface traction fields are suggested, from which stress intensity factors are evaluated. Copyright © 1997 Elsevier Science Limited. All rights reserved (Keywords: fretting fatigue, square punch, crack initiation site)
A significant decrease in fatigue life occurs along with fretting phenomenon ]-3. High crack growth rate is usually found at the early stage of fretting fatigue z3. Structures and components under fretting conditions may suffer a premature failure. Therefore, it is necessary to investigate the initiation and growth behaviour of a fretting fatigue crack. Many experimental studies have been done to investigate the behaviour of fretting fatigue crack. A fatigue testing machine was usually used with pads that exerted a compressive load on the specimen. A cylinder or a ball was widely used for the pad so that Hertzian contact could be formed. However, in engineering practice, a contact of flat surfaces is unavoidable. A pad of flat surface was sometimes used in the shape of a bridge 3'4. The bridge-shaped pad is beneficial for measuring the contact forces. But it is rather difficult to analyse the crack since the number of possible sites of crack initiation increases, i.e. the interaction of the cracks should be considered. It has been generally regarded that the fretting fatigue crack initiated at the boundary of the contact ] 3. However, some authors 4 found that the crack initiated at the stick and slip boundary. The stress condition of fretting fatigue is sophisticated. Many theoretical works have also been carried out. The classical elastic solutions were often consulted such as Hertzian contact, the rigid punch problem, and
the Flamant solution etc. 5 As for the mixed mode crack problem of fretting fatigue, the stress intensity factors (SIFs) were widely used 3'6-8. Since the compressive stress on the contact surface is an important parameter of fretting fatigue, the effect of the traction field on SIFs has been studied 8'9. The dominance of K~ or Kn during fretting fatigue cracking was also argued ~o.t ]. In this paper, the results of the fretting fatigue experiment with a punch-shaped pad are evaluated. Since the bridge-shaped pad produces multiple cracks, the punch-shaped pad is used to confine the number of crack initiation sites. Since it is worthwhile to know the effects of fretting parameters on the crack behaviour, the pad lengths and the contact forces are varied during the experiment. The angle between the crack and the direction of contacting normal force, termed as an obliquity hereafter, is investigated. The initiation sites and the growth behaviour of the cracks are also characterized. The applicability of the solution of punch problem is studied for analysis. The crack initiation sites are predicted with the concept of fracture mechanics. By suggesting the surface traction fields, the SIFs of the oblique crack are evaluated. The obliquity dependence of K1 and KI] is discussed.
EXPERIMENTAL
Materials and fretting fatigue specimens The fretting fatigue specimens, the pads and loading bars (pad assembly) were made from 2024-T4 alumi-
* Present address: Korea Atomic Research Institute, Taejon, 301313, Korea
243
244
H.-K. Kim and S.-B. Lee 180
60--
M5
(a)
Trailing Edge ~ i' ~- Leading Edge Pad foot length (b)
/-M5
~3/
~-~25
The fatigue loading was sinusoidal with a frequency of 7 Hz. The bulk stress amplitude was fixed to be 60 MPa with the stress ratio of R = 0 for all the tests except one case. The only different amplitude applied was 8 6 M P a with R = 0 for the 5 mm pad. This was conducted to change the tangential stress for comparing with the 2 mm pad. The tangential forces were measured during the fretting fatigue test. It was also assumed that the tangential stresses were uniformly distributed on the contact surface a priori. The maximum value of nominal tangential stress, q~oX..... occurred at maximum bulk stress. The fretting fatigue test conditions for each specimen are summarized in
Table 1. M5~ ,I
(e)
Figure 1 Geometries of (a) specimen, (b) pad and (c) loading bar num alloy whose yield strength and Poisson's ratio are 304 MPa and 0.32, respectively. Figure 1 shows the geometry of the specimen and the pad assembly. Pads of 2 mm, 5 mm and 10 mm were machined to have sharp edges. The specimens and the pad assemblies were machined from the asreceived bar of 30 mm dia. The surfaces of the specimens as well as the contact areas of the specimens and the pads were polished with fine emery paper.
Fretting fatigue test The normal force was applied on the contact surface of the specimen by extending two loading bars which were assembled with the pads by nuts. The specimen with loading bars and pads was installed in the fatigue testing machine. The tangential stress was formed on the contact surfaces during the fatigue loading being applied to the specimen. The normal and tangential forces were measured by strain gauges attached to the loading bars and the pads, respectively. The fretting fatigue crack was monitored by a CCD camera. The actual images were magnified 30 times in the camera and stored in a personal computer with a programmed interval of time. These stored images were analysed after the test with image processing program. Figure 2 shows the schematic diagram of the experimental setup. During the application of normal force to the specimen, it was assumed that the stresses were uniformly distributed on the contact surface a priori. The designed nominal stresses, p . . . . . were 66.7 MPa or 37.6 MPa.
Figure 2 Schematic diagram of the experimental setup
The normal and tangential force calibration It is important to know the normal and tangential forces on the contact surface to analyse the behaviour of the fretting fatigue crack. However, it is difficult to measure the forces directly during the test since those are exerted on the contact surface. In the present experiment, the deformations of the loading bars and the pads were used for measuring the normal and tangential forces, respectively. Therefore, the strain values of the loading bars and the pads were calibrated before fretting fatigue tests. The calibration method has been published separately 12, so the details are not reproduced here. The idea was to use the strain value of the calibration specimens, the loading bars and the pads as a correlating parameter. The relationships between normal force and loading bar strain and tangential force and pad strain were obtained respectively for each test condition. Typical examples of the calibration results for the 2 mm pad are given in Figure 3. The linearities between the forces and loading bar or pad strain were found to be quite good in every calibration test. The calibration results enabled one to perform in situ measurements of the normal and tangential forces during the tests. RESULTS
Slip regimes and crack initiation The contact surfaces were investigated by an optical microscope with 80 times magnification. The slip region was covered with oxidized wear debris coloured black so the boundary of stick and slip zone could be distinguished easily. All the tests with the 10 and 5 mm pads showed a partial slip regime, while a gross slip regime was shown for the 2 mm pad. Figures 4 and 5 show the contact surface of the partial slip and that of the gross slip, respectively. As for the specimen of the partial slip, the slip region was found mainly at a leading portion of the pad. No slip was found at the trailing portion of it (here, the terms of 'leading' and 'trailing' are defined as given in Figure 1). However, it was observed that two narrow bands of slip region formed along the specimen axis as shown in Figure 4. No symptom of slip was observed outside the bands. This implies that the normal and tangential stresses on the contact area are not uniform along the axial and the thickness directions of the specimen. As for the specimen of gross slip, the width of slip region was also narrower than the specimen thickness.
Crack behaviour of A12024-7"4 under fretting fatigue Table 1
245
Fretting fatigue experimental conditions
Specimen No.
Bulk stress range (MPa)
Pad length/ specimen thickness (mm/mm)
L10P1 L10P2 L5PI L5PIa L5P2 L2P I
0 0 0 0 0 0
10/3 10/3 5/3 5/3 5/3 2/3
~ ~ ~ ~ ~ ~
120 120 120 172 120 120
Designed p . . . . (MPa)
Actual contact Designed q~o~ ,,,~ length/width (MPa) (mm/mm)
Re-calculated p ..... (MPa)
Re-calculated q . . . . .n o r a (MPa)
66.7 36.5 67.7 67.2 36.9 66.8
16.3 9.42 32.45 37.78 19.93 39.1
130.78 61.83 153.86 96.00 93.81 89.07
31.96
80
-20
0
20
40
~" 7o
• Normal stress o Tangential stress
6O
/ //
13.
60
.~,) 9,~ /
15.96 73.75 53.40 50.67 52.13
t:ati gue direction Leading ~ I, Trailing edge ~ :: edge
Pad strain (x 10-6) -40 9O
9.56/1.60 9.27/1.91 4.78/1.38 4.95/2.12 5.00/1.18 2.00/2.25
20 (ID
~SSpecimen ~thickness
10 ~"
~n 50.
~
40
E 0 Z
30
!
o 0 "/
g
,It
L
lmm
Crack
O"
"13
20.
t
Crack
.lo~=
Figure 5
-20
Resultantly, the actual width was always narrower than the specimen thickness (3 mm), even though the foot width (5 mm) was wider than that in the present experiment. The actual contact area has been measured with the definition of a rectangle surrounded by the
Contact surface view of gross slip (L2PI)
100 0
Figure 3
5'0 "160 1gO 200 250 360 Loading bar strain (x 10-6)
Calibration results of contact stresses for the 2 m m pad
Leading edge Fatigue direction • Trailing edge
Crack
•
/
:~V- ~__~ p l a n e ~ . ~ ~ ~ .~ r
-
Figure4
Contact surface view of partial slip (L5PI)
Measuredcrack initiation site
i
(Specimen i thickness
H.-K. Kim and S.-B. Lee
246
outer-most boundary of slip region. Then, the nominal values of the normal and tangential stresses were recalculated according to the actual contact area. The results are also given in Table 1 with the designations of Pnom and qmaXom for comparison with the designed values. The fretting fatigue crack initiated exactly at the boundary of stick and slip zone when a specimen revealed a partial slip regime. However, in the case of gross slip regime, the crack initiated at the leading edge of the pad, i.e. the leading edge of slip region. On the other hand, the crack on the contact surface met the surface of specimen backwards (to the direction of trailing edge of the pad) as shown in Figures 4 and 5. As for the specimens of partial slip, the slip region had the shortest length at the middle of contact (the mid plane of the specimen). It increased outwards to the specimen surface. Therefore, the crack had a curved contour at its initiation site on the contact surface. All other specimens with partial slip regime showed the same phenomenon. This means that the distance between the edge of pad foot and the crack initiation site observed from the surface of the specimen is not the length of slip region or crack initiation site in the middle of contact width. In Table 2, the measured lengths of slip regions (or distances between the leading edge of the pad and the crack initiation site) in the middle of contact width are given for the specimens of partial slip regime. As for gross slip regime, the slip region is designated as the actual contact length/width in Table 1. However, the crack initiation site for it is zero in Table 2 since the crack initiated at the edge of slip region.
Oblique crack growth All the cracks behaved as a typical fretting fatigue crack. They grew obliquely in the direction of normal force initially. After the crack grew obliquely to some extent, it changed the direction (re-directed) to that of normal force as shown in Figure 6. After the crack redirected, it behaved like a mode I crack. No branch or kink crack was found during the oblique growth. The crack length was measured along the crack line on the specimen\ surface during the oblique growth. The length of oblique crack is given in Table 2 for each specimen together with the measured obliquity. The obliquity of each specimen was nearly constant and found to be within the range of 10-27 ° . The largest obliquity of 27 ° has been found in a specimen of gross slip regime (L2P1). Therefore, the oblique portion of the fretting fatigue crack can be modelled as a single surface oblique edge crack. After the crack re-directed to mode I, the distance between the speciTable 2
Pad
Fatigue direction (a)
(b) Pad
Crack length at mode change Mode I Crack ---[Obliquity Figure6 Typical view of oblique crack and mode change: (a) L5P1; and (b) L2P1
men edge and the crack tip was defined as a crack length. However, the behaviour of the crack after redirection is not the main concern of this study. The specimens were to be failed with monotonic tension when the re-directed cracks grew to some extent. The fractured surface could be divided into three parts of different features as shown in Figure 7. Those were the initial oblique and flat surface by fretting effect, the perpendicular and flat one after redirection and the perpendicular but shear surface due to monotonic tension. The crack front inside the specimen was examined. It has been found that the crack front of the initial oblique and flat surface was almost straight. However, the crack front of the perpendicular and fiat surface showed a thumb-nail contour, a typical shape of plain fatigue crack. Therefore, the oblique crack length measured from the surface of specimen is regarded as that along the mid plane of specimen. The straight front of oblique crack implies that the crack growth rate near and at the specimen surface was relatively higher than
Summary of experiment and analysis results
Specimen No.
Crack obliquity (°)
Measured crack initiation site (mm)
Estimated crack initiation site (ram)
Crack length at mode change (mm)
Crack length at min. growth rate (mm)
/x for estimating slip region
LIOPI L10P2 L5P1 L5Pla L5P2 L2P 1
10 16 17 16 10 27
0.18 0.05 0.32 0.10 0.15 0
0.23 0.07 0.28 0.15 0.14 0.05
1.91 1.49 1.77 1.10 1.24 1.15
1.29 0.83 1.19 0.45 0.73 0.75
0.65 0.85 0.78 >1 >1 >1
Crack behaviour of A12024-7-4 under fretting fatigue
247
10-3
I
LIOP1 L10P2
" ~ 10-4 R
bl
ooo o
C
1.91 mm
v
Slant view Front view a : Oblique crack b : Mode I crack c : Final fiacture
Z
10-5: 1.49 mm (LIOP2)
"o
chan~leto mode I
Figure 7 Typical view of fracture surface 10-6_ 10-1
the plain fatigue condition. This can be explained from the slip regions of outer sides of the contact area (see Figure 4). The contact (i.e. normal and tangential) stresses are concentrated on the edges of contact area. So, the fretting effect is higher on them. From our experiment, if the contact stresses are larger, the slip region are larger for the specimens of partial slip regime. Therefore, the outer slip regions can explain that the crack growth near and at the surface of the specimen is faster than the plain fatigue.
101
Figure 9 Relation of a and da/dN (10 mm pad)
10-3 • L5P1 • L5Pla z~ L5P2 10-4.
Crack initiation life and growth rate Figure 8 shows crack length vs fatigue cycle of the specimens, LIOP2, L5P1 and L2P1. The specimen of gross slip (L2P1) had the shortest initiation life. For the specimens of partial slip (L10P2 and L5P1), the initiation life decreased when the size of slip region increased. When L10P2 is compared with L5P1, the size of slip region depends on the contact stresses. But the contact stresses of L2P1 is smaller than L5P1. From this result, the condition of gross slip is predominant for decreasing the crack initiation life. When the conditions of partial slip were formed in our experiment, the initiation life decreased as the contact stresses increased. The plots of crack growth rate vs crack length are given for each pad foot in Figures 9-11. The crack growth rate is high at first and decreases as the crack grows during the early stage of fretting fatigue. The crack growth rate then increases again after a certain crack length at which the crack growth rate reaches the minimum. Therefore, the shape of the curves looks like a well. In Figures 9-11, the crack length where each crack re-directs to mode I is also marked. It is certain that the minimum crack growth rate occurred
100
a (rnm)
to
#
1.10 mm • (L5Pla)-~'~
~,
I.~7mm " ~ --# \ ~L,P,,
CIr~.^~- ..=\
E E Z 10-s-
1.24 mm (L5P2)
•
"o
chanoe to mode I 10-6 10 -1
1()o
101
a (mm)
Figure 1O Relation of a and da/dN (5 mm pad)
10-3. •
L2P1
• qt 10 -4. o
",& @
E E
4II |
3-
o LIOP2 • L5P1
2
3 N (xl0S)
4
1.15 ram_
Z lO-S.
/,,/
g
Figure 8
~ ~
chanae to mode I 10-6 10-1
5
Relation of crack length (a) and cycle (N)
1()0
a (mm) Figure 11
Relation of a and da/dN (2 mm pad)
101
248
H.-K. Kim and S.-B. Lee
during the obli_que growth of a crack. Hereafter, the range of decreasing crack growth rate is referred as stage I and that of increasing crack growth rate as stage II. The typical crack growth rate is shown schematically in Figure 12. The decrease of the crack growth rate of stage I is due to the decrease of the effect of contact stresses as the distance from the contact surface increases. This explanation is drawn from Figures 9-11 and also from Table 2. The crack length at which crack growth rate is the minimum is found to be longer as the contact stresses increase. The bulk stress has effects on driving the crack competitively to the contact stresses during the oblique growth. When the bulk stress governs the crack growth completely, the crack changes its direction to a mode I. This is evident from the results of different bulk stresses. In the case of the 5 mm pad (see Figure 10), the minimum crack growth rate occurs at a shorter crack length as the bulk stress is larger. By comparing the results of L5P2, L 5 P l a and L2P1, the crack growth rate and the shape of well are found to be similar provided that the contact stresses are similar. This enables one to predict the behaviour of fretting fatigue crack growth as long as the contact stresses are known. From this viewpoint, it is meaningful to investigate the effect of the contact stresses on the shape of well (the behaviour of crack growth rate). When the crack growth rates of L10P2, L5P1 and L2P1 are compared, it is found that the larger the contact stresses are (see Table 1), the narrower the width of well is. L5P1 shows the narrowest well, which was suffered from the largest contact stresses among the specimens. In the contact stresses, the tangential stress is more effective on the well shape. This explanation is plausible if we compare L5P1 and L10P1. From Table 1, the normal stress of L5P1 is comparable with that of LIOP1. However, the difference in tangential stress between L5P1 and L10PI is comparatively large. Again the specimens of the similar width of well (L5P2, L 5 P l a and L2P1) are found to have similar tangential stresses. Therefore, the increase of the tangential stress narrows the width of the well. DISCUSSION When an elastic punch with square edges presses an elastic half space in normal direction, the stick and
slip boundary of partial slip regime can be evaluated as follows: 13
K'(m)lK(m) = ~3/Ix where K(m) is the complete elliptic integral of the kind with argument, m and K'(m) = K(~/1- m2). argument m is equal to c/l, where c and l are the
(1)
first The half length of the stick and the contact length, respectively. /z is a friction coefficient and /3 is the Dunders parameter defined as 1 [/(1 2Vl)/Gll - / ( 1 2v2)/G2]] /3 ~ 2 L i-i i --Pl)/G1} +-{(1 ~ P 2 ) ~ 2 - j -
-
(2)
where v is Poisson's ratio and G is the shear modulus, the subscripts 1 and 2 designate two different materials. When the materials of the pad and the specimen are the same, /3 is zero and so is the right-hand-side of Equation (1). Then the solution of Equation (1) is m = 1, which means that the contact surfaces are completely adhered. Therefore, the slip regions found in the present experiment were due to the tangential force from the pad, which was produced by bulk stress exerted on the specimen. The actual contact surfaces are assumed to be flat when the normal and the bulk force are applied to the specimen. If there is no slip on the contact surface by normal and tangential forces, the normal and the tangential components of surface traction, p(x) and q(x) are obtained as follows in plane strain condition 5. f'
~rEdux(X),~,
7r(l - 2v)
I
q(S) d s = - 2 i l - v ) X--S
flp(S)ds~r(1-2v) x-s
- 2(1-v)
ux(x) = const.,
p(x)
q(x)
2(1
v2)
~
(j)
1rEduz(x) 2 ( l - v 2)
uz(x) = const.
dx
(4) (5)
where x and z are the axes of the tangential and the normal directions of the contact line, respectively. For the present analysis, x-axis is the direction of specimen axis at mid plane with the origin at the centre of the contact line. E is Young's modulus, l and v are the same parameters defined in Equations (1) and (2). ux(x) and uz(x) are the relative displacements of contact line in tangential and normal directions along the x-axis, respectively. The solutions of Equations (3) and (4) follow C
p(x) - 1r\ lZ~_x 2 (P c o s 0 - Q sin0) C
Stage I
o) o ..a
q(x) - rr\/lZ~_ x 2 (P sin0 + Q cos0) SWidthtiT" F~-of well
z
(6)
Min.crac growth rate
C,rack length at Cracklengthai re'odechange min.growthraii (iblique cracklength) Log crack length Figure 12 Schematic view of crack growth rate
(7)
where C = 2 ( 1 - v)/,J~3-4v, P and Q are the normal and the tangential forces per unit thickness of the contact surface, respectively. 0 is defined as o l n
(' +
i-~/
whence ~ = 27r In (3 - 4v).
Equations (3) and (4) are valid when the specimen can be regarded as a half plane. From the present experiment, the largest value of contact length/specimen width was 1/3. There would be an argumentation about the applicability of the equations. This problem has been considered for Hertzian contact, where the
Crack behaviour of A12024-T4 under fretting fatigue assumption of a half plane was resulted to be valid if the ratio of contact length to specimen width was less than about 1/2 for frictional contact t4. Even though the present experiment was conducted with an elastic punch rather than Hertzian contact, the specimen was assumed as a half plane for analysis. Figure 13 shows the shapes of p(x) and q(x) of L5P1 typically, p(x) goes to positive infinity at both ends of contact line. However, q(x) becomes positive and negative infinity at the leading and the trailing end of contact line, respectively. The negative values of q(x) means an opposite direction. The behaviour of q(x) at the trailing end was found to depend on the ratio of Q/P from tentative calculations. If Q/P became larger, q(x) went to positive infinity rather than negative one. However, q(x) always decreased to negative infinity with the present experimental conditions. The negative regions were extremely narrow for every condition. The elastic punch cannot sustain the infinite stress but yields in practice. Since p(x) and q(x) are a normal and a tangential component, each is comparable with the yield and the shear yield strength of A1 2024T4, respectively. The stress singularity of p(x) and q(x) at contact ends resembles the crack problem even though the order of singularity is <0.5 ~5. In order to estimate the crack initiation site on the contact line, the mid plane of the specimen has been analysed since the plane strain condition was formed along the mid plane. The punch and the half space of the same material may well be assumed to be a single body if contacted with pressure and no slip exists in between. So the sharp edge of the punch is regarded as a virtual crack tip from fracture mechanics point of view. Irwin's concept of plastic zone size ahead of crack tip was considered. The virtual crack with the yield zone ahead of it is assumed as a virtual effective crack as Irwin has proposed (of course, Irwin used two times of the yield zone). Since the crack initiated at the boundary of stick and slip zone for the specimens of partial slip, the shear yield strength and q(x) of Equation (7) were chosen as the governing parameters for obtaining the crack initiation site or slip region for partial slip regime. If q(x) is larger than shear yield strength, the
Trailing edge
249
pad and the specimen are thought to yield in the vicinity of contact edge. But the regions of local yield of each body are different because of the difference of dimensions which affects the stress field. In addition to that, the tangential traction fluctuates owing to fatigue loading cycle. Therefore, the slip is anticipated on the yield region. Tresca yield criterion was used for shear yield strength, so was 152 MPa. The elasticperfectly plastic behaviour was assumed. Table 2 shows the comparison of measured crack initiation sites and the estimated ones by the present analysis. It is shown that the present analysis is able to estimate the actual crack initiation sites quite well. On the other hand, it is generally accepted that the slip occurs when q(x) exceeds Ixp(x), where /z is a friction coefficient5,~6 or sometimes defined as Q/pi,7. Amonton's law of friction has been widely used. In Figure 14, the estimated slip regions where q(x) exceeds ~p(x) are shown for the specimens of partial slip. Since it is partial slip, the term of slip regions in Figure 14 is equivalent to the crack initiation sites. As for the values /x, Q/P was used, which was calculated from p . . . . qm%~.... and the actual contact length. Large differences are found if compared with the actual values as well as the results from the present analysis. Other calculations were tried to find o u t / z values with which the measured slip region could be estimated. The results are also given in Table 2. tx has values between 0.65 and > 1. This result cannot be accepted soundly because of the same condition of contact surfaces in the present ~experiment. It is not feasible either because /x is resulted to be >1 for L5P2 and L5Pla which revealed a partial slip regime. As for the specimen of gross slip (L2P1), it is quite acceptable that the crack initiates at the boundary of slip region. Since the slip is produced by relative displacement of the contact surfaces, the boundary of slip region and non-contact area where the displacement discontinuity occurs must be a preferable site for the crack initiation. Nevertheless, the crack initiation site of the specimen, L2P1 was also found to be estimated by the present analysis. The initiation site of L2P1 was predicted to be 0.05 mm inwards from the leading edge of the slip region, which was very close to the actual value of zero. Therefore, the present analysis can be used for estimating the crack initiation
Leading edge
50O - -
5-
p(x)
........ q(x)
400
4~
13.
t3"y
E
g
300
•
o L10P2
•
z~ L5P2
•
o L10P1
•
v
L5P1
3200
'
~.
~" loo
v
LU
Q.
1
-100
O
2-
. . . . .
-2
-1
0
1
2
Contact length (mm) Figure 13 Normal (p(x)) and tangential (q(x)) tractions on the contact line by punch (L5P1)
0.0
Perfect estimation
0.1
0.2
0.3
Measured (mm) Figure 14 Comparison of measured and predicted crack initiation site or slip region for the specimens of partial slip: solid symbols, present theory; hollow symbols, application of /~ = Q/P
250
H.-K. Kim and S.-B. Lee
10-
site of fretting fatigue regardless of the slip regime, and also the slip region of the partial slip regime. As a result, the surface traction field is proposed as follows:
C
p(x)
- 7r\l_~---:,l~_x ~
vv vvv= . , v ~ ~(:p
.
v~v
(P' cosO - Q' sinO)
p(x)
a Kr L10P2 z~~ • KII, L10P2 t~ KI, L5P1
f
• Kit, L5P1
cOD
O'y
p(x)>-o-y (8)
C q(x) - 7 r / 2 ~ _ x2 (P' sinO + Q' cosO)
,~ K r L5P2
[]
v
o
•=u~AAv
q(x)<% •
Ty
KtrL5P2
where o~,, and ~-y are the yield and the shear yield strength of the material. P' and Q' are the modified normal and tangential forces per unit thickness which compensate the area cut by ~ry and ~-yfor force equilibrium. The typical shape of the surface tractions are given in Figure 15 together with the crack initiation site by the present theory. It is necessary to know the SIFs to evaluate the integrity of the material with a fretting fatigue crack. Even, it is possible to control the crack if the effect of fretting parameters, e.g. the obliquity, the tractions etc. on the SIFs are characterized. Since it is a mixed mode crack problem, it is valuable to obtain the dominant driving force of a fretting fatigue crack in this sense. The surface traction profile is quite important to obtain the SIFs of a fretting fatigue crack precisely. In this paper, the traction profile is suggested as Equations (8) and (9). Therefore, the SIFs of an oblique crack of each specimen were calculated to find out the behaviour of the crack driving forces. The crack was modelled as a continuous distribution of dislocations. This method has been verified to have a good performance. The details of the method could be consulted by Nowell and Hills tT. The SIFs by this method has been studied also for a surface oblique crack with uniform surface tractions and bulk stress 18. Therefore, the details of the calculation procedure will not be re-written in this context. The difference from the previous study ~8 is the numerical integration procedure. If uniform surface tractions are assumed, those can be excluded from the integration of the Flamant solution5. However, the surface tractions of Equations (8) and (9) were included in the integrand for the present analysis, which increased the calculation time.
~VV
0
q(x)~> ~,. (9)
0.0
Figure16
1.0 1.5 a (mm)
0.5
2.0
2.5
SIFs of LIOP2, L5PI and L5P2
Maximum values of the loading parameters were plugged in for calculation. The calculation at a mean value (60 MPa) of bulk stress was also carried out. But the values of K] were always found to be negative with such values. If K1 is <0, the crack is regarded to be closed. It has been accepted that Kj] did not drive the crack solely due to the roughness of the crack surfaces if the crack is closed. Therefore, the maximum SIFs can be regarded as SIF ranges in this paper. Figure 16 shows some results of SIFs against the oblique crack length for the specimens of partial slip. No K, values are zero. The values of Kj are always larger than those of K.. Both K~ and Kll increase as the oblique crack grows. The values of K] is much larger than those of Kit for L5P2 when compared with L10P2 and L5P1. Therefore, K~ played a key role in driving the crack of L5P2. However, it cannot be regarded to be so for L10P2 and L5P1. Even though K I is also larger than Kit for those specimens, K, may not be neglected when the ratio of K[~/K~ is concerned. Figure 17 shows the ratio of KJK~ for the specimens in Figure 16. So, K. is also regarded to be an effective driving force for L10P2 and L5P1. If we remind the crack obliquity of each specimen (see Table 2), the ratio of K,/KI becomes larger as the obliquity is larger. In other words, the role of K, in driving the crack becomes larger as the obliquity is larger. Therefore,
100,
(~ Y
Y ~
• LIOP2 • L5P1 • L5P2
Contact region T
60
Y
......
,01
......................... -
Estimated crack initiation site (Slip region for partial slip regime) 13""
'
crack~
211
..
0.0
Obliquity
~~vv~vvv • 0.5
1.0 a (rnm)
Figure 15 Surface traction field and estimated crack initiation site suggested by the present theory
Figure 17
K.IK~ of LIOP2, L5P1 and L5P2
1.5
2.0
Crack behaviour of A12024-7"4 under fretting fatigue the dominant driving force for the oblique crack growth is regarded to depend on the obliquity as anticipated in the previous study TM. Since the SIFs are sensitive to the crack obliquity, the accurate prediction of crack obliquity is necessary to argue the driving force of a crack. This effort was made by Faanes ~ recently. Faanes advocated mode I growth of fretting fatigue crack because the fretting fatigue crack grew along the direction of Ku being close to zero from their experiment and analysis. However, the crack obliquity of present experiment could not be predicted by this theory. Further, the crack obliquity of partial slip regime showed the pad dependence. By comparing the results of the 10 and 5 mm pads (see Tables 1 and 2), the obliquity has an opposite tendency between the two pads with the increase of the contact stresses. Therefore, it was thought that there would remain arguments of the dominant crack driving force in fretting fatigue. Another difference between Faanes' ~ and the present analysis is the normal and tangential traction fields. Faanes assumed the uniform fields from the inspection of the contact surfaces, which were not expected in the present experiment. Since the traction fields affect the SIFs definitely, it is necessary to obtain more plausible traction fields according to the specific conditions. Therefore, some discrepancies may well occur between the studies. However, it is inevitable to obtain more precise SIFs to perform an analysis of fretting fatigue with LEFM, which was also pointed out by Faanes. CONCLUSIONS From the experimental and the analytical results of fretting fatigue tests with the punch-shaped pad, the following conclusions are drawn. 1. If the materials in contact are elastically the same, the crack initiation site can be predicted well by the present theory regardless of slip regime. 2. The crack initiates at the boundary of stick and slip regions for partial slip regime, while at the boundary of slip region for gross slip regime. 3. The oblique crack in the early stage of fretting fatigue has a straight front because it grows faster at or near the surface of specimen than plain fatigue due to the outer slip region on the contact surface. 4. The minimum crack growth rate occurs during the oblique crack growth. 5. If the contact (normal and tangential) stresses are similar, the crack growth rates and the well shapes are similar. 6. In the case of partial slip, the higher the tangential stress on the contact surface becomes, the larger
251
the size of slip region and the shorter the crack initiation life becomes. 7. The dominant crack driving force is dependent of the crack obliquity.
ACKNOWLEDGEMENT Mr Bong-Hoon Lee performed the fretting fatigue experiments. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Troshchenko, V.T., Tsybanev, G.V. and Khotsyanovsky, A.O. Fatigue Fract. Engng Mater. Struct. 1994, 17, 15 Sato, K., Fujii, H. and Kodama, S. Wear 1986, 107, 245 Faanes, S. and Fernando, U.S. Fati,~ue Fract. Engng Mater. Struct. 1994, 17, 939 Nakazawa, K., Sumita, M. and Maruyama, N. Fatigue Fract. Engng Mater. Struct, 1994, 7, 751 Johnson, K.L. 'Contact Mechanics', Cambridge University Press, Cambridge, 1989 Hills, D.A. and Comninou, M. Int. J. Solids Struct. 1985, 21,721 Tanaka,K., Mutoh, Y., Sakoda, S. and Leadbeater, G. Fatigue Fract. Engng Mater. Struct. 1985, 8, 129 Nix, K.J. and Lindley, T.C. Fatigue Fract. Engng Mater. Struet. 1985, 8, 143 Edwards, P.R. 'Fretting Fatigue' (Ed. R.B. Waterhouse), Applied Science Publishers, Essex, UK, 1981, Chap. 3 Kuno, M., Waterhouse, R.B., Nowell, D. and Hills, D.A. Fatigue Fract. Engng Mater. Struct. 1989, 12, 387 Faanes, S. Engng Fraet. Mech. 1995, 52, 71 Lee, B.H., Kim, H.K. and Lee, S.B. Exp. Tech., in press Spence, D.A. Proc. Camb. Phil Soc. 1973, 73, 249 Nowell, D. and Hills, D.A. Int. J. Solids Struct. 1988, 24, 105 Dunders, J. J. Elasticity 1972, 2, 109 Chang, F.K., Comminou, M. and Barber, J.R. Int. J. Mech. Sci. 1983, 25, 803 Nowell, D. and Hills, D.A.J. Strain Analyxis 1987, 22, 177 Kiln, H.K. and Lee, S.B. 'Proc. Int. Conf. Mechanics of Solids and Materials Engineering' Singapore, June 1995, pp. 829-834
APPENDIX
Sample calculation of crack initiation site (in case of L5P1) From Equation (7), the x coordinate value of the intersecting point of q(x) and the shear yield strength (152 MPa) is obtained as follows: 152 MPa -
C
(Psin0 + Qcos0)
(A 1)
IT\!,/I 2 - X 2
where the variables to be plugged in are P = 153.86 × 4.78 MPa-mm, Q = 73.75 MPa-mm, l = 4.78/2 mm and v = 0.32. Equation (A1) was solved numerically (NewtonRaphson method). To obtain the crack initiation site, the solution (x value) of Equation (A l) was subtracted from /.
ELSEVIER
PIh 0142-1123(96)00055-2
Crack initiation and growth behaviour of AI 2024-T4 under fretting fatigue Hyung-Kyu Kim* and Soon-Bok Lee Korea Advanced Institute of Science and Technology, Science Town, Taejon, 305701, Korea (Received 17 April 1996; revised 15 July 1996; accepted 22 July 1996) A fretting fatigue experiment is conducted with AI 2024-T4 pad and specimen. A pad has a rectangular foot for simulating punch pressure on the contact surface of the specimen. Fretting fatigue cracks are monitored and analysed by image processing. The crack initiation sites and obliquities are examined. The behaviour of the crack growth rate is also shown. An analysis is carried out along the mid plane of specimen to deal with a plane strain condition. The elastic solution of square punch is utilized to estimate the crack initiation site with assuming the elastic-perfectly plastic behaviour of material. The predictions are in good agreement with the experimental data. The surface traction fields are suggested, from which stress intensity factors are evaluated. Copyright © 1997 Elsevier Science Limited. All rights reserved (Keywords: fretting fatigue, square punch, crack initiation site)
A significant decrease in fatigue life occurs along with fretting phenomenon ]-3. High crack growth rate is usually found at the early stage of fretting fatigue z3. Structures and components under fretting conditions may suffer a premature failure. Therefore, it is necessary to investigate the initiation and growth behaviour of a fretting fatigue crack. Many experimental studies have been done to investigate the behaviour of fretting fatigue crack. A fatigue testing machine was usually used with pads that exerted a compressive load on the specimen. A cylinder or a ball was widely used for the pad so that Hertzian contact could be formed. However, in engineering practice, a contact of flat surfaces is unavoidable. A pad of flat surface was sometimes used in the shape of a bridge 3'4. The bridge-shaped pad is beneficial for measuring the contact forces. But it is rather difficult to analyse the crack since the number of possible sites of crack initiation increases, i.e. the interaction of the cracks should be considered. It has been generally regarded that the fretting fatigue crack initiated at the boundary of the contact ] 3. However, some authors 4 found that the crack initiated at the stick and slip boundary. The stress condition of fretting fatigue is sophisticated. Many theoretical works have also been carried out. The classical elastic solutions were often consulted such as Hertzian contact, the rigid punch problem, and
the Flamant solution etc. 5 As for the mixed mode crack problem of fretting fatigue, the stress intensity factors (SIFs) were widely used 3'6-8. Since the compressive stress on the contact surface is an important parameter of fretting fatigue, the effect of the traction field on SIFs has been studied 8'9. The dominance of K~ or Kn during fretting fatigue cracking was also argued ~o.t ]. In this paper, the results of the fretting fatigue experiment with a punch-shaped pad are evaluated. Since the bridge-shaped pad produces multiple cracks, the punch-shaped pad is used to confine the number of crack initiation sites. Since it is worthwhile to know the effects of fretting parameters on the crack behaviour, the pad lengths and the contact forces are varied during the experiment. The angle between the crack and the direction of contacting normal force, termed as an obliquity hereafter, is investigated. The initiation sites and the growth behaviour of the cracks are also characterized. The applicability of the solution of punch problem is studied for analysis. The crack initiation sites are predicted with the concept of fracture mechanics. By suggesting the surface traction fields, the SIFs of the oblique crack are evaluated. The obliquity dependence of K1 and KI] is discussed.
EXPERIMENTAL
Materials and fretting fatigue specimens The fretting fatigue specimens, the pads and loading bars (pad assembly) were made from 2024-T4 alumi-
* Present address: Korea Atomic Research Institute, Taejon, 301313, Korea
243
244
H.-K. Kim and S.-B. Lee 180
60--
M5
(a)
Trailing Edge ~ i' ~- Leading Edge Pad foot length (b)
/-M5
~3/
~-~25
The fatigue loading was sinusoidal with a frequency of 7 Hz. The bulk stress amplitude was fixed to be 60 MPa with the stress ratio of R = 0 for all the tests except one case. The only different amplitude applied was 8 6 M P a with R = 0 for the 5 mm pad. This was conducted to change the tangential stress for comparing with the 2 mm pad. The tangential forces were measured during the fretting fatigue test. It was also assumed that the tangential stresses were uniformly distributed on the contact surface a priori. The maximum value of nominal tangential stress, q~oX..... occurred at maximum bulk stress. The fretting fatigue test conditions for each specimen are summarized in
Table 1. M5~ ,I
(e)
Figure 1 Geometries of (a) specimen, (b) pad and (c) loading bar num alloy whose yield strength and Poisson's ratio are 304 MPa and 0.32, respectively. Figure 1 shows the geometry of the specimen and the pad assembly. Pads of 2 mm, 5 mm and 10 mm were machined to have sharp edges. The specimens and the pad assemblies were machined from the asreceived bar of 30 mm dia. The surfaces of the specimens as well as the contact areas of the specimens and the pads were polished with fine emery paper.
Fretting fatigue test The normal force was applied on the contact surface of the specimen by extending two loading bars which were assembled with the pads by nuts. The specimen with loading bars and pads was installed in the fatigue testing machine. The tangential stress was formed on the contact surfaces during the fatigue loading being applied to the specimen. The normal and tangential forces were measured by strain gauges attached to the loading bars and the pads, respectively. The fretting fatigue crack was monitored by a CCD camera. The actual images were magnified 30 times in the camera and stored in a personal computer with a programmed interval of time. These stored images were analysed after the test with image processing program. Figure 2 shows the schematic diagram of the experimental setup. During the application of normal force to the specimen, it was assumed that the stresses were uniformly distributed on the contact surface a priori. The designed nominal stresses, p . . . . . were 66.7 MPa or 37.6 MPa.
Figure 2 Schematic diagram of the experimental setup
The normal and tangential force calibration It is important to know the normal and tangential forces on the contact surface to analyse the behaviour of the fretting fatigue crack. However, it is difficult to measure the forces directly during the test since those are exerted on the contact surface. In the present experiment, the deformations of the loading bars and the pads were used for measuring the normal and tangential forces, respectively. Therefore, the strain values of the loading bars and the pads were calibrated before fretting fatigue tests. The calibration method has been published separately 12, so the details are not reproduced here. The idea was to use the strain value of the calibration specimens, the loading bars and the pads as a correlating parameter. The relationships between normal force and loading bar strain and tangential force and pad strain were obtained respectively for each test condition. Typical examples of the calibration results for the 2 mm pad are given in Figure 3. The linearities between the forces and loading bar or pad strain were found to be quite good in every calibration test. The calibration results enabled one to perform in situ measurements of the normal and tangential forces during the tests. RESULTS
Slip regimes and crack initiation The contact surfaces were investigated by an optical microscope with 80 times magnification. The slip region was covered with oxidized wear debris coloured black so the boundary of stick and slip zone could be distinguished easily. All the tests with the 10 and 5 mm pads showed a partial slip regime, while a gross slip regime was shown for the 2 mm pad. Figures 4 and 5 show the contact surface of the partial slip and that of the gross slip, respectively. As for the specimen of the partial slip, the slip region was found mainly at a leading portion of the pad. No slip was found at the trailing portion of it (here, the terms of 'leading' and 'trailing' are defined as given in Figure 1). However, it was observed that two narrow bands of slip region formed along the specimen axis as shown in Figure 4. No symptom of slip was observed outside the bands. This implies that the normal and tangential stresses on the contact area are not uniform along the axial and the thickness directions of the specimen. As for the specimen of gross slip, the width of slip region was also narrower than the specimen thickness.
Crack behaviour of A12024-7"4 under fretting fatigue Table 1
245
Fretting fatigue experimental conditions
Specimen No.
Bulk stress range (MPa)
Pad length/ specimen thickness (mm/mm)
L10P1 L10P2 L5PI L5PIa L5P2 L2P I
0 0 0 0 0 0
10/3 10/3 5/3 5/3 5/3 2/3
~ ~ ~ ~ ~ ~
120 120 120 172 120 120
Designed p . . . . (MPa)
Actual contact Designed q~o~ ,,,~ length/width (MPa) (mm/mm)
Re-calculated p ..... (MPa)
Re-calculated q . . . . .n o r a (MPa)
66.7 36.5 67.7 67.2 36.9 66.8
16.3 9.42 32.45 37.78 19.93 39.1
130.78 61.83 153.86 96.00 93.81 89.07
31.96
80
-20
0
20
40
~" 7o
• Normal stress o Tangential stress
6O
/ //
13.
60
.~,) 9,~ /
15.96 73.75 53.40 50.67 52.13
t:ati gue direction Leading ~ I, Trailing edge ~ :: edge
Pad strain (x 10-6) -40 9O
9.56/1.60 9.27/1.91 4.78/1.38 4.95/2.12 5.00/1.18 2.00/2.25
20 (ID
~SSpecimen ~thickness
10 ~"
~n 50.
~
40
E 0 Z
30
!
o 0 "/
g
,It
L
lmm
Crack
O"
"13
20.
t
Crack
.lo~=
Figure 5
-20
Resultantly, the actual width was always narrower than the specimen thickness (3 mm), even though the foot width (5 mm) was wider than that in the present experiment. The actual contact area has been measured with the definition of a rectangle surrounded by the
Contact surface view of gross slip (L2PI)
100 0
Figure 3
5'0 "160 1gO 200 250 360 Loading bar strain (x 10-6)
Calibration results of contact stresses for the 2 m m pad
Leading edge Fatigue direction • Trailing edge
Crack
•
/
:~V- ~__~ p l a n e ~ . ~ ~ ~ .~ r
-
Figure4
Contact surface view of partial slip (L5PI)
Measuredcrack initiation site
i
(Specimen i thickness
H.-K. Kim and S.-B. Lee
246
outer-most boundary of slip region. Then, the nominal values of the normal and tangential stresses were recalculated according to the actual contact area. The results are also given in Table 1 with the designations of Pnom and qmaXom for comparison with the designed values. The fretting fatigue crack initiated exactly at the boundary of stick and slip zone when a specimen revealed a partial slip regime. However, in the case of gross slip regime, the crack initiated at the leading edge of the pad, i.e. the leading edge of slip region. On the other hand, the crack on the contact surface met the surface of specimen backwards (to the direction of trailing edge of the pad) as shown in Figures 4 and 5. As for the specimens of partial slip, the slip region had the shortest length at the middle of contact (the mid plane of the specimen). It increased outwards to the specimen surface. Therefore, the crack had a curved contour at its initiation site on the contact surface. All other specimens with partial slip regime showed the same phenomenon. This means that the distance between the edge of pad foot and the crack initiation site observed from the surface of the specimen is not the length of slip region or crack initiation site in the middle of contact width. In Table 2, the measured lengths of slip regions (or distances between the leading edge of the pad and the crack initiation site) in the middle of contact width are given for the specimens of partial slip regime. As for gross slip regime, the slip region is designated as the actual contact length/width in Table 1. However, the crack initiation site for it is zero in Table 2 since the crack initiated at the edge of slip region.
Oblique crack growth All the cracks behaved as a typical fretting fatigue crack. They grew obliquely in the direction of normal force initially. After the crack grew obliquely to some extent, it changed the direction (re-directed) to that of normal force as shown in Figure 6. After the crack redirected, it behaved like a mode I crack. No branch or kink crack was found during the oblique growth. The crack length was measured along the crack line on the specimen\ surface during the oblique growth. The length of oblique crack is given in Table 2 for each specimen together with the measured obliquity. The obliquity of each specimen was nearly constant and found to be within the range of 10-27 ° . The largest obliquity of 27 ° has been found in a specimen of gross slip regime (L2P1). Therefore, the oblique portion of the fretting fatigue crack can be modelled as a single surface oblique edge crack. After the crack re-directed to mode I, the distance between the speciTable 2
Pad
Fatigue direction (a)
(b) Pad
Crack length at mode change Mode I Crack ---[Obliquity Figure6 Typical view of oblique crack and mode change: (a) L5P1; and (b) L2P1
men edge and the crack tip was defined as a crack length. However, the behaviour of the crack after redirection is not the main concern of this study. The specimens were to be failed with monotonic tension when the re-directed cracks grew to some extent. The fractured surface could be divided into three parts of different features as shown in Figure 7. Those were the initial oblique and flat surface by fretting effect, the perpendicular and flat one after redirection and the perpendicular but shear surface due to monotonic tension. The crack front inside the specimen was examined. It has been found that the crack front of the initial oblique and flat surface was almost straight. However, the crack front of the perpendicular and fiat surface showed a thumb-nail contour, a typical shape of plain fatigue crack. Therefore, the oblique crack length measured from the surface of specimen is regarded as that along the mid plane of specimen. The straight front of oblique crack implies that the crack growth rate near and at the specimen surface was relatively higher than
Summary of experiment and analysis results
Specimen No.
Crack obliquity (°)
Measured crack initiation site (mm)
Estimated crack initiation site (ram)
Crack length at mode change (mm)
Crack length at min. growth rate (mm)
/x for estimating slip region
LIOPI L10P2 L5P1 L5Pla L5P2 L2P 1
10 16 17 16 10 27
0.18 0.05 0.32 0.10 0.15 0
0.23 0.07 0.28 0.15 0.14 0.05
1.91 1.49 1.77 1.10 1.24 1.15
1.29 0.83 1.19 0.45 0.73 0.75
0.65 0.85 0.78 >1 >1 >1
Crack behaviour of A12024-7-4 under fretting fatigue
247
10-3
I
LIOP1 L10P2
" ~ 10-4 R
bl
ooo o
C
1.91 mm
v
Slant view Front view a : Oblique crack b : Mode I crack c : Final fiacture
Z
10-5: 1.49 mm (LIOP2)
"o
chan~leto mode I
Figure 7 Typical view of fracture surface 10-6_ 10-1
the plain fatigue condition. This can be explained from the slip regions of outer sides of the contact area (see Figure 4). The contact (i.e. normal and tangential) stresses are concentrated on the edges of contact area. So, the fretting effect is higher on them. From our experiment, if the contact stresses are larger, the slip region are larger for the specimens of partial slip regime. Therefore, the outer slip regions can explain that the crack growth near and at the surface of the specimen is faster than the plain fatigue.
101
Figure 9 Relation of a and da/dN (10 mm pad)
10-3 • L5P1 • L5Pla z~ L5P2 10-4.
Crack initiation life and growth rate Figure 8 shows crack length vs fatigue cycle of the specimens, LIOP2, L5P1 and L2P1. The specimen of gross slip (L2P1) had the shortest initiation life. For the specimens of partial slip (L10P2 and L5P1), the initiation life decreased when the size of slip region increased. When L10P2 is compared with L5P1, the size of slip region depends on the contact stresses. But the contact stresses of L2P1 is smaller than L5P1. From this result, the condition of gross slip is predominant for decreasing the crack initiation life. When the conditions of partial slip were formed in our experiment, the initiation life decreased as the contact stresses increased. The plots of crack growth rate vs crack length are given for each pad foot in Figures 9-11. The crack growth rate is high at first and decreases as the crack grows during the early stage of fretting fatigue. The crack growth rate then increases again after a certain crack length at which the crack growth rate reaches the minimum. Therefore, the shape of the curves looks like a well. In Figures 9-11, the crack length where each crack re-directs to mode I is also marked. It is certain that the minimum crack growth rate occurred
100
a (rnm)
to
#
1.10 mm • (L5Pla)-~'~
~,
I.~7mm " ~ --# \ ~L,P,,
CIr~.^~- ..=\
E E Z 10-s-
1.24 mm (L5P2)
•
"o
chanoe to mode I 10-6 10 -1
1()o
101
a (mm)
Figure 1O Relation of a and da/dN (5 mm pad)
10-3. •
L2P1
• qt 10 -4. o
",& @
E E
4II |
3-
o LIOP2 • L5P1
2
3 N (xl0S)
4
1.15 ram_
Z lO-S.
/,,/
g
Figure 8
~ ~
chanae to mode I 10-6 10-1
5
Relation of crack length (a) and cycle (N)
1()0
a (mm) Figure 11
Relation of a and da/dN (2 mm pad)
101
248
H.-K. Kim and S.-B. Lee
during the obli_que growth of a crack. Hereafter, the range of decreasing crack growth rate is referred as stage I and that of increasing crack growth rate as stage II. The typical crack growth rate is shown schematically in Figure 12. The decrease of the crack growth rate of stage I is due to the decrease of the effect of contact stresses as the distance from the contact surface increases. This explanation is drawn from Figures 9-11 and also from Table 2. The crack length at which crack growth rate is the minimum is found to be longer as the contact stresses increase. The bulk stress has effects on driving the crack competitively to the contact stresses during the oblique growth. When the bulk stress governs the crack growth completely, the crack changes its direction to a mode I. This is evident from the results of different bulk stresses. In the case of the 5 mm pad (see Figure 10), the minimum crack growth rate occurs at a shorter crack length as the bulk stress is larger. By comparing the results of L5P2, L 5 P l a and L2P1, the crack growth rate and the shape of well are found to be similar provided that the contact stresses are similar. This enables one to predict the behaviour of fretting fatigue crack growth as long as the contact stresses are known. From this viewpoint, it is meaningful to investigate the effect of the contact stresses on the shape of well (the behaviour of crack growth rate). When the crack growth rates of L10P2, L5P1 and L2P1 are compared, it is found that the larger the contact stresses are (see Table 1), the narrower the width of well is. L5P1 shows the narrowest well, which was suffered from the largest contact stresses among the specimens. In the contact stresses, the tangential stress is more effective on the well shape. This explanation is plausible if we compare L5P1 and L10P1. From Table 1, the normal stress of L5P1 is comparable with that of LIOP1. However, the difference in tangential stress between L5P1 and L10PI is comparatively large. Again the specimens of the similar width of well (L5P2, L 5 P l a and L2P1) are found to have similar tangential stresses. Therefore, the increase of the tangential stress narrows the width of the well. DISCUSSION When an elastic punch with square edges presses an elastic half space in normal direction, the stick and
slip boundary of partial slip regime can be evaluated as follows: 13
K'(m)lK(m) = ~3/Ix where K(m) is the complete elliptic integral of the kind with argument, m and K'(m) = K(~/1- m2). argument m is equal to c/l, where c and l are the
(1)
first The half length of the stick and the contact length, respectively. /z is a friction coefficient and /3 is the Dunders parameter defined as 1 [/(1 2Vl)/Gll - / ( 1 2v2)/G2]] /3 ~ 2 L i-i i --Pl)/G1} +-{(1 ~ P 2 ) ~ 2 - j -
-
(2)
where v is Poisson's ratio and G is the shear modulus, the subscripts 1 and 2 designate two different materials. When the materials of the pad and the specimen are the same, /3 is zero and so is the right-hand-side of Equation (1). Then the solution of Equation (1) is m = 1, which means that the contact surfaces are completely adhered. Therefore, the slip regions found in the present experiment were due to the tangential force from the pad, which was produced by bulk stress exerted on the specimen. The actual contact surfaces are assumed to be flat when the normal and the bulk force are applied to the specimen. If there is no slip on the contact surface by normal and tangential forces, the normal and the tangential components of surface traction, p(x) and q(x) are obtained as follows in plane strain condition 5. f'
~rEdux(X),~,
7r(l - 2v)
I
q(S) d s = - 2 i l - v ) X--S
flp(S)ds~r(1-2v) x-s
- 2(1-v)
ux(x) = const.,
p(x)
q(x)
2(1
v2)
~
(j)
1rEduz(x) 2 ( l - v 2)
uz(x) = const.
dx
(4) (5)
where x and z are the axes of the tangential and the normal directions of the contact line, respectively. For the present analysis, x-axis is the direction of specimen axis at mid plane with the origin at the centre of the contact line. E is Young's modulus, l and v are the same parameters defined in Equations (1) and (2). ux(x) and uz(x) are the relative displacements of contact line in tangential and normal directions along the x-axis, respectively. The solutions of Equations (3) and (4) follow C
p(x) - 1r\ lZ~_x 2 (P c o s 0 - Q sin0) C
Stage I
o) o ..a
q(x) - rr\/lZ~_ x 2 (P sin0 + Q cos0) SWidthtiT" F~-of well
z
(6)
Min.crac growth rate
C,rack length at Cracklengthai re'odechange min.growthraii (iblique cracklength) Log crack length Figure 12 Schematic view of crack growth rate
(7)
where C = 2 ( 1 - v)/,J~3-4v, P and Q are the normal and the tangential forces per unit thickness of the contact surface, respectively. 0 is defined as o l n
(' +
i-~/
whence ~ = 27r In (3 - 4v).
Equations (3) and (4) are valid when the specimen can be regarded as a half plane. From the present experiment, the largest value of contact length/specimen width was 1/3. There would be an argumentation about the applicability of the equations. This problem has been considered for Hertzian contact, where the
Crack behaviour of A12024-T4 under fretting fatigue assumption of a half plane was resulted to be valid if the ratio of contact length to specimen width was less than about 1/2 for frictional contact t4. Even though the present experiment was conducted with an elastic punch rather than Hertzian contact, the specimen was assumed as a half plane for analysis. Figure 13 shows the shapes of p(x) and q(x) of L5P1 typically, p(x) goes to positive infinity at both ends of contact line. However, q(x) becomes positive and negative infinity at the leading and the trailing end of contact line, respectively. The negative values of q(x) means an opposite direction. The behaviour of q(x) at the trailing end was found to depend on the ratio of Q/P from tentative calculations. If Q/P became larger, q(x) went to positive infinity rather than negative one. However, q(x) always decreased to negative infinity with the present experimental conditions. The negative regions were extremely narrow for every condition. The elastic punch cannot sustain the infinite stress but yields in practice. Since p(x) and q(x) are a normal and a tangential component, each is comparable with the yield and the shear yield strength of A1 2024T4, respectively. The stress singularity of p(x) and q(x) at contact ends resembles the crack problem even though the order of singularity is <0.5 ~5. In order to estimate the crack initiation site on the contact line, the mid plane of the specimen has been analysed since the plane strain condition was formed along the mid plane. The punch and the half space of the same material may well be assumed to be a single body if contacted with pressure and no slip exists in between. So the sharp edge of the punch is regarded as a virtual crack tip from fracture mechanics point of view. Irwin's concept of plastic zone size ahead of crack tip was considered. The virtual crack with the yield zone ahead of it is assumed as a virtual effective crack as Irwin has proposed (of course, Irwin used two times of the yield zone). Since the crack initiated at the boundary of stick and slip zone for the specimens of partial slip, the shear yield strength and q(x) of Equation (7) were chosen as the governing parameters for obtaining the crack initiation site or slip region for partial slip regime. If q(x) is larger than shear yield strength, the
Trailing edge
249
pad and the specimen are thought to yield in the vicinity of contact edge. But the regions of local yield of each body are different because of the difference of dimensions which affects the stress field. In addition to that, the tangential traction fluctuates owing to fatigue loading cycle. Therefore, the slip is anticipated on the yield region. Tresca yield criterion was used for shear yield strength, so was 152 MPa. The elasticperfectly plastic behaviour was assumed. Table 2 shows the comparison of measured crack initiation sites and the estimated ones by the present analysis. It is shown that the present analysis is able to estimate the actual crack initiation sites quite well. On the other hand, it is generally accepted that the slip occurs when q(x) exceeds Ixp(x), where /z is a friction coefficient5,~6 or sometimes defined as Q/pi,7. Amonton's law of friction has been widely used. In Figure 14, the estimated slip regions where q(x) exceeds ~p(x) are shown for the specimens of partial slip. Since it is partial slip, the term of slip regions in Figure 14 is equivalent to the crack initiation sites. As for the values /x, Q/P was used, which was calculated from p . . . . qm%~.... and the actual contact length. Large differences are found if compared with the actual values as well as the results from the present analysis. Other calculations were tried to find o u t / z values with which the measured slip region could be estimated. The results are also given in Table 2. tx has values between 0.65 and > 1. This result cannot be accepted soundly because of the same condition of contact surfaces in the present ~experiment. It is not feasible either because /x is resulted to be >1 for L5P2 and L5Pla which revealed a partial slip regime. As for the specimen of gross slip (L2P1), it is quite acceptable that the crack initiates at the boundary of slip region. Since the slip is produced by relative displacement of the contact surfaces, the boundary of slip region and non-contact area where the displacement discontinuity occurs must be a preferable site for the crack initiation. Nevertheless, the crack initiation site of the specimen, L2P1 was also found to be estimated by the present analysis. The initiation site of L2P1 was predicted to be 0.05 mm inwards from the leading edge of the slip region, which was very close to the actual value of zero. Therefore, the present analysis can be used for estimating the crack initiation
Leading edge
50O - -
5-
p(x)
........ q(x)
400
4~
13.
t3"y
E
g
300
•
o L10P2
•
z~ L5P2
•
o L10P1
•
v
L5P1
3200
'
~.
~" loo
v
LU
Q.
1
-100
O
2-
. . . . .
-2
-1
0
1
2
Contact length (mm) Figure 13 Normal (p(x)) and tangential (q(x)) tractions on the contact line by punch (L5P1)
0.0
Perfect estimation
0.1
0.2
0.3
Measured (mm) Figure 14 Comparison of measured and predicted crack initiation site or slip region for the specimens of partial slip: solid symbols, present theory; hollow symbols, application of /~ = Q/P
250
H.-K. Kim and S.-B. Lee
10-
site of fretting fatigue regardless of the slip regime, and also the slip region of the partial slip regime. As a result, the surface traction field is proposed as follows:
C
p(x)
- 7r\l_~---:,l~_x ~
vv vvv= . , v ~ ~(:p
.
v~v
(P' cosO - Q' sinO)
p(x)
a Kr L10P2 z~~ • KII, L10P2 t~ KI, L5P1
f
• Kit, L5P1
cOD
O'y
p(x)>-o-y (8)
C q(x) - 7 r / 2 ~ _ x2 (P' sinO + Q' cosO)
,~ K r L5P2
[]
v
o
•=u~AAv
q(x)<% •
Ty
KtrL5P2
where o~,, and ~-y are the yield and the shear yield strength of the material. P' and Q' are the modified normal and tangential forces per unit thickness which compensate the area cut by ~ry and ~-yfor force equilibrium. The typical shape of the surface tractions are given in Figure 15 together with the crack initiation site by the present theory. It is necessary to know the SIFs to evaluate the integrity of the material with a fretting fatigue crack. Even, it is possible to control the crack if the effect of fretting parameters, e.g. the obliquity, the tractions etc. on the SIFs are characterized. Since it is a mixed mode crack problem, it is valuable to obtain the dominant driving force of a fretting fatigue crack in this sense. The surface traction profile is quite important to obtain the SIFs of a fretting fatigue crack precisely. In this paper, the traction profile is suggested as Equations (8) and (9). Therefore, the SIFs of an oblique crack of each specimen were calculated to find out the behaviour of the crack driving forces. The crack was modelled as a continuous distribution of dislocations. This method has been verified to have a good performance. The details of the method could be consulted by Nowell and Hills tT. The SIFs by this method has been studied also for a surface oblique crack with uniform surface tractions and bulk stress 18. Therefore, the details of the calculation procedure will not be re-written in this context. The difference from the previous study ~8 is the numerical integration procedure. If uniform surface tractions are assumed, those can be excluded from the integration of the Flamant solution5. However, the surface tractions of Equations (8) and (9) were included in the integrand for the present analysis, which increased the calculation time.
~VV
0
q(x)~> ~,. (9)
0.0
Figure16
1.0 1.5 a (mm)
0.5
2.0
2.5
SIFs of LIOP2, L5PI and L5P2
Maximum values of the loading parameters were plugged in for calculation. The calculation at a mean value (60 MPa) of bulk stress was also carried out. But the values of K] were always found to be negative with such values. If K1 is <0, the crack is regarded to be closed. It has been accepted that Kj] did not drive the crack solely due to the roughness of the crack surfaces if the crack is closed. Therefore, the maximum SIFs can be regarded as SIF ranges in this paper. Figure 16 shows some results of SIFs against the oblique crack length for the specimens of partial slip. No K, values are zero. The values of Kj are always larger than those of K.. Both K~ and Kll increase as the oblique crack grows. The values of K] is much larger than those of Kit for L5P2 when compared with L10P2 and L5P1. Therefore, K~ played a key role in driving the crack of L5P2. However, it cannot be regarded to be so for L10P2 and L5P1. Even though K I is also larger than Kit for those specimens, K, may not be neglected when the ratio of K[~/K~ is concerned. Figure 17 shows the ratio of KJK~ for the specimens in Figure 16. So, K. is also regarded to be an effective driving force for L10P2 and L5P1. If we remind the crack obliquity of each specimen (see Table 2), the ratio of K,/KI becomes larger as the obliquity is larger. In other words, the role of K, in driving the crack becomes larger as the obliquity is larger. Therefore,
100,
(~ Y
Y ~
• LIOP2 • L5P1 • L5P2
Contact region T
60
Y
......
,01
......................... -
Estimated crack initiation site (Slip region for partial slip regime) 13""
'
crack~
211
..
0.0
Obliquity
~~vv~vvv • 0.5
1.0 a (rnm)
Figure 15 Surface traction field and estimated crack initiation site suggested by the present theory
Figure 17
K.IK~ of LIOP2, L5P1 and L5P2
1.5
2.0
Crack behaviour of A12024-7"4 under fretting fatigue the dominant driving force for the oblique crack growth is regarded to depend on the obliquity as anticipated in the previous study TM. Since the SIFs are sensitive to the crack obliquity, the accurate prediction of crack obliquity is necessary to argue the driving force of a crack. This effort was made by Faanes ~ recently. Faanes advocated mode I growth of fretting fatigue crack because the fretting fatigue crack grew along the direction of Ku being close to zero from their experiment and analysis. However, the crack obliquity of present experiment could not be predicted by this theory. Further, the crack obliquity of partial slip regime showed the pad dependence. By comparing the results of the 10 and 5 mm pads (see Tables 1 and 2), the obliquity has an opposite tendency between the two pads with the increase of the contact stresses. Therefore, it was thought that there would remain arguments of the dominant crack driving force in fretting fatigue. Another difference between Faanes' ~ and the present analysis is the normal and tangential traction fields. Faanes assumed the uniform fields from the inspection of the contact surfaces, which were not expected in the present experiment. Since the traction fields affect the SIFs definitely, it is necessary to obtain more plausible traction fields according to the specific conditions. Therefore, some discrepancies may well occur between the studies. However, it is inevitable to obtain more precise SIFs to perform an analysis of fretting fatigue with LEFM, which was also pointed out by Faanes. CONCLUSIONS From the experimental and the analytical results of fretting fatigue tests with the punch-shaped pad, the following conclusions are drawn. 1. If the materials in contact are elastically the same, the crack initiation site can be predicted well by the present theory regardless of slip regime. 2. The crack initiates at the boundary of stick and slip regions for partial slip regime, while at the boundary of slip region for gross slip regime. 3. The oblique crack in the early stage of fretting fatigue has a straight front because it grows faster at or near the surface of specimen than plain fatigue due to the outer slip region on the contact surface. 4. The minimum crack growth rate occurs during the oblique crack growth. 5. If the contact (normal and tangential) stresses are similar, the crack growth rates and the well shapes are similar. 6. In the case of partial slip, the higher the tangential stress on the contact surface becomes, the larger
251
the size of slip region and the shorter the crack initiation life becomes. 7. The dominant crack driving force is dependent of the crack obliquity.
ACKNOWLEDGEMENT Mr Bong-Hoon Lee performed the fretting fatigue experiments. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Troshchenko, V.T., Tsybanev, G.V. and Khotsyanovsky, A.O. Fatigue Fract. Engng Mater. Struct. 1994, 17, 15 Sato, K., Fujii, H. and Kodama, S. Wear 1986, 107, 245 Faanes, S. and Fernando, U.S. Fati,~ue Fract. Engng Mater. Struct. 1994, 17, 939 Nakazawa, K., Sumita, M. and Maruyama, N. Fatigue Fract. Engng Mater. Struct, 1994, 7, 751 Johnson, K.L. 'Contact Mechanics', Cambridge University Press, Cambridge, 1989 Hills, D.A. and Comninou, M. Int. J. Solids Struct. 1985, 21,721 Tanaka,K., Mutoh, Y., Sakoda, S. and Leadbeater, G. Fatigue Fract. Engng Mater. Struct. 1985, 8, 129 Nix, K.J. and Lindley, T.C. Fatigue Fract. Engng Mater. Struet. 1985, 8, 143 Edwards, P.R. 'Fretting Fatigue' (Ed. R.B. Waterhouse), Applied Science Publishers, Essex, UK, 1981, Chap. 3 Kuno, M., Waterhouse, R.B., Nowell, D. and Hills, D.A. Fatigue Fract. Engng Mater. Struct. 1989, 12, 387 Faanes, S. Engng Fraet. Mech. 1995, 52, 71 Lee, B.H., Kim, H.K. and Lee, S.B. Exp. Tech., in press Spence, D.A. Proc. Camb. Phil Soc. 1973, 73, 249 Nowell, D. and Hills, D.A. Int. J. Solids Struct. 1988, 24, 105 Dunders, J. J. Elasticity 1972, 2, 109 Chang, F.K., Comminou, M. and Barber, J.R. Int. J. Mech. Sci. 1983, 25, 803 Nowell, D. and Hills, D.A.J. Strain Analyxis 1987, 22, 177 Kiln, H.K. and Lee, S.B. 'Proc. Int. Conf. Mechanics of Solids and Materials Engineering' Singapore, June 1995, pp. 829-834
APPENDIX
Sample calculation of crack initiation site (in case of L5P1) From Equation (7), the x coordinate value of the intersecting point of q(x) and the shear yield strength (152 MPa) is obtained as follows: 152 MPa -
C
(Psin0 + Qcos0)
(A 1)
IT\!,/I 2 - X 2
where the variables to be plugged in are P = 153.86 × 4.78 MPa-mm, Q = 73.75 MPa-mm, l = 4.78/2 mm and v = 0.32. Equation (A1) was solved numerically (NewtonRaphson method). To obtain the crack initiation site, the solution (x value) of Equation (A l) was subtracted from /.