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Crack growth rate of an inclined surface crack
Engineering Fracture Mechanics Vol. 43, No. 5, pp. 815-825, 1992 Printed in Great Britain.
CRACK
GROWTH
0
RATE OF AN INCLINED CRACK
0013-7944/92 $5.00 + 0.00 I!392 F’ergamon F’ress Ltd.
SURFACE
ZHAOJING ZENG and SHUHO DA1 Department of Mechanical Engineering, Nanjing Institute of Chemical Technology, 5 New Model Road, Nanjing, Jiangsu 21000!9, P.R.C. Abstract-Investigation of surface crack growth in 16MnR steel specimens with crack inclination angle b = 0”, 30” and 45” and under the condition of biaxial loading ratio P,: P, = 0, 0.5 and 1 is reported in this paper. A formula of the stress intensity factor for surface cracked cruciform specimens loaded biaxially is calibrated using the finite element method and the line-spring boundary element method. The view that the effects of both the mode I and the mode III crack driving forces on fatigue crack growth rate cannot be neglected is proposed. In order to estimate the crack growth rate of an inclined surface crack correctly, the projection method is used and the modified Paris equation, which contains the normalized mode III stress intensity factor, is developed. The experiment data are sorted out group by group and classified in the light of the crack inclination angle and the biaxial loading ratio. Linear regression analyses and statistical tests were carried out. Consequently, the equations of the crack growth rate for inclined surface cracks in the thickness and length directions are obtained.
NOMENCLATURE a C
5
CO CC
da/dN dc/dN dc,, /dN fF :K AK; K, IK, rmli Kin IKin mar L xx 4 % ; PX PY P,: P, f $X t W xx2 B
maximum depth of surface crack (m) half of surface crack length (m) projected size of the half length of inclined surface crack in the plane perpendicular to the principal stress (m) Paris equation coefficient at the deepest point Paris equation coefficient at the surface intersection crack growth rate in the thickness direction (mm/cycle) crack growth rate in the length direction (mm/cycle) crack growth rate of inclined surface crack in the length direction (mm/cycle) modification coefficient in the formula of SIF for a cruciform specimen factor in Newman-Raju equation mode I stress intensity factor (SIF) of surface crack (MPafi) range of mode I SIF at the deepest point (MPafi) range of mode I SIF at the surface intersection (MPa&) normalized mode I SIF normalized mode III SIF corrected sum of squares of the x values, Lx, = Z;, , (xi - l/n Xl_, xi)2 Paris equation exponent at the deepest point Paris equation exponent at the surface intersection degree of freedom the number of load cycles load in the X-direction (N) load in the Y-direction (N) biaxial loading ratio square of complete elliptical integral of the second kind uniform tension stress (MPa) square root of the mean of residual sum of squares thickness of the thinned central zone of cruciform specimen (m) characteristic width of the thinned central zone of cruciform specimen (m) uncorrected sum of squares of the x values crack inclination angle
INTRODUCTION SURFACE CRACKS are likely to occur in engineering structures, such as pressure vessels and offshore platforms. Assessment of surface cracks is of great importance in structural reliability analyses. A substantial amount of research on surface cracks was devoted to studying the fatigue growth behavior of mode I surface cracks in uniaxial tensile loading. Somewhat less emphasis was placed on studying the growth of mode I surface cracks in plates under pure bending. In many engineering components, especially in thin-walled pressure vessels, however, the surface crack is usually 815
816
Z. J. ZENG and S. H. DA1
subjected to cyclic biaxial loading. The crack orientations are not limited to horizontally oriented mode I cracks. The only research work on inclined surface cracks under cyclic biaxial loading has, to the authors’ knowledge, been reported by Tu and Dai [l]. In ref. [l] the initial depth to half of the surface length a/c, the aspect ratio, of the surface crack was about 0.7. Surface cracks of this kind are of the balanced crack type [2]. Surface cracks with small aspect ratio a/c belong to shallow-long surface cracks, which are common defects observed in structural components. The Pressure Vessel Research Committee (PVRC) has taken the subject of randomly oriented surface flaws in regions subjected to a biaxial cyclic stress state as a long range plan for pressure vessel research [3]. Consequently, further research on the fatigue behavior of shallow-long surface cracks under biaxial loading is required. The present authors have performed research work on shallow surface cracks with various crack inclination angles under different biaxial loading ratios. In our research work, uniaxial loading was treated as a special case of biaxial loading with the biaxial loading ratio P.,: P+ = 0. The variation pattern of the macro fatigue-broken surfaces has been described elsewhere. This paper will report the results on the fatigue growth rate of the surface cracks. TEST SPECIMEN
AND TEST PROCEDURE
Chinese pressure vessel steel, 16MnR, was used in this investigation. In order to compare the influence of different biaxial loading ratios and different crack inclination angles on the features of crack propagation, the experimental scheme was arranged as given in Table 1. The configuration of specimens in Group I is rectangular and is shown in Fig. 1. The configuration of cruciform specimens in Groups II and III is illustrated in Fig. 2, where the specimen characteristic width W is equal to 120 mm; t is equal to 7 mm in the present tests. A surface crack is schematically plotted in the central area of the two kinds of specimens. The crack inclination angle, /I, of the surface crack is measured counter-clockwise and is calculated in the direction of the x-axis. The starter surface crack was produced with the aid of a spark discharge machine. The plane of the surface crack was measured as being semi-elliptical in shape with length 2c = 24 mm and the maximum depth a = 2.5 mm or so. In order to obtain an effective sharp fatigue crack, the fatigue precracking stage was arranged before the formal test process. The fatigue tests under the condition P,: P, = 0 were performed in MTS 880, an electrohydraulic servo material system. The maximum applied load P,.,,, was taken as 140 kN and the minimum applied load Pyminas zero. The crack trajectories in the frontal surface of the specimen were measured by means of a traveling microscope. The fatigue tests under biaxial loading were conducted on the biaxial fatigue test machine, which is an electro-hydraulic servo system with four actuators. In the y-direction, the maximum applied load was taken as P,.,,, = 156.8 kN and the minimum applied load was taken as Pymin= 0 in the tests. In the x-direction, the applied cycling load was set according to the required biaxial loading ratio. The crack trajectory was monitored employing a double slideway traveling microscope. The fatigue tests were conducted under load control mode. The specimens were loaded cyclically in sinusoidal wave form at a frequency of Table 1. Fatigue experimental scheme in groups
Group no.
Biaxial loading ratio P,: Py
Specimen configuration
Crack inclination angle and specimen no. ---___-----~~~ 45’ 0’ 30’
I
0
rectangular specimen
SN.1 SN.2 SN.3
SN.4 SN.5 SN.6
SN.7 SN.8
II
0.5
cruciform specimen
BN.1 BN.2 BN.3
BN.4 BN.5 BN.6
BN.7 BN.8
III
1
cruciform snecimen
BN.10
BN.11
BN.9 BN.12
Crack growth rate of an inclined surface crack
817
X t I
y1
._._.J!J-._._._
g
___I~ 7
Fig. 1. Rectangular specimen details (dimensions in mm).
10 Hz. The shape and depth of the growing fatigue crack were monitored using the beachmarking t~hnique. In the test process, the number of fatigue cycles during which subcritical crack growth took place and the number of marking cycles applied to obtain visible marks of the crack front were recorded for every stage. After the fatigue tests were completed, the fatigue-broken surfaces were cut out from the specimens. The fati~e-broken surfaces were heated in an electric furnace to make the contours more distinct. CALCULATION METHOD OF CRACK GROWTH RATE OF INCLINED SURFACE CRACK
The Paris crack growth equation has been used to describe the fatigue growth rate for a surface crack by most researchers. The Chinese Code of Pressure Vessels Defect Assessment, CVDA-1984, also stipulates the adoption of the Paris equation in the assessment of fatigue crack growth. This is mainly because of the simplicity of the form of the Paris equation and its s~tabi~ty for engineering calculations. The Paris equation is employed in our research work to describe the fatigue growth rate of surface cracks. For fatigue growth of surface cracks under the mode I condition, i.e. the crack inclination angle fl = 0” or the biaxial loading ratio Px: Py = 1, the Paris equation in the depth direction (at the deepest point) is:
(1)
12
-
Fig. 2. Cruciform specimen details.
7 -
818
Z. J. ZENG and S. H. DA1
and in the length direction (at the surface intersection)
it is:
dc/dN = C,(AK,,)“c.
(2)
For the rectangular specimen loaded uniaxially, the ranges of mode I SIF, AK,, and AK,, , are calculated by using the Newman-Raju equation for semi-elliptic surface cracks [4]. The equation is of the form: K,= s,,/C~alQJ.F
(3)
where the calculation formulas for the factors F and Q can be found in ref. [4]. For the cruciform specimen loaded biaxially, however, the Newman-Raju equation cannot be used directly, because the specimen shape and the loading condition are different from those of the rectangular surface-cracked specimen and do not tally with the application condition of the Newman-Raju equation. Consequently, a calibrated equation of the SIF for the surface-cracked cruciform specimen is proposed in this paper. The form of the calibrated equation is taken as the form of the Newman-Raju equation. The calibrated equation of the SIF for the surface-cracked cruciform specimen is Ki =f9,IWt)~~~a/Q)~~
(4)
where F and Q are calculated according to the Newman-Raju equation [4], and f is correlated to geometrical parameters and the loading condition of the cruciform specimen. The modification coefficientfis calibrated based on a series of values of SIFs for a mode I surface crack at the deepest point obtained from the finite element method and the line-spring boundary element method [S]. The values off are listed as follows: when P,:P), = 0.5 when P,x:P.y = 1.
0.9203 i 0.8833
f=
(5)
For an inclined surface crack, the crack growth rate is significantly affected by the crack inclination angle and the biaxial loading ratio. For instance, the dependence of the crack depth, a, on the number of load cycles for the rectangular specimens, SN.3 (p = O’), SN.6 (p = 30”) and SN.8 (fl = 45’7, is shown in Fig. 3. For the three specimens loaded uniaxially, the specimen size, the initial crack size and the value of applied force were the same. However, the numbers of load cycles, N, expended in penetrating the wall of the specimen were quite different from each other. Analogously, the dependence of the crack depth, a, on the number of load cycles for the cruciform specimens, BN.3 (fl = O’), BN.4 (/? = 30”) and BN.8 (/? = 457, is shown in Fig. 4. These specimens were loaded under the condition of P,:P,= 0.5. The specimen size, the initial crack size and the applied load were the same for all three specimens. However, the numbers of load cycles, N, were quite different from each other. In the authors’ opinion, the retarded crack growth is correlated with the appearance of angle deflection, which is observed in experimental research [6] and is illustrated in Fig. 5. For an inclined
SN.3
SN.6
SN.8
7r
21 0
I
4
I 8
I 12
I
I
16
20
I 24
I
I
28
32
NW14)ky~e~
Fig. 3. a-N curve for specimens loaded uniaxially.
I 36
819
Crack growth rate of an inclined surface crack
BN.4
21 0
i 4
I 8
I 12
BN.8
I 16
I 20
t 24
N(xld)(cycle) Fig. 4. n-N curve for specimens loaded biaxially.
crack under the load biaxiality of unequal tension-tension, several deflected cracks are along the crack front at the beginning of crack propagation. The creation of the deflected crack leads to the crack propagating in an unfavorable path. The crack propagation process of the inclined surface crack in the depth direction consists of each deflected crack propagating and coalescing. As a result, the fatigue crack growth rate of the inclined surface crack slows down. The authors deem that the angle deflection is caused by mutual action upon the mode I and mode III crack driving forces. The larger the mode III crack driving force and the smaller the mode I crack driving force, the slower the crack growth rate. The variation features of the normalized mode I and mode III SIFs in terms of the biaxial loading ratio for various parameter values of the crack inclination angle have been derived by the present authors [7]. The variation features of the normalized mode I and mode III SIFs are ~hemati~lly illustrated in Figs 6 and 7. Obviously, when the biaxial loading ratio is fixed, the mode I crack driving force decreases and the mode III crack driving force increases as the crack inclination angle increases (Figs 6 and 7). Therefore the crack growth rate is faster when the crack inclination angle is equal to zero, and the crack growth rate is slower when the crack inclination angle is equal to 45”. This inference is consistent with the test results plotted in Figs 3 and 4. Comparing the a-Ncurve for /I = 30” or 45” in Fig. 3 with that for the equal crack inclination angle in Fig. 4, we observe that when the crack inclination angle is equal to 30” or 45”, the number of load cycles expended in penetrating the wall thickness under the loading condition of P,:Py= 0 is much more than that with the equal crack inclination angle under the loading condition of P,:P,= 0.5. The reason why this phenomenon occurs can also be explained from Figs 6 and 7. When the crack inclination angle is fixed, the mode I crack driving force increases and the mode III crack driving force decreases as the biaxial loading ratio increases. Therefore, the crack growth rate is slower under the condition of P,:P y= 0 and faster under the condition of P,:P,, = 0.5. In conclusion, the variation of the crack inclination angle and the biaxial loading ratio causes the variation of crack driving forces; so the crack growth rates of surface cracks are not equal even surface
created
Previousc&k plane Fig. 5. Angle deflection and deflected crack.
820
Z. J. ZENG and S. H. DA1
Px : Py Fig. 6. Variation of the normalized mode I SIFs vs the biaxial loading ratio, for parametric values of crack inclination angle.
if their initial crack sizes are the same. Hence, if the fatigue life of an inclined surface crack obtained from a test under uniaxial loading is applied indiscriminately, like the fatigue life of an engineering structural component having an inclined surface crack under biaxial tension, the wrong conclusion on the unsafe side may be drawn. Therefore, when the fatigue data are sorted out, the effect of both the mode I and the mode III crack driving forces on fatigue crack growth rate cannot be neglected. In order to consider the effect of the mode I crack driving force, which decreases as the crack inclination angle increases, on the crack growth rate, the crack projection method is used to deal with the problem of the inclined surface crack. In the crack projection method, the surface crack is projected to the plane perpendicular to the principal stress. Thus, as the crack inclination angle increases, the projected size of the crack length decreases, and the projected size of the crack depth remains unchanged; as a result, the value of the mode I SIF calculated by means of projected size decreases. Therefore, the crack projection method properly reflects the physical facts that the mixed mode crack propagates in the way of mode I being dominant and that the mode I crack driving force decreases as the crack inclination angle increases. In order to consider the effect of the mode III crack driving force on the crack growth rate, we use the modified Paris equation to describe the fatigue propagation in the depth direction:
where&i&~
max
is the normalized mode III SIF, which can be obtained from Fig. 7.
I
/
I
0
0.5
1.0
P)( : Py Fig. 7. Variation of the normalized mode III SIFs vs the biaxial loading ratio, for parametric values of crack inclination angle.
Crack growth rate of an inclined surface crack
821
The correction term, (1 - Ku1/Ku, max) , in eq. (6) is dependent on the biaxial loading ratio and the crack inclination angle. It can be seen from eq. (6) that the larger KIII/K,IImax,the slower the crack growth rate. So, the modification to the Paris equation is consistent with the opinion mentioned above. For the surface crack loaded in pure mode I, K,i,/K,,,,, is equal to zero, and under this condition, eq. (6) is the same as eq. (1). Hence, eq. (1) is a special case of eq. (6). The crack growth rate in the length direction is correlated using the following equation: dc,/dN = C,(AK,,)“c.
(7)
in eq. (7), because the value of K,,, /K,,, maxis equal There is no correction term, (1 - K,,,/K,,,_), to zero at the surface intersection of an inclined surface crack. It is evident for the mode I surface crack that cp is equal to c. Hence eq. (2) can be regarded as a special case of eq. (7). In eqs (6) and (7), the ranges of SIF, AK,, and AK,,, are computed according to eq. (3) for uniaxial loading or according to eq. (4) for biaxial loading; the projected crack size is taken as the calculated crack size. Before calculating the crack growth rate, the experimental data should be treated. Reference [8] has studied the effect of the cycles consumed in beachmarking. The conclusion summarized in ref. [8] is that under the beachmarking condition of keeping the maximum load unchanged, beachmarking does not significantly affect the crack growth rate and neglecting the number of cycles consumed in beachmarking makes calculation simple. Hence we sort out the a-N and c-N relations by employing the method of neglecting the number of cycles consumed in beachmarking. The a-da/dN relation and c-dc/dN or c,-dc,/dN relation are obtained by using the incremental polynomial method of four points. FATIGUE
GROWTH
RATE
The experimental data are sorted out group by group and classified in the light of the crack inclination angle and the biaxial loading ratio. The fatigue growth rate of a surface crack is calculated according to the above-mentioned method. Linear regression analyses are performed. The data of fatigue growth rate for each group are plotted in log-log plots of daldNl(1 - KM/Ku, max) against A&, and dc,/dN against AK,,. Tables 2 and 3 list the resulting data group by group. The experimental data for specimen BN. 1 are deleted from group E in both tables, because there are less beachmarks on the fatigue-broken surface. Some observations can be obtained from Tables 2 and 3. The Paris equation exponents corresponding to the groups with the same biaxial loading ratio are close to each other; the Paris equation coefficients corresponding to the groups with the same biaxial loading ratio are also close to each other in Table 2. What we want to know is whether the difference between the Paris equation exponents or the difference between the Paris equation coefficients results from the difference of crack inclination angle or from fluctuation of the random factors. It is known that the Paris equation exponent m, is the regression coefficient for the samples’ regression equation and log C, is the regression constant for the samples’ regression equation. By performing statistical tests [9] for the sample data of Groups A (/.I = O’), B (/I = 30”) and C (/I = 45”) listed in Table 2, it can be considered at the 95% confidence level that there is no significant difference between the Paris equation exponent values for the populations corresponding to Groups A, B and C, respectively; there is also no significant difference between the Paris equation coefficient values for the populations corresponding to Groups A, B and C, respectively. Therefore, making use of the data in Group D in Table 2, the crack growth rate in the thickness direction (at the deepest point) for the surface crack loaded uniaxially can be expressed as: da/dN = 2.913 x 10-‘1(AK,,)4~“1(1- K,,,/K,,,,,,).
(8)
By performing statistical tests for the sample data listed in Table 3, a similar conclusion can be obtained. Therefore, making use of the data in Group D of Table 3, the crack growth rate in the length direction for surface cracks loaded uniaxially can be expressed as: dc,/dN = 4.444 x 10-‘“(AK,,)3~“6.
(9)
822
Z. J. ZENG and S. H. DA1 Table 2. Resulting data for surface cracks in the thickness direction
Group
Specimen no.
A
SN.1 SN.2 SN.3
0
0
4.148
4.093 x lo-”
0.1297
0.1431
51.45
23
B
SN.4 SN.5 SN.6
30”
0
4.110
4.857 x IO-”
0.06920
0.1843
84.99
39
c
SN.7 SN.8
0
4.070
4.455 x lo-”
0.1162
0. I 107
43.26
21
0
4.241
2.913 x IO-”
0.1058
0.4797
0”
0.5
4.093
1.245 x IO--‘”
0.1317
0.02405
21.64
If
30”
0.5
3.223
1.467 x 1O-9
0.1785
0.2310
65.97
34
BN.7 BN.8
0.5
4.165
6.207 x lo-”
0.05541
0.08779
42.88
23
H
BN.2 to BN.8
0.5
4.686
1.622 x lo-”
0.6330
0.3569
I
BN.9 BN.10 BN.ll BN.12
1
4.414
2.350 x lo-”
0.06495
0.1289
SN.1 D
179.7
83
SF.8 E F
G
BN.2 BN.3 BN.4 BN.5 BN.6
130.5
68
72.77
36
Table 3. Resulting data for surface cracks in the length direction Group A
Specimen no. SN.1 SN.2 SN.3
OU
0
3.583
6.531 x IO-‘0
0.2017
0.5428
43.42
23
SN.4 SN.5 SN.6
30”
0
3.501
4.533 x lo-‘0
0.1018
0.6726
73.94
39
SN.7 SN.8
45’
0
3.592
1.909 x lo-‘*
0.1607
0.3832
40.09
21
SN.l
0” 0
3.516
4.444 x lo-I0
0.2413
1.600
SE.8
I
&
157.4
83
BN.2 BN.3 BN.3
0
0.5
3.850
3.299 x IO-‘*
0.1731
0.1311
18.73
11
BN.4 BN.5 BN.6
30”
0.5
3.654
8.152 x lo-‘0
0.1457
0.5692
55.52
34
BN.7 BN.8
45”
0.5
3.545
5.671 x IO-”
0.1064
0.2897
38.96
23
BN.2
0”
B:8
&
OS
3.541
8.568 x lo-”
0.1912
1.002
BN.9 BN.10 BN.11
45” 0” 30”
1
3.643
7.044 x 10-10
0.08743
0.4951
113.2
61.63
68
36
Crack growth
rate of an inclinedsurfacecrack
823
Plots of the crack growth rate for Group D are presented in Figs 8 and 9. In a similar way, by performing statistical tests on the sample data in Groups E (/I = 0’) and F (B = 30”) and Groups E and G (/I = 45”), respectively, a conclusion similar to the case of uniaxial loading can be drawn. Hence, under the condition of P,:P,, = 0.5, by making use of the data in Group H listed in Table 2, the crack growth rate in the thickness direction can be expressed as: da/dN = 1.622 x 10-“(AK,,)4~6*6(1- K,,,/K,,, ,,).
(10)
Under the condition of P,:P,, = 0.5, by making use of the data in Group H listed in Table 3, the crack growth rate in the length direction is: dc,/dN = 8.568 x 10-‘“(A$,)3~24’.
(11)
In eqs (8)-(1 l), the proper crack inclination angle ranges from 0” to 45”, and the projected size in the plane perpendicular to the principal stress is chosen as the crack size needed in the calculation. The value of Ku1/Ku, _ is taken from Fig. 7 according to the biaxial loading ratio and the crack inclination angle. Plots of the crack growth rate for Group H are presented in Figs 10 and 11. Under the condition of P,:P,, = 1, the crack plane coincides with the plane perpendicular to the principal stress. Therefore, no matter what the value of the crack inclination angle is, by making use of the data in Group I listed in Tables 2 and 3, the crack growth rate in the thickness direction is: da/dN = 2.350 x 10-“(AK,,)4,4’4
(12)
and the crack growth rate in the length direction is: dc/dN = 7.044 x 10-‘“(AK,,)3@3.
(13)
Plots of the crack growth rate for Group I are presented in Figs 12 and 13.
FSN.l - SN.8
l-
-
SN. 8
SN.l-
J !L 1
20
I
AKla
30
40
( MPa q/m 1
Fig. 8. Crack growth rate for specimens with P,: P, = 0 and /I = O-450, in the thickness direction.
Jc___10
20
30
40
AKk(MPaVm)
Fig. 9. Crack growth rate for specimens with P,: P, = 0 and /3 = O-45”, in the length direction.
824
Z. J. ZENG and S. H. DA1
;
BN.2 - BN.8
Iti/ 10
AKla ( MPa grn 1
20
30
40
1
AKlc ( MPa Jm
Fig. 10. Crack growth rate for specimens with P,: P,. = 0.5 Fig. 11. Crack growth rate for specimens with P,:P,, = 0.5 and ELI = O-45”, in the thickness direction. and j? = O-45”. in the length direction,
BN.9 - BN.12
t: _ BN.9 - BN.12
20
AKia ( MPaJ m 1 Fig. 12. Crack growth rate for specimens with P,: P, = 1 and j? = O-45”, in the thickness direction.
AKlc ( MPa Jm
30
40
)
Fig. 13. Crack growth rate for specimens with P,r:Py =I and j = O-45”, in the length direction.
Crack growth rate of an inclined surface crack
825
CONCLUSIONS An experimental and theoretical investigation of surface crack growth in 16MnR steel specimens with crack inclination angle /I = O”, 30” and 45”, and under the condition of biaxial loading ratio P,:Py= 0,0.5 and 1, is reported in this paper. The main conclusions are summarized below. 1. A surface-cracked cruciform specimen for fatigue experiments under biaxial loading was designed. An SIF equation for a surface-cracked cruciform specimen loaded biaxially was calibrated using the finite element method and the line-spring boundary element method. 2. When the biaxial loading ratio was fixed (P,: P,, = 0 or OS), the number of load cycles expended in penetrating the wall of the specimen with crack inclination angle /I = 45” was much more than that with crack inclination angle /I = 30”. When the crack inclination angle was fixed (B = 30” or 45”), the number of load cycles expended in penetrating the wall of the specimen under the condition of P,:P, = 0 was much more than that under the condition of P,:P,, = 0.5. Therefore, if the fatigue life of an inclined surface crack obtained from a test under uniaxial loading is applied indiscriminately, like the fatigue life of an engineering structural comnponent having an inclined surface crack under biaxial tension, the wrong conclusion on the unsafe side may be drawn. 3. The crack growth rate is affected by crack driving forces. In the cases of different crack inclination angles and different biaxial loading ratios, increase/decrease of the mode I crack driving force, and decrease/increase of the mode III crack driving force lead to increase/decrease of the crack growth rate. In order to estimate the crack growth rate correctly, the projection method, in which the surface crack is projected to the plane perpendicular to the principal stress, is used to deal with the problem of an inclined surface crack; the modified Paris equation is used to describe the fatigue propagation in the depth direction. The modified Paris equation contains the normalized mode III SIF, which is dependent only on the crack inclination angle and the biaxial loading ratio. 4. The experimental data were sorted out group by group and classified in the light of the crack inclination angle and the biaxial loading ratio. Linear regression analyses were then carried out. By performing the statistical tests on samples, we obtained the conclusion at the 95% confidence level that under the condition of fixed biaxial loading ratio, there is no significant difference between the Paris equation exponent values for the populations in the thickness direction, and there is also no significant difference between the Paris equation coefficient values for the populations in the thickness direction, in spite of the difference of crack inclination angle. The situation is the same in the length direction. Consequently, the crack growth rate equations for inclined surface cracks in the thickness and length directions are obtained. REFERENCES [l] S. T. Tu and S. H. Dai, The fatigue growth of inclined surface cracks under biaxial loading. Znt. J. Press. Vess. Piping 41, 141-157 (1990). [2] S. Chen and Z. Cui, Surface crack growth behavior under tensile cyclic loading. Znt. J. Farigue 10, 43-47. [3] PVRC, The long-range plane for pressure vessel research (8th edn). WRC Bulletin, No. 327, pp. 9-14 (1987). [4] J. C. Newman, Jr. and I. S. Raju, An empirical stress-intensity factor equation for the surface crack. Engng Frocrure Mech. 15, 185-192 (1981). [5] Z. J. Zeng and S. H. Dai, Line-spring boundary element method for a surface cracked plate. Engng Fracture Mech. 36, 855-858 (1990). [6] Z. J. Zeng, A theoretical and experimental investigation of fatigue behavior for surface cracks with shallow-long shape. Ph.D. dissertation, Nanjing Institute of Chemical Technology (1990) [in Chinese]. [7] Z. J. Zeng and S. H. Dai, Stress intensity factors for an inclined surface crack under biaxial stress state. Compur. Structures (submitted). [8] S. Chen, Z. Lu, D. Zhang and Z. Cui, Experimental investigation of surface crack growth behavior under tensile cyclic loading. Mech. Practice 9(3), 34-39 (1987) [in Chinese]. [9] D. Dong, J. Yang, M. Su and G. Gu, Statistical Methods in Experimental Research. Chinese Measure Press, Beijing (1987) [in Chinese]. (Received 26 September 1991)
CRACK
GROWTH
0
RATE OF AN INCLINED CRACK
0013-7944/92 $5.00 + 0.00 I!392 F’ergamon F’ress Ltd.
SURFACE
ZHAOJING ZENG and SHUHO DA1 Department of Mechanical Engineering, Nanjing Institute of Chemical Technology, 5 New Model Road, Nanjing, Jiangsu 21000!9, P.R.C. Abstract-Investigation of surface crack growth in 16MnR steel specimens with crack inclination angle b = 0”, 30” and 45” and under the condition of biaxial loading ratio P,: P, = 0, 0.5 and 1 is reported in this paper. A formula of the stress intensity factor for surface cracked cruciform specimens loaded biaxially is calibrated using the finite element method and the line-spring boundary element method. The view that the effects of both the mode I and the mode III crack driving forces on fatigue crack growth rate cannot be neglected is proposed. In order to estimate the crack growth rate of an inclined surface crack correctly, the projection method is used and the modified Paris equation, which contains the normalized mode III stress intensity factor, is developed. The experiment data are sorted out group by group and classified in the light of the crack inclination angle and the biaxial loading ratio. Linear regression analyses and statistical tests were carried out. Consequently, the equations of the crack growth rate for inclined surface cracks in the thickness and length directions are obtained.
NOMENCLATURE a C
5
CO CC
da/dN dc/dN dc,, /dN fF :K AK; K, IK, rmli Kin IKin mar L xx 4 % ; PX PY P,: P, f $X t W xx2 B
maximum depth of surface crack (m) half of surface crack length (m) projected size of the half length of inclined surface crack in the plane perpendicular to the principal stress (m) Paris equation coefficient at the deepest point Paris equation coefficient at the surface intersection crack growth rate in the thickness direction (mm/cycle) crack growth rate in the length direction (mm/cycle) crack growth rate of inclined surface crack in the length direction (mm/cycle) modification coefficient in the formula of SIF for a cruciform specimen factor in Newman-Raju equation mode I stress intensity factor (SIF) of surface crack (MPafi) range of mode I SIF at the deepest point (MPafi) range of mode I SIF at the surface intersection (MPa&) normalized mode I SIF normalized mode III SIF corrected sum of squares of the x values, Lx, = Z;, , (xi - l/n Xl_, xi)2 Paris equation exponent at the deepest point Paris equation exponent at the surface intersection degree of freedom the number of load cycles load in the X-direction (N) load in the Y-direction (N) biaxial loading ratio square of complete elliptical integral of the second kind uniform tension stress (MPa) square root of the mean of residual sum of squares thickness of the thinned central zone of cruciform specimen (m) characteristic width of the thinned central zone of cruciform specimen (m) uncorrected sum of squares of the x values crack inclination angle
INTRODUCTION SURFACE CRACKS are likely to occur in engineering structures, such as pressure vessels and offshore platforms. Assessment of surface cracks is of great importance in structural reliability analyses. A substantial amount of research on surface cracks was devoted to studying the fatigue growth behavior of mode I surface cracks in uniaxial tensile loading. Somewhat less emphasis was placed on studying the growth of mode I surface cracks in plates under pure bending. In many engineering components, especially in thin-walled pressure vessels, however, the surface crack is usually 815
816
Z. J. ZENG and S. H. DA1
subjected to cyclic biaxial loading. The crack orientations are not limited to horizontally oriented mode I cracks. The only research work on inclined surface cracks under cyclic biaxial loading has, to the authors’ knowledge, been reported by Tu and Dai [l]. In ref. [l] the initial depth to half of the surface length a/c, the aspect ratio, of the surface crack was about 0.7. Surface cracks of this kind are of the balanced crack type [2]. Surface cracks with small aspect ratio a/c belong to shallow-long surface cracks, which are common defects observed in structural components. The Pressure Vessel Research Committee (PVRC) has taken the subject of randomly oriented surface flaws in regions subjected to a biaxial cyclic stress state as a long range plan for pressure vessel research [3]. Consequently, further research on the fatigue behavior of shallow-long surface cracks under biaxial loading is required. The present authors have performed research work on shallow surface cracks with various crack inclination angles under different biaxial loading ratios. In our research work, uniaxial loading was treated as a special case of biaxial loading with the biaxial loading ratio P.,: P+ = 0. The variation pattern of the macro fatigue-broken surfaces has been described elsewhere. This paper will report the results on the fatigue growth rate of the surface cracks. TEST SPECIMEN
AND TEST PROCEDURE
Chinese pressure vessel steel, 16MnR, was used in this investigation. In order to compare the influence of different biaxial loading ratios and different crack inclination angles on the features of crack propagation, the experimental scheme was arranged as given in Table 1. The configuration of specimens in Group I is rectangular and is shown in Fig. 1. The configuration of cruciform specimens in Groups II and III is illustrated in Fig. 2, where the specimen characteristic width W is equal to 120 mm; t is equal to 7 mm in the present tests. A surface crack is schematically plotted in the central area of the two kinds of specimens. The crack inclination angle, /I, of the surface crack is measured counter-clockwise and is calculated in the direction of the x-axis. The starter surface crack was produced with the aid of a spark discharge machine. The plane of the surface crack was measured as being semi-elliptical in shape with length 2c = 24 mm and the maximum depth a = 2.5 mm or so. In order to obtain an effective sharp fatigue crack, the fatigue precracking stage was arranged before the formal test process. The fatigue tests under the condition P,: P, = 0 were performed in MTS 880, an electrohydraulic servo material system. The maximum applied load P,.,,, was taken as 140 kN and the minimum applied load Pyminas zero. The crack trajectories in the frontal surface of the specimen were measured by means of a traveling microscope. The fatigue tests under biaxial loading were conducted on the biaxial fatigue test machine, which is an electro-hydraulic servo system with four actuators. In the y-direction, the maximum applied load was taken as P,.,,, = 156.8 kN and the minimum applied load was taken as Pymin= 0 in the tests. In the x-direction, the applied cycling load was set according to the required biaxial loading ratio. The crack trajectory was monitored employing a double slideway traveling microscope. The fatigue tests were conducted under load control mode. The specimens were loaded cyclically in sinusoidal wave form at a frequency of Table 1. Fatigue experimental scheme in groups
Group no.
Biaxial loading ratio P,: Py
Specimen configuration
Crack inclination angle and specimen no. ---___-----~~~ 45’ 0’ 30’
I
0
rectangular specimen
SN.1 SN.2 SN.3
SN.4 SN.5 SN.6
SN.7 SN.8
II
0.5
cruciform specimen
BN.1 BN.2 BN.3
BN.4 BN.5 BN.6
BN.7 BN.8
III
1
cruciform snecimen
BN.10
BN.11
BN.9 BN.12
Crack growth rate of an inclined surface crack
817
X t I
y1
._._.J!J-._._._
g
___I~ 7
Fig. 1. Rectangular specimen details (dimensions in mm).
10 Hz. The shape and depth of the growing fatigue crack were monitored using the beachmarking t~hnique. In the test process, the number of fatigue cycles during which subcritical crack growth took place and the number of marking cycles applied to obtain visible marks of the crack front were recorded for every stage. After the fatigue tests were completed, the fatigue-broken surfaces were cut out from the specimens. The fati~e-broken surfaces were heated in an electric furnace to make the contours more distinct. CALCULATION METHOD OF CRACK GROWTH RATE OF INCLINED SURFACE CRACK
The Paris crack growth equation has been used to describe the fatigue growth rate for a surface crack by most researchers. The Chinese Code of Pressure Vessels Defect Assessment, CVDA-1984, also stipulates the adoption of the Paris equation in the assessment of fatigue crack growth. This is mainly because of the simplicity of the form of the Paris equation and its s~tabi~ty for engineering calculations. The Paris equation is employed in our research work to describe the fatigue growth rate of surface cracks. For fatigue growth of surface cracks under the mode I condition, i.e. the crack inclination angle fl = 0” or the biaxial loading ratio Px: Py = 1, the Paris equation in the depth direction (at the deepest point) is:
(1)
12
-
Fig. 2. Cruciform specimen details.
7 -
818
Z. J. ZENG and S. H. DA1
and in the length direction (at the surface intersection)
it is:
dc/dN = C,(AK,,)“c.
(2)
For the rectangular specimen loaded uniaxially, the ranges of mode I SIF, AK,, and AK,, , are calculated by using the Newman-Raju equation for semi-elliptic surface cracks [4]. The equation is of the form: K,= s,,/C~alQJ.F
(3)
where the calculation formulas for the factors F and Q can be found in ref. [4]. For the cruciform specimen loaded biaxially, however, the Newman-Raju equation cannot be used directly, because the specimen shape and the loading condition are different from those of the rectangular surface-cracked specimen and do not tally with the application condition of the Newman-Raju equation. Consequently, a calibrated equation of the SIF for the surface-cracked cruciform specimen is proposed in this paper. The form of the calibrated equation is taken as the form of the Newman-Raju equation. The calibrated equation of the SIF for the surface-cracked cruciform specimen is Ki =f9,IWt)~~~a/Q)~~
(4)
where F and Q are calculated according to the Newman-Raju equation [4], and f is correlated to geometrical parameters and the loading condition of the cruciform specimen. The modification coefficientfis calibrated based on a series of values of SIFs for a mode I surface crack at the deepest point obtained from the finite element method and the line-spring boundary element method [S]. The values off are listed as follows: when P,:P), = 0.5 when P,x:P.y = 1.
0.9203 i 0.8833
f=
(5)
For an inclined surface crack, the crack growth rate is significantly affected by the crack inclination angle and the biaxial loading ratio. For instance, the dependence of the crack depth, a, on the number of load cycles for the rectangular specimens, SN.3 (p = O’), SN.6 (p = 30”) and SN.8 (fl = 45’7, is shown in Fig. 3. For the three specimens loaded uniaxially, the specimen size, the initial crack size and the value of applied force were the same. However, the numbers of load cycles, N, expended in penetrating the wall of the specimen were quite different from each other. Analogously, the dependence of the crack depth, a, on the number of load cycles for the cruciform specimens, BN.3 (fl = O’), BN.4 (/? = 30”) and BN.8 (/? = 457, is shown in Fig. 4. These specimens were loaded under the condition of P,:P,= 0.5. The specimen size, the initial crack size and the applied load were the same for all three specimens. However, the numbers of load cycles, N, were quite different from each other. In the authors’ opinion, the retarded crack growth is correlated with the appearance of angle deflection, which is observed in experimental research [6] and is illustrated in Fig. 5. For an inclined
SN.3
SN.6
SN.8
7r
21 0
I
4
I 8
I 12
I
I
16
20
I 24
I
I
28
32
NW14)ky~e~
Fig. 3. a-N curve for specimens loaded uniaxially.
I 36
819
Crack growth rate of an inclined surface crack
BN.4
21 0
i 4
I 8
I 12
BN.8
I 16
I 20
t 24
N(xld)(cycle) Fig. 4. n-N curve for specimens loaded biaxially.
crack under the load biaxiality of unequal tension-tension, several deflected cracks are along the crack front at the beginning of crack propagation. The creation of the deflected crack leads to the crack propagating in an unfavorable path. The crack propagation process of the inclined surface crack in the depth direction consists of each deflected crack propagating and coalescing. As a result, the fatigue crack growth rate of the inclined surface crack slows down. The authors deem that the angle deflection is caused by mutual action upon the mode I and mode III crack driving forces. The larger the mode III crack driving force and the smaller the mode I crack driving force, the slower the crack growth rate. The variation features of the normalized mode I and mode III SIFs in terms of the biaxial loading ratio for various parameter values of the crack inclination angle have been derived by the present authors [7]. The variation features of the normalized mode I and mode III SIFs are ~hemati~lly illustrated in Figs 6 and 7. Obviously, when the biaxial loading ratio is fixed, the mode I crack driving force decreases and the mode III crack driving force increases as the crack inclination angle increases (Figs 6 and 7). Therefore the crack growth rate is faster when the crack inclination angle is equal to zero, and the crack growth rate is slower when the crack inclination angle is equal to 45”. This inference is consistent with the test results plotted in Figs 3 and 4. Comparing the a-Ncurve for /I = 30” or 45” in Fig. 3 with that for the equal crack inclination angle in Fig. 4, we observe that when the crack inclination angle is equal to 30” or 45”, the number of load cycles expended in penetrating the wall thickness under the loading condition of P,:Py= 0 is much more than that with the equal crack inclination angle under the loading condition of P,:P,= 0.5. The reason why this phenomenon occurs can also be explained from Figs 6 and 7. When the crack inclination angle is fixed, the mode I crack driving force increases and the mode III crack driving force decreases as the biaxial loading ratio increases. Therefore, the crack growth rate is slower under the condition of P,:P y= 0 and faster under the condition of P,:P,, = 0.5. In conclusion, the variation of the crack inclination angle and the biaxial loading ratio causes the variation of crack driving forces; so the crack growth rates of surface cracks are not equal even surface
created
Previousc&k plane Fig. 5. Angle deflection and deflected crack.
820
Z. J. ZENG and S. H. DA1
Px : Py Fig. 6. Variation of the normalized mode I SIFs vs the biaxial loading ratio, for parametric values of crack inclination angle.
if their initial crack sizes are the same. Hence, if the fatigue life of an inclined surface crack obtained from a test under uniaxial loading is applied indiscriminately, like the fatigue life of an engineering structural component having an inclined surface crack under biaxial tension, the wrong conclusion on the unsafe side may be drawn. Therefore, when the fatigue data are sorted out, the effect of both the mode I and the mode III crack driving forces on fatigue crack growth rate cannot be neglected. In order to consider the effect of the mode I crack driving force, which decreases as the crack inclination angle increases, on the crack growth rate, the crack projection method is used to deal with the problem of the inclined surface crack. In the crack projection method, the surface crack is projected to the plane perpendicular to the principal stress. Thus, as the crack inclination angle increases, the projected size of the crack length decreases, and the projected size of the crack depth remains unchanged; as a result, the value of the mode I SIF calculated by means of projected size decreases. Therefore, the crack projection method properly reflects the physical facts that the mixed mode crack propagates in the way of mode I being dominant and that the mode I crack driving force decreases as the crack inclination angle increases. In order to consider the effect of the mode III crack driving force on the crack growth rate, we use the modified Paris equation to describe the fatigue propagation in the depth direction:
where&i&~
max
is the normalized mode III SIF, which can be obtained from Fig. 7.
I
/
I
0
0.5
1.0
P)( : Py Fig. 7. Variation of the normalized mode III SIFs vs the biaxial loading ratio, for parametric values of crack inclination angle.
Crack growth rate of an inclined surface crack
821
The correction term, (1 - Ku1/Ku, max) , in eq. (6) is dependent on the biaxial loading ratio and the crack inclination angle. It can be seen from eq. (6) that the larger KIII/K,IImax,the slower the crack growth rate. So, the modification to the Paris equation is consistent with the opinion mentioned above. For the surface crack loaded in pure mode I, K,i,/K,,,,, is equal to zero, and under this condition, eq. (6) is the same as eq. (1). Hence, eq. (1) is a special case of eq. (6). The crack growth rate in the length direction is correlated using the following equation: dc,/dN = C,(AK,,)“c.
(7)
in eq. (7), because the value of K,,, /K,,, maxis equal There is no correction term, (1 - K,,,/K,,,_), to zero at the surface intersection of an inclined surface crack. It is evident for the mode I surface crack that cp is equal to c. Hence eq. (2) can be regarded as a special case of eq. (7). In eqs (6) and (7), the ranges of SIF, AK,, and AK,,, are computed according to eq. (3) for uniaxial loading or according to eq. (4) for biaxial loading; the projected crack size is taken as the calculated crack size. Before calculating the crack growth rate, the experimental data should be treated. Reference [8] has studied the effect of the cycles consumed in beachmarking. The conclusion summarized in ref. [8] is that under the beachmarking condition of keeping the maximum load unchanged, beachmarking does not significantly affect the crack growth rate and neglecting the number of cycles consumed in beachmarking makes calculation simple. Hence we sort out the a-N and c-N relations by employing the method of neglecting the number of cycles consumed in beachmarking. The a-da/dN relation and c-dc/dN or c,-dc,/dN relation are obtained by using the incremental polynomial method of four points. FATIGUE
GROWTH
RATE
The experimental data are sorted out group by group and classified in the light of the crack inclination angle and the biaxial loading ratio. The fatigue growth rate of a surface crack is calculated according to the above-mentioned method. Linear regression analyses are performed. The data of fatigue growth rate for each group are plotted in log-log plots of daldNl(1 - KM/Ku, max) against A&, and dc,/dN against AK,,. Tables 2 and 3 list the resulting data group by group. The experimental data for specimen BN. 1 are deleted from group E in both tables, because there are less beachmarks on the fatigue-broken surface. Some observations can be obtained from Tables 2 and 3. The Paris equation exponents corresponding to the groups with the same biaxial loading ratio are close to each other; the Paris equation coefficients corresponding to the groups with the same biaxial loading ratio are also close to each other in Table 2. What we want to know is whether the difference between the Paris equation exponents or the difference between the Paris equation coefficients results from the difference of crack inclination angle or from fluctuation of the random factors. It is known that the Paris equation exponent m, is the regression coefficient for the samples’ regression equation and log C, is the regression constant for the samples’ regression equation. By performing statistical tests [9] for the sample data of Groups A (/.I = O’), B (/I = 30”) and C (/I = 45”) listed in Table 2, it can be considered at the 95% confidence level that there is no significant difference between the Paris equation exponent values for the populations corresponding to Groups A, B and C, respectively; there is also no significant difference between the Paris equation coefficient values for the populations corresponding to Groups A, B and C, respectively. Therefore, making use of the data in Group D in Table 2, the crack growth rate in the thickness direction (at the deepest point) for the surface crack loaded uniaxially can be expressed as: da/dN = 2.913 x 10-‘1(AK,,)4~“1(1- K,,,/K,,,,,,).
(8)
By performing statistical tests for the sample data listed in Table 3, a similar conclusion can be obtained. Therefore, making use of the data in Group D of Table 3, the crack growth rate in the length direction for surface cracks loaded uniaxially can be expressed as: dc,/dN = 4.444 x 10-‘“(AK,,)3~“6.
(9)
822
Z. J. ZENG and S. H. DA1 Table 2. Resulting data for surface cracks in the thickness direction
Group
Specimen no.
A
SN.1 SN.2 SN.3
0
0
4.148
4.093 x lo-”
0.1297
0.1431
51.45
23
B
SN.4 SN.5 SN.6
30”
0
4.110
4.857 x IO-”
0.06920
0.1843
84.99
39
c
SN.7 SN.8
0
4.070
4.455 x lo-”
0.1162
0. I 107
43.26
21
0
4.241
2.913 x IO-”
0.1058
0.4797
0”
0.5
4.093
1.245 x IO--‘”
0.1317
0.02405
21.64
If
30”
0.5
3.223
1.467 x 1O-9
0.1785
0.2310
65.97
34
BN.7 BN.8
0.5
4.165
6.207 x lo-”
0.05541
0.08779
42.88
23
H
BN.2 to BN.8
0.5
4.686
1.622 x lo-”
0.6330
0.3569
I
BN.9 BN.10 BN.ll BN.12
1
4.414
2.350 x lo-”
0.06495
0.1289
SN.1 D
179.7
83
SF.8 E F
G
BN.2 BN.3 BN.4 BN.5 BN.6
130.5
68
72.77
36
Table 3. Resulting data for surface cracks in the length direction Group A
Specimen no. SN.1 SN.2 SN.3
OU
0
3.583
6.531 x IO-‘0
0.2017
0.5428
43.42
23
SN.4 SN.5 SN.6
30”
0
3.501
4.533 x lo-‘0
0.1018
0.6726
73.94
39
SN.7 SN.8
45’
0
3.592
1.909 x lo-‘*
0.1607
0.3832
40.09
21
SN.l
0” 0
3.516
4.444 x lo-I0
0.2413
1.600
SE.8
I
&
157.4
83
BN.2 BN.3 BN.3
0
0.5
3.850
3.299 x IO-‘*
0.1731
0.1311
18.73
11
BN.4 BN.5 BN.6
30”
0.5
3.654
8.152 x lo-‘0
0.1457
0.5692
55.52
34
BN.7 BN.8
45”
0.5
3.545
5.671 x IO-”
0.1064
0.2897
38.96
23
BN.2
0”
B:8
&
OS
3.541
8.568 x lo-”
0.1912
1.002
BN.9 BN.10 BN.11
45” 0” 30”
1
3.643
7.044 x 10-10
0.08743
0.4951
113.2
61.63
68
36
Crack growth
rate of an inclinedsurfacecrack
823
Plots of the crack growth rate for Group D are presented in Figs 8 and 9. In a similar way, by performing statistical tests on the sample data in Groups E (/I = 0’) and F (B = 30”) and Groups E and G (/I = 45”), respectively, a conclusion similar to the case of uniaxial loading can be drawn. Hence, under the condition of P,:P,, = 0.5, by making use of the data in Group H listed in Table 2, the crack growth rate in the thickness direction can be expressed as: da/dN = 1.622 x 10-“(AK,,)4~6*6(1- K,,,/K,,, ,,).
(10)
Under the condition of P,:P,, = 0.5, by making use of the data in Group H listed in Table 3, the crack growth rate in the length direction is: dc,/dN = 8.568 x 10-‘“(A$,)3~24’.
(11)
In eqs (8)-(1 l), the proper crack inclination angle ranges from 0” to 45”, and the projected size in the plane perpendicular to the principal stress is chosen as the crack size needed in the calculation. The value of Ku1/Ku, _ is taken from Fig. 7 according to the biaxial loading ratio and the crack inclination angle. Plots of the crack growth rate for Group H are presented in Figs 10 and 11. Under the condition of P,:P,, = 1, the crack plane coincides with the plane perpendicular to the principal stress. Therefore, no matter what the value of the crack inclination angle is, by making use of the data in Group I listed in Tables 2 and 3, the crack growth rate in the thickness direction is: da/dN = 2.350 x 10-“(AK,,)4,4’4
(12)
and the crack growth rate in the length direction is: dc/dN = 7.044 x 10-‘“(AK,,)3@3.
(13)
Plots of the crack growth rate for Group I are presented in Figs 12 and 13.
FSN.l - SN.8
l-
-
SN. 8
SN.l-
J !L 1
20
I
AKla
30
40
( MPa q/m 1
Fig. 8. Crack growth rate for specimens with P,: P, = 0 and /I = O-450, in the thickness direction.
Jc___10
20
30
40
AKk(MPaVm)
Fig. 9. Crack growth rate for specimens with P,: P, = 0 and /3 = O-45”, in the length direction.
824
Z. J. ZENG and S. H. DA1
;
BN.2 - BN.8
Iti/ 10
AKla ( MPa grn 1
20
30
40
1
AKlc ( MPa Jm
Fig. 10. Crack growth rate for specimens with P,: P,. = 0.5 Fig. 11. Crack growth rate for specimens with P,:P,, = 0.5 and ELI = O-45”, in the thickness direction. and j? = O-45”. in the length direction,
BN.9 - BN.12
t: _ BN.9 - BN.12
20
AKia ( MPaJ m 1 Fig. 12. Crack growth rate for specimens with P,: P, = 1 and j? = O-45”, in the thickness direction.
AKlc ( MPa Jm
30
40
)
Fig. 13. Crack growth rate for specimens with P,r:Py =I and j = O-45”, in the length direction.
Crack growth rate of an inclined surface crack
825
CONCLUSIONS An experimental and theoretical investigation of surface crack growth in 16MnR steel specimens with crack inclination angle /I = O”, 30” and 45”, and under the condition of biaxial loading ratio P,:Py= 0,0.5 and 1, is reported in this paper. The main conclusions are summarized below. 1. A surface-cracked cruciform specimen for fatigue experiments under biaxial loading was designed. An SIF equation for a surface-cracked cruciform specimen loaded biaxially was calibrated using the finite element method and the line-spring boundary element method. 2. When the biaxial loading ratio was fixed (P,: P,, = 0 or OS), the number of load cycles expended in penetrating the wall of the specimen with crack inclination angle /I = 45” was much more than that with crack inclination angle /I = 30”. When the crack inclination angle was fixed (B = 30” or 45”), the number of load cycles expended in penetrating the wall of the specimen under the condition of P,:P, = 0 was much more than that under the condition of P,:P,, = 0.5. Therefore, if the fatigue life of an inclined surface crack obtained from a test under uniaxial loading is applied indiscriminately, like the fatigue life of an engineering structural comnponent having an inclined surface crack under biaxial tension, the wrong conclusion on the unsafe side may be drawn. 3. The crack growth rate is affected by crack driving forces. In the cases of different crack inclination angles and different biaxial loading ratios, increase/decrease of the mode I crack driving force, and decrease/increase of the mode III crack driving force lead to increase/decrease of the crack growth rate. In order to estimate the crack growth rate correctly, the projection method, in which the surface crack is projected to the plane perpendicular to the principal stress, is used to deal with the problem of an inclined surface crack; the modified Paris equation is used to describe the fatigue propagation in the depth direction. The modified Paris equation contains the normalized mode III SIF, which is dependent only on the crack inclination angle and the biaxial loading ratio. 4. The experimental data were sorted out group by group and classified in the light of the crack inclination angle and the biaxial loading ratio. Linear regression analyses were then carried out. By performing the statistical tests on samples, we obtained the conclusion at the 95% confidence level that under the condition of fixed biaxial loading ratio, there is no significant difference between the Paris equation exponent values for the populations in the thickness direction, and there is also no significant difference between the Paris equation coefficient values for the populations in the thickness direction, in spite of the difference of crack inclination angle. The situation is the same in the length direction. Consequently, the crack growth rate equations for inclined surface cracks in the thickness and length directions are obtained. REFERENCES [l] S. T. Tu and S. H. Dai, The fatigue growth of inclined surface cracks under biaxial loading. Znt. J. Press. Vess. Piping 41, 141-157 (1990). [2] S. Chen and Z. Cui, Surface crack growth behavior under tensile cyclic loading. Znt. J. Farigue 10, 43-47. [3] PVRC, The long-range plane for pressure vessel research (8th edn). WRC Bulletin, No. 327, pp. 9-14 (1987). [4] J. C. Newman, Jr. and I. S. Raju, An empirical stress-intensity factor equation for the surface crack. Engng Frocrure Mech. 15, 185-192 (1981). [5] Z. J. Zeng and S. H. Dai, Line-spring boundary element method for a surface cracked plate. Engng Fracture Mech. 36, 855-858 (1990). [6] Z. J. Zeng, A theoretical and experimental investigation of fatigue behavior for surface cracks with shallow-long shape. Ph.D. dissertation, Nanjing Institute of Chemical Technology (1990) [in Chinese]. [7] Z. J. Zeng and S. H. Dai, Stress intensity factors for an inclined surface crack under biaxial stress state. Compur. Structures (submitted). [8] S. Chen, Z. Lu, D. Zhang and Z. Cui, Experimental investigation of surface crack growth behavior under tensile cyclic loading. Mech. Practice 9(3), 34-39 (1987) [in Chinese]. [9] D. Dong, J. Yang, M. Su and G. Gu, Statistical Methods in Experimental Research. Chinese Measure Press, Beijing (1987) [in Chinese]. (Received 26 September 1991)