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The effects of simultaneous overload and spot heating on crack growth retardation in fatigue
0013-7944/93 $6.00+ 0.00 0 1993BergamonPress Ltd.
Engineering Fracture Mechanics Vol. 44, No. 4, pp. 561-572, 1993
Printed in Great Britain.
THE EFFECTS OF SIMULTANEOUS OVERLOAD AND SPOT HEATING ON CRACK GROWTH RETARDATION IN FATIGUE B. D. CHEN,t J. R. GRIFFITHSI
and Y. C. LAM?
Departments of tMechanica1 and IMaterials Engineering, Monash University, Victoria 3168, Australia Abstract-A new method for the retardation of fatigue crack growth, spot heating at the crack tip combined with an overload, is described in this paper. A theoretical analysis indicates that the plastic zone size and the crack tip residual compressive stress region are larger using this method than is the case for simple overload. It is therefore suggested that fatigue crack retardation will be greater for the new method than for a simple tensile overload. Experimental investigations with single-edge cracked steel specimens were carried out. For the conditions investigated, crack arrests were observed after the application of the method, and the retardation effect was, indeed, far greater than that obtained with simple overload alone.
INTRODUCTION A GROWING fatigue crack can be arrested or retarded by introducing suitable compressive residual stresses to the structure. Such stresses have been achieved using the methods of overload [l], localized or spot heating near the crack tip [2] and, more recently, heating of the whole cross-section with a load equal to the maximum load experienced in service [3]. However, there are some limitations to the practical use of these methods. For the overload procedure, the magnitude of an overload is generally limited since it must be below the collapse or failure load of the structures. On the other hand, the spot heating procedure is difficult to control as the precise heating position has to be determined; a wrongly positioned spot might cause crack growth acceleration. Finally, the procedure of heating the entire cross-section whilst the component is under maximum fatigue load could lead to general yielding as the yield stress in the section decreases when the temperature increases. In the present investigation, a new procedure is proposed whereby spot heating is applied in conjunction with an overload. The proposal has two merits: first, it avoids the difficulties associated with previous methods and, second, the retardation of fatigue crack growth is particularly marked. Furthermore, the method can also be applied to notches or other stress concentrators-it is not limited to cracked structures. In the proposed method spot heating is applied to a component at the same time as an overload is applied. The overload is removed after the component has cooled down to its original temperature. Figure 1 demonstrates the application of a circular hot spot applied to a single-edge cracked mild steel specimen. The hot spot must be large enough to cover the crack tip and its monotonic plastic zone but, otherwise, its location is not critical. An overload can be safely applied using this procedure since the fracture toughness increases with an increase in temperature (especially for steels). In addition, the chances of general yielding are negligible as most of the load will be withstood by the colder part of the structure beyond the hot spot. CRACK GROWTH
RETARDATION
Consider a structure which contains a crack which is growing under the influence of a fatigue load. When a static overload is applied to the component a larger monotonic plastic zone is formed; thus a larger reversed plastic zone will be generated on unloading. When spot heating is applied while the overload is maintained, an even larger monotonic plastic zone will be produced. When unloading after the component has cooled down, therefore, a larger reversed plastic zone will be produced compared with that caused by an overload. The distributions of residual stresses using the proposed procedure and other procedures are illustrated in Fig. 2 (neglecting any residual tensile stresses induced by spot heating). Curves A, B, C and D are the distributions of stresses 567
B. D. CHEN et a!.
?
I e
v
Y
”
I t
t
Applied
t
load
Fig. 1. Illustration of the application of a hot spot to the crack tip.
on loading and curves 1, 2, 3 and 4 are the residual stress distributions on unloading for the following four cases: constant amplitude loading, after applying an overload, after spot heating carried out at maximum load and after spot heating carried out in conjunction with an overload, respectively.
w
Distance ahead of the Crack Tip
Fig. 2. Streaa distributions using various procedures. d,. d;, d:, d: are monotonic plastic zones and 4, di, d:, dp arc reverse plastic zones under constant amplitude loading, after an overload, aRer spot heating with maxhnum fatigue load and after spot heating with an overload respectively.
Effect of spot heating and overload on crack growth
569
The fact that the yield stress usually decreases with increasing temperature means that spot heating applied in conjunction with an overload produces the largest monotonic and reversed plastic zones. As residual compressive stress retards fatigue crack growth, the increase in the compressive region (i.e. the reversed plastic zone) will cause increased retardation. In addition, the larger monotonic plastic zone will increase the amount of crack closure as it passes to the wake of the crack. This will also increase retardation. With spot heating, it is possible to cause residual tensile stress on cooling if the temperature is high enough for plastic flow to occur in the spot. Thus, the beneficial residual compressive stress as analysed above would be counteracted. However, for this method, it is not necessary for the applied- temperature to be high; a temperature of 300°C is adequate and, even for low strength steels, this will not cause a significant residual tensile stress. EXPERIMENTAL
INVESTIGATIONS
Specimens AS 1444/X1320H plain carbon-manganese boiler plate steel was used for this investigation. Tensile and fatigue test specimens were cut from the plate such that the longitudinal axis of the specimen was aligned with the rolling direction. The fatigue specimens were of the single-edge-notch type and were 75 mm wide and 3 mm thick. All the specimens were annealed for 1: hr after machining to relieve the residual stresses due to rolling and machining. Tensile tests Tensile tests were carried out at temperatures from room temperature to 500°C. Figure 3 shows the dependence of the yield stress and tensile strength on temperature; 300°C was chosen as the spot heating temperature as the yield stress is significantly lowered with negligible decrease in the tensile strength. Moreover, as explained above, at 300°C it is not expected that significant plastic yielding will occur during spot heating. Spot heating Spot heating was performed in the way described by Formby and Griffiths [4]. Two copper cylinders each of 30 mm diameter were mounted in a jig so that they could be pressed against the fatigue sample from opposite faces whilst it was mounted in the fatigue machine. The copper blocks were wound with electrical resistance heating wires and their temperatures measured with thermocouples inserted into drilled holes. During spot heating the cylinders were pressed against the sample with a hydraulic ram to a force of 5.9 kN (a contact pressure of 8.3 MPa) for l~min and then removed.
Fig. 3. Tensile strength and yield stress versus temperature.
B. D. CHEN et
al.
Stress Intensity Factor Ronge nK(MPam”‘) Fig. 4. Baseline fatigue data.
Fatigue tests
A series of fatigue tests on single-edge cracked specimens was conducted to test the theoretical predictions of retardation. The tests were as follows. (1) Baseline data. A specimen was tested under a constant amplitude load range of OS-24 kN. The result is shown in Fig. 4, where the line of best fit for the data can be represented by the equation: &
= 3.1 x 10-9(AK)3
where the units are mm/cycle and MPa mli2. (2) Spot heating alone. A specimen was tested at cyclic constant stress intensity factor range, AK, of 20 MPa ml/’ and load ratio, R = 0.02. (The crack length was monitored with a microscope and the load conditions were adjusted for every 0.5 mm increment in crack length to maintain constant AK and R.) The test was interrupted after the crack had grown by 7 mm and a semi-circular hot spot (Fig. 1) was applied whilst the sample was held at the minimum load. The fatigue test was then continued and, as shown in Fig. 5, a slight decrease in crack growth rate was observed, with a minimum crack growth rate of approximately 1.3 x 10e5 mm/cycle as compared with the baseline crack growth rate of 2.5 x lo-‘mm/cycle. No retardation had been expected for this test since compressive residual stresses were not introduced in the vicinity of the crack tip. A possible explanation of the slight reduction in the growth rate is the formation of an oxide layer on the crack surface, which might cause a decrease in the effective stress intensity factor range by enhancing crack closure. (3) Overload. Two tests were performed to examine the effect on crack growth of applying an overload alone. One specimen was tested with AK = 18 MPa ml/’ and with an overload ratio R = 1.25 (0 is defined as the ratio of the maximum overload stress intensity factor to the maximum steady state stress intensity factor). A second specimen was tested at AK = 20 MPa ml/’ and n = 1.5. The value of R was close to zero for both tests. The results are given in Fig. 6. The crack growth rates decreased from the baseline growth rates of 1.5 x 10m5 and 2.3 x 10e5 mm/cycle for the two experiments to a minimum of 7 x low6 mm/cycle. (4) Overload with spot heating at 300°C. Two tests were performed at AK = 20 MPa m’12.For one specimen spot heating was applied at Q = 1 (i.e. no overload) and for the other at S2= 1.25. The results, presented in Figs 7 and 8, show that for both samples the crack growth rate decreased and that, finally, crack arrest occurred (crack arrest was defined as a growth rate of less than 5 x lo-’ mm/cycle). The crack only resumed growing when AK was increased. For the sample
571
Effect of spot heating and overload on crack growth
lo-‘k
$A1
I
I
spot heating applied
1
-LI;_-----;-7*--T-r-r .+.. . :#
q10-‘r 8 B er 5 lo3 -
6 -5 e U
lo-’ 5
I I 10 15 Crack Length a (mm)
.
Fig. 5. ElTectof spot heating alone on crack growth. (--) indicates the crack growth rates predicted based on constant amplitude loading [eq. (l)].
“,,-i , 5
, 1
10 15 Crack Length a(mm)
Fig. 6. Effect of single overload alone on crack growth. (- - -) and (- -) indicate the growth rates predicted based on constant amplitude loading [eq. (l)] for AK = 18 MPa rntn and AK = 20 MPa rn’o respectively.
with R = 1 (Fig. 7) the crack only continued growing when AK was increased to 22 MPa ml/‘. For the case of R = 1.25 (Fig. 8), it was necessary to increase AK to 24 MPa m1’2;in fact, for this sample, AK was finally increased to 26 MPa ml” as shown in the figure. DISCUSSION It can be seen in Fig. 6 that after a single overload with no spot heating the minimum growth rate only occurred after the crack had grown some way into the plastic zones. This behaviour is consistent with previous reports (see, e.g., refs [5-8]). For an overload with spot heating a similar phenomenon was noted: the minimum growth rate, in this case complete arrest, did not occur immediately after treatment (see Figs 7 and 8). Indeed, Fig. 8 shows that after restarting the
Crork Length a (mm) Fig. 7. Et&t of spot heating with maximum fatigue load (n = I) on crack growth. (--) indicates the growth rates predicted based on constant amplitude loading [eq. (l)]. (1) indicates crack arrest occurmd for a given AK.
Fig. 8. E&t of spot heating with an overload (Q = 1.25) on crack growth. (--) indicates the crack growth rates predicted based on constant amplitude loading [q. (l)]. (1) indicates crack arrest occurred for a given AK.
572
B. D. CHEN et al.
crack after the first arrest, by raising AK from 20 to 22 MPa m112,a second arrest occcurred. This suggests that the full retardation effect is only felt after the crack has grown by some distance. Of particular interest is the result for 52= 1 (in which heating is applied at the maximum load). It is clear that the thermal effect is similar to that of an overload-and, indeed, is more potent than an overload for the particular conditions examined. In this sense, the technique proposed in this paper has successfully combined the effect of a “thermal overload” with a mechanical overload. However, we are not able, at present, to compare the relative ~~tudes of the effects for the two processes, As a pre~~na~ indication we use the proposal of Williams and Lam [9]that the severity of crack arrest can be measured by the increase in AK necessary for re-initiation of crack growth. With this criterion, when spot heating is applied with no overload (a = I), AK had to be increased from 20 to 22 MPa m”2 (i.e. by lo%), whereas for Sz= 1.25 an increase from 20 to 24 MPa rn1j2 (i.e. 20%) was required. This shows that spot heating with an overload is more effective than spot heating with no overload. CONCLUSIONS
The results show that for the testing conditions considered, either spot heating alone at low temperature (3OO’C)or overloads alone with small overload ratio (Q = 1.25 or 1.5) caused negligible retardation. However, when the two procedures were combined, i.e. spot heating applied with an overload, sign&ant retardation even to the extent of crack arrest could be achieved even with a low heating temperature and overload ratio. Acknowfedgement-The support of a Monash Graduate Scholarship for B. D. Chen is Statefully acknowledge&
REFERENCES J. Schijve and D. Broek, Aircruft Engng 34, 314 (1962). J. D. Harrison, Br. WeldingJl. 12, 258 (1965). Y, C. Lam and J. R. G&ii&s, Theor. appl. &cture Mech. 14, 37 (1990). C. L. Formby and J. R. GriiBths, Proc. Int. Conf. on ResidualStressesin WehiedConstructionsat& their E&c&, p. 359. The Welding Institute, London (1977). [5] J. C. M&i&n and R. W. Hertzberg, ASTM STP 434,89 (1968). [6] D. M. Corbly and P. F. Packman, &tgttg Fructttre Mech. S, 479 (1973). [A C. Bathiaaand M. Vancon, Engng Fracture Mech, 10,409(1978). [S] N. Ranganathan, J. Petit and 8. V. Bouchet, .&gng Fracture Mech. $1, 775 (1979). [9] J. F. Williams and Y. C. Lam, Theor. appl Fracture Mech. 6, 21 (1986). (lo] K. N. Raju, V. Ningiah and B. V. S. Rao, Int. J. Fracture Med. 8, 99 (1979). [I] [2] [3] [4]
(Received 28 February 1992)
Engineering Fracture Mechanics Vol. 44, No. 4, pp. 561-572, 1993
Printed in Great Britain.
THE EFFECTS OF SIMULTANEOUS OVERLOAD AND SPOT HEATING ON CRACK GROWTH RETARDATION IN FATIGUE B. D. CHEN,t J. R. GRIFFITHSI
and Y. C. LAM?
Departments of tMechanica1 and IMaterials Engineering, Monash University, Victoria 3168, Australia Abstract-A new method for the retardation of fatigue crack growth, spot heating at the crack tip combined with an overload, is described in this paper. A theoretical analysis indicates that the plastic zone size and the crack tip residual compressive stress region are larger using this method than is the case for simple overload. It is therefore suggested that fatigue crack retardation will be greater for the new method than for a simple tensile overload. Experimental investigations with single-edge cracked steel specimens were carried out. For the conditions investigated, crack arrests were observed after the application of the method, and the retardation effect was, indeed, far greater than that obtained with simple overload alone.
INTRODUCTION A GROWING fatigue crack can be arrested or retarded by introducing suitable compressive residual stresses to the structure. Such stresses have been achieved using the methods of overload [l], localized or spot heating near the crack tip [2] and, more recently, heating of the whole cross-section with a load equal to the maximum load experienced in service [3]. However, there are some limitations to the practical use of these methods. For the overload procedure, the magnitude of an overload is generally limited since it must be below the collapse or failure load of the structures. On the other hand, the spot heating procedure is difficult to control as the precise heating position has to be determined; a wrongly positioned spot might cause crack growth acceleration. Finally, the procedure of heating the entire cross-section whilst the component is under maximum fatigue load could lead to general yielding as the yield stress in the section decreases when the temperature increases. In the present investigation, a new procedure is proposed whereby spot heating is applied in conjunction with an overload. The proposal has two merits: first, it avoids the difficulties associated with previous methods and, second, the retardation of fatigue crack growth is particularly marked. Furthermore, the method can also be applied to notches or other stress concentrators-it is not limited to cracked structures. In the proposed method spot heating is applied to a component at the same time as an overload is applied. The overload is removed after the component has cooled down to its original temperature. Figure 1 demonstrates the application of a circular hot spot applied to a single-edge cracked mild steel specimen. The hot spot must be large enough to cover the crack tip and its monotonic plastic zone but, otherwise, its location is not critical. An overload can be safely applied using this procedure since the fracture toughness increases with an increase in temperature (especially for steels). In addition, the chances of general yielding are negligible as most of the load will be withstood by the colder part of the structure beyond the hot spot. CRACK GROWTH
RETARDATION
Consider a structure which contains a crack which is growing under the influence of a fatigue load. When a static overload is applied to the component a larger monotonic plastic zone is formed; thus a larger reversed plastic zone will be generated on unloading. When spot heating is applied while the overload is maintained, an even larger monotonic plastic zone will be produced. When unloading after the component has cooled down, therefore, a larger reversed plastic zone will be produced compared with that caused by an overload. The distributions of residual stresses using the proposed procedure and other procedures are illustrated in Fig. 2 (neglecting any residual tensile stresses induced by spot heating). Curves A, B, C and D are the distributions of stresses 567
B. D. CHEN et a!.
?
I e
v
Y
”
I t
t
Applied
t
load
Fig. 1. Illustration of the application of a hot spot to the crack tip.
on loading and curves 1, 2, 3 and 4 are the residual stress distributions on unloading for the following four cases: constant amplitude loading, after applying an overload, after spot heating carried out at maximum load and after spot heating carried out in conjunction with an overload, respectively.
w
Distance ahead of the Crack Tip
Fig. 2. Streaa distributions using various procedures. d,. d;, d:, d: are monotonic plastic zones and 4, di, d:, dp arc reverse plastic zones under constant amplitude loading, after an overload, aRer spot heating with maxhnum fatigue load and after spot heating with an overload respectively.
Effect of spot heating and overload on crack growth
569
The fact that the yield stress usually decreases with increasing temperature means that spot heating applied in conjunction with an overload produces the largest monotonic and reversed plastic zones. As residual compressive stress retards fatigue crack growth, the increase in the compressive region (i.e. the reversed plastic zone) will cause increased retardation. In addition, the larger monotonic plastic zone will increase the amount of crack closure as it passes to the wake of the crack. This will also increase retardation. With spot heating, it is possible to cause residual tensile stress on cooling if the temperature is high enough for plastic flow to occur in the spot. Thus, the beneficial residual compressive stress as analysed above would be counteracted. However, for this method, it is not necessary for the applied- temperature to be high; a temperature of 300°C is adequate and, even for low strength steels, this will not cause a significant residual tensile stress. EXPERIMENTAL
INVESTIGATIONS
Specimens AS 1444/X1320H plain carbon-manganese boiler plate steel was used for this investigation. Tensile and fatigue test specimens were cut from the plate such that the longitudinal axis of the specimen was aligned with the rolling direction. The fatigue specimens were of the single-edge-notch type and were 75 mm wide and 3 mm thick. All the specimens were annealed for 1: hr after machining to relieve the residual stresses due to rolling and machining. Tensile tests Tensile tests were carried out at temperatures from room temperature to 500°C. Figure 3 shows the dependence of the yield stress and tensile strength on temperature; 300°C was chosen as the spot heating temperature as the yield stress is significantly lowered with negligible decrease in the tensile strength. Moreover, as explained above, at 300°C it is not expected that significant plastic yielding will occur during spot heating. Spot heating Spot heating was performed in the way described by Formby and Griffiths [4]. Two copper cylinders each of 30 mm diameter were mounted in a jig so that they could be pressed against the fatigue sample from opposite faces whilst it was mounted in the fatigue machine. The copper blocks were wound with electrical resistance heating wires and their temperatures measured with thermocouples inserted into drilled holes. During spot heating the cylinders were pressed against the sample with a hydraulic ram to a force of 5.9 kN (a contact pressure of 8.3 MPa) for l~min and then removed.
Fig. 3. Tensile strength and yield stress versus temperature.
B. D. CHEN et
al.
Stress Intensity Factor Ronge nK(MPam”‘) Fig. 4. Baseline fatigue data.
Fatigue tests
A series of fatigue tests on single-edge cracked specimens was conducted to test the theoretical predictions of retardation. The tests were as follows. (1) Baseline data. A specimen was tested under a constant amplitude load range of OS-24 kN. The result is shown in Fig. 4, where the line of best fit for the data can be represented by the equation: &
= 3.1 x 10-9(AK)3
where the units are mm/cycle and MPa mli2. (2) Spot heating alone. A specimen was tested at cyclic constant stress intensity factor range, AK, of 20 MPa ml/’ and load ratio, R = 0.02. (The crack length was monitored with a microscope and the load conditions were adjusted for every 0.5 mm increment in crack length to maintain constant AK and R.) The test was interrupted after the crack had grown by 7 mm and a semi-circular hot spot (Fig. 1) was applied whilst the sample was held at the minimum load. The fatigue test was then continued and, as shown in Fig. 5, a slight decrease in crack growth rate was observed, with a minimum crack growth rate of approximately 1.3 x 10e5 mm/cycle as compared with the baseline crack growth rate of 2.5 x lo-‘mm/cycle. No retardation had been expected for this test since compressive residual stresses were not introduced in the vicinity of the crack tip. A possible explanation of the slight reduction in the growth rate is the formation of an oxide layer on the crack surface, which might cause a decrease in the effective stress intensity factor range by enhancing crack closure. (3) Overload. Two tests were performed to examine the effect on crack growth of applying an overload alone. One specimen was tested with AK = 18 MPa ml/’ and with an overload ratio R = 1.25 (0 is defined as the ratio of the maximum overload stress intensity factor to the maximum steady state stress intensity factor). A second specimen was tested at AK = 20 MPa ml/’ and n = 1.5. The value of R was close to zero for both tests. The results are given in Fig. 6. The crack growth rates decreased from the baseline growth rates of 1.5 x 10m5 and 2.3 x 10e5 mm/cycle for the two experiments to a minimum of 7 x low6 mm/cycle. (4) Overload with spot heating at 300°C. Two tests were performed at AK = 20 MPa m’12.For one specimen spot heating was applied at Q = 1 (i.e. no overload) and for the other at S2= 1.25. The results, presented in Figs 7 and 8, show that for both samples the crack growth rate decreased and that, finally, crack arrest occurred (crack arrest was defined as a growth rate of less than 5 x lo-’ mm/cycle). The crack only resumed growing when AK was increased. For the sample
571
Effect of spot heating and overload on crack growth
lo-‘k
$A1
I
I
spot heating applied
1
-LI;_-----;-7*--T-r-r .+.. . :#
q10-‘r 8 B er 5 lo3 -
6 -5 e U
lo-’ 5
I I 10 15 Crack Length a (mm)
.
Fig. 5. ElTectof spot heating alone on crack growth. (--) indicates the crack growth rates predicted based on constant amplitude loading [eq. (l)].
“,,-i , 5
, 1
10 15 Crack Length a(mm)
Fig. 6. Effect of single overload alone on crack growth. (- - -) and (- -) indicate the growth rates predicted based on constant amplitude loading [eq. (l)] for AK = 18 MPa rntn and AK = 20 MPa rn’o respectively.
with R = 1 (Fig. 7) the crack only continued growing when AK was increased to 22 MPa ml/‘. For the case of R = 1.25 (Fig. 8), it was necessary to increase AK to 24 MPa m1’2;in fact, for this sample, AK was finally increased to 26 MPa ml” as shown in the figure. DISCUSSION It can be seen in Fig. 6 that after a single overload with no spot heating the minimum growth rate only occurred after the crack had grown some way into the plastic zones. This behaviour is consistent with previous reports (see, e.g., refs [5-8]). For an overload with spot heating a similar phenomenon was noted: the minimum growth rate, in this case complete arrest, did not occur immediately after treatment (see Figs 7 and 8). Indeed, Fig. 8 shows that after restarting the
Crork Length a (mm) Fig. 7. Et&t of spot heating with maximum fatigue load (n = I) on crack growth. (--) indicates the growth rates predicted based on constant amplitude loading [eq. (l)]. (1) indicates crack arrest occurmd for a given AK.
Fig. 8. E&t of spot heating with an overload (Q = 1.25) on crack growth. (--) indicates the crack growth rates predicted based on constant amplitude loading [q. (l)]. (1) indicates crack arrest occurred for a given AK.
572
B. D. CHEN et al.
crack after the first arrest, by raising AK from 20 to 22 MPa m112,a second arrest occcurred. This suggests that the full retardation effect is only felt after the crack has grown by some distance. Of particular interest is the result for 52= 1 (in which heating is applied at the maximum load). It is clear that the thermal effect is similar to that of an overload-and, indeed, is more potent than an overload for the particular conditions examined. In this sense, the technique proposed in this paper has successfully combined the effect of a “thermal overload” with a mechanical overload. However, we are not able, at present, to compare the relative ~~tudes of the effects for the two processes, As a pre~~na~ indication we use the proposal of Williams and Lam [9]that the severity of crack arrest can be measured by the increase in AK necessary for re-initiation of crack growth. With this criterion, when spot heating is applied with no overload (a = I), AK had to be increased from 20 to 22 MPa m”2 (i.e. by lo%), whereas for Sz= 1.25 an increase from 20 to 24 MPa rn1j2 (i.e. 20%) was required. This shows that spot heating with an overload is more effective than spot heating with no overload. CONCLUSIONS
The results show that for the testing conditions considered, either spot heating alone at low temperature (3OO’C)or overloads alone with small overload ratio (Q = 1.25 or 1.5) caused negligible retardation. However, when the two procedures were combined, i.e. spot heating applied with an overload, sign&ant retardation even to the extent of crack arrest could be achieved even with a low heating temperature and overload ratio. Acknowfedgement-The support of a Monash Graduate Scholarship for B. D. Chen is Statefully acknowledge&
REFERENCES J. Schijve and D. Broek, Aircruft Engng 34, 314 (1962). J. D. Harrison, Br. WeldingJl. 12, 258 (1965). Y, C. Lam and J. R. G&ii&s, Theor. appl. &cture Mech. 14, 37 (1990). C. L. Formby and J. R. GriiBths, Proc. Int. Conf. on ResidualStressesin WehiedConstructionsat& their E&c&, p. 359. The Welding Institute, London (1977). [5] J. C. M&i&n and R. W. Hertzberg, ASTM STP 434,89 (1968). [6] D. M. Corbly and P. F. Packman, &tgttg Fructttre Mech. S, 479 (1973). [A C. Bathiaaand M. Vancon, Engng Fracture Mech, 10,409(1978). [S] N. Ranganathan, J. Petit and 8. V. Bouchet, .&gng Fracture Mech. $1, 775 (1979). [9] J. F. Williams and Y. C. Lam, Theor. appl Fracture Mech. 6, 21 (1986). (lo] K. N. Raju, V. Ningiah and B. V. S. Rao, Int. J. Fracture Med. 8, 99 (1979). [I] [2] [3] [4]
(Received 28 February 1992)