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Crack instability in a multiple-cracked system
Int. J. Pres. Ves. & Piping 20 (1985) 55-64
Crack Instability in a Multiple-cracked System E. Smith Joint Manchester University/UMIST Department of Metallurgy and Materials Science, Grosvenor Street, Manchester MI 7HS, Great Britain (Received: 31 October, 1984)
ABSTRACT The paper analyses the behaviour of a solid containing two symmetrically situated deep cracks, and with tension of the small remaining ligament. The results are used to support the view that when a piping system fabricated from a ductile material, such as 304 stainless steel, contains more than one crack, then the presence of other cracks has an insignificant effect on the condition for instability of growth of a given crack.
INTRODUCTION Against the background of the technological problem of the cracking of 304 stainless steel Boiling Water Reactor piping, the theoretical analysis of Tada et al.l showed that circumferential growth of a through-wall crack should be stable for the pipe run lengths that are typical of a reactor coolant system. Their approach, based on the tearing modulus procedure developed by Paris and co-workers, 2 assumed the material to be elastic-perfectly plastic, with the plastic deformation being confined to the cracked cross-section which is fully yielded. This cross-section thus behaves like a plastic hinge with a region of tension immediately adjacent to the crack tips and above the neutral axis, and a region of compression below the neutral axis. It has now become apparent that if a crack forms, by a stress corrosion mechanism, in one part of a piping system, then there is a real possibility that there will be similar cracks in other parts of 55 Int. J. Pres. Ves. & Piping 0308-0161/85/$03.30 ~ Elsevier Applied Science Publishers
Ltd, England, 1985. Printed in Great Britain
56
E. Smith
the system. With this situation in mind, an earlier analysis by the author 3 has extended the approach o f T a d a et al. 1 and shown that, when there are several circumferential cracks together with their associated plastic hinges in a stainless steel piping system, then the tendency towards instability of circumferential growth of any given crack is no greater than for the case where the given crack exists in isolation. Instability was considered from the basis that though there is a moment reduction at a cross-section containing a non-growing crack, as another crack grows under a fixed displacement applied to the system, the plastic rotation recovery at a nongrowing crack cross-section can be ignored in comparison with the elastic recovery of the system. This paper focuses on this particular point, and uses the results from an analysis of the behaviour of a solid containing two symmetrically situated deep cracks and with tension of the small remaining ligament, to show that if the tension is due to a load applied at a point away from the ligament and there is a small load reduction AP below the general yield load, then the elastic displacement recovery is proportional to AP, and the plastic displacement recovery is proportional to (AP) 3. It is thereby demonstrated, for a material with a high resistance to crack growth such as 304 stainless steel, that, when there are several such cracks and their associated fully plastic ligaments, the tendency towards fracture instability of a given ligament is essentially unaffected by the presence of the other cracks. This result is therefore supportive, and indeed important back-up, to the conclusion reached in Ref. 3.
T H E O R E T I C A L ANALYSIS Figure 1 shows the model representing the plane strain deformation of a solid containing two symmetrically situated deep cracks, and with tension of the small remaining ligament which is fully yielded; the ligament width and thickness are respectively 2L and B, while the solid width is 2h. The displacement is A at a great distance D/2 (i.e. D >>L) from the ligament, while P is the magnitude of the load associated with this displacement. This model simulates the behaviour of a growing crack, but, in order to assess the effect of a non-growing crack on this growth behaviour, it is supposed that the system is connected with another similar system containing a non-growing crack. In other words the overall system is subjected to a total displacement Av and the displacement of the second
Crack instability in a multiple-cracked system
57
P,A
<
<
Y
Y
Y
Y 2L--
2h--
:~
~,-
Fig. 1. The plane strain deformation of a solid containing two symmetrically situated deep cracks. A displacement A is applied to the solid at a great distance from the ligament, while P is the corresponding load.
system is A, with AT -- A, + A. The stability of the system will now be assessed for a fixed value of the total displacement AT (see Fig. 2). The analysis is based on the Dugdale-Bilby-Cottrell-Swinden (DBCS) representation of plastic deformation, 4'5 whereby yield is confined entirely to the planes containing the cracks. (In this respect the model is analogous to that used by Tada et al. 1 and also to that used by the author in Ref. 3, where plastic deformation is confined to the cracked crosssection of a pipe.) The tensile stress within the plastic zones between the crack tips is assumed to have the value Y which is representative of the material's tensile yield stress. For the particular configuration in Fig. l, the magnitude of the J-integral is given by the relation (see Appendix eqns (A. 7) and (A. 8)): J = JEL + JPL + JGY =
4(1 -- v2)y2L 4(1 -- v2)y2L (2In 1rE -~ rtE 2 - 1) + YAov
(1)
where E is Young's modulus, and v is Poisson's ratio. The first term is the elastic component JEL due to the application of the limit load P = 2BLY, presuming that there is no plasticity; the second term is the additional
E. Smith
58
P, A T
D
Non-growing cracks
D
Growing cracks
L
I~
J
2h
"-1
Fig. 2. A composite system consisting of a section containing growing cracks, and another section with non-growing cracks. The ligaments in both sections are assumed to
be fully yielded.
(plastic) contribution JPL when the general yield state is just attained; while the third term J~v is due to the post-general yield plastic deformation; A~v is the post-general yield plastic contribution to the displacement A. The total load point displacement AT = A, + A, assumed to remain fixed during crack growth, is given by the expression (see Appendix eqns (A. 15) and (A. 16)): A = A, + AEL,NCR+ AEL,Ca + ApL + AGy
= A, 4
(1-vg)LDY 8(1-_v2)LYln(2h ~ hE ¢" nE \nL/ -~
8(1 -- vz)YL
nE
(1 - ln2) + A~v
(2)
where the first term A, is the displacement associated with the section containing the non-growing cracks; the second term AEL,NCRis the elastic contribution to A assuming that there is no crack and no plastic deformation; the third term AEL,CR is the elastic contribution to A due to the crack's presence again assuming that there is no plastic deformation; the fourth term ApL is the additional (plastic) contribution to A when the
59
Crack instability in a multiple-cracked system
general yield state is just attained; while the fifth term AGv has already been defined. The original Paris-Tada-Zahoor=-Ernst approach 2 is based on the assumption that the J-crack growth resistance curve is specific for a given material, with the tearing modulus TMaT being related to the slope of the J-crack growth curve: E dJ (3)
TMAT -- y2 dc
where 6c is an increment of crack length. Instability under a displacement control condition (i.e. d AT/dC = 0) is presumed to occur when TApP > TMAT, TApp being related to J by an expression similar to eqn (3), and J being given by eqn (1). It follows from eqns (1), (2) and (3), noting that fc = - ilL, that the condition for crack growth instability is E/dA,
dAELNCR
dAELCR
d;
di
dApLX
E //dJEL
dA~_~
+ (4)
after elimination of AGv. Of the various terms on the left-hand side of eqn (4), eqns (1) and (2) show that with the possible exception of the terms involving A. and AEL,NCa,i.e. the displacement associated with the nongrowing crack section and the elastic displacement in the crack's absence for the growing crack section, all the terms will be ~ 10 if realistic values are assumed for h and L. Consequently if the material has a high crack growth resistance, and 200 has been suggested1 as being an appropriate value for the tearing modulus TMAT for 304 stainless steel, then it is only the terms involving A, and AEL,NCRthat need be considered, and the instability condition given by eqn (4) simplifies, by use of eqn (2), to Y \ dL
+
_ - Y dL
+
h
> TMA x
(5)
Provided that the tearing modulus of the material exceeds ,-~ 100, neglect of the other terms is justified and the ensuing instability condition (eqn (5)) should be accurate. Even if the tearing modulus is as low as 50, neglect of the other terms should still give a usable instability criterion. The next stage in the analysis involves a consideration of the term involving A, in the instability criterion of eqn (5) i.e. the term concerned with the displacement associated with the non-growing crack section.
E. Smith
60
Now as the cracks grow in the growing crack section, there will be a reduction in the load P. This produces a reduction in both the elastic and plastic contribution to A,. If the load reduction is AP, the reduction in the elastic contribution can be obtained by use ofeqn (A. 16) in the Appendix. The reduction is
(1-v2)DAP 8 ( 1 - v 2 ) A P , {2h~ 2BhE + ~ffE mt~£ )
(6)
the first term being due to elastic behaviour with no crack present, and the second term due to elastic behaviour associated with the crack's presence. As regards the reduction in the plastic contribution to A,, this stems from the formation of a compressive zone (stress-Y) in the immediate vicinity of the crack tips. The magnitude of this reduction is readily obtained from eqn (A. 14) in the Appendix, with Y replaced by 2Y and 2 now being equal to AP/4BLY. Remembering that AP is small, this reduction is equal to
8(1-v:)YL 1( AP ~3 )E 3 \4BLYf
(7)
Because the reduction in the elastic contribution is proportional to AP (see expression (6)), while the reduction in the plastic contribution is proportional to AP 3 (see expression (7)), the latter can be ignored in the limit as AP-~ 0. It immediately follows from expression (6) that the first term on the left-hand side of eqn (5) becomes E dA, (1 Y dL -
-
v2)D 8(1 - v2) In ( 2 h ' ] h +
(8)
Using the same arguments as those used earlier for the growing crack, one can ignore the second term on the right-hand side of eqn (8), i.e. that associated with the elastic contribution due to the crack's presence. It then follows from eqns (5) and (8) that the crack instability condition becomes 2(1
v2)D
-
h
>
TMA T
(9)
which is the same instability condition as for the case where the nongrowing crack does not exist. In other words the presence of the nongrowing crack has no effect on the instability criterion for the first crack, and the instability criterion may therefore be determined from the basis that the second crack does not exist in the system.
Crack instability in a multiple-cracked system
61
DISCUSSION The preceding section's analysis has clearly shown that provided a material's crack growth resistance is sufficiently great, and this should be the case with 304 stainless steel, then the tendency towards failure instability of a given ligament, when there are two symmetrically situated deep cracks, should not be affected by the presence of other cracks and their associated plastic ligaments. (By implication, the same conclusion holds when the other ligaments are not fully yielded.) In the context of the behaviour of circumferential through-wall cracks in 304 stainless steel pipes, the work described in this paper provides valuable support for the conclusion reached by the author in Ref. 3. There it was shown that if several circumferential through-wall cracks, together with their associated plastic hinges, are present in a stainless steel piping system, then the tendency towards instability of circumferential growth of a given crack is no greater than for the case where the given crack is present in isolation. A basic assumption in the analysis in Ref. 3 was that the plastic rotation recovery at a non-growing crack cross-section can be ignored, and this assumption is vindicated by the preceding section's analysis. It is important to appreciate that a prime reason why this conclusion is likely to be valid is that the material's crack growth resistance is presumed to be high and in this context a lower limiting value of 50 for the material's tearing modulus has been identified. If the tearing modulus falls below this value, neglect of the various terms in the preceding section's analysis cannot be justified and the situation must then be assessed in a completely different light. Indeed, in this new type of situation, the tearing modulus approach is unlikely to be valid as has been demonstrated elsewhere ;6 one will then have to resort to some alternative approach, such as, for example, an approach based on the maintenance of a constant crack tip opening angle at the tip of a propagating crack.
ACKNOWLEDGEMENTS The work described in this paper is part of the Electric Power Research Institute Program on Pipe Cracking. The author thanks Drs T. U. Marston and D. M. Norris for valuable discussions in this problem area.
62
E. Smith
REFERENCES 1. Tada, H., Paris, P. C. and Gamble, R. M., Proceedings oJ' the Twelfth National Symposium on Fracture Mechanics, ASTM STP 700, July 1980, p. 296. 2. Paris, P. C., Tada, H., Zahoor, A. and Ernst, H., Proceedings of Symposium on Elastic Plastic Fracture, ASTM STP 668, March 1979, p. 5. 3. Smith, E., The instability of growth of a through-wall crack in a piping system when other cracks are present, Int. J. Pres. Ves. & Piping, 20 (1985), this issue. 4. Dugdale, D. S., J. Mechs. Phys. Solids, 8 (1960), p. 100. 5. Bilby, B. A., Cottrell, A. H. and Swinden, K. H., Proc. Roy. Soc., A262 (1963), p. 304. 6. Smith, E., Int. J. Fracture, 17 (1981), p. 373. 7. Bilby, B. A., Cottrell, A. H., Smith, E. and Swinden, K. H., Proc. Roy. Sot., A279 (1964), p. 1. 8. Paris, P, C., Ernst, H. and Turner, C. E., Fracture Mechanics." Tweff?h Confi'rence, ASTM STP 700, July 1980, p. 338. 9. Rice, J. R., Mechanics and Mechanisms o[' Crack Growth, Proceedings of Conference at Cambridge, M. J. May Ed., British Steel Corporation Physical Metallurgy Centre Publication, 1974, p. 14.
APPENDIX In analysing this paper's model, the results ~ for a periodic system of coplanar cracks in an infinite solid will be used, the solid being cut along vertical surfaces so as to give the model of a solid of total width 2h containing two symmetrically situated cracks of depth c, the solid deforming under plane strain conditions due to the application of an applied tensile stress a (this cutting procedure is exact for the analogous Mode I II model). Using the DBCS representation of yield,4"5 the results 7 for the contained yield situation show that s and J are given respectively by the expressions
I
sin
na
_ sin2 nc
tan (A.1)
s = 2ec t.
sin [ ~ ) and J=
8(1 -- v2)h Ysin e ~./2 cos Z In [sin (Z + ~ ) ] n2E J~, x/1 - sin2~sin2z
dz
(A.2)
Crack instability in a multiple-cracked system
63
where (a - c) = R p is the size of the plastic zone at each tip and o~= na/2h is given by the expression sin(n~-~)
(~__y) cos
sin
= sin ~b
(A. 3)
For the special case where the cracks are deep in comparison with the solid width, when the model reduces to the case where a load P is applied at a point a great distance away from the ligament (thickness B), eqns (A.1), (A.2) and (A.3) give s = 2eL22
(A.4)
and J=
4(1-v2)y2L
nE
[(1 + 2)ln (1 + 2 ) + ( 1 - 2)In(1 - 2 ) ]
(A.5)
where 2 = P / 2 B L Y ; 2 is small for small-scale yielding conditions while 2 = 1 at general yield. For the general yield state, the parameter s is given by the relation (A.6)
s = 2eL
while the J-integral can be separated into an elastic component JEL (the value of J assuming there to be no plasticity), i.e. JEL
--
4(1 - v 2 ) y 2 L nE
(A. 7)
and an additional (plastic) contribution JPL, given by eqns (A.5) and (A.7) with 2 = 1, as 4(1
JPL -----
- vZ)YZL
nE
(21n 2 - 1)
(A.8)
The load point displacement A can be separated into respectively elastic and plastic components AEL and ApL. Then using the same dimensional arguments as Paris et al.8 for the contained yield situation, JPL can be expressed in the form JPL _ _ ).g(2) 2LY
2
I ~g(e) de
Jo
(A.9)
64
E. Smith
with AEL ~ Lg(2). Thus if g(2) = dI(2) d2, eqn (A.9) reduces to dI d2
21 2
JpL(2)
(A.lO)
LY2
which integrates to give I=~
22 f~ JPL(e)de
(A.1 1)
e3
Since eqn (A.5) gives the value of the J-integral, and because the elastic component JEL is 4(1 -JEL =
v2)y2L22 roE
(A. 12)
it follows that the plastic component JPL is given by the expression
JPL~-
4(1 -
v2)y2L
~E
[(1 + 2)ln (1 + 2 ) + ( 1 - 2)ln(1 --2)--22]
(A.13)
whereupon substitution in eqn (A. 11) gives I. It then follows that
LdI
ApL = Lg(2) -- d2 -
4(1
vz)YL
-
roE
[22-(1 + 2)In(1 + 2 ) + ( 1 - 2 ) l n ( 1 - 2 ) ]
(A.14)
At general yield (2 = 1), ApL is given by the expression ApL --
8(1 -
v2)yL
rtE
(1 - In 2)
(A. 15)
while the elastic component AEL of the load point displacement is given 9 by the expression ApL=AELNcR+AELcR '
"
-
(I-v2)LDY hE
+
8(1-vZ)LYln(2h ~ ~E
\~LJ
(A.16)
where AEL,NCRis the contribution assuming there is no crack, and AEL,CRis the contribution due to the crack's presence.
Crack Instability in a Multiple-cracked System E. Smith Joint Manchester University/UMIST Department of Metallurgy and Materials Science, Grosvenor Street, Manchester MI 7HS, Great Britain (Received: 31 October, 1984)
ABSTRACT The paper analyses the behaviour of a solid containing two symmetrically situated deep cracks, and with tension of the small remaining ligament. The results are used to support the view that when a piping system fabricated from a ductile material, such as 304 stainless steel, contains more than one crack, then the presence of other cracks has an insignificant effect on the condition for instability of growth of a given crack.
INTRODUCTION Against the background of the technological problem of the cracking of 304 stainless steel Boiling Water Reactor piping, the theoretical analysis of Tada et al.l showed that circumferential growth of a through-wall crack should be stable for the pipe run lengths that are typical of a reactor coolant system. Their approach, based on the tearing modulus procedure developed by Paris and co-workers, 2 assumed the material to be elastic-perfectly plastic, with the plastic deformation being confined to the cracked cross-section which is fully yielded. This cross-section thus behaves like a plastic hinge with a region of tension immediately adjacent to the crack tips and above the neutral axis, and a region of compression below the neutral axis. It has now become apparent that if a crack forms, by a stress corrosion mechanism, in one part of a piping system, then there is a real possibility that there will be similar cracks in other parts of 55 Int. J. Pres. Ves. & Piping 0308-0161/85/$03.30 ~ Elsevier Applied Science Publishers
Ltd, England, 1985. Printed in Great Britain
56
E. Smith
the system. With this situation in mind, an earlier analysis by the author 3 has extended the approach o f T a d a et al. 1 and shown that, when there are several circumferential cracks together with their associated plastic hinges in a stainless steel piping system, then the tendency towards instability of circumferential growth of any given crack is no greater than for the case where the given crack exists in isolation. Instability was considered from the basis that though there is a moment reduction at a cross-section containing a non-growing crack, as another crack grows under a fixed displacement applied to the system, the plastic rotation recovery at a nongrowing crack cross-section can be ignored in comparison with the elastic recovery of the system. This paper focuses on this particular point, and uses the results from an analysis of the behaviour of a solid containing two symmetrically situated deep cracks and with tension of the small remaining ligament, to show that if the tension is due to a load applied at a point away from the ligament and there is a small load reduction AP below the general yield load, then the elastic displacement recovery is proportional to AP, and the plastic displacement recovery is proportional to (AP) 3. It is thereby demonstrated, for a material with a high resistance to crack growth such as 304 stainless steel, that, when there are several such cracks and their associated fully plastic ligaments, the tendency towards fracture instability of a given ligament is essentially unaffected by the presence of the other cracks. This result is therefore supportive, and indeed important back-up, to the conclusion reached in Ref. 3.
T H E O R E T I C A L ANALYSIS Figure 1 shows the model representing the plane strain deformation of a solid containing two symmetrically situated deep cracks, and with tension of the small remaining ligament which is fully yielded; the ligament width and thickness are respectively 2L and B, while the solid width is 2h. The displacement is A at a great distance D/2 (i.e. D >>L) from the ligament, while P is the magnitude of the load associated with this displacement. This model simulates the behaviour of a growing crack, but, in order to assess the effect of a non-growing crack on this growth behaviour, it is supposed that the system is connected with another similar system containing a non-growing crack. In other words the overall system is subjected to a total displacement Av and the displacement of the second
Crack instability in a multiple-cracked system
57
P,A
<
<
Y
Y
Y
Y 2L--
2h--
:~
~,-
Fig. 1. The plane strain deformation of a solid containing two symmetrically situated deep cracks. A displacement A is applied to the solid at a great distance from the ligament, while P is the corresponding load.
system is A, with AT -- A, + A. The stability of the system will now be assessed for a fixed value of the total displacement AT (see Fig. 2). The analysis is based on the Dugdale-Bilby-Cottrell-Swinden (DBCS) representation of plastic deformation, 4'5 whereby yield is confined entirely to the planes containing the cracks. (In this respect the model is analogous to that used by Tada et al. 1 and also to that used by the author in Ref. 3, where plastic deformation is confined to the cracked crosssection of a pipe.) The tensile stress within the plastic zones between the crack tips is assumed to have the value Y which is representative of the material's tensile yield stress. For the particular configuration in Fig. l, the magnitude of the J-integral is given by the relation (see Appendix eqns (A. 7) and (A. 8)): J = JEL + JPL + JGY =
4(1 -- v2)y2L 4(1 -- v2)y2L (2In 1rE -~ rtE 2 - 1) + YAov
(1)
where E is Young's modulus, and v is Poisson's ratio. The first term is the elastic component JEL due to the application of the limit load P = 2BLY, presuming that there is no plasticity; the second term is the additional
E. Smith
58
P, A T
D
Non-growing cracks
D
Growing cracks
L
I~
J
2h
"-1
Fig. 2. A composite system consisting of a section containing growing cracks, and another section with non-growing cracks. The ligaments in both sections are assumed to
be fully yielded.
(plastic) contribution JPL when the general yield state is just attained; while the third term J~v is due to the post-general yield plastic deformation; A~v is the post-general yield plastic contribution to the displacement A. The total load point displacement AT = A, + A, assumed to remain fixed during crack growth, is given by the expression (see Appendix eqns (A. 15) and (A. 16)): A = A, + AEL,NCR+ AEL,Ca + ApL + AGy
= A, 4
(1-vg)LDY 8(1-_v2)LYln(2h ~ hE ¢" nE \nL/ -~
8(1 -- vz)YL
nE
(1 - ln2) + A~v
(2)
where the first term A, is the displacement associated with the section containing the non-growing cracks; the second term AEL,NCRis the elastic contribution to A assuming that there is no crack and no plastic deformation; the third term AEL,CR is the elastic contribution to A due to the crack's presence again assuming that there is no plastic deformation; the fourth term ApL is the additional (plastic) contribution to A when the
59
Crack instability in a multiple-cracked system
general yield state is just attained; while the fifth term AGv has already been defined. The original Paris-Tada-Zahoor=-Ernst approach 2 is based on the assumption that the J-crack growth resistance curve is specific for a given material, with the tearing modulus TMaT being related to the slope of the J-crack growth curve: E dJ (3)
TMAT -- y2 dc
where 6c is an increment of crack length. Instability under a displacement control condition (i.e. d AT/dC = 0) is presumed to occur when TApP > TMAT, TApp being related to J by an expression similar to eqn (3), and J being given by eqn (1). It follows from eqns (1), (2) and (3), noting that fc = - ilL, that the condition for crack growth instability is E/dA,
dAELNCR
dAELCR
d;
di
dApLX
E //dJEL
dA~_~
+ (4)
after elimination of AGv. Of the various terms on the left-hand side of eqn (4), eqns (1) and (2) show that with the possible exception of the terms involving A. and AEL,NCa,i.e. the displacement associated with the nongrowing crack section and the elastic displacement in the crack's absence for the growing crack section, all the terms will be ~ 10 if realistic values are assumed for h and L. Consequently if the material has a high crack growth resistance, and 200 has been suggested1 as being an appropriate value for the tearing modulus TMAT for 304 stainless steel, then it is only the terms involving A, and AEL,NCRthat need be considered, and the instability condition given by eqn (4) simplifies, by use of eqn (2), to Y \ dL
+
_ - Y dL
+
h
> TMA x
(5)
Provided that the tearing modulus of the material exceeds ,-~ 100, neglect of the other terms is justified and the ensuing instability condition (eqn (5)) should be accurate. Even if the tearing modulus is as low as 50, neglect of the other terms should still give a usable instability criterion. The next stage in the analysis involves a consideration of the term involving A, in the instability criterion of eqn (5) i.e. the term concerned with the displacement associated with the non-growing crack section.
E. Smith
60
Now as the cracks grow in the growing crack section, there will be a reduction in the load P. This produces a reduction in both the elastic and plastic contribution to A,. If the load reduction is AP, the reduction in the elastic contribution can be obtained by use ofeqn (A. 16) in the Appendix. The reduction is
(1-v2)DAP 8 ( 1 - v 2 ) A P , {2h~ 2BhE + ~ffE mt~£ )
(6)
the first term being due to elastic behaviour with no crack present, and the second term due to elastic behaviour associated with the crack's presence. As regards the reduction in the plastic contribution to A,, this stems from the formation of a compressive zone (stress-Y) in the immediate vicinity of the crack tips. The magnitude of this reduction is readily obtained from eqn (A. 14) in the Appendix, with Y replaced by 2Y and 2 now being equal to AP/4BLY. Remembering that AP is small, this reduction is equal to
8(1-v:)YL 1( AP ~3 )E 3 \4BLYf
(7)
Because the reduction in the elastic contribution is proportional to AP (see expression (6)), while the reduction in the plastic contribution is proportional to AP 3 (see expression (7)), the latter can be ignored in the limit as AP-~ 0. It immediately follows from expression (6) that the first term on the left-hand side of eqn (5) becomes E dA, (1 Y dL -
-
v2)D 8(1 - v2) In ( 2 h ' ] h +
(8)
Using the same arguments as those used earlier for the growing crack, one can ignore the second term on the right-hand side of eqn (8), i.e. that associated with the elastic contribution due to the crack's presence. It then follows from eqns (5) and (8) that the crack instability condition becomes 2(1
v2)D
-
h
>
TMA T
(9)
which is the same instability condition as for the case where the nongrowing crack does not exist. In other words the presence of the nongrowing crack has no effect on the instability criterion for the first crack, and the instability criterion may therefore be determined from the basis that the second crack does not exist in the system.
Crack instability in a multiple-cracked system
61
DISCUSSION The preceding section's analysis has clearly shown that provided a material's crack growth resistance is sufficiently great, and this should be the case with 304 stainless steel, then the tendency towards failure instability of a given ligament, when there are two symmetrically situated deep cracks, should not be affected by the presence of other cracks and their associated plastic ligaments. (By implication, the same conclusion holds when the other ligaments are not fully yielded.) In the context of the behaviour of circumferential through-wall cracks in 304 stainless steel pipes, the work described in this paper provides valuable support for the conclusion reached by the author in Ref. 3. There it was shown that if several circumferential through-wall cracks, together with their associated plastic hinges, are present in a stainless steel piping system, then the tendency towards instability of circumferential growth of a given crack is no greater than for the case where the given crack is present in isolation. A basic assumption in the analysis in Ref. 3 was that the plastic rotation recovery at a non-growing crack cross-section can be ignored, and this assumption is vindicated by the preceding section's analysis. It is important to appreciate that a prime reason why this conclusion is likely to be valid is that the material's crack growth resistance is presumed to be high and in this context a lower limiting value of 50 for the material's tearing modulus has been identified. If the tearing modulus falls below this value, neglect of the various terms in the preceding section's analysis cannot be justified and the situation must then be assessed in a completely different light. Indeed, in this new type of situation, the tearing modulus approach is unlikely to be valid as has been demonstrated elsewhere ;6 one will then have to resort to some alternative approach, such as, for example, an approach based on the maintenance of a constant crack tip opening angle at the tip of a propagating crack.
ACKNOWLEDGEMENTS The work described in this paper is part of the Electric Power Research Institute Program on Pipe Cracking. The author thanks Drs T. U. Marston and D. M. Norris for valuable discussions in this problem area.
62
E. Smith
REFERENCES 1. Tada, H., Paris, P. C. and Gamble, R. M., Proceedings oJ' the Twelfth National Symposium on Fracture Mechanics, ASTM STP 700, July 1980, p. 296. 2. Paris, P. C., Tada, H., Zahoor, A. and Ernst, H., Proceedings of Symposium on Elastic Plastic Fracture, ASTM STP 668, March 1979, p. 5. 3. Smith, E., The instability of growth of a through-wall crack in a piping system when other cracks are present, Int. J. Pres. Ves. & Piping, 20 (1985), this issue. 4. Dugdale, D. S., J. Mechs. Phys. Solids, 8 (1960), p. 100. 5. Bilby, B. A., Cottrell, A. H. and Swinden, K. H., Proc. Roy. Soc., A262 (1963), p. 304. 6. Smith, E., Int. J. Fracture, 17 (1981), p. 373. 7. Bilby, B. A., Cottrell, A. H., Smith, E. and Swinden, K. H., Proc. Roy. Sot., A279 (1964), p. 1. 8. Paris, P, C., Ernst, H. and Turner, C. E., Fracture Mechanics." Tweff?h Confi'rence, ASTM STP 700, July 1980, p. 338. 9. Rice, J. R., Mechanics and Mechanisms o[' Crack Growth, Proceedings of Conference at Cambridge, M. J. May Ed., British Steel Corporation Physical Metallurgy Centre Publication, 1974, p. 14.
APPENDIX In analysing this paper's model, the results ~ for a periodic system of coplanar cracks in an infinite solid will be used, the solid being cut along vertical surfaces so as to give the model of a solid of total width 2h containing two symmetrically situated cracks of depth c, the solid deforming under plane strain conditions due to the application of an applied tensile stress a (this cutting procedure is exact for the analogous Mode I II model). Using the DBCS representation of yield,4"5 the results 7 for the contained yield situation show that s and J are given respectively by the expressions
I
sin
na
_ sin2 nc
tan (A.1)
s = 2ec t.
sin [ ~ ) and J=
8(1 -- v2)h Ysin e ~./2 cos Z In [sin (Z + ~ ) ] n2E J~, x/1 - sin2~sin2z
dz
(A.2)
Crack instability in a multiple-cracked system
63
where (a - c) = R p is the size of the plastic zone at each tip and o~= na/2h is given by the expression sin(n~-~)
(~__y) cos
sin
= sin ~b
(A. 3)
For the special case where the cracks are deep in comparison with the solid width, when the model reduces to the case where a load P is applied at a point a great distance away from the ligament (thickness B), eqns (A.1), (A.2) and (A.3) give s = 2eL22
(A.4)
and J=
4(1-v2)y2L
nE
[(1 + 2)ln (1 + 2 ) + ( 1 - 2)In(1 - 2 ) ]
(A.5)
where 2 = P / 2 B L Y ; 2 is small for small-scale yielding conditions while 2 = 1 at general yield. For the general yield state, the parameter s is given by the relation (A.6)
s = 2eL
while the J-integral can be separated into an elastic component JEL (the value of J assuming there to be no plasticity), i.e. JEL
--
4(1 - v 2 ) y 2 L nE
(A. 7)
and an additional (plastic) contribution JPL, given by eqns (A.5) and (A.7) with 2 = 1, as 4(1
JPL -----
- vZ)YZL
nE
(21n 2 - 1)
(A.8)
The load point displacement A can be separated into respectively elastic and plastic components AEL and ApL. Then using the same dimensional arguments as Paris et al.8 for the contained yield situation, JPL can be expressed in the form JPL _ _ ).g(2) 2LY
2
I ~g(e) de
Jo
(A.9)
64
E. Smith
with AEL ~ Lg(2). Thus if g(2) = dI(2) d2, eqn (A.9) reduces to dI d2
21 2
JpL(2)
(A.lO)
LY2
which integrates to give I=~
22 f~ JPL(e)de
(A.1 1)
e3
Since eqn (A.5) gives the value of the J-integral, and because the elastic component JEL is 4(1 -JEL =
v2)y2L22 roE
(A. 12)
it follows that the plastic component JPL is given by the expression
JPL~-
4(1 -
v2)y2L
~E
[(1 + 2)ln (1 + 2 ) + ( 1 - 2)ln(1 --2)--22]
(A.13)
whereupon substitution in eqn (A. 11) gives I. It then follows that
LdI
ApL = Lg(2) -- d2 -
4(1
vz)YL
-
roE
[22-(1 + 2)In(1 + 2 ) + ( 1 - 2 ) l n ( 1 - 2 ) ]
(A.14)
At general yield (2 = 1), ApL is given by the expression ApL --
8(1 -
v2)yL
rtE
(1 - In 2)
(A. 15)
while the elastic component AEL of the load point displacement is given 9 by the expression ApL=AELNcR+AELcR '
"
-
(I-v2)LDY hE
+
8(1-vZ)LYln(2h ~ ~E
\~LJ
(A.16)
where AEL,NCRis the contribution assuming there is no crack, and AEL,CRis the contribution due to the crack's presence.