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APPLICATIONS OF TWO P A R A M E T E R APPROACHES IN ELASTIC-PLASTIC F R A C T U R E MECHANICS
NOEL P. O'DOWD
Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BX, United Kingdom
Abstract--Two parameter approaches to elastic-plastic fracture mechanics were introduced to remove some of the conservatism inherent in the one parameter approach based on the J integral [J. R. Rice, J. appl. Mech. 35, 379 (1968)] and to account for observed "size effects" on fracture toughness. It was shown in refs IC. F. Shih, N. P. O'Dowd and M. T. Kirk, Constraint Effects in Fracture, ASTM STP 1171, 2 (1993)], [J. W. Hancock, W. G. Reuter and D. M. Parks, Constraint Effects in Fracture, ASTM STP 1171, 21 (1993)] and [J. D. G. Sumpter and A. T. Forbes, Proc. TWI /EWI / IS Int. Conf. Shallow Crack Fracture Mechanics and Applications, Cambridge, U.K. (1992)], that much of the dependence of fracture toughness on specimen geometry could be explained by two parameter fracture theories based on T or Q.
This paper briefly reviews the two parameter J Q theory and examines some of the pertinent issues with regard to the application of the theory. In particular, the use of existing Q solutions for characterising constraint in real materials is investigated. Interpolation schemes for Q based on the T stress and pure power law solutions are also discussed. Two parameter toughness curves for failure by cleavage and ductile tearing are presented.
1. INTRODUCTION
ELASTIC--PLASTIC fracture mechanics using the J integral [1] is based on the concept of J dominance, whereby the near tip stress and strain states are set by J. Recently two parameter approaches have been applied to situations where J dominance does not hold, (see e.g. refs [2], [3] and [4]). There is general agreement that the applicability of the J approach is limited to so-called high constraint crack geometries. For example, when moderate size tensile crack geometries are loaded to general yield, the J annulus is smaller than physically relevant length scales and the zone of finite strains (see e.g. refs [5] and [6]). In low constraint geometries the near tip stress distribution can be significantly lower than the high constraint J dominant state.
For a power law hardening material where the plastic strain is given by
Ep/¢o = ~(~/~o)" , (1)
it has been shown in refs [7] and [8] that under proportional loading the first term in the expansion of the near tip elastic-plastic is given by the HRR singularity
t~u_ ( j ,]L,(,+,) tr0 \ ~ / flu(0; n). (2)
Here I, is a constant which depends only on the hardening exponent n. Figure 1 provides a measure of the ability of the HRR singularity to characterise the crack
tip fields in four crack geometries, shallow and deeply cracked center cracked tension specimens, and shallow and deeply cracked bend bars. The ratio between the hoop stress obtained from a finite element elastic-plastic analysis and the HRR field at a distance r = 2J /a o from the crack tip is plotted vs load as measured by J/aao. It is clear that apart from the deeply cracked bend bar the fields deviate very quickly from the HRR singularity e.g. in the shallow cracked tension geometry at J/atr o = 0.1 the near tip field is only ca 60% of the HRR field. Thus, if the fracture toughness depends strongly on the hoop stress, a fracture criterion based on J and the HRR field would be expected to be very conservative for this geometry. Indeed, fracture toughness values well above the JJc toughness have been measured in center cracked tension specimens [4].
It was shown in refs [9] and [10] that a two parameter description, using J and a constraint parameter Q, fully characterises the near tip stress and strain states in a range of crack tip geometries. The parameter Q is determined from a finite element analysis and is the difference between the actual hoop stress and reference field hoop stress. More will be said of the choice of
Fig. 1. Comparison of H R R field and near tip fields obtained from by a finite element analysis at r/(J/ao) = 2 for (a) and (b) center cracked panel in uniaxial tension (ccp); (c) and (d) three point bend
bar (tpb).
reference field later. It was shown that Q is a measure of hydrostatic stress so that the near tip fields may be represented as
O ' i j : ( f f i j ) R E F " F a a o ~ i j , for r > J / a o , [0[ <1t/2. (3)
It was argued in ref. [17] that the weak coupling between deformation and stress triaxiality under plastic deformation provides support for this representation of the fields. Since plastic flow is incompressible, the superposition of a purely hydrostatic stress state induces only an elastic volume change. Therefore, provided that boundary conditions are satisfied, a hydrostatic stress Q a o
which induces only minimal effects on the deformation state can be imposed on the stress field. This implies that, in the forward sector, deformation and stress triaxiality are weakly coupled. (In the back sector, due to the traction free crack faces, the addition of a hydrostatic stress term will not satisfy the boundary conditions.) It follows that near tip deformation and stress triaxiality must be scaled by two parameters which are effectively independent. At a fixed deformation level as characterised by J a range of stress states, differing by what is essentially a hydrostatic stress, can exist ahead of the crack.
The elastic T stress, the second term of the asymptotic series for the stress field in a linear elastic material [11], has also been used to characterise constraint (see e.g. refs [12], [13], [14] and [15]). Under small and moderate-scale yielding the T and Q approaches are essentially equivalent and this will be discussed later.
2. J-Q T H E O R Y
The plane strain family of crack tip fields for the full range of stress triaxiality is written as
aij = ao ,0; , Eij = eogq ,0; , u i = ha ,0; . (4)
Two parameter approaches in elastic-plastic fracture mechanics 447
(a) ,.o t 3.0-
t g 2.0-
1.0-
0.0 0.0
. . . . I . . . . I ' ' ' ' 1 . . . . I . . . .
1.0 2.0 3.0 4.0 5.0
g
0.10
(b) 0.08-
0 . 0 6 ¸
,to 0.04-
0.02 -
0.00
0.0
ccp ...... t p b r = 4d/ao . . . . . ccp03)
0.2 0.4 0.6 0.8 1.0
O/rr Fig. 2. Full field stresses and effective plastic strain for three different geometries at Q ~ -0 .5 . For the uniaxially loaded center cracked tension geometry (ccp) J/(aao)= 0.0015, for the three point bend bar (tpb) J/(aao) = 0.01 and for the biaxially loaded center cracked tension geometry [ccp(B)] J/(aao) = 0.03.
The form in eq. (4) constitutes a one parameter family of self-similar solutions. It is independent of material description and holds strictly under contained yielding conditions where the remote fields are given by the first two terms of the Williams expansion. However, the form of eq. (1) can also hold for finite width crack geometries. Figure 2 shows the stress and strain distributions for three geometries: a shallow crack center cracked panel under uniaxial and biaxial loading (B = a~/a~' = 0.5) and a shallow cracked three point bend bar. Each of the geometries has a different J level, but identical Q level (Q = -0.48) . It is clear that when distances are normalised appropriately the fields collapse onto a single distribution. A precise definition of Q is dealt with next.
Through full field analysis it was shown that the following form provides an approximate, but robust, description of the near tip fields over physically significant distances:
aij=(aij)REF-]-Qffo(~ij, for r >J/ao, 101 <re/2. (5)
The first term is a high triaxiality reference distribution and the second term is a constant hydrostatic term, i.e. independent of distance and angle. Two fields have been considered as the reference field: (aij)nRR the HRR singularity field [eq. (1)] and (aij)ssv which is the solution to the standard small-scale yielding problem driven by the elastic K-field. The latter reference field can be regarded as the field pertaining to a long crack in an infinitely large body. For the linear elastic power law hardening material, studied in refs [9] and [10], eq. (5) provides a good representation of the fields with either (ai~)HRR or (a~j)ssv as the reference field. It was shown in ref. [16] that the second term in eq. (5) actually represents the effect of the four higher order terms in the elastic-plastic near tip fields.
A range of crack tip fields for a linear elastic power law hardening material with hardening exponent n = l0 is presented in Fig. 3. These were generated by carrying out a boundary layer analysis with the remote fields given by the first two terms of the elastic crack tip field [11], i.e.
aFj , , ~ f j ( O ) + T6,,6,j. (6)
By varying the level of T/go, different near tip distributions corresponding to different levels of constraint are generated. Figure 3(a) plots hydrostatic stress vs distance normalised by J/ao, Fig. 3(b) provides the angular variation of the hydrostatic stress at r = 4J/a o and Fig. 3(c) provides the angular variation of the shear stress at r = 4J/go. The reference T = 0 field is indicated. It is clear that eq. (5) provides a good representation of the near tip fields with the T = 0 field as the reference field.
The Q values for many different geometries have been presented in ref. [17]. Figure 4 shows the dependence of Q on distance for a shallow crack specimen in tension and bending at different amounts of loading. Here, the remote load P is normalised by the limit load for the geometry P0.
448 N.P. O'DOWD
&), 4.o-
3.0-
Ib 2.0-
1.0"
0.0
0.0
.... I .... I .... I .... I ....
l.O 2.0 3.0 4.0 5.0
7
(b)
,b"
• r 4 J / a o ~
0.0 0.2 0.4 0.6 0.8 1.0
e / ~ r
C) 2.0
1.5"
l.O"
I~ ~ o.5. 0.0 ~
~.5.
-I.0 0.0 0.2 0.4 0.6 0.8 1.0
Fig. 3. Two parameter family of crack tip fields generated by modified boundary layer analysis for n = 10. (a) Hydrostatic stress vs normalised distance, (b) angular variation of hydrostatic stress at r/(J/ao) = 4
and (c) angular variation of shear stress at r/(J/%) = 4.
It is noted that for a wide range o f loading Q shows very weak dependence on distance, thus verifying the form o f eq. (5).
3. J - Q T H E O R Y A N D R E F E R E N C E F I E L D S
3.1• R a m b e r g - O s g o o d mater ia l
The asymptot ic analyses to date have been based on a R a m b e r g - O s g o o d material and ,/2 deformat ion theory. In uniaxial tension the material deforms according to
E/E0 = ~r/~r0 + ~(~/~0)". (7)
Two parameter approaches in elastic-plastic fracture mechanics 449
This material does not have an explicit yield point and the ratio tr0/c0 gives the initial slope of the stress-strain curve.
It has been suggested [18] that only two higher order terms are required to fully represent the crack tip fields in different geometries for Ramberg--Osgood materials. The amplitude of these two terms are not independent and the full field may be represented in the form:
f J ~U'n+l)ffr'~-l/¢"+l) r Sl 2 2 r ~2 3
The first term in eq. (8) is the HRR singularity; the exponents st, s2 and the dimensionless functions al.~ and alJ have been provided in ref. [18]. The amplitude A 2 in eq. (8) is not determined by the asymptotic analysis and must be obtained numerically. Note that distances in eq. (8) are normalised by a characteristic dimension L, rather than J/ao.
It is clear that eq. (8) does not agree with eq. (5), the equation for a linear elastic power law hardening material when ffHR R is the reference field. As mentioned earlier, up to five terms were required in ref. [16] to describe the near tip fields for this material. The discrepancy between the two results is explained by the different material relations employed in each case. For n > 3.7 the effects of elasticity do not enter the first five terms of the asymptotic series, thus the form of solution for a Ramberg-Osgood material should be the same as that for the linear elastic power law hardening material examined in refs [9, 10, 17]. However, the amplitudes of the individual terms will depend on the constitutive description and this will influence whether three, four or more terms are required to describe the near tip fields. For example, the near tip distribution for T = 0 obtained from the Ramberg-Osgood material with n = 10 and 0t = 1 is compared with the linear elastic power law hardening model in Fig. 5(a). The difference field with the HRR field as the reference field for both material descriptions is provided in Fig. 5(b). Clearly, while a constant term works well for the linear power law material
(a)
(b)
0.5
CY
0 . 0 -
-0.5.
- 1 . 0 "
-1.5"
P/Po = 0.2 m m , , , ~ , n
P/Po = 1.4
ccp
-2.0 .... i .... i .... f .... i .... ~ ....
0.0 1.0 2.0 3.0 4.0 5.0 6.0
r
0.5
0 . 0 o
-0.5-
-1.0-
-I.5-
P~o = 0.2
P~o=l .5
t pb ° 2 , 0 . . . . I . . . . [ . . . . I . . . . I . . . . . . . .
0.0 1.0 2.0 3.0 4.0 5.0 6.0
r
Fig. 4. Dependence o f Q on distance [F = r/(J/oo)] for shallow cracked center cracked tension geometry (a) and three point bend bar (b) at different load levels. Here P0 is the limit load for the geometry.
450 N . P . O ' D O W D
it does not provide a good representation of the difference field for the Ramberg-Osgood material. Similarly, in Fig. 5(c) the difference field for T/g o = - 0 . 5 is plotted for both materials with the same conclusion.
However, this difficulty with the reference field and choice of material description is avoided if we take the small-scale yielding solution as the reference solution. Figure 5(d) shows the difference field for both material descriptions using the corresponding small-scale yielding field as the reference field. Note that the Q values are very similar in both cases and both are independent of distance. The representation of eq. (5) has the advantage that it can be employed regardless of the material description and, as will be discussed later, has been found to be a good description of the fields even for materials which do not display power law hardening. If a high constraint measure J,c is to be used as the reference toughness then it makes sense to use SSY (T = 0) field as the reference distribution, as this is the field which the specimen experiences rather than the HRR field (even for a Ramberg-Osgood material). In other words the high constraint J~c measurement should correspond to the stress state of the reference field.
For a Ramberg-Osgood material, to convert from an A2 description to a Q description simply involves adjusting the reference field--they are equivalent. However, using an A2 representation is not recommended for the linear elastic power law hardening material as the three terms do not provide an adequate representation of the fields. For this material either the H RR or SSY field may be used as the reference solution in eq. (5) as in both cases the difference field is constant. Indeed, we can write eq. (5) as
oij = ( O ' i j ) H R R "Jr- ao(n)trofij + Qtrot}ij , for r > J/~ro, 10l < r~/2, (9)
. . . . i . . . . . . . I ~ . . . . I . . . . . 1,5
0.0 1,0 2.0 3.0 4.0 5.0 0.0
g
T / a o = - 0 . 5
- - l i nea r -power l aw
- - - R a m b e r g - O s g o o d
(d) . . . . I . . . . I . . . . I . . . . I . . . .
1.0 2 . 0 3 . 0 4 . 0
Fig. 5. B o u n d a r y l ayer resul ts fo r R a m b e r g - O s g o o d ma te r i a l a n d l inear elastic p o w e r l aw mater ia l . (a) T = 0 fields c o m p a r e d wi th H R R field, (b) dif ference fields wi th H R R field as reference field for T = 0, (c) d i f ference fields wi th H R R field as reference field for T / a o = - 0 . 5 a n d (d) difference fields wi th SSY
( T = 0) field as reference field for T/tr o = - 0 . 5 .
5.0
Two parameter approaches in elastic-plastic fracture mechanics
where the first two terms, corresponding to the T = 0 solution of Fig. 5(a), represent the reference field. For n -- 10, a0 = - 0 . 2 as seen in Fig. 5(b).
3.2. A S T M A-302 B plate material
Size effects on the resistance curve for this material were investigated in ref. [19]. The tensile stress/strain curve is shown in Fig. 6. In this section we investigate whether a two parameter field based on eq. (5) can be used to characterise the crack tip fields for the A-302 material. The reference field used is the small-scale yielding T -- 0 solution which must be determined by a finite element analysis. The finite element program ABAQUS was employed and the actual tensile stress/strain relationship was used in the material description. Initially, a modified boundary layer analysis was carried out with T/go- - -0 .5 , 0 and 0.5, and the stress fields obtained from the finite element analysis are presented in Fig. 7. Figure 7(a-c) gives the angular variation of the hoop, radial and shear stress at r = 4J/go and Fig. 7(d-f) gives the radial variation. It is clear that the fields for T/ao = - 0 . 5 and 0 differ from the reference T = 0 distribution by a fixed amount and obey the form of eq. (5). Thus, it is concluded that eq. (5) is not limited to power law hardening behavior, but also holds for the more general constitutive behavior. That this should be the case may have been anticipated from the incompressibility of the plastic flow which gives rise to crack tip states differing only by a hydrostatic stress as discussed in Section 1.
Finally, two finite width geometries were examined: the edge crack specimen in tension and bending. In Fig. 8, the loss of constraint with increasing load is seen for both specimens. This behavior is consistent with what was observed in refs [9, 10] for power law hardening materials.
Though not presented here, calculations were also carried out using a linear hardening curve to represent the material behavior. Again it was observed that the response follows the form of eq. (5), the stress distributions differing by a hydrostatic stress term from the reference T = 0 distribution.
4. USE OF EXISTING Q SOLUTIONS
Q values have been provided in the literature in refs [9, 10, 17]. The next section deals with how these Q solutions may be used for materials with different elastic and plastic properties.
4.1. Effect o f elastic properties on Q values
The constitutive law used in refs [9, 10, 17] has the form in uniaxial tension:
/ % = a / a 0 g < g o
= (g - gO~go + (g/go) ~ g > ~0. (10)
This may be compared with eq. (7) for the Ramberg-Osgood material. Here a0 plays the role of a yield stress and % is the yield strain. Since Q is a measure of stress difference, provided stress
452 N, P. O ' D O W D
quantities are normalised by a0 and strain quantities by Co, the Q vs load curve depends only on n and not on % or %. Thus, plotting Q against tr°~/a 0 or J/(a%%) should produce a material independent curve (for a fixed n). However, this is strictly true only when Q is independent of distance. Since Q is evaluated at distance r = 2J/tr o and since J at a fixed value of a/tro is proportional to %, if Q depends on distance then different values of % will give rise to different values of Q for the same load level. Q is also expected to depend on Poisson's ratio, v.
These features are illustrated in Fig. 9 for a shallow cracked bend bar which has a Q value independent of distance over a wide range of plastic deformation and a deeply cracked bend bar where Q is dependent on distance at higher loads. In all cases the small-scale yielding solution was taken as the reference field. Four values of Poisson's ratio were analysed, v = 0.2, 0.3, 0.45 and
3 .0
2.0-
I~ I.o-
0.O
-1.0
0.0
- ° - ° . . .
" ' " ' , , T - - 0 ~ . . ......... T=-0.5
(~) . . . . p . . . . . . . . . . . . I . . . .
0.2 0.4 0.6 0.8 1.0
o/Tr
,o IL, . :-o5 . . . . . T = 0 . 5 3 o , , , . . - - - < .... iiii!i--..:...!
1.0 . . . . I ' ' ' ' I ' ' ' ' I ' ' ' ' I . . . .
0.0 1.0 2.0 3.0 4.0 5.0
F F
Fig. 8. Loss of constraint for ASTM A-302 steel as loading is increased in tension (a) and bending (b).
0.49, and three values of E0 = 0.001, 0.002 and 0.0033. Figure 9(a and b) shows the dependence on Poisson's ratio which is seen to be weak. Using the v = 0.3 curve is probably acceptable for most metals.
Figure 9(c and d) shows the dependence on E0. Here J is normalised by ago which was the normalisation used in ref. [17] (they considered a fixed E0=0.002). Note the quite strong dependence on E0 for the shallow cracked bend bar. Surprisingly, the deeply cracked bend bar shows a weak dependence on ~0. If J is normalised by a%go as shown in Fig. 9(d and e) the dependence on E0 for the shallow bend bar is removed. However, Q for the deeply cracked bend bar now depends on e0. That this dependence is a consequence of the distance dependence of Q is illustrated in Fig. 10. Here Q is plotted at different distances from the crack tip for a fixed E0. [? = r/(J/go)] Note the strong dependence of Q on distance for J/(ago) > 0.02 in the deeply cracked bend bar. Thus, if Q is evaluated for two different values of ~0 at the same remote stress level [which implies the same value of J/(ago~o) ] we are evaluating Q at different physical distances from the crack tip and this leads to a different Q value.
O'Dowd and Shih have presented Q values in ref. [17] for e0 = 0.002; here Q was plotted vs J/(aao). If these data are to be used for other values of E0 the x-axis should be reinterpreted as J/(aao 500%). However, care should be taken in using this normalisation for deeply cracked bend bars or other geometries showing strong Q dependence on distance.
4.2. Effect of plastic properties on Q values
4.2.1. Ramberg-Osgood material. The Ramberg-Osgood material defined via eq. (7) has the additional material parameter ~. However, the three parameters ~0, g0, and ~ are not unique for a given material response. Equation (7) may be rewritten as follows
, - ~ + , (11) E0 g0
where
~ ;= E0/e l/l"-° and a;--- a0/e l,'l'-I) (12)
Thus, this allows existing power law solutions (with ~ implicitly equal to 1) to be used for any Ramberg-Osgood relation by modifying the reference stress and strain appropriately. There will be slight differences in Q values obtained for a Ramberg-Osgood material and a power law material as discussed earlier.
4.2.2. A S T M A-302 B plate material. The ability of power law solutions to predict loss of constraint in real materials is now assessed. Q values obtained for the A-302 steel are compared with values obtained from a power law hardening material and a Ramberg-Osgood material using two different material representations.
454 N. P. O ' D O W D
O-q O.5
J , . ~ ,o=ooos I | , . 0 ~
0 t 0 -1.0 -1.0.
. ~ o t . . . . , . . . . , . . . . I - ~ ® , . . . . -
Fig. 9. Dependence o f Q on elastic quantities, v and E l % for shallow and deeply cracked bend bar.
0.5
~o --- 0 . 0 0 3 Y= 1
~ v = o.o5 (a)
-ZC . . . .
0.001
0.0-
-0.5-
-1.0-
' ' ' " 1 . . . . 0.010 0.100 1.000
J/ao,
::1 0:0003
-2-01 . . . . , , ' ~ I ~ D , , , 0.001 O.OlO O.lO0 1.000
J/ao,
Fig. 10. Dependence o f Q on normalised distance [f = r / (J#r o)] for shallow and deeply cracked bend bar.
Two parameter approaches in elastic-plastic fracture mechanics 455
The different material laws used are shown in Fig. 11, "a" corresponds to the actual stress-strain relationship, "b" a Ramberg-Osgood material with E0 = 0.004, a0 = 450, ~ = 1.24 and n = 8.5. (The latter were the constants used in ref. [19] to fit the data. This curve provided the best fit to the post yield behavior, but a poor fit to the elastic part of the stress-strain curve.) Curve "c" corresponds to a Ramberg-Osgood material with E0 = 0.002, a0 = 450, ~ = 1 and n = 11. This provided a good fit to the material data in the elastic and elastic-plastic regimes. Finally curve "d" is for the linearly elastic power law hardening material with E0 = 0.002, a0 = 450, and n = 11.
The two specimens discussed in the previous section were examined for all four material descriptions. In Fig. 12(a and b) the J values obtained are plotted against remote load. Here J is normalised by ago, where a is the crack length. The trends are the same for both the tension and bend specimen. The crosses indicate the values obtained from the EPRI J estimation scheme using the true elastic modulus and taking n = 8.5 in the power law plastic relationship. Ramberg-Osgood materials "c" and the flow theory material "d" provide a good estimate of J, though at high loads the discrepancy is quite large. (It is disguised somewhat by the use of the log scale. At P/Po = !.2, the J value for the tension specimen obtained from the " t rue" material description is 0.1, from the power law material 0.06 and from the Ramberg-Osgood material "c" it is 0.07.) Ramberg-Osgood material "b" does not provide a very accurate measure of the true J value except at very high load. This is because this material does not match the true response well in the elastic regime. The EPRI estimation scheme provides a good representation of the behavior over the full range of loading.
Bearing in mind that the J values obtained for the different material representations vary somewhat, the constraint as measured by Q is next plotted against J in Fig. 12(c and d). Again J is normalised by ago. In each case Q is calculated relative to the reference T = 0 field for the appropriate material. It is seen that apart from curve "b", the J - Q curve is reasonably material insensitive, with the linear elastic power law hardening material (material "d") providing a lower bound for the tension and bend geometry. It is interesting that despite the fact that material "b" more closely models the hardening behavior at high load, materials, "c" and "d" provide a better estimate of the loss of constraint for both the tension and bend geometry--suggesting that accurate modeling of the elastic portion of the curve is important even at high load. Figure 12 (e and f) plots Q vs load. In this case material "b" provides a reasonable estimate of the true Q value. The results presented in this section give some confidence in the use of existing Q solutions for different material response provided the material response is reasonably closely represented by the power law behavior.
5. Q ESTIMATES UNDER SMALL-SCALE AND FULLY YIELDED CONDITIONS
5.1. Estimates based on the elastic T stress
While Q provides an exact specification of the near tip field, it must be obtained from a full field finite element analysis. The one-to-one relationship between T and Q under small-scale
800.0
600.0-
400,0-
200.0- . . . . . . . . . . . . C
. . . . . . . d
A 302-B Plate
0.0 . . . . I . . . . L . . . . I . . . . I . . . .
0.00 0.02 0.04 0.06 0.08 O. 10
true strain
Fig. 11. Ramberg-Osgood and linear elastic power law fits to A-302 steel. (The meaning of the designations a, b, c and d is explained in the text.)
456 N. P. O 'DOWD
0.I000,
o.o5oo-:
0.0100'
0.0050-
0.0010'
0.OOO5 -:
0.0001
0.2
0.0'
-0.2'
-0.4,
(a) .. "~ o ~ ° . ~ "
f f C
f -
, / ~ EPRI . . . . I . . . . I . . . . I . . . . I . . . . I . . . .
0.0 0.2 0.4 0.6 0.8 l.O 1.2
P/Po
O.lOCO.
o.o5oo-"
0.0100,
o.co~o"
0•0010
o.ooo5-
0.0001 0.0
so,
/ . a ~ . . . . . . . d
' - ' ' 1 ' ' ' ' I ' ~ ' I . . . . I . . . . I . . . .
0•2 0•4 0•6 0•8 1.0 1 •2
P/Po
-0.6'
- 0 , 8 ,
-I,0 0.0001
<° > ii :i ) . . . . I . . . . I . . . .
0.0010 0.0100 O•lO00
Y
0.2
0•0-
-0.2
-0.+
-0.6'
" O . S "
-!.0 0.0001
...... b x: : ,~ , ",q...~
............ C "'~,~
(d) d
. . . . I . . . . I . . . .
0.0010 0•0100 0.1000
J
0.2 0.2
0.0"
-0.2-
.0.4 -
.0.6 -
.0.8 -
-I.0
0.0
. . . . . . 2~g2"'i.~........"
............ c ~;J"£)...~'"
(e) . . . . I . . . . I . . . . I . . . . I . . . . I . . . .
0.2 0.4 0.6 0.8 1.0 1.2
0,0"
-0.2-
-0•4-
.0.6"
-O.S"
-I •0
0•0
• , * , % .
. . . . . . b
. . . . . . . . . . . . c
( f ) . . . . . . . d
. . . . I . . . . I . . . . I . . . . I . . . . I . . . .
0.2 0.4 0•6 0•8 1.0 1.2
P ~ o P ~ o
Fig. 12. Results for A-302 steel with four different material descriptions (a) J for edge cracked bar in tension; (b) J for edge cracked bar in bending; (c) Q vs J for edge cracked bar in tension; (d) Q vs J for edge cracked bar in bending; (e) Q vs load for edge cracked bar in tension and (f) Q vs load for edge
cracked bar in bending.
yielding conditions suggests the use of T as a constraint measuring parameter• Since T is an elastic quantity it can be determined much more easily than Q. The ability of T to characterise constraint is assessed in this section.
A relationship between T and Q under contained yielding may be obtained from a modified boundary layer analysis, where the remote stress is given by the elastic K - T field. The T - Q curve (from ref. [17]) is shown in Fig. 13 for a power law material with various hardening exponents. This curve was fitted by a cubic polynomial such that
Two parameter approaches in elastic-plastic fracture mechanics 457
The parameters a~ have been tabulated in ref. [17]. More recently, Ainsworth and O'Dowd [20] have proposed the relationship
Q = T/go T/go < O
Q = 0.5T/tro T/go > 0, (14)
which provides a simple relationship independent of n. This is in fact equivalent to what has been suggested by Du and Hancock [14] for perfectly plastic materials. This fit is also shown in Fig. 13.
The T stress is directly proportional to applied load so we can write
T = fl(a/W)g ~, (15)
where cr ~ is a representative stress amplitude and fl is a dimensionless parameter which depends on geometry. Values of fl for various geometries have been tabulated by Sham [21], and Leevers and Radon [22]. Figures 14 and 15 assess the ability of the T stress to quantify constraint in different geometries. Similar comparisons have been made in refs [10, 17, 20 and 23]. In Fig. 14 the hoop stress from a finite element analysis, divided by the hoop stress obtained from eq. (9) with the Q value obtained using both eq. (13) and (14) in conjunction with eq. (15) is plotted for shallow cracked tension and bend specimens for an n = 10 material. These data may be compared with that in Fig. 1. The ratio goo/gffo is plotted vs normalised J in Fig. 14(a, c and e) and against normalised load in Fig. 14(b, d and f). Figure 14(a and b) is for the shallow notched bar in uniaxial tension. Figure 14(c and d) is for the shallow notched bar in biaxial tension with B = crx~/g~.= 0.5. Figure 14(e and f) is for a shallow notched bar in bending. If the ratio goo/g~o is greater than one, then the T approach is non-conservative and predicts a greater loss of constraint than is actually the case. It is observed that the T approach using eq. (13) is non-conservative for the shallow cracked tension geometry for both uniaxial and biaxial loading. The benefit of using the T stress over J alone is quite significant for the bend geometry--at J/aao = 1 the stress as predicted using J and the HRR field is twice the actual stress while the stress predicted by the T approach is 1.25 times the actual stress. It is seen in Fig. 14(a) that using eq. (14) removes the non-conservatism for the center cracked tension specimen. The quadratic term in eq. (13) was responsible for the high negative Q values in the T stress approximation for the uniaxially loaded tension specimen. By including only a linear term in the approximation for Q, Q remains lower at high load and closer to the actual value. However, the T approximation is still non-conservative for the biaxially loaded panel.
J 1.0
Q O.5
...... n=3 o---nr.O . ~ n --10
n z ~
o!s ,!o ~ ~/oo
~-1.0 -O.5 . . ~ 0
,' "'~f ~ . (5.2)
Fig. 13. Variation of Q with T/a o under contained yielding for a range of hardening moduli. Equation 04) is indicated by the solid line.
Fig. 14. Comparison of hoop stress based on T stress using eqs (13) and (14) and near tip fields obtained from a finite element analysis at r/(J/%)= 2 for three geometries.
Figure 15 provides a comparison between the Q and T stress approximation for six geometries using eq. (14) for n = 10. In this figure, Q is plotted against remote load normalised by limit load P0 for each geometry. Since Q in eq. (14) is directly proportional to T/a o which is directly proportional to load via eq. (15) we can write
Qr=f lT(P/Po) T/tro <O
= 0.5flr(P/Po) T/tro > 0, (16)
where the notation Q r is used to indicate that the Q value was obtained using a T stress approximation. As stated previously the T stress estimates are conservative for all geometries
5.2. Q Est imates based on pure power law solutions
As discussed in ref. [10] the value of Q at fully plastic conditions can be obtained from pure power law hardening solutions of the type used in simplified engineering fracture analysis (Shih et al. [24], Kumar et al. [25]).
. . . . I . . . . I . . . . I . . . . I . . . . I . . . .
0.25 0.50 0.75 1.00 1.25 1.50
P/Po
0.0"
(~ -0.5-
-l.o'
-1.5
0.00
(d) ecpB =0.5
a/W=0.8
Two parameter approaches in elastic-plastic fracture mechanics 459
examined, except the biaxially loaded panel. For an n = 5 material eqs (13) and (14) give very similar estimates for Q and similar trends to that shown in Fig. 15 are observed.
0.25 0.50 0.75 1.00 1.25 1.50
P/Po
0.5
0.0-
-0.5-
-1.0"
-1.5 0.00 1.50
(e) tpb a/W =0.1
0 o
. . . . I . . . . I . . . . I . . . . I . . . . I ' '
0.25 0.50 0.75 1.00 1.25
P/Po
0.5
(0 0 . 0 ' C"~- r~ r ' n n ~ 0 0 0 O 0 0 0 0 0 0
-1.0. tpb a /W = 0.5
- 1 . 5 . . . . i . . . . i . . . . I . . . . J . . . . i . . . .
0.00 0.25 0,50 0.75 1.00 1.25 1 .$0
P/Po
-0.5.
Fig. 15. Comparison of Q estimates based on T stress using eq. (14) (circles) and numerically determined Q values (solid line) for six geometries for n = 10.
460 N.P. O'DOWD
In uniaxial tension a pure power law material deforms according to
E tEo = ~(a tao)". (17)
The solution of a traction boundary value problem based on eq. (17) and involving only a single load parameter which is increasing monotonically has the form:
ai:/ao = (P/Po)Oi: (x/L; geometry, n), (18)
where L is a characteristic crack dimension and P0 is the limit load Oij and (i: are dimensionless functions of spatial position x and depend on the crack geometry and strain hardening exponent. With the H R R field as the reference field in eq. (5), eq. (18) implies that for a power law material
Q = (P/Po)~Q(geometry, n), (19)
i.e. Q is directly proportional to applied load. The notation /~Q is used to be consistent with the previous section. For Q based on the small-scale yielding field eq. (19) is modified slightly so that Q is given by
Q P = (P/Po)~Q (geometry, n) - a0 (n), (20)
where a0 was defined earlier and is - 0 . 2 for n = 10. Values of ]~Q can be obtained from the linear elastic power law solutions at P/Po > 1.0, where the power law response will dominate. Note that eqs (19) and (20) strictly only apply when Q is evaluated at a fixed distance ahead of the crack. However, Q is evaluated at a distance, r = 2J/ao, which varies with load. Equation (19) or (20) will still apply so long as Q is independent of distance over the load range of interest.
The overall Q response can be obtained by interpolating between the small-scale yielding solution based on T and the fully plastic solutions. Figure 16 shows a proposed interpolation scheme for three geometries. In each case the approximate Q solution was generated using:
The final equation is simply a linear interpolation between Q based on the T solution at P/Po = 0.8 and Q based on a power law solution at P/Po = 1.2. In each case a quite good fit is obtained. For the three point bend bar no at tempt was made to fit the region P/Po > 1.3 as the Q values are strongly dependent on distance here and thus the power law fit is not relevant.
6. GENERATION OF J-Q T O U G H N E S S CURVES FOR CLEAVAGE A N D TEARING
A simple cleavage toughness locus may be constructed based on the attainment of a critical hoop stress at a critical distance ahead of the crack. From eq. (5) the hoop stress field takes the following form:
fro \~Eoaoi, r / 022(0 = 0) + ao(n) + Q. (22)
Assuming that the critical distance r~ is within the J - Q annulus and applying the fracture criterion we obtain a fracture toughness curve:
, ( 2 3 ) J c = J i c l ~c/~Q-a where Jtc is the toughness measured in a Q = 0 geometry, e.g. a deeply cracked bend bar. This curve is a modified version of the relation in ref. [10] where the H R R field was used as the reference field. Note that rc does not appear explicitly in the Jc(Q) toughness curve; the only additional material quantities required are ac/tr o and n the hardening exponent.
I f the effects of crack tip blunting are taking into account this will limit the choice of a c. For an n = 10 material the finite strain analyses of refs [9, 10] indicate that for Q < - I the near tip stress cannot exceed 3a 0. Thus if ac > 3tr 0, cleavage will not occur for Q < - 1. For n = 5 the maximum hoop stress for Q < - 1 from ref. [9] is 4.8a 0.
Two parameter approaches in elastic-plastic fracture mechanics 461
0.5
0.0.
-0.5-
-I.0-
-1.5-
-2.0
0,0 I . . . . I . . . . I
0.5 1.0 1.5
P/Po
0.5
0.0 L
-0.5.
-1.0-
-1.5-
-2.0
0.0
(b)
f fW=O.1
I . . . . I . . . . I
0.5 1.0 1.5
P/Po
0.5
0.0,
-0.5.
-I.0-
-1.5-
-2.0
0.0
(c)
i . . . . I . . . . I
05 1.0 1.5
P~o
Fig. 16. Proposed interpolation scheme for Q using T stress and power law approximations. Solid lines are finite element results, dashed lines are from eq. (21).
Figure 17 shows the cleavage data of Sumpter and Forbes [4], and a cleavage toughness curve with n = 5 and ~rc/~ 0 = 4.25. The predicted curve correctly captures the trend of the data. Note that only two fracture tests are actually required to construct this toughness curve--a high constraint Q = 0 geometry to give J~c and any geometry with non-zero Q value to obtain ~c/~0. However, whether such a simple model can be used to reliably predict fracture toughness is debatable. It is perhaps most useful in illustrating trends and extrapolating existing data.
Next, failure by a ductile tearing mode is examined. Only initiation is dealt with here; stable ductile tearing has been examined in refs [26] and [27]. Crack opening profiles obtained for the power law hardening material at three different levels of T stress are shown in Fig. 18. The profiles
EFM 52/~-F
462 N.P. O'DOWD
025
0.2.0
0.15
0.10
0.05
A ~ o/W ,- 0.113 o ~ ~ / W , , 0 . ' r ¢
. . . . . . . . . R g l l -.it" 't.
÷ 't" ¢'t' , ~ ' ~ " " ~ ,t, * &
& 4" ~oLO°~°°]~ "
. . . . . . . . . . . . . ¢¢
• I. ! I 0.00 . . . . . . . . . ' ' ' ' . . . .
0.5 0.0 -0.5 -I.0 -I~
Q
Fig. 17. Data of Sumpter and Forbes [4] for mild steel at -50°C. Dashed line is toughness based on critical cleavage stress, a / a c = 4.25.
are all p lo t ted on the same scale. No te tha t crack opening d i sp lacement is somewha t higher for the negat ive T stress than for the zero and posi t ive T stresses. In ref. [9] the re la t ionship
J 6, = d ( n , Q ) - - (24)
o- 0
was p roposed . The crack opening d isp lacement , 6,, is defined as the y d isp lacement at the po in t where lines emana t ing f rom the crack tip at 45 ° to the init ial crack face intersect the de fo rmed crack face. This re la t ion general ises an earl ier result [28]. The dependence o f d on Q, ob ta ined f rom full
( T = 0
- 1
(c)
Fig. 18. Crack opening profiles from boundary layer analysis (a) T / a o = 0; (b) T / a o = --1; (c) T / a o = 1.
Two parameter approaches in elastic-plastic fracture mechanics 463
. . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . .
0.2 0.4 0.6 0.8 1.0 1.2 1.4
-Q
Fig. 19.(a) Dependence of crack tip opening displacement on Q. (b) Cleavage and tearing toughness curves using eqs (23) and (26).
field solutions, is shown in Fig. 19(a) for n = 10. (For Q = 0, d(10) ~ 0.5 which is consistent with the result of ref. [28].) Note that for negative Q, d is almost linear with Q. This relationship will hold strictly under small-scale yielding conditions and is expected to retain validity as fully yielded conditions are reached. If a ductile fracture criterion based on a critical crack opening displacement, 6c, is postulated, the criterion for fracture is:
6c = d(n, Q) Jc. (25) G o
For an n = 10 material, taking fie = 0.5J~c/ao, where J~c is the J value at which ductile tearing initiates when Q = 0, and rearranging leads to the simple relationship
Jc/J~c = 1/[2d(n, Q)]. (26)
Thus, Fig. 19(a) can be rearranged in the form of a toughness curve as shown by the dotted line in Fig. 19(b). Note that ductile toughness decreases slightly with increasing negative Q. The ductile tearing data of ref. [29] show that fracture initiation is very weakly dependent on constraint as measured by T/ao, but that the tearing modulus increases with decreasing constraint after some amount of growth.
A cleavage toughness locus is also included in Fig. 20(b) to indicate the competition between cleavage and ductile tearing. This curve uses eq. (23) with n = 10 and a¢/ao = 4. In the plot J is normalised by a different value for the cleavage and tearing mode, i.e. normalised by J~c for the tearing curve and by J~c for the cleavage curve. The ratio J~c/J~c has been taken arbitrarily to be 4. It would appear to be impossible to obtain J~c for tearing and cleavage for the same material--either it will fail by cleavage or tearing at Q = 0 giving either J~c or J~c; the other must be obtained from a theoretical model or by extrapolating experimental data.
7. THREE DIMENSIONAL EFFECTS--PLANE STRESS VS PLANE STRAIN
All the data presented so far have applied to plane strain conditions--real cracks are three dimensional, and the stress and strain state will vary through the thickness of the specimen. However, close to the crack front and away from the intersection of the crack front with the external surface of the body, the out-of-plane strain components are negligible compared to the in-plane singular terms so plane strain conditions prevail and thus the plane strain J -Q field is still applicable for three dimensional geometries. The Q values presented here and in the earlier work are applicable to through cracks in thick plates. For thin plates or elliptical shaped flaws the magnitude of J and Q will vary along the crack front and must be determined by a three dimensional analysis. An average measure of constraint was introduced in ref. [2] for three dimensional geometries: lfo Q,ve = -L Q(s) ds, (27)
1 i . 0 . . . . I . . . . I . . . . I ' ' ' ' I ' ' ' '
0.0 1.0 2.0 3.0 4.0 5.0
r
3.0
2.~-
It~ 2.o-
1.5-
SSY
............ P/Po = 0 .4
"-::il P o.o.6 • "P/Pof f i I .O
( b ) 1.0 . . . . I . . . . I ' ' ' ' 1 . . . . I ' ' ' '
0.0 1.0 2.0 3.0 4.0 5.0
r
5.0 t SSY
~.~, ............ P/Po ffi 0.4
. . . . . . P/Po = 0.6 a'°" 1"°
........... 22~ ~.
2.0
1.0 0.0 1.0 2.0 3.0 4.0 5.0
r
3.0
2.5-
®
I~ 2.o
1.5.
SSY
............ P/Po ffi 0.4
. . . . . . P/Po = 0.6
. . . . . . . P/Po ffi 1.0
( d ) 1.0 . . . . i . . . . J . . . . i . . . . i . . . .
0.0 1.0 2.0 3.0 4.0 5.0
r
Fig. 20. H o o p stresses ahead of the crack in plane strain and plane stress; (a) and (b) edge cracked tension, (c) and (d) edge cracked bend bar.
where s measures distance along the crack front and L is the crack length. The overall constraint for thick test geometries, Q .... is expected to be nearly identical to Q evaluated by plane strain analysis; Qave for thin specimens will be smaller than Q for plane strain. Dodds e t al. [30] have examined surface semi-elliptical cracks under biaxial and uniaxial loading. They found that Q values are reasonably independent of distance at different points along the crack front, but the magnitude of Q depends on location along the crack front.
It is often assumed that failure starts at the mid-plane since we expect the stresses to be most severe here. However, this may not necessarily be the case since both J and Q are varying through the thickness. Figure 20 provides plane strain and plane stress near tip fields for an edge cracked specimen in tension and bend. Under small-scale yielding conditions it is has been shown by Hutchinson [7] that the plane strain and plane stress solutions differ by a factor of ca 2 for n = 10 and this is reflected in the lower initial plane stress distribution. Note, however, that there is no loss of constraint as load is increased for the plane stress specimen, while the plane strain specimen shows a continuous reduction in constraint. If we consider a three dimensional crack with conditions varying from plane strain at the center to plane stress at the surface the condition may arise where the stress state is more severe away from the mid-plane. Indeed, for biaxial loading of semi-elliptical surface cracks Dodds e t al. [30] have found that the critical location for fracture is at ~b = 17 °, where ~ measures distance along the crack front, with ~b = 90 ° corresponding to the mid-plane.
8. DISCUSSION
This paper has examined constraint in elastic-plastic materials. It has investigated a two parameter representation of the near tip fields where the second parameter, Q, is a measure of the hydrostatic stress relative to a high triaxiality reference field. Although this representation was
Two parameter approaches in elastic-plastic fracture mechanics 465
developed for power law hardening materials it was observed that it can also apply for real materials--the particular material examined in the paper was an ASTM A-302 B steel. It was also observed that existing power law solutions may be used to estimate Q in real materials, though some care must be taken in specifying the material response. The application of the J-Q theory may be summarised as follows:
• A set of experiments should be carried out with geometries of different constraint to generate a Jc(Q) toughness curve such as in Fig. 18. It is recommended that true values of Q be obtained from an elastic-plastic finite element analysis using the actual material stress/strain response for the test geometries. Models such as described above may be used to extrapolate for data not obtainable from the experiments.
• In practise, estimates obtained using the elastic T stress and power law solutions, e.g. eq. (21), may be employed to determine Q values (provided such estimates are conservative). If the point (J, Q) lies below the toughness curve Jc(Q) then the structure is deemed to be safe.
The J(Q) fracture concept can be incorporated quite simply into existing procedures for fracture assessment such as the R6 procedure [31] and British Standards Document, PD6493 [32]. A framework for including constraint effects in existing procedures is discussed in ref. [20].
Acknowledgements--Helpful discussions with Dr J, D. Sumpter of the DRA, Dunfermline and Dr R. A. Ainsworth of Nuclear Electric plc are acknowledged. The computations reported were performed on the a DEC 5000/340, funding for which was provided by Nuclear Electric, plc. Some of the finite element analyses were carried out with the ABAQUS code which was made available under academic licence from Hibbitt, Karlsson and Sorensen, Inc., Providence, RI, U.S.A.
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Mechanics. American Society for Testing and Materials, Philadelphia (1994). [27] N. P. O'Dowd, Computational Mater. Sci. 3, 207 (1994). [28] C. F. Shih, J. Mech. Phys. Solids 29, 305 (1981). [29] J. W. Hancock, Proc. TWI /EWI / IS Int. Conf. Shallow Crack Fracture Mechanics Test and Applications. Cambridge,
U.K. (1992). [30] R. H. Dodds, C. F. Shih and T. L. Anderson, Int. J. Fracture 64, 101 (1993). [31] I. Milne, R. A. Ainsworth, A. R. Dowling and A. T. Stewart, Int. J. Press. Vess. Piping 32, 3-104 (1988). [32] British Standards Institution, Guidance on methods for assessing the acceptability of flaws in welded structures,
