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A numerical study of two integral type elasto-plastic fracture parameters under cyclic loading
Engineering Fracture Mechanics Printed in Great Bntain.
Vol. 26. No. 5, pp. 741-752.
A NUMERICAL ELASTO-PLASTIC
OOL-7944/87 $3.tm+.oa Pergamon Journals Ltd.
1987
STUDY OF TWO INTEGRAL TYPE FRACTURE PARAMETERS UNDER CYCLIC LOADING
A. K. WONG and R. JONES Aeronautical Research Laboratories, P.O. Box 4331, Melbourne, Victoria 3001, Australia Abstract-This paper examines the behaviour of two elasto-plastic fracture parameters which characterise the severity of a crack-tip field under large scale yielding. One may be regarded as a local crack-tip parameter which associates closely to the stress/strain fields at the crack-tip whilst the other may be considered as a far-field parameter. Several centre-cracked and edge-cracked plates were analysed under various loading and unloading sequences, and it was found that such parameters can be of significant use in explaining the effects of prior plastic deformation in fracture problems.
1. INTRODUCTION IN LINEAR elastic fracture mechanics, the stress intensity factor K has been universally used to characterise the severity of stresses near the tip of sharp cracks. However, many real fracture problems come under the category of ‘ductile fracture’, in which there may be substantial yielding around the crack-tip prior to crack growth. In such cases, analysis using linear elastic fracture mechanics does not suffice, and the significance of the stress intensity factor is lost. For a nonlinearly elastic material, a useful parameter known as the J-integral has been widely adopted for characterising the crack-tip stress-strain field. First derived by Eshelby[ I], and later reformulated by Cherepanov[2] and Rice[3], the J-integral is defined as
J=
Wn 1
(1)
where F is any simple path enclosing the crack-tip which starts and ends on the free surfaces of the crack (see Fig. l), W is the strain energy density, T is the traction vector on F, u is the displacement vector, xi and x2 are the local coordinates which are parallel and perpendicular to the crack-axis respectively, and n, dS = dx,. It can be shown that for an isotropic elastic (linear or non-linear) solid under isothermal and two-dimensional conditions, J is path-independent, and for self-similar crack growth, it equals the energy released per unit growth. As such, following Griffith’s theory on critical energy release rate, there exists a critical value Jcr which governs incipient fracture, and is an intrinsic property of the material. Although this is true strictly for elastic materials, the concept of using a critical value Jcr for predicting crack growth has been extended to elasto-plastic solids. Provided the yielding is of small scale (i.e. the yielding zone is small compared to the crack length), the J-integral has been
Crock
-
Fig. 1. Integration path for J. 741
142
A. K. WONG and R. JONES
found to be a useful fracture parameter, and standards (such as those recommended by the ASTM) for measuring Jcr have been established. However, these standards give varying results[4], and it is the present authors’ opinion that they are not entirely satisfactory. Under large scale yielding, .I cannot be considered as the energy release rate, nor can it be interpreted as the potential-energy difference between identical and identally loaded bodies whose crack lengths differ by an infinitesimal amount. Furthermore, under an arbitrary loading history, and for an incremental theory of plasticity, the physical meaning of J and its property of path-independence are lost[5]. In an attempt to extend the usefulness of the J approach into this regime, various integrals, all resembling more or less J, have been proposed as possible fracture parameters. Some of these are presented in the following section.
2. INTEGRAL
TYPE PARAMETERS
McClintock[6] suggested that for non-elastic materials, the strain energy density W is not an appropriate quantity to influence fracture, and subsequently, the W-term in the integrand was dropped to yield a modified integral J’. To circumvent the problem of path-dependertcy, a limit was taken so that the path collapsed onto the crack surfaces. That is,
J’ = hi
s -T
+dS
'ax,
r.
(2)
3
where E is a characteristic radial length of the path F,. Unfortunately, for a blunt crack with traction-free crack surfaces, J’ vanishes. Since it is expected that an elasto-plastic crack-tip parameter should be able to characterise both sharp and blunt cracks as ductile fracture invariably begins with the blunting of the crack-tip, it is therefore unlikely that J’ can serve as a crack-tip parameter. Kishimoto et ~I.[73 proposed a similar integral 3, where
in which Fp is the boundary Whilst J^ does not have zone is usually unclear, and Blackburn[8] proposed
of the process zone. the same problem as J’, the location of the boundary to the process would be expected to vary with loading conditions. an integral J* such that
J* = hi
I,..&i,
1 - T$
2 ‘1axj
dS. 1
(4)
By Green’s Theorem, eq. (4) (for a general 3-D solution) becomes
J* =
- T.g
dS 1
(5)
where F is any path enclosing the crack-tip, Vr and V, are the volumes enclosed by F and E respectively (see Fig. 2).
Elasto-plastic fracture parameters
743
J* as given by eq. (5) is considered to be path-independent in the sense that r can be arbitrarily chosen. However, it must be pointed out that due to the non-vanishing volume integral (area integral for the 2-D case) over Vr - V, as E + 0, the significance of the path-independence for J* cannot be compared to that for J. Because the calculation of J involves only a contour integral, the path can be located well away from the crack-tip region, thus avoiding any singularity, or just large inaccuracies in evaluating the integrand. For J* on the other hand, the limiting volume integral over Vr - V, would inevitably require the integration of terms around the crack-tip where it is most prone to numerical errors. One desirable feature of J* is that for linear elastic materials, it can be seen from eq. (5) that it is identical to the well-proved parameter J. However, for an elasto-plastic solid, the replacement of W by Oi,~aUi/axj)/2 seems to lack any physical basis as it disregards the role of the e - E relationship, and its ability to characterise a crack-tip field as suggested by Blackburn et aI.[9] and Batte et al.[lO] may therefore be somewhat dubious. More recently, Atluri er aI.[5], and Kuang[l l] have suggested an incremental approach to the J-integral for elasto-plastic materials. These authors proposed an incremental integral AJ, (AT,* in [S]) such that
AJ, = li_i
Aa dS, s rr
(6)
which, by Green’s Theorem, becomes
AJ, =
A@dS + li_i sr
(7)
where A@ = Awn, - A[T*(au/axr)], and cij, &iiare the mean stress and strain components over the load increment. The status of the crack-tip fields at the end of a load sequence P,, P,, . . . , PN is then postulated to be governed by
J, = ~AJ,. 1
It can be easily shown that for a deformation theory of plasticity, the volume integral in eq. (7) vanishes so that J, = J. Under an incremental theory however, this volume term does not necessarily vanish, and could therefore possibly account for the degree of accumulative damage due to local yielding of the crack-tip. A second incremental integral AT, (referred to as AJs here for consistency) was also introduced by Atluri et a/.[53 where
Fig. 2. Integration path for 5,.
744
A. K. WONG and R. JONES
AJs =
AadS,
(9)
ss in which S is the external boundary of the body. Again applying expressed as a path-independent integral, viz.,
AJs=iAOdS+i_V I
Green’s
Theorem,
{Asijg-Aoijz}dV. I-
AJ, may be
(10)
The above authors showed J, to be a measurable property as it corresponded to the incremental difference in the area under the load-displacement curves of two identical and identically loaded specimens. However, the interpretation of this to be the difference in potential energy may be questionable as, for an elasto-plastic material under load, the potential energy (i.e. the stored energy which is available to do work) would depend on the unloading characteristics of the material. Nevertheless, the fact that J, is a measurable property is appealing, and thus warrants further investigation. The purpose of this paper is to investigate the usefulness of the J, and J, integrals as elastoplastic crack-tip parameters for large scale yielding and under arbitary loading histories. To this end, several simple cracked specimens were modelled using a finite elements program, and J, and J, were evaluated and monitored under various loading-unloading situations. 3. NUMERICAL
IMPLEMENTATION
OF Js AND J,
Equation (7) shows that AJ, (and hence J,) is path-independent in the same sense as for J*. Consequently, it suffers from the same problem in that a volume integral over the crack-tip region needs to be performed. For an arbitrary load history, the nature of the stress/strain fields in this region is unknown so that the values computed in the crack-tip elements will be prone to errors. As a result, J, as calculated by eq. (7) would be inaccurate. To circumvent this problem, it is proposed here that the direct application of eq. (6) should be used in preference to eq. (7). The limiting process is simply achieved by means of extrapolation of values of AJ for a family of geometrically similar contours of diminishing size. An additional advantage of this method is that this process can provide some indication on the accuracy of the solution. Since it is known that J must be path-independent under a deformation theory or under a small-scale yielding, a simple comparison of values of J for different paths during initial loading can suggest that either mesh refinement is necessary or simply that the results for certain paths may not be dependable and should thus be excluded from being used to extrapolate J,. The evaluation of J, using the path-independent form [i.e. eq. (lo)] does not involve any singular terms as the region of integration is outside the crack-tip region. However, the use of eq. (10) has no advantage over the direct application of eq. (9). In fact, it is clear that the use of eq. (10) would increase the amount of computation necessary, and hence possibly decrease the accuracy. Consequently, eq. (9) is adopted in this work. The proposed method was implemented by modifying parts of the finite element package PAFEC. The modifications essentially involved the up-grading of the original capability of a linearelastic J-integral calculation performed on a single default path, to allow for a non-linear O--E relationship and to perform the integration over a number of user-defined paths. Currently, the extrapolation procedure is performed by a separate program using a kth order polynomial fitted through values of J for a family of N(k < N) concentric and geometrically similar paths located close to the crack-tip region. For the problems considered, k = 2 and N = 5 or 6 seemed to be reasonable. 4. TEST PROBLEMS The program was used to analyse both a centre-cracked plate and an edge-cracked plate under mode I conditions. The specimens have a width to height ratio of one and a crack-length to width ratio of 0.25. Because of symmetry, only one-quarter of the centre-cracked plate, and one-half of
Elasto-plastic
Pam
1
fracture
145
parameters
2 3 4
Fig. 3. Mesh I, J, is evaluated
on path
1; paths 2, 3 and 4 are used for J,.
the edge-cracked specimen were modelled. Two-dimensional isoparametric elements, comprising 6-noded triangular and 8-noded trapezoidal elements were used. A meshing scheme similar to that used by Blackburn et al.[9] (referred to as Mesh I herein), as well as a refined scheme (Mesh II) were adopted (see Figs 3 and 4, where the heavy lines denote chosen integration paths). The material was assumed to be bi-linear with an elastic modulus E = 204.5GPa, and a post-yield modulus E’ = E/l 1. The Poisson’s ratio was taken to be 0.3 and the strain at yield was taken to be 0.083%, corresponding to a yield stress by = 169.7 MPa. Various loading and unloading sequences were performed, and the results are presented below. Case 1. Monotonic loading of a centre-cracked plate under Plane strain To determine the adequacy of the meshes used, the centre-cracked plate was loaded to r~ = 0.960, over 31 load increments. At each load increment, the J-integral was calculated on the designated paths and the limiting integral J, extrapolated. It was found that for both mesh schemes, there was a general increase in J with an increase in path size, so that J, and J, formed an envelope containing all possible J values. It is clear from Fig. 5 that the envelope size is smaller for the finer mesh, indicating that at least part of the variation in J, or non path-independence, was due to numerical errors. Also shown in Fig. 5 is Js obtained by Blackburn et al.[9] for the same problem using a mesh similar to Mesh I. Whilst their result agrees well with the present analysis up to a load of approximately e = 0.8a,, considerable discrepancies exist for higher loads. A comparison of the plastic zones between the two analyses revealed a substantial difference at peak load (see Fig. 6). It appears that the program used in [9] gives a “softer” post-yield behaviour, and could be accounted for by possible differences in yield criterion, plastic stress-strain laws, or the load increment size used. In view of the above, it was concluded that all subsequent problems would be modelled with Mesh II, and that the maximum load would be limited to 0.70,. Case 2. Non-monotonic loading of a centre-cracked plate under The centre-cracked plate of Case I was subjected to a increase in load from 0 to 0.70,; a total unload followed by a as {c/e,.: 0, 0.7, 0, 0.7)). Plane stress conditions were assumed.
plane stress loading sequence consisting of an reload back to 0.7a, (denoted here J, and J, during the load sequence
746
A. K. WONG
path
2
1
\
and R. JONES
see Fig.
4b for
detail
b)
3
Fig. 4. (a) Mesh II, J, is evaluated
4
on path
5
678
1. (b)Crack-tip
region for Mesh II, paths 2-7 are used for J,.
are shown in Fig. 7. On the load-up segment, J was found to be relatively path-independent, giving a J, and J, which were almost identical. Upon unloading and reloading however, the two quickly deviated from one another. Because the plastic region did not reach the boundary at peak load [see Fig. 8(a)], J, returned to zero as the load was removed. It is also interesting to note that the reloading curve for J, was for all practical purposes identical to the unloading curve, indicating that Js was unable to detect the small amount of reverse yielding which took place during unloading [see Fig. 8(b)]. J, on the other hand, being closely associated to the crack-tip stress/strain fields, unloaded to a negative value, and on reloading, J, followed a path which was distinctly higher than the unloading curve. This shows J, to be much more sensitive to any plastic deformation experienced near the crack-tip, although it is not as yet entirely clear what is inferred by a negative
747
Elasto-plastic fracture parameters
0
01
-
Mesh
I icoarse)
- ---
Mesh
II
---
Blackbum
02
03
(fine1 et a\.[91
04
05
06
07
08
09
a/a,
Fig. 5. Js and J, for Case 1. Present Analysis Plastic zone a/$ 1 0.67 2 0.88 3 0.96
Blackburn et aL[91 Plastic zone a/o, 4 0.687 5 0.875 6 0.962
Fig. 6. Plastic zones for Case 1.
Case 3. Cyclic loading of a centre-cracked plate under plane strain It was seen in Case 2 that after one load-unload cycle, JI showed a substantial gain in value
upon reloading to peak load. One question which immediately comes to mind is: would ..!,continue to increase at peak load as the number of cycles is increased? In an attempt to find an answer to this question, the same plate was subjected to the load sequence. {C/C,: 0,0.7,0,0.7,0,0.7}. Because plane strain conditions were assumed in this case, thereby implying additional restraint to the structure, both Js and J,, as well as the size of the plastic region were lower than for the previous
748
A. K. WONG
and R. JONES
Fig. 7. J, and J, for Case 2.
Plastic zme
at peak l;d
/i
Fig. 8. Plastic zones for Case 2.
case (see Figs 9 and 10). It is also noted that the difference between the JS and J, curves for the load-up segment was more substantial in this case, indicating a lesser degree of path-independence for the J-integral. However, the shape of the Js and J, curves remained similar to the plane stress case. The most interesting feature was that whilst there was a large increase in J, from the first peak load to the second peak load, a much smaller increase was found between the second and the third peaks. This suggests that the major contribution to any accumulated damage due to plastic flow around the crack-tip is during the first occurences of yielding and reverse yielding. Case 4. Non-constant magnitude loading of a centre-cracked plate under plane stress In Cases 2 and 3, it was found that after reloading, J, reached a higher value at peak load after initial loading, whilst Js returned to essentially the same peak value. However, a feature common to both the Js and J, reloading curves is that, towards peak load, they both tended to a
749
Elasto-plastic fracture parameters
-I
0
01
02
04
03
05
06
Fig. 9. JS and J, for Case 3.
Plastic z&e
at peak load
Plastic zone after unloading
i
Fig. 10. Plastic zones for Case 3.
lower gradient compared with their respective initial load-up curves. Since one of the reasons for this work is to study the effects of prior plastic deformation (or cold working) on the residual strength of cracked components, it seems appropriate to ask whether these lower gradients would, if the load were to increase beyond the initial peak load, result in a less severe (in fracture mechanics terms) situation than if there had been no prior plastic deformation. To investigate this possibility, the centre-cracked plate of Case 2 was subjected to the load sequence {c/cry: 0,0.6, 0, 0.7). Figures 11 and 12 show respectively the variation in Js and J, for this loading sequence. Also shown in these figures are the curves for the monotonic loading to 0.70, (shown as dashed lines). The results on the reloading segment were somewhat surprising as both the Js and J, curves showed a rapid change in gradient when the load surpassed the initial peak load (0.60,), and managed to remain slightly above the dashed curves. This implies that both parameters indicated that no beneficial effects were achieved in this case, and if anything, a slight accumulation of damage was evident.
A. K. WONG and R. JONES
Fig. 11. J, for Case 4.
-,I 0
01
02
03
04
05
06
(
o/a,
Fig. 12. J, for Case 4.
Case 5. Edge-cracked plates
Using the same meshing scheme and material properties as for previous cases, one half of an edge-cracked plate was modelled and studied under both plane stress and plane strain conditions. A loading sequence {g/c,: 0, 0.5, 0, OS} was applied. The plots of Js and J, are presented in Figs 13 and 14. Again, the shapes of the J, and J, curves were similar to those shown by the centrecracked plate problems. It is noted that, like the centre-cracked plate, the difference between Js and J, during the initial load-up segment was more significant for plane strain than for plane stress. This could be attributed to the tri-axial stress state of the plane strain case, which caused a greater departure of the incremental theory from the deformation theory, and thus the possibility of non path-independency of the J-integral even under monotonic loading. 5. CONCLUSION
Two possible elasto-plastic fracture parameters, Js and J, have been examined numerically for several centre-cracked and edge-cracked plate problems under various loading conditions. The findings from these test cases may be summarised as follows: (1) J, appeared to be more sensitive in response to local yielding and reverse yielding around the crack-tip than J,. This is as expected as J, may be regarded as a local crack-tip parameter whereas Js is essentially a far-field parameter. (2) It is expected that the errors in the calculation of J, will be less than those for J, since the stress/strain fields away from the crack-tip are more amenable to calculation.
Elasto-plastic
2-
-l
-
Jt
----
Js
I
0
fracture
I
01
I 03
02
751
parameters
I 04
(
a/u,
Fig. 13. J, and J, for Case 5 (plane stress).
L
Js
-
----
Jt
I-
-I
I
0
I 01
I 02
I 03
I 04
a/a,
Fig. 14. J, and J, for Case 5 (plane strain).
(3) For a constant amplitude cyclic stressing sequence, both parameters indicated that the most damaging effects were experienced during the first load-unload cycle. This is shown by the relatively small deviations (particularly for J,) between the loading and unloading curves after the first cycle. (4) In Case 4, it was seen that prior tensile stressing did not serve any beneficial purpose in terms of strengthening the structure for subsequent tensile loadings at higher magnitudes. It may also be concluded from the behaviour of the J, curves (in all cases) that the initial plastic deformation cannot be beneficial for subsequent tensile loading at lower levels as J, for reloading was invariably higher than for the initial load-up curve. On the other hand, the interpretation of J, for lower subsequent loads is somewhat more difficult. This arises from the fact that J, can become negative over a substantial part of the loading range. As a specific example, for the plane strain edge-cracked plate in Case 5, J1 first became negative on unloading at 0 = 0.260, (which is approximately half of the load range), and on reloading, J, returned to the positive regime at 0 = 0.2a,, and crossed over (or became greater than) the load-up curve at 0 = 0.320,. It is not clear as yet what is implied by a negative J,. In the linearly elastic case, J (and hence J,) represents the energy available for crack growth, so that a negative value would infer a favourable situation. In the elasto-plastic case, such generalisation may not be made, and the meaning of a
752
A. K. WONG and R. JONES
negative J, is open to speculation. Blackburn et al.[9] interpreted a similar parameter J* to have a strengthening effect when unloaded to a negative value. If this interpretation is taken for J,, then for the above-mentioned edge-cracked plate problem, it seems reasonable to postulate that the initial loading to OSa, would be beneficial for subsequent cyclic loadings provided their maximums do not exceed 0.320,. However, if it is the J, range [i.e. J,(max) - J,(min)] which is important in determining failure, then J, would predict no beneficial effects being achieved by the pre-stressing. The above discussion shows the potential for such parameters in the analyses of elasto-plastic crack, problems. It however also highlights the difficulties in interpreting these parameters due to an incomplete knowledge of their physical significance. It is apparent that J, and J, convey different information about the stress/strain fields at the crack-tip, and no definitive statement on their suitability as elasto-plastic fracture parameters can be made without experimental validation. Indeed, work is currently being undertaken by the present authors towards this goal. It must also be mentioned that although there is a tendency to concentrate on Js for 2-D problems as J, is a measurable quantity, it is generally thought that the near-tip field plays a dominant role in 3-D problems[12]. Hence, it is important that an in-depth understanding be obtained for parameters such as J,. Another promising elasto-plastic crack-tip parameter is the strain energy density factor of Sih[ 133, and its load-cycle response, as well as its relationship to the integral type parameters studied in this work, will be the subject of a subsequent paper. Acknowledgements-This work was carried out as part of a Commonwealth Advisory Aeronautical Research Council Cooperative project on ductile fracture.
REFERENCES Cl1 J. D. Eshelby, The continuum theory of lattice defects, Solid PI G. P. Cherepanov, Prkl. Mat. Mekh. 25,476-488 (1967).
State Physics. 3, Academic Press. (1956).
c33 J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. appl. Mech. 34, 379-386 (1968). c41P. W. Beaver, A microgridding and replicating techniques for determining local plastic strain distribution under static and cyclic loading. Proc. Symp. Australian Fracture Group, pp, 12-21 (1985). PI S. N. Atluri, T. Nishioka and M. Nakagaki, Recent studies of energy integrals and their applications. Adu. Fracture Res. 1, 181-209 (1984). C61F. A. McClintock, Plasticity aspects of fracture, in Fracture, Vol. 3. (Edited by H. Liebowitz). Academic Press, New York (1971). Engng Fracture Meeh. 13, 841-850 PI K. Kishimoto, S. Akoi and M. Sakata, On the path inde~ndent integral-f. (1980). PI W. S. Blackburn, Path independent integrals to predict onset of crack instability in an elastic plastic material. lnt. J. Fracture Mech. 8, 343-346 (1972). E91W. S. Blackburn, A. D. Jackson and T. K. Hellen, An integral associated with the state of a crack tip in a non elastic material. lnt. J. Fracture 13, 183-199 (1977). Cl01A. D. Batte, W. S. Blackburn, A. Elsender, T. K. Hellen and A. D. Jackson, A comparison of the J* integral with other methods of yield fracture mechanics. Int. J. Fracture 21, 49-66 (1983). 11112. B. Kuang, Some problems in, the J-integral. Engng Fracture Me&. 19, 1161-1165 (1984). Cl21J. C. Newman, Jr and I. S. Raju,‘;Prediction of fatigue crack growth patterns ahd lives in three-dimensional cracked bodies. Ado. Fracture Res. 5, 1597-1608 (1984). El31G. C. Sih, A special theory of crack propagation, in Methods of Analysis nnd Solutions to Crack goblets (Edited by G. C. Sih). Wolters-Noordhoff (1972). (Receioed 11 April 1986)
Vol. 26. No. 5, pp. 741-752.
A NUMERICAL ELASTO-PLASTIC
OOL-7944/87 $3.tm+.oa Pergamon Journals Ltd.
1987
STUDY OF TWO INTEGRAL TYPE FRACTURE PARAMETERS UNDER CYCLIC LOADING
A. K. WONG and R. JONES Aeronautical Research Laboratories, P.O. Box 4331, Melbourne, Victoria 3001, Australia Abstract-This paper examines the behaviour of two elasto-plastic fracture parameters which characterise the severity of a crack-tip field under large scale yielding. One may be regarded as a local crack-tip parameter which associates closely to the stress/strain fields at the crack-tip whilst the other may be considered as a far-field parameter. Several centre-cracked and edge-cracked plates were analysed under various loading and unloading sequences, and it was found that such parameters can be of significant use in explaining the effects of prior plastic deformation in fracture problems.
1. INTRODUCTION IN LINEAR elastic fracture mechanics, the stress intensity factor K has been universally used to characterise the severity of stresses near the tip of sharp cracks. However, many real fracture problems come under the category of ‘ductile fracture’, in which there may be substantial yielding around the crack-tip prior to crack growth. In such cases, analysis using linear elastic fracture mechanics does not suffice, and the significance of the stress intensity factor is lost. For a nonlinearly elastic material, a useful parameter known as the J-integral has been widely adopted for characterising the crack-tip stress-strain field. First derived by Eshelby[ I], and later reformulated by Cherepanov[2] and Rice[3], the J-integral is defined as
J=
Wn 1
(1)
where F is any simple path enclosing the crack-tip which starts and ends on the free surfaces of the crack (see Fig. l), W is the strain energy density, T is the traction vector on F, u is the displacement vector, xi and x2 are the local coordinates which are parallel and perpendicular to the crack-axis respectively, and n, dS = dx,. It can be shown that for an isotropic elastic (linear or non-linear) solid under isothermal and two-dimensional conditions, J is path-independent, and for self-similar crack growth, it equals the energy released per unit growth. As such, following Griffith’s theory on critical energy release rate, there exists a critical value Jcr which governs incipient fracture, and is an intrinsic property of the material. Although this is true strictly for elastic materials, the concept of using a critical value Jcr for predicting crack growth has been extended to elasto-plastic solids. Provided the yielding is of small scale (i.e. the yielding zone is small compared to the crack length), the J-integral has been
Crock
-
Fig. 1. Integration path for J. 741
142
A. K. WONG and R. JONES
found to be a useful fracture parameter, and standards (such as those recommended by the ASTM) for measuring Jcr have been established. However, these standards give varying results[4], and it is the present authors’ opinion that they are not entirely satisfactory. Under large scale yielding, .I cannot be considered as the energy release rate, nor can it be interpreted as the potential-energy difference between identical and identally loaded bodies whose crack lengths differ by an infinitesimal amount. Furthermore, under an arbitrary loading history, and for an incremental theory of plasticity, the physical meaning of J and its property of path-independence are lost[5]. In an attempt to extend the usefulness of the J approach into this regime, various integrals, all resembling more or less J, have been proposed as possible fracture parameters. Some of these are presented in the following section.
2. INTEGRAL
TYPE PARAMETERS
McClintock[6] suggested that for non-elastic materials, the strain energy density W is not an appropriate quantity to influence fracture, and subsequently, the W-term in the integrand was dropped to yield a modified integral J’. To circumvent the problem of path-dependertcy, a limit was taken so that the path collapsed onto the crack surfaces. That is,
J’ = hi
s -T
+dS
'ax,
r.
(2)
3
where E is a characteristic radial length of the path F,. Unfortunately, for a blunt crack with traction-free crack surfaces, J’ vanishes. Since it is expected that an elasto-plastic crack-tip parameter should be able to characterise both sharp and blunt cracks as ductile fracture invariably begins with the blunting of the crack-tip, it is therefore unlikely that J’ can serve as a crack-tip parameter. Kishimoto et ~I.[73 proposed a similar integral 3, where
in which Fp is the boundary Whilst J^ does not have zone is usually unclear, and Blackburn[8] proposed
of the process zone. the same problem as J’, the location of the boundary to the process would be expected to vary with loading conditions. an integral J* such that
J* = hi
I,..&i,
1 - T$
2 ‘1axj
dS. 1
(4)
By Green’s Theorem, eq. (4) (for a general 3-D solution) becomes
J* =
- T.g
dS 1
(5)
where F is any path enclosing the crack-tip, Vr and V, are the volumes enclosed by F and E respectively (see Fig. 2).
Elasto-plastic fracture parameters
743
J* as given by eq. (5) is considered to be path-independent in the sense that r can be arbitrarily chosen. However, it must be pointed out that due to the non-vanishing volume integral (area integral for the 2-D case) over Vr - V, as E + 0, the significance of the path-independence for J* cannot be compared to that for J. Because the calculation of J involves only a contour integral, the path can be located well away from the crack-tip region, thus avoiding any singularity, or just large inaccuracies in evaluating the integrand. For J* on the other hand, the limiting volume integral over Vr - V, would inevitably require the integration of terms around the crack-tip where it is most prone to numerical errors. One desirable feature of J* is that for linear elastic materials, it can be seen from eq. (5) that it is identical to the well-proved parameter J. However, for an elasto-plastic solid, the replacement of W by Oi,~aUi/axj)/2 seems to lack any physical basis as it disregards the role of the e - E relationship, and its ability to characterise a crack-tip field as suggested by Blackburn et aI.[9] and Batte et al.[lO] may therefore be somewhat dubious. More recently, Atluri er aI.[5], and Kuang[l l] have suggested an incremental approach to the J-integral for elasto-plastic materials. These authors proposed an incremental integral AJ, (AT,* in [S]) such that
AJ, = li_i
Aa dS, s rr
(6)
which, by Green’s Theorem, becomes
AJ, =
A@dS + li_i sr
(7)
where A@ = Awn, - A[T*(au/axr)], and cij, &iiare the mean stress and strain components over the load increment. The status of the crack-tip fields at the end of a load sequence P,, P,, . . . , PN is then postulated to be governed by
J, = ~AJ,. 1
It can be easily shown that for a deformation theory of plasticity, the volume integral in eq. (7) vanishes so that J, = J. Under an incremental theory however, this volume term does not necessarily vanish, and could therefore possibly account for the degree of accumulative damage due to local yielding of the crack-tip. A second incremental integral AT, (referred to as AJs here for consistency) was also introduced by Atluri et a/.[53 where
Fig. 2. Integration path for 5,.
744
A. K. WONG and R. JONES
AJs =
AadS,
(9)
ss in which S is the external boundary of the body. Again applying expressed as a path-independent integral, viz.,
AJs=iAOdS+i_V I
Green’s
Theorem,
{Asijg-Aoijz}dV. I-
AJ, may be
(10)
The above authors showed J, to be a measurable property as it corresponded to the incremental difference in the area under the load-displacement curves of two identical and identically loaded specimens. However, the interpretation of this to be the difference in potential energy may be questionable as, for an elasto-plastic material under load, the potential energy (i.e. the stored energy which is available to do work) would depend on the unloading characteristics of the material. Nevertheless, the fact that J, is a measurable property is appealing, and thus warrants further investigation. The purpose of this paper is to investigate the usefulness of the J, and J, integrals as elastoplastic crack-tip parameters for large scale yielding and under arbitary loading histories. To this end, several simple cracked specimens were modelled using a finite elements program, and J, and J, were evaluated and monitored under various loading-unloading situations. 3. NUMERICAL
IMPLEMENTATION
OF Js AND J,
Equation (7) shows that AJ, (and hence J,) is path-independent in the same sense as for J*. Consequently, it suffers from the same problem in that a volume integral over the crack-tip region needs to be performed. For an arbitrary load history, the nature of the stress/strain fields in this region is unknown so that the values computed in the crack-tip elements will be prone to errors. As a result, J, as calculated by eq. (7) would be inaccurate. To circumvent this problem, it is proposed here that the direct application of eq. (6) should be used in preference to eq. (7). The limiting process is simply achieved by means of extrapolation of values of AJ for a family of geometrically similar contours of diminishing size. An additional advantage of this method is that this process can provide some indication on the accuracy of the solution. Since it is known that J must be path-independent under a deformation theory or under a small-scale yielding, a simple comparison of values of J for different paths during initial loading can suggest that either mesh refinement is necessary or simply that the results for certain paths may not be dependable and should thus be excluded from being used to extrapolate J,. The evaluation of J, using the path-independent form [i.e. eq. (lo)] does not involve any singular terms as the region of integration is outside the crack-tip region. However, the use of eq. (10) has no advantage over the direct application of eq. (9). In fact, it is clear that the use of eq. (10) would increase the amount of computation necessary, and hence possibly decrease the accuracy. Consequently, eq. (9) is adopted in this work. The proposed method was implemented by modifying parts of the finite element package PAFEC. The modifications essentially involved the up-grading of the original capability of a linearelastic J-integral calculation performed on a single default path, to allow for a non-linear O--E relationship and to perform the integration over a number of user-defined paths. Currently, the extrapolation procedure is performed by a separate program using a kth order polynomial fitted through values of J for a family of N(k < N) concentric and geometrically similar paths located close to the crack-tip region. For the problems considered, k = 2 and N = 5 or 6 seemed to be reasonable. 4. TEST PROBLEMS The program was used to analyse both a centre-cracked plate and an edge-cracked plate under mode I conditions. The specimens have a width to height ratio of one and a crack-length to width ratio of 0.25. Because of symmetry, only one-quarter of the centre-cracked plate, and one-half of
Elasto-plastic
Pam
1
fracture
145
parameters
2 3 4
Fig. 3. Mesh I, J, is evaluated
on path
1; paths 2, 3 and 4 are used for J,.
the edge-cracked specimen were modelled. Two-dimensional isoparametric elements, comprising 6-noded triangular and 8-noded trapezoidal elements were used. A meshing scheme similar to that used by Blackburn et al.[9] (referred to as Mesh I herein), as well as a refined scheme (Mesh II) were adopted (see Figs 3 and 4, where the heavy lines denote chosen integration paths). The material was assumed to be bi-linear with an elastic modulus E = 204.5GPa, and a post-yield modulus E’ = E/l 1. The Poisson’s ratio was taken to be 0.3 and the strain at yield was taken to be 0.083%, corresponding to a yield stress by = 169.7 MPa. Various loading and unloading sequences were performed, and the results are presented below. Case 1. Monotonic loading of a centre-cracked plate under Plane strain To determine the adequacy of the meshes used, the centre-cracked plate was loaded to r~ = 0.960, over 31 load increments. At each load increment, the J-integral was calculated on the designated paths and the limiting integral J, extrapolated. It was found that for both mesh schemes, there was a general increase in J with an increase in path size, so that J, and J, formed an envelope containing all possible J values. It is clear from Fig. 5 that the envelope size is smaller for the finer mesh, indicating that at least part of the variation in J, or non path-independence, was due to numerical errors. Also shown in Fig. 5 is Js obtained by Blackburn et al.[9] for the same problem using a mesh similar to Mesh I. Whilst their result agrees well with the present analysis up to a load of approximately e = 0.8a,, considerable discrepancies exist for higher loads. A comparison of the plastic zones between the two analyses revealed a substantial difference at peak load (see Fig. 6). It appears that the program used in [9] gives a “softer” post-yield behaviour, and could be accounted for by possible differences in yield criterion, plastic stress-strain laws, or the load increment size used. In view of the above, it was concluded that all subsequent problems would be modelled with Mesh II, and that the maximum load would be limited to 0.70,. Case 2. Non-monotonic loading of a centre-cracked plate under The centre-cracked plate of Case I was subjected to a increase in load from 0 to 0.70,; a total unload followed by a as {c/e,.: 0, 0.7, 0, 0.7)). Plane stress conditions were assumed.
plane stress loading sequence consisting of an reload back to 0.7a, (denoted here J, and J, during the load sequence
746
A. K. WONG
path
2
1
\
and R. JONES
see Fig.
4b for
detail
b)
3
Fig. 4. (a) Mesh II, J, is evaluated
4
on path
5
678
1. (b)Crack-tip
region for Mesh II, paths 2-7 are used for J,.
are shown in Fig. 7. On the load-up segment, J was found to be relatively path-independent, giving a J, and J, which were almost identical. Upon unloading and reloading however, the two quickly deviated from one another. Because the plastic region did not reach the boundary at peak load [see Fig. 8(a)], J, returned to zero as the load was removed. It is also interesting to note that the reloading curve for J, was for all practical purposes identical to the unloading curve, indicating that Js was unable to detect the small amount of reverse yielding which took place during unloading [see Fig. 8(b)]. J, on the other hand, being closely associated to the crack-tip stress/strain fields, unloaded to a negative value, and on reloading, J, followed a path which was distinctly higher than the unloading curve. This shows J, to be much more sensitive to any plastic deformation experienced near the crack-tip, although it is not as yet entirely clear what is inferred by a negative
747
Elasto-plastic fracture parameters
0
01
-
Mesh
I icoarse)
- ---
Mesh
II
---
Blackbum
02
03
(fine1 et a\.[91
04
05
06
07
08
09
a/a,
Fig. 5. Js and J, for Case 1. Present Analysis Plastic zone a/$ 1 0.67 2 0.88 3 0.96
Blackburn et aL[91 Plastic zone a/o, 4 0.687 5 0.875 6 0.962
Fig. 6. Plastic zones for Case 1.
Case 3. Cyclic loading of a centre-cracked plate under plane strain It was seen in Case 2 that after one load-unload cycle, JI showed a substantial gain in value
upon reloading to peak load. One question which immediately comes to mind is: would ..!,continue to increase at peak load as the number of cycles is increased? In an attempt to find an answer to this question, the same plate was subjected to the load sequence. {C/C,: 0,0.7,0,0.7,0,0.7}. Because plane strain conditions were assumed in this case, thereby implying additional restraint to the structure, both Js and J,, as well as the size of the plastic region were lower than for the previous
748
A. K. WONG
and R. JONES
Fig. 7. J, and J, for Case 2.
Plastic zme
at peak l;d
/i
Fig. 8. Plastic zones for Case 2.
case (see Figs 9 and 10). It is also noted that the difference between the JS and J, curves for the load-up segment was more substantial in this case, indicating a lesser degree of path-independence for the J-integral. However, the shape of the Js and J, curves remained similar to the plane stress case. The most interesting feature was that whilst there was a large increase in J, from the first peak load to the second peak load, a much smaller increase was found between the second and the third peaks. This suggests that the major contribution to any accumulated damage due to plastic flow around the crack-tip is during the first occurences of yielding and reverse yielding. Case 4. Non-constant magnitude loading of a centre-cracked plate under plane stress In Cases 2 and 3, it was found that after reloading, J, reached a higher value at peak load after initial loading, whilst Js returned to essentially the same peak value. However, a feature common to both the Js and J, reloading curves is that, towards peak load, they both tended to a
749
Elasto-plastic fracture parameters
-I
0
01
02
04
03
05
06
Fig. 9. JS and J, for Case 3.
Plastic z&e
at peak load
Plastic zone after unloading
i
Fig. 10. Plastic zones for Case 3.
lower gradient compared with their respective initial load-up curves. Since one of the reasons for this work is to study the effects of prior plastic deformation (or cold working) on the residual strength of cracked components, it seems appropriate to ask whether these lower gradients would, if the load were to increase beyond the initial peak load, result in a less severe (in fracture mechanics terms) situation than if there had been no prior plastic deformation. To investigate this possibility, the centre-cracked plate of Case 2 was subjected to the load sequence {c/cry: 0,0.6, 0, 0.7). Figures 11 and 12 show respectively the variation in Js and J, for this loading sequence. Also shown in these figures are the curves for the monotonic loading to 0.70, (shown as dashed lines). The results on the reloading segment were somewhat surprising as both the Js and J, curves showed a rapid change in gradient when the load surpassed the initial peak load (0.60,), and managed to remain slightly above the dashed curves. This implies that both parameters indicated that no beneficial effects were achieved in this case, and if anything, a slight accumulation of damage was evident.
A. K. WONG and R. JONES
Fig. 11. J, for Case 4.
-,I 0
01
02
03
04
05
06
(
o/a,
Fig. 12. J, for Case 4.
Case 5. Edge-cracked plates
Using the same meshing scheme and material properties as for previous cases, one half of an edge-cracked plate was modelled and studied under both plane stress and plane strain conditions. A loading sequence {g/c,: 0, 0.5, 0, OS} was applied. The plots of Js and J, are presented in Figs 13 and 14. Again, the shapes of the J, and J, curves were similar to those shown by the centrecracked plate problems. It is noted that, like the centre-cracked plate, the difference between Js and J, during the initial load-up segment was more significant for plane strain than for plane stress. This could be attributed to the tri-axial stress state of the plane strain case, which caused a greater departure of the incremental theory from the deformation theory, and thus the possibility of non path-independency of the J-integral even under monotonic loading. 5. CONCLUSION
Two possible elasto-plastic fracture parameters, Js and J, have been examined numerically for several centre-cracked and edge-cracked plate problems under various loading conditions. The findings from these test cases may be summarised as follows: (1) J, appeared to be more sensitive in response to local yielding and reverse yielding around the crack-tip than J,. This is as expected as J, may be regarded as a local crack-tip parameter whereas Js is essentially a far-field parameter. (2) It is expected that the errors in the calculation of J, will be less than those for J, since the stress/strain fields away from the crack-tip are more amenable to calculation.
Elasto-plastic
2-
-l
-
Jt
----
Js
I
0
fracture
I
01
I 03
02
751
parameters
I 04
(
a/u,
Fig. 13. J, and J, for Case 5 (plane stress).
L
Js
-
----
Jt
I-
-I
I
0
I 01
I 02
I 03
I 04
a/a,
Fig. 14. J, and J, for Case 5 (plane strain).
(3) For a constant amplitude cyclic stressing sequence, both parameters indicated that the most damaging effects were experienced during the first load-unload cycle. This is shown by the relatively small deviations (particularly for J,) between the loading and unloading curves after the first cycle. (4) In Case 4, it was seen that prior tensile stressing did not serve any beneficial purpose in terms of strengthening the structure for subsequent tensile loadings at higher magnitudes. It may also be concluded from the behaviour of the J, curves (in all cases) that the initial plastic deformation cannot be beneficial for subsequent tensile loading at lower levels as J, for reloading was invariably higher than for the initial load-up curve. On the other hand, the interpretation of J, for lower subsequent loads is somewhat more difficult. This arises from the fact that J, can become negative over a substantial part of the loading range. As a specific example, for the plane strain edge-cracked plate in Case 5, J1 first became negative on unloading at 0 = 0.260, (which is approximately half of the load range), and on reloading, J, returned to the positive regime at 0 = 0.2a,, and crossed over (or became greater than) the load-up curve at 0 = 0.320,. It is not clear as yet what is implied by a negative J,. In the linearly elastic case, J (and hence J,) represents the energy available for crack growth, so that a negative value would infer a favourable situation. In the elasto-plastic case, such generalisation may not be made, and the meaning of a
752
A. K. WONG and R. JONES
negative J, is open to speculation. Blackburn et al.[9] interpreted a similar parameter J* to have a strengthening effect when unloaded to a negative value. If this interpretation is taken for J,, then for the above-mentioned edge-cracked plate problem, it seems reasonable to postulate that the initial loading to OSa, would be beneficial for subsequent cyclic loadings provided their maximums do not exceed 0.320,. However, if it is the J, range [i.e. J,(max) - J,(min)] which is important in determining failure, then J, would predict no beneficial effects being achieved by the pre-stressing. The above discussion shows the potential for such parameters in the analyses of elasto-plastic crack, problems. It however also highlights the difficulties in interpreting these parameters due to an incomplete knowledge of their physical significance. It is apparent that J, and J, convey different information about the stress/strain fields at the crack-tip, and no definitive statement on their suitability as elasto-plastic fracture parameters can be made without experimental validation. Indeed, work is currently being undertaken by the present authors towards this goal. It must also be mentioned that although there is a tendency to concentrate on Js for 2-D problems as J, is a measurable quantity, it is generally thought that the near-tip field plays a dominant role in 3-D problems[12]. Hence, it is important that an in-depth understanding be obtained for parameters such as J,. Another promising elasto-plastic crack-tip parameter is the strain energy density factor of Sih[ 133, and its load-cycle response, as well as its relationship to the integral type parameters studied in this work, will be the subject of a subsequent paper. Acknowledgements-This work was carried out as part of a Commonwealth Advisory Aeronautical Research Council Cooperative project on ductile fracture.
REFERENCES Cl1 J. D. Eshelby, The continuum theory of lattice defects, Solid PI G. P. Cherepanov, Prkl. Mat. Mekh. 25,476-488 (1967).
State Physics. 3, Academic Press. (1956).
c33 J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. appl. Mech. 34, 379-386 (1968). c41P. W. Beaver, A microgridding and replicating techniques for determining local plastic strain distribution under static and cyclic loading. Proc. Symp. Australian Fracture Group, pp, 12-21 (1985). PI S. N. Atluri, T. Nishioka and M. Nakagaki, Recent studies of energy integrals and their applications. Adu. Fracture Res. 1, 181-209 (1984). C61F. A. McClintock, Plasticity aspects of fracture, in Fracture, Vol. 3. (Edited by H. Liebowitz). Academic Press, New York (1971). Engng Fracture Meeh. 13, 841-850 PI K. Kishimoto, S. Akoi and M. Sakata, On the path inde~ndent integral-f. (1980). PI W. S. Blackburn, Path independent integrals to predict onset of crack instability in an elastic plastic material. lnt. J. Fracture Mech. 8, 343-346 (1972). E91W. S. Blackburn, A. D. Jackson and T. K. Hellen, An integral associated with the state of a crack tip in a non elastic material. lnt. J. Fracture 13, 183-199 (1977). Cl01A. D. Batte, W. S. Blackburn, A. Elsender, T. K. Hellen and A. D. Jackson, A comparison of the J* integral with other methods of yield fracture mechanics. Int. J. Fracture 21, 49-66 (1983). 11112. B. Kuang, Some problems in, the J-integral. Engng Fracture Me&. 19, 1161-1165 (1984). Cl21J. C. Newman, Jr and I. S. Raju,‘;Prediction of fatigue crack growth patterns ahd lives in three-dimensional cracked bodies. Ado. Fracture Res. 5, 1597-1608 (1984). El31G. C. Sih, A special theory of crack propagation, in Methods of Analysis nnd Solutions to Crack goblets (Edited by G. C. Sih). Wolters-Noordhoff (1972). (Receioed 11 April 1986)