J-INTEGRAL AND HRR FIELD ASSOCIATED WITH LARGE CRACK EXTENSION R. H. DRINNON Boeing Commercial Airplane Company, P.O. Box 3707, Seattle, WA 98124, U.S.A. A. S. KOBAYASHI Department of ~~h~icai
~n~n~~ng,
Unive~ty
of Was~ngton,
Seattle, WA 98195, U.S.A.
metric Moire with line density of 39.4 lines/mm (1000 lines/in.), was used to study the path independency of the J-integral and the adequacy of the HRR field displacement associated with large crack extensions up to 12 mm in 2024-O aluminum SEN specimens. The J-integral was not path ind~nd~t after a small stable crack growth of about 1.5 mm in this specimen. The HRR u- and v-displacements were also computed by using either the near- or far-field J-integral values. Despite some scatter, the results showed that for a small stable crack growth, the measured ~~jspla~rnent agreed better with the covalent linear elastic fracture mechanics (LEFM) displacements while the measured c-displacement nearly coincided with the HRR u-displacement, which was obtained by using the near-field J-values.
INTRODUCTION
past three years, Kang and Kobayashi[ 1,2] and Dadkhah and co-workers@-5J have the path independence of the J-integral[q and the validity of the HRR field[7,8] in an elastic-plastic field surrounding a s~tiona~ and a stably growing crack. For small stable crack growth of l-2 mm in aluminum alloy specimens, the J-integral was found to be path inde~ndent within ex~~rnen~l errors. These measured ~-integral values also differed substantially with the J-values which were obtained by using approximation formulas[c)]. The HRR field, however, ceased to be valid during the early part of plastic yielding since the measured crack tip displacement parallel to the crack did not conform with the corresponding HRR displacement. The measured crack tip displacement perpendicular to the crack, however, agreed with the corresponding WRR displacement. The above finding casted serious doubt on the popular use of the J-integral as a single parameter characte~~tion of ductile fracture where J represented, for a power hardening material, the strength of the crack tip plastic singularity of the HRR field. The invalid HRR field eliminated the physical link of J with the crack tip. Due to the dire consequences of the above finding, the associated crack tip strain fields were also determined and compared with the cor~s~nding HRR strain field. Such strain analysis reduced possible experimental error due to rigid body displacements, Both Dadkhah et al.[S] and Kang and Liu[lO] showed that the normal strain ~r~ndicular to the crack agreed well with the corresponding HRR strain but the normal strain parallel to the crack deviated substantially from its related HRR strain. In addition, Kang and Liu showed that for a region of 1S-2 mm from the crack tip and beyond, the elastic portion of the normal strain parallel to the crack is equal in magnitude with the plastic portion thus accounting, in part, for the discrepancy cited above. Motivated by these experimental findings, the computational mechanics community started to reinvestigate the anomaly in the HRR field. This, despite the two decades of intensive numerical analysis on the elastic-plastic crack tip field in which attention was inevitably focused only on the ‘dominant stresses and strains’, namely those ~~ndicular to the crack. Recent numerical analyses of a stationary crack under large-scale plastic yielding by Zhang and Ravi-Chandar[l l] are in q~~tive agreement with the above findings. The experimental analysis cited above was confined to a small crack extension. While previous numerical analyses[l2] have shown that the J-integral is no longer path independent under larger crack extension, of the order of 5 mm, experimental ve~fi~tion of such results, needless to mention DURING THE
studied
experimentally
R. H. DRINNON and A. S. KOBAYASHI
/
Moire grating 39.4 lines/nun
3.75 in.1
Material: 2024-Osheet aluminum alloy Thickness: 0.8mm (1132 inch)
e
m 50.8 mm (2.0 in.)
Fig. 1. Single edge notch specimen.
verification of the HRR field, is lacking. The purpose of this paper is to provide the missing experimental analysis.
J-evaluation The procedure for evaluating the J-integrat using the u- and u-displacement field obtained from Moire interferometry has been amply documented in previous papers[l-51. In essence, the procedure involves: (1) determining the power hardening stress-strain relation of the material; (2) determining the u- and u-displacement fields of the fracture specimen at various stages of stable crack growth; and (3) finally computing the J-integral along contours which enclose the crack tip. In the present study, Moire interferometry was replaced with geometric Moire due to the excessive large-scale yielding involved in the 2024-O aluminum alloy specimens.
Figure 1 shows the single edge notched @EN), 2024-O aluminum alloy specimen used in this study. The uniaxial stress-strain relation and its power hardening material constants of this 2024-O Table 1. 2024-O aluminum material properties Young’s modulus Poisson’s ratio Yield stress a n
74.2 GPa (10,800 ksi) 0.334 67 MPa (0.7 ksi) I.0 4
J-integral and HRR fieId
(a
1
u-field
1bf v-field Fig. 2. Typical u- and v-field Moire fringe patterns.
687
J-integral and HRR field
689
J kPA-m -
Near Field
-f---
far Field
+
Crack extenebn Fig. 3. J vs h
Daokhah
Aa
curve for 2024-O aluminum.
aluminum ahoy were determined previously[3] and are shown in Table 1. The machined initial crack of 19 mm length was sharpened by chevron notching it with a razor blade. Crossed Moire gratings of 39.4 lines/mm (1000 lines/in.) were bonded to the specimen using Micro-Measurements PC& adhesive.
The SEN specimen was loaded at a constant displacement rate of 0.05 cm/mm in a tabletop Instron testing machine to assure stable crack growth during testing The specimen grating was illuminated with two floodlamps. A photolastic Moire projector with a 39.4lines/mm reference grating and a Nikon 35 mm camera with a Nikkor 200 mmfi4 lens were used to record the u- and u-displacement fields for each incremental crack extension.
0 f)=fiOdegrees -
LEFM -lo!sw botmd
LEFM -upper bound -‘-‘” HRR --field J ---HRR-farfieIdJ
r, mm Fig. 4. u-displacement vs r for 2024-O ahninum:
Au = 10.0 mm.
690
R. H. DRINNON
and A. S. KOBAYASHI
-
LEFM -upper bound
-.-*- HRR -near field J ---HRR-farfield I
IO
1, mm
Fig. 5. Displa~m~nt
vs r for 2024-O aluminum; da = fO.Omm.
RESULTS Two specimens were loaded until a stable crack growth of about 12 mm was obtained. Figure 2 shows typical U- and v-field Moire fringe patterns together with some of the contours used in J-integral evaluation. Figure 3 shows the J-integral values which were obtained along the three 6.4 x 6.4, 12.7 x 12.7, and 19.1 x 19. I mm contours surrounding the crack tip in the two specimens. Due to the large scatter in J-values under a modest crack extension in excess of 2mm, a material-dependent J-resistance curve could not be drawn through these points. For comparison, the J-values obtained for the same material by Dadkhah and Kobayashi[3] are also shown in this figure. The results of Fig. 3 show that the J-integral is path independent for a small crack extension of about 2 mm (6% of the original remaining ligament). The J-integral progressively loses its path independency with increasing crack extension. These results are in qualitative agreement with those of ref. [12].
0
1
Cl
SpecImenNo.
1
m
SpecImenNo
2
NearFteld
-
HRR -
--
HRR - Far Field J
J
-‘-.-
LEFM - Lower Bound
- --
LEFM - Upper Bound
2
3
4
5
6
7
8
9
IO
,,
,2
Aa. mm
Fig. 6.
u vs
An for 2024-O aluminum; @= 60”. r = 3 mm.
,3
,Q
,5
691
J-integral and HRR field
Figures 4 and 5 are log-log plots of the u- and u-displacement variations with radial distance, t, from the crack tip, which extended 10 mm, at a polar angle of 8 = 60”. Also shown are the corresponding HRR displacements which were computed using either the near- or the far-field J-values. The displacement components for linear elastic fracture mechanics (LEFM), which are shown in Figs 4 and 5, were obtained by equating the J-value to the LEFM strain energy release rate from which the corresponding stress intensity factor and hence LEFM crack tip displacements were computed. The measured u-displacement approached that of LEFM for smaller radial distance, r, while the u-displacement coincided somewhat with the corresponding HRR displacement at r > 3 mm. Figures 6 and 7 show the variations in the u- and u-displacements, which were measured at r = 3 mm and 0 = 60”, with crack extension. Also shown are the corresponding HRR and LEFM displacements. Despite the scatter in measured data, these figures show that the u-displacement is closer to the LEFM u-displacement while the a-displacement follows the HRR u-displacement. These results are in qualitative agreement with previous findings[l-51. Figures 8 and 9 show typical normal strains variations, c, and L,.,,,with radial distance at 8 = 45” and 0”, respectively. Also shown are the corresponding HRR strains computed by using the far- and near-field J-values. The strain components are not influenced by rigid body displacements or the experimental uncertainty in the crack tip location and thus provide an alternate evaluation on the validity of the HRR field. These results show that both err and eYY agree with the corresponding HRR strain components at r > 10 mm which is beyond the region for an asymptotic solution, such as the HRR field, to be valid. In the valid region, say r < 3 mm, the measured and HRR strains of cYYappear to be in reasonable agreement with each other. The measured and HRR strains of erx differ, however, particularly for 8 = 45” of Fig. 8. The caustics generated by the large plastic deformation in the region of r < 1 mm prevented strains being measured in this closed-in region.
DISCUSSION
AND CONCLUSIONS
The J-integral is not path independent for crack extension larger than 2 mm. The v-displacement agreed with the HRR u-displacement for about 7 mm of crack extension. The u-displacement did not agree with the HRR u-displacement field. The measured c,,,,agreed with the corresponding
0 n
SpecImenNo. I Specimen NO. 2
-
HRR - Near Field J
--
HRR - Far Field J
-*-*- LEFM - Lower Bound - - - LEFM - Upper Bound
0
2
3
4
5
6
7
6
9
IO
II
12
Al&mm
Fig. 7. v vs Aa for 2024-Oaluminum; 0 = 60”, r = 3 mm.
13
I4
15
692
R. H. DRINNON
and A. S. KOBAYASHI
0
.*.__
5
10
15
Exx-exp Exx-HRR
2.5
20
Distance from crack tip, r (mm)
0
Eyy-exP
..*.-
5
f5 10 D&awe from awk tip, T bnml
E~~HRR
20
25
Fig. 8. Variations of t, and c,,, in 2024-O aluminum; Aa = 3.2 mm, @= 45 ~.
HRR strain for crack extension of about 7 mm. The measured 6, 7 however, did not agree with the corresponding HRR strain, While most of the results obtained in this large crack extension study were not unexpected, these results complement previous findings[l-51 on small crack extensions of Aa -=z3mm. One unexpected fmding is that the measured normal strains far away, i.e. r > lOmm, from the crack tip coincided with the computed HRR strains. Such results would have been expected in the far-field domain of a semi-infinite crack in an infinite domain, but this agreement occurred in the mid-region of the remaining ligament in the finite width SEN specimen.
693
J-integral and HRR field
0.06 r
04
0.0s 0.04 0.03 0.02 0.01 I
a~“..“‘.‘-‘.-..‘...“‘...J 5 0 Distance%m
20
25
crack 8, r (mm)
Fig. 9. Variations of L, and cYpin 2024-O ahrminum; u = 3.2 mm, 6 = 0”.
,&knowiedgemems-This research was funded by the QtBce of Naval Research under Contract
[email protected] authors are indebted to Dr. Yapa Rajapakse for his en~u~~ent and support during the course of this investigation.
REFERENCES B. S-J. Kang and A. S. Kobayashi, J-estimation prccedure based on Moire interferometry data. ASME J. Press. Vess. Tech&. 110, 291-300 (1988). B. S.-J. Kang and A. S. Kobayashi, Stable crack growth aluminum tensile specimens. J&p. Me& 27,234-245 (1987). M. S. Dadkhah and A. S. Kobayashi, HRR field of a moving crack, an experimental analysis. Engng Fracture Mech. 34, 253-262 (1989). M. S. Dadkhah, and A. S. Kobayashi, Further studies in the HRR field of a moving crack, an experimental anaiysis. J. Plasticity6, 633-650 (1990). M. S. Dadkhah, A. S. Kobayashi and W. L. Morris, J-R curves of four aluminum alloys. Frucfure MechanicsTwenty-third Symposircm, ASTM (to be published). J. R. Rice, Path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME J. appt. Mech. 35, Series E, 378-386 (1968). J. W. Hutchinson. Singular behaviour at the end of a tensile crack in a hardening material. J. Mech. Phys. So/& 16, 13-31 (1968). J. R. Rice and G. F. Rosengren, Plane strain deformation near a crack tip. J. Mech. Phys Solids 16, I-12 (1968). C. F. Shih, M. D. German and V. Kumar. An engineering approach for examining crack growth and stability in flawed structures. Inr. J. Press. Vess. Pipings 9, 159-196 (1981). B. S.-J. Kang and Q.-K Liu, Crack tip blunting, initation and growth anatysis of aluminum SEN specimens by Moire interferometry. Fracture Mechanics- Twenty-rhirdSymposium, ASTM (to be publish~).
694
R. I-I. DRINNON
[l I] Y. Zhang and K. Ravi-Chandar, A finite element Part I: Dominance of the HRR field. J. Mech. [12] C. F. Shih, R. G. deLorenzi and W. R. Andrews, Fracture (Edited by J. D. Landes, J. A. Begley
and A. S. KOBAYASHI
investigation into the foundation of elastic-plastic fracture mechanics. Phys Solids (to be published). Studies on crack initiation and stable crack growth, in Elastic-Plastic and G. A. Clarke). ASTM STP 668, 65-120 (1979).
(Received
15.Januar.v 1991)