F. P. Chiang, MEMBER SPIET. V. HareeshB. C. LiuS. LiState University of New York at Stony BrookLaboratory for Experimental Mechanics ResearchStony Brook, New York 11794 -2300
Abstract. This paper examines the experimental u1, u2, and u3 displace-ments obtained in the vicinity of a plastically deformed crack tip in differentwork hardening materials in the background of the Hutchinson -Rice-Rosengren (HRR) field equations. It is shown that the two -dimensionalplane stress solution breaks down in the immediate vicinity of the cracktip because of the three -dimensional nature of the deformations due tofinite plate thickness. This "inner limit" seems to vary from 0.75 to 1.5times the thickness with directional dependence. Some light is also shedon the "outer limit" of HRR equations, beyond which the theory ceasesto be valid under the influence of the surrounding elastic field or thephysical boundary of the plate or a combination of both.
Subject terms: photomechanics; moire methods; elastic - plastic fracture; work hard-ening; three -dimensional deformations.
1. INTRODUCTIONLately there has been considerable interest in experimentallyinvestigating the nature of the prevailing displacement, strain,and stress fields in the vicinity of a deformed crack tip. Moreoften than not, the crack tip fields are described by Westergaard'sequations under elastic or small scale yielding conditions andby the Hutchinson -Rice -Rosengren (HRR) equationsl'2 whenyielding is significant and comparable to the uncracked specimenlength. These are simple plane stress or plane strain modelsbased on classical 2 -D assumptions. However, in practice, oneoften uses materials with finite thickness that are approximatedto either plane stress or plane strain conditions. It is well knownthat the existing field near the deforming crack tip is highlycomplex and three -dimensional. This results in the breakdownof 2 -D assumptions in a region close to the crack tip. Rosakisand Ravi Chandar3 investigated these 3 -D effects using the methodof caustics in elastic situations. They observed that plane stress
assumptions are valid beyond about one -half of the plate thick-ness. Chiang and Hareesh4 have used a combined moire methodto study deformation fields near a plastically deformed crack tipin a low hardening aluminum. Results suggest that there is azone of 3 -D effects around the crack tip that could vary from0.75 to 1.5 times the plate thickness with directional dependence.Recently, Zehnder et al.5 have used the method of reflectioncaustics and finite elements to imply that HRR field equationsunder plane stress conditions have an outer limit of validity.Their results show that this limit seems to occur at about a thirdof Irwin's plastic zone length factor. Kang et aí.6'7 have alsoobserved, in their displacement fields obtained from white lightinterferometry, that the HRR field prevails near the crack tip.Numerical calculations performed by Shih8 show that J- dominancein plane strain conditions prevails over short distances of 1% to7% of the uncracked ligament length, depending upon whetherthe crack is under pure tension or pure bending. It is anticipatedto prevail over longer lengths when the crack is under planestress.*
In this study we further investigate the inner (3 -D zone) andouter limits of HRR displacement and strain descriptions in sin-gle edge notch (SEN) samples of materials with various hard-ening indices. The sample geometry chosen is such that thematerial could globally be assumed to be under plane stressloading.
2. EXPERIMENTAL TECHNIQUEA combined in -plane and out -of -plane moire method4 is used inthis study to simultaneously obtain all three components of dis-placement Ili , u2, and u3 along the xl, x2, and x3 directions,respectively. The experimental arrangement is shown in Fig.1(a), and the specimen geometry is shown in Fig. 1(b). To begin
F. P. Chiang, MEMBER SPIE T. V. Hareesh B. C. Liu S.LiState University of New York at Stony Brook Laboratory for Experimental Mechanics Research Stony Brook, New York 11794-2300
Abstract. This paper examines the experimental u 1r u 2, and u3 displace ments obtained in the vicinity of a plastically deformed crack tip in different work hardening materials in the background of the Hutchinson-Rice- Rosengren (HRR) field equations. It is shown that the two-dimensional plane stress solution breaks down in the immediate vicinity of the crack tip because of the three-dimensional nature of the deformations due to finite plate thickness. This "inner limit" seems to vary from 0.75 to 1.5 times the thickness with directional dependence. Some light is also shed on the "outer limit" of HRR equations, beyond which the theory ceases to be valid under the influence of the surrounding elastic field or the physical boundary of the plate or a combination of both.
Lately there has been considerable interest in experimentally investigating the nature of the prevailing displacement, strain, and stress fields in the vicinity of a deformed crack tip. More often than not, the crack tip fields are described by Westergaard's equations under elastic or small scale yielding conditions and by the Hutchinson-Rice-Rosengren (HRR) equations 1 ' 2 when yielding is significant and comparable to the uncracked specimen length. These are simple plane stress or plane strain models based on classical 2-D assumptions. However, in practice, one often uses materials with finite thickness that are approximated to either plane stress or plane strain conditions. It is well known that the existing field near the deforming crack tip is highly complex and three-dimensional. This results in the breakdown of 2-D assumptions in a region close to the crack tip. Rosakis and Ravi Chandar3 investigated these 3-D effects using the method of caustics in elastic situations. They observed that plane stress
assumptions are valid beyond about one-half of the plate thick ness. Chiang and Hareesh4 have used a combined moire method to study deformation fields near a plastically deformed crack tip in a low hardening aluminum. Results suggest that there is a zone of 3-D effects around the crack tip that could vary from 0.75 to 1.5 times the plate thickness with directional dependence. Recently, Zehnder et al. 5 have used the method of reflection caustics and finite elements to imply that HRR field equations under plane stress conditions have an outer limit of validity. Their results show that this limit seems to occur at about a third of Irwin's plastic zone length factor. Kang et al. 6 '7 have also observed, in their displacement fields obtained from white light interferometry, that the HRR field prevails near the crack tip. Numerical calculations performed by Shih8 show that J-dominance in plane strain conditions prevails over short distances of 1% to 7% of the uncracked ligament length, depending upon whether the crack is under pure tension or pure bending. It is anticipated to prevail over longer lengths when the crack is under plane stress.*
In this study we further investigate the inner (3-D zone) and outer limits of HRR displacement and strain descriptions in sin gle edge notch (SEN) samples of materials with various hard ening indices. The sample geometry chosen is such that the material could globally be assumed to be under plane stress loading.
2. EXPERIMENTAL TECHNIQUE
A combined in-plane and out-of-plane moire method4 is used in this study to simultaneously obtain all three components of dis placement ui, U2, and us along the xi, \2, and xs directions, respectively. The experimental arrangement is shown in Fig. l(a), and the specimen geometry is shown in Fig. l(b). To begin
with, the SEN specimen is printed with a cross grating of pitchp using a photoengraving process. Typical values of p in thisstudy are 50.8 p.m and 25.4 µm.
A He -Ne laser beam is first expanded by a microscope ob-jective and then collimated by a field lens before it impingesupon a line grating. The regular and diffracted wavefronts arecollected by a second field lens, which forms the diffractionspectrum of the grating at its focal plane in the form of a seriesof equally spaced bright dots called diffraction orders. All theorders are blocked by a mask except the ± 1 orders, which arecollected by a third lens to form two nearly collimated beamswith an angle 2a between them to impinge upon the specimen.Within the intersecting beams there exists a standing wave ofperiod (or pitch)
where X is the wavelength.When a specimen is placed in this field, it is covered by two
patterns. One is the projected grating from the standing wavewith a pitch
X9 2sinacosß
(2)
where ß is the angle between the normal to the plate and theoptical axis of the projection system. Another is the photograph-ically printed pattern on the specimen surface. This entire opticalfield on the specimen is recorded before load is applied (mastergrating) and after load is applied (deformed grating). Severaldeformed gratings are recorded, corresponding to different loadlevels. An enlargement of one such recording is shown in Fig.1(c). Each of the deformed gratings contains the informationcorresponding to all three displacement components. All thesecan be delineated in the form of contour fringes of displacementcomponent through a process called optical spatial filtering.9 Asystem for performing filtering is shown schematically in Fig. 2.A superimposed pair of master grating and deformed grating isplaced in a coherent light field to display the spatial frequencycontent of the recordings at the Fourier transform plane of thefirst field lens. By appropriately selecting the diffraction orderat this plane to be collected by the second lens, one can obtaincontour fringes of u t , u2, and u3. Their governing equations are
ut =Np ,
u2=N'p ,
N"),u3 4tanasinß
where N, N', and N" = 0, ±1, ±2, ±3, etc.
(3)
(4)
(5)
3. EXPERIMENTSThree aluminum alloys of different hardening indices (n) wereused in this study: A12024 -0 (n ~ 3), A15052 -H32 (n ~ 7), andA16061 -T6 (n 18). We calibrated the materials by using asimple tension test and fitting a Ramberg -Osgood type of curve,i.e.,
CHIANG, HAREESH, LIU, LI
Specimen
Diffraction Spectrum \ /
vVX7-^v*xLine Grating
^ specklegram
Camera
(a)
O (t) O
—-U5|——
o I o090
^(b)
Fig. 1. (a) Experimental setup, (b) Specimen geometry (dimensions in millimeters), (c) Enlarged recorded grating.
with, the SEN specimen is printed with a cross grating of pitch p using a photoengraving process. Typical values of p in this study are 50.8 jxm and 25.4 jxm.
A He-Ne laser beam is first expanded by a microscope ob jective and then collimated by a field lens before it impinges upon a line grating. The regular and diffracted wavefronts are collected by a second field lens, which forms the diffraction spectrum of the grating at its focal plane in the form of a series of equally spaced bright dots called diffraction orders. All the orders are blocked by a mask except the ± 1 orders, which are collected by a third lens to form two nearly collimated beams with an angle 2a between them to impinge upon the specimen. Within the intersecting beams there exists a standing wave of period (or pitch)
q' =2sina
(1)
laser beai
transform lens
reconstruction lens
image plane
Fig. 2. Schematic for optical spatial filtering.
where \ is the wavelength.When a specimen is placed in this field, it is covered by two
patterns. One is the projected grating from the standing wave with a pitch
2sinacosp(2)
where p is the angle between the normal to the plate and the optical axis of the projection system. Another is the photograph ically printed pattern on the specimen surface. This entire optical field on the specimen is recorded before load is applied (master grating) and after load is applied (deformed grating). Several deformed gratings are recorded, corresponding to different load levels. An enlargement of one such recording is shown in Fig. l(c). Each of the deformed gratings contains the information corresponding to all three displacement components. All these can be delineated in the form of contour fringes of displacement component through a process called optical spatial filtering. 9 A system for performing filtering is shown schematically in Fig. 2. A superimposed pair of master grating and deformed grating is placed in a coherent light field to display the spatial frequency content of the recordings at the Fourier transform plane of the first field lens. By appropriately selecting the diffraction order at this plane to be collected by the second lens, one can obtain contour fringes of m, U2, and us. Their governing equations are
ui = Np u2 = N'p
u3 =N"\
4tanasin(3
(3)
(4)
(5)
where N, N', and N" = 0, ± 1, ±2, ±3, etc.
3. EXPERIMENTSThree aluminum alloys of different hardening indices (n) were used in this study: A12024-0 (n * 3), A15052-H32 (n « 7), and A16061-T6 (n « 18). We calibrated the materials by using a simple tension test and fitting a Ramberg-Osgood type of curve, i.e.,
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n
>Eo
E=á +a¡v
1\1io io
OPTICAL ANALYSIS OF HRR FIELD
(6)
to the data points. In the above equation, vo is the yield stress,E0 is the yield strain, n is the hardening index, and a is a materialconstant.
SEN samples were made from 3.17 mm thick sheets withrolling direction parallel to the loading axis of the specimen.Initially, notches 12 mm long were cut into each specimen bythe electrodischarge machining (EDM) process. Fatigue crackswere grown in these EDM samples by subjecting them to sin-usoidal tensile -to- tensile cyclic loads. The cyclic load levelswere kept to a small fraction of the yield stress (typically 30%of io), and notches were extended to about 14.5 mm. Thesesamples were then subjected to monotonically increasing load,and deformed gratings were recorded at different load levels.The filtering process mentioned in the previous section resultedin the typical fringe patterns shown in Fig. 3. Here all threedisplacement contours are shown for the three materials chosenat the indicated load levels. Each fringe represents a displace-ment value of one pitch.
4. EXPERIMENTAL RESULTS COMPARED TO HRREQUATIONS
HRR singularity descriptions 1,2 are widely accepted and usedin the study of plastically deformed cracks in work hardeningmaterials. The dominant strain -stress relationship based ondeformation theory of plasticity for a hardening material isgiven by
31 (7)
where E is Young's modulus and sÿ is the stress deviator of a;i.Here ie is the effective stress given by ae = V(3 /2)(s;is;i)Using the stress function approach, Hutchinson, Rice, and Ro-sengren derived plane stress field equations for strain and dis-placement components as
E¡J(r, 9) = aEoCN(n +1)E,J(e, n) , (8)
u ¡(r, e) = aEorCn/(n+1)û
¡(e, n) , (9)
where
C= J
aiaEaInr
Under plane stress conditions, the out -of -plane displacement u3can be expressed in terms of average u3 direction strain:
2u3E33 -
h= -(ER + Egg), (10)
where h is the thickness of the undeformed sheet. In the aboveequations, J is Rice's path -independent contour integral aroundthe crack tip, given by
and In is a constant dependent on hardening index n. Terms @iiand ûi are functions of the polar coordinate O and n. Since thecrack growth is controlled largely by local phenomena at thecrack tip, Shih10 has hypothesized that the crack tip openingdisplacement (CTOD) bt should be a measure of the damage atthe crack tip. He has shown that CTOD is related to J by
st = (aEo)1/n Dnio
where Dn is a function of n. Tabulations of all these constantsare available in Ref. 11.
Figure 4 compares typical normalized displacements u2 andu3 obtained from experimental results with those calculated bythe HRR equations. Here 8t is used for normalizing the dis-placements, and plate thickness h is used for the polar coordinater. Experimental St were measured by counting the total numberof fringes between the lower and upper lips of the crack up toabout 1 to 2 mm behind the visible crack tip position, wherethe crack lips are parallel. Plate thickness h is used to normalizer to bring out the 3 -D effects around the crack tip, where plane
to the data points. In the above equation, a0 is the yield stress, e0 is the yield strain, n is the hardening index, and a is a material constant.
SEN samples were made from 3.17 mm thick sheets with rolling direction parallel to the loading axis of the specimen. Initially, notches 12 mm long were cut into each specimen by the electrodischarge machining (EDM) process. Fatigue cracks were grown in these EDM samples by subjecting them to sin usoidal tensile-to-tensile cyclic loads. The cyclic load levels were kept to a small fraction of the yield stress (typically 30% of a0), and notches were extended to about 14.5 mm. These samples were then subjected to monotonically increasing load, and deformed gratings were recorded at different load levels. The filtering process mentioned in the previous section resulted in the typical fringe patterns shown in Fig. 3. Here all three displacement contours are shown for the three materials chosen at the indicated load levels. Each fringe represents a displace ment value of one pitch.
4. EXPERIMENTAL RESULTS COMPARED TO HRR EQUATIONSHRR singularity descriptions l ' 2 are widely accepted and used in the study of plastically deformed cracks in work hardening materials. The dominant strain-stress relationship based on deformation theory of plasticity for a hardening material is given by
AI2024-0 (CTX/CTO = 2.04): m, u2 = 0.0508 mm/fringe, u3 = 0.036 mm/fringe
AI5052-H32 (<rJ&0 = 0.88); u 1f u2 = 0.0254 mm/fringe, u3 = 0.032 mm/fringe
where E is Young's modulus and Sy is the stress deviator of CTJJ. Here ae is the effective stress given by ae = V(3/2)(sySij). Using the stress function approach, Hutchinson, Rice, and Ro- sengren derived plane stress field equations for strain and dis placement components as
m(r, 6) =
where
C =
, n)
, n)
(8)
(9)
Under plane stress conditions, the out-of-plane displacement can be expressed in terms of average U3 direction strain:
>- 4e) . (io)n
where h is the thickness of the undeformed sheet. In the above equations, J is Rice's path-independent contour integral around the crack tip, given by
J = I Wdx2 <TijnjUi,Xlds , 'r
where F is the path of integration,
Material <TO kg/mm2
Al 2024-0 5.0Al 5052-H32 17.0Al 6061-T6 28.0
e0 « n
0.00067 1.9 3.050.0023 3.75 7.30.004 1.22 18.0
Fig. 3. Crack tip displacement fields.
W = J o-jjdeij ,
and In is a constant dependent on hardening index n. Terms ly and ui are functions of the polar coordinate 6 and n. Since the crack growth is controlled largely by local phenomena at the crack tip, Shih 10 has hypothesized that the crack tip opening displacement (CTOD) 8t should be a measure of the damage at the crack tip. He has shown that CTOD is related to J by
5t = (cte0) 1/n Dn - (H)
where Dn is a function of n. Tabulations of all these constants are available in Ref. 11.
Figure 4 compares typical normalized displacements U2 and ua obtained from experimental results with those calculated by the HRR equations. Here 8t is used for normalizing the dis placements, and plate thickness h is used for the polar coordinate r. Experimental 8t were measured by counting the total number of fringes between the lower and upper lips of the crack up to about 1 to 2 mm behind the visible crack tip position, where the crack lips are parallel. Plate thickness h is used to normalize r to bring out the 3-D effects around the crack tip, where plane
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c°mE
Ñ °fa) ;ô
$
o
°Eóz
0.0 2.3 3.0
Ir/h)
CHIANG, HAREESH, LIU, LI
5.0
0
- HRR equationso Experimental
0.0
(r/h)
Fig. 4. Normalized displacements compared with plane stress HRRequations for (a) u2 and (b) u3.
stress assumptions break down owing to the finite plate thicknessand crack tip blunting. In these figures, note that in the immediatevicinity of the crack tip there are marked deviations betweenexperimental measurement and theory. The out -of -plane dis-placement u3 is finite at the crack tip, unlike the theoreticalprediction. To estimate the size of this region of dominant 3 -Deffects, we chose the boundary of the region to be at length r,where the difference between the HRR model and the experi-mental measurements of normalized displacements is ±0.025or more. Several experiments were done with three differentmaterials, and results were plotted. In each case, the boundaryof the 3 -D effects was measured using the above criterion alongdiscrete directions ahead of the crack tip. Figure 5 is a compositeof all such measurements. Despite the experimental scatter, thesedata points seem to have a regular trend showing that the 3 -Dzone size is about 0.75h to 1.5h with direction dependence. Thisforms the "inner limit" of the HRR equations, beyond whichthe 2 -D model seems to agree with the experimental observa-tions.
Figures 6(a) and 6(b) are logarithmic plots of fringe order(equivalent to the opening displacement component u2) versusr along different directions from the crack tip for two differentmaterials. The solid line represents the slope [1/(n + 1)] of theseexperimental curves based on the HRR equations. Note that ineach of these displacement plots there seems to be a region ofconstant slope (approximately that indicated by the solid lines)in a region where r ranges from about 5 mm to 13 to 15 mm.Outside this region, the slopes differ from those indicated bythe HRR equations. In Fig. 6(c), the log of the opening strain£22 is plotted against log (r) along A = 0° (xi axis). The HRRfield predicts a slope of -n /(n + 1) for the strain component,
A: a. = 1.9ao (A12024-0)13: a: = 0.85ao (A15052-H32)C: = 0.69ao (A16061-T6)D: a. = 0.88ao(A15052-432)E: a: = 0.670.0 (A16061-T6)
A
(r /h) = 1
Fig. 5. Estimated 3 -D zone around the crack tip.
which is represented by the solid lines. The two types of trianglesare the strain values calculated from the fringe patterns shownin the figures. Here again, there is a region of constant slope.In these logarithmic plots, the initial deviation in the slope, asexplained earlier, is due to the 3 -D effects near the crack tip.Deviation beyond r = 13 to 15 mm can be looked at as the"outer limit" of the HRR fields, where the singularity descrip-tions cease to dominate owing to the boundary effects of thespecimen or the elastic field surrounding it.
The results obtained from the A15052 -H32 sample, for whichthe outer limit is at about 14 mm, are in fair agreement withthose of Zehnder et al.5 for a steel with similar hardening index(n ^ 8.5). Their investigations showed that the outer limit ofthe HRR field is at about a third of Irwin's plastic zone lengthfactor rp, given by
2
rp =1
Q
and the stress intensity factor K, given by
(12)
K = com rra' , (13)
where a' is the corrected crack length to account for the devia-tions from small scale yielding assumptions,12 c is a constant toaccount for geometrical effects of the specimen,13 and v. is thefar -field applied stress. Equations (12) and (13) give a value ofrp /3 equal to 12.5 mm, which is close to the value of 14 mmshown by plots 6(b) and 6(c) and where the curves begin tochange slope. However, when similar calculations were doneon the A12024 -0 results, rp /3 turned out to be 53 mm, althoughthe plots still show slope deviation at around 13 to 14 mm. Thiscould be because A12024 -0 is a soft variety of aluminum andthe applied far -field stress (2.04a.) had already resulted in netsection yielding of the specimen. Thus, the above approach ofestimating plastic zone length is unrealistic in this case. How-ever, in both cases, for the same specimen geometry, the outerlimit seems to occur at about 0.2b, where b is the uncrackedligament length of the sample.
CHIANG, HAREESH, LIU, LI
(a)
A: a
2.0 3.0
(r/h)
(b)
—— HRR equations
7 Experimental
(r/h)
Rg. 4. Normalized displacements compared with plane stress HRR equations for (a) 112 and (b) u3.
stress assumptions break down owing to the finite plate thickness and crack tip blunting. In these figures, note that in the immediate vicinity of the crack tip there are marked deviations between experimental measurement and theory. The out-of-plane dis placement us is finite at the crack tip, unlike the theoretical prediction. To estimate the size of this region of dominant 3-D effects, we chose the boundary of the region to be at length r, where the difference between the HRR model and the experi mental measurements of normalized displacements is ±0.025 or more. Several experiments were done with three different materials, and results were plotted. In each case, the boundary of the 3-D effects was measured using the above criterion along discrete directions ahead of the crack tip. Figure 5 is a composite of all such measurements. Despite the experimental scatter, these data points seem to have a regular trend showing that the 3-D zone size is about 0.75h to 1.5h with direction dependence. This forms the "inner limit" of the HRR equations, beyond which the 2-D model seems to agree with the experimental observa tions.
Figures 6(a) and 6(b) are logarithmic plots of fringe order (equivalent to the opening displacement component ui) versus r along different directions from the crack tip for two different materials. The solid line represents the slope [l/(n 4-1)] of these experimental curves based on the HRR equations. Note that in each of these displacement plots there seems to be a region of constant slope (approximately that indicated by the solid lines) in a region where r ranges from about 5 mm to 13 to 15 mm. Outside this region, the slopes differ from those indicated by the HRR equations. In Fig. 6(c), the log of the opening strain 622 is plotted against log (r) along 6 = 0° (xi axis). The HRR field predicts a slope of n/(n+ 1) for the strain component,
= 1.9a0 (AI2024-0)= 0.85a0 (AI5052-H32)
C: a = 0.69 a0 (AI6061-T6) D: a = 0.88 a0 (AI5052-H32) E: a = 0.67a0 (AI6061-T6)
Fig. 5. Estimated 3-D zone around the crack tip.
which is represented by the solid lines. The two types of triangles are the strain values calculated from the fringe patterns shown in the figures. Here again, there is a region of constant slope. In these logarithmic plots, the initial deviation in the slope, as explained earlier, is due to the 3-D effects near the crack tip. Deviation beyond r = 13 to 15 mm can be looked at as the "outer limit" of the HRR fields, where the singularity descrip tions cease to dominate owing to the boundary effects of the specimen or the elastic field surrounding it.
The results obtained from the A15052-H32 sample, for which the outer limit is at about 14 mm, are in fair agreement with those of Zehnder et al. 5 for a steel with similar hardening index (n « 8.5). Their investigations showed that the outer limit of the HRR field is at about a third of Irwin's plastic zone length factor rp , given by
-M-) IT \OV
and the stress intensity factor K, given by
K = CCToc
(12)
(13)
where a' is the corrected crack length to account for the devia tions from small scale yielding assumptions, 12 c is a constant to account for geometrical effects of the specimen,13 and a«> is the far-field applied stress. Equations (12) and (13) give a value of rp/3 equal to 12.5 mm, which is close to the value of 14 mm shown by plots 6(b) and 6(c) and where the curves begin to change slope. However, when similar calculations were done on the A12024-0 results, rp/3 turned out to be 53 mm, although the plots still show slope deviation at around 13 to 14 mm. This could be because A12024-0 is a soft variety of aluminum and the applied far-field stress (2.04a0) had already resulted in net section yielding of the specimen. Thus, the above approach of estimating plastic zone length is unrealistic in this case. How ever, in both cases, for the same specimen geometry, the outer limit seems to occur at about 0.2b, where b is the uncracked ligament length of the sample.
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1912021-01 rtrd.n1 Ind.., 3
log (r)
(a)
o
2ó
OPTICAL ANALYSIS OF HRR HELD
, ... 45'
E:1
1115052yO211a-d.ni.,g Ind.x, 7
s.1ó' id.
log (r)
(b)
Fig. 6. Log (u2) vs log (r) for (a) A12024 -0 and (b) A15052 -H32. (c) Log (e22) vs log (r).
5. CONCLUSIONSUsing a combined in -plane and out -of -plane moire method, wehave been able to map out the complete displacement field arounda plastically deformed crack tip. Comparison of the normalizeddisplacements near the crack tip shows that there is a regionaround the crack tip where plane stress approximations do nothold owing to the strong 3 -D effects. This zone is estimated tobe from 0.75 to 1.5 times the thickness of the sheet with direc-tional dependence. From the logarithmic plots of u2 displacementand 622 strain, we estimate the outer limit of plane stress HRRfields to be at about 13 or 14 mm. This measures out to about0.2 times the uncracked ligament length.
6. ACKNOWLEDGMENTSThis work is supported by ONR Mechanics Division (Y. Ra-japkse, Scientific Officer) Contract N0001482K0566 and NSFSolid and Geo- Mechanics Program (K. Thirumalai, ProgramDirector) Grant MEA8403912.
7. REFERENCES1. J. W. Hutchinson, "Singular behavior at the end of a tensile crack in a
hardening material," J. Mech. Phys. Solids 16, 13 -31 (1968).2. J. R. Rice and G. F. Rosengren, "Plane strain deformation near a crack
tip in a power law hardening material," J. Mech. Phys. Solids 16, 337-347 (1968).
3. A. J. Rosakis and K. Ravi Chandar, "On crack stress state: an experimentalevaluation of three dimensional effects," Rept. SM84 -2, Graduate Aero-nautical Labs., California Institute of Technology (1984).
4. F. P. Chiang and T. V. Hareesh, "Three dimensional crack tip deformation:an experimental study and comparison to HRR field," Tech. Rept. 481,College of Engineering and Applied Sciences, SUNY at Stony Brook,(1986); Int. J. Fracture 36, 243 -257 (1988).
5. A. T. Zehnder, A. J. Rosakis, and R. Narasimhan, "Measurement of theJ integral with caustics: an experimental and numerical investigation," Rept.SM86 -8, Graduate Aeronautical Labs., California Institute of Technology(1986).
6. B. S. J. Kang, A. S. Kobayashi, and D. Post, "Stable crack growth inaluminum tensile specimens," Tech. Rept., Univ. of Washington (1986).
7. A. S. Kobayashi and B. S. J. Kang, "Stable crack growth in aluminumtensile specimens," Exp. Mech. 27(3), 523 -526 (1986).
8. C. F. Shih, "J- dominance under plane strain fully plastic conditions: theedge crack panel subjected to combined tension and bending," Int. J. Frac-ture 29, 73 -84 (1985).
9. F. P. Chiang, "Techniques of optical spatial filtering applied to the pro-cessing of moiré fringe patterns," Exp. Mech. 6(11), 523 -526 (1979).
10. C. F. Shih, "Relationship between the J- integral and the crack openingdisplacement for stationary and extending cracks," J. Mech. Phys. Solids29, 305 -326 (1981).
11. C. F. Shih, "Tables of HRR singular field quantities," Rept. MRL E -147,Materials Research Lab., Brown Univ. (1983).
(c)
12. M. F. Kanninen and C. H. Popelar, Advanced Fracture Mechanics, OxfordEngineering Science Series 15, Oxford University Press, New York (1985).
13. D. P. Rooke and D. J. Cartwrite, Compedium of Stress Intensity Factors,Her Majesty's Stationery Office, London (1975).
F. P. Chiang: Biography and photograph appear with the Guest Ed-itorial on p. 595.
T. V. Hareesh graduated in 1980 from Ban-galore University, India, with a bachelor's de-gree in mechanical engineering. In 1982 hereceived his master's degree in mechanicalengineering from the Indian Institute of Sci-ence, after which he worked as a scientist inthe Gas Turbine Research Establishment,Bangalore, India. Since 1983 he has been as-sociated with the Laboratory for Experimen-tal Mechanics Research, State University ofNew York at Stony Brook, where he received
his Ph.D. degree in 1988. His research interests include optical andnumerical methods for studying linear and nonlinear fracture of ma-terials, NDE, material fatigue, and automation.
Bao -chen Liu graduated from the Depart-ment of Engineering Mechanics of TsinghuaUniversity, Beijing, China, in 1959 IBS de-gree) and 1963 (Ph.D. degree). During 1963to 1972, she worked on elastic -plastic anal-ysis and experimental mechanics of shellsand plates. Since 1975, she has been usingoptical methods in the field of fracture me-chanics and engineering structure stressanalysis. About 60 of her technical papers inthese areas have been published. She is now
an associate professor at Tsinghua University and a visiting professorin the Department of Mechanical Engineering, State University ofNew York at Stony Brook.
Shen Li received the BS degree from the De-partment of Mechanical Engineering at Bei-jing Institute of Posts & Communications,Beijing, China, in 1982. From 1982 to 1986,she was an assistant engineer for the Re-search Institute of Postal Science & Tech-nology, Beijing, China. Now she is a graduatestudent in the Department of Mechanical En-gineering, State University of New York atStony Brook.
Fig. 6. Log (u2 ) vs log (r) for (a) AI2024-0 and (b) AI5052-H32. (c) Log (£22) vs log (r).
Itf
5. CONCLUSIONSUsing a combined in-plane and out-of-plane moire method, we have been able to map out the complete displacement field around a plastically deformed crack tip. Comparison of the normalized displacements near the crack tip shows that there is a region around the crack tip where plane stress approximations do not hold owing to the strong 3-D effects. This zone is estimated to be from 0.75 to 1.5 times the thickness of the sheet with direc tional dependence. From the logarithmic plots of 112 displacement and 622 strain, we estimate the outer limit of plane stress HRR fields to be at about 13 or 14 mm. This measures out to about 0.2 times the uncracked ligament length.
6. ACKNOWLEDGMENTSThis work is supported by ONR Mechanics Division (Y. Ra- japkse, Scientific Officer) Contract N0001482KQ566 and NSF Solid and Geo-Mechanics Program (K. Thirumalai, Program Director) Grant MEA8403912.
7. REFERENCES1. J. W. Hutchinson, "Singular behavior at the end of a tensile crack in a
hardening material," J. Mech. Phys. Solids 16, 13-31 (1968).2. J. R. Rice and G. F. Rosengren, "Plane strain deformation near a crack
tip in a power law hardening material," J. Mech. Phys. Solids 16, 337 347 (1968).
3. A. J. Rosakis and K. Ravi C bandar, "On crack stress state: an experimental evaluation of three dimensional effects," Rept. SM84-2, Graduate Aero nautical Labs., California Institute of Technology (1984).
4. F. P. Chiang and T. V. Hareesh, * Three dimensional crack tip deformation: an experimental study and comparison to HRR field," Tech. Rept. 481, College of Engineering and Applied Sciences, SUNY at Stony Brook, (1986); Int. J. Fracture 36, 243-257 (1988).
5. A. T. Zehnder, A. J. Rosakis, and R. Narasimhan, "Measurement of the J integral with caustics: an experimental and numerical investigation," Rept. SM86-8, Graduate Aeronautical Labs., California Institute of Technology (1986).
6. B. S. J. Kang, A. S. Kobayashi, and D. Post, "Stable crack growth in aluminum tensile specimens," Tech. Rept., Univ. of Washington (1986).
7. A. S. Kobayashi and B. S. J. Kang, "Stable crack growth in aluminum tensile specimens," Exp. Mech. 27(3), 523-526 (1986).
8. C. F. Shin, "J-dominance under plane strain fully plastic conditions: the edge crack panel subjected to combined tension and bending," Int. J. Frac ture 29, 73-84(1985).
9. F. P. Chiang, "Techniques of optical spatial filtering applied to the pro cessing of moire fringe patterns," Exp. Mech. 6(11), 523-526 (1979),
10. C. F. Shin, "Relationship between the J-integral and the crack opening displacement for stationary and extending cracks," J. Mech. Phys. Solids 29, 305-326 (1981).
11. C. F. Shin, "Tables of HRR singular field quantities," Rept. MRL E-147, Materials Research Lab., Brown Univ. (1983).
12. M. F. Kanninen and C. H. Popelar, Advanced Fracture Mechanics, Oxford Engineering Science Series 15, Oxford University Press, New York (1985).
13. D. P. Rooke and D. J. Caitwrite, Compedium of Stress Intensity Factors, Her Majesty's Stationery Office, London (1975). 3
F. P. Chiang: Biography and photograph appear with the Guest Ed itorial on p. 595,
T. V. Hareesh graduated in 1980 from Ban galore University, India, with a bachelor's de gree in mechanical engineering. In 1982 he received his master's degree in mechanical engineering from the Indian Institute of Sci ence, after which he worked as a scientist in the Gas Turbine Research Establishment, Bangalore, India. Since 1983 he has been as sociated with the Laboratory for Experimen tal Mechanics Research, State University of New York at Stony Brook, where he received
his Ph.D. degree in 1988. His research interests include optical and numerical methods for studying linear and nonlinear fracture of ma terials, NDE, material fatigue, and automation.
Bao-chen Liu graduated from the Depart ment of Engineering Mechanics of Tsinghua University, Beijing, China, in 1959 (BS de gree) and 1963 (Ph.D. degree). During 1963 to 1972, she worked on elastic-plastic anal ysis and experimental mechanics of shells and plates. Since 1975, she has been using optical methods in the field of fracture me chanics and engineering structure stress
» W^Kiilll analysis. About 60 of her technical papers in these areas have been published. She is now
an associate professor at Tsinghua University and a visiting professor in the Department of Mechanical Engineering, State University of New York at Stony Brook.
Shen Li received the BS degree from the De partment of Mechanical Engineering at Bei jing Institute of Posts & Communications, Beijing, China, in 1982. From 1982 to 1986, she was an assistant engineer for the Re search Institute of Postal Science & Tech nology, Beijing, China. Now she is a graduate student in the Department of Mechanical En gineering, State University of New York at Stony Brook.
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