OOIJ-7*M/Í9/OMÍ-0«17/M2.<KI»
£»/««""» Fnctun Uickvücí VoL II. pp. 117429
© Pcrpnoo Prtii U ¿ , 1979. Prálcd in Grut Brilain
_z). Vol. III,
II- Revs. 17,
STRESS-INTENSITY FACTORS FOR A WIDE
RANGE OF SEMI-ELLIPTICAL SURFACE
CRACKS IN FINITE-THICKNESS PLATES
'(fonnathn
977).
I. S. RAJU and J. C. NEWMAN, Jr.
NASA-Langley Research Center, Hampton, Virginia, U.S.A.
Abstract—Surface cracks are among the more common flaws in aircraft and pressure vessel components.
Accurate stress analyses of surface-cracked components are needed íor reliable prediction of their crack
growth rates and fracture strengths. Several calculations of stress-intensity factors for semi-elliptical
surface cracks subjected to tensión have appeared in the literature. However, some of these solutions are in
disagreement by 50-100%.
In this paper stress-intensity factors for shallow and deep semi-elliptical surface cracks in plates
subjected to tensión are prcsenled. To verífy the accuracy of the three-dimensional finite-element models
employed, convergence was studied by varying the number of degrees of freedom in the models from 1500
to 6900. The 6900 degrees of freedom used here were more than twice the number used in previously
reported solutions. Also, Ihe stress-intensity variations in the boundary-layer región at the intersection of
the crack with the free surface were investigated.
NOTATION
a
b
•
c
F
h
K
Q
S
I
x,y,z
V
*
depth of surface crack
half-width of cracked píate
half-length of surface crack
stress-intensity boundary-correction factor
half-length of cracked píate
stress-intensity factor (Mode ])
shape factor for an elliptical crack
applied uniform stress
píate thickness
Cartesian coordínate*
Poisson's ratio
parametríc angle of the ellipse
1
INTRODUCTION
SURFACE CRACKS are among the more common flaws in aircraft and pressure vessel components.
Accurate stress analyses of these surface-cracked components are needed for reliable prediction of their crack growth rates and fracture strengths. Exact solutions to these difficult
problems are not available; therefore, approximate methods must be used. For a semi-elliptical
surface crack in a finite-thickness píate subjected to tensión (Fig. 1), Browning and Smith[l],
Kobayash¡[2] and Smith and Sorensen[3] used the alternating method and Kathiresan[4] used
the fínite-element method to obtain the stress-intensity factor variations along the crack front
for various crack shapes. For a deep semi-elliptical surface crack (with alt - 0.8 and ale = 0.2)
subjected to tensión, the stress-intensity factors obtained by Smith and Sorensen[3],
Kobayash¡[2] and Kath¡resan[4] disagreed by 50-100%. The reasons for these discrepancies are
not well understood.
This paper presents stress-intensity factors for shallow and deep semi-elliptical surface
cracks in plates subjected to uniform tensión. To test the validity of the present analysis, two
crack configurations, both embedded in a large body subjected to uniform tensión, were
analyzed: (1) a circular (penny-shaped) crack and (2) an elliptical crack. These results are
compared with exact solutions from the literature [5]. To verify the accuracy of the solutions
for surface cracks in finite-thickness plates, convergence was studied by varying, from 1500 to
6900, the number of degrees of freedom in the finite-element models. The 6900 degrees of
freedom used here were more than twice the Iargest number used previously [4]. These models
were composed of singular elements around the crack front and isoparametríc (linear strain)
elements elsewhere. Mode I elastic stress-intensity factors were calculated by using a nodalforcé method.
817
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818
I. S. RAJU and J. C. NEWMAN, Jr.
Fig. 1. Surface crack ¡n a píate subjected to tensión.
The stress-intensity varíations in a boundary-Iayer región at the intersection of the crack
with the free surface were aiso investigated for a typical semi-circular surface crack. Here the
finite-element models used were highly refined in the boundary-layer región.
The computations reported herein were conducted on a unique computer called the
STAR-100 at the NASA Langley Research Center.
THREE-DIMENSIONAL ANALYSIS
Several attempts[6-8] have been made to develop special three-dimensional finite-elements
that account for the stress and strain singularities caused by a crack. These elements had
assumed displacement or stress distríbutions that simúlate the square-root singularity of the
stresses and slrains at the crack front. Such three-dimensional singularity elements [6,9} were
also used herein for the analysis of finite-thickness plates containing embedded elliptical or
semi-elliptical surface cracks (see, for example, Fig. 1).
Finite-element ¡dealhation
Two types of elements (isoparametric and singular) were used in combination to model
elastic bodies with embedded elliptical cracks or semi-elliptical surface cracks. Figure 2 shows
a typical finite-element model for an embedded circular crack in a large body (hla - b¡a = 5).
S Singularity element
I Isoparametric eleznent
SU
(a) Specimen model.
I I
(b) Element pattern around crack
front on the i
plañe.
Fig. 2. Finite-element idealization for an embedded circular (penny-shaped) crack.
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Stress-intensity factors of semi-elliptical surface cracks infinite-thicknessplates
819
This model idealizes one eight of the body. Various numbers of wedges were used to form the
desired conñguration. Figure 2(a) shows a typical model with eight wedges. Each wedge is
composed of elements that are identical in pattern to that shown in the </> = constant plañe. The
arrangement of the elements around the crack front is shown in Fig. 2(b). The isoparametric
(linear strain) elements (denoted as /) were used everywhere except near the crack front.
Around the crack front each wedge contained eight "singularity" elements (5) in the shape of
pentahedrons. The "singularity" elements had square-root terms in their assumed displacement
distribution and, therefore, producéd a singular stress ñeld at the crack front. Details of the
formulation of these types of elements are given in Refs. [6] and [9] and are not repeated here.
The finite-element model for the embedded elliptical or semi-elliptical surface crack was
obtained from the finite-element model for the circular crack by using an elliptic transformation. This transformation was needed because the stress-intensity factors must be
evaluated from either crack-opening displacements or nodal forces along the normáis to the
crack front. If (x, y, z) are the Cartesian coordinates of a node in the circular-crack model and
(x',y\z') are the coordinates of that same node in the elliptical-crack model, then the
transformation is
for
e crack
lere the
led the
lements
nts had
' of the
9] were
itical or
-<•!
c
0)
and
' = y, z
for JC and z not at the origin. Figure 3 shows how circular ares and radial unes in the JC, z plañe
of the circular-crack model are transformed into ellipses and hyperbolas, respectively, in the
x', z' plañe of the elliptical-crack model using eqns (1). Because eqns (1) are not valid at the
origin, a circle of very small radius, a/1000, was used near the origin in the circular crack model.
The small circle maps onto an extremely narrow ellipse ¡n the x', z' plañe. The use of the small
circle avoids ill-shaped elements near the origin in the elliptical-crack model. Figure 4 shows a
typicalfinite-elementmodel of a finite píate containing an elliptical crack. The transformation
reduced the ble ratio; therefore, in order to maintáin ble 5* 4, additional elements were added
along the x'-axis to elimínate the influence of píate width.
t model
I shows
•/a = 5).
-z1
(a) circular crack.
(b) Elliptic crack.
Fig. 3. Circle to ellipse transformation.
Stress-intensity factor
The stress-intensity factor is a measure of the magnitude of the stresses near the crack
front. Under general loading, the stress-intensity factor depends on three basic modes of
deformation (tensión and in- and out-of-plane shear). But here only tensión loading was
considered and, therefore, only Mode I deformations oceurred. The Mode I stress-intensity
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820
I. S. RAJU and J. C. NEWMAN, Jr.
(a) Specimen model.
(b) Element pattern.on y'» 0 plañe.
Fig. 4. Finite-element idealization for a semi-elliptical surface crack,
factor, K, at any point along an elliptical or semi-elliptical crack in a finite píate is taken to be
where S is the applied stress, a ís the crack depth, Q is the shape factor for an ellipse and is
given by the square of the complete elliptic integral of the second k¡nd[5]. The boundarycorrection factor, F(a¡t, ale, <p), is a function of crack depth, crack length, píate thickness and
the parametnc angle of the ellipse. The present paper gives valúes for F as a function of alt
and <f> for ale - 0.2, 0.4, 0.6, 1.0 and 2.0. The alt valúes ranged from 0.2 to 0.8.
The stress-intensity factors from the finite-element models.of embedded elliptical. and
semi-elliptical surface cracks were obtained by using a nodal-force method, details of which are
given in the appendix. In this method, the nodal forces normal to the crack plañe and ahead of
the crack front are used to evalúate the stress-intensity factor. In contrast to the crack-opening
displacement method[6], this method requires no prior assumption oí either plañe stress or
plañe strain. For a surface crack in a finite píate, the state of stress varíes from plañe strain in
the interior of the píate to plañe stress at the surface. Thus, the crack-opening displacement
method could yield erroneous stress-intensity factors.
RESULTS AND DISCUSSION
In the following sections a circular (penny-shaped) crack and an elliptical crack completely
embedded in a large body subjected to uniform tensión were analyzed using the finite-element
method. The calculated stress-intensity factors for these crack configurations are compared
with the exact solutions to verify the validity of the present finite-element method.
For a semi-circular and semi-elliptical surface crack in a finite-thickness píate, convergence
of the stress-intensity factors was studied while the number of degrees of freedom in the
finite-element models ranged from about 1500 to 6900. The stress-intensity varíations in a thin
boundary-Iayer región at the intersection of the crack with the free surface were also
investigated. The stress-intensity factor varíations along the crack front for semi-circular
(a¡c= 1) and semi-elliptical (ale = 0.2, 0.4, 0.6, 1.0 and 2.0) surface cracks were obtained as
functions of alt with hlc = ble == 4. Whenever possible, these stress-intensity factors are
compared with results from the literatura.
Exact solutions
In this section a comparison is made between the stress-intensity factors calculated from the
finite-element analysis and from the exact solution[5] for an embedded circular (penny-shaped)
crack (a/c= 1) and an embedded elíptica! crack (o/c = 0.2) in an infinite body. In the finiteelement model, h and b were taken to be large enough that the free boundary would have
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i
Stress-intensity factors of semi-ellipticál surface cracks infinile-lhicknesspiales
821
a negligible effect on stress intensity. The boundary correction on stress intensity for a circular
crack in a cylinder of radius b with b¡a = S is about l%[10]. Therefore, to simúlate a
large body thefinite-elementmodel was assigned the dimensions lila = bla = 5, along with 3078
degrees of freedom. The calculated stress-intensity factors along the crack front from the
circular crack model were about 0.4% below the exact solution.
The embedded elliptical crack {ale = 0.2) model, with /i/c-ble = 4 and //a = 5, had 3348
degrees of freedom. The influence of finite boundaries on stress intensity was estimated to be
about 1%. The finite-element analysis gave stress-intensity factors along the crack front
generally within \% of the exact solution, except ¡n the región of sharpest curvature of the
ellipse, where the calculated valúes were about 3% higher than the exact solution. Further
refinement in the mesh size in this área gave more accurate stress-intensity factors.
. Because the present method yielded stress-intensity factors for completely embedded
circular and elliptical cracks generally within 0.4-1% of the exact solutions, the method was
considered suitable for analyses of more complex configurations, provided that enough degrees
of freedom were used to obtain good convergence.
Approximate solutions
To verify that the finite-element meshes used for the circular and elliptical crack models
were sufficient to analyze cracked plates with free boundaries, through-the-thickness cracks
with crack length-to-width ratios ranging from 0.2 to 0.8 were analyzed under plañe strain
assumptions. The meshes used here were exactly the same as those which oceur on the <f> = -nll
plañe of the circular and elliptical crack models. These meshes were then used to model the
center-crack tensión specimen. For crack length-to-width ratios (elb) ranging from 0.2 to 0.6,
thefinite-elementresults were within 1.3% of the approximate solutions given in Ref. [10]. For
elb - 0.8, thefinite-elementresult was 2% below the solution given in Ref. [10]. These results
indícate that the mesh pattern along any plañe has enough degrees of freedom to account for
the ¡nfluence of free boundaries on the stress-intensity factor.
n to be
(2)
and is
mdary:ss and
of a/í
al. and
ich are
lead of
pening
ess or
rain in
:ement
•J1
Convergence
'
.
In the previous section the mesh pattern along any <f> = constant plañe was found to be
sufficient to account for the influence of free boundaries even for very deep cracks. In this
section, the mesh pattern in the angular direction, <f>, ¡s studied. Figures 5 and 6 show the results
of a convergence study on the stress-intensity factors for a semi-circular surface crack and a
semi-elliptical surface crack, respectively, in'a finite-thickness píate. The larger numbers of
degrees of freedom are associated with smaller wedges ¡n the </>-direction and a more accurate
representation of the crack shape.
•
pletely
lement
npared
rgence
in the
a thin
e also
ircular
led as
rs are
smthe
haped)
finite1 have
Degrees of freedom
1500
O 2439
Q 4317
O 6195
Fig. 5. Convergence of stress-intensity faclors for a deep semi-circular surface crack (Q = irJ/4; o// = 0.8).
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822
I. S. RAJU and J. C. NEWMAN. Jr.
2.0
1.6
1.2
s
fF
Degrees of freedom
.8
.4
0
A 1692
^2727
Q 4797
O 6867
,l
2c
25
.5
.75
'
1
i*
Fig. 6. Convergence of stress-intensiiy fadors for a deep semi-elliptical surface crack (Q =1.104; alt = 0.8;
ale = 0.2).
For the semi-circular crack (Fig. 5), hla = tya = 5 and alt = 0.8. This configuration was
chosen because the cióse proximity of the back surface to the crack front was expected to
cause difficulty in achieving convergence. The models were composed of either 2, 4, 8 or 12
equal wedges. The number of degrees of freedom ranged from 1500 to 6195. The two ñnest
models (4317 and 6195 degrees of freedom) gave stress-intensity factors within about 1% of
each other.
Figure 6 shows the convergence study for a semi-elliptical surface crack (ale = 0.2). The h¡c
and ble ratios were equal to four and, again, alt was chosen as 0.8. The number of degrees of
freedom ranged from 1692 to 6867. The eight-wedge model with 4797 degrees of freedom gave
results within about 1% of those from the finest model.
The results shown in Figs. 5 and 6 demónstrate rapid convergence of the solution and
indícate that the eight-wedge model may be used to obtain stress-intensity factors as a function
of alt and ale. However, further consideration of the stress-intensity varíations in the
boundary-layer región at the intersección of the crack with the free surface must be given.
Boundary-layer effect
Hartranft and Sih[ll] proposed that the stress-intensity factors in a very thin "boundary
layer" near the free surface drop off rapidly and equal zero at the free surface. To investígate
the boundary-layer effect, a semi-circular surface crack in a large body was considered. Three
differenlfinite-elementmodels with 8,10 and 14 wedges were analyzed. The eight wedge model,
shown ¡n Fig. 2(a), had eight equal wedges. The other models had nonuniform wedges and were
obtained by refining the eight-wedge model near the free surface. The smallest wedge angle for
the ten- and fourteen-wedge models were ÍT/32 and TT/128, respectively. The stress-intensity
factors obtained from the three models are shown in Fig. 7. The máximum stress-intensity
factor obtained from the eight-wedge model at the free surface was slightly less (1.4%) than the
peak valué obtained from the fourteen-wedge model. These results also show that the stress
intensities near the free surface were affected by the refinements and drop off rapidly as
proposed by Hartranft and Sih. However, the stress-intensity distributions in the interior
(4> > 7r/16) were unaffected by the refinements. These results strongly suggest that the stress
intensities in the interior are insensitive to whatever is happening in the boundary layer. An
estímate for the size of the boundary layer was obtained from Ref. [11], assuming that the crack
was a through crack of length, la, and this estímate is also shown in Fig. 7.
In view of the highly localized boundary-layer effect, the eight-wedge'model was used
subsequently to obtain stress-intensity factors as a function of ale and alt. The stress-intensity
factor obtained at the free surface with the eight-wedge models were interpreted as an average
stress-íntensity factor near the free surface.
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i
Stress-inlensity factors of semi-elliptical surface cracks in finite-thickness plates
823
1.25
e
8 Wedge model
o 10 Wedge model
1.20
O 14 Wedge model
1.15
1.10
1.05
1.00
.25
.5
2*
.75
n
= 0.8;
Fig. 7. Effects of mesh refínement near the free surface on the distribution of stress ¡ntensity for
semi-circular surface crack (Q = ir'/4; alt = 0.2).
.tion was
jected to
, 8 or 12
wo finest
ut \% of
. The hlc
:grees of
lom gave
tion and
function
s in the
ven.
ioundary
vestigate
d. Three
e model,
ind were
ingle for
intensity
intensity
than the
te stress
pidly as
interior
le stress
iyer. An
le crack
Semi-circular surface crack in afinite-thicknesspíate
Figure 8 shows and Table 1 presents the stress-intensity factors for a semi-circular surface
crack in a finite-thickness píate as a function of the parametric angle, <f>, and the crack
depth-to-plate thickness ratio, alt. Near the intersection of the crack and the free surface, the
stress-intensity factor increases more rapidly with alt than at the deepest point (<£ = irU). For
each valué of alt, the stress-intensity factor calculated from the eight-wedge model is largest at
the free surface. "
For a semi-circular surface crack with alt - 0.55, the stress-intensity factors calculated by
Browning and Smhhfl], who used the alternating method, were about 1-3% below the present
results for various valúes of <f>.
Semi-elliptical surface crack in afinite-thicknesspíate
The stress-intensity factors for a semi-elliptical surface crack (ale - 0.2,0.4, 0.6 or 2.0) in a
finite-thickness píate as a function of the parametric angle, <f>, and the crack depth-to-plate
thickness ratio, alt, are given in Table 1.
1.6 r
a/t
1-2
T
M
.4
.25
as used
ntensity
average
.5
.75
ÍÉ
n
Fig. 8. Distribution of stress-intensity factors along crack front for a semi-circular surface crack (Q =
Supplied by The British Library - "The world's knowledge"
I. S. RAJU and J. C. NEWMAN. Jr.
824
Table 1. Boundary correction factors, F, for semi-elliptical
surface cracks subjected to tensión (c = 0.3; F-
J
all
ale
Wv
0.2
0.4
0.6
0.8
0.2
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1.0
(D.6I7
).65O
D.754
3.882
().99O
.072
.128
.161'
.173
0.724
0.775
0.883
1.009
1.122
1.222
1.297
1.344
1.359
0.899
0.953
1.080
1.237
1.384
1.501
1.581
1.627
1.642
1.190
1.217
1.345
1.504
1.657
1.759
1.824
1.846
1.851
0.4
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1.0
().767
().78I
().842
().923
).998
.058
.103
.129
.138
0.896
0.902
0.946
1.010
1.075
1.136
1.184
1.214
1.225
1.080
1.075
1.113
1.179
1.247
1.302
1.341
1.363
1.370
1.318
1.285
1.297
1.327
1.374
1.408
1.437
1.446
1.447-
0.6
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1.0
().916
().919
().942
().982
.024
.059
.087
.104
.110
1.015
1.004
1.009
1.033
1.062
1.093
1.121
1.139
1.145
1.172
1.149
1.142
1.160
1.182
1.202
Í.2I8
1.227
1.230
1.353
1.304
1.265
1.240
1.243
1.245
1.260
1.264
1.264
1.0
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1.0
.174
.145
.105
.082
1.067
.058
.053
.050
1.049
1.229
1.206
1.157
1.126
1.104
1.088
1.075
1.066
1.062
1.355
1.321
1.256
1.214
1.181
1.153
1.129
1.113
1.107
1.464
1.410
1.314
1.234
1.193
1.150
1.134
1.118
1.112
2.0
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1.0
().82I
().794
().74O
().692
().646
((.599
(1.552
((.512
(1.495
0.848
0.818
0.759
0.708
0.659
0.609
0.560
0.519
0.501
0.866
0.833
0.771
0.716
0.664
0.610
0.560
0.519
0.501
0.876
0.839
0.775
0.717
0.661
0.607
0.554
0.513
0.496
Figure 9 shows the stress-intensity factors for a semi-elliptical surface crack (ale = 0.2) as a
function of the parametric angle and the crack depth-to-plate thickness ratio. For each valué of
alt, the máximum stress-intensity factor oceurs at the deepest point (<f> = w/2). Also, the
máximum stress-intensity factor i.s larger for larger valúes of all.
Figure 10 shows the stress-intensity distributions for various semi-elliptical surface cracks
as a function of ale with a/í = 0.8. For smaller valúes of ale, the normalized stress-intensity
factor at 4> - W2 is larger. The máximum normalized stress-intensity factor oceurred at (f> = .irl2
for ale « 0.4 but oceurred at <f> = 0 for a¡c ^1.0.
Figures. 11 and 12 shows stress-intensity factors obtained by several investigators for a
semi-elliptical surface crack in afinite-thicknesspíate. Figure 11 shows the results for a surface
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Stress-intensity factors of semi-elliptical surface cracks infinite-thicknesspiales
825
2.0
Fig. 9. Disiríbution of stress-intensity factors along crack front for a semi-clliptical surface crack {Q =
1.104; a/c = 0.2).
2.0
r
Fig. 10. Distríbution of slress-intensity factors along crack front for semi-elliptical surface cracks {alt =
0.8).
).2) as a
/alue of
Iso, the
: cracks
itensity
s for a
surface
crack with ale = 0.2 and alt = 0.8. Smith and Sorenson[3] and Kobayashi[2] used the alternating method and Kathiresan[4] used the finite-element method to obtain stress-intensity factor
variations along the crack front. These three solutions disagree by as much as 50-100%. The
reasons for these discrepancies are not well understood. The present results, shown as solid
symbols, are considerably higher than the previous solutions[2-4]. The results from Smith and
Sorensen[3] are generally closer to the present results, though 10-25% lower.
Figure 12 shows a comparison of the máximum stress-intensity factors obtained by several
investigators for a semi-elliptical surface crack as a function of alt. The máximum stressintensity factors oceurred at 4> = v\2. The present results are shown as solid symbols. The open
symbols show the results from Smith and Sorensen[3], Kobayashi[2,12] and Rice and
Levy[13]. The results from Rice nad Levy, obtained from a line-spring model, are about 3.5%
below the present results over an alt range from 0.2 to 0.6. For a//>0.6, the Rice-Levy
solution shows a reduction in stress intensity. The dash-dot curve shows the results of an
approximate equation proposed by Newman[14] for a wide range of ale and ají ratios.
Supplied by The British Library - "The world's knowledge"
826
I. S. RAJU and J. C. NEWMAN. Jr.
1.6
G
. B
1.2
<
.8*
[^
^
#
0
•
0
• 4
1
Present rosults
Smith and Sorensen 13)
Kobayashi |2]
Kathiresan (41
_1
.5
.25
1
1
.75
Fig. 11. Comparison of slress-intensity faclors for a deep semi-elliptical surface crack (Q = 1.104; a// = 0 8 '
ole = 0.2).
2.2
•
O
2.0
O
A
Q
1.8
——-»
Present resulta
Rice and Levy [13]
Smith and Sorengen 13)
Kobayashi 121
Kobayashi [12]
Newnan
/
/
1.0
.2
.4
.8
Fig. II. Comparison of the máximum stress-intensity factor for a semi-elliptical surface crack as a function
of all (<f> = ir/2; (?= 1.104; ale = 0.2).
Newman's equation is within ±5% oí the present results over an a¡t range from 0.2 to 0.8.
Newman's equation gives a good engineering estímate for the máximum stress-intensity factor.
CONCLUDING REMARKS
A threé-dimensional finite-element eiastic stress analysis was used to calcúlate stressintensity factor varíations along the crack front for completely embedded elliptical cracks in
large bodies and for semi-elliptical surface cracks (crack depth-to-half crack length ratios were 0.2
to 2.0) in finite-thickness platas.- Three-dimensional singularity elements were used at the
crack front. A nodal forcé method which requires no prior assumption of either plañe stress or
plañe strain was used to evalúate the stress-intensity factors along the crack front.
Completely embedded circular and elliptical cracks were analyzed to verify the aecuracy of
thefinite-elementanalysis. The stress-intensity factors for these cracks were generally about
0.4-1% below the exact solutions. However, for the elliptical crack the calculated stressintensity factors in the región of sharpest curvature of the ellipse were about 3% higher than the
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Stress-inlensity factors of semi-elliptical surface cracks infínite-thicknessplates
= 0.8;
827
exact solution. The numbers of degrees of freedom in the embedded crack models were about
3000. A convergence study on stress-intensity factors for semi-elliptical surface cracks in
finite-thickness plates showed that convergence was achieved for both the semi-circular and the
semi-elliptical surface crack with about 4500 degrees of freedom.
For the semi-circular surface crack the máximum stress-intensity factor occurred near the
intersection of the crack with the free surface. On the other hand, for the semi-elliptical surface
crack (crack depth-to-crack half length ratio of 0.2), the máximum stress-intensity factor
occurred at the deepest point. For both the semi-circular and semi-elliptical surface crack the
stress-intensity factors were larger for larger valúes of crack depth-to-plate thickness ratio.
For the semi-circular surface crack, the stress-intensity factors calculated by Browning and
Smith[l] using the alternating method, agreed generally within about 3% with the present results.
However, for semi-elliptical surface cracks (crack depth-to-plate thickness ratio of 0.2) Smith
and Sorensen[3) using the alternating method gave stress-intensity factors in considerable
disagreement (10-25%) with the present results. For semi-elliptical surface cracks the results
from Rice and Levy[13] for crack depth-to-plate thickness ratios less than or equal to 0.6 and
an approximate equation proposed by Newman[14] were in good agreement with the present
results.
The stress-intensity factors obtáined herein should be useful in correlating fatigue crack
growth rates as well as fracture toughness calculations for the suríace-crack conñgurations
considered.
REFERENCES
tion
.2 to 0.8.
ly factor.
[1] W. M. Browning and F. W. Smith, An analysis for complex three-dimensional crack problems. Developments in
Theoretical and Applied Mechanics, Vol. 8, Proc. Sth SECTAM Con}. (1976).
(2) A. S. Kobayashi, Surface flaws in plates in bending. Proc. Ylth Annual Meeting of the Soc. of Engng ScL, Austin,
Ttxas(1975).
[3] F. W. Smith and D. R. Sorensen, Mixed mode stress ¡ntensity factors for semi-elliptical surface cracks. NASA
CK-134684 (1974).
[4] K. Kathiresan, Three-dimensional linear elastic fracture mechanics analysis by a displacement hybrid finite clement
model. Ph.D. Thesis, Georgia Institute of Technology (1976).
[5] A. E. Green and I. N. Sneddon, The distribution of stress in the neighborhood of a fíat elliptical crack in an elastic
solid. Proc Cambridge Phü Soc. ü (1959).
(6) D. M. Tracey, Finite elemcnt for three-dimensional elastic crack analysis. Nucí. Engr. and Design 2í (1974).
[7] R. S. Barsoum, On the use of isoparametric finite elemente in linear fracture mechanics. Int. J. Num. Melh. Engr. 10(1)
25-37 (1976).
[8] S. Atlurí and K. Kathiresan, An assumed displacement hybrid finite clement model for three-dimensional linear
fracture mechanics analysis. Proc. ttth Annual Meeting of the Soc. Engr. Science, University of Texas, Austin, Texas
(1975).
[9] I. S. Raju and J. C. Newman, Jr., Three-dimensionalfinite-elementanalysis offinite-thicknessfracture specimens.
MS/irND-8414(1977).
[10] H. Tada, P. C. París and G. R. Irwin, The Stress Analysis of Cracks Handbook, p. 28.1. Del. Research Corp., C.
(1973).
[11] R. J. Hartranft and G. C. Sih, An approximate three-dimensional theory of plates with application to crack problems.
Int. J. Eng. Sci. 8(8), 711-729 (1970).
[12] A. S. Kobayashi, Crack opening displacement in a surface flawed píate subjected to tensión or píate bending.
Presented at the Ind Int. Conf. Mech. Behaiior of Materials, Boston, MA (1976).
(13) J. R. Rice and N. Levy, The part-through surface crack in an elastic píate. Trans. ASME J. Appl. Mech. Paper No.
71-APM-20 (1970).
[14] J. C. Newman, Jr., Fracture analysis of surface- and through-cracked sheets and plates. Engng Fracture Mech. 4,
667-689 (1973).
(Receked 18 July 1978; received for publication 4 September 1978)
e stress:racks in
; were 0.2
d at the
stress or
uracy of
lly about
i stressthan the
APPENDK
Determination of stress-intensity factors using the nodal-force method
The nodal-force method, developed in Ref. [9] for evaluating stress-intensity factors is presented herein. In this method,
the nodal forces normal to the crack plañe and ahead of the crack front are used to evalúate the stress-intensity factor. In
contras! to the crack-opening displacement method [6], this method requires no prior assumption of either plañe stress or
plañe strain.
Consider an elliptic surface crack idealized by severa! wedges as shown in Fig. 13. Each wedge was composed of
various number of elements. The nodal forces (F¡) normal to the y' = 0 plañe along the junction (dashed curvé) between
wedges í and / + 1 are assumed to be contnbuted from the normal stresses acting over one-half of the wedges on either
side of the junction (shaded arca). For clarity, the shaded área has been enlarged (R 4 a). The dashed curves are
hyperbolas normal to the crack front. Nodal forces along these hyperbolas are needed to evalúate the stress-intensity
factors.
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828
1. S. RAJU and J. C. NEWMAN. Jr.
x'
Fig. 13. Nodal forces on >•' = 0 plañe and at the junction between wedges i and í + I.
The normal stress a, at any point in the shaded área is assumed to be given by
(3)
where r is the distance normal to the crack front. In general, K and A\ are functions of the coordínate along the crack
front. However, in the present analysis X and A'¡ were assumed to be constant over the shaded área. This assumplion is
justificó when Ihe wedges are ihin or the actual X variations across a wedge are small.
The total forcé acting in the /-direction over the shaded área is given by
f
(4)
Jo
where U is the average width of the shaded área (/„ ««(fc + f,.i)/2). Substituting eqh (3) into eqn (4) gives
F, = /„ ( ^ 2 V « + A,R>I2+A2R
(5)
where Ai represents conslants. Using onty Ihe first term on Ihe righthand side of eqn (5), X is replaced by the apparent
stress-intensity factor Kff and is evaluated as
v
(6)
1.2Sr
1.20
1.15
»P
1.10
1.05
1.00
.04
.08
.12
.16
JR
Fig. 14.'Apparent stress-intensity factors for semi-circular surface crack (a/f = 0.2) as a function of Ría
and ihe parametric angle, <t>.
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Stress-intensity factors of semi-elliptical surface cracks in finite-thickness plates
829
The forcé F, was obtained from Ihe ñnite-element analysis and was determined as follows. For example, let nodes 1-4 in
Fig. 13 be the nodes lying on the junction between wedges í and / + I. Node 4 is a distance R from the crack front. The
nodal forces F¡ (/ = 1-4) are obtained from the finile-clement analysis. Thus, the total forcé acting over a distance R is
F, = F¡ + F2 + F¡ + F't
(7)
where F',, the nodal forcé at node 4, is calculated using the elements that are connected to node 4 and are inside the shaded
área. For various valúes of R, the number of nodal forces contributing to F, would also vary.
The total forcé F, obtained from the finite-element analysis is substituted into eqn (6), and the apparent stress-intensity
factor is evaluated. The procedure is repeated for varíous valúes of R. Equating eqns (5) and (6) gives
where B, represents constants. From eqn (8), a plot of Kap vs R is linear for small valúes of R and the intercept at R = 0, at
Ihe crack front, gives the stress-intensity factor at a particular location along the crack front. The procedure is repeated at
various locations along the crack front.
Figure 14 shows the apparent stress-intensity factor for a typical semi-circular surface crack obtained by the
nodal-force method plotted against the distance Ría for varíous valúes of the parametnc angle, 4>- If the Kop valué nearest
the crack front (R = 0) is neglected, the relationship between Kar and Ría are nearly linear, as cxpected. Therefore, the
Kcf valué closest to the crack front was neglected. For each valué of 4>, a straight line was fitted (by the method of least
squares) to the Kaf results to evalúate the stress-intensity factor. The intercept of these lines at Ría = 0 gives the
stress-intensity factor.
(3)
ng the crack
issumption is
(4)
the apparent
.'
(6)
-I
Ría
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