A closed form expression for the weight function for a strip with a single edge crack is presented. The expression is valid for relative crack lengths from zero to unity. It is based on the assumption that the shape of an opened edge crack can be approximated by a conic section. The results agree well with published values for weight functions, stress intensity factors, and crack mouth opening displacements

Orange, T. W.

English

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V	 A
NASA Technical Memorandum 83478
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Wide Range Weight functions for the
Strip with a Single Edge Crack
(MAZA —Tf1-83478) WIDE LANGE WEIGHT FUNCTIQNS
	 N84-11512
FOR THE STRIP WITH A Slk(;LE EDGE CRACK
(MASA) 12 p HC A02/8P AU 1
	 CSCL 20K
Unc ids
'J/39 42356
Thomas W. Orange
Lewis Research Center
Cleveland, Ohio
p
Prepared for the
Sixteenth National Symposium on Fracture Mechanics
sponsored by the American Society for Testing and Materials
Columbus, Ohio, August 15-18, 1983
T
WIDE RANGE WEIGHT FUNCTIONS FOR THE STRIP WITH A SINGLE EDGE CRACK
Thomas W. Orange
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
SUMMARY
This paper presents a closed-form expression for the weight function for
a strip with a single edge crack. The expression is valid for relative crack
a	 lengths from zero to unity. It is based on the assumption that the shape of
an opened edge crack can be approximated by a conic section. The results agree
well with published values for weight functions, stress intensity factors, and
crack mouth opening displacements.
INTRODUCTION
Bueckner's weight function concept (refs. 1 and 2) has proven to be a use-
ful tool for elastic crack analysis. The weight function is a function of spe-
cimen (or structure) and crag. onfigurations only and is independent of the
load system. If an expression for the weight function is available, the stress
intensity factor for any load system may be computed from the stress distribu-
tion in the.uncracked body by simple integration. In presenting this concept
Bueckner used the example of a strip with a single edge crack, which is a com-
mon test-specimen configuration and which also has application to real-world
structural problems. He presented (ref. 2) a simple expression for the weight
function for this configuration which is useful for relative crack lengths from
zero to one-half.
Rice (ref. 3) showed that weight functions can be calculated from crack
surface displacements and stress intensity factors. Thus, if the stress inten-
sity factor and the complete crack displacement solution are available for a
given cracked-body configuration under one loading condition (say, simple ten-
sion), the weight function can be computed. This configuration and loading is
usually called the reference solution.
At about the same time, this author (ref. 4) proposed that the opening
shape of an edge crack could be approximated by a conic section. In the pres-
ent paper, the conic section approximation is used to develop a closed-form
expression for the weight function which is valid for all values of relative
crack length, i.e., from zero to unity.
ANALYSIS
The generalized equation for the conic section (ref. 4) and the notation
used are shown in figure 1. The conic section coefficients which are shown
(m - -1, 0, 10, W) correspond respectively to an ellipse, a parabola, a hy-
perbola, and a pair of straight lines. Using relations given in reference 4,
the opening displacement (n)'may be written as
UMWNAL PAGIE 18
OF POOR QUALM
112	 112	 112
n = -Er (*)
	
KI x	 (1 + ^ a)
	
(1)
where E' is the effective elastic modulus, KI is the opening-mode stress in-
tensity factor, and m is the conic section coefficient. Rice (ref. 3) showed
that the weight function ;h) may be written as
h
	
E  
an	
(2)
=	 as
In comparing weight function formulations, note that Bueckner ' s form M(x) and
Rice's form h(x) are related by
M(x) V2—.  h(x)
Now a general expression for the weight function may be determined by differ-
entiating equation ( 1). In doing so, note that the coordin:; it system in fig-
ure 1 is located at the crack tip. If the crack extends an amount da, the
coordinate system moves a like amount (i.e., dx = da). After performing the
required differentiation, Bueckner's reduced form of the weight function (N(x))
may be written as
112	 1/2
N(a) -x	 M (a) = 2 a(Y da + 7) +1] (1 + 2 a)C
x x a dm m	 m	 m x 112
a [a (T UT T ) 7
where a = a/W is the relative crack length and Y = K I /a VaT is the dimen-
sionless stress intensity factor coefficient. Thus the weight function can be
computed if we know how the conic section and stress intensity factor coeffi-
cients vary with relative crack length for one form of loading, say, simple
tension.
Values of the conic section coefficient are compuated as follows. Equa-
tion (5) of reference 4 gives the relationship between stress intensity factor,
crack mouth displacement, and conic section coefficient as
E'no /aa = 2YU2 + m)/*11/2
Thus if stress intensity and mouth displacement are known, the conic section
coefficient can readily be determined. Since the tip radius of an opened crack
is proportional to the square of the stress intensity factor, this approach is
equivalent to fitting a conic section to the opened crack shape at the mouth
and at the tip.
In order to determine values of the conic section coefficient, values for
the stress intensity and crack mouth displacement were taken from the litera-
1	 ture. Stallybrass (ref. 5) gives the stress intensity factor and crack opening
displacement for an edge crack in a half-plane as
x
2
ORIGMAL R^^;E I$
OF POOR QUALITY
Y - 0.7930 2 - 1.988
P no/aa - 0.7930 x 242/(*2 - 4) - 2.908
from which the conic section coefficient at x - 0 is
m - (8 - *2 M W2 - 4) - -0.3185
Keer and Freedman (ref. 6) give stress intensity and opening displacement val-
ues for various edge cracks in finite plates. However, their results at
x - 0.85 appear to be inaccurate and were ignored (their value for the stress
intensity coefficient is Y - 42.9, whereas Benthem and Koiter (ref. 7) give
Y - 33.0). Benthem and Koiter (ref. 7) treat the edge-dam in a half-plane,
which corresponds to the major limit of the single edge crack. From their
work, the limit values (x - 1) of the conic section coefficient and its slope
are
lima+l [m(1 - x)l - 3.104
lima, ld[m(1 - x)] /dx - 8.034
Values of the conic section coefficient were computed from the results of ref-
erences F to 7 using equation (5) of reference 4, which was given earlier.
A polynomial in terms of relative crack length was derived from these
coefficients as follows. First, the conic section coefficients were put into
finite form and a polynomial fit by the method of least squares. Then the
polynomial coefficients were adjusted by trial and error to provide the correct
magnitude and slope at each extreme. The resulting polynomial is
m(1 - x) - -0.3185 + 0.3185 x + 1.3765 x2 + 6.2475 x3 -8.82 x4 + 4.3 x 5	 (4)
The polynomial is everywhere finite, has the correct magnitudes and slopes at
the extremities, is within 1 percent of the reference values, and is shown in
figure 2.
Finally a wide-range expression for the stress intensity coefficient for
tension is needed, and Benthem and Koiter's (ref. 6) should suffice. Since
only tabular values are given in reference 6, the equation was rederived as
Y - V_w(1 + 20 0 - 
0-312 
[21.006 + 87.065 x +
L	 2 1/2
-(395.38 + 3618.5 x + 7584.88 x )	 (5)
The term in brackets is the coefficient tabulated in reference 7 and agrees
with the tabulations within 0.001. Now having wide-range expressions for the
conic section and stress intensity coefficients, the calculation of weight
functions is straightforward.
3
OMQML PAGE 09
a` POOR QUALITY
Once the weight function has been determined, the stress intensity factor
for any loading can be determined by simple integration, as
K I	
2/* 0	
p (x/ a ) N(x/a) x
-112 dx	 (6)
where p(x/a) is the stress distribution over the crack location in the un-
tracked body.
RESULTS AND DISCUSSION
Weight function values computed from equations (3) to (5) and those given
by Bueckner (refs. 1 and 2) are presented in table I. The conic-section weight
functions are within 4 percent of Bueckner's for relative crack lengths of one-
half and less. As a further check, stress intensity coefficients for tension
and pure bending were computed by integrating the conic-section weight func-
tions. These results are presented in table II, and are within 1.8 percent
(tension) and 2.7 percent (bending) of those given by Benthem and Koiter
(ref. 7) over the entire range of relative crack lengths from zero to unity.
Crack mouth displacement coefficients are presented in table III and will be
discussed later.
An alternate approach to the approximation of weight functions was pre-
sented by Petroski and Achenbach (ref. 8). They prescribe the crack opening
displacement in the form
n =
	
°	 4 KI	 a 112 x1/2 + G(a) a-1/2 x3/2	 (7)
E'	 a Ira
in which the term G(a) is determined by requiring that the stress intensity
factor, when determined by integration as in equation (6), be identical to the
value used in equation (7). In other words, equations (6) and (7) must be
self-similar. The approximate weight function is then determined using equa-
tion (2). This approach is advantageous if crack mouth displacements are not
available for the reference problem. However, if such displacements are avail-
able, th3 conic section approximation yields more accurate displacement coef-
ficients, as might be expected. This can be seen in table III. The coeffi-
cients computed from equation (1) are within 0.7 percent of Keer and Freedman's
(ref. 6) up to a = 0.65 and within 2.1 percent at a = 0.80. As mentioned
earlier, the results in (ref. 6) for a = 0.85 are probably inaccurate. For
comparison, the interpolation equations of Srawley and Gross (ref. q ) and Tada
(ref. 10) give crack mouth displacement coefficients of 150.1 and 149.8, re-
spectively, at a = 0.85.
Even though these results may be satisfactory, some further attention
should be paid to the matter of crack opening shape. Consider Stallybrass's
solution (ref. 5) for the edge crack in a half-plane, which was mentioned
earlier. Note that the conic section coefficient is an exact value, indepen-
dent of his numerical computations. This might lead one to expect that the
crack opening shape is an exact conic section. However, this is not the case,
as can be seen in figure 3. Here the displacements from the conic section
4
qtr'`
model are compared with Bueckner's (ref. 1) and the earlier results of Wiggles-
worth (ref. 11). Although they seem small in figure 3(a), the differences are
significant, as can be seen in figure 3(b).* The conic-section displacements
agree with Wigglesworth's at the crack tip and mouth, but are larger in be-
tween. They are smaller than Bueckner's at the tip and mouth but, again, are
larger in between. Similar small but consistent differences can also be found
in the weight functions themselves (see table I). The conic-section weight
functions are almost everywhere larger than Bueckner's for x < 0.40 and
smaller for x < 0.45. These small unidirectional errors do not average out
when the weiqht function is integrated to determine a stress intensity factor.
For example, the stress intensity coefficient at x • 0, when calculated by in-
tegration, is 1.1413 (table II). This compares with the value 1.122 from which
the conic section coefficient was calculated. The conic section model is ob-
viously not self-similar.
In figure 3(b), the conic-section model and Petroski and Achenbach's model
provide at best onl y a rough approximation of the crack profile. Both models
are essentially two-parameter representations. It seems doubtful that any two-
parameter representation could closelj match the Bueckner or Wigglesworth crack
profiles. Although more elaborate models could be constructed, the effort may
not be warranted. Either method may be adequate for most engineering applica-
tions. The choice of method will most likely depend on the information avail-
able for the reference problem.
SUMARY OF RESULTS
A closed-form expression is given for thi weight function for a strip with
a single edge crack. The expression is valid for relative crack lenghts from
zero to unity. Computed weight functions are within 4 percent of reference
values for relative crack lengths of one-half and less. Computed etress in-
tensity coefficients are within 1.8 percent (tension) and 2.7 percent (bending)
of reference values for relative crack lengths from zero to unity. Crack mouth
opening displacement coefficients are within 2.1 percent of reference values
for relative crack lengths of 0.8 and less.
This paper and (ref. 8) each present methods for developing approximate
weight functions. Each assumes a form for the crack opening displacements.
Neither method appears to have an overwhelming advantage, and the choice may
depend on the problem to be solved and the available information.
*The special form of the displacement coefficient used in figure 3(b)
de:-.erves comment. It can be shown that
( x/a)-1/2 Wn /ca) Ix/a=0 =1M K  /o ,ra
Thus the form used in figure 3(b) is particularly useful since the inter-
cept at x/a - 0 is proportional to the stress intensity factor and the
intercept at x/a - 1 is the crack mouth displacement. In this way, two
significant features of the opening shape may be clearly seen on the same
plot.
Gy
REFERENCES
1. Bueckner, H. F., "A Novel Principle for the Computation of Stress Intensity
Factors," Zeitschrift fUir An ewandte Mathematik and Mechanik, Vol. 50,
No. 9, 1970, pp. 529-546.
2. Bueckner, H. F., "Weight Functions for the Notched Bar," Zeitschrift fur
An ewandte Mathematik and Mechanik. Vol. 51, 1971, pp. 97--M9.
3. Rice, J. R.,	 n erns ona ournal of Solids and Structures, Vol. 8, No.
6, June 1972, pp. 751-75F.-
4. Orange, T. W., in Stress Analysis and Growth of Cracks Proceedings of the
1971 National SI ,o-M—mon Practure mechanics, Part,
er can SoCl ety for Testing and materi al s, Philadelphia, i97Z, pp. 71-78.
5. Stallybrass, M. P., International Journal of Engineering Science, Vol. 8,
No. 5, 1970, pp. 351- 	 .
6. Keer, L. M. and Freedman, J. M., International Journal of Engineeri _
ence Vol. 11, No. 12, Dec. 1973, pp. 1265-1275.
7.hem, J. P. and Koiter, W. T., in Methods of Anal sis and Solut = is of
Crack Problems: Recent DevelopmentsFractureMechanics; Theory ^^
Methods  o o vinCrack rob-Items,	 ., Noordhoff internaffonal
P-071shing, Leiden, The Netherlands, 1973, pp. 131-178.
8. Petroski, H. J. and Achenbach, J. D., Engineering Fracture Mechanics, Vol.
10, No. 2, 1978, pp. 257-266.
9. Srawley, J. E. and Gross, B., in Cracks and Fracture. Proceedings of the
Ninth Symposium on Fracture Mechan cs ASTM STP-601 , American soci ety or
Testing and materials, Philadelphia, 1976, pp. 590-579.
10. Tada, H., Paris, P. C., and Irwin, G. R., The Stres: .•",nal sis of Cracks
Handbook, Del Research Coro., Hellertown, a.,
11.igA —geTsworth, L. A.: "Stress Distribution in a Notched Plate," Mathe-
matika, Vol. 4, June 1957, pp. 76-96.
6
ORIGINAL 	 ,j
OF DOOR QUALITY
TABLE I. - WEIGHT FUNCTIONS FOR THE EDGE-CRACKED STRIP
(a) Bueckner (taken from second table in [2] except as noted).
a/W x/a
0.1 1	 0.2 1	 0.3 1	 0.4 1	 0.5 0.6 1	 0.1 1	 0.8 1	 0.9 1	 1.0a
NIX/4)
O.OUb 1.0736	 1.1407 1.2079 1.2797 1.3590 1.4468 1.!	 24 1.6433 1.7453 1.8424
05 1.0659	 1.1361 1.2115 1.2928 1.3806 1.4751 1.5756 1.6806 1.7874' 1.9161
.10 1.0818	 1.1688 1.2614 1.3601 1.4652 1.5768 1.6942 1.8161 1.9398 2.0697
.15 1.1061 . 1.2176 1.3348 1.4581 1.5881 1.7248 1.3679 2.0158 2.1656 2.326420 1.139( 1.2841 1.4358 1.5945 1.7606 1.9343 2.1151 2.3020 2.4934 2.6889
.?5 1.177.	 1.3623 1.5554 1.7569 1.9670 2.1859 2.4132 2.6477 2.8874 3.1637
.30 1.225!	 1.4610 1.7077 1.9653 2.2338 2.5132 2.8033 3.1032 3.4107 3.765035 1.283L	 1.5814 1.8949 2.2233 2.6655 2.9424 3.2962 3.6811 4.0772 4.5178
.40 1.3501	 1.7224 2 1162 2.5306 2.9650 3.4190 3.8918 4.3820 4.8867 5.4635
.4 1.4348
	 1.9024 2.4012 2.9296 3.4864 4.0705 4.6809 5.3160 5.9737 6.6653
.50 1.53761 2.1237 2.7550 3.42881 4.1430 4.8956 5.6848 6.5086 7.3649 8.2149
a This column cal.ulated from eqn. (2.5) of [1] except for a.0.
b This row caicula,ed from eqns. (4.32) and (4.35) of [2].
(b) This report, Eqs. (3) to (5).
a/W x/a
0.1 0.2 0.3 0.4 0.5 0.6 1	 0.7 1	 0.9 1	 0.9 1.0
N(x/a)
0.00 1.0768 1.1548 1.2343 1.3152 1.3975 1.4815 1.5670 1.6542 1.7431 1.8338
.05 1.0836 1.1686 1.2549 1.3426c 1.4318 1.5225 1.6147 1.7087 1.8043 1.9018
.10 1.1004 1.2022 1.3056 1.4104 1.5169 1.6250 1.7349 1.8465 1.9600 2.0755
.15 1.1242 1.2505 1.3787 1.5090 1.6415 1.7762 1.9132 2.0526 2.1943 2.3387
.20 1.1452 1.2924 1.4417 1.5932 1.7470 1.9033 2.0620 2.2233 2.3d73 2.6949
.25 1.1911 1.3875 1.5894 1.7967 2.0095 2.7%79 2.4519 2.6815 2.9169 3.1581
.30 1.2352 1.4793 1.7322 1.9440 2.2646 2.5440 2.8323 3.1293 3.4352 3.7499
.35 1.2882 1.5906 1.9069 2.2372 2.5811 2.9385 3.3093 3.6932 4.0903 4.5002
.40 1.3521 1.7262 2.1218 2.5382 2.9752 3.4320 3.9085 4.4040 4.9183 5.4509
.45 1.4300 1.8933 2.3885 2.9144 3.4701 4.0545 4.6667 5.3058 15.9 712 6.662050 1.5263 2.1019 2.7242 3.3910 4.1001 4.8497 5.6382 6.4641 7.3261 8.2230
cLargest difference, *3.85 percent.
nURIGINAL PAGE 1-5
Of POOR QUALITY
TABLE 1I. - STRESS INTENSITY FACTOR COEFFICIENTS
a/W Uniform tension,
(K	 /0,1 18) 	 (1 - 1) 3/2/(1	 + 21)
Pure bending,
(K I /o0.a)	 (1 - 1)3/2
Ref. [7] This report Ref.	 [1] This report
0.00 1.122 1.1413 1.122 1.1413
.05 .961 .9780 S90 1.0107
.10 .849 .8631 .8)0 .9122
15 .767 .7768 .811 .8328
.20 .703 .7096 .747 .7667
.25 .653 .6558 .693 .7108
.30 .611 .6121 .649 .6633
35 .517 .5761 .610 .6224
.40 .548 .5457 .577 .5889
.45 .523 .5205 .549 .5562
.50 .501 .4984 .523 .5290
.55 .482 .4790 .501 .5047
60 .464 .4618 .431 .4830
.65 .449 .4466 .463 .4635
.70 .435 .4329 .446 .4461
15 .423 .4207 .432 .4306
80 .411 .4097 .418 .4167
.85 .401 .3999 .406 .4043
.90 .391 .3908 .394 .3932
.95 .382 .3823 .184 .3831
1.00 .374 .37"
TABLE III. - CRACK MOUTH
DISPLACEMENT COEFFICIENTS
a/w E'no/ca
Ref.	 [6] lRef.	 [8) Eq.	 (1)
0.05 2.967 2.878 2.965
.10 3.107 3.011 3.116
.15 3.336 3.238 3.354
.20 3.667 3.562 3.689
.?J 4.707 4.607 4.739
.40 6.549 6.468 6.589
.50 9.925 9.938 9.972
.60 16.68 16.92 16.70
.65 22.73 23.29 22.69
80 81.26 82.15 79.55
.85 168.9 184.6 140.7
D 
-A
1.0
.8
.6
p
Wp .43J
m
ORUNAL Pi-%GE iF
OF POOR QUALITY
.2	
Ind 2? m O 2m m/a)2
0	 .2	 .4	 .6	 .8	 1.0
NORMALIZED DISTANCE FROM CRACK TIP, xla
Figure 1. - ^unit sections.
3	 O STALLYBRASS 151	 t(EXACTIA BENTHEM & KOITER 171
p KEER & FREEDMAN 161
	
E 
2	 -a 3185 + a 3185h + L 3765A2
+ 6.2475A3
 - 8.82X4 + 4.3X5_
O
,^ 1
v_
U
F-
W 0H
V
	
E .	 OV
	'	
_1l	 I	 t	 1	 I	 7
	
0	 .2	 .4	 .6	 .8	 1.0
RELATIVE CRACK LENGTH, A - WW
Figure 2. - Conic section coefficient for tension load.
.oc
z
w
4R
z
WUg
a
0
c^z
zWd
YU
Q
d'U
`TJ
OF POOR QUALITY
3
CONIC SECTION, EQ. (1)^
BUECKNER I11-^`
A
	 WIGGLE SWORTH 1111
0
Z
W
z 2
WU
a
c
c^
z
z
1
0
YUQ
U
01	 f	 1	 1	 1	 f
(a) Conventional form.
3.2
CONIC SECTION, EQ. (1)
BUECKNER I11
3.1	 %	 PETROSKI & ACHENBACH 181
i i— WIGGLESWORTH fill
3.0
2.
2.
0	 .2	 .4	 .6	 .8	 1.0
NORMALIZED DISTANCE FROM CRACK TIP,
xia
(b) Special form.
Figure 3. - Crack opening displacement for
the edge crack In a half-plane.


Wide range weight functions for the strip with a single edge crack

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