Mode 1 stress intensity coefficients were obtained for an edge-cracked disk (round compact specimen). Results for this plane elastostatic problem, obtained by a boundary collocation analysis are presented for ratios 0.35 less than A/D less than 1, where A is the crack length and D is the disk diameter. The results presented are for two complementary types of loading. By superposition of these results the stress intensity factor K sub I for any practical load line location of a pin-loaded round compact specimen can be obtained

Gross, B.

English

NASA Technical Reports Server

by Bernard Gross
Lewis Research Center
Cleveland, Ohio 44135 A UG 1Si7
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NASA TECHNICAL	 NASA TM 73692
MEMORANDUM
(NASA-TM- 73692)
 MODE I ANALYSIS OF A
CRACKED CIRCULAR DISK SUBJECT TO A COUPLE
AND A FORCE (NASA) 15 p HC A02/MF A01
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MODE I ANALYSIS OF A CRACKED C I RCULAR D I SK
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https://ntrs.nasa.gov/search.jsp?R=19770020487 2020-03-22T09:43:46+00:00Z
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1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
NASA TM 73692
4. Title and Subtitle 5. Report Date
MODE I ANALYSIS OF A CRACKED CIRCULAR DISK SUBJECT
6. Performing Organization CodeTO A COUPLE AND A FORCE
7. Authors) 8. Performing Organization Report No.
Bernard Gross E-9228
10. Work Unit No.
9. Parformins Organization Name and Address
National Aeronautics and Space Administration 11. Contract or Grant No.
Lewis Research Center
Cleveland, Ohio 44135 13. Type of Report and Period Covered
Technical Memorandum12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration 14. Sponsoring Agency Code
Washington, D. C. 	 20546
15. Supplementary Notes
16. Abstract
Movie I stress intensity coefficients were obtained for an edge-cracked disk (Round Compact
Specimen).	 Results for this plane elastostatic problem, obtained by a boundary collocation
analysis are presented for ratios 0.35 < A/D < 1, where A is the crack length and D is
the disk diameter. 	 The results presented are for two complementary types of loading. 	 By
superposition of these results the stress intensity factor K I, for any practical load line loca-
tion of a pin-loaded round compact specimen can be obtained.
17. Key Words ISuggested by Author(s)) 18. Distribution Statement
Fracture mechanics Unclassified - unlimited
Stress intensity factors STAR Category 39
Boundary collocations
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22.	 Price*
Unclassified Unclassified
For sale by the National Technical Information Service, Springfield, Virginia 22161
rw
NASA-C-168 (Rev. 10-75)
MODE I ANALYSIS OF A CRACKED CIRCULAR DISK SUBJECT
TO A COUPLE AND A FORCE
by Bernard Gross
Lewis Research Center
SUMMARY
Mode I stress intensity coefficients were obtained for an edge-
cracked disk (Round Compact Specimen). Results for this plane elasto-
static problem, obtained by a boundary collocation analysis are presented
for ratios 0.35 < A/D < 1, where A is the crack length and D is the
disk diameter. The results presented are for two complementary types-of
loading. By superposition of these results the stress intensity factor
KI , for any practical load line location of a pin-loaded round compact
specimen can be obtained.
N	 INTRODUCTIONN
wThe round compact specimen hedge-cracked disk) is currently being
considered by ASTM Comaittee E-24 on Fracture Testing for incorporation
into ASTM Standard Method of Test E 399 on Plane Strain Fracture Tough-
ness of Metallic Materials. In reference 1 values of the mode I stress
intensity coefficients r  and rM for round compact specimens are
given for the range of ratio of crack length to specimen diameter, A/D,
from 0.65 to 0.9 (where crack length A here is the distance from the
crack tip to the specimen circumference). The coefficients rp and r,,,
apply to two complementary types of specimen loading; r  to a nominal
uniform distribution of stress across the net section, and rM to a
nominal bending stress distribution (fig. 1). While these types of load-
ing are impractical in themselves, the two coefficients can be combined
to represent any practical case of loading of the specimen by a pair of
equal and opposite forces normal to the crack (pin loading). The appro-
priate value of the stress intensity factor is then obtained from this
combination of coefficients. The approach used in reference 1 was to
obtain values of r  and rn by the boundary collocation method for a
set of ring segment specimens (ref. 2) with successively diminished ratios
of internal to external radius to a minimum ratio of 0.07. These results
were then extrapolated to estimate values for the limit case of zero
internal radius, corresponding to the edge-cracked disk.
W2
I.
To extend and assess the results of reference 1, a more direct
approach has now been taken which employs boundary value stress functions
and derivatives appropriate to the cracked disk. The present results
cover the range of ratio of crack length to specimen diameter, A/D, from
0.35 to 1. These results are given both in tabular form and in the form
of interpolation equations obtained by multiple linear and nonlinear re-
gression analysis. It should be appreciated that there is a limitation
on the applicability of the present results to practical pin-loaded
cracked disk specimens. When the crack tip is close to the load line
there will be a significant difference between the actual distribution
of loading forces and that assumed in the present model, and the values
of the corresponding stress intensity factors will be significantly dif-
ferent. It is estimated that this effect will be negligible when the
distance of the crack tip from the load line exceeds 0.15 D.
LIST OF SYMBOLS
A	 crack length measured from crack tip to the circum-
ference
a	 crack length measured from the crack tip to load line
B	 specimen thickness
D	 disk diameter
KI
	Mode I stress intensity factor KI = KIM when ap = 09
and KI = KIP when a  = 0
L	 load line location
M	 resultant moment at nominal neutral axis of specimen
P	 applied pin load
W= (D/2+L)	 distance measured from load line to circumference of
K	 specimen
^	 r(op + M ^ A (1-Q 	A/D stress intensity coefficient r = I' when a  = 0,)
and r = r  when aP = 0
aM=6M/[B(D-A) 2 ]	 component of fictitious normal net stress at the crack
tip due to M
ap=P/[B (D-A)]	 component of fictitious normal net stress due to load P
X	 stress function.
w•
3
APPROACH
As shown in figure 1, the cracked round specimen has a diameter D
and is loaded through pins by a couple Mo and opposed tensile forces
P normal to the crack with load line offset a distance L from the
center of the specimen. Calculations were performed for each of two
complementary types of loading: case 1, nominal uniform tension and
case 2, nominal bending. For case 1, provided the ratio a/D (=A/D
+L/D-1/2) is at least 0.15, and regardless of the load line location L,
the computed stress intensity factor KIp is a function of two vari-
ables: nominal uniform stress a  and crack length ratio A/D. This
case is obtained by adjusting the boundary load condition along %V
to produce at the nominal neutral axis position (D-A)/2, a resultant
moment of zero, and force equal to the applied load P.
For case 2, provided the ratio a/D is at least 0.15, and regard-
less of the load line location L the computed stress intensity factor
KIM is a function of two variables: nominal bending stress at the
crack tip oM, and crack length ratio A/D. This case is obtained by
applying only a cour le at boundary G a
Stress intensity coeff icienty rp , r  and r are defined as
KI
p vp fTl - A D)
r	 -"IM
M aM A 1 - A/ D)
KI
r=
(Qp + aM) A 1 - A/D)
For the general case (fig. 1), there are two independent variables L/D
and A/D. As an illustration, for a disk with load P and couple o
at load line location L:
P
o P = B(D-A)
and
I
(1)
to
(2)
(3)
«.M
4
3[20+P(2L+A)]
M	 B (D - A) 2
By superposition we have
KI = KIP + KIM
Applying equation (1) and equation (2) we obtain
^F	 oM
r= vp + a rP+ a
^ 
rM
Replacing a  and aM by their equivalent definitions we obtain
P(D - A)rp + 3[2M + P(2L + A)]rM
r = P(D - A) + 3[2K 0  + P(2L + A) ]
If at loadline L. a load P is applied through pins and the moment
H = 0, after algebraic manipulation of equation (4) we obtain a more
common form
KB B ^= 1 -
A
 A D r p + 
3(	 + .(2L/D)_ r	 (5)( / }	 1	 (A/D)	 M
RESULTS AND DISCUSSION
As shown in figure 1, the resultant solution of the cracked disk
problem is obtained by combining two complementary types of loading.
The first type of loading is based on a nominal constant mid-net
section stress a  = P/[B(D-A)] resulting in the stress intensity coef-
ficient r  as a function of a  and A/D. The second solution is
based on a nominal pure bending stress at the mid net section where
aM=3[P(A+2L)+M0]/[B(D-A)2] from which the stress intensity coefficient
r  as a function of a  and A/D is obtained.
Pp
 and r  are obtained using the boundary collocation method
with 60 boundary stations and an overdetermined system of eq ,:ations as
detailed in reference 3. Twenty equally spaced boundary stations were
taken along the straight portion of the boundary GB and 40 stations
equally spaced were taken along the curved boundary
	 QC (fig. Al).
iph"
I
	
5
The appropriate stress function boundary conditions for both cases are
given in the appendix. The limit values for r  and r M ae A/D-l-1 were
obtained from reference 4. The values of r  and rM given in table I
were obtained for a load line offset L = 0.35D. The values given in
table II of r were obtained directly by boundary collocation with the
load line L = D/4.
It has been shown (ref. 5) that the pin loaded holes can have a
significant effect on the stress intensity factor. It is therefore
estimated that the results presented herein will apply when the distance
from the crack tip to the load line exceeds 0.15 D.
Table II (eq. (4) with M  = 0) contains a comparison of the present
results with those of references 1, 6, and 7, for a pin loaded cracked
disk with load line at L = D/4. Excellent agreement is obtained. In-
cluded in this table are the stress intensity coefficients of the stand-
ard rectangular compact specimen (ref. 8). A geometric comparison be-
tween the standard rectangular and round compact specimen is given in
figure 2. Fitting functions for r  and rM were obtained by linear
and nonlinear least squares best fit regression analyses respectively.
These functions are:
2	 3	 4
rM = 5.611 - 21.246 D + 38.149 (D) - ';? - ? 59 (A + 10.320 (A)
and
r	 0.501 +	
0.02104 A/D) - 0.525
p	 [ (A/D) - 0.525 [ (A/D) - 0.0798] + 0.
These functions are considered to be accurate to less than a percent of
the computed solution in the range 0.35 < A/D < 1. Clearly, the accuracy
of application is dependent upon how well the assumed model boundary
conditionG approximate the real boundary conditions.
EXAMPLE
Two examples now follow which will demonstrate the use of table I
in conjunction with equation (4) and equation (5).
Examp le 1
For a pin loaded disk with load P at load line location L = D/4,
to find the value of KB A /P where W = 0.75 D and a/W = 0.4. Thus
Ob
a • A - 0.25 D, A/D • 0.55 and from table I, r  = 0.509 and r  • 1.064.
From equation (5) since M  = 0, we have KB *w7P	 3/4 KB 4/7DP • 7.618.
There is a small difference between this value and the value 7.613
given in table II. The values in table II are obtained by direct appli-
cation of the boundary conditions for this loading and the value 7.618
in the above example is obtained by combination.
Example 2
Given a disk with crack length A. load P, and couple M  at
L : 0, to find the resultant stress intensity coefficient r. From
table I we obtain r  and r  and using equation (4) we have
P(D - A) rp + 3(2M0 + PA)rM
r	 PD + 6Mo + 2PA
i
7I--
APPENDIX
The results presented here were obtained by boundary collocation
analysis of a homogeneous isotropic specimen under plane elastostatic
conditions. This method is described in detail in reference 3. The
necessary boundary conditions to be satisfied by the stress function
and its normal derivative (fig. Al) are as follows
Along 0 0
i	 -sin a sin 0 cos a cos 0	 o sin 20 -
-
20 cos 2a 
1)2a + sin 2a
	
2a - cos 2a	 2 {2a cos 2a sin 2a
+P cos	
• -sin 3a	 + a cos a - sin a cos`a
p	 2a + sin 2a	 2a - sin 2a
- Pp sin	 a sin a + cos a sin 3a +	 Cos 3a2a + sin 2a	 2a - sin 2a
and
U = U co s ( o + a) - ^ sin (o + a)ax
Along B ^C
X = P 
Z (cos 0 + sin 2a) + Pe - Mo
and
aR = P cos 0
The symbols p, ^ , a, M  , @, and a are defined in figure Al.
p
8REFERENCES
1. Gross, Bernard: Mode I Stress Intensity Factors for Round Compact
Specimens. NASA TM X-73472, 1976.
2. Gross, B.; and Srswley, J. E.: Analysis of Radially Cracked Ring
Segments Subject to Forces and Couples. NASA TM X-71842, 1975.
3. Gross, B.: Some Plane Problem Elastostatic Solutions for Plates
Having a V-Notch. Ph.D. Thesis, Case Western Reserve Univ., 1970.
(See also NASA TM X-69370.)
4. Srawley, J. E.; and Gross, B.: Side Cracked Plates Subject to Com-
bined Direct and Bending Forces. Cracks and Fracture. Am. Soc.
Test. Mater. Spec. Tech. Publ. 601, 1974, pp. 559-579.
5. Newman, J. C., Jr.: Stress Analysis of the Compact Specimen In-
cluding the Effects of Pin Loading. Fracture Analysis. Am. Soc.
Test. Mater. Spec. Tech. Publ. 560, 1974, pp. 105-121.
6. Feddern, Von Gunter; and Macherauch, Eckard: Eine Neu* Probenform
fur Bruchmechanische Experiments (A New Specimen Geometry for RIC
Measurements). Z. Metallkde, Bd. 64 9 Heft 12, Dec. 1973, pp. 882-884.
7. Private cou mication with D. F. Mowbray and W. R. Andrews, General
Electric Company, Schenectady, N. Y., May 1976.
8. Srawley, J. E.; and Gross, B.: Stress Intensity Factors for Bend
and Compact Specimens. Eng. Fract. Mech., vol. 4, no. 3, 1972,
pp. 587-5890
i 
I 
44 
f 
, 
TABLE I. - VALUES Of STRESS INTENSITY COEFFICIENTS 
a 
rp AND rK FOR THE EDGE CRACKED DISK 
WHEN aiD> 0.15 
AID rp rK 
0.35 0.2587 1.6157 
.40 .3857 1.4172 
.45 .4538 1.2695 
.50 .4901 1.1551 
.55 .5090 1.06~7 
.60 .5184 • 9f:So~ 
.65 .5228 .9255 
.70 .5246 .8717 
.75 .5251 .8252 
.80 .5251 .7845 
.85 .5249 .7487 
.90 .5242 .7168 
.95 .522 .682 
a1.00 .521 .663 
Limit values ~bta1ned from ref. 4. 
T 
a/W Direct
collo-
cation
solid
disk
Collocation
ring seg-
ment
Ref.	 1
Experi-
mental
results
Ref. 6
Finite
elements
Ref.	 7
Compliance
results
Ref.
	
7
Standard
rectangular
compact
specimen
Ref. 8
KIBA/W-/P
0.200 4.745 ------ ------ 4.750
.267 5.472 ------ ------ ------ ------
.300 !._!! ------ __i___ _M___
------ 5.844
.333 6.365 ------ 6.244 ------ ------
.400 7.613 ------ 7.299 ------ _!!!M 7.333
.467 9.203 ------ 8.948 ------ ------
.500 ------ ------ 10.185 10.286 9.631
.533 11.378 11.926 11.314 ------ ______ _!____
.600 14.532 14.820 14.721 14.738
 14.566 13.62
.667 19.368 19.518 19.680 ------ ------
.700 ------ ------ ---___ 22.653 22.798 21. .56
.733 27.411 27.490 26.879 ------ ----^ -_-
.800 42.773 42.745 ------ 42.542 --•---- 41.07
.867 79.542 79.523 ------ ------ ------
.933 226.
•
.l
10
TABLE II. - COMPARISON OF PRESENT STRESS INTENSITY PARAMETER
KIB ,4VW`/P AS A FUNCTION OF a/W WITH 'THOSE RESULTS OF
REFERENCES 1 9 6 9 7 9
 AND 8 (AS SHOWN IN FIG. 2
FOR LID • 0.25 AND W = 0.75 D)
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Mode I analysis of a cracked circular disk subject to a couple and a force

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