The mode 1 stress intensity factors were computed for round compact specimens by the boundary collocation method. Results are presented for ratios A sub T/R sub 0 in the range 0.3 to 0.8, where A sub t is the distance from the specimen center to the crack tip for a specimen of diameter 2R sub 0

Gross, B.

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{4
NASA TECHNICAL NASA TMX-73472
MEMORANDUM
a
a
:-
N
(NASA-TM -X-73472)	 MOLE 1 STRESS INTENSITY. N77-105y7
ti FACTORS FOR ROUND COMPACT SPtCIME 	 (	 )
7 p HC A02/MF A01	 CSCI. 20K Unclas
,.. G3/39	 08955
r
f. MODE I STRESS INTENSITY FACTORS FOR
ROUND COMPACT SPECIMENS
by Bernard Gross;T
Lewis Research Center
Cleveland, Ohio 44135
TECHNICAL PAPER to be presented at
Task Group E-24. 01. 12 of Committee E-24 of the
American Society for Testing and Materials
Philadelphia, Pennsylvania, October 5, 1976
iMODE I STRESS INTENSITY FACTORS FOR ROUND COMPACT SPECIMENS
by Bernard Gross*
ABSTRACT
Mode I stress intensity factors K were computed for round compact
specimens by the boundary collocation method. Results are presented for
ratios A
t 
/R in the range 0.3 to 0.8, where A is the distance frora the
specimen center to the crack tip for a specimen of diameter 2R o.
INTRODUCTION
ASTM Standard Method of Test E 399 for Plane Strain Fracture
Toughness of Metallic materials is presently confined to two rectangular
shaped specimens: bend or compact 11].
In order to economize on test specimen stock and machining for
testing of structures such as discs and solid cylinders, a round compact
specimen is an obvious substitute for the rectangular compact specimen
[2-5]. For this reason, and to evaluate existing experimental and finite
element results on the round compact specimen, an independent analytical
study was made. The method employed is quite versatile in that a range of
load line locations can be assigned and results derived by the superposi-
tion of two independent solutions obtained by the boundary collocation
method with 60 boundary stations and an overdetermined system of equations
as detailed in Ref. 6.
APPROACH
Figure 1(a) shows a cracked round compact specimen, of diameter
2 R , loaded through pins by opposed tensile forces P, normal to the
cracK. The line of action of the load is offset by the distance X from
the center of the specimen. The analytical solution to this configuration
is based on the model shown in Fig. 1(b). There are two independent vari-
ables Xo /Ro and A /Roy where At is the distance measured from the cen-^
ter of the model to the  crack tip.
The analysis follows that of a previous paper on cracked ring
segments [7], in that the loading of the specimen is characterized by the
statically equivalent combination of resultant force P, chosen to act
through the mid-net section, and complementary couple M, as shown in
Fig. 2. For the cracked ring segment there is an additional independent
parameter Ri/Ro , where Ri is the inner radius. The approach taken here
was to estimate the limit of the stress intensity coefficients (defined in
Ref. 7) rp and rM as Ri/Ro->O, in the expectation that these limits
would represent appropriate values for the round compact specimen,
It is necessary to consider the limitations of applicability of
the present results to practical specimens which are loaded by opposed
tensile forces acting through pins. If the crack tip is sufficiently close
to the load line, the difference between the actual distribution of loading
forces and that assumed in the present analytical model cannot be neglected.
Guidance in this respect was obtained from Ref. 8, which deals with the
_ASTM Standard E399 rectangular compact specimen. For crack length to width
ratios less than 0.3, the difference between KI obtained from an assumed
simple boundary load condition as used here and one which models the lo-
calized pin loading, becomes significant. Gne may compare the two speci
*Lewis Research Center, Cleveland, Ohio, U.S.A.
I
i
`ter
so
r co
w
_	 .r	 ..
F
F
! men types on the basis of the same relative crack lengths (a/W for the
rectangular compact specimen and 	 At /R0	for the round compact specimen).
This is shown in Fig. 3 where a rectangular compact specimen is superim-
posed on a round compact specimen.	 Both specimens have the same relative
crack length ratio of 0.3.	 From this representation and the results of
Ref. 8, it is surmised that the analytical results presented will apply to
the load line locations	 Xo > 0, and	 A/Rgreater than 0.3.t	 0
€4 RESULTS AND DISCUSSION
For the round compact specimen, as shown in Fig, 1, there are two
independent variables, X0 /Rand the relative crack length to outer radius
•
0
ratio, At/Ro .	 With load
	
P	 applied at	 X0 , the mid net section nominal
bending moment is	 M - P(Xo + (Ro + A )/2).	 The ratio	 aM/ap	 algebrai-
1)/(7.cally reduces to 	 3(2X /RO + At /Ro +	 - At/Ro).
The values o?	 I'p	 and	 rM	 were obtained using the boundary col-
location technique with 60 boundary stations and an overdetermined system
of equations as detailed in Ref. 6.	 The results for the limit case as
Ri-►O, were based on examination'of the trends shown in Figs. 4 and 5. 	 From
this output the estimated limit values in Table I of 	 rp	 and	 rM	 were
obtained.
Applying the principle of superposition to the stress intensity
factor	 K, the equation relating the stress intensity coefficients 	 I', rp
and
	
rM	 where	 P	 KI/(ap + oM)Ar(1 - At/Ro) and is derived as
ap
	 amT = rp + rM	 (1)
ap + aM
	
	 aM
Through algebraic manipulation one obtains
ap	1 - At/R0 e
F	 (2)
ap + a	 2(3X0/Ro + At/Ro + 2)
and
aM	 3(2X0/Ro + At/Ro + 1)
_	 (3)
_+Go	 7m) 2(3X0 /R0 + At /R0 + 2)
As shown in Fig. 4,IM	converges rapidly with decreasing 	 Ri/Ro	ratios.
In contrast, rp, Fig. 5 is not as well behaved as 	 TM.	 However, this un-
certainty in	 rp	 is not critical since the coefficient of this term in
Eq.	 (1) is small compared with the coefficient of 	 rM	 (i.e., the specimen
is primarily deformed in bending).
Table II shows a comparison of the present results obtained a
through superposition with the finite element results of Ref. 5 and thef
experimental results of Refs. 2 and 5.	 Very good agreement is obtained.
In Ref. 8, for the standard rectangular compact specimen, it was
determined that the pin loaded holes can have a significant effect on the
stress intensity factor. 	 It is inferred from those results that the ana-
lytical results presented here will apply to load line locations 	 Xo > 0,
and
	
At /Ro	 values greater than 0.3.
REPRODUCIBILITY Cp TUN
ORIGINAL 
'AGE TS Poft2
I4
5
4
NUMERICAL RESULTS
A specific example of using Eq. (1) to obtain the stress inten-
sity coefficient r follows.
For Xo = Ro /2, one obtains from Eq. (2)
a 
	
1 - At/Ro
(ap + aM)	 (7 + 2 At/Ro)
and from Eq.- (3)
am
	 6 + 3 At/Ro
aP + aM (7 + 2 At/Ro)
For At NO = 0.5, from Table I rp =0.635 and rM = 1.02. Thus,
P = 0.0625(0.635) + 0.9375(1.02) = 0.996
CONCLUSIONS
An analytical solution to the model of the round compact speci-
men (an obvious substitute for the rectangular compact specimen for testing
round product forms) has been obtained by the boundary collocation method.
Through application of the principle of superposition, the solution for a
wide range of load line locations can be obtained. Very good agreement was
obtained when compared with available experimental and finite element
results for ratios of At/Ro > 0.3.
Round compact specimens can be used to economize on specimen
stock and machining for structures such as discs and solid cylinders.
REFERENCES
1. 1975 Annual Book of ASTM Standards. Part 10-Metals-Mechanical, Fracture
and Corrosion Testing; ASTM (1975), pp. 561-580.
2. V. G. Feddern, E. Macherauch: Z. Metallkde, vol. 64, 1973, pp. 882-884.
3. W. Schutz, W. Oberparleiter: AGARD No. 176, Jan. 1974, pp. 370-394.
4. J. H. Underwood, D. P. 'Kendall: Watervliet Arsenal Rpt. INT-TR-74041
(1974)
5. D. F. Mawbray, W. R. Andrews: Stress Intensity Calibration for A Com-
pact Specimen With a Round Profile, General Electric Technical Infor-
mation Series, Jan. 1975.
6 B. Gross, A. Mendelson: Int. J. of Fracture Mechanics, Sept. 1972,
vol. 8, no. 3.
7. B. Gross, J. E. Srawley: NASA TM X-71842, 1976.
8. J. C. Newman, Jr: pt. 2, ASTM STP 560, ASTM Phila., Pa., 1974 pp. 105
121.
3
TABLE I. - ESTIMATED LIMIT
VALUES OF r  AND r  AS
R i o/R -► 0
At
Ro
r r 
0.3 0.950 1.41
.4 .720 1.17
.5 .635 1.02
.6 .597 .909
.7	 ' .573 .825
.8 .555 .760
A
h
Ln
c.a
i
W
At Present Experimental Experimental Finite element
-- collocation results,	 [2] results,	 [5] results,	 [5]
Ro results, r r r
r
0.25 1.46 1.44
.30 1.37 1.30
.40 1.14 1.13 1.12 1.13
.50 .996 1.00
.55 .936 .929
.60 .894- .874
.70 .816 .812
.80 .755
4
r.
j'
}
r.
i
TABLE II. - COMPARISOM OF PRESENT RESULTS WITH THOSE OF REFS. 2 AND
5 FOR Xo/Ro
 0.5
a -At =. 3
W Rp
W -Ro =1
i°.
i
i
^p*
f	
_	
CoPo
QM lpp	 +	 "m rM
	
NET SECTION	 NOMINAL NEUTRAL 	
_
NEUTRALAXI S-7	 p	 AXIS POSITION	 p/Ro+ Atl	 )
THRU NET SECTION-
	
2/ Ro+At
/P	 j	 P	 /	
MII P X°+ 2
Y
	
	
AT NEUTRAL	 AT NEUTRAL
Rti
	
0-At	 X°	 1 
AXIS POSITION	 AXIS POSITION
Alt Z 	 Mo=PX°	 At	
I	 P-0	 M=MII	 t
	
t--	
—	 t I 	 At	 — At	 P=O
Xo	
R°	 = PX	
!	 °P	
°MMM° 	 °	 1^
P	 Pp	 Ro	 p	 R°	 M
(a) RADIALLY CRACKED SPECIMEN 	 (b) RADIALLY CRACKED	 CASE 1	 CASE 2
	
WITH APPLIED P AT X0.	SPECIMEN WITH APPLIED
LOAD P AND MOMENT Mo
	 figure 2. - Application of superposition to round compact specimen loaded through pins at distance
 AT X = 0.	 Kl	 tf
where r _
Figure L - Load modelling of the round compact specimen.
	
(op+aM) At(1- At1Ro
^^	 1
Figure 3. - Round compact specimen superimposed on the
rectangular compact specimen for estimating the pin loaded
hole effect
e
1.50
1.40
1.30
1.20
V
v
1.10
N
Z
Z
N 1.00
H
IYN
90
.80
r
A"'o
0.3
A w A w A	
. 44	 ,
.5
.6
i
0	 .05	 .10	 .15	 .20	 25	 .30	 .35	 AD
RATIO OF INNER RADIUS TO OUTER RADIUS, Ri/Ro
Figure 4, - Convergence of rm with decreasing R IIRo ratios for A t/R D
 ratios in the range	 iof 0.3 to 0.8.
i
{
a
{
i
{
ul
4t7
^
y^
u
ri
.95 _
	
. 90	
AtlRo
0.3
.85
zW
U
,75
U
N
	
.70	
4
W
65
.5
.60
.^6
.55
.8
.50
0	 .05	 .10	 .15	 .20	 .25	 .30	 .35	 .40
RATIO OF INNER RADIUS TO OUTER RADIUS, RilRo
Figure 5, - Convergence of rp with decreasing R i lRo ratios for A t1Ro ratios in the
range of 0, 3 to 0.8.
I
i


Mode 1 stress intensity factors for round compact specimens

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