A set of equations describing certain fracture mechanics parameters for chevron-notch bar and rod specimens are presented. They are developed by fitting earlier compliance calibration data. The difficulty in determining the minimum stress intensity coefficient and the critical crack length is discussed

Bubsey, Raymond T.

Orange, Thomas W.

Pierce, William S.

Shannon, John L., Jr.

English

NASA Technical Reports Server

NASA Technical Memorandum !__04367
===
/
--? \.
Analysis of Some Compliance
CalibratiOn Data for Chevron-Notch
Bar and Rod Specimens
(NASA-TM-I04367) ANALYSIS OF SOME
COMPLIANCE CALIBRATION DATA FOR
CHEVRON-NOTCH BAR ANO ROD SPECIMENS (NASA)
14 p CSCL 20K
G3/39
N91-22594
Unclas
00II69I
Thomas W. Orange, Raymond T. Bubsey, William S. Pierce,
and John L. Shannon, Jr.
Lewis Research Center
Cleveland, Ohio
Prepared for the
Symposium on Chevron-Notch Fracture Test Experience
sponsored by the American Society for Testing and Materials
Indianapolis, Indiana, May 6, 1991
https://ntrs.nasa.gov/search.jsp?R=19910013281 2020-03-19T18:00:18+00:00Z

ANALYSIS OF SOME COMPLIANCE CAUBRATION DATA
FOR CHEVRON-NOTCH BAR AND ROD SPECIMENS
Thomas W. Orange, Raymond T. Bubsey," William S. Pierce," and John L. Shannon, Jr.
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
SUMMARY
This paper presents a set of equations describing certain fracture mechanics
parameters for chevron-notch bar and rod specimens. They are developed by fitting
earlier compliance calibration data. The difficulty in determining the minimum stress
intensity coefficient and the critical crack length is discussed.
INTRODUCTION
This paper presents a set of equations describing certain fracture mechanics
parameters for chevron-notch bar and rod specimens. They are developed by fitting
previously reported experimental compliance calibration data. Their use will facilitate
the testing and analysis of both brittle metals and the tougher ceramics. The
equations present the various parameters in forms suitable for determining fracture
toughness from maximum load, for determining the crack-extension resistance curve
(R-curve), and for setting instrument sensitivities. The data encompass the entire
range of the specimen geometries most commonly used.
We first discuss briefly the background of the chevron-notch specimens and the
experimental data to be used. Then we present a more extensive discussion on some
particular characteristics of the chevron-notch specimens and their practical
application. The fitted equations are presented and their fitting accuracies are
discussed. Finally, problems in determining the minimum stress intensity coefficient
and the critical crack length are discussed.
SYMBOLS
a
ao
am
B
C
C'
D
Crack length (measured from load line)
Distance from load line to tip of chevron
Crack length at which Y* is minimum
Specimen thickness
Specimen compliance, C = EBV/P
Compliance derivative, dC/do=
Diameter (rod), D = B
"Retired.
E
K,
P
V
W
y,
a
%
_m
Elastic (Young's) modulus
Opening-mode stress intensity factor
Plane-strain fracture toughness for chevron-notch specimens
Applied load
Crack mouth opening displacement
Width
Dimensionless stress intensity factor for a crack in a chevron notch,
K_W'/2/P
Minimum value of Y" as a function of e
a/w
ao/W
a./W
BACKGROUND
The chevron-notch specimens are fairly recent additions to the field of fracture
mechanics. Consequently they do not have the same historical background of
extensive stress intensity and displacement analysis as do the more common
specimen types. But, like the earliest specimen types, we can develop useful
expressions using experimental compliance data.
Compliance data for the chevron-notch bar [1] and rod [2] specimens were
previously reported. In each paper, one fitted equation was presented relating the
minimum stress intensity factor to the initial crack length and to the specimen
dimensions. A later paper [3] reported additional data for specimens having smaller
initial crack lengths and also revised the previous equations to cover the wider range
of crack lengths. But those equations alone are not sufficient for all analyses and
tests involving high-toughness ceramics. To make them more complete and useful, a
new set of generalized equations are presented in this paper. These equations are
developed by fitting curves to the existing data. They are usable over a wide range of
specimen dimensions.
CHARACTERISTICS OF CHEVRON-NOTCH SPECIMENS
For most common fracture test specimens, the dimensionless stress intensity
factor (Y) increases continually with increasing relative crack length (a/14/). But due to
the wedge shape of the un-notched material in the chevron-notch specimen, the
corresponding factor Of) reaches a minimum, denoted Y'm, as the crack length
reaches a value denoted am. The values of Y'm and =m are functions of specimen
dimensions and notch geometry only and are independent of material properties.
If the material being tested has a crack growth resistance curve which increases
rapidly to a relatively constant plateau (known as a "flat" R-curve), instability will occur
at a =ar, and P=Pr,=c" Then the fracture toughness (K_,) can be calculated from
-P'== (Z)
and no other test measurements are necessary.
For some materials (even some ceramics), however, the R-curve does not reach a
plateau but continues to increase with increasing crack extension (a "rising" R-curve).
For such materials eqn. (1) does not apply and it may be desirable to determine the
complete R-curve. In this case ASTM Test Method E 561 [4] may be used for
guidance. If crack mouth opening displacement (CMOD) is measured during the test
(as in E561) and appropriate compliance relations are available, one can calculate the
instantaneous crack length. From crack length and load, one can calculate the crack
extension resistance as
Kz = r -.ar_/2 (2)
A plot of crack extension resistance against crack advance is the R-curve.
Experimental
PROCEDURE
The experimental procedure is described in detail in Refs [1,2]. The complete data
are presented in Ref [5]. At least three replicate tests for each crack length were
averaged to obtain the data reported here. For each specimen, 7 to 15 crack lengths
(depending on the initial crack length) were tested.
Basic data reduction
Analysis of the data is based on the following equation [1]
g -a o da
(3)
and its derivative with respect to a. In [1-3] the logarithms of the basic compliance
data (C = EBV/P) were fit with a fourth degree polynomial in (=. The fitted curve was
differentiated and the values of Y* calculated from Eq 3.
In Ref [1] the reported values of Y*r, and o=m corresponded to the minimum of that
fitted curve. In Ref [2], Y*m and %n were determined in the same way, but the data
range was restricted to seven points symmetrical about the value of (=mfound by the
first fitting. Ref [3] used still another procedure. Seven points were selected by the
previous criterion. Then a fourth degree polynomial was fit to the logarithms of the
3 P
compliance derivatives. That second polynomial was used to calculate Y*m and _zm-
In the process of verifying these calculations, some general concerns arose
concerning procedures for determining Y*m and am. These will be discussed later.
Development of generalized equations
The following expressions are useful for computing the plane strain fracture
toughness Kh, when the material has a relatively "flat" R-curve.
== = _ + _=0 * ,=-==o* A3=_ (4)
and
I,." - Bo * %=° * B.=_ , _3=_o (s)
These were developed by first fitting third-degree polynomials in eo for each specimen
type (bar or rod) and each value of W/B. Then the coefficients of the intermediate
polynomials were in turn fit to a second-degree polynomial in W/B to produce the final
forms of Eqns (4) and (5). Values of the coefficients for Eqns (4) and (5) are given in
Tables 1 and 2, respectively.
An expression for determining the relative crack length a as a function of
measured displacements is
= =Co *qu*c=_ .c_r] 3 .c,_ (6)
where U is the Saxena and Hudak form [6]
U
1
The coefficients for Eqn (6) are given in Table 3. This equation lends itself to
computer-controlled fracture toughness testing since the subcritical crack growth can
be determined from automated load and displacement data acquisition.
When the relative crack length c=is known, the stress intensity factor Y* and the
dimensionless compliance EBV/P can be computed from the following expressions:
y° = eDo • _= . z_== . z_=' + D4== (7)
and
i
P
(8)
The coefficients for Eqns (7) and (8) are given in Tables 4 and 5, respectively.
DISCUSSION
Generalized equations
Eqn (4) fits the calculated values of am within 0.013W for the bar specimens and
within 0.006W for the rod specimens. Eqn (5) fits the calculated values of Y*m within
1.0% for the bar specimens and within 2.7% for the rod specimens.
Within the ranges of a and a o specified in Tables 3 to 5, Eqn (6) fits the measured
values of a within 0.003W for the bar specimen and within 0.002W for the rod
specimen; Eqn (7) fits the calculated values of Y* within 2.9% for the bar specimen and
within 2.1% for the rod specimen; and Eqn (8) fits the measured values of EBV/P
within 1.4% for both the bar and the rod specimen.
Table 3 of Ref [7] gives values of Y*m and a critical slope ratio r=. That ratio is the
ratio of the compliances corresponding to am and %. For specimens with W/B = 2.0,
the values of Y*m computed from Eq (5) for both the bar and rod specimens are within
0.6% of those in Ref [7]. The critical slope ratio computed from Eq (8) is within 1% for
the bar specimen but is 7.8% low for the rod specimen.
Problems in determining Y'rn and a m
The method of data analysis used in Ref [3] was not given explicitly and could not
be determined directly from archival records. In attempting to verify the numerical
analysis (by duplication), several methods were tried. Each produced a significantly
different value for a m for the same data set, and this is a problem that should be
discussed.
The problem is inherent in the chevron-notch specimen. It is due to the same
characteristic that makes it desirable, namely the fact that Y* has a minimum. For
example, assume that we have a function f such that
EBV/P = f(cx)
where f includes the data transform (if any) and a fitting function.
the derivative of Eq (3) and eliminating non-zero terms we have
0 - 1 f/(¢=) - f//(¢.) (9)
Substituting this into
where f' and f' are the first and second derivatives and a m is the root of this equation.
5
Unlike simpler specimens, we need to determine the second derivative as well. This
presents a strong challenge to the analyst.
Fig 2, from Ref [1], shows the typical variation of Y" with = for different values of
a o. Experimental compliance data would be expected to scatter about these lines. It
is apparent from this figure that for a short initial crack (say, =o=0.2) Y*m will be
relatively insensitive to the method of curve fitting but =m will be very sensitive.
However, for a long initial crack (say, =o=0.5) the opposite will be true.
Thus if the primary objective of the test is to determine Kb,, the initial crack length
should be short. This is the case in Ref [7]. However, a long initial crack length is
preferable if the critical crack length is important for, say, fractographic purposes.
It should be pointed out that numerical analyses (i.e., finite element or boundary
integral methods) are subject to the same problem, although to a lesser degree.
Discrete pairs of Or, =o) for several initial crack lengths must be fitted with a function
to calculate a minimum. Three pairs are required, more would be preferred.
CONCLUSIONS
The equations presented here are in forms suitable for several purposes in fracture
testing with chevron-notch specimens. They encompass the range of specimen
geometries most commonly used and provide a good fit with the basic compliance
data. The inherent difficulty in determining the critical crack length from compliance
measurements is discussed.
REFERENCES
[1]
[2]
[3]
Munz, D., Bubsey, R. T., and Srawley, J. E., "Compliance and Stress Intensity
Coefficients for Short Bar Specimens with Chevron Notches," International
Journal of Fracture, Vol. 16, No. 4, August 1980, pp. 359-374.
Bubsey, R. T., Munz, D., Pierce, W. S., and Shannon, J. L, Jr, "Compliance
Calibration of the Short Rod Chevron-Notch Specimen for Fracture Toughness
Testing of Brittle Materials," International Journal of Fracture, Vol. 18, No. 2,
February 1982, pp. 125-133.
Shannon, J. L., Jr., Bubsey, R. T., Pierce, W. S., and Munz, D., "Extended
Range Stress Intensity Factor Expressions for Chevron-Notched Short Bar and
Short Rod Fracture Toughness Specimens," International Journal of Fracture,
Vol. 19, No. 3, July 1982, pp. R55-R58.
!
[4]
[s]
[6]
[7]
Standard Practice for R-Curve Determination, ASTM E 561-86, Annual Book of
ASTM Standardsl Vol. 03.01, 1990.
Bubsey, R. T., Orange, T. W., Pierce, W. S., and Shannon, J. L, Jr., "Closed-
Form Expressions for Crack-Mouth Displacements and Stress Intensity Factors
for Chevron-Notched Short Bar and Short Rod Specimens Based on
Experimental Compliance Measurements," NASA TM 83796, 1991.
Saxena, A., and Hudak, S. J., Jr., "Review and Extension of Compliance
Information for Common Crack Growth Specimens," International Journal of
Fracture, Voi. 14, No. 5, October 1978, pp. 453-468.
Standard Test Method for Plane-Strain (Chevron-Notch) Fracture Toughness of
Metallic Materials, ASTM E 1304-89, Annual Book of ASTM Standards, Vol.
03.01, 1990.
7
Table ]--Coefficients for Eq I
Specimen Coeff. Expression
Bar
Ao -0.]]0 +0.354(WB) -O.088(WB) z
A1 0.268 +].628(WB) -0.400(WB) 2
Az 1.637 -6.358(WB) +1.872(WB) 2
A3 0.075 +4.462(W/B) -].508(W/B) 2
Rod
Ao 0.147 +o.oe9(WB) -O.026(WB) 2
A+ 0.358 +].]50(W/B) -O.O96(W/B) a
A2 2.860 -5.1gO(WB) +0.770(W/B) z
A3 -3.610 +5.]O0(WB) -O.800(W/B) z
Range: ].5_(g/B)_2.0, O_o_O.S
Table 2--Coefficients for Eq 5
Specimen Coeff. Expression
Bo
Bar B1
82
83
-17.03 +29.94(W/B) -5.0(W/B) 2
-]16.00 +]4].60(W/B) -29.6(W/B) 2
1131.00 -1304.00(W/B) +342.0(W/B) 2
-1351.00 +1654.00(W/B) -443.2(W/B) 2
Rod
Bo 5.47 + 6.2g(WB) +2.46(W/B) 2
B+ -65.g3 + 72.62(WB) -5.62(WB) 2
B2 622.00 -659.80(g/B) +]46.IO(W/B) 2
B5 -541.40 +62g.]o(g/B) -135.20(W/B) z
Range: ].5_(W/B)s2.0, 0_-_0.5
8
Table 3--Coefficients for Eq 6
Specimen Coeff. Expression
CO
Bar C1
WB=I.5 C2
C]
C4
3.09 -24.12% +57.12% 2
-]09.30 +1 227.00% -2 876.00% z
1 908.00 -22 216.00% +51 286.00% 2
-14 900.00 +168 580.00% -381 240.00% 2
41 390.00 -451 059.00% +987 080.00% 2
.... --------w .................. U----B--" .... W"
Co 2.08
Bar C1 -63.31
W/B=2.0 Cz 1 086.00
C3 -9 327.00
C4
-8.74% +16.93% z
+540.00% -1 019.00% z
-11 296,00% +20 043.00% z
+98 493.00% -158 690.00% 2
28 430.00 -284 970.00% +366 330.00% 2
...... . ....... . .... -- ....... i.w.------mmw.--mu ...... m.----wo. ...........
Co 0.672 +4.85% -23.93% z
Rod C1 25.670 -361.90% +1 624.00% z
W/B=1.5 Cz -858.000 +9 512.00% -39 580.00_02
C3 9 219.000 -105 260.00% +411 440.00% 2
C4 -35 145.000 +417 050.00% -1 550 300.00% 2
CO O. 896 +7.24% - 26.5% z
Rod C1 21.800 -590.40% +2 087.0_o 2
W/B=2.0 Cz -1 192.000 +17 166.00% -58 980.0% z
Cs 16 772.000 -2]3 330.00% +713 640.0% z
C4 -78 837.000 +961 870.00% -3 146 400.0% z
Range: 0.18_<%_0.22, _<a_0.8
t
Tab]e 4--Coefficients for Eq 7
Specimen Coeff. Expression
Bar
Do 3.329 +1.02_ +78.21% z
O, -0.812 -58.080=O -334.40=O 2
Dz -2.061 +265.260% +461.40% z
D3 4.350 -417.120% -156.1_o z
D4 0.349 +219.80_ -65.5_o 2
W/B--] .S
N/B=2.0
! i i !! I i i ! i i li j i! !1 ! lllil !il i i! iii !!1 I_ / _O Oil iililll I! !1 I I .....
DO 4.308 -4.757= o +83.77O.o z
Bar D 1 -6.529 -19.19_ o -358.7_o z
Dz 16.630 +172.00_ o +483. ]O=Oz
D3 -22.170 -313.000=O - 151.1_o z
D4 13.220 +173.700=O -72.71=O z
DO -2.28 +106.3=O -567.0=O z +1 062=O3
Rod D1 29.61 -582.5=O +3 100.0=O z -5 830<Zo3
t,//B=I.5 Dz -60.17 +1 167.0=O -6 070.0=O z +11 589=O3
D3 52.60 -1 022.0% +5 051.0=O z -9 869=O3
D4 -15.00 +337.80= o -1 517.0% z +3 042% 3
Rod
DO 2.19 +41.9_ o -263.7=O 2 +749._o 3
D1 -1.41 -128.2_ o +945._o z -3 532.0% 3
Dz 22.30 +38.71=O -691._o z +5 784._o 3
D3 -40.95 +179.10=O -703.0=O 2 -3 646.0=O 3
D4 22.92 - 13]. 30% +741.2=O 2 +607.4.% 3
W/B=2.0
Range: 0.1__<0.35 (bar), 0.1__0.40 (rod), _0.8
10
Table 5--Coefficients for Eq 8
Spectmen Coeff. Expression
Eo 2.850 -6.48% +61.56% z
E1 1.177 +26.59% -349.30% z
Ez 9.650 -8.37% +708.90% z
E3 -]6.240 -62.60% -597.00% z
E4 10.450 +56.82% +167.90% z
Bar
W/B=].5
Bar
W/B=2.0
Eo 3.885 -17.75a o +94.97% z
EI -5.160 +123.20% -624.20% z
Ez 34.270 -324.50% +] 562.00% z
E3 -52.330 +386.80% -1 756.00% 2
E4 27.950 -173.20% +741.40% z
Rod
W/B=1.5
Eo 3.91 -23.18% +138.4% z -91.84% 3
E1 -10.01 +237.70% -1 356.0% z +] 325.00% 3
Ez 51.60 -758.80% +4 284.0% z -4 777.00% 3
E3 -74.66 +969.60a o -5 480.0% 2 +6 516.60% 3
E4 37.83 -433.00% +2 464.5% 2 -3 043.00% 3
Rod
W/B=2.0
Eo 2.92 +0.28% +26.67% 2 +33.27% 3
E1 1.68 +28.52% -336.00% z +111.40% 3
E2 20.59 -135.84% +1 157.00% 2 -918.40% 3
E3 -39.16 +218.10% -1 581.00% 2 +1 581.00% 3
E4 22.64 -115.78% +756.60% 2 -837.20% 3
Range: 0.1_<e_0.35 (bar), 0.1_<e_0.40 (rod), a_<o_;0.8
11
(a)rod (b)bar
FIG. I - Chevron-notch rod and bar specimens.
4O
Stress
Intensity
Factor, 0
y-
30
2O
I_ I I I I I t I I
0 .I .2 .3 .4 .5 .6 .7 .8
Crack length, a
FIG. 2 - Typical variation of stress intensity factor with crack length for chevron-notch
specimens [I]. Arrows denote minima. "
12

Natkmal/Ummttalm and
$pem/¢Z_a_arVaeon
1. Repo,-I No. 2. Government Acceu_ No.
NASA TM - 104367
4. Tdb and SuM_
Analysis of Some Compliance Calibration Data for Chevron-Notch
Bar and Rod Specimens
Report Documentation Page
7./_x(s)
Thomas W. Orange, Raymond T. Bubsey, William S. Pierce, and
John L. Shannon, Jr.
"g. Performing Organizatkm Name and Address
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135 - 3191
e. PerformingC_mizaeonCode
8. Perfomdng OrganlzaEon Report No.
E -6167
10. Work Unit No.
505 -63- 52
11. Contract e¢ Grw'd No.
13. Type of Report and Period Covered
Technical Memorandum
14. Spormodng/tOency Code
12. $pontmringAgencyNameandAddrus
National Aeronautics and Space Administration
Washington, D.C. 20546 - 0001
15. Supplementary Notes
Prepared for the Symposium on Chevron-Notch Fracture Test Experience sponsored by the American Society for
Testing and Materials, Indianapolis, Indiana, May 6, 1991. Raymond T. Bubsey and William S. Pierce, retired.
Responsible person, Thomas W. Orange, (216) 433-3301.
lB. /_bttmct
This paper presents a set of equations describing certain fracture mechanics parameters for chevron-notch bar and rod
specimens. They are developed by fitting earlier compliance calibration data. The difficulty in determining the mini-
mum stress intensity coefficient and the critical crack length is discussed.
17. KeyWords(SuggestedbyAuthor(s))
Specimens; Compliance; Calibration; Stress intensity
factors; Critical length; Fracture mechanics; Crack
opening displacement
18. Distribution Sttbl.'rmnt
Unclassified - Unlimited
Subject Category 39
19. ,._curityClamdf. (oflheml:x_t) 20. SecudtyClmmif. (ol_islmge) 21. No. ofpagtm
Unclassified Unclassified 14
NASAVOnMIrateOCT== *For sale bythe NationalTechnicalInformationService,Springfield,Virginia 22161


Analysis of some compliance calibration data for chevron-notch bar and rod specimens

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