U•l MUTNIIINT OF €lMIMulr

AD-A024 711

THEORETICAL ANALYSIS OF ATJS GRAPHITE DISKS

FRACTURED BY SPINNING

CALIFORNIA UNIVERSITY

PREPARED FOR

OFFICE OF NAVAL RESEARCH

MARCH 1976

1< _ •

IWO-

UCLA-ENG-7624MARCHi 1976

THEORETICAL ANALYSIS OF ATJS GRAPHITE DISKSFRACTURED BY SPINNING

REPROOUC16 ByNATIONAL TECHNICAL

INFORMATION SERVICE SB ADRU.S. DEPARTMENT Of COMMEKESB.S TO R

SICUP,[ CLASKIFICAVtOs Of THIS PAGE fUt&P Dods Soverod)

REPORT DOCUflAENTATI VO PAGE - 2tM k M 'HR' EPORT f~umerm r-it-Sloss 40 I1. KIPI16CTALG UMWi

UCLA-ENG-7624- YEV goTaPAOOVf04 TI1'L.F (and 5-.,('til) I TP FR P R E10 O E E

TH50ORETICAL ANALYSIS OF ATJS GRAPHITE DISKS Technical 2975-1976FRACTUR~ED BY SPINNING 4.PERFagOR"I 111c REPORT "UNDERf

7 AUT~ioRw) 6. CONTRACT ONt 31ANT MUM@SIR(.)

S. B. Batdorf N00014.-76-C-0445

S. PERFRIoNGi~i ORIGANIZATIOM NAME AND A00RES1 - iSrAjEKT.IOCT TSchool of Engineering and Applied SciencaU~niversity of CaliforniaLosAngelenoCalifornia _________

11. CONYROLILING OPPrICE NAME ANO A0014E39 12 PRPORT DATI

March 1976Department of Navy 13. #4UMERN OF PACKS

Q_ MONI 0 ING AOINCY NAME & AOORIESS(II different hem C~ftw6N~d Otob.) III. 6CURTYCLASS. (01Iflis ft*41

Office of Naval Research - Branch Office Unclassified1030 E. Green Ot. Is* 7 AI -oow-N/wt OON A GIN$Pasadena, Ca. 91101 Cu~

16. DISTRIOUTION STATEMENT (of I~ile Report)

Distribution is unliimited

11 4 -'eM, AT2~i. *'i rgtbe4 ahes6e o o in f*ffg0.it fi w m w ev

Ill. SUPPLEMENTARY NOTES

19. KEY WOmOG (C~gernetm~ gono~ rewwoo I N8t**M neesm Ida"i~ W000 bI7a w ea")Fracture Grapl it. FractureBrittle Fracture Strength TheorilesFracture StatisticsPolyaxial Stress-

Strain Relations ______

30 A&STRACY (Ca"00wo an revereet sta it wego..wv &V Idm uii' by block amnmbeeIn a recent experiment~l investigavt:on of ATJS disks spun to failure,

maximium st~ains at fracture vere compared vith predictionsa based on theWeller-OASIS and Jones-Nelson polyaxial stress-strain relations. In additio:uthe fracture data wore analyzed to determine approximate fiiilura vs. straincurvs fo a umbe of tres raios

The present paper shows that improved strain predictions are obtainedusing a stress strain relaticm proposed by the present author. The paper

DO, AN~ 1473 COITION OF Ia NOV66 S OSSOLE TES/N012 F 04401SECURITY CLA8,FIC^TIOM OF VNIS PA4E ( .)acm NI Nu.,"

_ScumITV CLASIFICAYOOW OF THIS PAOGUMR aeM ER..•dM)

'also esown hoy the statisttcal theory of fro. ture under combined stresses can

ooe mployed to deduce the unmaxial. as vwell as improved biaxial stress or strainfailure relations for the particular graphite used.

59t UNITY Cý-AASIIFIlCATIOW Ofr THIS PDAG(f1tenh DO&€ Vnfer*, 0

Iid

I lit UCLA-DIG.-7624March 1976

II

THEORETICAL ANALYSIS OF ATJS GRAPHITE DISKSFRACTURED BY SPINNING

S. B. Batdorf

Ssonaored by the Department of the NavyOffic" of Naval Research

Contract No. N00014-76-C-0445

BCo-Sponsored hy

Office of Naval Research andAir Force Office of Scientific Research

Reproduction in who1 or in part is permitted for

any purpose of the United States Government

School of Engineering a&ad Applied ScienceUniversity of CaliforniaLoa Angeles, California

"•~

ABSTRACT

In a recent experimental Investigation of ATJS disks spun to failure,

maximum strains at fracture were compared with predictions based on the Weiler-

OASiS and Jones-Pelson polyaxlal strese-strain relations. In addition the

fracture data were analyzed to determine approximate Alurt vs. #train curves

for a number oi stress ratios.

Tha present paper ahows that inproved strain predictions are obtained

using a stress strain relation propused by the present author. The paper also

shows how the statistical theory of fracture under combined stresses can be

employed to deduce the uniaxial af - ell as impioved biaxial stress or striin

failure relations for the particular graphite used.

KM

Table of Contents

Page

Live of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .. . I

Kxpertam tmal Data .*.. . . . . . . . . . . . . . . . . . . . . . 2

Stresa Analysis of Disk ............. ......................... 5

fracture Stacisutics. .......... ..... .......................... ... 10

Conclusions ..................... .............................. .. 37

References ........................ ... ... ........................... 38

14

k V

List of Figures

Page

Figure 1 Median Streoe-.Strain Curves for AT.J-S Billetsin With-Grain Tension at Roma Temperature ..... . 3

F Figure 2 Mesaured and Predicted Kaxisum Strains in Disks . . . . 4

Figure 3 Ogivee for Failure Strain, Calculated for aVolwvle of 0.07 Cubic Inch ........... ............... 7

figure 4 Comparison of Uniaxial and Equibiaial Witi-GrainStresa Strain ker:ponse of ATJS Gr:.phite. ........... 9

Figure 5 Predictions of Witt,-Grain Stress--Strain Responsein Equal (1:1) Bia: '.1 lensiora, ATJ-S ....... .. 11

Figure 6 omputed Stress Distribution in Rotating ATJS Disks 13

Figure 7 Computed Strain Dlstributio,a i1 Rotalinq ATIS Disks . 14

Figutre 8 Maximi= Stralis Predicted in Four-Inch DinmeterATJ-S Disks Compared to Jortner's 'rest Data ...... .. 15

Figure 9 Variation of k with Biaxiality ........................ 18

FiguLe 10 Frobabijiiy of Failure of spinning AT.1S (raphirm i)usks. if

Figure 1U Comparison Between Computed Points and AnalyticalApproximation at W - 4200 rad./sec .............. ... 21

Figure 12 Theoretical Analysis of Spinning ATJS Graphite Disks. 22

Figure 13 Cowparison of Pf( ) Data for 20 Select-ed StandardTensile Specimens with Pf(f) Deduccd from Disk Data .)4

Figure 14 Fracture Locations in Standard Tavsile Coupons .... 25

Figure 15 Theoretica.ly Determined Probability of Fractureof 0.07 in . ATJS Graphite Specimens in Uninxialand Various Biaxial Tensions .... ............. .. 27

Figure 16 Probability of Failure of 0.07 in 3. Specimens ofATJS Graphite in Biaxial Loading Corresponding tothat in Complete Disk aad in Inner 0.65" .......... .... 28

Figure 17 Biaxial Disk FaiLujv Da,A. Mterial Within x-0.65 inch. 30

Figure 11 Probability if Fracture of Spinning ATJS Graphite DisksBs a Function of Center Strain .... ............ .. 31

vii

Figure 19 Probability of Failure versus Strain Curves Obtainedfrom Strea3 Distribution in Tuble 2 and TheoreticalNtaxial Fractur, Statistics of Table 3. Circles andTriangles Obtained from Fracture Date of Table I andStrain Distribution of Table 2 ................. ... 33

Viture 20 Fracture Frequency as a Function of Radial Location 34

Vitt

INTRTWODUCTION

Refractory materials are frequent1y used in reentry vehicle noeatips t-o

combat tho seere hasting c'onditlonr encounteved. The thetil stresses

therl v induced Introduce r risk of fracture. To calculate the surviv, ,rob-

ability. it Ito necessary to "Q able to determine the stroamoa and strains

induced by the combioed thermal and aerodynamic loads, to find the probability

of fracture In each volum e slment, and fr(m the latter to estimate the our-

rival probabilit-" of the entire nomettp. imwnig the things needed tc accon-

plish this task are sufficiently accurate conatitutivv laws for the matsrials

employed, and a knowledge of the fracture statistics of the material as

affected by specimen size and the (generally biaxlal or triaxial) stress state

4-,olved.

In a recent experimental study to shed light these matters, A*I.S graph-

its disks and bars were tested to destruction by spinning. The centrifugal

forces produced biaxial stres.eu which varied with the nomitisn In the dfak.

I Since only strains (not atressei) could be measured direc lv, such tests fur-

nish &. litated but neverthelesa usef,'i means of checking the adequacy of

various proposed polyaxial streus-strain relations. In addition, the expert-

ments provide data on the statistics -if fracture under polyaxial loading con-

ditions.

In Reference 1, strains measured at the center of the disk were checked

against theoretical predictions of two dif."erent theoretical constitutive laws.

In turned out that one theory generally overestimated the strains, while thK

other undersstimated them. The fracture results were given in terms of prob-

ability of failure vs. strain for biaxially loaded specimens of the size used

I,- th- stan4ard tnr-ilz L-..4 0.0/ cubic inches). imitj4 data were given on

the effect of biaxiality on fracture, but no correlation with theory was

atterpted.

1

The primary objective of the present Paper is (L. show hov the untaxiel

strese-*train date and Lmiaxlal fracture statistics can be generalized with

the aid of theor to stress analyze the rotating disks and also account for

the oshirred failure statistits. It is shown• that agiree~nt betwaeen theory/

and expetiment is good for both types of problem.

A aecond objective is to investigate the opplicabilit7 of weakest link

thoory and its aLsociated v.lme affect. In one recent study' it was con-

cluded that larger p..cimons break at a lowwr strobe iii attotd with theory,

but the quantitative agreement between theory and exper1betit left something

to be desired. In another, no volume offect was foul. The present

investigatiovi supports weakest link theory and the implled vnlum. eflect

quantitatively am ell as qu"litativslty.

fXIT.RIIrAl.fl DATA

Ina ox.,ortmentai set-tin, pra(raJtre, and ,Jats tedauct~in tirchniqouss are

dest~rihud in detail in Reference 1. rti-fly. 'it ATIS disks were 0.4•" thic:k

and 4" In dlmniter, and were cut so t..at the matei ial was Isotropic and

exhibited with-grain properties in the plane of the disk. The with-grain

stress-*train relation is showna in rig. 1. The average specific gravity of

reg sraphite w•, 1.83, and the Poison coefficient was 0.1.

At the center of the rotating disk the radial strain i is aq'al to ther

circumferential strain c0, Figure 2 shove how this strain varies with speed

of rotation for an elastic disk and also for disks obeying the Weiler-OAUSIS

amd JX)no-Nolson constitutive relations. Albao shown is Owe range of measured

valuvi of thiv strain at the tiwL of disk fracture. The axpeo'cmental data

exhibit considerable scatter aa a result of variations in material roperties,

but coe theoretical calculation almost always underestimated the strain, while

the other nearly always overestimatwd It.

2

S• ( • " " • = • ' • ... • . . . ., -• • •'• • • •. - . • ..... . .... . . . ... .....

r..

�t �is 4

6

E -1 �) � � 2/

4

- /

/.M

1,7/'I! /

'72P-

//I -it /7

4

F, 9 .,, I �da.�' Sw,�, S;�..n A Ii S tI.IIet, �n �S�h (2rhn

00 at Roon� T.rnp..Aue. 4A'�eptscI V,�n- R.� I

Al-a-----

ATJ- GRAPHCTIE, 70&F

0000 / JONES-NELSON

/I• WEILER-OASIS

5000

RANGE OFROTATIONAL

4000 DATA

I- / /,,

, ELASTIO'; SOLTOS!

~~ =. 1.~l85 x1jC6psi10O9

f/

0 10,000 73,000 30,000 40,000 50,000

RPM

Fielure 2, Mmsured and Predicte Maximum Strains in Disks (Takenfron', Rot. 1)

4

-LZL .• • . . ... .... .. .•l P . .. . .' . . .... • ... • -

Table 1 summarizes the fracture data. For each specimen the rotation

"speed at failure is listed, together with the center strain And the maximum

principal strain it the location of the crack causing the failure, as cal-

culated by a method described in Peference 1. These data were processed by a

metiod also discussed in R,•Zerence 1 to give the probability of failure vs.

maximum principal strain for specimens having a volume of 0.07 cubic inchea,

By considering deparately the cases of fractures originating at radii less

than 0.65", 1.1", and 2" respectively, an approximate measure of the effect

of biaxiality was obtained. The biaxial results are displayed in Fig. 3.

Also shown are uniaxial data obtained from standard test specimens.

STRESS ANALYSIS C! DISK

The stress analysis of a loaded body requires the siwultaneous satisfac-

tion of the equations of equilibrium, mompatibility, and the conetitutive

"equations for the material. Because of the radial symmetry of the problem

under ccnsideration, derivatives ia the tangential direction vanish and the

equations of stress equilibrium reduce to the single equation

"d 22

d) r - -

with the boundary condition a (2) - 0. p is material dansity, and W ii

angular speed in radians/second. The equations of compatibility also reduce

to a single equation;

d- (rr) C (2)dr (r0 =r

The above two equations in four unknowns can be solved when we add two stress-

strain equationa involving the same unknowns.

In Reference 4 a stress-strain relation for ATJ-S graphite was proposed

that becomes in the biaxial case

5

TABLE I

BIAXIAL DISK DATA ORDERED BY INCREASING MAXI4UMPR.INCIPAL STRAIN AT FAILURE

Ivm .. 4e.1 3 n.

3ME1.6aa Hpe. . 334 Ile

i.514 3214.3 2.2, 50 33, 000 . 3,

2.335l 131o I 33,5 37t,041 I. 41

,I3 4-4 2 .930 7, 000 1 ..

2,.6,1 i3-2 a. 7110 6,000 0 65

Z, 460 12-.2 2,650 36.500 0

2.964 A41-3 3, 000 33,2OO 0.4

3.000 113-S 3.100 to 9,240 6

3,072 IR1-4 3,200 36.400 0

3,315 532.2tz 3,300 35.S 00 I3 I

3. 119 IR31- 3 3,o300 39.000 0 5

3, 136 3Z2Z-S 3. zoo 37,600 0.5

3. 137 IZR4-2 3. 150 35, ?00 0. 2

3, 179 lit 3- 3 3,400 40, 300 0. S5

3, 214 9R3-4 3. 260 37,600 0.4

3.230 n2341 3, 550 39,400 1.0

1. 3"2 52R-4 3.420 40. 00n 0 8

3, 283 IZZ-4 3, 350 39,000 0 5

3,256 93).3 3. n'0 3,5800 0 3

3,293 3134-4 3. 360 39,700 0 5

3. 33Z 913-Z 3.400 40.400 0. 5

3, 360 I2R3- 3 3.6810 38.,400 0. zS

3. 3)9 8114-.' 3,540 4. K100 . .7

3 3139 9R3).S 3.530 33.1700 0 65

3.445 9-R -4 3,480 40,000 0 )s

3.446 9R 1-2 3.460 39. 600 0 2

3.45 8 11-5 3, SG) 40.400 0.4

),471 8R323 ',900 40,400 1 I

3, SO I -S 3,4 0 41.000 0 6

3.363 i.t•,.-3 3,.00 40,400 0 65

3,577 3Z-5S 3.650 33.800 0 S

3.600 9OR I-3 3, 150 40. 1O0 0 7

3. 66Z 9m i -3 3.700 40,400 0.

3, 7•05 854- 1 3.900 42, 400 0. 75

3.707 383-3 4.030 41,000 0 93

3,952l 933-1 4. 2S0 43,000 0 9

3,"961 kz4-s 4, 170 4/. 300 0. 71

3,91 52I-I 4,040 41. 000 0 40

,.051 i33-4 4, 22O 40.400 0.7

4,413 Iz33-z 4,S50 42.000 0

6

ATJ-8 GRAPHITE

DATA FROM 30 ROTATINW DISKS, FOR V 0.07 IN3

SYMBOL MATERIAL CONSIDERED MINIMUM BIAXIALSTRESS RATIO

go 0 ENTlRF, DISK, R-2" 0I VOLUME WITHIN 1.1" Rý 0.7 - 0.85AO VOLUME WITHIN 0.65" FR. 0.9-. 0.95

90/

seo

5o0i

F 40 y'•- DATfA FROM 32

A .L ,STAfVDA.•STENSILE

0. 20 O"TESTS

44

S710

NOMINAL / -IAXIAL

STRESS RATIO//°• :

50/0

0 0.96

~~A 20TET

02 < 0.70 o.1: 0.

2000 3000 4000 5000

MAXIMUM PRINCIPAL STRAIN, MICROSTRAIN

Figure 3. Ogiva for Failure Strain. Cakwuated for a Volume of 0.07 Cubic Imnh(Adsw d from Raif. 1).

t ~7

- 00.1•c • (I +F) -- •-e(3)

-- Y r + -•+ F)()

where

0+r " 1 2C 1 (5)

P

In Eq. (5) o0 is the elastic limit str~ss in the with-grai-n direction andr

P K is a constant to be adjusted to give good agreement with the uniaxial

stress strain curve near the fracture point.

The stress-strain curves shown in Fig. 1 do not exhibit an elastic limit,

so for simplicity of analysis it is assumed equal to zero. As a result, Eq.

(5) becomes

K1 2 2F ,a - _ C + aA (a

r

rK 2 (6)

when or approaches zero. The choice E - 1.5 x 103 kBi and K2 - 0.048 (kai)"'r2

leads to good a, reement with the uniaxial stress strain curve, as is apparent

from Fig. 1. Thus our choice of biaxial stress-strain relations becomes

Or ( + r0 o + ) 6 .7 x 10 -5 J (7)

e 6.7 x 10- or +. 150 1 + 0.048 a+ 0r (8)

In the above equations stresses are expressed in ksi.

ID Fig. 4 the equibiaxial stress-strain behavior implied by equations (7)

and (8) is ccmpared with the uniaxial behavior. For comparison, the corre-

sponding curv s as calculatei in the Weiler-OASIS and Jones-Nelson appzoachus

• ii

E 1500 ksi

5

4/

3

EOkBIAX2 ~UNIAX .

.001 .002 .003 .0(A .006

STRAI N

Figure 4. Coonpaerlon of Unlaxial and Equlblaxliid Wilh-Grain Strass StrainReap... of ATJS Graphite, Using Th a- of Reheraian 4.I

9

are given in rig. 5. Whereas in the WeLler-OASIS treatmert the equibiar curve

Is generally higher than the uniaxial and in the Jones-Nelson treatment It is

s~bstantially lower, Eqs. (7) and (8) imply the two curves are very close

together, especially at the stresses at which most of the fractures occurred.

Equations (1) and (2), (7) and (8) were solved simultaneously by itera-

tion for a series of values of w, and the results are given in Table 2. The

way in which stress and strain vary v-'th radius is shown in Figs. 6 and 7

respectively. Figure 8 shows the manner in which the maximum strain varies

with spe3d of rotation and compares the result with the experiment4l data from

Reference 1. The present calculation agrees significantly better with the

data than the calculations based on the Weiler-OASIS and Jotes-Nelscn

approaches.

FRACTURE STATISTICS

Stati-sticlts theories of fracture are usually formulated in terms ot

stresses. The simplest treatment involves the uise of tVaibull's 2-parameter

5,form which anssumes for the probability of fracture in uniform uniaxial ten-

sion

Sf M 1 - exp[-Vk o0a] (9)

In this squation V is the volume, and the parameters k and m are chosen for0

optimum fit to the experimental data. Figure 3 contains experimental data on

probability of fracture of standard tensile test specimzns as a function of

strain, but the data apply to highev strains than those at which the majority

of disk fractures occurred.

There is no reliable way of extrapolating the standard tensile specimen

data to the strain levels at i•hich most of the disks fractured, expecially

since the data as presented in Fig. 3 do not conform well to any of the

10

S~/

/

WEI LER-OASIS

BIAXIAL4 UNIAXIAL -"

S~JONES-NELSON•K BIAXIAL

/, -- UNIAXIAL

2 2

0 I .i

0 1000 2000 3000 4000 5000

MICROSTRAIN

Figure 5. Predictions of With-Gri'n Streu-Strain Response in Equal (1:1)

Biaxiel Tension, ATJ-S (Taken from Ref 1 ).

TABLE 2

STRESSES AND STRAINS TANROTATING ATJS DISKS

W(Rad.Iuec.)

vig . t. it F.d'f n *T 0

I 1O I -t1.''9F-2 IL 147 "", :'a o#0

:2A &A '-' '.tt clr -3 1:6 fl 3.600-0

. 1.1 o n t q u - 12 9 - 3I S 7 + o3 6 2 F 1

'4M

2

RADIUS, IN.

Figure 6. CamPuted Stress Distribution in Rotating ATJS Disks.

13

54

2 eo

1 2

RADIUS, IN.

ui,~, T. Conuted Stivun Distr~bmtikm in Roawtqn ATJS M~ks.

14

I I PR~ESENTI ~flLUTION,.r

.004

.003

ELASTICM2 ER 1.6 10 PS

PO R 0- 115LATIC

WO LER--10SIPSI

~00

.02 10 20 30 40SIFso

I 1LASTIC

VORwr -08 Mmxvnm. 4rmPdtdi o-nhDsmoAJ s

Con~eed ToJortn's To Date

15m.SXOS

theoretical formulations in general use, Accordingly. we hall ue] loy theory

to deduce from the disk fracture data vat the uniaxitl fracture statistics

in the deslrgd li-rees range must be The procedure followed here ie to assume

the uniaxial statistics obey Eq. (9) and then apply the statist.ical theo!-y for

biaxiel stresses to determine what values of a tnd k lead to the observed0

disk fracture scatistics.

Mien it is assumed ttmat the uniaxial tensile 'racture statistics obey

Eq. (9) and that fracture is determined solely by the component of stress

normal to the crack plane, the btaxial tensile statistics become (Ref. 6)

Pt=I -ex4 [-Vk(r-i -a) 0] (10)

where o1 and 02 are the principal stresses with (2 0< a1 and

k (02) a)k 0

Table 3 lists the values of this ratio for a number of value@ of m and a2//l,

and Fig. 9 contains curves for selected values of a.

The above considirations can be used to calculate the probabllItv of

survival of the rotating disks of Referince 1 as a function of the speed of

rotation. The applicable equation im

P (W) -1- e xp f O , dV1

I - exp - k c" C

where

0 k

16

'I

Table 3

Valueo of k(ar/";,7)/ko

2 3 5 7 1' 20

(o /• ) 0 1.0 1.0 1.0 i.0 1.0 1.0

0.2 1.173 1.153 1.135 1.129 1.124 1.122

0.4 1.427 1.400 1.353 1.331 1.311 1.302

0.6 1.760 1.793 1.758 1.710 1.646 1.615

0.8 2.113 2.377 2.549 2.584 2.017 2.399

0.9 2.410 2.756 3.188 3.430 3.664 3.653

0.95 2.536 2.970 3.593 4.026 4.678 5.113

0.98 2.614 3.106 3.867 4.455 5.518 6.588

F1.0 2.667 3.200 4.063 4.774 6.204 7.975

17

120

"II

I U .

II

10 TC)

II4N C

1 A

• o

2

mII

002 0.4 0.6 0. 1.0

02101 k

Figs, •. Variation of k Wib, Saxiality.

11

I .i. "--

The k ratio and the a ratio appearing in the integrai in Eq. (13) are

both virtually independeat of the rate of rotation. Accordingly we evaluate

the integr&l after m is determined, using the stress ratios for the rotation

speed in Table 2 coming closest to a failure probability of 502, i.e.,

w - 4200 rad./sec.

The experimental data in Table 1 can be used to determine Pf a a func-

tion of the rate of rotation. A plot of failure probability vs. rotation

speed is given in Fig. 10. We fine the valnes of m and C k to fit thesemao

data as follows. For two selected valuer of w for which amax appears in

Table 2 we require that Eq. (12) gives the experimental r;,eults of Fig. 10.

It then follows that

f 1) [OIa x (Wl)(m! -] (14)Zn[1-F f PWU2)

A~s--.% P at 3800 r•a.,'ec. is 0.1 and F at 4200 is 0.64, we obtain m - 12.ff

Theu using Eq. (12) we find that k C - 1.43 x 19-8. Pf(w) is now completelyom

defined, and is plottc.d in Fig. 10.

To find k we must evaluate Cm. This can be dove as follows. Figure 11

shows ao/ax as a function of r for w - 4200 rad/sec. To a close approxima-

tion the ratio is given byi2

O-- = 1 - 0.086 r (15)

ameX

Figure 12a shows ar/la as a function of r. Combining this result with the

data in Table 2 for m - 12, we obtain the results shown in Fig. 12b for

k 2 )/k . Evaluating the integrnl in Eq. (13) with the aid of Eq. (15)

and Fig. 12 we obtain C 3.4, from which it follows that ki is 4.2 x 10-9.

Thus we conclude that in the stress range within which disk fractures occurred,

simrle tensile fracture obeys the equation

19

* EXPL. PTS.

.8 Pf 1 - EXP 1-1.43 x108 4OAAx:1)

.6

Pf

f*.4

.2

030 32 34 36 38 40 42 44

RPM x 10-3 Pf

lr. Wro 10. Probaility of FalNure of Splnning ATJS Graphite Disks.

20

00 r/UA 1.019 r2

.6

.21

RADIUS, IN.

Fipw II. Compuison Between Conpuled Point eW nd Auyfled AprfoximntoonAt wo 4200 nd./sec.

Im 12. w-4200 rid/rn

.8 )

.6 OR /go

.4

.2-

0 a 1 2

r, in.

6

4

3[

2N

4.A(b) 1 2

r, In

.6 r[aq(r)/v8 (O)I2.4

.2

0 C 1 2r, In.

Figue 12. TheometuI Andyib of Spimdtq A15 Grp~lte Dkks.

22

ei

P f -l-exp [.- V4. 2X10-9 a 12] (16)

lefore proceeding further it is appropriate to eoxaine the compatibility

of the result just obtained with the results of standard tensile tests at a

higher stress range. This is done in Fig. 13, which shows what Sq. (16)

implies for specimens having a volume of 0.07 cubic inches. The stresse*

have been converted into strains using Eq. (8) for the uniaxial case, i.e.,

5 (l + 0.048 a) (8')

Also shown for comparison are fracture data for 20 selected standard test

specimens.

The rationale for this selection is apparent from FI! 14 which shows

that the fracture origins tended to concentrate heavily at the ends of the

test section (i.e. the region of uniform cross section), and even outside

the test section. To compensate somewhat for this anomaly, only fractures

occurring at least 0.05" inside the test section were included.

The smooth solid curve through these data looks like a reasonablt

extrapolation of the disk data, except for a slight vertical offset. However

if the lowest point were eliminated, the resulting smooth curve would be

approximately that shown dashed in the figure, which has virtually no vertical

discontinuity. Thus we conclude that the disk data are compatible, within

normal experimental scatter, with the data obtained using standard test

specimtna.

Assuming, then, that the uniax'_el behavior of ATJS in the disk fracture

stress range is given with satisfactory accuracy by Eq. (16), we use this

relation to predict theoretically the three disk failure curves of Fig. 3.

This can be done as follows:

23

0

C I

S20 SELECTED N--MEANSTANDARDTENSILE ITESTS 0

O I!MENIF

/ LUWEST DATAPOINT IS DELETED

DISKRESULT

.003 .004 .006 .008

STRAIN

Firm 13. Com•e•on of Pf( Deb for 20 seleW kStandd T1ei aeupamom

UNIFORM

G1AGE SECTION..TANGENT POINT"

1.00" . OPTICAL STRAINI I (GAGE LENGTH

15 --

DISTRIBUTION10- OF

FAILURE5- - LOCATIONS40 SPECIMENS

BILr- I

BIL9Y• iSo• * e oo •

8 8 o FAILURE

I I LOCATIONSIN SPECIMENS

19 FROM EACH BILLET

12 o.8r I

Flgure 14. Fracture Locations in Standard Tensile Coupons (Taken from Ref. 1).

25

Since the link data represent results for a degree of biaxiality that

is averaged over the region under consideration (i.e., r S rI) we evaluate

the quantity k(r1r) that represents k(Or / 0,m) averaged over this region.

This k must cbey the relation

1 I m( 1r dV _r1 k(Or Com dV (17)

or

k 212) k (1-0.086 r2)12 r dr

kk (18)

k0 fr1 (1-0.086 r 2 ) r dr

The denominator can be integrated analytically by elementary means,

yielding

r (1n.ORA 2)1.2 T r- n-L17 r1-(1-_onAA8 2

The numerator cAn be evaluateO umerically with the aid of Fig. 12. The

result is

(O.65")Ik° 0- 5.1

•(i. l")Iko - 4.00

k(2")/k - 3.4

Using these values of k instead of k in Eq. (9) and converting to

strains, we obtain the curves shown in Fig. 15 for rI 0.65", 1.1", and 2".

Comparing this result with the experimental data portrayed in Fig. 3 some sig-

nificant differences are apparent (see Fig. 16). We now address the questions

of what these differences are and how to resolve the discrepancies. For sim-

plicity, attention will be limited to r 1 0.65" and rI - 2".

26

20 '

10 BIAX

r15 0.66* "

2

1 •UNIAX

0.5

0.01

.0026 .003 .0035 ,004

STRAIN

Figure 16. Theoretically Detemnined Probability of Fraciwe of 0.07 in 3 . ATJ8Graphit Speimewt in Uniaxial and Vawious Biaxial Tansions.(Circl•s Found by Applying Eq. (20) to the Smoothed Data inFigure 17).

27

THEORETICALPREDICTION

JORTNERTS

RESULTS0 r, - 0.65"

5 O r, - 2.0" Z 0.65"'2'f'2

Pf1

0.5

0.1 1./

0.10

0.01 .. .. ... . . . . ..__ _ _ _ _' . ...

.0025 .03 .0036 .01k

STRAOM1

Figure 16. Probai~ty of Fv,,re of 0.07 0. SpeIN•mm of ATJS Graphite inBiaxial Loading CornePanding to thot i Complete 0* and inInner 0.85". Pfamnt Results Comparud with Thowe of Janma.

28

The first differepce noted is that the curves just ralculated are nearly

st~raight lines on probability paper. %¶heraas the curves of Fig. 3 are quite

Irregular The reason for this turns out to le that in effect, Reference 1

processes bar graph data with the aid of the equation

Pf(C) -) Ps(C) (20)

where AN is the number of failures in a chosen strain interval and M is the

equivalent number of specimens of 0.07 cubic inches contained In the total

volume of ATJS that van strained to at least the value E. If the equation

is applied to smoothed data (se& Fig. 17) wo obtain for Pf(u) for the case

rI . 0.65" the circles shown in Fig. 15. ThVes - lts are smoother and also

closer to our theoretical calculation.

It is difficult to carry out an analgous treatment for r 1 - 2" because

the large scatter in the test data makes the smoothing process very uncertain.

Accordingr~ly -4c rx-• L Lo a ditferent approach.

From Table I we can construct the experimental plot of Pf (Cmax ) shown int ma

Fig. 18. We ther. assume that an average k exists for probability o' fracture

as a function of strain level which satisfies the equation

Pf()- I. - exp [- f i ' £ dv] (21)

It cw, be verified from Table 3 that for w - 4200 rad./sec., the. strair ratio

is closely approximately by

€0 2- 1- 0.105 r (22)

max

Inserting this relation In Eq. (21) we obtain

Pt (•x - eirp - 2rt EC02 r dr(l--0.105 r2) (23)

t max

ý9!

BIAXIAL DISK DATA30 CONSIDERING MATERIAL WITHIN 0.66" RADIUS ONLY

.,'1•,....-/ •"" "'--"Ni

20

5 9

2*

9/ \0

"SMOOTHEN a6/

DATA" i yIN 3

0 1000 2000 3000 4000 0)MAXIMUM PRINCIPAL STRAIN, si

F*"ur 17. Bli-s.iW Vhsk Falilure Do'm, Nlatoilal M lh ms x -0.06 Inch.(AhfW,,W fromh 1.tf. 1)

30

0pin 1 - xp (-5.38 X 1021 9A4

.9

.4

3 4

f .0x10

Fivx 18 Prba~ityof rocureof onnng TJSGrahi"OM9as Fuwtin o Cow Svin

31I

...... . .... , .7" 3

Carrying out the in. ral and fitting the •'s of Fig. 18 at the points

correspouding to w m 3800 rad.,'sec. and 4200 rad./sec. we obtain m' - 9.4

andk' -7.7 x 10 2

The theoretical curve corresponding to these values is shown in Fig. 18,

and the fit is very sati3factory, Accoreingly, we compute che probability of

frfact~re as a function of utrain for biaxially stressed 0.07 cubic inch

specimens as deduced from data on the complete disks to be

Pf( ) - 1 - exp [-0.07 x 7.7 x 1022 e9] (24)

A milar treatment of the inner 0.65" .f the dis1' leads to m' - 9.6,

',- .2 x 10 23 . These rerults, shown as circles and triangles in Figure 19,

turn out to be in excellent agreement with the theoretical prediction.

As a final check on the theory we calculate the frequency of fracture

as function of distance from the center of the disk. The relative frequency

is given by the integrand of the equation

P e- x 27 t a1 2 r dr(1-0.086 r (25)

L 0

Accordingly the suitably normalized function

F(r) - Kr(l-0.086 r2) 1 2 k( , 1)/k (26)

is plotted in Fig. 20. Also shown are the experimental data, as a plot of

fracture location vs. rate of rotation, and the corresponding bar graph. The

theoretical prediction is seen to be in excellent agreement with the tar graph.

DISCUSSION

As mentioned earlier, Reference 1 emp'oyed the principal strains at the

locations of the fracttre origins to re+uce the disk failure data. No use was

mae' of the statistical theory of fracture under biaxial loading. As a result,

32

40

30

20

2.0"*

10Pt

2

05

02

01I

.0025 .003 .0035 .004STR/A IN

Figure 1W. Probability of Fai'ure versus Strain CL rves Obtained from Strme Distribution inTable 2 ead Thinoreticel Biaxial Fracti,,. Statistics of lable 3. Circles andTriangles Obtained from Fracture Doti, of Table 1 and Strain Distribution ofTable 2.

33........16 j ......

42 0

* ** 0• •9 TEST DATA

40 40 2COINCIDENT* POINTS

o0

X 38

36

34

32 1" 2" "

FRACTURE LOCATION

1212

10 --- r k (oa/cO, 12)uo (NORMALIZED)

8

6 /4

0 1"2

r

Figure 20. Fracture Frequency a a Function if Radial L•cation.

34

A°

S...... i.

all that could be obLained was the probability of fracture vs. principal

strain for several biaxial stress ratios that zould only be approximately

identified.

Application of the statistical theory Jf fracture together with the

0Q computed stress distribution has made it possible to deduce from the disk

failure data the uniaxial failure statistics as well as the biaxial failure

statistics for any stress ratio. This renders the results directly useful

for structural e-plications. Agreement between theory and experiment was

very good.

The computed stress distribution and the stat ' ,:cal theory of fracture

under biaxial loading were also applied to determine the probability of frac-

ture origin as a function of radial distance from the disk center, again

obtaining very good agreement with experiment. Since the experiment did not

Drovide a basis for ehneking polyannvY franrt-ire svtiat-s4 and ntveea AR4-

:ribution individually it cannot be asserted that these were unambiguously

confirmed. But the fact that the theory of biaxial fracture statistics and

"the assumed polyaxial stress strain law combined to give so accurately the

overall fracture statistics of complete disks or portions thereof as well as

the radial distribution of the fracture origins and the strain at tht disk

center, Is rather persuasive indirect evilence of the accuracy of both theo-

retical inputs.

The experimental data and the success oi the theoretical analysis can

also be regarded as evidence supporting weakest link theory as applied to

graphite fracttqre. This is of particular interý t in the light of findings

in two earlier studies e4---,"d more specifically at looking into this ques-

tion. In both studiee attention was focussed on the volume effect. According

to weakest link thecry, the probability of fracture takes the form

35

Ff - I - (-Vf(4))

where fi!o) is a monotonic function, the form of which dependr an the particu-

lar theory. Thus for a chosen value of Pf, the product Vf(a) is constant,

and this relation determines the volume scaling. The earlier study2 led to

the conclusions that Welbull theory is qualitatively correct in predicting

a lover mean fracture stress and smaller dispersion for larger specimens,

but that the Weibull parameters are somewhat dependent on specimen volume.30

The second study3 led to conclusions contradicting weakest link theory. It

was reported in fact that "no size dependence of critical stress was found

in the data obtained."

The present findings are compatible with those of the first study, but

at variance with those in the second. A discussion of possible reasons for

the negative findings in the second study is beyond the scope of the present

paper. However it is apparent even without any anatlysis that them sraenat

in quotes is not supported by the data in the ATJS fracture study of Reference

1. Here the maximum strains in the large disks at fracture ranged from

0.0023 to 0.0046 whereas in the small standard tensile specimens they rang.d

from 0.0038 to 0.0057.

The applicability of weakast link theory to the graphite disks is

intuitively evident from the fact that in 3lmost all cases it was possible

to identify the flaw that initiated the fracture. In addition, the assump-

tion of weakest link t0 ory led to excellent detailed egreemeui with the

experimental data. It may be that although not planned with that in mind,

the test data of Reference 1, interpreted as in the present paper, constitutes

one of the most definitive confirmations of weakest link theory available.

36

CONCLUSIONS

1. Nsasured maximum strains in ATJS disks at various rates of rotation

were in excellent agreement with calculations based on a polyaxial stress

strain relation proposed by the present author.

2. The corresp-nding stress analysis coupled to the theory of biaxial

fracture statistics led to very good agrement with all the observed

disk fracture data.

P 3. The agreement between theory and experiment strongly supports the

thesis that veakest link theory is applicable to the fracture of ATJS

graphite.

4. The success of the analysis illustrates how theory can be used for

the purpose of enhancing the accuracy, consistency and practical uspful-

ness of experimental results.

37

371

-RUIRENCES

1. Jortner, Julius "Biazial Tensile Fracture of ATJ-5 Graphite," Technical j

Report No. AFbL-TR-74-262, March, 1975. Air Force Materials Laboratozy,

WriSht-Patterson Airforce Base, Ohio.

2. Pears, C.D., and Starrett, H. Stuart, "An Experimental Study of the

Weibull Volme Theory." Technical Report No. AFML-TR-66-228, March, 1968.

Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio.

3. Mausal, G.K., Duckworth, W.H., and Niesz, D.E., "Strength-Size Relation-

ships in Ceramic Materials: Invest~gation of Pyroceram 9606," Technical

Report No. 3, Office of Naval Research, November, 1974.

4. Batdorf, S.B., "A Polyaxial Stress-Strain Law for ATJS Graphite," SAMSO

Report SAMSO-T.-75-39, January, 1975.

5. Batdorf, S.B., "Observations on Approximate and Exact Treatmients of

Fracture Statistics for Polyaxial Stress States," SAMSO Report SAMSO-TR-

75-289, December, 1975.

38