- a~
THE VELOCITY BEHAVIOR
OF A GROWING CRACK
by
ERNEST NEIL DULANEY
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
Is\NT. TEc
DEGREE OF MASTER OF
SCIENCE
LINDGREN
at the
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY
34h-I.
/9(q
Signature of Author..........*0
Department of Geology and Geophysics
May 25, 1959
Certified by...........-.
...
..
...
........
Thesis Supervisor
Accepted by.. - .......
.............
Chairman, Departmental Committee
on Graduate Students
?LLx V.4 OcITr
RaVIaxc
Of A :v IS
@OSACI
W
l DAas er
;aeo
rle
nAeSe
te
t
Dparteunt et QGeoloAegy M tA ophyes
Asn prial
ae
Sor the 4
tault
NIIewjin K. F.
flmot
ot asgy
1a aPAlie to a
*a
3ileat t theaclLre&meB
of te nqrutiamt1ee
the priniplo
ta 1.
oneravati
nLeek pr"pIlattxl in as
Latinatte plate of vaaishn%6 thiokaes with a uatoera
tnosLo ap4li1
tiqe
veleita
ask
a&t iaLiuty. A rtiaement or
*vr that ust V *tt yiUls * *1
0e the
a+geth
itial .ak
14
The pr~Oi
t data taken oin shee*t
athastteal
aetb; e trie
vlooeity behavior
is eompare
ot jAkelethlm
aebeflate
*
The
h*
npariSna af the data wit, t4
thqr
is taoeoncluive benonsus
antsk
Oerng Pr
"it
oesk
6Ist wa. nt
onstant sAt beoau'O at ot er
oftectsena as Astle
flew.
It is
oatluetd that vaMusfAoa of vet4ity. bealor pnrtteted
/ Mtt* thnSy oust a*ait eatul
veloeity tQties on a brittle
rebal ae
as 4as.
enerA 4*. t
sns14sdenti fs of saca an oreariaet
Ttale, Aseitant
rfessor o9
Ge~olg
*ha"A ol Costa
b
atnboA*atitio
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statnmnt *t tM £flin6
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tuwta Qneek
10
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15
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3.
4.
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23
figure ?
Otof Ve"Lotty Y
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Luegth for Two
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ns
27
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otuer
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W
um
seornastena ae
THE VELOCITY BEHAVIOR OF A GROWING CRACK
I. Introduction
1.
Statement of the Problem
The problem investigated in this paper is that of
theoretically predicting the behavior of the velocity of
a propogating crack in a brittle material from the time
of onset of rupture until terminal velocity is approached.
The crack will be considered to grow from an originally
static one at an interior region of an infinitely
thin plate, infinite in the other two directions.
A
uniform tensile stress is applied at infinity normal
to the long axis of the crack.
The tensile stress
is increased until the crack becomes unstable and starts
to grow in its own plane.
The stress is held fixed
at this critical value during propogation of the crack.
The crack is
approximated by an ellipse with a very
small minor axis.
See Fig. 1.
2. Previous Work
Some work has been done on predicting the terminal
velocity of rupture.
The agreement of the predicted
terminal velocities with those found experimentally is
apparently not conclusive according to some authorities,
App oilaaft
ellipa with a
by a
la7
safl
maieo axis the sack grows In the positive sat negative
tnos
u ta
se4n* tasO due to the aetion t th wa
U, appli
at lutiat
.
--
x--i-~~----i--
00
-c'
C----
FIG I
jfh"cs~,k.,,_.~~.l~l*~d~,,
,._~~~_r~*CblC,~:riC. r91~.LI i\*1~'
8
VI*LIYL*1I~C~II
.I .II-/
e.g.,
Schardin 1 .
Schardin feels that terminal crack
velocity is a function of an effective free surface
energy and is not simply related to other parameters.
In support of this, he points to the evident lack of
success of other proposed prediction theories.
Stroh 2 feels that a propogating crack is similar
to the phenomenon of Rayleigh waves and hence inertial
effects would determine terminal velocity.
Yoffe 3 has calculated the stress field about a
moving elliptical hole of constant length in a simple
two-dimensional geometry.
She tabulates the stress
field at the head of the hole as a function of velocity
and shows that the direction across which maximum tensile
stress occurs branches noticeably at approximately
0.5 the speed of transverse waves in the material.
Cracks are observed to branch at roughly this velocity;
if the branching velocity is necessarily the limiting
one, then the limiting velocity would be controlled by
inertial effects.
Irwin 4 concludes that inertial effects control the
limiting velocity and states that Irwin and W'ells in a
personal communication calculated that terminal velocity
should be 0.5 the speed of transverse waves.
Irwin
presents a brief table of data showing excellent
correlation with properly performed experiments on
various materials.
Mott 5 demonstrates on theoretical grounds that
terminal velocity should be equal to a constant times
the square root of Young's i odulus divided by the square
root of the density.
Schardin presents a plot of this
quantity versus terminal velocity for several types of
glass.
The data points substantially deviate from a
straight line.
The developments of this paper are a
direct extension of Mott's work in which he applied
the principle of conservation of energy to a moving
crack of the simple two-dimensional geometry which we
are considering here.
Lacking precise information as
to the manner in which terminal velocities were
measured in the glasses, the apparent disagreement
with Mott's theory is attributed to experiments
inadvertantly performed incorrectly relevant to the
criterion presented in Irwin and discussed in this
paper.
Mott and later Roberts and Wells 6 were primarily
concerned with the terminal velocity of cracks while
apparently overlooking that Mott's approach will also
yield the behavior of the velocity from zero to terminal
velocity.
The following study is directed toward pre-
diction of the entire velocity behavior.
II. Analysis of the Behavior of a Growing Crack
1. Review of Mott's Approach
Consider Fig. 1.
The ellipse represents a crack
in an infinitely thin plate growing equally in the
positive and negative x-directions.
a represents a
uniform plane tension applied at infinity.
Mott applied
the conservation of energy to the growing crack.
This
can be expressed as the sum of three terms F+S+K=constant.
The terms represent the following three energies
associated with the crack:
S= the stored elastic energy in the plate in the
presence of the crack minus the stored elastic
energy in the plate in the absence of the crack
with the same uniform stress at infinity in
both cases.
F= free surface energy of the crack.
K= kinetic energy of the time varying displacement
field associated with the moving crack.
When properly evaluated, the kinetic energy term, K,
accounts for elastic disturbances radiating energy from
the region of the crack.
If the half length of the crack is c as shown in
Fig. 1, and T is the free surface energy for unit area,
then the free surface energy, F, is given by F=4cT.
Evaluation of terms S and K depend upon the stress
field about a moving crack.
It
is
appropriate to note
that Yoffe's expression cannot be used in the evaluation
of these terms.
We require integration of the stress
and displacement fields over all space where the
approximation of substituting a moving elliptical hole
of constant length for a crack is no longer valid.
In
In order to evaluate terms S and K we must use the
expressions for the fields associated with a standing
crack due to Inglis 7 primarily because these are all
that are available.
Roberts and Wells, and Irwin offer
brief discussions of the validity of using the standing
crack expressions; howeter, the most convincing demonstration to date of the validity of using Inglis's
expressions is a series of photographs of the photoelastic effect associated with a moving crack presented
by Wells and Post8 .
The photographs show that the
dynamic stress field and static stress field do not
noticeably differ significantly.
Assuming that the
static crack representations are valid, we have Mott's
expressions for S and K.
S = -io ca /E
and
K = 0.5kc2
62 o'Z/E'
where E = Young's modulus, k = a numerical factor less
than 1 determined by Roberts and Wells, q = density and
c = the time derivative of c, i.e.,
t'e velocity of
propagation of the crack.
The resulting equation for the conservation of
energy in a growing crack is
1kqc
2
_?
EZ
- no' c a + 4cT = constant
E
(1)
Mott then differentiated with respect to c to remove
the unknown constant.
Because he was primarily
investigating terminal velocity, Mott allowed
E_ to
equal zero to facillitate the analysis, which then leads
c=)T1
(1-
O
+o
when for 4T we have 4T * 2x' e/S twUOrttifh's
eiterion at eo is te
initial orafk length
Lortu%.tAo
of 0belP and 4all!
2h
2.
rupture
The value of th,e
known nuereal constant k mse
not doeaiMaS by 4*ott
Mott preseated an integral
tosselation of k which R berts and .411s evluatoe4 for
ratio equal to 0.25.
oAseoao
i
its value is sOhk thatVSF
deae,
Me
&*
's
*
oj.3
theory as evaluatea
w Roberts and
by
ojaYWr vu£iyij .
the approxtastioa used by
a
and wells of setting
0
(2)
oett *Ad later Roberts
Vad when &
vly
A
io
is vry near tenminal veleetty.
4
k is so evaluate4
4shoa
Sq. (2) shots that
# 0 in tetianee of the assurptios
Sq. (1) sy
to the
be solveW
rigorously without appealhag
The solution of Sq.
rsmqniation above
so obtaiaea is
lapLe4 in
coirnet withia the traework assaptions
Sq. (1).
oritrion in
(1)
Let us replace
4? by the Griffith
0o*
q. (1) and call the conatasnta
then have
e.
wwo
t
sa a
3
13
3
e
.
Se
for the
et
poiat ou
ell
at
a
that a bemaary
4, is
elaoet,
f O.
0oe% law
Applyr~athias bousary ooadition to q. (3)
enorg
therott
sheen s Sq. (4) we have
value of
iflGriff ith
balsaae equation of a gw
NO
*oo
a3
nfly tor
4.
t
*
ay be seld alsbie
sq. ()
(4)
I .tt*'
ea
+ ta
te
it14s
3
3
Lubtitatia
onaitioan
r+
a
a
4 vsfws (L.o)
S~t
aoefdin
etof the velotty behavior
(6) is the paietie
to Mott
Sq. C6) ditters tro
a 0, i
ar
(4)
the behavior pretditet
then.
(lfi
a square root sign is St.
4,
ustuag thesppmaatioa
that felnt b
that the te
he Rea4tta1ok
As the *rack g1ow
4azei
without
by
) i
Sqt.
(6) eppears
(2).
aM 4fow t9 n
itt
ow]la
0 /* will approah
seor ant the terminal veleTity wil be Yft/
us
Let
releity.
Vt1 the trail
i
.
7)
(V
(V/)
S*
l O
ts
AMen & t pletted saant 1/* a stmaLot line nVa
Inae
is O
L
~h6 vloetity axs
in Pig. 2.
as s)he
atOeeept of the
s1Vv anA the eosiproeal erask lawth ass isato ept
. Tis plot when appliet to v*teiity sessutaelts
provides a sonveatesoteetk of the thenBr
0s eonolteA
that th
prope solution
vele lty shoult sake
tak
that e
neagy equation pretis
tOMett's
an hyprpebolle asymptotio approach to teoyinal velo lty.
Figs ) Is a pot tf
reasfs a taSq. (7) hea
the
psetited velocity nure.
AM the.
AARM"anica
n lnnet
0.
In a4dition to the asumed vali4dity of stanUing
cak
stress and displasement tiolts when applLt
to a
oving erack the following sonaitioas aust be set bv the
phrstrcl situation Aesoribed by the athomatics of the
above
analysi7s
a.
The surtae eneay per untt length of the
Wrack Is constate
b.
No work tS addse
at the bouadaries hdur
propagation of the
racak*
This
orrespouts to
constant strain at infinity ratuer than costant
stress but is
a valid assumption for an
ianfiiteo plate as dise seed by 4obef s a~nda
IS-
-
&lls.
The vleitsp (4) axis intce"pt is the tegnaal
veloity (V
a
VW
V
t to te reniproc&d length of the
leasgth ax"isatt
initiatag e4a
othe v44it
o
vool7
OIC:
e
(1/ )
e
is givn by
veloeity oal makes a
e
lrack
while the inase
t
(1.).
Th
asyaptotte approach to V*
, ;:tO
so
4
l
n n~
i
Z,,
)
eaprth!oVl
----- ----;.-:--II~-,
I
II
I
_~~-~-~i-rczL
~---~- ~~n-~qlzs~Cg~
VT
C
o
o
Co
FIG 2
VT
C
oo
C
I1
FIG 3
17
Ill(^
912s
n-*
infinite plate as discussed by Roberts and Wells.
c. The elastic parameters of the material are
constant during propagation.
In particular
this includes Young's modulus.
d. The fracture is brittle; i.e., no plastic flow.
e. The plate is infinite.
The effects of a physical situation not complying
with some of the above conditions are discussed in the
following section.
III.
Comparison of Theory with Measurements of Crack
Velocities in Polymethylmethacrylate
1.
Description of the Experiments
In order to test the theory developed above,
measurements of crack velocity in polymethylmethacrylate
were made available by Professor E. Orowan from studies
of Mr. J. Neimark to be published elsewhere.
Rupture was initiated in sheets measuring roughly
6 ft. by 2 ft. by 1/4 inch at a sawed notch a fraction
of a centimeter to two centimeters long cut halfway
along one of the six foot sides.
Tension was applied
at the ends by means of a configuration of clamps
suspended on knife edges.
When rupture occurred, the
passage of the arack across the face of the sheet was
observed photographically on a cathode ray oscilloscope
as the crack broke evenly spaced lines of printed
circuit paint placed perpendicularly to the path of the
crack.
From these data velocity of the crack versus
orack.
a trnoae data velocity of the track vorsus
kr
e sett
t .* d e of
dst-nc~e froa
obtaiaon.
wP~a
tosoe veloaity versus distansoe curvos are
xof
j Qised b
.eal ones teoutat
4%ecieas are thu
p n
apte
ecLeas 10l and lk
l
wijth
velocity with4in the
toerisal
naioa of the
e de
Fing. 4.
e e
reseted
apvQoaz a
1
of the
ow
a4 11f branchod betwen 35 and 4c selatenrS
0t" azd Ilk are vorl u.aoth and btoe
rtAkuhtort
ho
owk& in
rti:iatin
crack aoac the
te
surface o
te
63 cntimeters.
total width of the Plate beia11
projreasivoly
usA before fAkimag t e swrface is so
Uth
r ,:u
e~4 that sas1
brncia
A
(up to %ao ceatisetOre in teoAth)
cracks occur.
asumptioa
ls probably a goo
It
LmieS an itaoroao lu of
Vatt tibo rouieghnin
*
o
surface energy por unit erack length
otive
Ater branching,
*4
of the cracks becoes proroessively *aootker ad
te
other
eolatively rueri
aaintaias
a
the other specitas In til.
esaller cha neo inlufa tee
propagation.
edto of the plate is rneibd
ero oAly a Oi.htly treted crack aurtaes
t,,and
earlier 2alr
iiotS
4 exhibit * amuch
charater turiau
By the time the ap proaczia
amBnch as is
surface.
not applicable afte.r brahFbnt.
theory is alearly
thse sOalos
'atea
n sLamIs
of ViIe
0LP antA
rack.
19
P at a much
e lestathanna g
Fhese are sa
prrenet
ot Pntonr onsa's 4a,
n for se.eatsrd
enat sen peteae
.
.um
Lath
to thoe pe
. beS
meseeOe
ea
5Ples
6
10 P
5
4
3P
104
CM
SEC
3
2
I-
5
10
15
20
FIG 4
25
30
3
40
CM
flger 5 aM Ftr se s
The or4ks braa hed whoj* th* earve
ieat.
Thor7
ppaiats that when atteetie eu;rtan mnry per muIt
anot length Sia
in the apecilans,
ases with craek lngth, as it
does
the voolity whea plotted a&anst
seiproal eoack length ihoul& not be straight Itneos
as it
is
ver a easiterable rarg
A eeabiSatia of eofteets
ho"OTC
in thee ePrs.
auses a straight line plot
9
91A
wo
---v
tp
dOl
e
oet* that te
a
range contrwr
Tho eeks
hOes 0.
lflmaa.ardwmanneAmma
0 *s
are not sttigb t lices over
to the pr*etioa
AA seot brsh
of wsple theor.
withi"A the wit4th of the
6
7P
4-
3
2
4
10
cm
sec
Oi
.02
cm
.03
0 4 .0 5
.06
O,
.0 8
.0 9
.10
I
c
FIG 7
,", =,
,,,,,
r ,.
..
.
I il
mI
nlnm
n ..
. .
.
. n --
II .. .
...
.
... . . . ..
.
.
.
. .
.
ay Wae
ltW
Sim*e 44t n
sheeMt.
ilO the
Aalel fin
n A wia t"suhe
t .M.
ot the
6
5
4P
4
3P
10 ' -
cm
Sec
L
I
.01
.02 .0 3.04
cmCmT
L
L
L-I*,±-
0506 .07 .08 .09 .10
1
00
F;G 8
:6-~:
4
I6
:9j-~jt
on the character of the surfaces when the studies are
published.
A brief discussion of some reasons for
roughening may be found in Irwin.
3. Comparison with Theory
The comparison with theory is best done by plotting
velocity versus inverse crack length.
Figs. 4 and 5 show
such plots for specimens 10P and 11P.
The small circles
are not data points but are points used to redraft
the velocity curves.
The points for transcribence
were selected at 5 centimeter intervals.
The theoretically predicted curve would be a
straight line between co and V T = 0.38V7
.
c
0
in this case would be the length of the sawed notch
plue a dmall
tack ghat grew from the notch while the
specimen was brought up to load.
The value of VT
obtained from Young's modulus could not be determined
because Young's modulus varied according to strain
rate, etc.
Hence the best determination of VT
according to Mott's theory is obtained by extrapolating
the straight line regions of the curves which lie
approximately bwtween 10 eentimeters and 35 to 40
centimeters.
When the extrapolation is done as in
Fig. 5 and 6 we see that the terminal velocities lie
between 5 x 104 and 6 x 104 cm/sec.
The crack velocities never reached this value of
VT but branched first.
Branching might have occurred
too soon because of the influence of the approaching
edge.
At the point of branching the cracks were
2/3 of the way across the sheets and it is to be expected that the approaching boundary would have some
influence on the stresses near the crack.
Nhether or
not the edge caused branching cannot be answered here;
however, we can say that if the edge was not influential
in branching then the terminal velocity as predicted
by Mott's theory was not reached.
Since Yoffe's calculations show that a continually
increasing branching of the stress field at the head
of the crack would occur at crack velocities below
VT, the question arises whether or not the terminal
velocity predicted by Mott's theory will ever be
reached or will branching occur instead.
to VT is hyperbolic.
Therefore, barring infinite crack
lengths, to say that VT is
approximation.
The approach
"reached" must involve some
The theory states that when the crack
length is 100c0, the velocity is
c = 1000c o , c = 0.9 99VT.
= O. 9 9VT, when
Hence VT can never be
*reached' before branching occurs or the edge of the
plate is
reached.
At the present time it
cannot be
said at what fraction of VT branching will occur.
If
the extrapolated value of VT is
taken as that
predicted by Mott's theory, then it will be seen that
in Fig. 5 and 6 the measured velocities are everywhere
less than those predicted.
The straight line portions
of 10P and llP behave as if the cracks originated at an
effective c0 approximately 5 times the c o observed.
the
behave at I
0? and 11
o
this 4Ans*n
pesj betwen Pre410tt
it
eerio
froa the region of the
aek.
elt
rol
oul
for by a
00aoctd
ha04inUg
Ion"e
tfAis reovat
1earfas
oal4 be
per talt
u
,ner
these
of Losee.
*egy
elatie e
ts
of Vshe plate 4r plastic floew i
of the oreek.
possibly be Wrsatest d4ratn
int, w o
oid
the arly stagee of
La seaest.
ably acceleraton
prs
iadtative losse
loe;a e
rdtitive
teels that th
-rh
arnm
woee sonha ow rmovet
e to raiatiave Losses o
sould be
to otber rgfi
rgia
at4 obnrved veloc4ity
by oeter typue
laacgth of the cras k o
observet,
the e
aprotiatsely, 5 ties
ottfotive
1r wact
0;t444ate4d at a
rask
will be assAuM
satil la order to
The l oss of euer~ due to ;Iastie 4tozraioa
probabt
taportant In these QaasQ*eants beenuse
do*es ezaibit plnsti
aollothkiAZtherlat
tVo effet
is aneta
on.
s
t"# work i.i*volvf
4he qualitativ
eCet
easared velocstise Less t.a at t
evens7 ssea
this o
Qarves 1WP e
of
the aous of f1w
an be glvea beosnS
andinre
4eoerAtioa
aaa4js4s
o quafatittiv
for slow staain ratOea.
teora
oIs
he case. se
la th
w-alt be to ae
reAete4.
p4
flii,
Ia
6,
S"
7,
lndk plot as stra1,ht Ahns over
3o
.
is
swa of their ranag
La te
otteoltiVe srtfse ener
L a
ffeset as
t e.
unit length aU a fation
(O) was vieblty an
In the spOOiLsenOS above,
(1)
t a
o
(004 ) wno
(O)
tnt
The
*ettiv length.
Ss
that apear
ent.
at lrea
be evaluated as tollws
iaeresi% ftlntloa of o
t oa
~er
estrfae energy ter
T(s) is
ieoeabie
otin*r
the a
e
*p t
t*en t * e
as s ene
Mr
the laaSS
io obt8iLe4 ftI
t (s)S)4 e 4s
"a
is outsastnt.
hen
0 toa *a
Destsnato to a the value ott teen
s Grittith otrtron e have
(e)sl e .
Then Ie (s)
sat ia alst sh
) Oms
().
*as
T
a(*)
s
Lrways
1.,
s
(
o tofor
>, ewe have
the result tree Sq. (5) who ae >> e
q. (5)
o%
5 AW
say be uest to evalsate a
it
ekaaagI% surfsee
(a
en"y per malt orak lagth it
1ie14 a hyperbola for e.
marten energy will .till
e
peia
A
10P
lP are hyerbol*e Over ausco
et
Aeapite the notieoeable ohange La ettetive
txeir rate
eoFerg with 0 aOk l enth.
sartes
nes.
(16(e) e
serked continual ~tserase La trm o
o
4. (9) sh es that
T(e) *
other otteoots are also present ea
tor the oters
nae that
this
aeither these ourvee
fit the eleple theory of 4et tnive
for brittle material
99#tstfl&4ML
>.
4reasqnereiaBn
t be'I#
4o o4nelue that seataeas 10V and Ik
fit the
ourve ~reLcteA by M lott*s tbery over a cosiderable
ange but t
e
sade at the effet of
a formulation 1
*hAnsing srtfaoe enOrgy, such as
arvesl ano
finger
it the
t
probably pastie tflow
leoq
s observ
the
nd oth*Vr Otffoots
are laportant also,
The aeasure4 veloeties of the other tour
speeotee
to"e
when plottet against re,4proeJ orck iecgf
Arvfd ltu
a
a stari~
t oe
over their entire na.je rather than
y
as th ory
mi
setes
Roe of tat speebers tit &tt*s tieory.
C Ii
be du* ts effects that are ?roporties of
eterral s camy or de
t 4 effeeCts frts
This
Lat.sti
nich brtttle
uatrlals would suafer, too.
leeits t±at
ueld 4cease brittle 4ubstsa.ces s
3Z
early stages of the crack growth
(b)edge effects when working with cracks initiated
from an edge
(c)addition of work at the ends under tension
during propagation of the crack.
(d)dynamic unloading of the sample so that it
is
not moving in the same stress field that it
would move in if a uniform tension was applied
at infinity on an infinitely long plate.
In the above list (b) refers to edge effects of the plate
perpendicular to the direction of the crack while(c)
and (d) refer to the parallel edges.
Effects that presumably would be important in
plastic materials only are:
(a)changing surface energy of the crack as it grows
(b)plastic deformation particularly in the region
of the crack.
The effect in (a) may be a result of (b)ll .
We tentatively conclude that primarily due to
plastic flow, plastic materials should require a greater
ratio of crack length to initiating crack length before
terminal velocity is approached or branching occurs
than would brittle
IV
materials.
Suggested Future Experiments
A searching test of Mott's theory must await
careful velocity measurements, such as those furnished
by Professor Orowan, performed on a brittle substance.
Glass is suggested.
The surface energy of a crack in
glass does not noticeably change by important factors
as the crack growsnnor would plastic flow be expected.
In order to eliminate the effect of the edge at
the beginning of rupture it would be best to start
from a crack in the middle of a large plate.
However,
the edge effect is probably very small after the crack
is a few centimeters long.
This could be evaluated
by experiment.
Whether rupture is initiated at an edge or in the
middle,
the plate should be of sufficient width that by
the time the crack has grown to 50 or 100 times the
length of the initiating crack no more than half the
plate should be traversed by the crack.
Hence initiating
crack lengths should be kept as short as possible.
If elastic disturbances radiated by the moving
crack are important,the edge to which the uniform
tension is applied will have to be far enough away from
the crack so that stress waves will not be reflected
by these edges and returned to the region of the crack
before terminal velocity is approached and branching
occurs.
If it is felt after expetiment that stress waves
reflected from the edge that the crack is approaching
could be causing deviations from the theoretical velocity
values, then the plate will have to be made wider or the
size of the initiating crack further reduced.
Design of velocity measurements to conform to the
3t
above conditions require an estimate of accumulated time
as a function of the ratio c/c o
The accumulated time
.
may be obtained from a first time integral of the
velocity expression.
do = VT(1-c )
dt
-C
C
Sdt=
So
VT
c -c
-o
0
The integral is singular at c= c o
do
.
This is an expression
of the fact that the crack is in unstable equilibrium
at c = co, hence the crack may stay at length co for an
infinite amount of time or no amount of time.
Accumulated
time may be calculated between the length c and co(l+6)
where 6 is any positive number greater than 1 and
c>co(l+6).
That is, after the crack has started to
move, the time between any two points may be calculated.
When the integral is so formulated we have
t
0
[accumulated between c=co(1+6) and c>co(i+6)
1
do
Sc
0c(1+6)
C-C
The integration yields
t = c
1 (1+6)(n-l)+lnn(l+6)-l]
--
,
VT
where ~i'= co (1+6)
0
'3S-
6
3
=
If 6 << 1 and n >> 1 then
t = co [n+ln n),
n large, 6 small.
V
It is not possible to state how much time expires
between c
0
and c (1+6).
We should like to have an
0
estimate of this time because presumably stress waves
are emitted by the crack as soon as it starts to grow
from c .
The only simple way out of this difficulty
is to assume some average velocity when calculating the
time to grow from c o .
The above equations can be used
for calculating the time between observations of the
cracks as a function of crack length.
Placement of conducting strips could be arranged
so that approximately equal time intervals would appear
on an oscilliscope or timing between camera exposures
could be adjusted so that the crack had moved approximately equal distances between exposures, thus
obtaining maximum resolution in both cases.
U.
bAfla
it.
j
4o.-VjtqA4
AMtM
4~*rSo,12, CORLwN I
4 L2.0-149,
ti.
is
Ar,
a,3,ot*#
S.
Ma,, jj
6.
4wjcottt
Ma
19,
fAtfra
#*'JA
"nt.%:n
AsZ1
As.
o~ ZE$'
pnon
ittitt Unsk, S;h44
.jk "ThO ;anias
flr /, fl* (1951).
Y.. aA *stels*
Aobrt0o,
4
VoeIntty of
1913).)
ktotat
8*
, 4*
of1l,4tq Doi
#
Z4ntao AES
(19*>.
ZoS
PAna
~,
Wfl,
~xs9vC
J*. -io tot
90
*o
(oaaaajo
c~
;I, * 4*j TA"S.4 Afto ;i40COe841*
- Orlom A,
1t35t 19l).
IC?
,#
?M
aas poiatot out tb iOtsor?.
;#*t*saor
ournA
TA*
Ma"o
nrtkt ttat plastic flow Aatt
Sacroas. th# bri-ttl*aas* in tho oafl4, star.v -o
oa4'
growt n*&sia,
a
this "iogin.
37
oa*th @nck surftaf
in