8/11/2019 Basic Fracture Mechanic
1/8
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Lesson
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-
Assuring
Component
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Lesson
5
-
Testing
Design
ConcePts
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49\V-
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20
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PENTON
EDUCATION
DIVISION
PENTON
BUILDING
CLEVELAND,
OHIO
44113
A
ffiAS[C
COURSE
Eru
FRACTTI
RE
M
ECHAIUECS
Pkfr/,6ld
Prepared
bY
CARL
C.
OSGOOD
Member.
Technical
AdvisorY
Staff
Astro-Electronics
D
iv-
RCA
princeton,
N.
J.
for
MACHINE
DESIGN
Magazine
CONTENTS
'
t :rt
r
'
lt-_l__-:--tl
I-esson
I
-
VifLaL
rs rtautulc
lYlcuiiairiu)'
Lesson
2
-
Building
on
the F-undamentals
Lesson
3
-
Applying
Fracture
Mechanics
to Design
15
Rt
c
-{//-
'-ji
--/'
.
I
i
Copyrlqht
l97l
by
Th.
Prnton
h'rbtlthlng
Co'.
Clcvtlrn
8/11/2019 Basic Fracture Mechanic
2/8
.8
0,
basic
course
in
racture
mecha,nics
What
ls
Fracture
Mechanics?
tsuilding
on
the
Fundamentals
Applying
Fracture
Mechanics
to Design
Assuring
Gomponent
Ufe
I
esttng
Design
Concepts
Lesson
1
D
n
D
n
What
is
Fracture
Mechanics?
Fnacrtrnrs can
occur
in
pipelines, pressure
vessels,
ship
hulls,
and
aircraft
structures
at
stresses
below
the
yield
strength
of the
structural
materials.
Such
failures
have often
come
as
a surprise
to
both
user
and
designer
because
all
t h
e
conventional
strength-of-materials
rules
had
been
followed.
Oh-
viously,
something
was
overlooked-
8/11/2019 Basic Fracture Mechanic
3/8
sa
psetend
that
you're
the
captain
of a
Liberty ship,
pounding
in
a
heavy
sea
in
the
North
Atlantic
during
the winter
of
lg4l..
fhe
ship's so new
that
rust has
barely
had
time
to
set
in.
5uddenly,
you
feel
the
deck .trembte,
hear
the
snap
of
steel
cracking,
and
see the
bow
drift
away.
You've
iust
observed
what happens
when sq:called ductile
metals
don't
behave
in
a
ductile
mannsl-which
is
what
fracture
mechanics
is
all
about.
Here
is
the
first
part
of a
five-part course
that
will
explain
how
the
concept
of
fracture
mechanics evolved, how
to
recognize
designs
to
which
it
should
be
applied,
and
how to
put
the
concept
to
work.
Nomenclature
a
=
Half crack length,
in-
E
=
Modulus
of elasticity,
psi
G
=
Crack driving
force,
lb
K
=
Stress-intensity
factor,
psi-in-)l
Kl"
=
Critical
stress-intensity factor
for simple
cracking.
Psi-in-*Kr
=
Stress-conceatration
factor
(local
stress/gross
stress)
Q
=
Proportional constant
for
stress-lntensity-factor
equation
R
=
Resistance
to
crack
propagation,
lb
r
=
Polar
coordinate
(radius),
ln
T
=
Surface
tension,
ln
-Ib
t
=
Sheet
thict
ness,
ia-
U'=
PotenUal
energy, iL-Ib
x
=
Factor
of
lVestergaard
equetion
)
=
Factor
of
Westergaard
cquation
Zt
=
'rfr/estcrgaard
functloa
a
=
Angular
polar
coordinate, deg
p
=
Radius of crack 6p,
ln-
o
=
Tensile
stress,.
psi
r
=
Sbear stress,
Dsl
'Subscript
c
identifies
tle decrea-se
iu
fotential
eoergl'
due
to deformation;
o,
the
potential
energy
prior
to
inducing
a
crack;
and
T,
tle increase
in
potential
energy
due to
e
new
surface-
'
tPrime
indicates first
differeotial
lSubscript
x'
y,
and
ry
indicate
tbe
direction
or
plane Iu
which
a stress lies.
account
for minute
cracks in materials.
The frac-
ture-mechanics
design
parameters
are:
a
tough-
ness
(enerry-absorption)
factor, applied stress,
size
of
crack, operating
temperature, and
t}te
"state
of
st1g5s"-qrfuigfi
is
often
far
more
complor t]an
that
usually
assumed.
This
concept
can be
applied
to
any
solid material,
although
most
testing
to
date
has
been concentratd
on
metals-
Reoson
for Doubt
Early
work on
the
effects
of notches
and holes
on
local
stresses accounted
for these
discontinui-
r.h
/r1
o
Jr r
3
in.
l'
L0.00t
\
Lo.
305
rl
|
2
4
5
810
20
40
60 80100
200
400
0istorxe
Aheod
ot
t{olch
Tip
lt0-3
in.)
.Fig-
l-Effect
of
crack-tip
radiu ,.
pr
on
stress eround
thc
crack
in err
infioite
plate
C-urvcs
iadicete
dret
sre$-concentration
fectors
K,
near
a
small
cra&
are
lerge,
but not as
lerge
es
drorc
dcrived
from
the
cooven-
tionel
gcooctric
apprech
1
rRclcrrocs
t.bglrtc{
rt
cr}d
oC
lgao-
._400
;
zoo
v,
o
G
t00
o$Q
, 60
6a9
o
o
J
2n
0
Gl0
0.r o.2
0.3
Tilckntg
(in)
t-1
Yield
Srenl
(ot
0-27J
tl
V
/
t,
rocl
n
:islon
C,R
-200
-r00
0
t00
200 300
400
Temperoture
(F)
Ei1,* z-C-omparison
of
coovenrionel
mcclreoice-l
propcr'
ricl
of a
tyiicat
low-alloy
steel
I'ith
is
resistrnce
to
cncking.
t5
.E
Es
ro
cv
9
t 5
Eto
Es5
lr,
0
?0
u
-Cl
ar-
lo
E-
0
200
0.4
E
_o
l5O
:o
vt=
8/11/2019 Basic Fracture Mechanic
4/8
ties
in
a
stress
field,
but
these
early
theories
were
unable
to
treat
fine
cracks-
For
instance,
ap.plying
conventional
stress-concentration
factors
to.
f
i
n
e
cracks
resulted
in
large,
unrealistic
values;
these
values
have.-
been
subs,equently
modified
by
frac-
ture-mechanics
concepi,
Fig-
1.
Also,
the
purely
geometric
approach
did
not
consider
strain'
load-
ing
rate,
and
toughness-
However,
these
theories
did
point
out
certain
inconsistencies
in
understanding
a
material's
me-
chanicalpropertiesthroughconventionalstrength-
of-materials
concepts-
In
conventional
design
theories,
elongation
is
often
considered
to
be
a
qualitative
measure
of
a
metal's
ability
to
yield
upon
loading,
thus
redistributing
the
sLresses
-
and
reducing
stress
concentrations
to
tolerable
values'
HowevJr,
evaluating
a
material's
fracture
resist-
ance
by
ils
ability
to
elongate
is
faulty
bgcause
elongation
is
not
directly
related
to
two
very
rm-
portint
design parameters: operating
temperature
Ld
section
thickness,
Fig.
2-
Simila.rly,
elongation
of
steel
remains
essentially
constant
with
temper-
ing
temperature,
Fig-
3;
ihis
fact
has
led
to
the
fal-se
conclusion
thal
tempering
to
the
maximum
Vield
strength
is
desirable
and
does
not
appreciably
reduce
toughness.
Actually,
using
yield
strength
as
a
guidJ
to
selectin
a
tempering
temperature
produles
the
minimum
notch
(crack)
strength'
Fie-
g.
Notch
strength
is
a
function
of
operating
temperature
and,
for
many
steels'
falls
rapidly
just
below
room
temperature'
Thus'
conventional
iesign
thinking
compounds
the
danger
of
abntpt'
brittle
fracturJ
of
a
material thought
to
be ductile
on
the
basis
of
its
room-temperature
eiongation'
Two
APProoches
The
techriolory
of
fracture
mechanics
is
pres-
.nUy
based
ott
t*o
approaches
that
are
some-
what
complementary
and
give
generally
similar
re-
sults.
one
"ppro"ih
grerr/
from
the
recognition
,trJ
bie"
d"ls''l
o
s-"
ptaiti
ei
tv
a
.'
lemPe-Ttqre-'st
The
other,
*or"
*iiiuc"r
ipproich'
was
derived
from
consideraiiont
ot
:lr
tgggt-f-teld
'and
'the
ptasticzol._1 hq 9.s--oJ-cg9
8/11/2019 Basic Fracture Mechanic
5/8
to
be
the most
fracture
resistant-
Results
of transition-temperature
tests
cannot
be
expressed
directly
in load-carrying
terms.
However,
a
fracture-analysis
diagram
(FAD)
--derived
from
laboratory
tests
and
service
failures-relates
ap-
plied
stiess,
defect
size,
and
temperature,
Fig. 5.
This
diagram
is
applicable
to the structural
grades
of
steel
(between
y2 and
2-in-
thick)
that
have
a
pronounced
ductile-to-brittle
transition
behavior.
Using
the FAD and
knowing
one
of
the
transition
temperatures
(most
commonly
NDT), the
com-
bination
of
stress
and defect
size
that will
cause
catastrophic
failure
can
be approximated
for
a
giv-
en
material.
Below
the
CAT
curve,
the
probability
of
crack
propagation
is small. Above the
CAT, the
constant
fiaw-size
cdrves show the
stress-tempera-
ture
combinations
that
will
intiate fracture
For
high-enerry
tear materials,
applying
the
FAD
:t'*
i?
::"#::';,,1'
liL,
ffff
'
:l
.
T,',flHH
tion
temperatures (preferably
NDT).
Then,
if
you
know
the
applied
stress,
you
can estimate
the
size
of
defect
required
to
start a brittle
fracture.
Con-
versely,
if
you
know
the operating
temperature
relative
to NDT and
the
size
of
a
defect
that
exists
in a
structure,
you
can
estimate the
stress t}at
will
start
a fracture-
Certain limirations
restrict the
general
use
of
the
FAD. Because
it
is
based
on
the stmctural
grades
of
steel
that
have an
abrupt ductiletcr.,b;ti,l;
X;:i.
tion, its use
should
be
restricted
to
these materials.
It is
not applicable
to
the ultrahigh-strength
steels
or to
aluminum
and
titanium alloys
because
t}ese
materials
do
not
have
this
abnrpt transition
be-
havior.
Stress-Analysis
Approach:
The
analytical
apr
proach
to
fracture
mechanics is
based
on
the
stress
field
in
the
vicinity
of
the
ever-present crack.
The
most
useful
way
of determining
this
stress
is
from
the
relationship
between
stress
ahd
changes
in
the
surface
and
potential
energiei
of
a.part-
This
approach
to
fracture
mechanics
assumes
that
a
part
contains
only
one
crack
with
one
plas-
tic
zone at its
tip.
There is
no
need
to consider
multiple
cracks
because
failure is
as
complete
when
a
part
fractures
into
two
pieces
as
it
is
into
three.
The
stress analysis for
fracture
mechanics
be.
Froclure
opPeoronce
Encrgy
T, Tz
Tr
T{
Temptrolrre
-+
ig.,{-Tnnsi
tion ternpcnnrrcs
and
correspondia.B
casrq.y
..vels
obrined from
Cherpy- V-notch
-
inpect
ierc
7r,
is
the trznsition
rcrnp&rnrrc derermincd
bi
a fixed
levei
of
impact cnergy
Er
(e
lcvel
rher can
often
bc estiaritJ
from
producer"
dea)-
The
fncrure-appc2nocc
tnnsidoa
tempcnture
(FATT),
Tz
(arbi,;erily
akea
*
507o
brittle
znd
50Vo
shczr
frzcture), ir
uscd
lo
comparc
thc
fr:cnrre
resisance
of
differcnt
mete.ie '
Ts
ic
the
aridpoint
tear-
pcrenrier-
with
corre-tpondiag
:"qtgy
lcvel
Ea. A[.*r
?r,
fracnrre
is entirdy
rhear.
r00
I
I
.o
Lz
o
a
st
,5
et
s
5
509
la
g
CD
gYs
rtt
=
l5O
C\
o
=
t00
E
o
c
O
;50
:s
c
o
o
b
c
o
fa,
250
200
4
34vs
r6vs
14
Ys
400
600
800
1,000
.
Tempering
Temperolure
(Fl
NoT+lzc
liq.
3--How
remnerlng
rcrnpccrure
affccs
ud
notch
strength
of
a
it"iot.ir
iii.t-t
Fis.
5-Fncnrre
analysis
diegram
(FAD).
Tt-.c
OIT
cutrve
is
the
critical
strcs-tempcrature
rcletionrhip
ftor
stopping
a
nrnning
craclc- Orher curves above
rre-
CrtT
cJrve represcnt
consnnt flew sizes and
indicete
combine-
tions
of
stress
and
tempcfilrure
that
rr.ill
cause
frecnrre.
1,200
elontedon
ield
ttrength
(ot
0.2%l
Froclure
Stress
NOT
NDI+60
Tenperolurr
(Fl
8/11/2019 Basic Fracture Mechanic
6/8
gins
wift
the
theory
of elasticity,
but
includes
the
efts:t
of
crack-tip
plasticity.
C;'acx-SrABrt.rry
(GnrrrrrH) TecHNreuei
Con-
sider
an
infinite
sheet
of elastic
material,
subject
to
uniform
biaxial
stress
q
at infinity,
into
which
a cr;rck'of
length
2a ii
introduced-
Potential
enerry
U of
the
system
is:
g'=Uo-Uq-FU1
(l)
where
Uo
is
the
potential
energy
prior
to
introduc-
ing
the
crack,
U.
is the
decrease
in
potential enerry
due
to
deformation
(strain
enerry
and
boundary
forct
work),
and U.
is
the
increase
in
surface
ener
ilJ
due to
the new
surface.
The
potential
ener-
W
r'f
deformation
is:
r o2o2t
tt
E
whr,re
t is
the
sheet
thickness
and
E
is
the
m@ulus
of
elasticily
of
the
material.
The
surface
energ/
is
;imply the
surface
tension
of
the
material,
T,
timci
the
new
crack
surface
area.
Ur
=
latT
Combining
Equations
1,2,
and 3
gives:
t&o2t
Ur=Uo-T*4atT
E
The
minimum
potentiat enerry
\*ith
respect
to
crack
size is
a stable
equilibrium
position,
whereas
the
ma:rimum
potential
energy
is
unstable'
Thus,
the
change
in
the
potential
enerry
with
respect
to
crack
sii-e
(the
first
derivative
of
Equation
4)
is
the erack
driving
force,
G, and
the
resistance,
R"
t&a
G=-=27-R
Cnncr-Tlp
Srness-FlElo
AppnoecH:
The
most
common
fracture
is
the
direct
opening
of a
crack'
where
thp
motion
of the
crack
face
is
normal
to
the
plane
of
the
face
(ModeJ)
,
Fig-
7-
The elastic
streis
field
near
the crack
till
can
be
described
by
the
theory
of
elasticity
(i.e.,
plane
stress
or
plane
strain).
The
analysis
of
the
simple
(Mode
I)
crack
is applicable
to
more
complex
cases (lvlodes
II
and
III)
where
shear
motion
occurs
between
the
crack
faces.
In
general,
a
plane
extensional
linear
elasticity
problem can
be
solved
by
finding
a stress
function
ihat
satisfies
a
suitable
biharmonic
equation
such
that
the
resulting
stresses
and/or
displacements
also
safisfy
the
given
boundary
conditions
of
a
problem.
Forthesimplecrack,theWestergaardfunction'
in
the
form
z
-
x
*
iy,
has
proved
(both
mathe'
matically
and
by
experience)
to
"fi "
this
apprcach
to
fracture
mechanics-
Using
this
function,
t-he
stresses
ctst
oyt
and
r'
are
expresed by:
c,
-
fReallZ
-
Y[Imag]Z'
(6)
.*
o,=
fReallZ*YUmag)Z'
';
--
-Y[Real]Z'
where
function
z
(z)
will
give
stresses
that'
auto'
matically
satisfy
the
elastic
theory.
Thus,
oniy
fur.ction
z@)
that
satisfy
the
boundary
conditions
of
a crack
problem
must
be
found
Near
a crack
tip,
the
adjacent
crack
surfaces
are
:|-::::-f:::,
d :t".f :g
the
character
of
z(z)
in
that
vicinity.
Assuming
a coordinate
origin
at
the
right
end
oi a
crack
parallel
to
the
x-axis,
Fig.
8,
z
be'
comes:
;_
tG)
z*
where
f
(z)
must
approach
a
real
constant'at
the
origin.
-Tttus,
r3,
arrd
c,
appro?ch
zero
at
the
crack
siiriouc
(i,iie
su.-i"ii
;i
siress-frle)
-
However,
S
character
of
f
(z)
at any
distance
from
the
crack
tip
is
unspecified
and
can
be
adiusted
to
solve
-iny
configurations
of
simple
symmetrical
cracks
Therefore,
in
the crack
tiP
region:
(2)
(4)
(o
E
Equation
5
describes
the unstable
condltion
(un-
stairie
-irccause
this
is
a
riiaidtrrtiur
l*irtii
ior
a
craek
in
a
perfectly
brittle
material-
Because
plas-
ticity
and
other
effects
in
the
crack-tip
region
in-
crease
a
material's.
resistance
to
crack
extension,
R
=
2I
is
only
the approximate
resistance
to
crack
propagation.
For
ductile
materials'
t h e
additive
ternrs
are
large
compared
to
2T.
However,
frac-
ture
in
brittle
materials
can
be
assumed
to
be
an
enetry-nxts
exchange
where
the rate
of
elastic
enurry
available,
G,
opposes
the
material's
dissipa-
tion
rate,
R.
Rapid
extension
of
a crack
occurs
when
G
exceeds
R.
(As
a
rule
of
thumb,
tough
materials
have Gs
greater
than
50
ft-lb
per
in-:;
brittle
materials
have
Gs
smaller
than this
value.)
The relation
between crack
stobility
and
energy
rote
is
shown
in Fig.
6. The
available
energy
rate
increases
with
applied
stress. Therefore,
upon
load-
ing.
the
material
state
(brittle
or
ductile)
at any
time
is the
first
intersection
of
the
G
and
R
curves,
and
load
can
be
lncreased
until
the
instability
point
"
I
o,*o
where K1
is
a
real
constant.
If
polar
coordinates
(a
d)
are
fixed at
the
origin,
the stresses become:
(8)
t/
2rz
r,=
KI
.cos
t
(t
tiEz\
r
3t
\
-sin_.16_
I
(9)
22
c f r
3,
\
.
cos-. I I *sin-.
Jin-
I
2
\
2
2t
K1
ty=:
{
2rr
i(r
0
,tu=:
'
3in-
2rr
2
03c
cos-. cos-
aa
hr*
8/11/2019 Basic Fracture Mechanic
7/8
The
factor
K1,
the
crack-tip
stress-intensit5r
fac_
tor,
is
a
constant
in
that
vicinity
and
depen-ds
on
the
body
configuration
and
the
mode
of
loading.
(The
stress-intensity
factor
should
not
be
confused
with
stress-concentration
factors,
Kr,
which
are
a
function
of
part
geometry.)
Because
stresses
in
Iinear
erasticity
depend
liae.
arly on
applied
load,
Equation
g
implies
that
stress-
intensity
factors
must
be,
in
part,
composed of
the
load-
Examining
the
dimensions
of Equation
g
thow-s
that
K, must
also
contain
length
paiameters
for
the
bodies
invorved,
incruding
J.".t
rength
cr-
Thus,
I(r=
e.
o,\l
From
Equation
L-the
enerry
criteria
for
crack
stability
in
brittle
materials-instability
occurs
at
a constant
stress-field
intensity
at
tjre
crack tip.
The
stress
fields
surrounding
tjre
tips
of
simpie
cracks
have
the
same
distribution
and
airrer
galy
in
their
intensity
from
one
case
to
another.
tLri,
unstable crack
grofih
takes
place
when
intensity
Kr reaches
a
critical
value,
Kr".
This value
is
a
material property
and
reflects
a
material's
abifiry
to
withstand
a
given
stress
at a
crack-tip.
This
value
can
be
applied
to
cracks
where
a
small
zone
of plasticity
is
present.
For
a
given
material,
plasticity
always
disturbs
the
elasti.lfiua
equation
(Equation
9)
in
the
same.way,
and
con-
sequently,
failure
occurs
at
an
..apparent"
field
intensit5r,
K1.
Also,
K,
=
Kr" can
-be
used
as
a
"brittle
fracture"
criterion
for
ductile
materials,
if
the
material's
resistance
curve
is
shaped
like
the
one
shown
in
Fig.
6.
This
curve
is iuitable
for
most high-strength
materials,
but
it
is
often iaap
propriatc
for
low
strength
and
Ngh-toughness
ma-
terials-
A summary
of
the stress-intenslry
factors
is
given
in
Table
l.
FrxprNc
Srnrss-h.rreNslTy
Fectons:
Tbe
strex
intensity
factbr,
K,
can
be
ob^iair.ed
for;a:-;-;=
;;.-i
shapes
from
tJre
theory
of
elasticity.
The
-o"i
O-
rect
method
is
to
compute
K
from
the
westergaard
O2
Hotf
Croct
trngth
Fig-
fi-{reck
ertension
cnergy
G
and
_R
vcrsus
cnck
tcngth
_foi
a?i
iJeally
Vhen
stress
is
low,
G
8/11/2019 Basic Fracture Mechanic
8/8
sress
function,
z,
which
must
be
estimated,
on
die
basis
of
experience.
For
specialists
in
the
ileor|
of
elasticity,
guessing
tl,e
functions
is
noc
difficult.
But
for
the
less experienced,
techniques
for
finding
K
for_
two
general
c"ses are
presented
below.
Case
l:
For
the
Griffith
shape
(an
infinite
sheet
with
an
edge
crack
on
the
x-axis,
extending
-a I x
S
o, gdth uniform
bia-.cial
stress,
o, at
in-
nrury)
the
stress
function
is:
oz
L_
(ll)
vv4
where
z
=
x
+
iy,
with
the
origin at
the
center
of
the
crack-
Boundary
stresses,
from
Equation
6
Srei o,
=
s,
=
ol
Txt
=
0, aS
lZl
*
co.
On the
CraCk
surface:
o,
=
r,,
=
0.
If
the
right-hand crack
lip
is
moved
to
the
origin,
z
become
s z
*
a-
o(z
*
o)
(z
* 2a)-n
z*
K,
becomes:
j(12)
Kr
=
o(rra)'A
(13)
which
is
the
constant
of
proportionality
for
Equa-
tion
10.
Case
2:
For
an
infinite sheet having
a
centrally
located
crack
with
a
pair
of
equal
and
opposite
wedge
forces,
P
(force
per
unit
thickness),
at
the
center
of the
crack
prylng
it open,
the
stress
func-
tion
is:
Pa
L-
Toble
I
-stress-lntensity
Foctors
for
Tensile Looding
B:slc Equation
Ky2
-
Qo%a
Modlbdng
Faitor
Q.
Crack
tlrough
infinite
Platej
Iuternal
circular
crack:
Internal
elliptical
cradc
Long
surface
crack (shallow):
Eliptical
surface
craclc
frf
g-J
o
I
l[(62-a\7c21
a
a
o
a
o
I
4lrrz
L/Q2
t.2
1.2162
sinzd dd
L_
From
rz ffi
Whea
this
is
compared
with
Equations
7
and
the
orign
is
relocatd at the crack
becomes:
where
d
is
the
angular
polar
coordinate
Cir-cular
cracls
c
=
crack
radius
Eliptical
cracls:
a
=
Ienglh
of semiminor
axis
c
=
lengtb
of semimajor
axis
Questions
l.
What
parameter distinguishes
tough
materials
from brittle
materials?
2.
On
a
conventional
stress-strain
plot
showing
both
elastic
and
plastic
aclion, which
of
the
following
is
true?
a-
BrittJe
materials
can.
exist
at
any
yield
point
level;
b.
Brittle materials can exist at
any ultimate
. fracture
lqvel-
'
3. Distinguish between
the
stress{oncentration
factor,
Kr,
and
the
stress-intensity
factors, K,
Ro
K1,
K1", etc.
'
RrrrrExcEs
L Y.
\FcEr rnd
S
-kres-"Qrttlcrl
Apprrts.l
of
frcturr
Xcclralc+"
rrprbt troEB srrapostuEo
oD
ltrctur.
Toutlacsr
TsdLDa r-Dd Itl Appllcrttoor,
ASn{,
STP No. 3EL 19Ea
2-
J. L
qhrn;oD,
r.
rud
\tr.
F.
Bmr.l.
Jn-"Protltg
tB ftrctutt
X.cb.lDlcr,"
llacEtxt
DErcx,
ldlr. 5,
19?0.
3.
E- t. Wcucl,
j
rt
-"f,seJrrccrlaf
Mctbodr tor tbc D.:taE
a d.
Sehcuor ol {rtcrlrlr
Atallst
l:leturc."
W*tlaglottsr
Res-
Irb..,
AD
tO10O5, Ju,nr
2{,
196{L
{.
J.
L SbrDloq,
Jr.-"tr:rstun
t{ccbrllcs," }[IcErNr
DtE
to}r,
EcpL
2a. 196?.
5. P.
C. Prdr
r, d
C.
C. Sif"Strcss
Alrtfie
ol Craclr." ttlB
Totglts4tt.
?cttl*g
aad,
Itt Lr?ltcotlo t,
A.gT a,
gIP.
38L
Answers
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(actog
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r{1it'r
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cllsFaloererp
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puE
ssarts
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Jo
slJnpord are
faqa
'dp
4oere
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ssarls
aql aqFosap
pue
'stoleey
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arp
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rlllou
aql
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ssa4s
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ssaJls
uoll3as ssor8
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aql
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oltu
ssaluolsuaulp
aql
sl
,X
'g
('araq pa:ou8J
s1
fltult
leuoluodo:d
pue
luld
plal/(
uaaryrlaq
uoll
-Jul?slp
,(uV)
';(11ec1rse1d
ur:ogap
?ou
ll /v\
ielral
-etu
anllrq
e
'uorllur;ap
.,(g
'arul
sl
q
luauralels
'Z
..'urf,
Jad 8Ja
ro
r'ul
.Iad
q1-11
u1 ,t11ecrd.(1 'earE
tlun
rad
IloA\
'I
)
fl
P
(r4)
and
8,
tip, Kr
(1s)
stress field
dimin-
and running cracks
K1
-
(ra)'A
In
this situation,
the crack:tip
ishes
as
the
crack
gets
longer,
are
sometimes
self-arresting.
Stress-intensity
factors
can
also
be
computed
from
stress-concentration
factors
as
the
notch
radius,
p,
appnlaches
zro.
The
maxirnum
stress
occurs
on
the
notch
and is
proportional
to
the
stress
field
around
iL
Thus:
K1
=
cc.".rpll
(16)
where
C
is a
constanl
For an
ellipticat
hole,
taking
the
limit
as
p
-+
0
gives
C
-
ox/2.
Thus:
.
tta
K1
=
lim-
---
'
glarr
p+0
2
This
may
be used
to
I
crack
pta
(
l7)
lvlodeetermine
K,
for
any
Next lesson
further
of
fracture
mechanics.
I
I
explains the
fundamentals