Embed Size (px)
Citation preview
1
The
10th
Joi
nt D
OD
/NA
SA/F
AA
Con
fere
nce
on A
ging
A
ircra
ftA
pril
16-1
9, 2
007
Num
eric
al S
trip
-Yie
ld C
alcu
latio
nof
CTO
D
Joac
him
Bee
kR
oyce
For
man
NA
SA J
ohns
on S
pace
Cen
ter
Hou
ston
https://ntrs.nasa.gov/search.jsp?R=20070017318 2018-06-10T22:52:10+00:00Z
2
Out
line
CTO
D b
ackg
roun
d
Usi
ng B
ound
ary
Ele
men
ts to
cal
cula
te c
rack
face
di
spla
cem
ents
Theo
ryP
ract
ical
pro
cedu
reE
xam
ple
case
s
Sum
mar
y an
d fu
ture
pla
ns
3
CTO
D b
ackg
roun
d:pl
astic
zon
e si
zes
Irwin
(195
8)LE
FM g
ives
σ~1
/√r;
how
ever
: rea
l mat
eria
ls y
ield
Cra
ck b
ehav
es a
s if
it w
ere
long
er: a
eff=
a+ρ
Pla
stic
zon
e si
ze e
stim
ated
from
stre
ss re
dist
ribut
ion
Dug
dale
(196
0)Y
ield
ing
conf
ined
to n
arro
w s
trip
ahea
d of
cra
ck
(the
“stri
p yi
eld”
mod
el)
Stre
sses
at “
effe
ctiv
e” c
rack
tip
(a+ρ
) are
fini
teY
ield
zon
e lo
adin
g ne
utra
lizes
stre
ss s
ingu
larit
y du
e to
rem
ote
load
ing
plas
tic z
one
size
est
imat
ed fr
om s
ettin
g K(
a+ρ)
=0
4
CTO
D b
ackg
roun
d:pl
astic
zon
e si
zes,
con
t’d
Kno
wle
dge
of ρ
enab
led
deriv
atio
n of
exp
licit
CTO
D
expr
essi
onC
ompl
ex-v
aria
ble
anal
ysis
use
d (n
o fu
ll el
astic
-pla
stic
ana
lysi
s)E
last
ic-p
last
ic b
ehav
ior m
odel
ed b
y su
perp
ositi
on o
f 2 e
last
icso
lutio
ns
Wel
ls (1
963)
CTO
D is
pro
porti
onal
to o
vera
ll te
nsile
stra
in, e
ven
afte
r gen
eral
yi
eldi
ng CTO
D b
ecam
e w
idel
y ac
cept
ed a
s a
usef
ul fr
actu
re c
riter
ion
whe
n ef
fect
s of
the
crac
k tip
pla
stic
zon
e ar
e im
porta
nt
5
CTO
D b
ackg
roun
d:so
me
calc
ulat
ion
met
hods
Dug
dale
’sm
odel
Bas
ed o
n th
in in
finite
pla
te, p
lane
stre
ss, r
emot
e te
nsio
nE
xten
sion
s to
oth
er in
finite
geo
met
ries
limite
d to
a fe
w p
artic
ular
ca
ses
Arb
itrar
y fin
ite g
eom
etrie
s re
quire
tailo
r-m
ade
elas
tic s
olut
ions
Wei
ght f
unct
ion,
gre
en’s
func
tion,
col
loca
tion
met
hods
Dev
elop
ed fo
r par
ticul
ar fi
nite
geo
met
ries
Pot
entia
lly h
eavy
com
puta
tiona
l bur
den
(e.g
. ref
eren
ce s
olut
ions
)
Fini
te e
lem
ents
Gen
eral
-pur
pose
, but
als
o se
vere
com
puta
tiona
l tol
l W
here
beh
ind
the
crac
k tip
to m
easu
re C
TOD
?1st
node
, 2nd
node
, 45°
inte
rcep
t, or
som
e pr
escr
ibed
dis
tanc
eR
e-m
eshi
ng b
urde
n fo
r ana
lyse
s of
mul
tiple
load
s or
cra
cks
6
Usi
ng B
ound
ary
Elem
ents
to c
alcu
late
cr
ack
face
dis
plac
emen
ts: t
heor
y
Dire
ct a
pplic
atio
n of
con
vent
iona
l BE
Ms
to fr
actu
re
prob
lem
s le
ads
to m
athe
mat
ical
ly d
egen
erat
e fo
rmul
atio
nC
ause
: geo
met
ric p
roxi
mity
of c
rack
sur
face
sIn
form
atio
n ab
out c
rack
face
trac
tions
is lo
stC
an c
ircum
vent
by
deve
lopi
ng a
dditi
onal
inte
gral
equ
atio
n fo
r cr
ack
face
trac
tions
One
app
roac
h:D
eriv
e cr
ack
face
trac
tion
equa
tion
from
dis
plac
emen
t eqn
via
Stra
in-d
ispl
acem
ent r
elat
ions
, Hoo
ke’s
law
, lim
iting
pro
cess
Res
ultin
g eq
uatio
n co
ntai
ns h
yper
sing
ular
kern
elR
equi
res
spec
ial i
nter
pret
atio
n; c
halle
ngin
g to
eva
luat
e nu
mer
ical
ly
7
Usi
ng B
ound
ary
Elem
ents
to c
alcu
late
cr
ack
face
dis
plac
emen
ts: t
heor
y (c
ont’d
)
Bet
ter a
ppro
ach
(Pro
f. M
eare
t al,
Uni
vof
Tex
as):
Hyp
ersi
ngul
arity
avoi
ded
by e
limin
atin
g th
e of
fend
ing
term
s in
the
disp
lace
men
t equ
atio
n be
fore
deriv
ing
tract
ion
equa
tion:
A
ppro
pria
te c
hoic
e of
stre
ss fu
nctio
n fo
r the
stre
ss k
erne
l In
tegr
atio
n by
par
ts to
obt
ain
a “m
odifi
ed” d
ispl
acem
ent e
quat
ion
Cra
ck fa
ce tr
actio
n eq
uatio
n is
then
der
ived
as
befo
re
(stra
in-d
ispl
acem
ent r
elat
ions
, Hoo
ke’s
Law
, lim
iting
pro
cess
)
Oth
er p
ract
ical
ben
efits
: sm
all m
esh
size
and
fast
sol
utio
n tim
es
This
is th
e ba
sis
of N
AS
GR
O’s
BE
com
pone
nt
8
Usi
ng B
ound
ary
Elem
ents
to c
alcu
late
cr
ack
face
dis
plac
emen
ts: t
heor
y (c
ont’d
)
Gra
dien
ts o
f rel
ativ
e cr
ack
face
dis
plac
emen
ts Δ
DD
escr
ibed
by
disl
ocat
ion
dens
ity fu
nctio
n A
Ais
app
roxi
mat
ed b
y fu
nctio
ns c
onta
inin
g th
e re
quis
ite s
ingu
larit
yA j
are
noda
l qua
ntiti
es in
th
e ve
ctor
of u
nkno
wns
so
lved
by
NA
SB
EM
Tech
niqu
e im
plem
ente
d in
N
AS
BE
M to
inte
grat
e A
ΔDis
sum
of c
ontri
butio
ns
from
eac
h cr
ack
elem
ent
betw
een
the
tip a
nd th
e po
int o
f int
eres
t
9
NA
SBEM
is N
ASG
RO
’sB
ound
ary
Elem
ent A
naly
sis
mod
ule
NA
SG
RO
is a
n an
alys
is s
oftw
are
suite
with
four
dis
tinct
mod
ules
:Fr
actu
re m
echa
nics
an
d fa
tigue
cra
ck
grow
th a
naly
sis
(NA
SFL
A s
erie
s)Fr
actu
re a
nd fa
tigue
cr
ack
grow
th m
ater
ial
prop
erty
dat
abas
e;
fittin
g of
exp
erim
enta
l da
ta (N
AS
MA
T)2D
bou
ndar
y el
emen
t st
ress
ana
lysi
s an
d st
ress
inte
nsity
fact
or
calc
ulat
ion
(NA
SB
EM
)Fa
tigue
cra
ck
form
atio
n (in
itiat
ion)
an
alys
is (N
AS
FOR
M)
10
NA
SGR
O h
isto
ry
1980
s:N
AS
A/F
LAG
RO
dev
elop
men
t ini
tiate
d to
pro
vide
frac
ture
con
trol
anal
ysis
for m
anne
d sp
ace
prog
ram
sN
AS
A F
ract
ure
Con
trol M
etho
dolo
gy P
anel
form
ed to
sta
ndar
dize
m
etho
ds a
nd m
onito
r NA
SA
/FLA
GR
O d
evel
opm
ent
1990
s:N
AS
A In
tera
genc
y W
orki
ng G
roup
(NA
SA
, DoD
, FA
A, E
SA
) for
med
to
prov
ide
guid
ance
for N
AS
A/F
LAG
RO
dev
elop
men
tA
dditi
onal
NA
SA
, FA
A, U
SA
F su
ppor
t for
agi
ng a
ircra
ft
2000
s:N
AS
A a
nd S
outh
wes
t Res
earc
h In
stitu
te®
sign
Spa
ce A
ct A
gree
men
t fo
r joi
nt N
AS
GR
O d
evel
opm
ent
NA
SG
RO
Indu
stria
l Con
sorti
um fo
rmed
by
Sw
RI;
mem
bers
incl
ude
gove
rnm
ent a
genc
ies
and
indu
stria
l rep
rese
ntat
ives
11
Exam
ple
of ty
pica
l NA
SBEM
use
:O
rbite
r fee
dlin
eflo
wlin
er
Fatig
ue c
rack
s in
flow
liner
(LH
2su
pply
to S
SM
E)
1’ Ø
, 8-1
2’ L
Bel
low
s w
ithin
gi
mba
lling
join
tsFl
owlin
ers
insi
de
bello
ws
to
smoo
th fl
owC
once
rn:
Eng
ine
failu
re d
ue
to d
ebris
Loss
of
mis
sion
or
vehi
cle
NA
SB
EM
use
d to
get
Kvs
a
12
Usi
ng N
ASB
EM to
cal
cula
te C
TOD
: pr
oced
ure
Use
NA
SB
EM
to c
onst
ruct
mod
el“M
athe
mat
ical
” cra
ck c
onsi
stin
g of
Phy
sica
l cra
ck a
Coh
esiv
e lo
ad z
one ρ
App
lied
load
ing
Coh
esiv
e yi
eld
load
ing
Follo
win
g D
ugda
le’s
idea
Pla
stic
zon
e is
siz
ed s
o th
at
Kdu
e to
coh
esiv
e lo
adin
g ca
ncel
s K
due
to a
pplie
d lo
adin
g:K σ
y =
-Kσ
13
Usi
ng N
ASB
EM to
cal
cula
te C
TOD
: pr
oced
ure
(con
t’d)
For a
giv
en y
ield
stre
ss σ
Y, a
chie
ve
K(a+ρ)
= K
σy+
K σ=
0by
set
ting
the
plas
tic z
one
size
ρan
d ite
ratin
g on
the
rem
ote
stre
ss σ
adva
ntag
e: n
o ne
ed to
rem
esh
whi
le it
erat
ing
or b
y se
tting
the
rem
ote
stre
ss σ
and
itera
ting
on ρ
adva
ntag
e: C
TOD
obt
aine
d fo
r sp
ecifi
c va
lues
of σ
CTO
D v
alue
is g
iven
by
crac
k fa
ce
disp
lace
men
t at t
ip o
f phy
sica
l cra
ck a
14
Usi
ng N
ASB
EM to
cal
cula
te C
TOD
: re
sults
Mes
hQ
uadr
atic
bou
ndar
y el
emen
ts, l
inea
r cra
ck e
lem
ents
Sm
alle
r mes
h si
ze th
an o
ther
BE
M fo
rmul
atio
nsTy
pica
l err
or <
3% w
ith 2
0 el
emen
ts o
r les
s pe
r bou
ndar
y or
cra
ckC
rack
face
load
ing
disc
ontin
uity
requ
ires
a fin
er m
esh
Fast
resu
lts (e
xam
ple
case
s ru
n in
2-3
sec
onds
)
Con
figur
atio
ns s
tudi
edC
ente
r cra
ck in
fini
te a
nd in
finite
she
ets
Edg
e cr
ack
in fi
nite
she
etC
rack
s fro
m h
oles
in in
finite
she
ets
Per
iodi
c cr
acks
in in
finite
she
et3-
hole
tens
ion
spec
imen
15
CTO
D v
erifi
catio
n ca
se:
repr
oduc
ing
Dug
dale
’sre
sult
Firs
t ver
ifica
tion
case
Rep
rodu
cing
Dug
dale
’sm
odel
re
ally
sho
uld
wor
k!
Dug
dale
mod
el c
onsi
sts
ofC
ente
r cra
ck in
infin
ite p
late
Rem
ote
unifo
rm te
nsio
n C
ohes
ive
yiel
d st
ress
on
crac
k fa
ces
near
cra
ck ti
ps
16
CTO
D v
erifi
catio
n ca
se:
repr
oduc
ing
Dug
dale
’sre
sult,
con
t’d
Res
ults
virt
ually
iden
tical
ov
er w
ide
rang
e of
σ/σ
Y
4 si
gnifi
cant
dig
itsno
n-un
iform
erro
r du
e to
man
ually
ite
ratin
g to
K=0
σ/ σ
YC
TOD
/(σYa
/E)
BEM
CTO
D/(σ
Ya/E
)D
ugda
leEr
ror
(%)
0.16
0.08
100.
0813
0.37
0.24
0.18
450.
1854
0.49
0.32
0.33
620.
3362
00.
400.
5397
0.53
970
0.48
0.81
000.
8050
0.62
0.56
1.14
691.
1470
0.01
0.64
1.58
901.
5890
00.
722.
1740
2.17
400
0.80
2.99
002.
9900
00.
884.
2650
4.26
400.
020.
967.
0620
7.04
900.
18
17
CTO
D v
erifi
catio
n ca
se:
edge
cra
ck
Ver
ifica
tion
case
: C(T
) spe
cim
en w
ith W
=3, a
=1P
last
ic z
one
size
and
CTO
D w
ere
calc
ulat
ed (ρ
is s
how
n he
re)
Exc
elle
nt a
gree
men
t with
col
loca
tion
resu
lts b
y N
ewm
an &
Mal
l, an
d al
so T
erad
a (b
oth
1983
)
0.00
00
0.20
00
0.40
00
0.60
00
0.80
00
1.00
00
1.20
00
1.40
00
0.00
0.10
0.20
0.30
0.40
0.50
P/σ
Y
ρ
BE
New
man
& M
all
18
CTO
D v
erifi
catio
n ca
se:
1 cr
ack
from
a h
ole,
infin
ite p
late
Ver
ifica
tion
case
: 1 c
rack
from
a h
ole
in a
n in
finite
pla
te
unde
r rem
ote
unia
xial
load
ing
Pla
stic
zon
e si
ze c
alcu
late
dE
xcel
lent
cor
rela
tion
to a
naly
tical
resu
lts b
y R
ich
(com
plex
-va
riabl
e an
alys
is w
ith c
onfo
rmal
map
ping
, 196
8)
19
CTO
D v
erifi
catio
n ca
se:
2 cr
acks
from
a h
ole,
infin
ite p
late
Ver
ifica
tion
case
: pla
stic
zon
e si
ze s
tudi
ed fo
r var
ious
val
ues
of a
/RD
iffer
ence
bet
wee
n N
AS
BE
M a
nd R
ich
< 2.
5%La
rger
ρin
sm
all c
rack
s du
e to
hig
her s
tress
con
cent
ratio
n at
hol
eS
olut
ion
appr
oach
es D
ugda
leso
lutio
n fo
r lar
ge a
/R
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00 0.0
00.2
00.4
00.6
00.8
01.0
0σ/σ Y
ρ/c
a/R=0
.5 (B
E)a/R
=0.5
(Rich
)a/R
=1 (B
E)a/R
=1 (R
ich)
a/R=2
(BE)
a/R=2
(Rich
)
20
CTO
D v
erifi
catio
n ca
se:
perio
dic
crac
ks in
an
infin
ite s
heet
Pra
ctic
al c
onsi
dera
tions
:H
ow to
mod
el a
n in
finite
num
ber o
f cra
cks?
7 cr
acks
see
ms
a go
od a
ppro
xim
atio
n --
idea
take
n fro
m li
tera
ture
on
mod
ellin
gla
rge
arra
ys o
f fus
elag
e fa
sten
ers
BE
com
pare
d to
Tad
a (W
este
rgaa
rdst
ress
func
tion,
197
4)
21
CTO
D v
erifi
catio
n ca
se:
perio
dic
crac
ks in
an
infin
ite
shee
t, co
nt’d
Exc
elle
nt c
orre
latio
n fo
r pla
stic
zon
e si
ze v
app
lied
stre
ss
22
CTO
D v
erifi
catio
n ca
se:
perio
dic
crac
ks in
an
infin
ite s
heet
, co
nt’d
Exc
elle
nt c
orre
latio
n fo
r CTO
D v
pla
stic
zo
ne s
ize
23
CTO
D c
alcu
latio
ns:
cent
er c
rack
in fi
nite
-wid
th s
heet
BE
com
pare
d to
te
st d
ata
(For
man
, 19
66)
Test
s on
0.0
20”
AM
350C
RT
stee
l sh
eet f
or to
ughn
ess
varia
tion
with
sp
ecim
en s
ize
Pla
stic
zon
e si
zes
wer
e m
easu
red
phot
ogra
phic
ally
NA
SB
EM
com
pare
s w
ell w
ith te
st d
ata
24
CTO
A c
alcu
latio
ns
Usi
ng N
AS
BE
M a
s a
fract
ure
pred
icto
r:C
rack
tip
open
ing
angl
e (C
TOA
) has
bee
n no
ted
by m
any
to b
e a
usef
ul fr
actu
re c
riter
ion
CTO
A is
cal
cula
ted
at ~
0.04
” (1
mm
) beh
ind
the
crac
k tip
Com
paris
ons
to a
naly
tical
resu
lts o
n pr
evio
us p
ages
wer
e fo
r CTO
D a
t cra
ck ti
p (“δ 5
”) –
not a
pra
ctic
al lo
catio
n fo
r rea
l m
easu
rem
ents
CTO
A =
2 *
tan-1
(CTO
D/2
x), w
here
x is
dis
tanc
e be
hind
cra
ck ti
p
Com
paris
ons
for
M(T
) spe
cim
en: A
l 707
5-T6
, Al 2
024-
T81
3-ho
le te
nsio
n sp
ecim
en: A
l 707
5
25
CTO
A c
alcu
latio
ns:
cent
er c
rack
in fi
nite
-wid
th s
heet
BE
com
pare
d to
test
da
ta (F
orm
an, 1
966)
M(T
) spe
cim
ens,
0.0
60”
shee
tA
l 707
5-T6
, 202
4-T8
1
Idea
was
to s
ee if
ca
lcul
ated
CTO
D o
r C
TOA
was
reas
onab
ly
cons
tant
ove
r cra
ck
size
a
look
s go
od fo
r 707
5le
ss s
o fo
r 202
4
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
01
23
45
a
CTOA70
75-T
6
2024
-T81
26
CTO
A c
alcu
latio
ns:
3-ho
le te
nsio
n sp
ecim
en
3-ho
le te
nsio
n (T
HT)
spe
cim
en s
imul
ates
Kfo
r a
crac
ked
stiff
ened
pan
elK
curv
e ta
ken
from
AS
TM S
TP 8
96 (1
985)
27
CTO
A c
alcu
latio
ns:
3-ho
le te
nsio
n sp
ecim
en
CTO
A c
alcu
late
d fo
r fai
lure
load
s ta
ken
from
AS
TM ro
und-
robi
n on
ex
perim
enta
l and
pre
dict
ive
fract
ure
anal
ysis
met
hods
(AS
TM
STP
896
)K
and
cal
cula
ted
CTO
A s
how
that
TH
T is
a c
ompl
ex c
onfig
urat
ion
Wor
k is
in p
rogr
ess
28
Sum
mar
y
Man
y ex
istin
g m
etho
ds to
cal
cula
te C
TOD
can
be
cost
ly a
nd
com
plic
ated
, or a
pply
onl
y to
par
ticul
ar c
onfig
urat
ions
A n
ew n
umer
ical
met
hod
for c
alcu
latin
g C
TOD
was
in
vest
igat
edN
AS
GR
O’s
Bou
ndar
y E
lem
ent m
odul
e N
AS
BE
M w
as a
dapt
ed to
ca
lcul
ate
disp
lace
men
ts a
t any
poi
nt o
n th
e cr
ack
Dem
onst
rate
d fo
r a n
umbe
r of c
rack
con
figur
atio
ns:
finite
and
infin
ite d
omai
nsce
nter
and
edg
e cr
acks
com
plex
cas
es w
ith s
ever
al c
rack
s an
d ho
les
Gre
at a
ccur
acy
at m
inim
al c
ompu
tatio
nal c
ost
29
Futu
re
Stil
l a w
ork
in p
rogr
ess:
CTO
A in
vest
igat
ed …
mor
e w
ork
need
s to
be
done
Is K
cco
rrec
ted
for D
ugda
lepl
astic
zon
e si
ze a
bet
ter
fract
ure
crite
rion
than
Kc(a
)alo
ne, o
r Kc
corr
ecte
d fo
r Irw
in p
last
ic z
one
size
? M
ulti-
site
dam
age
issu
es to
be
inve
stig
ated
Stra
in h
arde
ning
--ea
sy to
mod
el
CTO
D c
apab
ility
is c
urre
ntly
stil
l a re
sear
ch to
olTu
rn c
apab
ility
into
pro
duct
ion-
leve
l too
lIm
plem
ent a
utom
atio
n of
CTO
D c
alcu
latio
nsM
anua
l mes
hing
and
con
verg
ence
to K
=0fo
r mul
tiple
cra
cks
is
tedi
ous!
