of introduction to theory ofintroduction theory technical...
TRANSCRIPT
![Page 1: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/1.jpg)
Intr
oduc
tion
toTh
eory
ofIn
trod
uctio
n to
The
ory
of
Elas
ticity
2011
Sum
mer
Ken
go N
akaj
ima
Tech
nica
l & S
cien
tific
Com
putin
g I (
4820
-102
7)S
emin
ar o
n C
ompu
ter S
cien
ce (4
810-
1204
)
![Page 2: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/2.jpg)
elas
t2
•Th
eory
of E
last
icity
yy
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
![Page 3: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/3.jpg)
elas
t3
Theo
ryof
Ela
stic
ityTh
eory
of E
last
icity
•C
ontin
uum
Mec
hani
cs,S
olid
Mec
hani
csC
ontin
uum
Mec
hani
cs, S
olid
Mec
hani
cs•
Ela
stic
Mat
eria
lTh
eory
ofE
last
icity
Ela
stom
echa
nics
–Th
eory
of E
last
icity
, Ela
stom
echa
nics
![Page 4: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/4.jpg)
elas
t4
Wha
tis
Ela
stic
Mat
eria
l?W
hat i
s E
last
ic M
ater
ial ?
•D
efor
mat
ion
is p
ropo
rtion
al
to lo
ad
–H
ooke
’s la
w–
Exa
mpl
e
Load
•S
prin
gkx
= -m
g•
Met
al, F
iber
, Res
in
–If
load
is re
mov
ed, d
efor
mat
ion
goes
to0
Def
orm
atio
n
goes
to 0
.•
Orig
inal
sha
pe
![Page 5: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/5.jpg)
elas
t5
If lo
ad (d
efor
mat
ion)
incr
ease
s,
()
mat
eria
l is
not e
last
ic a
ny m
ore
•Y
ield
–Y
ield
poin
tY
ield
poi
nt–
Ela
stic
lim
it
Load
Yiel
d P
oint
•In
elas
tic•
Pla
stic
Def
orm
atio
n
![Page 6: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/6.jpg)
elas
t6
Def
orm
atio
n do
es n
ot g
o to
0 w
ith
gre
mov
ed lo
ad, a
fter e
last
ic li
mita
tion.
•
Initi
al s
hape
is n
ot
reco
vere
dan
ym
ore
reco
vere
d an
y m
ore.
•P
erm
anen
t def
orm
atio
n
Load
Yiel
d P
oint
Def
orm
atio
nP
erm
anen
t D
efor
mat
ion
Def
orm
atio
n
![Page 7: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/7.jpg)
elas
t7
Theo
ry o
f Ela
stic
ity c
over
s …
yy
•U
pto
Yie
ldP
oint
,Ela
stic
Up
to Y
ield
Poi
nt, E
last
ic
Lim
itatio
n–
Sm
alld
efor
mat
ion
Sm
all d
efor
mat
ion
–In
finite
sim
al th
eory
•S
hape
doe
s no
t cha
nge
Load
Sap
edo
esot
ca
ge
–Li
near
•P
last
ic/In
elas
tic⇒
Non
linea
r•
Pla
stic
/Inel
astic
⇒N
onlin
ear
–M
ore
inte
rest
ing
part
of re
sear
chE
lti
iti
it
titi
li
i
Def
orm
atio
n
•E
last
icity
is m
ore
impo
rtant
in p
ract
ical
eng
inee
ring
–To
con
trol l
oad/
defo
rmat
ion
belo
w e
last
ic li
mita
tion
is
it
tim
porta
nt–
Pla
stic
/Inel
astic
: Acc
iden
t con
ditio
n
![Page 8: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/8.jpg)
elas
t8
•Th
eory
of E
last
icity
yy
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
![Page 9: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/9.jpg)
elas
t9
Stre
ss(1
/6)
Stre
ss (1
/6)
•If
exte
rnal
forc
eis
elas
ticbo
dyth
ebo
dyde
form
s•
If ex
tern
al fo
rce
is e
last
ic b
ody,
the
body
def
orm
s,
and
resi
sts
agai
nst e
xter
nal f
orce
by
inte
rnal
forc
e ge
nera
ted
byin
term
olec
ular
forc
esge
nera
ted
by in
term
olec
ular
forc
es.
•D
efor
mat
ion
of th
e bo
dy re
ach
stea
dy s
tate
, whe
n t
lfd
it
lfb
ld
exte
rnal
forc
e an
d in
tern
al fo
rce
are
bala
nced
.•
Ext
erna
l For
ce–
Sur
face
forc
e–
Bod
y fo
rce
•E
xter
nal/I
nter
nal f
orce
s ar
e ve
ctor
s.
![Page 10: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/10.jpg)
elas
t10
Stre
ss(2
/6)
Stre
ss (2
/6)
•A
nel
astic
body
inun
derb
alan
ced
cond
ition
with
•A
n el
astic
bod
y in
und
er b
alan
ced
cond
ition
with
ex
tern
al fo
rces
at “
n” p
oint
s.
P
P n-1
P 1
P n
P 2
n
![Page 11: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/11.jpg)
elas
t11
Stre
ss(3
/6)
Stre
ss (3
/6)
•If
we
assu
me
anar
bitra
rysu
rface
Sin
tern
alfo
rce
•If
we
assu
me
an a
rbitr
ary
surfa
ce S
, int
erna
l for
ce
betw
een
part-
Aan
d pa
rt-B
acts
on
thro
ugh
surfa
ce S
.
P
P n-1
P 1
P n
AB
P 2
n
S
![Page 12: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/12.jpg)
elas
t12
Stre
ss(4
/6)
Stre
ss (4
/6)
•C
onsi
ders
mal
lsur
face
ΔSon
surfa
ceS
ofpa
rtA
•C
onsi
der s
mal
l sur
face
ΔS
on s
urfa
ce S
of p
art-A
, an
d re
sulta
nt fo
rce
vect
or ΔF
Ifi
idd
df
•If p
is c
onsi
dere
d as
ave
rage
d fo
rce
per a
rea
ΔF/ΔS
with
infin
itesi
mal
ΔS,
pis
cal
led
“stre
ss
t”
vect
or”
P n-1
ΔF
SS
ΔΔ=
→Δ
Fp
0lim
P n
AΔ
S
n
S
![Page 13: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/13.jpg)
elas
t13
Stre
ss(5
/6)
Stre
ss (5
/6)
•S
tress
: For
ce V
ecto
r per
Uni
t Sur
face
p–
Pos
itive
for e
xten
sion
, neg
ativ
e fo
r com
pres
sion
•O
na
surfa
ceO
n a
surfa
ce–
Nor
mal
: Nor
mal
stre
ss)
–P
aral
lel:
She
arst
ress)
–P
aral
lel:
She
ar s
tress)
•“Y
ield
Stre
ss” i
s an
impo
rtant
des
ign
para
met
er.
P n-1
ΔF
SS
ΔΔ=
→Δ
Fp
0lim
P n
AΔ
S
n
S
![Page 14: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/14.jpg)
elas
t14
Stre
ss(6
/6)
Stre
ss (6
/6)
•S
tress
com
pone
nts
inor
thog
onal
coor
dina
te•
Stre
ss c
ompo
nent
s in
orth
ogon
al c
oord
inat
e sy
stem
9t
i3D
–9
com
pone
nts
in 3
D–
norm
al s
tress
σh
t–
shea
r stre
ssτ
⎫⎧
{}
⎪⎪ ⎬⎫
⎪⎪ ⎨⎧
=yz
yyx
xzxy
x
τσ
ττ
τσ
σ⎪ ⎭
⎪ ⎩z
zyzx
στ
τ
![Page 15: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/15.jpg)
elas
t15
•Th
eory
of E
last
icity
yy
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
![Page 16: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/16.jpg)
elas
t16
Gov
erni
ngE
quat
ions
inTh
eory
ofG
over
ning
Equ
atio
ns in
The
ory
of
Ela
stic
ity
•E
quili
briu
m E
quat
ions
•C
ompa
tibili
ty C
ondi
tions
–D
ispl
acem
ent-S
train
p•
Con
stitu
tive
Equ
atio
ns–
Stre
ss-S
train
–S
tress
-Stra
in
2Dl
•2D
exa
mpl
e
![Page 17: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/17.jpg)
elas
t17
Equ
ilibr
ium
dy
yyxyx
∂∂+
ττ
dyyy
y∂∂
+σ
σdy
yyxyx
∂∂+
ττ
dyyy
y∂∂
+σ
σq Equ
atio
nsi
Xi
dxdy
dxxxy
xy∂∂
+τ
τ
y∂
dxdy
dxxxy
xy∂∂
+τ
τ
y∂
in X
-axi
sIn
finite
sim
alG
xσ
y
xyτ
dxxx
x∂∂
+σ
σG
xσ
y
xyτ
dxxx
x∂∂
+σ
σ
Infin
itesi
mal
Ele
men
ty
y
yxτy
y
yxτ
⎞⎛
∂σx
yσ
zx
yσ
z
11
⎞⎛
∂
×⋅
−×
⎟ ⎠⎞⎜ ⎝⎛
∂∂+
dydy
dxx
xx
x
τ
σσ
σ
01
11
=×
⋅⋅
+×
⋅−
×⎟⎟ ⎠⎞
⎜⎜ ⎝⎛∂∂
++
dydx
Xdx
dxdy
yyx
yxyx
ττ
τB
ody
Forc
e
0=
+∂∂
+∂∂
Xy
xyx
xτ
σin
X-d
irect
ion
![Page 18: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/18.jpg)
elas
t18
dyyyx
yx∂∂
+τ
τdy
yyy
∂∂+
σσ
dyyyx
yx∂∂
+τ
τdy
yyy
∂∂+
σσ
Equ
ilibr
ium
dx
dydx
xxyxy
∂∂+
ττ
y∂
dxdy
dxxxy
xy∂∂
+τ
τ
y∂q Equ
atio
nsi
Yi
Gx
σy
xyτ
dxxx
x∂∂
+σ
σG
xσ
y
xyτ
dxxx
x∂∂
+σ
σin
Y-a
xis
Infin
itesi
mal
y
y
yxτy
y
yxτ
Infin
itesi
mal
Ele
men
t ⎟⎞⎜⎛
∂σx
yσ
zx
yσ
z
11
⎞⎛
∂
×⋅
−×
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∂∂+
dxdx
dyy
yy
yσ
σσ
01
11
=×
⋅⋅
+×
⋅−
×⎟⎟ ⎠⎞
⎜⎜ ⎝⎛∂∂
++
dydx
Ydy
dydx
xxy
xyxy
ττ
τB
ody
Forc
e
0=
+∂∂
+∂∂
Yx
yxy
yτ
σin
Y-d
irect
ion
![Page 19: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/19.jpg)
elas
t19
dyyyx
yx∂∂
+τ
τdy
yyy
∂∂+
σσ
dyyyx
yx∂∂
+τ
τdy
yyy
∂∂+
σσ
dxdy
dxxxy
xy∂∂
+τ
τ
y∂
dxdy
dxxxy
xy∂∂
+τ
τ
y∂
Mom
ent a
roun
dZ
iG
xσ
y
xyτ
dxxx
x∂∂
+σ
σG
xσ
y
xyτ
dxxx
x∂∂
+σ
σZ-
axis
atpo
int-G
y
y
yxτy
y
yxτ
atpo
intG
xy
σz
xy
σz xy
xyxy
dxdy
dxdy
dxx
ττ
τ×
×+
××
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∂∂+
21
21
yxyx
yxdy
dxdy
dxdy
yτ
ττ
=×
×−
××
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∂∂+
−
⎠⎝
02
12
1
yxxy
yτ
τ=
∴⎠
⎝∂
22
![Page 20: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/20.jpg)
elas
t20
Equ
ilibr
ium
Equ
atio
nsin
2DE
quili
briu
m E
quat
ions
in 2
D
0=
+∂∂
+∂∂
Xy
xxy
xτ
σ
0=
+∂∂
+∂∂
∂∂
Yx
y
yx
xyy
τσ
∂∂
xy
![Page 21: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/21.jpg)
elas
t21
Equ
ilibr
ium
Equ
atio
nsin
3D
6
Inde
pend
ent S
tress
Com
pone
nts
⎫⎧
ττ
{}
⎪⎪ ⎬⎫
⎪⎪ ⎨⎧
=yz
yxy
zxxy
x
τσ
ττ
τσ
σzy
yz
yxxy
ττ
ττ
==
⎪ ⎭⎪ ⎩
zyz
zxσ
ττ
xzzx
ττ
=
0=
+∂∂
+∂∂
+∂∂
Xz
yx
zxxy
xτ
τσ
0=
+∂∂
+∂∂
+∂∂
∂∂
∂
Y
zy
x
yzy
xyτ
στ
0=
+∂
+∂
+∂
∂∂
∂
Z
zy
x
zyz
zxσ
ττ
0+
∂+
∂+
∂Z
zy
x
![Page 22: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/22.jpg)
elas
t22
Wha
tis
“Stra
in”?
Wha
t is
Stra
in ?
•S
olid
Mec
hani
cs•
Sol
id M
echa
nics
–Lo
ad –
Def
orm
atio
nS
•S
tress
–Lo
ad/F
orce
per
uni
t sur
face
•S
train
–R
ate
of D
efor
mat
ion/
Dis
plac
emen
tp
![Page 23: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/23.jpg)
elas
t23
Stra
in:R
ate
ofD
ispl
acem
ent
Stra
in: R
ate
of D
ispl
acem
ent
•N
orm
alst
rain
Nor
mal
stra
in LΔL
LLΔ=
ε
•S
hear
stra
inΔΔx
xΔγ
LL
=γ
![Page 24: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/24.jpg)
elas
t24
Stra
in-D
ispl
acem
ent
p•
Dis
plac
emen
t in
3D: (
u, v
, w)
f2D
Ifi
iti
lEl
t•
for 2
D In
finite
sim
al E
lem
ent
–B
efor
e D
efor
mat
ion:
P, Q
, R, A
fter D
efor
mat
ion:
P’,
Q’,
R’
R’
),
(:P
yx
RR’
),
(:R
),
(:Q
dyy
xy
dxx
++
dQ’
yu
∂∂
/
dy
P’
Q’
)(
),
(:P'
dv
du
d
vy
ux
∂∂
++
y
dxP
Q)
(R
'
),
(:Q
'
dv
dd
u
dxxv
vy
dxxu
udx
x
∂∂
∂∂+
+∂∂
++
+x
v∂
∂/
xz
dxQ
),
(:R
'dy
yv
dyy
dyy
ux
∂+
++
∂+
+
![Page 25: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/25.jpg)
elas
t25
Nor
mal
Stra
in -
Dis
plac
emen
tp
•PQ
⇒P’
Q’
()
dxu
xdx
uu
dxx
∂−
⎭⎬⎫
⎩⎨⎧+
− ⎟ ⎠⎞⎜ ⎝⎛
∂∂+
++
()
xudxx
ε x∂∂
=⎭⎬
⎩⎨⎠
⎝∂
=
RR’
u∂
dQ’
yu
∂∂
/vxu
ε x
∂∂∂=
dy
P’
Q’
wyvε y
∂∂∂=
y
dxP
Q
xv∂
∂/
zwε z
∂∂=
xz
dxQ
![Page 26: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/26.jpg)
elas
t26
She
ar S
train
-D
ispl
acem
ent
p
RR’
xvyu
xy∂∂
+∂∂
=γ
dQ’
yu
∂∂
/yw
zvx
y
yz∂∂
+∂∂
=
∂∂
γdy
P’
Q’
zuxw
yz
zx∂∂
+∂∂
=
∂∂
γy
dxP
Q
xv∂
∂/
zx
∂∂
xz
dxQ
![Page 27: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/27.jpg)
elas
t27
Com
patib
ility
Con
ditio
nsp
y
∂∂
∂2
22
•2D
yx
xy
xyy
x
∂∂∂
=∂∂
+∂∂
γε
ε2
2
2
2
2
•3D
∂∂
∂∂
∂∂
∂∂
∂γ
εε
γε
εγ
εε
22
22
22
22
2
xz
zx
zy
yz
yx
xy
zxx
zyz
zy
xyy
x
∂∂∂
=∂∂
+∂∂
∂∂∂
=∂∂
+∂∂
∂∂∂
=∂∂
+∂∂
γε
εγ
εε
γε
ε2
22
22
2,
,
⎞⎛
∂∂
∂∂
∂γ
γ2
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∂∂+
∂∂+
∂∂−
∂∂=
∂∂∂
zy
xx
zy
xyzx
yzx
γγ
γε2
2
⎞⎛
∂∂
∂∂
∂2
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∂∂+
∂∂+
∂∂−
∂∂=
∂∂∂
xz
yy
xz
yzxy
zxy
γγ
γε2
2
⎞⎛
∂∂
2
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∂∂+
∂∂+
∂∂−
∂∂=
∂∂∂
yx
zz
yx
xzyz
xyz
γγ
γε2
2
![Page 28: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/28.jpg)
elas
t28
Con
stitu
tive
Eqn
’s: S
tress
-Stra
in(1
/3)
q(
)•
You
ng’s
Mod
ulus
ES
tress
Stra
in:P
ropo
rtion
al–
Stre
ss-S
train
: Pro
porti
onal
–P
ropo
rtion
ality
: E (d
epen
ds o
n m
ater
ial)
EE
xx
xx
σε
εσ
==
,
Poi
sson
’sR
atio
xy
νεε
−=
•P
oiss
on’s
Rat
ioν
–B
ody
defo
rms
in Y
-and
Z-
dire
ctio
nsev
enif
exte
rnal
forc
eis
σxε
dire
ctio
ns, e
ven
if ex
tern
al fo
rce
is
in X
-dire
ctio
n.–
Poi
sson
’sra
tiois
prop
ortio
nalit
yfo
rx
σx
–P
oiss
ons
ratio
is p
ropo
rtion
ality
for
this
late
ral s
train
.•
depe
nds
on m
ater
ial
σp –
Met
al: 0
.30
–R
ubbe
r, W
ater
: 0.5
0 (in
com
pres
sibl
e)Ex
xy
σν
νεε
−=
−=
![Page 29: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/29.jpg)
elas
t29
Con
stitu
tive
Eqn
’s: S
tress
-Stra
in(2
/3)
q(
)•
Effe
ct o
f nor
mal
stre
ss c
ompo
nent
s in
3 d
irect
ions
()
(σ x
,σy,σ
z)
–ac
cum
ulat
ion
of e
ach
stra
in c
ompo
nent
()
{}
zy
xz
yx
xE
EE
Eσ
σν
σσ
νσ
νσ
ε+
−=
−−
=1
()
{}
()
{}
xz
yx
zy
y
zy
xx
EE
EE
EE
EE
σσ
νσ
σν
σν
σε
+−
=−
−=
1(
){
}
()
{}
yx
zy
xz
z
xz
yy
EE
EE
EE
EE
σσ
νσ
σν
σν
σε
+−
=−
−=
1(
){
}y
xz
zE
EE
E
![Page 30: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/30.jpg)
elas
t30
Con
stitu
tive
Eqn
’s: S
tress
-Stra
in(3
/3)
q(
)•
She
ar s
train
com
pone
nts
do n
ot d
epen
d on
l
tt
Thti
lno
rmal
stre
ss c
ompo
nent
s. T
hey
are
prop
ortio
nal
to s
hear
stre
ss τ
.
ττ
τ
–La
tera
l Ela
stic
Mod
ulus
: G
GG
Gzx
zxyz
yzxy
xyτ
γτ
γτ
γ=
==
,,
E ()
ν+=
12
EG
![Page 31: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/31.jpg)
elas
t31
Stre
ss-S
train
Rel
atio
nshi
pp
⎪⎪⎫
⎪⎪⎧ ⎥⎥⎤
⎢⎢⎡−
−−
−
⎪⎪⎫
⎪⎪⎧x
x
σσν
νν
νεε
00
01
00
01
()
⎪⎪⎪⎪ ⎬⎪⎪⎪⎪ ⎨ ⎥⎥⎥⎥
⎢⎢⎢⎢
+−
−=
⎪⎪⎪⎪ ⎬⎪⎪⎪⎪ ⎨
zy
zy
Eτσσ
νν
νν
ν
γεε
00
12
00
00
00
10
00
11
()
()
()
⎪⎪⎪⎪ ⎭⎪⎪⎪⎪ ⎩⎥⎥⎥⎥ ⎦
⎢⎢⎢⎢ ⎣
++
⎪⎪⎪⎪ ⎭⎪⎪⎪⎪ ⎩
yzxy
yzxyE
ττν
νγγ
12
00
00
00
12
00
00
00
12
00
0
()
⎪ ⎭⎪ ⎩⎥ ⎦
⎢ ⎣+
⎪ ⎭⎪ ⎩
zxzx
τν
γ1
20
00
00
![Page 32: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/32.jpg)
elas
t32
Stra
in-S
tress
Rel
atio
nshi
pp
⎤⎡−
νν
ν0
00
1
⎪⎪⎪⎫
⎪⎪⎪⎧ ⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−−
⎪⎪⎪⎫
⎪⎪⎪⎧
yx
yx
εε
νν
νν
νν
σσ
10
00
10
00
1
()(
)(
)
()
⎪⎪⎪⎪ ⎬
⎪⎪⎪⎪ ⎨ ⎥⎥⎥⎥
⎢⎢⎢⎢−
−+
=
⎪⎪⎪⎪ ⎬
⎪⎪⎪⎪ ⎨xyz
xyzE
γε
ν
νν
ντσ
02
11
00
00
00
21
210
00
21
1(
)
()
⎪⎪⎪ ⎭⎪⎪⎪ ⎩ ⎥⎥⎥⎥ ⎦
⎢⎢⎢⎢ ⎣−
−
⎪⎪⎪ ⎭⎪⎪⎪ ⎩
zxyz
zxyz
γγ
ν
ν
ττ
21
210
00
00
02
12
00
00
⎦⎣
2
[]
D
{}
[]{} ε
σD
=
•In
com
pres
sibl
e M
ater
ial(ν~
0.50
): S
peci
al
Trea
tmen
t Nee
ded
![Page 33: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/33.jpg)
elas
t33
Som
e A
ssum
ptio
ns in
this
Cla
ssp
•Is
otro
pic
Mat
eria
lp
–U
nifo
rm E
, and
ν(~
0.30
)–
CFR
P (C
arbo
n Fi
ber R
einf
orce
d P
last
ics)
()
•O
rthot
ropi
c
![Page 34: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/34.jpg)
elas
t34
Fini
te-E
lem
ent M
etho
d•
Dis
plac
emen
t-bas
ed F
EM
p–
Dep
ende
nt V
aria
ble:
Dis
plac
emen
t•
Gen
eral
ly u
sed
appr
oach
–Th
is c
lass
ado
pts
this
app
roac
h•
Stre
ss-b
ased
FE
M–
Dep
ende
nt V
aria
ble:
Stre
ss
![Page 35: of Introduction to Theory ofIntroduction Theory Technical ...nkl.cc.u-tokyo.ac.jp/11s/english/intro/elast.pdf · Elasticity 2011 Summer Kengo Nakajima Technical & Scientific Computing](https://reader031.vdocuments.mx/reader031/viewer/2022022502/5aae0baa7f8b9a25088bc30b/html5/thumbnails/35.jpg)
elas
t35
1D P
robl
em•
Ext
ensi
on o
f 1D
trus
s el
emen
tel
emen
t –
only
def
orm
s in
X-d
ir.U
nifo
rmse
ctio
nala
rea
AF
–U
nifo
rm s
ectio
nal a
rea
A–
You
ng’s
Mod
ulus
E0@
X0
Ext
erna
lFor
ce–
u=0@
X=0,
Ext
erna
l For
ce
F@X=
L0
=+
∂X
xσ
ux
∂=
εx
xEε
σ=
∂xx
x∂
xx
•D
ispl
acem
entb
ased
FEM
•D
ispl
acem
ent-b
ased
FE
M
0=
+ ⎟⎞⎜⎛
∂∂
Xu
E0
+ ⎟ ⎠⎜ ⎝
∂∂
Xx
Ex