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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
)
elas
t2
•Th
eory
of E
last
icity
yy
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
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
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
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
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
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
elas
t8
•Th
eory
of E
last
icity
yy
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
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.
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
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
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
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
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
elas
t15
•Th
eory
of E
last
icity
yy
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
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
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
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
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
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
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
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
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
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
∂+
++
∂+
+
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
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
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
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
−=
−=
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
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
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
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
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
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
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
