28
Introduction to Theory of Elasticity Kengo Nakajima Information Technology Center The University of Tokyo
Intr
oduc
tion
to T
heor
y of
El
astic
ity
Ken
go N
akaj
ima
Info
rmat
ion
Tech
nolo
gy C
ente
rTh
e U
nive
rsity
of T
okyo
elas
t2
•Th
eory
of E
last
icity
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
elas
t3
Theo
ry o
f Ela
stic
ity•
Con
tinuu
m M
echa
nics
, Sol
id M
echa
nics
•E
last
ic M
ater
ial
–Th
eory
of E
last
icity
, Ela
stom
echa
nics
elas
t4
Wha
t is
Ela
stic
Mat
eria
l ?
•D
efor
mat
ion
is p
ropo
rtion
al
to lo
ad
–H
ooke
’s la
w–
Exa
mpl
e•
Spr
ing
kx =
-mg
•M
etal
, Fib
er, R
esin
–If
load
is re
mov
ed, d
efor
mat
ion
goes
to 0
.•
Orig
inal
sha
pe
Def
orm
atio
n
Load
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
poi
nt–
Ela
stic
lim
it
•In
elas
tic•
Pla
stic
Def
orm
atio
n
Load
Yiel
d P
oint
elas
t6
Def
orm
atio
n do
es n
ot g
o to
0 w
ith
rem
oved
load
, afte
r ela
stic
lim
itatio
n.
•In
itial
sha
pe is
not
re
cove
red
any
mor
e.•
Per
man
ent d
efor
mat
ion
Def
orm
atio
n
Load
Per
man
ent
Def
orm
atio
n
Yiel
d P
oint
elas
t7
Theo
ry o
f Ela
stic
ity c
over
s …
•U
p to
Yie
ld P
oint
, Ela
stic
Li
mita
tion
–S
mal
l def
orm
atio
n–
Infin
itesi
mal
theo
ry•
Sha
pe d
oes
not c
hang
e
–Li
near
•P
last
ic/In
elas
tic⇒
Non
linea
r–
Mor
e in
tere
stin
g pa
rt of
rese
arch
•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
impo
rtant
–P
last
ic/In
elas
tic: A
ccid
ent c
ondi
tion
Def
orm
atio
n
Load
elas
t8
•Th
eory
of E
last
icity
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
elas
t9
Stre
ss (1
/6)
•If
exte
rnal
forc
e is
ela
stic
bod
y, th
e bo
dy d
efor
ms,
an
d re
sist
s ag
ains
t ext
erna
l for
ce b
y in
tern
al fo
rce
gene
rate
d by
inte
rmol
ecul
ar fo
rces
.•
Def
orm
atio
n of
the
body
reac
h st
eady
sta
te, w
hen
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)
•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 1
P 2
P n
P n-1
elas
t11
Stre
ss (3
/6)
•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 1
P 2
P n
P n-1
AB
S
elas
t12
Stre
ss (4
/6)
•C
onsi
der s
mal
l sur
face
S
on s
urfa
ce S
of p
art-A
, an
d re
sulta
nt fo
rce
vect
or F
•If p
is c
onsi
dere
d as
ave
rage
d fo
rce
per a
rea
F/S
with
infin
itesi
mal
S,
pis
cal
led
“stre
ss
vect
or”
P n
P n-1
A
S
S
F
SS
Fp
0lim
elas
t13
Stre
ss (5
/6)
•S
tress
: For
ce V
ecto
r per
Uni
t Sur
face
–P
ositi
ve fo
r ext
ensi
on, n
egat
ive
for c
ompr
essi
on•
On
a su
rface
–N
orm
al: N
orm
al s
tress）
–P
aral
lel:
She
ar s
tress
）
•“Y
ield
Stre
ss” i
s an
impo
rtant
des
ign
para
met
er.
P n
P n-1
A
S
S
F
SS
Fp
0lim
elas
t14
Stre
ss (6
/6)
•S
tress
com
pone
nts
in o
rthog
onal
coo
rdin
ate
syst
em–
9 co
mpo
nent
s in
3D
–no
rmal
stre
ss
–sh
ear s
tress
zzy
zx
yzy
yx
xzxy
x
σ
elas
t15
•Th
eory
of E
last
icity
–Ta
rget
–S
tress
–G
over
ning
Equ
atio
ns
elas
t16
Gov
erni
ng E
quat
ions
in T
heor
y of
E
last
icity
•E
quili
briu
m E
quat
ions
•C
ompa
tibili
ty C
ondi
tions
–D
ispl
acem
ent-S
train
•C
onst
itutiv
e E
quat
ions
–S
tress
-Stra
in
•2D
exa
mpl
e
elas
t17
Equ
ilibr
ium
Equ
atio
nsin
2D
00
Yx
y
Xy
x
xyy
xyx
elas
t18
Equ
ilibr
ium
Equ
atio
nsin
3D
6
Inde
pend
ent S
tress
Com
pone
nts
000
Zz
yx
Yz
yx
Xz
yx
zyz
zx
yzy
xy
zxxy
x
zyz
zx
yzy
xy
zxxy
x
σ
xzzx
zyyz
yxxy
elas
t19
Wha
t is
“Stra
in” ?
•S
olid
Mec
hani
cs–
Load
–D
efor
mat
ion
•S
tress
–Lo
ad/F
orce
per
uni
t sur
face
•S
train
–R
ate
of D
efor
mat
ion/
Dis
plac
emen
t
elas
t20
Stra
in: R
ate
of D
ispl
acem
ent
•N
orm
al s
train L
L
LL
•S
hear
stra
in
L
x
Lx
elas
t21
Con
stitu
tive
Eqn
’s: S
tress
-Stra
in(1
/3)
EE
xx
xx
,
•Y
oung
’s M
odul
usE
–S
tress
-Stra
in: P
ropo
rtion
al–
Pro
porti
onal
ity: E
(dep
ends
on
mat
eria
l)
x
x
xy
•P
oiss
on’s
Rat
io
–B
ody
defo
rms
in Y
-and
Z-
dire
ctio
ns, e
ven
if ex
tern
al fo
rce
is
in X
-dire
ctio
n.–
Poi
sson
’s ra
tio is
pro
porti
onal
ity fo
r th
is la
tera
l stra
in.
•de
pend
s on
mat
eria
l–
Met
al: 0
.30
–R
ubbe
r, W
ater
: 0.5
0 (in
com
pres
sibl
e)Ex
xy
elas
t22
Con
stitu
tive
Eqn
’s: S
tress
-Stra
in(2
/3)
yx
zy
xz
z
xz
yx
zy
y
zy
xz
yx
x
EE
EE
EE
EE
EE
EE
111
•E
ffect
of n
orm
al s
tress
com
pone
nts
in 3
dire
ctio
ns（ x
,y,
z）
–ac
cum
ulat
ion
of e
ach
stra
in c
ompo
nent
elas
t23
Con
stitu
tive
Eqn
’s: S
tress
-Stra
in(3
/3)
GG
Gzx
zxyz
yzxy
xy
,,
•S
hear
stra
in c
ompo
nent
s do
not
dep
end
on
norm
al s
tress
com
pone
nts.
The
y ar
e pr
opor
tiona
l to
she
ar s
tress
.
–La
tera
l Ela
stic
Mod
ulus
: G
12
EG
elas
t24
Stre
ss-S
train
Rel
atio
nshi
p
zxyzxyzyx
zxyzxyzyx
E
12
00
00
00
12
00
00
00
12
00
00
00
10
00
10
00
1
1
elas
t25
Stra
in-S
tress
Rel
atio
nshi
p
zxyzxyzyx
zxyzxyzyx
E
21
210
00
00
02
121
00
00
00
21
210
00
00
01
00
01
00
01
21
1
D
D
•In
com
pres
sibl
e M
ater
ial(~
0.50
): S
peci
al
Trea
tmen
t Nee
ded
elas
t26
Som
e A
ssum
ptio
ns in
this
Cla
ss•
Isot
ropi
c M
ater
ial
–U
nifo
rm E
, and
(~
0.30
)–
CFR
P (C
arbo
n Fi
ber R
einf
orce
d P
last
ics)
•O
rthot
ropi
c
elas
t27
Fini
te-E
lem
ent M
etho
d•
Dis
plac
emen
t-bas
ed F
EM
–D
epen
dent
Var
iabl
e: D
ispl
acem
ent
•G
ener
ally
use
d ap
proa
ch
–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
t28
1D S
tatic
Lin
ear-
Ela
stic
Pro
blem
•E
xten
sion
of 1
D tr
uss
elem
ent
–on
ly d
efor
ms
in X
-dir.
–U
nifo
rm s
ectio
nal a
rea
A–
You
ng’s
Mod
ulus
E–
u=0@
X=0,
Ext
erna
l For
ce
F@X=
L
F
0
Xxx
xux
x
xE
•D
ispl
acem
ent-b
ased
FE
M
0
Xxu
Ex
