of Introduction to Theory ofIntroduction Theory Technical...

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