11.2 Governing Equations11.3 Weak Formulations

제품설계분석 (II) 제 11 장Plane Elasticity

* 유한변형 (Finite deformation)

𝜀𝑥=𝜕𝑢𝜕 𝑥

𝜀𝑥𝑦=12 (𝜕𝑣𝜕 𝑥 + 𝜕𝑢

𝜕 𝑦 )전제조건

1) 선형 탄성 변형

( 미소변형 + 선형 탄성체 )

2) 평면 변형률 조건 , 평면 응력 조건

3) 경계치 문제 →약형→유한요소수식화

𝜎=𝐸 𝜀=𝐸 (𝜀) ∙𝜀

§ 11.2 Governing Equations

11.2.1 Plane Strain

( , )

( , )

0

x

y

z

u u u x y

u v v x y

u w

Strain field

Displacement field

<A hollow cylindrical member with

internal and external applied loads>

, , 2

0

y yx xxx yy xy

xz yz zz

u uu u

x y y x

• 적합성 조건• 선형 탄성해석에서는

“미소변형이론” 사용 “ 유한변형이론”

Stress-Strain relationship for an orthotropic material

11 12

21 22

66

0

0

0 0 2

xx xx

yy yy

xy xy

c c

c c

c

1 1211

21 12 21

2 2122

12 12 21

12 12 22 21 11 66 12

(1 )

(1 )(1 )

(1 )

(1 )(1 )

,

Ec

Ec

c c c c G

13 23

31 2

0,xz yz zz xx yyEE E

σ Cε

For an isotropic material1 2 3

12 21 13 23

12

,

,

2(1 )

E E E E

EG G

* From Chapter 8

𝐶=[𝑎11 𝑎12 0𝑎21 𝑎22 00 0 𝑎00 ]+𝑎00𝑢− 𝑓 =0 𝑖𝑛Ω

11.2.2 Plane Stress

0, 0xz yz zz

1 211 22

12 21 12 21

12 12 22 21 11 66 12

,(1 ) (1 )

,

E Ec c

c c c c G

<A thin plate in a state of plane

stress>

11.2.3 Summary of equations

(1) Equations of motion (diff.

equations) xyxxx x

xy yyy y

f ux y

f ux y

,ij j i if u T D σ f u

0

, , ,

0

xxx xT

yyy y

xy

f ux y

f u

y x

D σ f u

(2) Boundary Conditions

Essential B.C.

Suppressible (Natural)

B.C.

*x xu u *

y yu u uor on

*x xx x xy y xt n n t *

y xy x yy y yt n n t tor on𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛

Ω

Γ

𝑖𝑛Ω

0

→ �̈�={�̈�𝑥

�̈�𝑦}

(3) Strain-Displacement Relations

1, ,

2y yx x

xx yy xy

u uu u

x y y x

0

ˆ ˆ0

2

xxxx

yy yy

xyy x

u

u

ε Du DΨu Bu

(4) Stress-Strain Relations

(Constitutive eqns.)

ˆ σ Cε CDu CBu11 12

21 22

66

0

0

0 0

xx xx

yy yy

xy xy

c c

c c

c

* 이산화 (Discretization)

𝑢=𝜳 �̂�❶

❷

❸

❹

𝑇=𝜓 �̂� 𝑢

T D σ f u T D CDu f u

(5) Vector form in summary

ˆ σ Cε CDu CBu

11 12 66

66 12 22

y yx xx xx x xy y x y

y yx xy xy x yy y x y

u uu ut n n c c n c n

x y y x

u uu ut n n c n c c n

y x x y

Diff. eqns. of

displacement

Traction

0

0

xxx yx

yyy y x

xy

n nt

t n n

t Nσ NCDu

−𝑫𝑇𝑪𝑫𝒖= 𝑓 −𝜌 �̈�

§ 11.3 Weak Formulations

11.3.2 Principle of virtual displacements in vector form

Internal error vector inTI ε D CDu f u

inT D CDu f u

Boundary error vector* onB t ε t t

Weight functions or variation of ux

y

w

w

w ˆ ˆ( )

x

y

u

u

u Ψu Ψ u

*

0

( ) ( )

t

t

T Te I e B

T T Te e

I h d h d

h d h d

w ε w ε

w D CDu f u w t t

𝑢=𝜳 �̂�

*

*

0 ( ) ( )

( ) ( )

t

t

T T T

T T T T

d d

d d d

w D σ f u w t t

w D σ w f u w t t

11.3.3 Weak form*0 ( ) ( )

t

T T T Td d d

w D σ w f u w t t

( ) ( )

( ) ( )

[ ( ) ( )] ,

u t

xy xy yyT T xxx y

y yx xx xx y xy x xy y yy xx xy xy yy

xx x xx y xy y x xy y yy

d w w dx y x y

w ww ww w w w d d

x y x x y y

wn w w n w w d

x

w D σ

,( )

{ , } ( )

t t

xxy y x

yy

xy

x T T T Tx y

y

w w wd

y x y

tw w d d d d

t

Dw σ w t w D σ

*0 ( )

t

T T T Td d d

w D CDu w f u w t

11.4 유한요소모델

u=𝜳 �̂�=[𝜓1 0 𝜓 2 0 ⋯ 𝜓𝑛 00 𝜓1 0 𝜓 2 ⋯ 0 𝜓𝑛

]{�̂�𝑥

1

�̂�𝑦1

⋮�̂�𝑥

𝑛

�̂�𝑦𝑛}

𝜀={𝜕𝑢𝑥

𝜕 𝑥𝜕𝑢𝑦

𝜕 𝑦𝜕𝑢𝑥

𝜕 𝑦+𝜕𝑢𝑦

𝜕𝑥}=[

𝜕𝜕 𝑥

0

0𝜕𝜕 𝑦

𝜕𝜕 𝑦𝜕𝜕 𝑥

]{𝑢𝑥

𝑢𝑦}=𝑫𝑢=𝑫𝜳 �̂�=𝑩�̂�

𝑩=[𝜕𝜓1

𝜕 𝑥0

𝜕𝜓 2

𝜕 𝑥0 ⋯

𝜕𝜓𝑛

𝜕 𝑥0

0¿¿

𝜕𝜓 1

𝜕 𝑦

𝜕𝜓1

𝜕 𝑦0

𝜕𝜓 2

𝜕 𝑦⋯ 0

𝜕𝜓𝑛

𝜕 𝑦𝜕𝜓1

𝜕 𝑥𝜕𝜓 2

𝜕 𝑦𝜕𝜓 2

𝜕 𝑥⋯

𝜕𝜓𝑛

𝜕 𝑦𝜕𝜓𝑛

𝜕𝑥]

Weak Form*( ) ( ) ( ) 0

t

T T Tw u d w f u d w t d

D C D

*ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) 0t

T T Tw u d w f u d w t d

B C B

*ˆˆ ˆ ˆ ˆ( ) 0t

T T T T T Tw d u w f u d w t d

B CB

ˆTw G = 0

*ˆˆ 0t

T T Td u f d ud t d

TG = B CB

ˆ ˆu u M K F +Q

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆe e e e e eu u M K F +Q

T d

M = d

TK = B CB

T f d

F = * 0

t

T t d

Q =