Motion and Stress Analysis by Vector Mechanics

Edward C. Ting

Professor Emeritus of Applied MechanicsPurdue University, West Lafayette, IN

National Central University, ChungLi, Taiwan

a computer framework

for the study of a multi-component structural system with

• component motion • component interactions: connection, contact, collision, penetration• geometrical changes: deformation, displacement, fragmentation, collapse

• stress distribution• behavior and material property changes

a physics approach of mechanics

motion analysis and VFIFE

* vector mechanics ---- particle mechanics * discrete description * intrinsic finite element ---- physical structural element

example: a rod in plane motion

Newton’s law

1. displacement is a motion

1,2,3

, 1,2,3

2. ,

3. 0,

j j j rj jr

rj jr r j

m

x F S

S S

m1

m2

m3

m1

m3

F3

S32F2

F1S21

S12

S23

y

x

x2

m2

analytical mechanics:

For motion analysis, assume

1. rigid body, 2. functional description

ˆc

c jj

jj

x

m

I M

x x e

x F

e

1

xxc

x

01 1

cm

I M

x x

x F

1 1

1 1

1 1

ˆ

( sin 0)

x

I M

I mgd

x x e

pendulum problem: hinged at end 1

motion analysis1. general formulation:

2. complete formulation:

e

1

x

xc

1xd1

x1

1

1 1

21 1

ˆ

ˆ( , )

ˆ ˆ( )

dx

x ds

component

ds f x dx

x dF dS

x x e dS e

e

1ˆ, ( )s sdu dxA AE

stress analysis

assume: 1. deformable body, 2. Hooke’s law

e

1

1ˆdx

1ˆdx

1ˆf dx s ds

sx

1x

X1

dF

1. An approximation

→ separate motion analysis and stress analysis

► continuous bodies: motion--rigid body; stress--deformable body

► variables: motion--displacement; stress--deformation

► governing equations: motion--translation and rotation; stress--equilibrium

2. Described by continuous functions

→ discretization

computation based on analytical mechanics

1,2,3,j j j rj j

r

m

x F S

12 21 1

23 32 2

f

f

S S e

S S e

1 2 1 1

2 3 2 2

u

u

x x

x x

1 11

2 22

( )

( )

f AE u

f AE u

01 1x x

1 2e e

1. Newton’s law

2. behavior model

3. kinematics

4. Hooke’s law

5. pendulum: constraint conditions hinged end: straight rod:

vector mechanics

1

1

2

3

f2e2

-f2e2

f1e1

-f1e1

2

3

e1

e2

l1

l 2

properties:

1. structure: a set of particles

2. always a dynamic process

3. always deformable

advantages:

1. suitable for computation

2. a general and systematic formulation

3. explicit constraint conditions

development needs:

1. describe structural geometry: intrinsic finite element

2. kinematics: fictitious reversed motion

3. continuity requirements

4. mechanics requirements

5. material model: standard tests

elements:

plane rod, plane frame, plane solid, space rod, space frame, 3d membrane, 3d solid, 3d plate shell

V-5 research group:

e. c. ting, c. y. wang, t. y. wu, r. z. wang, c. j. chuang

motion analysis procedure: a simple rod structure

A

B

B

C0t

t

0x

xu

0P

P

x

y

discrete model: mass particles and structural elements

A

B

B

C0t

t

0P

P( )tu

A

B

B

C0t

t( )tu

vector form equation of motion

Am

B

C1f

t

( )tu

P

1f2f

2fB

2

1 22( ) ( ) [ ( ) ( )]

dm t P t t t

dt

uf f

A

B

C

0t

( )tu

at

bt

ct

ft

B

a bt t t

path element

1. element geometry remains unchanged

2. small deformation

discrete path:

kinematics and force calculation

1 material frame: configuration at

2. variable: nodal deformation

3. fictitious reversed motion to define deformation

4. infinitesimal strain and engineering stress

5. nodal forces: use finite element

6. internal forces are in equilibrium

at

reversed motion for nodal deformation

A

A

vB

vB

aB

tB

a

( )

te

ae

u

( )r u

du

l

al

l

f

f( )

d r

a al l

u u u

e

A

A

vB

tB

te

ae

l

al laf

fal

ae

aB

af

f

f

f

a

a

l l

l

ˆˆa

a a

f ff

A A

aE

ˆˆ ( ) aa a a a a a a a

a

l lf f A E A

l

f e e e

ˆtff e

governing equations

21 2

21 2

x x x

y y y

P f fudm

P f fvdt

1 11 11 1 1

1 11

cos

sinx a

a a ay a

f l lE A

f l

2 22 22 2 2

2 22

cos

sinx a

a a ay a

f l lE A

f l

2 2 21 1 1

2 2 22 2 2

( ) ( )

( ) ( )

l b u h v

l b u h v

a bt t t

,a a

a a

u uu ud

v vv vdt

difference equation (symmetrical case)

2

22 , a b

d v h vm P f t t t

dt l

aa a a

a

l lf E A

l

2 2 2( )l b h v , ,a a a

dvt t v v v

dt

2

1 1

( )2 2n

n n n n nn

h vtv P f v v

m l

n a

n a a aa

l lf E A

l

2 2 2( )n nl b h v

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

5

10

15

20

25

P*

M otion Analysis

Analytica l so l.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

-0 .50

-0.25

0.00

0.25

0.50

0.75

P*

M otion Ana lysis

Analytica l so l.

0 .00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

-0 .50

-0 .25

0.00

0.25

0.50

0.75

P*

M otion Ana lysis

Analytica l so l.

3* 1

3

PlP

AEh v

h

,

4 node plane solid element

x

y

1u

01

0t

t

02

3u

3au

3u

1au

1u

at

03 04

1a2a

3a

4a

1

2

3

4

estimate the rigid body motion

4

1a

2a

3a

4a

2

3

4

2

3

1

1

1ae

1e

4ae

3ae

2ae

4e

3e

2e1

4

1

1

4 jj

element translation

element rotation

u

fictitious reversed motion

1 ,1,1a

2a

3a

4a

2

3

4

2

3

4

nodal deformation

1( )( )r Ti i η R I x x

cos( ) sin( )

sin( ) cos( )

R

1 ,1,1a

2a

3a

4a

2

3

4

2

3

4

4η

4( )r η

4dη1

1 1

1 1

0

0

( ) , 2,3,4

dxdy

di ix T

di iy i

u u x xi

v v y y

R I

deformation coordinates to define independent variables

ˆˆ x Qx

ˆ ˆ , 1,2,3,4ˆ

di x

di y i

ui

v

Q

1 ,1a

2a

3a

4a

2 2ˆd u η

1e

2

3

4

x

y

2e

x

y

2dη

3v3u

4dη

4v

4u

1 1 2ˆ ˆ ˆ 0u v v

4 4

1 1

ˆ ˆ ˆ ˆ;i i i ii i

x N x y N y

1 2

3 4

1 1;

4 41 1

;4 4

N l s l t N l s l t

N l s l t N l s l t

4 4

1 1

ˆ ˆ ˆ ˆ;i i i ii i

u N u v N v

shape functions:

2 2 3 3 4 4 2 2 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ;x N x N x N x y N y N y N y

2 2 3 3 4 4 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ;u N u N u N u v N v N v

*ˆ ˆ n ε Bu

2 3 4

1

JB B B B

*2 3 3 4 5ˆ ˆ ˆ ˆ ˆ ˆ( )Tn u u v u vu

2 2

2

2 2

ˆ ˆ, ,

0

ˆ ˆ, ,

t s s t

s t t s

x N y N

x N x N

B

ˆ ˆ, , 0

ˆ ˆ0 , , , 3,4

ˆ ˆ ˆ ˆ, , , ,

t i s s i t

i s i t t i s

s i t t i s t i s s i t

y N y N

x N x N i

x N x N y N y N

B

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

x xa x

y ya y

xy xya xy

1 2U U

* * *ˆ ˆ ˆ( ) ( ) ( )a

T T Tn n a a aA

d dA u f u B σ σ

ˆ a σ E ε

*2 3 3 4 4

ˆ ˆ ˆ ˆ ˆˆx x y x yf f f f ff

* * *ˆ ˆ ˆa f f f

* *ˆ ˆˆ ˆ;a a

T Ta a a a a a aA A

d dA d dA f B σ f B σ

ˆ 2 2 2 3 3 3 3 4 4 4 42

ˆ 1 2 3 4

ˆ 1 2 3 4

1ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ0,ˆ

ˆ ˆ ˆ ˆ0,

ˆ ˆ ˆ ˆ0,

z y x y x y x

x x x x x

y y y y y

M f f y f x f y f x f yx

F f f f f

F f f f f

nodal forces

1 ,1a

2a

3a

4a

2

3

4

x

y

3ˆ

axf

3ˆ

ayf

4ˆ

axf

4ˆ

ayf

1ˆaxf

1ˆayf

2ˆ

axf2ˆ

ayf2ˆ

xf

2ˆ

yf

1ˆxf

1ˆ

yf

4ˆ

xf

4ˆ

yf3ˆ

xf

3ˆ

yf

1 ,1a

2a

3a

4a

2

3

4 4xf

4 yf

4ˆ

xf4

ˆyf

2

3

4

1

x

y

x

y

x

y

x

y

4xf

4 yf

4xf

4 yf1u

V

aV

V

ˆˆ

ˆix ixT

iy iy

f f

f f

RQ

stress

x

y

2

3

1

4

2

3

4

1

V

V

x

x

x

x

y

x

y

1u

xSxyS

yS

xS

xyS

yS

ˆ xˆ xy

ˆ y

xxy

y

ˆ ˆˆ ,

ˆ ˆx xy x xy

xy y xy y

S S

S S

σ S

ˆ ˆˆTS Q σQ

x xy

xy y

σ

Tσ RSR

Thank You