PDF Chapter 5: Basic finite element analysis of continua

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Computational Plasticity Spring, 2023

Chapter 5: Basic finite element analysis of continua

Myoung-Gyu Lee

TA: Gyu Jang Sim ([email protected])

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Ch. 5

In this chapter,

• Generalized formulation of finite element procedure is introduced

• We will apply continuum mechanics learned in Chapter 4 to the development of finite element formulations for two- and three- dimensional continua.

• Uppercase letters will relate to the initial coordinates, while lower-case relate to the current configuration.

• Throughout the total Lagrangian formulation, such a distinction is unnecessary since we will always be referring to the initial

configuration.

, , X Y Z

, , x y z

5.1 INTRODUCTION AND THE TOTAL LAGRANGIAN FORMULATION

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Ch. 5 5.1 INTRODUCTION AND THE TOTAL LAGRANGIAN FORMULATION

2

1 1 1

2 2 2

T T T

é - ù = é + ù+

ë û

= ë û

E F F I D D D D Where, F I D= + ^{[eq. 5.1]}

2

1 1

2 2

1 2

Td T Td

dE = F D+ dD F+ dD D

2 : _v2 0 _e ^T _v 0 _e

V = ò^S d^E dV -V = ò^{S E}d dV -V

(

^v² ^: ^t⁴ ^: ² ^: ^T^v

)

⁰

(

^T ^t² ^T ⁽ ^v²⁾

)

⁰

V dV dV

d =

ò

d^E ^C d^E +^S d d^{D D} =

ò

d^{E C} d^{E S}+ d d^E Review of chapter 4

[eq. 5.2]

[eq. 5.4]

[eq. 5.3]

Green strain

Virtual work principle

'2’ = second order tensor

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Ch. 4 4.4 GREEN’S STRAIN

4.4.1 Virtual work expressions using Green’s strain

• Conjugate of the Green strain is the 2^ndPiola-Kirchhoff stress

2) : 2

(or ^nd Piola Kir- chhoff stress t soen r S S

• Virtual work expression is given by:

0 2 : 2 0

T

i e v e v e

V V= -V =

ò

^{S E}d dV -V =

ò

^S d^E dV -V

2

1 1

2 1

2 2

T T T

é - ù = +

ë û é ù

= ë û+

E F F I D D D D

by ,

2

1

2 2 2

2

1

1 1

T

T T T

d

d d

d d d d d

=

é ù

= ë + û+ + +

D D

E D D D D D D D D



2

1 1

2 2

1 2

Td T Td

dE = F D+ dD F+ dD D

or

( )

(

^T _v ^T ( )_v

)

0 ( 2 : _v2 2 : ⁽ ( _v2)⁾) 0

V dV dV

d =

ò

d d^{S E} +^S d d ^E =

ò

d^S d^E +^S d d ^E

2

1 1

2 2

T T

v v v

dE = F Dd + dD F

[eq. 4.76]

[eq. 4.74]

[eq. 4.77] [eq. 4.79]

[eq. 4.80]

2

1 2

T T

v v v v

dE = ^éëdD +dD ^ùû+D Dd

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Ch. 4 4.4 GREEN’S STRAIN

( )

(

^T _v ^T ( )_v

)

0 ( 2 : _v2 2 : ⁽ ( _v2)⁾) 0

V dV dV

d =

ò

d d^{S E} +^S d d ^E =

ò

d^S d^E +^S d d ^E

( 2 )

( 1

) 2 ^T ^T ^T

v v v v

d d E = ^éëd dD D+d dD D ^ùû =d dD D

2 , 2 4 : 2

t t

dS C= dE dS = C dE

( )

(

^T^v ^t² ^: ^T^v

)

⁰

V dV

d =

ò

d^{E C} d^{E S}+ d d^{D D}

[eq. 4.81]

[eq. 4.82]

[eq. 4.83]

[eq. 4.80]

2

1

v 2 v v

T T

dE = ^éëdD +dDv ^ùû+D Dd

d2 is neglected for virtual change

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( , )^T

u = h

x h

u ^{[eq. 5.5]}

x y

x x

x y

y y

x x x

h x x

¶ ¶ ¶

æ ö é ù ¶æ ö æ ¶ ö ç ¶ ÷ ê¶ ¶ ú ç ¶ ÷ ç ¶ ÷ ç ÷ ê= ú ç ÷ = ç ÷

¶ ¶

¶ ¶ ¶

ç ÷ ê ú ç ÷ ç ÷

ç ÷ ç ÷

ç ¶ ÷ ê¶ ¶ ú ¶è ø è ¶ ø

è ø ë û

J 5.1.1 Element formulation

[eq. 5.6]

• Two-dimensional formulation where displacements are related to nodal values via shape functions

( , )^T v = h

x h

v ( , )^T

x = h

x h

x y = h( , )

x h

^T y

• Differentiation in two coordinates systems

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• For example,

1 1

(1,1) (1, 2)

(1,1) ( , )^T (1, 2) ( , )^T

u u u

x

x h

x h x h

- -

¶ ¶ ¶

= +

¶

+

=

¶ J J ¶

J h u J h u

u x

¶

[eq. 5.7]

• Vectorized form of can be obtained as:D

1 1

(1,1) (1, 2)

(2,1) (2, 2)

(1,1) (1, 2)

(2,1) (2, 2)

T T T

u x u y v x v y

x h

- -

é¶ ù ê¶ ú ê ú ê¶ ú

ê¶ ú é ù

= êê¶ úú = ê úë û =

ê¶ ú ê

é + ù

ê + ú

ê ú

ê + ú

ê ú

ú + ê¶ ú ê

ú û

û

¶ ú

êë ë

J h J h 0

J h J h 0 u

θ Gp

0 J h J h v

0 J h J h

• Incremental form is obtained by:

[eq. 5.8]

dθ G p= d ^{[eq. 5.9]}

x y

x x

x y

y y

x x x

h x x

¶ ¶ ¶

æ ö é ù ¶æ ö æ ¶ ö

ç ¶ ÷ ê¶ ¶ ú ç ¶ ÷ ç ¶ ÷

ç ÷ ê= ú ç ÷ = ç ÷

¶ ¶

¶ ¶ ¶

ç ÷ ê ú ç ÷ ç ÷

ç ÷ ç ÷

ç¶ ÷ ê¶ ¶ ú ¶è ø è ¶ ø

è ø ë û

J

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• Using eq. 5.8, Greens strain can be written in vector form as:

1 1

0 0

0 0 (

2 2 )

l nl l

u

u u v x

x x x u

v u v y

v

y y y

u v u u v v x

y x y x y x v

y é¶ ù ê ú

é ¶ ù é¶ ¶ ùê¶ ú

ê ¶ ú ê¶ ¶ ú ¶ê ú

ê ú ê ú

ê¶ ú

ê ¶ ú ê ¶ ¶ ú

= êê ¶ úú+ êê ¶ ¶ ê úú ¶úê ú = ê¶ ú ê¶ + ¶ ú ê¶ ¶ ¶ ¶ úê ú ê¶ ¶ ú ê¶ ¶ ¶ ¶ ê úú ¶

ë û ë û

+ = +

ê¶ ú ë û

E E E E A θ θ ^{[eq. 5.10]}

1 ( )

l nl 2

é ù

= + = êë + ú E E E H A θ θû

1 0 0 0 0 0 0 1 0 1 1 0

é ù

ê ú

= ê ú

ê ú

ë û

[eq. 5.11] H [eq. 5.12]

or where

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• Incremental form is:

( )d =d ( ) A θ θ A θ θ

( )

²

1 1

( ) ( )

2 2

l O

dE=dE + A θ θd + dA θ θ+ dθ

xx

xy yy

S S S

æ ö

ç ÷

=ç ÷

ç ÷

è ø

S

[eq. 5.13]

[eq. 5.14]

( )

²

l ( ) O

d d d

= E + A θ θ+ θ

[ ^{( )}] ^d ^O

( )

^d ²

= H A+ θ G p+ θ ^{[eq. 5.15]}

[ ⁽ ⁾ ]

( )

²

nl

l d O d

= + +

B

B A Gp G p p

 ^{[eq. 5.16]}

=

θ Gp ^{[eq. 5.8]}

v nl( ) v

dE = B p pd ^{[eq. 5.17]}

Bl  HG and

T T ( ) T

v nl dVo v e

d d

= ^p ò^{B p S} - ^{p q}

• Virtual work is:

0 T

v e

V = ò^{S E}d dV -V ^{[eq. 5.3]}

T

d v

= p g

[eq. 5.19]

• Out-of-balance force is:

( )

T ( ) T

nl dVo - e = dVo - e

= ò ò éë + ùû

g B p S q G H A θ S q

[eq. 5.18]

dθ G p= d

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5.1.2 The tangent stiffness matrix

T

V =dp gv

T T

t

v v

V d

d d d d d

= dg =

p p p p K p

(

^d ^T^v ^t^d ^T^{d d}^T^v

)

^dV⁰

= ò ^{E C E} +^{S D D} ^{[eq. 4.83]} _{( )}

v nl v

dE = B p pd ^{[eq. 5.17]}

( )

(

^T^v ^t² ^: ^T^v

)

⁰

V dV

d =

ò

d^{E C} d^E +^S d d^{D D} ^{[eq. 4.83]}

( ) ⁽ ⁾

1

0 0

) ) :

( (

t

T T T

v nl t nl dV v dV

d d d d d

= ò +ò

K

p B p C B p p S D D



[eq. 5.21]

[eq. 5.22]

( )

⁰

: d d^T_v dV

ò^S ^{D D}

• Second term is given by:

dθ G p= d ^{[eq. 5.9]}

Vectorized isdD dθ



T

v dVo

d d

= ò ^{θ S θ}

 ^

11 12

12 22

1 11

12 2

2 2

0 0 0 0 0 0

0 0

S S

éé ù é ù ù

êê ú ê ú ú

ë û

ê ú

= êêêë éêë ùúû éêë ùúûúúúû

[eq. 5.24] where S

(



)

t

T T

v dVo

d d

= ò

Kσ

p G SG p



1

t = t + t_σ

K K K

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PDF Chapter 5: Basic finite element analysis of continua

Chapter 5: Basic finite element analysis of continua

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Thank you!

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