Finite Strain Theory Detallado

8/18/2019 Finite Strain Theory Detallado

1/13

!"#!"#!$

!

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

3/+ / 45)*+6 ) 1789 +:;8? @98A? =AB789 A?9=:B C,D

E F?G:BH7I:9 ;7A>= G:B @98A? =AB789

J )9@98A?=8H7K =AB789 79L A>? 89@98A?=8H7K =AB789

A?9=:B M

!"#!"#!$ NNO"O !

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

!"#!"#!$ NNO"O $

+>? =?PQ?9


2/13

!"#!"#!$

$

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

!"#!"#!$ NNO"O O

+>? =?PQ?9


3/13

!"#!"#!$

O

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

)) +>? @98A? =AB789 A?9=:B C,D

J F?B8Y7I:9 :G C,D

!

$

O

\

!"#!"#!$ NNO"O &

dx

dy

!

"##

$

%&&= dX

[ ]

ds( )2

= dx( )2

+ dy( )2

ds( )2

= d x dy!" #

$dx

dy

!

"%%

#

$&&

dx dy!"

#$ = dX [ ]

T

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

)) +>? @98A? =AB789 A?9=:B C,D

J F?B8Y7I:9 :G C,D

!

$

O

\

!"#!"#!$ NNO"O _

&

_

`

a

dx

dy

!

"##

$

%&&= dX [ ]

ds( )2

= dx( )2

+ dy( )2

ds( )2= dX [ ]

T

dX [ ] = dX [ ]T

I [ ] dX [ ],

where I [ ] =1 0

0 1

!

"#

$

%&

d !s( )2

= d ! x d ! y"# $

%d ! x

d ! y

"

#&&

$

%''

= d ! X [ ]T

d ! X [ ]

ds( )2

= d x dy!" #

$dx

dy

!

"%%

#

$&&

dx dy!"

#$ = dX [ ]

T

d ! X [ ] = F [ ] dX [ ] (B:H K?


4/13

!"#!"#!$

\

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

J F?B8Y7I:9 :G C,D 37A L: A>? !"#$% :G , H?79d

!"#!"#!$ NNO"O a

1

2d !s( )

2

" ds( )2{ } = dX [ ]T E [ ] dX [ ]

E [ ] ! 1

2F [ ]

T

F [ ]" I [ ]#$ %

&

E [ ] ! 1

2 J

u + I [ ]

T

J u + I [ ]" I [ ]#$

%&

N8Y?9 , 79L Le 3A>? L8X?B?9


5/13

!"#!"#!$

&

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

, ,];79=8:9 :G C,D

!

$

!"#!"#!$ NNO"O c

J u =

!u

! x

!u

! y

!v

! x

!v

! y

"

#

$$$$$

%

&

'''''

E [ ] ! 1

2 J

u + I [ ]

T

J u + I [ ]" I [ ]#$

%&

E [ ] ! 1

2

"u

" x

"u

" y

"v

" x

"v

" y

+1 0

0 1

#

$%

&

'(

#

$

%%%%%

&

'

(((((

T

"u

" x

"u

" y

"v

" x

"v

" y

+1 0

0 1

#

$%

&

'(

#

$

%%%%%

&

'

(((((

) 1 0

0 1

#

$%

&

'(

#

$

%%%%%

&

'

(((((

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

, ,];79=8:9 :G C,D

O

\

!"#!"#!$ NNO"O !"

E [ ] ! 1

2 J

u + I [ ]

T

J u + I [ ]" I [ ]#$

%&

E [ ] ! 1

2

"u

" x+1

"v

" x

"u

" y

"v

" y+1

#

$

%%%%%

&

'

(((((

T

"u

" x+1

"u

" y

"v

" x

"v

" y+1

#

$

%%%%%

&

'

(((((

) 1 0

0 1

#

$%

&

'(

#

$

%%%%%

&

'

(((((

E [ ] !

1

2

"u" x

+1# $ %

& ' ( "u" x

+1# $ %

& ' ( +

"v" x

# $ %

& ' ( "v" x

# $ %

& ' ( ) 1

"u" x

+1# $ %

& ' ( "u" y

# $ %

& ' ( +

"v" x

# $ %

& ' ( "v" y

+1# $ %

& ' (

"u" y

# $ %

& ' (

"u" x

+1# $ %

& ' ( +

"v" y

+1# $ %

& ' (

"v" x

# $ %

& ' (

"u" y

# $ %

& ' (

"u" y

# $ %

& ' ( +

"v" y

+1# $ %

& ' (

"v" y

+1# $ %

& ' ( ) 1

*

+

,,

,,,

-

.

//

///

+>? H?7989S :G A>? A?BH= 89 , 8= 9:A 89AQ8IY?

)H7S89? V>7A A>? ;B:LQ


6/13

!"#!"#!$

_

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

( J:9Y?B=8:9 :G C,D A: CMD G:B =H7KK =AB789=

!

$

!"#!"#!$ NNO"O !!

E [ ] ! 1

2

"u" x

+1# $ %

& ' ( "u" x

+1# $ %

& ' ( +

"v" x

# $ %

& ' ( "v" x

# $ %

& ' ( ) 1

"u" x

+1# $ %

& ' ( "u" y

# $ %

& ' ( +

"v" x

# $ %

& ' ( "v" y

+1# $ %

& ' (

"u" y

# $ %

& ' (

"u" x

+1# $ %

& ' ( +

"v" y

+1# $ %

& ' (

"v" x

# $ %

& ' (

"u" y

# $ %

& ' (

"u" y

# $ %

& ' ( +

"v" y

+1# $ %

& ' (

"v" y

+1# $ %

& ' ( ) 1

*

+

,,,,,

-

.

/////

! [ ] " 1

2

#u# x$ % & ' ( ) +

#u# x$ % & ' ( )

#u# y

$ % &

' ( ) +

#v# x$ % & ' ( )

#u# y

$ % &

' ( ) +

#v# x

$ % &

' ( )

#u# y

$ % &

' ( ) +

#u# y

$ % &

' ( )

*

+

,,,,,

-

.

/////

=1

2 J u[ ]+ J u[ ]

T *+

-.

)G A>? L8=;K7


7/13

!"#!"#!$

`

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

))) F?G:BH7I:9 ;7A>= G:B@98A? =AB789

J:9=8L?B AV: L8X?B?9AL?G:BH7I:9=

/ F?G:BH7I:9 !

E F?G:BH7I:9 $

J ($ 7


8/13

!"#!"#!$

a

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

!"#!"#!$ NNO"O !&

C(!D

C($D

C($DC(!D

C(!DC($D

! x

! y

"

#$$

%

&''=

2 1

0 1

"

#$

%

&'

x

y

"

#$$

%

&''

2 1

0 1

!

"#

$

%& =

2 1

0 1

!

"#

$

%&

1 0

0 1

!

"#

$

%&

! x

! y

"

#$$

%

&''=

1 0

0 2

"

#$

%

&'

x

y

"

#$$

%

&''

1 0

0 2

!

"#

$

%& =

1 0

0 2

!

"#

$

%&

1 0

0 1

!

"#

$

%&

!! x

!! y

"

#$$

%

&''=

2 1

0 2

"

#$

%

&'

x

y

"

#$$

%

&''

!!! x

!!! y

"

#$$

%

&''=

1 0

0 2

"

#$

%

&'

x

y

"

#$$

%

&''

2 1

0 2

!

"#

$

%& =

2 1

0 2

!

"#

$

%&

1 0

0 1

!

"#

$

%&

2 2

0 2

!

"#

$

%& =

2 2

0 2

!

"#

$

%&

1 0

0 1

!

"#

$

%&

C(!D

C($D

C($DC(!D

C(!DC($DF 2[ ] F 1[ ] ! F 1[ ] F 2[ ]

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

)j )9@98A?=8H7K =AB789 79L A>?89@98A?=8H7K =AB789 A?9=:B CMD

/ )9@98A?=8H7K =AB789

F?G:BH7I:9 V>?B? A>?L8=;K7


9/13

!"#!"#!$

c

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

)j )9@98A?=8H7K =AB789 79LA>? 89@98A?=8H7K =AB789A?9=:B CMD 3


10/13

!"#!"#!$

!"

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2 -+./)*

J l>W


11/13

!"#!"#!$

!!

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2

-+./)*F +7WK:B =?B8?= ?];79=8:9 3


12/13

!"#!"#!$

!$

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2

-+./)*

!"#!"#!$ NNO"O $O

!&' ()*)+, -+./)* 0 )*()*)+,-)1/2

-+./)*

!"#!"#!$ NNO"O $\

! xx

=

"u

" x !

yy =

"v

" y

! xy =

1

2"

1 # "

2( ) =1

2

$u$ y

+$v$ x

% & '

( ) *

! xy =

1

2"

1 + "

2( ) =1

2

#v# x

$ #u# y

% & '

( ) *

(:B =H7KK 79SK?=^ q r A79q

4:=8IY? 79SK?= 7B? H?7=QB?L 7U:QA A>? [i7]8= Q=89S 7 B8S>A >79L BQK?' )9 3? 79SK? q$

8=


13/13

Documents

Finite Strain Theory Detallado