Lat. Am. j. solids struct. vol.12 número3

!

"

# $ $

%

&

%

&

'( )

* +,-%$ .

* /00-$

.

* /0++1

#

'#21

*

#2

3

'+,-,1$

'+,451 6

#2

'

* /0+0$

* /004$

* +,,7$ 8

* /0++$ 8

*

/0+51 2

* !

.

'/0+51

#2

9

.

:

'/00;1

#2

! " #!

< & ! *

= = *

2 * = *

# >

?

>@@ @+0 +;,0@+-4, 47/;++7;

2 /, 0+ /0+% 07 0, /0+% +0 0, /0+%

<

*

'

* /007$ !

.

* /0+%1

'

* /007$ !

.

* /0+%1

#2

*

%

6

'+,,%1

/<

.

'/00-1

8

&

'/0+/1

%

8

A

*

'#21

9

'#

* +,,0$

* /007$ 8

* /0+51*

' ))

1

'!

* /00/$

* /0041

#2

9

)

'&

& "

* /0+%1*

'/0071

%

2

/

%

*

B

*

/< #2

#2

'

1

*

3

#2

*

%

2

9

*

9

*

9

*

2

'

* +,,-1

$

%

& %

%

$

*

*

6

/ +

/ /*

'/0071

*

6

/ 5*

$'

( )

*

"

. +

*

(

)

> C ' * /0071

(

)

=

(

)

=

*

(

)

* #*

(

)

(

)

(

)

=

=

+

+

+

=

+

+

+

'+1

*

(

)

= = − = −

'/1

*

*

(

)

(

)

(

)

=

=

+

+

+

=

+

+

+

'51

*

θ

(

)

(

)

(

)

(

)

θ =

=

 _{+ − − − + − −} 

 

=

 _{+ − − + + − −} 

 

∑

∑

'%1

* D+* *%

θ θ

θ θ

     _{+ − −}   

     

 ₌   ₋ 

  _{ }   

  _− _ + − −   

     

       

';1

.

*

   

=_{ } =_{ }

'-1

$'$

+

#

,

6

9

*

δ

δ

δ

=

=

=

'41

= =

'71

*

>

δ = δ

',1

!9 7* '8

* /0+51*

'

*

/0071>

 

= _{+  }

'+01

(

)

θ

θ

θ

θ



−











=

_

_

=

=

_

_

















=

_

_

_

_

_

_

−

−

−

−

_

_

'

* /0071>

(

)

=

 

= −_ − − − _

 

= _− − − − _

+

∑

'+/1

(

)

=

*

(

)

=

=

+

=

+

$'-

+

(

,

#

,

*

*

*

*

*

(

)

{

}

ρ

=

_∫

ɺ

+

ɺ

'+51

*

ρ

E @

5

F*

ɺ

ɺ

8

9

*

_∂  _∂ 

   

= _{ }−_{ }

∂ ∂

   

 ɺ   

'+%1

*

*

>

=

ɺ

ɺ

_'+;1

=

'+-1

!9 '+-1*

!

*

*

*

'.

#

)* /00%1

(

)

 θ_θ θ_θ

= _{=  }

−

 

 

'+41

>

_∂   _{ = +} _∂     

ɺɺ ɺ ɺ

ɺ

'+71

θ

>

(

)

θ θ

θ

θ

θ

∂ ∂

₌

₌

∂ ∂

ɺ

ɺ

ɺ

_'+,1

!9 '+,1*

_θ

"

θ

>

θ θ θ

∂

₌

∂

∂

∂

∂

∂

'/01

>

(

)

θ   ∂ _{= =  } = ₋ 

∂ _{ }

'/+1

6

!9 /+

!9 /0

_θ

(

)

θ θ

θ

∂

=

=

+

=

∂

'//1

.

*

>

θ θ θ _∂  _{∂ ∂  }   _{= =}_{ } _∂ _{ ∂ ∂} _ _  

ɺ ɺ

_'/51

!9 +7 /5

!9 +%*

θ

 _



= + −_ _

 

ɺɺ ɺ ɺ ɺ ɺ

'/%1

$'-'

.

9

*

ɺ ɺɺ

(

ɺ ɺɺ

)

+

( )

− =

_'/;1

*

*

*

!9 /;

'#

* +,,41

(

)

( )

+ + +

+ +

−

=

=

=

ɺ ɺɺ

+

'/-1

+

9

*

!9 /;

₊

>

(

)

+ + +

∂

=

+

−

=

+

∂

'/41

∂

∂

=

+

∂

∂

'/71

6

!9 /4

!9 /-*

(

)

−

+

=

−

'/,1

*

(

)

(

)

(

)

β

γ

γ

β

β

β

β

β

β

β

+ + + + +

=

+





₋



_





_

=

+



_

−

+



_

−

−

−

_

_











−

=

−

−

−

ɺ

ɺ

ɺɺ

ɺ

ɺɺ

ɺɺ

ɺ

ɺɺ

'501

!9 50*

β

γ

=

*

'!9 501* !9

/-9

)

*

2

*

δ

= −

− _{+ +}

'5+1

*

*

γ

δ

δ

β

δ

β

+ + + + + +

=

+

=

+

=

+

ɺ

ɺ

ɺɺ

ɺɺ

'5/1

9

*

*

(

)

(

)

(

)

θ

θ

θ

∂

∂

∂

∂

∂



_

∂



_

=

=

=

=

+

−





∂

∂

ɺɺ

∂

ɺ

∂

ɺ

∂

ɺ

_

∂

ɺ

_

'551

∂

∂

(

)

θ = = ∂ ɺɺ

ɺɺ

_{'5% 1}

(

)

θ

(

)

θ θ _∂ _  _ = _{ }+ − +  ∂  

ɺ ɺ ɺ ɺ

_{'5% 1}

(

)(

)

(

)

θ θ θ  ∂ _    _  

= ɺ ∂ ɺ  + ɺ ɺ  − +

'5% 1

*

∂

∂

'+1'/1'51'%1';1'-1'41'71',1'+01'++1'+/1'+51'+%1'+;1'+-1'+41'+71'+,1'/01'/+1'//1'/51'/%1'/;1'/-1'/41'/71'/,1'501'5+1'5/1'551'5%1

=

'5;1

θ

θ

∂

∂

!9 '/0 //1

>

(

)

θ

θ θ θ θ

θ

θ

θ

∂

∂

∂

=

=

+

∂

∂

∂

'5-1

(

)

 − 

= _{=  }

 

6

!9 +0* 55

5%

!9 /7*

*

γ

β

β

=

+

+

+

'571

-

&/

% /0 &1%/

&

8

G

4 % '2/004 1

*

H - '

* +,4-1

H

-8

*

β

γ

-' /,

*

2

#

.

*

.

/

*

* (

%0

I( * 0 5

+4%0 @

5

*

ω

=

. /

*

9

ω

%000

/;

π

@ *

)

+ /! 5 *

50% 9

.

5

%*

J

K

*

.

5

%*

8

$# J

# K

.

;

-

J

K

:

*

/%0 9

.

;

-*

+

#(=

2

# > /%0 +% 5/ '+701 +4 %5 '+,41 . > 50% /0 /5 '+7;1 /% /; '+,,1

! % # #(= ' 1

# J

+*

'

8

1

*

#2

+*

/%0

*

'

!9 ';11

'

. 41

/

A

/

*

#2

# $

507 50, 5+5 5+%

D0 00;;

0 07--! ; 0 005+! ; 0 0+5-! ; 0 0+0+! ; 0 507%! ; 0 07+,! ; 0 +/+;! ; 0 05,0! ; D0 0;;

0 /0;7! % 0 /+/+! % 0 0;07! % 0 0%%;! % 0 /+45! % 0 0,7,! % 0 //%,! % 0 +0-;! % D0 5

0 0,5,! % 0 077/! % 0 0075! % 0 0+%! % 0 +,4-! % 0 0004! % 0 +,07! % 0 004;! %

! % #! . 4

-'$ /,

* $2

#

*

. 7

<

*

4700 @

5

*

/00

/+0

*

*

J

ω

=

D %000 *

ω

=

π

@

:

*

)

5 /! 5 *

/%0

6

*

K

.

,

.

.

,*

*

*

.

+0

+-0 9

9

9

5

'#9 $ *

2

# > +-0 7 ;+ '751 +/ /5 '+0;1

. > /%0 +/ +- '7-1 +- ;4 '+041

! % $#9 #(= ' 1

L

'

+1

# K

)

%

*

#2

*

9

*

#2

:

*

#2

*

.

*

)

#2

*

* : * . * 6 * '/0++1 <

. ! < +,-> +45 +7+

* & * & ! * '/0+/1 I ( 6 @ !

* 2 * :" "* & * * M & * '/0+01 #

# & & ! +,,> 5+,4 5/+/ * C M * '+,,-1 . * ( :

* M & * '/0041 # &

& ! +,-> +,0; +,+%

* M & * '/0071 % & 2 # 5;> %07N%+5

* M &* '/0071 M

! 4;> -4/ -7,

* C * 6 * & #* * * '+,,71 # #

& & ! +;%> +;+ +-+ * * : * M * '+,451

M & ! 4> /;; /4+

# * & * '+,,01 * *

# & & ! +7> +5+ +;0

# * & * '+,,41 8 . ! 6 6 > G /

3 * # *

%;;N%;-! * * . * 6 * '/0+51 H

M 6 6 < +5> +/;00-4 '/- 1

! * * ( * # * '/00/1 ! 9

# & & ! +,> 544; 57+0

. * # * '/00-1 6 H %> # & & !

+,;> ;5+-N;5%/

. * # * : * * '/00;1 >

# & & ! +,%> //7; /55; . . I * '/00-1

6 & ! * 45> ,-- ,40

. * * # )* & * '/00%1 6 M 6 G

/4;> %+;N//

)) * * '/00;1

M ! -%> +5;0 +54%

8 * * * M & * :" "* & * '/0++1 ! /< # & %7> +;5N+-+

8 * * * M & * :" "* & * '/0+51 5<

* # & & ! /-,> ;57 ;-; ( ) * M 6 * * 8 * '+,-%1 <

. < 8 2 .<8 +7N-% 2 ) ( " * & * & * & * '/0+%1

8 M 6 6 ++> +0+/ +05;

6 * 3 * * 6 * '+,,%1* ) *

# & # * * I * +%* ;%, ;7;

* 2 * * ( * 3 * ! * '+,4-1 M

& ! +0> +/++N+/+,

3 * I * '+,-,1 . * M