Lat. Am. j. solids struct. vol.11 número4

(1)

(2)

.

0   

0   

t      



0 0 .

t

  

   



  0 0

0

 7

4



10 0

12 8.85 10 

 

0 0 1

. c

 

(3)

, J E

J 

 V

( ) ( v ).

J E V B E B

t

  

     



 

2 2

0 0 2 0

t



  

   

 _ 

 ò 

2 2

0 0 2 0

t



  

   

 _ 

 ò 

2 2

2

2 2 2

1 1

r r r r 

  

   

  



r, ,



z



, , , . , ,

rr r rz rr z zz

s s_ s s s_ s_ s T T T_R, _, _Z

, ,

u v w

2 2

1 1

( ) ,

r

rr rz

rr R

s

s s u

s s T

r r z r t



 





 _ _ _ _ _ _ 

   

2 2 2

1

,

r z r

s s s s v

T

r r z r t

   

 



 _  _ _ _ _ 

   

2 2 1

.

z

rz zz rz

Z

s

s s s w

T

r r z r t

 _





  

    

(4)

0 0 0

11 12 13 ,

rr rr zz

s  e  e_  e

0 0 0

21 rr 22 23 zz,

s_  e  e_  e

0 0 0

31 32 33 ,

zz rr zz

s  e  e_  e

0

44 ,

rz rz

s  e

0

55 ,

z z

s_  e_

0

66 .

r r

s_  e_

ij

  ij

2

0 / / /

2

ij ij ij ij

t t

     

  ij

/

ij



/ /

ij



_ij.

1 , 2

rr

u e

r  



1 1

, 2

v u e

r r

 _



 

 _  _



 

1 , 2

zz

w e

z  



1 1

, 2

z

w v

e

r r z



 

 

 _  _

 

 

1

, 2

rz

w u

e

r z

 

 

 _  _

 

(5)

1 , 2

zz

w e

z  



1 1

, 2

r

w v

r  z

 

 

  _  _

 

 

1 1

, 2

u w

r z r



 

 

  _  _

 

 

1 ( )

.

z

rv u

r r 

 

 

  _  _

 

 



J



J







2

1 1

( ) ,

r

rr rz

rr

R

s

s s u

s s J

r r z r t



 





 _ _ _ _ _ _{ } 

   





2

2 2

1

,

r z r

s s s s v

J

r r z r t

   

 



   

     

   





2

2 1

.

z

rz zz rz

Z

s

s s s w

J

r r z r t

 _





 _ _ _ _ _{ } 

   

    r  0   z



0, 0,



0 i

     

0

 i

 

0



   ₀ 7

(6)

2 0

v

t t

   

   _   _  __

  

 

 

2 , 0

1

r

t  

 _ _{ }



2 , 0

1 t



  

 _ _{ }



2 0

1 .

z

t  

 _ _{ }



v

0,

u

0 w

( , ).

vv r z

0,

rr zz zr

e e_ e e 

1 , 2

z

v e

z



     __ _

1

. 2

r

v v e

r r





 

 _  _



 

1 , 2

r

(7)

0,



 

.

z

v v r r 

  



0,

rr zz rz

s s_ s s 

2 / / / 66 66 66 2

1

( ) ( ),

2

r

v v s

t t r r

   

  

   

  

2 / / / 55 55 55 2

1

( )( ).

2

z

v s

t t r

   

  

   

  

/

ij



/ /

ij



_ij.

  

0 v_{, 0, 0}

c t

  

 

  _ _



 

0, , 0

i

v z 

 

 _  _



 

0 v_{, 0, 0} _0, v_{, 0}

c t z

   

_  _ _ 

 _  _ _ _

 

2 2

2 0, _eH v, 0

z



  

   _ 

(8)

2 2

/ / / / / /

66 66 66 2 55 55 55 2 2

2

2 2

/ / / 2 2

66 66 66 2 2

1 1

( ) ( ) ( )( )

2 2

2 1

( ) ( )

2 e e 2

v v v

v

r t t r r z t t r

t

v v p v

H E

r t t r r z

             _  _   _ _  _  _  _   _ _ _ _ _ _ _ _  _  _{ }  _ _ _ _  _   _ _ _ _ __ _ _ _           e

 e

/ / / / / /

, ,

l l l

ij ij ij ij ij ij

C 



r C 



r C 



r ₀ m

r   ij  / ij



/ / ij



₀ r l m,

2 / / / 66 66 66 2

1 ( ) ( ), 2 l r v v s r

t t r r

              2 / / / 66 66 66 2

1 ( ) ( ), 2 l r v v s r

t t r r

        

  

2 2

/ / / / / /

66 66 66 2 55 55 55 2 2

0 2

2 2

/ / / 2 2

66 66 66 2 2

1 1 ( ) ( ) ( ) ( ) 2 2 2 1 ( ) ( ) 2 2 l l m l e e

v v v

r r

v

r t t r r z t t r

r t

v v p v

r H E

r t t r r z

                            _ _ _ _ _ _ _ _  _  _{ }  _ _ _ _  _   _ _ _ _ __ _ _ _           e

 e

( ) ( ) i z t v r e  

2

2 2

1 2

2 2

( 1) ( 1)

0

l

l l

r r r r r

  _ _ 

 _   _  _{   } _

(9)

2 / / / 2 2

2 0 55 55 55

1 / / / 2

66 66 66

2 ( )

, i

i

       

    

  

 

 

2 2

2 / / / 2

66 66 66

2 2

.

( )

eH eE

p

i

  _

    

 

 

 

 

 

 

0

l l2

0 l 0

l

2

2 2

1 1

( ) 0

r r r r

  _

 

    

 

2 2 2

1 2

    

( )

1 1

{ ( ) ( )} i z t

v PJ Gr QX Gr e  

/ // 2 ( )

66 66 66 0 1 0 1

2 2

{ } { ( ) ( ) { ( ) ( )

2 2

i z t

r

P Q

s i GJ Gr J Gr GX Gr X Gr e

r r

 

            

 

r

0

( )

r r r

s_ _  _

r

0

( )

r r r

(10)

r



0

(r)



0

( ) 0

r r

  

( ) 0

1 1

{ ( ) ( )} i z t

i PJ Gr QX Gr e

c

 

  _ 

   

( )

0

i z t

e     

2

2 2

1

0

r r r 

 _ _{  }

 

2

2 2

2 c



  

0( ) 0( ) RJ



r SX



r

  

0

J X₀

( )

0 0

{RJ (r) SX (r e)} izt

  

0 1 0 1

{ ( ) 2 ( )} { ( ) 2 ( )} 0 P G J G   J G Q G X G   X G 

0 1 0 1

{ ( ) 2 ( )} { ( ) 2 ( )} 0 P G J G   J G Q G X Ga  X G 

0 1 0 1

( ) 2 ( ) ( ) 2 ( )

0

( ) 2 ( ) ( ) 2 ( )

G J G J G G X G X G

G J G J G G X Ga X G





 



(11)

0 1 0 1

( ) 2 ( ) ( ) 2 ( )

0

( ) 2 ( ) ( ) 2 ( )

G J G J G G X G X G

     

     

 _  _

 

 

  



2 3

, ...

 

 

3 2

1 0



    

2 2

2 / / / 2 2 2

0 55 55 55

2

/ / / 2 66 66 66

2 ( )

2 2

eH eE

i p

i

 

        

    

 

   _   _

 

 

 







  

1 / c 

 

2 2

2 1

2 / / / 2

0 66 66 66

2 2

2

eH eE _p

c

c i

 



     

 

 

 

   

  _ _   

 

 

2 ,

k 

  55 55/ 55/ / 2

/ / / 2 66 66 66

, i

i

    

 

 

 

/ / / 2 2 66 66 66 0

0 2

i

c     



 



,E 

/ // 0

ij ij

(12)

3

0 0  0

    

2 2

2

2 66 2 55

2 0

0 66 66

2 2

eH eE _p

c

 

  

   

  

 

  

    

 _ _ _   _

 

 

 

1

2 2 2

2 0

55 2

2

0 66 66

[ ] ₂ ₂

2 [ ]

eH eE _p

c or

c

 

 

  



 _  

 

  

  

_   _

 

 

2

0 66 2 0

c  

,E 

/ //

55 66 0,

ij ij



















2 2

2

2 2

2 0

0

2 2

1

2 2

eH eE _p

c

 

 

  

  

 

  

    

 _ _ _   _

 

 

(13)

2 l

2 2

1

2 2

(3 )

3

( 0

r r r r

    _

 _  _{  } _

 

1 ( )r r   

2 2

2 1

2 2

1

0

r r r r

 

   

   _ _ 

  _ _

2 2

2

3    

1 2

( ) ( )

RJ_ r SX_ r

    





( )

1 1

1

{RJ ( r) SX ( r e)} i z t

r

 

 

    

1 1 1 1

/ / / 2 ( )

66 66 66

1 1 1 1

{ ( ) ( 2) ( )}

2

( ) 0

{ ( ) ( 2) ( )}

2

i z t r

R

rJ r J r

s i e

S

rX r X r

 

     

 



 

 _ _ _{  } _ 

 

     

_ _ _ _{  } _ 

 

 

1 1 1 1 1 1 1 1

{ ( ) ( 2) ( )} { ( ) ( 2) ( )} 0

2 2

R S

J J X X

 _  _   _  _ 

             

1 1 1 1 1 1 1 1

{ ( ) ( 2) ( )} { ( ) ( 2) ( )} 0

2 2

R S

J J X X



_



_



_



_



             

1 1 1 1 1 1 1 1

{ ( ) ( 2) ( )} { ( ) ( 2) ( )}

0

{ ( ) ( 2) ( )} { ( ) ( 2) ( )}

J J X X









   

           



(14)

1 1 1 1 1 1 1 1

{ ( ) ( 2) ( )} { ( ) ( 2) ( )}

J J J J

X X X X

     

 _  _   _  _ 

      _      

           

1 1 1 1 1 1 1 1 1 1 1

{ ( ) ( 2) ( )} { ( ) ( 2) ( )}

J J F J F J F

X X F X F X F

     

 _  _   _  _ 

   _   

     

1

F  

 

  



2 3

, ...

 

 

2 2 2

1 1

1

( 2)    _2 1 _(   2)



0



 

2 2

2

2 2 2

2 / / / 2

66 66 66

2 2

3 3 ,

( )

eH eE

p

i

  _

    

 

 

 

 

       

 

2 / / / 2 2

2 0 55 55 55

1 / / / 2

66 66 66

2 ( )

. i

i

       

    

  

 

 

2 _/ _{/ /} ₂

2

2 55 55 55

2 / / / 2

0 2 66 66 66

i c

c i



   









   

 

 

 _ _ 

 

 

/ / / 0

ij ij

(15)

1 1 1 1

3 1 3 3 3 1 3 3

{ ( ) ( 2) ( )} { ( ) ( 2) ( )}

J J J J

X X X X









                                     3

2 6 2 3 0

      2 2 2 2 1 66 2 2 3 ,

eH eE _p

  _         

   2 0 2 55 2

3 66 2 ,      

 



₂  ₃



at  ₁ 1

0 l 1 2 ₂ 2 3 55 2 01 66 2 c c       _ _  _ _       _{ }   _{ }      2

01 66/ 2 0

c  

/ //

55 66 0,

ij ij



















2 2 2 2

4 1 4 4 4 1 4 4

{ ( ) ( 2) ( )} { ( ) ( 2) ( )}

J J J J

X X X X









                                     2 2 2 2 2 2 2 3 ,

eH eE _p

(16)

02

2 2 2

4

2 2

2

1 c

c

  



_ _ 

_ _ 

 

 

 _{ } 

 _{ } 

 

 

02

2

0 2

c 

 

₀  ₅₅/₆₆

2 0

c

c ₀

c c

2 0

c

c ₀

(17)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1 1.5 2 2.5 3 3.5 4

Diameter/Wavelength

P

ha

se

V

e

loc

it

y

(18)