Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48

( ) ˆ ˆ ; H X H X =- ˆ ˆ ˆ XXY

( ) ( ) X Ξ X Ξ

( ) X Ξ

( ) X Ξ

2 ; cos sin

2 ; cos sin

H

H

; cos cos 2 q q

q q

+

q q

-

-

XXX

XYY

XXY

(13)

( ) X Ξ

( ) X Ξ

2 q q

2 q q

H

H

H

; cos sin

; cos sin

; cos 2 sin , q q

-

+

-

YXX

YYY

YXY

( ) ˆ ˆ ; H X H X = ˆ ˆ ˆ YXY

( ) X Ξ

( ) X Ξ

( ) X Ξ

2 q q

2 q q

H

H

; cos sin

; cos sin

; cos 2 sin

-

-

q q

+

XXX

XYY

XXY

(14)

( ) X Ξ

( ) X Ξ

( ) X Ξ

2 ; cos sin

2 ; cos sin

H

H

H

; cos cos 2 , q q

-

+

-

q q

q q

YXX

YYY

YXY

where

ˆ

ˆ

ˆ

ˆ X

é ê ë

ù ú û

é ê ë

ù ú û

(15)

cos = + R a

X R

cos = +X R a

cos , sin R q

cos , sin q

sin , q

sin a q +X

a

+

X

Ξ

1

1

1

1

and

¥

{

( ) ; å X Ξ

( n h k - n

) ( h k - Z - Z + Z Z ) 1 1 , ; g n - - n - -

H

, ; g

= Á

XXX

n

(16)

n

1

=

}

( ) ( 1 n h k

)

( p k - - Z ) , ; g n

n - -

n - -

2

1

, ; g

,

+ +

Z

ZZ

Z

n

-

¥

{

= Á å

( ) ; X Ξ

( n h k - n

) ( h k - Z - Z Z ) 1 , ; g n - -

n - -

1

H

p

, ; g

-

Z

XYY

n

(17)

n

1

=

}

( ) ( 1 n h k

)

( p k - + Z ) , ; g n

n - -

n - -

2

1

, ; g

,

- +

Z

ZZ

Z

n

-

¥

{

( ) ; å X Ξ

( n h k - = Â - Z + Z + Z Z ) ( h k - ) 1 1 , ; g , ; g n - - n - -

H

XXY

n

n

(18)

n

1

=

}

( ) ( 1 n h k

)

( p k - - Z ) , ; g n

n - -

n - -

2

1

, ; g

,

+ +

Z

ZZ

Z

n

-

¥

{

= Â å

( ) ; X Ξ

( n h k - n

) ( h k - Z + Z + Z Z ) 1 1 , ; g n - - n - -

H

, ; g

YXX

n

(19)

n

1

=

}

( ) ( 1 n h k

)

( p k - - Z ) , ; g n

n - -

n - -

2

1

, ; g

,

+ +

Z

ZZ

Z

n

-

¥

{

= Â å

( ) ; X Ξ

( n h k - n

) ( h k - Z + Z - Z Z ) 1 1 , ; g n - - n - -

H

, ; g

YYY

n

(20)

n

1

=

}

( ) ( 1 n h k

)

( p k - + Z ) , ; g n

n - -

n - -

2

1

, ; g

,

- +

Z

ZZ

Z

n

-

¥

( n h k - n {

( ) ; å X Ξ

) ( h k - Z + Z - Z Z ) 1 1 , ; g n - - n - -

H

, ; g

= Á

YXY

n

(21)

n

1

=

}

( ) ( 1 n h k

)

( p k - + Z ) , ; g n

n - -

n - -

2

1

, ; g

.

- +

Z

ZZ

Z

n

-

The symbols { } . Â and { } . Á mean real and imaginary value of the complex expression. The dislocation is continuously distributed along the ˆ / x e -axis in the interval ( ) ˆ 0,1 X Î . This process is modelled by introducing the density of the Burgers vector ( ) ˆ ˆ Y B X and ( ) ˆ ˆ X B X , which is in relation with Burgers vector ( ) ˆ ˆ , X Y b b = b as follows

( ) ˆ X

( ) ˆ X

b

b

d

d

( ) ˆ

( ) ˆ

ˆ Y

ˆ X d

B

B

(22)

,

,

X =

X =

ˆ X

ˆ X

ˆ Y

ˆ X

d

507