PSI - Issue 2_A

M. Nourazar et al. / Procedia Structural Integrity 2 (2016) 2415–2423 Author name / Structural Integrity Procedia 00 (2016) 000–000

2417

3

Y

Y

2

2

ζ

ζ

where are the shear modulus. Utilizing Eq. (1) in the absence of body forces, the basic equations which govern the anti-plane deformation behaviour of the medium can be expressed in a fixed Cartesian coordinate system are: X X Y ( ) = µ 0 µ and Y Y µ e Y ( ) = µ 0 e

2

2

2

W

W

W

Y C W =

1

2

1

∂

ζ

∂ ∂

∂

∂

(2)

,

+

+

2

2

2

2

2

2

∂

t

X

f

Y

f

∂

∂

X

Y e ζ 2

where is the characteristic elastic shear wave velocity for the material in the x direction. For the current problem of a crack propagating at constant velocity V along the X –direction, it is convenient to introduce the following Galilean transformation , , . X x Vt Y y V t x ∂ ∂ = + = = − ∂ ∂ (3) with x and y being a translating coordinate system, which is attached to the propagating crack. Therefore, Eq. (1) becomes independent of time and can be converted into y ( ) = ρ 0 ρ is the material mass density. Also 0 ρ 0 µ X X C =

2

2

w

w

y w

1

2

ζ

∂

∂

∂

2

(4)

0,

=

α

+

+

2

2

2

2

∂

x

f

y

f

∂

∂

) 2 2 x V C = − α . The traction-free on the half-plane boundary implies that: (1

( , , ) ( , ) w x y W X Y t = and

where

(5)

( , ) 0, = x h zy σ

Let a Volterra type screw dislocation with Bergers vector z b be situated at the origin of the coordinate system with the dislocation line 0 > x . The conditions representing the screw dislocation are

+

−

( ,0 ) + x

( ,0 ), x −

( ,0 ) ( ,0 ) − w x w x

b H x

( ),

σ

σ

=

=

(6)

z

zy

zy

Here, ( ) H x is the Heaviside step-function. The first Eq. (6) shows the multivaluedness of displacement while the second implies the continuity of traction along the dislocation line. To obtain a solution for the differential equation (4) subjected to the conditions (5) and (6), the complex Fourier transform is defined as follows:

1

+∞

+∞ −

∫

∫

( ) , e f x dx f x ( ) i x λ

i x e f λ

*

*

f

d λ λ

(7)

( ) λ

( )

=

=

2

π

−∞

−∞

where 1 = − i . Applying Fourier transform (7) to Eq. (4) leads to a second order ordinary differential equation for ( , ) * w y λ . Its solution is readily found to be

2 2 2 2 ζ ζ α λ f

2 2 2 2 ζ ζ α λ f

y

y

(

)

(

)

− + +

− − +

*

( , ) w y A e λ λ = ( )

( ) B e λ

y h

0

,

+

< <

1

1

(8)

2 2 2 2 ζ ζ α λ f

2 2 2 2 ζ ζ α λ f

y

y

(

)

(

)

− + +

− − +

*

( , ) w y A e λ λ = ( )

( ) B e λ

y − ∞ < <

0,

+

2

2

where ( ), ( ), A B i i λ λ

1,2 = i

are unknown. Application of conditions (5) and (6) to Eq. (8) gives the unknown

coefficients. Therefore, the expressions for displacement components become: