PSI - Issue 2_A

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

2419

5

3. Singular integral equation and solution The dislocation solutions obtained in Section 2 is utilized to analyze functionally graded orthotropic half-plane weakened by N arbitrary moving straight cracks. The distributed dislocation technique is an efficient means to carry out this task, see for instance Weertman (1996). The moving cracks configuration may be described in parametric form as:

0 x x l s y y = + = , i i i

,

(13)

i

N

1 1 s

1, 2,...,

=

− ≤ ≤

i

i

0

We consider local coordinate systems moving on the face of ith crack. The anti-plane traction on the face of the ith crack in terms of stress components in Cartesian coordinates becomes: 0 ( , ) nz i i x y σ τ = (14)

zj B is distributed on the infinitesimal segment i dl located at the face of

Suppose dislocations with unknown density the jth crack where the parameter

1 1 − ≤ ≤ p and prime denotes differentiation with respect to the relevant argument. The traction on the face of ith crack due to the presence of distribution of dislocations on the face of all N moving cracks yields:

N

1

∑∫

( ( ), ( )) x s y s

( , ) k s p l B p dp ( ) ,

i

1, 2, ..., , N

(15)

=

=

σ

n z

ij

j

zj

1

−

j

1

=

where from Eqs. (12), the kernel of integral equation is

( ( ) i j y s y p ζ −

( ))

( ( ) x s x p e ζ − ( ))

( k r f

( k R f

) ζ α

) ζ α

0 µ

y i

j

ij

ij

1

1

( ( ), ( ), ( ), k x s y s x p y p ( ))

(16)

[

],

=

−

ij

i

i

j

j

r

R

2

π

ij

ij

We substitute Eq. (14) and (16) into Eq. (15), becomes

( ( ) y s y p ζ −

( ))

N

( ( ) x s x p e ζ −

( k r 1

f

( k R 1

f

) α ζ

) α ζ

2 ( )) π

0 µ

i

j

1

1 = − ∑∫ j

ij

ij

y i

j

(17)

B p l dp

(

)] ( ) zj

,

[

−

=

τ

j

0

r

R

1

ij

ij

Since the singularity of stress fields for dislocation is of Cauchy type then the Eqs. (17) is Cauchy singular equations for unknown dislocation densities. Employing the definition of the dislocation density function, the equation for the crack opening displacement across the jth crack is

s

∫

−

+

(18)

( ) w s w s − ( )

( ) , l B p dp zj j

j

. 1,2,3,..., N

=

=

j

j

1

−

The displacement field is single-valued for the faces of cracks. Consequently, the dislocation density functions are subjected to the following closure requirements

1 ∫

(19)

( ) zj B p dp

j

N

0,

1, 2, 3,...,

=

=

1

−