PSI - Issue 2_A

M. Nourazar et al. / Procedia Structural Integrity 2 (2016) 3423–3431

3426

Author name / StructuralIntegrity Procedia 00 (2016) 000–000

4

0 ∫ ∞

( sg y ξ

)

− −

( G b g y x wz

G b

)

ξ

−

cosh( ) sg

sinh( ) sg ξ

e

ξ χ +

x wz

sg y h

s x

)) , ds

h y − < <

[

cosh( (

)) + −

]cos( (

σ

η

ξ

= −

+

−

zx

2

2

2 g y

2 ) ( − + − x ξ

2 ) ]

sgh

sgh

cosh(

)

sinh(

)

2

π

χ

−

2 [ ( π

η

2

2

0 ∫ ∞

( sg y ξ

)

− −

( gG b x x wz

gG b

)

−

η

cosh( ) sg

sinh( ) sg ξ

e

ξ χ +

x wz

sg y h

)) , s x ds η −

[

sinh( (

)) + −

]sin( (

= −

+

σ

zy

2

2 g y

2 ) ( − + − x ξ

2 ) ]

sgh

sgh

cosh(

)

sinh(

)

2

−

π

χ

2 [ ( π

η

2

2

(11)

h y − < <

ξ

2

where the Cauchy-type singularity at the dislocation location have been separated through a standard asymptotic analysis. All integrals in Eq. (11) decay reasonably fast as s → ∞ .

2. The Integral Equations

The formulation accomplished the previous section can be used to analyze layers containing multiple cracks. Let the substrate weakened by N embedded cracks. The cracks configurations may be expressed in parametric form as { } ( ), ( ), 1, 2,..., 1 1 i i i i x x s y y s i N s = = ∈ − ≤ ≤ (12) The movable orthogonal coordinate system n – s is chosen on the i-th crack such that the origin locates on the crack while the s axis remains tangent to the crack surface. The stress components on the surface of i-th crack in the Cartesian coordinates ( , ) x y become: ( , ) cos sin nz i i zy i zx i x y σ σ ϕ σ ϕ = − (13) where i φ is the angle between x and s axes. A crack is constructed by a continuous distribution of dislocations. Suppose dislocations with unknown density wz b are distributed on the infinitesimal segment 2 2 ( ) ( ) j j x y dt ′ ′ + at the surface of the j th crack where 1 1 t − ≤ ≤ and prime denotes differentiation with respect to the relevant argument. The traction components on the surface of the i th crack at a point with coordinates, ( ( ), ( )) i i x s y s , due to the presence of the above-mentioned distribution of dislocations on all N cracks, we obtain the following integral equations for the dislocation densities:

N

1

∑∫

2

2

j ′

j ′

( ( ), ( )) x s y s

ij wzj K s t b t

( , ) ( ) [ ( )] [ ( )] , x t y t dt +

1 1 , s

i

N

(14)

1, 2,...,

=

− ≤ ≤

=

σ

nz i

i

1

−

j

1

=

( ) wz b t . The kernel in

Equations (14) must be satisfied at every point along the crack and is to be solved for the unknown density

the above equation is

0 ∫ ∞

( j i sg y y −

)

gG sgy

sgy sgh

cosh( cosh(

) )

sinh( sinh(

)

+ −

χ χ

e

y

i

i

K

sg y h

)) s x x ds −

[

{

sinh( (

)) + −

}sin( (

=

ij

j

i

j

2

sgh

)

2

π

2

2

0 ∫ ∞

gG

x x −

(

)

G sgy

sgy sgh

sinh( cosh(

)

cosh( sinh(

)

+

χ

y

i

j

x

i

i

]cos( ) [ θ −

{

+

2

2 ) ( − + −

2

sgh

2

)

)

π

−

π

χ

( g y y

x x

)

2

2

i

j

i

j