Issue 41

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 41 (2017) 175-182; DOI: 10.3221/IGF-ESIS.41.24

ߙ = 3/2

ߨ and the angle corresponding to the line of the crack

ߴ =

ߨ /4. Therefore we have: 0.5445 ,   

For the considered geometry, the internal wedge angle is 2

is

 I

0.9085

II

I

II



 

g

g

0.543 ,

0.219

(2)





r

r

0

0 II



 

K

K K

K

0.7304 ,

1.087

I

I

II

where K I 0 , K II 0 are simply scaled versions of the Mode I and Mode II notch stress intensity factors K I , K II

, obtained from a ≪ a and lying along the x -axis. A distribution of edge dislocations of densities

calibration with the finite element method. Let us consider the crack to be of length c

B x , B y is added, so that the resulting integral equations which govern the problem are the following:

c

c

  

  

  (



 1

 1 ( ) )

 0  0 c

 0

 

 

 ( ) B F x xxy x

 ( , )



 ( ) B F x d ( , )

 



 t





 



(3)

d

x

x

(

)

y

yxy



x

G

2

c

  

  







(

1)

 1



 0

 

 





 



 ( ) B F x d ( , )

 

 ( ) B F x

 ( , )





 n

(4)

d

( ) x

( ) x

x

xyy

y

y

y

y



x

G

2

where G is the shear modulus, ߥ is the Poisson ratio. The ( ξ ) , can be found in [3]. The right ߪ ∞ (x) , obtained from the Williams solution, and the shear side terms of Eqs. (3)-(4) contain the far-field stresses and normal stresses applied to the crack surfaces, n (x) , respectively. Introducing the normalized variables t,s , with crack extremes ± 1, instead of x and  respectively, we can write the singular integral equations in the usual form, and express the unknown dislocation densities B x , B y as follows: ߪ t (x) and ߪ where we have chosen the fundamental form of the solution ߱ (s) to be singular at the crack tip s = 1 and bounded at s = - 1. A numerical solution of the integral equations is needed, and this can be achieved by means of the Gauss-Chebyshev quadrature described in [10]. The resulting 2N equations in the 2N unknown ߶ j (s i ) are the following:                                 1 1) ( 1 ( ) ( ) ( , ) ( ) ( , ) ( ) ( ) 2 i x i xxy k i i yxy k i k t k k i N y W s s F t s s F t s t t t s G (6) influence function F kij (x, ξ ) , connecting the stress component  j  ( ) ( ) s s  j      j x y , 1 1 ( ) ( ) s , j s s B s (5) ߢ = 3 - 4 ߥ is the Kolosov constant for plane strain, and ߪ ij (x) to a dislocation b k ߬ ∞ (x) and

     



i

    

  1 N i

 

  



  (

 1

1

)







   ( )





 n



 



W s

( ) s F t s xyy

i yyy k i s F t s

( ) k t

( ) k t

( )

( , ) k i

( , )

(7)

i

x i

y

t

s

G

2

k

i

where W(i) are the weight functions, s i tip are directly related to the value of ߶ j 2 G K are computed, whereas t k

are the integration points at which the unknown functions and the displacements are the collocation points at which we evaluate the stresses. Stress intensity factors at the crack

(s) for s = +1:



 2 ( c  j

 i I II j x  , ,

(8)

y

1),

,

i







(

1)

߶ j at end-points are not included in this quadrature method, thus Krenk’s interpolation is used [11].

Values of

ߪ

ߪ

Interface constitutive law In Eqs. (6)-(7), the stresses on the crack surface

t n are related to the relative displacements by means of a constitutive law which describes friction and roughness. According to this law, the crack surfaces are assumed to be and

177