PSI - Issue 28

Vera Petrova et al. / Procedia Structural Integrity 28 (2020) 608–618 Author name / Structural Integrity Procedia 00 (2019) 000–000

613

6

a

n

a 

( ) g t dt 

0

(for internal cracks),



n



n

2





.





n g x 

( )

[ ] [ ] u i v 



n

n

( 1) 

i

x

 

The number of equations N is equal to the number of cracks. The unknown functions ( ) n g x  contain the shear [ u n ] and normal [ v n ] displacement jumps on the n -th crack line, μ = E /2(1+ ν ) is the shear modulus, E is Young’s modulus, ν is Poisson’s ratio, κ = 3 - 4 ν for the plane strain state, and κ = (3 - ν )/(1+ ν ) for the plane stress state. An overbar (...) denotes the complex conjugate. The regular kernels R nk and S nk determine the geometry of the problem and can be found in Petrova and Schmauder (2020) or in Panasyuk et al. (1976). In Eq. (10) the functions p n are known functions determined by the load on the crack lines, Eq. (9). 3.2. Numerical solution The singular integral equations (10) are solved numerically based on the Gauss-Chebyshev quadrature. Different versions of this method are used for the solution, and the effectiveness of the method has been proven in many studies (Erdogan and Gupta, 1972). In the present work, the version described in Panasuyk et al. (1976) is applied. Eqs. (10) are rewritten in dimensionless form with the non-dimensionless coordinates / k t a   and / n x a   , where 2 a k is the length of the k -th crack. The unknown function ( ) n g   presents as

2 g u       , ( ) ( ) / 1

n

n

where the function ( ) n u  is a bounded continuous function in the segment [-1,1] and 2 1 / 1   is the weight function, which is taking into account the square root singularities at the crack tips. For an edge crack, the function ( ) n g   possesses a singularity less than 1 / 1   at the edge point η = – 1, this condition is accounted for as ( 1) 0 n u   . Using Gauss’s quadrature formulae for regular and singular integrals, the singular integral equations are reduced to the following system of N x M ( N – number of cracks, M – number of nodes) algebraic equations

1

1 1   M N m k  





( ) ( , ) k m nk m r u R 

( ) ( , ) k m nk m r u S         ( ) n r p

,

(11)

 

M

1 2 1 ( 1) ( ) tan 4 m n m m u M     M m 

M 

0

( ) 0 

n m u 





(for edge cracks) or

(for internal cracks),

1

m



( n =1, 2, …, N ; r = 1, 2, …, M -1),

2 1 cos 2 m M 

r

M 

( m = 1, 2, …, M ),

( r = 1, 2, …, M -1).

cos







 

m

r

M is the total number of discrete points of the unknown functions ( ) n u  on the segment [-1,1]. By applying the conjugate operation to the system (11), additional N x M equations are obtained, i.e. 2 N x M equations should be solved, where N is the number of cracks. The functions ( ) n u  are calculated by the interpolation formula:

2

1

1

M

M

M 



( )    n m u

( ) 

( ) ( )  

( ) 

u

r m r T T

u





(12)

n

n m

M

M

1

0

0

m

r

m





