PSI - Issue 3

Nataly Vaysfeld et al. / Procedia Structural Integrity 3 (2017) 526–544 Author name / Structural Integrity Procedia 00 (2017) 000–000

534

9

Realisation of the standard scheme of orthogonal polynomial methods leads to a system of linear algebraic equations with regard to the coefficients of the expansion (21)

1 0  N j n   

i

ij

ij

 

 

(22)

, g C h i   A B C mn mn mn j n i m i i m   

1, ,

0,1, 2,...

k

N m









m i m

1 

1 

2



ln 2,

0

m

 



T



d



  



   n

 A T i mn

where

, 

,

l

d

k

  

  

m

i

m

1



 

2

2

2

2 ,

1

m m



1

1











1

1





1 

1 

1 

1 

T

T





d

d





  

    ,   n

  

    ,   n

 B T ij mn

 C T ij mn

, 

, 

M

d

L

d

m

ij

m

ij

2

2

2

2

1

1

1

1

















1

1

1

1









1 

1 

T

T

 

 

     m

     m

, 

. 

g

g

d

h

h

d





im

i

im i

2

2

1

1





1

1





, i C so why one can search the solution of the

Right hand parts of the system (22) have the unknown constants

system as the linear combinations

 N

0

s

(23)

, C i  s i m

1, ,

0,1, 2, ...

N m

  i m i m

 





1

s



1 0  N j n   

where 0 

0

0

 s i m are the solutions of the

i

ij

ij

 

 

i m are the solutions of the system

,

k

   A B C mn mn mn jn i m g 





m i m

1 0  N j n   

1,     0,

i s

s

i

ij

ij

s

 

 

system

, h s N 1, ,  

.

k

   A B C mn mn mn j n i m i s 







 

m i m

i s

i s

After solving these systems and calculating the coefficients 0 

i m and  s

i m one must find the constants . i C Such

  ,  i t should be found from the system of the integral

grounds can be used with this aim. The functions equations(18) and should satisfy the conditions

 i

 i

  i d t dt G t t dt G t t dt which arise from the closeness condition of the crack   executed change of the variables are equivalent to the next ones     1 1 | 0,   i i  i  i  i  i            

  0.       i i i i These conditions with regard to the

1 

1 

  

 

2 1    i      n T

2   i      

       d

exp

exp

0,   i

1, , N

d



i

i n

2  

0

n



1

1





Taking into consideration formula ( 2.18.1.10 , Prudnikov et al (1983)) and representation (23) one gets the system of linear equation to find the constants i C

N



 

1       i 2 

2   i     

1 , 1, .  i N

  s

0

s

 C I i n n

I

 



i n n

1

0

0

s

n

n





