Issue 63

L. Nazarova et alii, Frattura ed Integrità Strutturale, 63 (2023) 13-25; DOI: 10.3221/IGF-ESIS.63.02

   ) σ x H f x ( , ( ),

  ) σ x H F x ( , ( )

(7)

xy

m

yy

y

    ( , ) ( , ) 0 , xx xy σ L y σ L y

where

x x x x

    1 0

m

 

m f x

( )

.

m

m { σ x y

The set

( , )}

of solutions of system (1)–(3), (7) is discrete Green’s function for the boundary value problem:

 1 m m m 2 ,...,

ij

 σ x H F x ( , )

 σ x H F x ( , ) ( )

( ),

xy

x

yy

y

   ) σ x H F x ( , ( ),

  ) σ x H F x ( , ( )

xy

y

yy

y

    ( , ) ( , ) 0 xx xy σ L y σ L y We represent the function ௫ as a sum m

   2 1 m m

x F x

m m C f x

( )

( )

(8)

with unknown coefficients C m which are determined by the LS method from the condition of the cost function minimum

2

 I m m   2     m

   

m





 C σ x y σ x y dxdy min  ( , ) ( , )

( C C , ...,

)

(9)

m

m

m

*

1

2

1

where

m σ x y

 m m xx yy

  ( , ) (1 )(

ν σ σ

) / 3

.

A system of linear equations with respect to C m can be derived from (9)

 m m  2 m

 K C Q nm m n

(10)

1



 *

m

n

n





 σ x y σ x y dxdy n m m ( , ) ( , ) , , ...,

K

( , ) ( , ) σ x y σ x y dxdy Q ,

.

mn

n

1

2

I

I

Solution of (10) together with (8) gives the desired distribution of the shear stresses σ xy ( x , H )= F x ( x ) at the boundary of the extraction sub-panel. The red dashed line (Fig. 6) shows the exact distribution of σ xy ( x , H ) at b =7 m, and the blue solid line shows the result of the inverse problem solution using the described procedure—the reconstructed function F x ( x ) which clearly indicates the existence of a weak zone in the section 6 ≤ x ≤ 8.

18