PSI - Issue 65

Anvar Chanyshev et al. / Procedia Structural Integrity 65 (2024) 56–65 Anvar Chanyshev / Structural Integrity Procedia 00 (2024) 000–000

62

7

      

 

1 y y      y y     1

 

  

y

1 u F x  x

2 F x 

... ,

  

 

1 

(20)

 

 

  

y

1 u G x  y

2 G x 

....

  

 

1 



1

1

The system of equations (18) under the constraints (19) constitutes a system of four equations for determining required functions 1 F , 2 F , 3 F , 4 F . Its solution, being a solution to a system of linear equations, exists and is unique. For an initially isotropic medium, the roots of the characteristic equation (11) coincide with the numbers i  . In this case, the Kolosov-Muskhelishvili formulas (Muskhelishvili 1977) are valid. As an example, let us solve the problem for an initially isotropic medium, with the following boundary conditions (Chanyshev at al 2011):

Ax

AH

0 y   ,

0 xy   ,

(21)

u

u

2 ,

,





x

y

2 x b 

2 x b 

2

where A , H , b are approximate parameters of the boundary displacements. Note, that the boundary function ( ) y y u u x  is an even function, ( ) x x u u x  is an odd one. The graphs of these functions are presented below (Fig. 2):

(b)

(a)

Fig. 2. Variation of functions (a): y y u u x  determined by formulas (21). The function y u describes the extrusion of the material along the y -axis, and the function x u describes the displacements of points to the left and right of the centre point during extrusion. In order to solve the problem, by analogy with the previous study, the complex Kolosov-Muskhelishvili potentials ( ) z  , ( ) z  , ( ) z  , ( ) z  are set up (Muskhelishvili 1977). It is assumed that the material of the half plane is an initially isotropic medium with elasticity constants E and  . The potential ( ) z  is defined here as up (Muskhelishvili 1977) ( ) x x u u x  , (b): ( )

2 ) Ab z iHz z b   2 2 2 2

2



( ) z  

,

(22)

2  

1 (

potential is equal to