PSI - Issue 40

A.I. Chanyshev et al. / Procedia Structural Integrity 40 (2022) 97–104 Chanyshev A.I. at al. / Structural Integrity Procedia 00 (2022) 000 – 000

98 2

1. Introduction There are various ways to diagnose the condition of a product: X-ray irradiation, MRI scanning, analysis of acoustic signals, pushing shock loads Stepanova et al. (2020), Bespal'ko et al. (2016), Berladir et al. (2016). In order to determine the SSS (stress strain state) mathematical models, which include the constitutive relations of the medium and establish links between stresses and strains at a point for various medium states, the equilibrium equations, and the conditions for deformation compatibility, are used Ambartcumyan (1982). In this case, it is necessary to know all information about the structure of the object under research Musheshvili (1966), Savin (1965). If there is no such information, then at any boundary of the object an additional survey of the displacement values is applied Vorovitch et al. (1979), Nazarov and Sher (1995). The complete set of inverse and ill-posed problems is given, for example, in Kabanihin (2018). Below it is proposed to diagnose a product, knowing only either the entire external boundary, or the geometry of one of its holes. So, in this formulation, the problem of determining all holes or solid inclusions and stress-strain state in the object is set. 2. Problem setting and methods to solve it There are statistical and dynamic problems in mechanics (mathematics). These problems, depending on the factors at the highest derivatives with respect to spatial variables, are subdivided into elliptic, parabolic, hyperbolic Korn and Korn (2003). To be more specific, let us consider the Laplace equation of the elliptic type:

2    

2 u u

0

(1)

 

2 x y

2

For equation (1), there are classical settings such as of Dirichlet, Neumann, Robin Godunov (1979). Let consider (1) to solve the problem for a circle of radius r a  when the function itself

(2)

2 ( )  

(Dirichlet problem),

r a u

 

and its normal derivative

( ) 2 a   

u r

 

(3)



(Neumann problem)

r a 

( )    - derivative of the function ( )   with respect to

(here coefficients “2” in (2), (3) are taken for convenience,

angle  ) are simultaneously specified on this boundary. Classically Gritsko (1976), such a setting is absurd because each of the statements (2) or (3) gives the unique solution, but only in the functions that are analytic in the circle with the radius r a  . That is, the function ( , ) u u x y  should not have any poles inside the circle, be limited everywhere, including the center of the circle 0 r  . But if we assume that inside the circle there are some poles to be found, then the classical solutions do not work, as the function ( , ) u u x y  is no longer analytical, and the task is to restore the function ( , ) u x y according to conditions (2), (3). The solution of this problem is shown below. Let's use the complex plane z x i y   . Equation (1) in complex variables , z z can be written as

2 u z z 

0

(4)



 