PSI - Issue 50

Kirill E. Kazakov et al. / Procedia Structural Integrity 50 (2023) 131–136 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

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including additive manufacturing, has made it possible to produce materials with functionally gradient properties when the properties continuously change inside the body. Similar properties near the outer surfaces of the material can also be acquired during the surface treatment of finished products. The hollow cylinders with such properties have attracted attention of many scientists and engineers, and therefore, in the recent years, many studies have been conducted related to the study of the stress-strain state of functionally gradient bodies under various operating conditions. In some works, such studies were carried out only using numerical methods. Another class of works was devoted to the approach according to which an inhomogeneous cylinder was replaced by a multilayer cylinder such that each of its layers is homogeneous and an analytical solution was constructed for each of these layers. Essentially, it was also a numerical method. The most interesting works from the point of view of fundamental results are the studies devoted to obtaining analytical solutions for problems of deformation of inhomogeneous cylindrical bodies. This paper discusses the general formulation of the problem of deformation of a radially inhomogeneous cylinder and shows the process of constructing a solution in the special case of specifying the inhomogeneity of the material. 2. Statement of the boundary value problem A circular cylinder made of an inhomogeneous elastic material is strained on a rigid shaft. A rigid circular coupling is put on the outside of the specified cylinder. The radii of the shaft, the cylinder and the coupling are known. Due to the difference in the radii, the cylinder is deformed. It is necessary to find the displacements, deformations, and stresses arising in the cylinder assuming that the Young's modulus changes along the radial coordinate according to the logarithmic law and the Poisson's ratio is constant. Consider the cases of plane-stressed and plane-deformed states. First, we obtain equations describing a mathematical model, in the general case of an axisymmetric problem, when both the Young's modulus and the Poisson's ratio depend on the radial coordinate. To do this, we write the basic equations in cylindrical coordinates: 1) the radial stress ( ) r r  and the hoop stress ( ) r   must satisfy the equilibrium equation:

( ) r

( ) r    

( )

dr d r r



0;

r





(1)

r

2) the radial strain ( ) r r  and the tangential strain ( ) r   are related to the radial displacements ( ) u r by

( )

( )

dr du r

r u r

( )

,

( ) r

;

r r





 



(2)

3) Hooke’s law, relating the strains to the stresses, can be written 3.1) in the case of plane stress:

( )

( )

E r

E r

(3)

( ) r

[ ( ) ( ) ( )], r r r 

( ) r    

[ ( ) ( ) ( )]; r r r 

r 





  

r

r







2

2

1

( ) r

1

( ) r









3.2) in the case of plane strain:

       r

  

       

  

1 ( ) 2 ( ) ( )[1 ( )] 2    r r r E r

1 ( ) ( )  r r  

1 ( ) 2 ( ) ( )[1 ( )] 2    r r r E r

1 ( ) ( )  r r  

(4)

( ) . r

( ) , r

( ) r

( ) r

( ) r

( ) r









r 





r 





Here ( ) E r and ( ) r  are the Young's modulus and the Poisson's ratio depending on the radial coordinate. Then from equations (1) – (4) it is possible to obtain the differential equation: