PSI - Issue 40

I.G. Emel’yanov et al. / Procedia Structural Integrity 40 (2022) 124–128 I. G. Emel’yanov / Structural Integrity Procedia 00 (2022) 000 – 000

126

3

ds dY

P Y f ij   , ( ,

1,2,....6)  i j ,

( 0 L s s s  

)

(1)

with boundary conditions

1 1 0 ( ) B Y s b  ,

2 2 ( ) B Y s b L 

(2)

Here , , , } , { , s r z s r z Y N N M u u   – vector-function of the required solution N r , N Z – radial and axial forces; u r , u Z – displacement; M s – meridional bending moment; S  – angle of rotation of the normal to the shell surface. Matrix elements P ij depend on the geometric and mechanical parameters of the shell, f – the vector depends on the loads applied to the shell. i B – given matrices; i b are the given vectors. Matrix elements P ij and the column vector f are in Shevchenko and Prokhorenko (1981). When solving the linear boundary value problem (1), the Runge-Kutta method with discrete orthogonalization and normalization of S.K. Godunov is used, Grigorenko and Vasilenko (1981). Since the muffle shell is operated at high temperatures, plastic deformations are possible. Taking into account the plastic deformation of the material, the problem of determining the stress state becomes nonlinear. The problem will be described by the same system of equations (1), and the relationship between stress and strain will be linearized by the method of additional strains. This relationship is presented in the form of Hooke's law, but with additional terms that take into account the dependence of the mechanical properties of the material on deformation and temperature, Babeshko and Shevchenko (2007), Babeshko and Shevchenko (2011), Shevchenko (1994). In this case, the three dimensional stress state of the shell will be compared with the one-dimensional state upon simple stretching of the sample

 * 

3   S

1

 H



,

,

(3)

3

where  and  are stresses and strains during simple tension of the sample, and *  deformation.Shear stresses S and shear strains and H for the shell are defined as

the coefficient of transverse

2

2

,

(4)

(1/ 3) ( 

)    

S



 

s

s





,

) ( 2      ) ( 2     

) ] 2

(1/ 6) [( 

H



s 

s









where s  and   are the meridian and circumferential stresses, respectively, and s  ,   ,   are the components of deformations along the meridian, circumference and normal to the shell surface. Muffle shell temperature during operation. For the structure under study, we will assume that the temperature along the body of the shell in the direction of the meridional and circumferential coordinates, spreads uniformly. Considering the small thickness of the shell, the temperature across the thickness will be the same. Consequently, the mechanical properties of the material are assumed to be the same for all points of the muffle shell. And as it warms up, the properties of the material will change in the same way at all points of the muffle shell, in proportion to one parameter – time t . Mechanical parameters of the muffle material. As mentioned above, the shell of the muffle is made of St3ps steel. It is known that the range of application of elements made of steel St3 is wide enough, Zubchenko (2003). But heating structures above 400 ° C leads to a strong drop in the yield strength and ultimate strength of this steel. Therefore, the mechanical properties of steel at temperatures above 500 ° C have not been studied.