PSI - Issue 47

Sergei Kabrits et al. / Procedia Structural Integrity 47 (2023) 513–520 S. Kabrits and E. Kolpak / Structural Integrity Procedia 00 (2019) 000 – 000

516 4

Eh

Eh

(

),

(

),

( = +

) ,

( = +

) ,

T

T

M D

M D

=

  +

=

  +

 

 

1

1

2

2

2

1

1

1

2

2

2

1

2 

2 

1

1

−

−

sin sin

3

0 r   − 0

r

Eh

(1 ),

(1 ),

,

,

D

1 

= +

 

= +

 

=

=

1

2

2

2

0 2 12(1 )  − s is the arc length of the undeformed meridian; r and z are the radial and axial coordinates of the median surface of the shell;  is the angle between the normal to the median surface and the axis of revolution; 1  and 2  are relative elongations of the meridian and parallel; 1  and 2  are the change in the curvature of the meridian and parallel; 1 T , 2 T , and n T are the resultant forces, 1 M and 2 M are the resultant moments; n q is normal pressure; E is the Young modulus, and  is the Poisson ratio. All the values with index “0” refer to an undeformed configuration . Solution of nonlinear boundary value problem (2) in a spatial variable – at each time step – is carried out by the method of multiple shooting (see Keller (1968), Kabrits and Kolpak (2015, 2016)). To discretize equations (1) in time, the Euler method is used. The convergence of the methods is checked by reducing the sampling steps both in spatial coordinate and in time. 4. Results and discussion To illustrate the behavior of the compound shell under the mechanochemical corrosion conditions, solutions of two problems are presented below. In these examples, corrosion kinetics constant were a i = a o = 0.0001 (m /year); m i = m o = a i /200 (m/(year MPa)). First, the effect of mechanochemical corrosion on the shell composed of a spherical lid (hemisphere) and a cylinder is considered. Both objects have the same radii ( R cyl = R sph ) and the same initial thicknesses ( h 0 = 0.02 R cyl ). It is important that this ensures the continuity of change in the angle of inclination of the tangent along the meridian of the shell (with the origin at the top of the hemisphere – Figure 2). r

Fig. 2. The shape of a quarter of the shell (left) and the relative stresses along the shell meridian at time t 2 (right).