# 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

135

5

(0.5(1 ~),1,2ln(

/ )) 0 r r r u U out out out

. / )) / )) (0.5(1 ~),1,2ln( (0.5(1 ~),1,2ln( / )) (0.5(1 ~),1,2ln( , / )) / )) (0.5(1 ~),1,2ln( (0.5(1 ~),1,2ln( / )) (0.5(1 ~),1,2ln( in 0 in 0 out 0 in 0 r r r r U r r r r r r U r r

in in r u U

C

1

/ )) / )) (0.5(1 ~),1,2ln( (0.5(1 ~),1,2ln( r r M r r U M

in 0

out 0

(0.5(1 ~),1,2ln(

/ )) 0 r r r u M out out out

in in r u M

C

2

/ )) / )) (0.5(1 ~),1,2ln( (0.5(1 ~),1,2ln( r r M r r U M

in 0

out 0

out 0

in 0

Substituting solution (19) into formulas (2), (3) or (4) we obtain expressions for deformations and stresses:

[1 0.5(1 ~) / ln( / )][

(0.5(1 ~),1,2ln( / ))

(0.5(1 ~),1,2ln( / ))] r r

r r CM

r r CU

( ) r

0

1

0

2

0

r

2

r

0.5 (1 ~) ( 0.5(1 ~),1,2ln( / )) C M

( 0.5(1 ~),1,2ln( / )) r r

r r CU

,

1

0

2

0

2

ln( / )

r r r

0

(0.5(1 ~),1,2ln( / ))

, (0.5(1 ~),1,2ln( / )) 0 r r

1 CM

r r CU

( ) r

0

2

2

r

(1 ~)[1 0.5/ ln( / )][

(0.5(1 ~),1,2ln( / ))

(0.5(1 ~),1,2ln( / ))] r r

r r CM

r r CU

(20)

( )

( )

0

1

0

2

0

r K r

r

2

r

[0.5 (1 ~) ( 0.5(1 ~),1,2ln( / )) C M

( 0.5(1 ~),1,2ln( / ))] r r

r r CU

,

1

0

2

0

2

ln( / )

r r r

0

(1 ~)[1 0.5~ / ln( / )][

(0.5(1 ~),1,2ln( / ))

(0.5(1 ~),1,2ln( / ))] r r

r r CU

r r CM

( )

( )

0

1

0

2

0

r K r

2

r

~[0.5 (1 ~) ( 0.5(1 ~),1,2ln( / )) C M

( 0.5(1 ~),1,2ln( / ))] r r

r r CU

.

1

0

2

0

2

ln( / )

r r r

0

4. Discussion and conclusions The obtained formulas (19) and (20) for displacements, strains, and stresses allow us to accurately analyze the processes occurring in an inhomogeneous elastic cylinder during its axisymmetric deformation. It can be shown that the averaging of elastic modules can lead to a significant error in the analysis of stress levels arising in an inhomogeneous cylinder. For example, in the case where the inner and outer radii are 20 mm and 30 mm, respectively, the Poisson's ratio is 0.35, and the Young’s modulus varies logarithmically from 2.5 GPa to 3.5 GPa, the radial strain graph has the form shown in Fig. 1 by the solid curve (internal tension is 0.2 mm, external tension is – 0.2 mm). If we average the value of the elastic modulus over the thickness, that is, take it equal to 3 GPa, then the strain levels change significantly on the radial strain graph (Fig. 1, dashed curve).

Fig. 1. Radial strain for inhomogeneous (solid curve) and homogeneous (dashed curve) cylinders.

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