PSI - Issue 28

H. Lutsenko et al. / Procedia Structural Integrity 28 (2020) 770–775 H. Lutsenko / Structural Integrity Procedia 00 (2020) 000–000

775

6

and transformations Z nk ( ζ ), Z ∗

nk ( ζ ) ≡ 0 fully defined, but for a full solution we need to find U nk ( ζ ), V nk ( ζ )

U ( a 1 ρ,φ, a 1 ζ ) 2 G V ( a 1 ρ,φ, a 1 ζ ) 2 G

U ( ρ,φ,ζ ) 2 G V ( ρ,φ,ζ ) 2 G

u r ( r , φ, z ) = u φ ( r , φ, z ) =

=

(23)

=

To find them we will use equations (12). If we take into account the equality (33) and variable replacement from (18), and notation

rY ( r , φ, z ) = a 1 ρ Y ( ρ, φ, ζ ) = a 1 Y ∗ ( ρ, φ, ζ )

, Y ( ρ, φ, ζ ) =

Y ( r , φ, z ) =

U ( r , φ, z ) V ( r , φ, z )

U ( ρ, φ, ζ ) V ( ρ, φ, ζ ) etc .

(24)

we can write down di ff erential equations

∇ Y ∗ ( ρ, φ, ζ ) = a 1 ρ −

ρ, φ, ζ )

1 ρ 2 Z ( ρ, φ, ζ ) a 1 Z ˙ (

(25)

The exact solution was obtained by applying of sequence of the integral transforms to each equation:

a 1 ν 2

U ∗

1 a ρ

n π φ 1

2 Z n ( ρ, ζ ) φ n , 1 ( ρ, ν ) d ρ, µ n =

, ν = ν 1

nk ( ζ ) =

k , n , k = 1 , 2 , ...

(26)

ν 2

φ 1 0

V ∗

1 a ρ Z ( ρ, φ, ζ ) φ n ∗ ( ρ, ν ) cos ( µ n φ ) d φ d ρ, µ n = n π φ 1

a 1 µ n

, ν = ν ∗ k , n , k = 1 , 2 , ...

nk ( ζ ) =

where

φ n ( ρ, ν ) = N µ ( a ν ) J µ ( νρ ) − J µ ( a ν ) N µ ( νρ ) a φ n ∗ = J µ ( νρ ) ν aN µ ( a µ ) − N a µ − N µ ( νρ ) ν aJ µ ( a ν ) − J µ ( a ν ) N µ ( a ν ) J µ ( ν ) − J µ ( a ν ) N µ ( ν ) = 0 , ν = ν 1 k ν J µ ( ν ) − J µ ( ν ) ν aN µ ( a µ ) − N a µ − ν N µ ( µ ) − N µ ν aJ µ ( a ν ) − J µ ( a ν ) = 0 , ν = ν ∗ k

(27)

Inverted transformations have the same form as (10), but with another eigenvalues. Thus, displacement transformations U ∗ nk ( ζ ) , V ∗ nk ( ζ ) and W ∗ nk ( ζ ) are found and defined by formulas (26), (20), (17). By inverting transformation formulas (10) we find originals, and by using of formulas (24), (23), (20) we find displacements, thus, we found the exact solution of the boundary problem.

References

Popov G,. Ya. 2002. Exact solutions of some mixed problems of uncoupled thermoelasticity theory for a finite hollow circular cylinder with a groove along the generatrix. Journal of Applied Mathematics and Mechanics 66(4), p.673-682 (2002). A. D. Kovalenko, 1965. An Introduction to Thermoelasticity [in Russian], Naukova Dumka, Kiev (1965) E. M. Kartashov, 2001. Analytical Method in the Theory of Thermal Conductivity of Solids [in Russian], Vysshaya Shkola, Moscow (2001). V. T. Grinchenko and V. V. Meleshko, 1981. Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kiev (1981). Popov G. Ya., 1999. Green functions and matrices of one-dimensional problems [in Russian] - Almaty: Rayan, 1999. 133 p.