PSI - Issue 28

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

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1.2. Transformation the thermoelasticity equation to a new form

We introduce the notation for displacements in the cylindrical coordinate system and notation for functions Z ( r , φ, z , t ) and Z ∗ ( r , φ, z , t )(Popov, G. Ya (2002)) 2 G || u r , u φ , u z || = || U , V , W || , Z Z ∗ = 1 r   rU rV ± V U ˙   (6) Taking into account the ratios (5), (6) and after some operation applying the dynamic thermoelasticity equations A. D. Kovalenko (1965) take the form ∆ W + µ 0 Z , + ω 2 cW = α µ T , ∆ Z + µ 0 ∇ Z + ω 2 cZ = α µ ∇ T ∆ Z ∗ + ω 2 cZ ∗ = 0 ∇ f ( r , φ, z ) = r − 1 r f ( r , φ, z ) + r − 2 f ˙˙ ( r , φ, z ) Z = Z + W , (7) where T = T ( r , φ, z , t ) - temperature, ∆ - Laplace operator in cylindrical coordinates, partial derivative with respect to r is denoted as ( ), partial derivative with respect to φ is denoted as ˙ , partial derivative with respect to z is denoted as ( , ). Functions U , V are expressed though Z , Z ∗ ∇ rU rV = 1 r   r 2 Z r 2 Z ∗ ∓ Z ∗ Z ˙   (8) If the equations (7) are solved, i.e. if the functions Z , Z ∗ determined by formulas (6) are found, then to find the functions U , V , that determine the displacements, we should solve the equations (8).

1.3. Applying of integral transforms method

We use integral transformations with respect to variables φ and ρ (with assumption, that φ 0 = 0)

X n ( r , z ) =

φ 1 0 X ( r , φ, z ) cos µ n φ d φ, µ n =

( n − 1) π φ 1

, n = 1 , 2 , ...

(9)

X nk ( ζ ) =

1 a X n ( ρ, ζ ) φ n , 0 ( ρ, ν ) d ρ, ν = ν 0

k , k = 1 , 2 , ...

with invert transformations

2 φ 1

X 1 ( r , z ) 2

( n − 1) π φ 1

∞ n = 2 X n ( r , z ) cos µ n φ, µ n =

X ( r , φ, z ) =

, n = 1 , 2 , ...

+

(10)

X n ( ρ, ζ ) = ∞ k = 1 X nk ( ζ )

φ n ( ρ,ν ) || φ n ( ρ,ν ) ||

2 , ν = ν k , k = 1 , 2 , ...