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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investigated in E. M. Kartashov (2001) and the theory of Green’s matrix was investigated by Popov G. Ya. (1999). The main goal of deriving of the exact solutions for the finite bodies is research of the resonance of the body, i.e. deriving of the disperssion equation and finding of the eigenfrequencies (T. Grinchenko and V. V. Meleshko (1981)). In this paper the formulas for the displacements and stresses were derived in explicit form.

Nomenclature

G

shear modulus

µ coe ffi cient of Poisson α T coe ffi cient of linear thermal expansion ω steady state frequency c density

1.1. Statement of the problem

We consider the steady state problem of incoherent thermoelasticity for a finite hollow elastic cylinder, which occupies the region in a cylindrical coordinate system ( r , φ, z )

a 0 ≤ r ≤ a 1 , φ 0 ≤ φ ≤ φ 1 , h 0 ≤ z ≤ h 1

(1)

On the inner and outer cylindrical surfaces ideal contact conditions are given

u r ( a i , φ, z , t ) = 0 , τ r φ ( a i , φ, z , t ) = τ rz ( a i , φ, z , t ) = 0; u φ ( r , φ i , z , t ) = 0 , τ φ r ( r , φ i , z , t ) = τ z φ ( r , φ i , z , t ) = 0; i = 0 , 1

(2)

Without limiting the generality of reasoning, we consider a dynamic normal load applying at the upper face of the cylinder, and there are no tangential stress there. σ z ( r , φ, h 1 , t ) = − p ( r , φ ) e i ω t , τ rz ( r , φ, h 1 , t ) = τ z φ ( r , φ i , h 1 , t ) = 0 (3) The bottom face of the cylinder is fixed u r ( r , φ, h 0 , t ) = u φ ( r , φ, h 0 , t ) = u z ( r , φ, h 0 , t ) = 0 (4) The cylinder is under steady state oscillations, so f ( r , φ, z , t ) = f ( r , φ, z ) e i ω t (5) We assume that the temperature field is already known. We note, that the problem posed in this way is equivalent to solving a similar boundary-value problem for dynamic inhomogeneous Lame equations with volume forces of a special form.