PSI - Issue 72

Alexander Kamenskikh et al. / Procedia Structural Integrity 72 (2025) 252–259

254

where σ ij are the components of the Piola – Kirchhoff stress tensor of the second kind, δ ij is the Kronecker symbol, u i are the components of the vector of unknown displacements, ρ is the density. The geometric relationships:   , , , , 1 , 2 ij i j j i k i k j u u u u    

(2)

where ε ij are the components of the Green strain tensor. the physical relationships:   th , ij ijkl kl kl C    

(3)

where C ijkl are the components of elastic constant tensor, th kl kl T    are the components of temperature strains tensor,

α kl are the coefficients of linear thermal expansion. and the following kinematic boundary conditions: 0, . i u u S   x

(4)

The formulation of the free vibration problem taking into account the preliminary stress-strain state is based on the linearization of relations (1) and (2) with respect to the state characterized by a small deviation from the initial equilibrium position due to a temperature load (Nedin et al., 2018; Kamenskikh et al., 2024). The solution is represented in the form of an exponent ( , ) ( )exp(i ), t t   u x u x where ( ) u x is a function depending only on the coordinates x , i is the imaginary unit,  is the natural frequency of vibration, t is the time. After transformations and a sequence of simplifications, we obtain the following variational equation written in matrix form (Kamenskikh et al., 2024; Kamenskikh et al., 2025):   T 2 T T T 1 1 1 1 1 0 1 1 0 1 d d d d s V V V V V V V V                  u u ε cε s X s s X s   T T T T T 1 0 1 1 0 1 1 0 0 1 d d d 0, V V V V V V             ε cS s s S cε s S cS s (5) where the subscripts “0” and “1” correspond to the initial equilibrium state and a small deviation from the latter, and all unknown quantities should be treated in terms of their coordinates (the wavy line at the top is omitted), V is the volume of the structure,   T 1 2 3 , , u u u  u is the displacement vector, c is the matrix of elastic constants  is the vector, containing the components of finite strain tensor (2):

1



 T

1 2 3 12 13 23 2           ε ε Ss , , , , , ,

(6)

T

3 u x x x x x x x x x                  ε , T 3 3 3 1 1 1 2 2 2 u u u u u u x x x x x x x x x                s , 3 3 1 2 1 2 1 2 1 2 3 2 1 3 1 3 2 , , , , , u u u u u u u u      1 2 3 1 2 3 1 2 3 , , , , , , , , u u u