PSI - Issue 52

732 Feifei Ren et al. / Procedia Structural Integrity 52 (2024) 730–739 Author name / Structural Integrity Procedia 00 (2023) 000–000 3 The analytical solution implements an exponential time harmonic term of the form: e i ( ξ r − ω t plate ) , where i is the imaginary unit, ξ is the wavenumber, ω = 2 π f is the angular frequency. The displacement field can thus be written as:

e e i ( ξ r − ω t plate )

u e

= N ( ς ) Q

(1)

N ( ς ) =   

  

N 1 0 0 N 2 0 0 N 3 0 0 0 N 1 0 0 N 2 0 0 N 3 0 0 0 N 1 0 0 N 2 0 0 N 3

(2)

=  U x 1 U y 1 U z 1 U x 2 U yz U zz U x 3 U y 3 U z 3  T

Q ( e )

(3)

In order to discretize the cross-section, 3-noded isoparametric quadratic line elements are utilized. These elements employ shape functions defined as follows: N 1 = ς 2 − ς 2 , N 2 = 1 − ς 2 , N 3 = ς 2 + ς 2 . Here, ς represents the variable in the local coordinate system that corresponds to the local coordinate of the line element. The values of ς are assigned as ς = − 1, ς = 0, ς = 1 at nodes 1 through 3, respectively. The strain and stress fields in each element can be determined using the following equations:

=  L x

∂ ∂ z 

∂ ∂ x

∂ ∂ y

(e) e i ( ξ r − ω t plate )

ε ( e )

u e

= ( B 1 + i ξ B 2 ) Q

+ L y

+ L z

(4)

( e ) ε ( e )

( e ) ( B

(e) e i ( ξ r − ω t plate )

σ ( e )

= C

= C

1 + i ξ B 2 ) Q

(5)

where C ( e ) represents the material sti

ff ness matrix of the element, and

    

    

    

    

    

    

1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0

0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0

L x =

, L y =

, L z =

(6)

B 1 = L x N , y + L z N ,z , B 2 = L x N

(7)

By utilizing a conventional finite element assembly approach on each element and implementing traction-free boundary conditions on the upper and lower surfaces of the plate, an eigenvalue problem can be established within