PSI - Issue 52

Carlos A. Galán Pinilla et al. / Procedia Structural Integrity 52 (2024) 20–27

22

Carlos A. Gala´n-Pinilla et al. / Structural Integrity Procedia 00 (2023) 000–000

3

Fig. 1. Schematic representation of the coated plate.

=   

N 1 ( η ) 0 N 2 ( η ) 0 ... N n ( η ) 0 0 N 1 ( η ) 0 N 2 ( η ) ... 0 N n ( η )  

u y u z 

u = 

 u n ( z ) = N ( η ) u n ( z )

(1)

The symbol N ( n ) denotes the shape functions with respect to the displacements. The nodal displacements are unknown functions at the y-coordinate. Using Eq. 1 for the case, the rectangular cross-section is in the ( x , y ) plane, and the wave propagation along the z axis with wavenumber k and frequency ω the deformation-displacement relation of the element is expressed in terms of the nodal displacements as:

where L y = 

0 0 1 0 1 0 

L z = 

1 0 0 0 0 1 

= Lu =  L z

∂ ∂ y 

T

T

ε =  ε y ε z ε yz  T

T

(2)

+ L y

u ;

∂ ∂ z

,

where the superindex T denotes the transpose and L is a two-dimensional di ff erential operator. The constitutive relations at a point are given by σ =  E ε , where  E is the complex modulus of elasticity used for the coating material, Eq. 4. The real part of the wave number represents the wave propagation, and the imaginary part describes the wave attenuation Bartoli et al. (2006). Following the Kelvin-Voigt model, the attenuation of waves varies linearly with frequency Zhang et al. (2021); therefore, the complex velocities of longitudinal and transverse waves are defined as a function of damping parameters K La and K Sa , which for viscoelastic material must first be calculated as follows,  V L , T = V L , T  1 + i K La , Sa 2 π  − 1 (3) where  V L , T are the longitudinal and transverse bulk wave complex apparent velocities for the viscoelastic material, V L , T the longitudinal and transverse bulk wave velocities. The complex modulus of elasticity  E and Poisson’s ratio  ν