PSI - Issue 52

Carlos A. Galán Pinilla et al. / Procedia Structural Integrity 52 (2024) 20–27 Carlos A. Gala´n-Pinilla et al. / Structural Integrity Procedia 00 (2023) 000–000

23

4

are defined in Eq.4,

1 2     C 2  C 2

T   

2 T  

2 T   

L − 2  C 2 T L −  C

3  C 2  C 2

L − 4  C 2 T L −  C

 E = ρ  C 2

 ;  ν =

(4)

With this approach the viscoelastic elasticity matrix  D of the coating in Eq. 5 is based on the complex velocities of the longitudinal and transverse waves, which are constant over the entire frequency range examined,  D =  E (1 +  ν )(1 − 2  ν )     1 −  ν  ν 0  ν 1 −  ν 0 0 0 1 − 2  ν 2     (5) The model is implemented by discretizing the thickness in the direction of the local coordinates η using a spectral element for the elastic material Ω 1 and a spectral element for the viscoelastic material Ω 2 (see fig. 1). The high-order shape functions are generated from the Lagrangian interpolation functions from the defining multiple nodes for a single finite element per material, with the Gauss-Lobatto-Legendre placement GLL Gravenkamp et al. (2021). For the spectral elements, two nodes are placed at the extremes, η = − 1andat η p + 1 = 1, and the nodes p − 1 corresponding to the internal points of the GLL -distributed elements, which are obtained when they satisfy the condition: L 0 p − 1 ( η ) = d d η P p ( η i ). Thus, P p denotes the Legendre polynomial of order p . The GLL quadrature points are calculated for an element of the required order as follows: For a one-dimensional element of order p with local coordinates η , from − 1 to 1, the internal nodes of the GLL quadrature are the roots of the Lotatto polynomial of order p − 1, which is defined as the first derivative of the Legendre polynomial P p ( η ) of order p . The derivative of the Legendre polynomial P p of order p − 1 yields the Lobatto polynomial L , so that the location of the nodes are the roots of the polynomial (1 − η ) 2 L ′ p ( η ). In this way, the Legendre polynomials for the given order can be obtained as L 0 ( η ) = 1, and L 1 ( η ) = η and L k ( η ):

2 k − 1 k

k − 1 k

L k ( η ) =

η L k − 1 ( η ) −

L k − 2

for k = 2 , 3 , 4 , ...

(6)

With the nodal positions defined under the GLL scheme, the N i shape functions are calculated using the Lagrange equation and local coordinates ( η,ξ ). The semi-analytic equation to estimate the couples ( ω,λ ), the foundation of the dispersion curves, is obtained using the principle of virtual work by inserting the kinetic and potential energies under stress-free boundary conditions. As a result, the generalized semi-analytic wave Eq. 7 of the SBFEM method is obtained. Starting from equilibrium in the ratio of the internal nodal forces q n in the finite element direction and in the scaling direction z , it can be assumed that:

1  E T

1  u − q n  − λ E T

0  E

T 1  u n − q n  − E 2  u n + ω

1  u n + E 1 E − 1

2 M

λ E 0 E 0 −

0  u 0 = 0

(7)