PSI - Issue 52

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

24

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

5

With this new equation derived with the SBFEM, the problem is linear organized in matrix form as follows:

1 0 Coe ffi cient matrix Z E − 1 0 E T 1 − E − 0 − E 1 E − 1 E 1 E − 1 0 E T 1 − E 2 + ω 2 M 0

q n

= λ u n

q n

u n

(8)

The equation 8 is a function of the coe ffi cient matrices (sti ff ness matrices), E 0 , E 1 , E2 and the mass matrix M 0 . In this work, they are solved numerically by Gaussian quadrature using the η i coordinates defined in Eq. 9 and 10 as integration points, as follows:

p + 1 i = 1

p + 1 i = 1

T 1( η i ) DB 1( η i ) |

T 2( η i ) DB 1( η i ) |

E 1 =

w i B

w i B

E 0 =

J | ,

(9)

J | ;

p + 1 i = 1

p + 1 i = 1

T 2( η i ) DB 2( η i ) |

T ( η i ) ρ N ( η i ) | J | ;

E 2 =

M 0 =

w i B

w i N

J | ;

(10)

Where the absolute value | J | of the Jacobian determinant and the matrices B 1 , B 2 are definided as

1 y ,η

B 1 = b 1 N ; B 2 =

b 2 N ,η ,

(11)

with N η as derivatives of the interpolation functions with respect to η .

3. Algorithm for Computing Dispersion Curves by SBFEM

Dispersion curves have been generated for an ASTM A 106 B steel plate of 6 mm thickness, Young’s modulus E = 212 GPa , ν = 0.29, ρ = 7750( kg / m 3 ), C t = 3184( m / s ), C l = 5987( m / s ) including two types of coatings: a layer of 2 mm of bitumen, ρ = 1200( kg / m 3 ), C l = 1900( m / s ), C s = 850( m / s ), attenuation coe ffi cients K La = 0 . 047( Np /λ ) and K Sa = 0 . 597( Np /λ ) and a single-layer fusion bonded epoxy coating of 2 mm , ρ = 800( kg / m 3 ), C l = 1900( m / s ), C s = 860( m / s ), attenuation coe ffi cients K La = 0 . 026( Np /λ ) and K Sa = 0 . 0869( Np /λ ) Hua and Rose (2010). The first step is to set the model parameters by defining the material properties, thicknesses, complex elastic modulus, and Poisson’s ratio, which are calculated with Eq. 4. The second step is meshing the plate and the cladding, where shape functions order can be independently selected for each spectral element. The third step is obtaining the global matrix concatenating every element matrix. The fourth step is the determination of the coe ffi cient matrices E 0 , E 1 , E 2 and M 0 using the Gaussian quadrature method. Finally, the linear eigenvalue problem of the Eq. 8 is solved to find the eigenvalues ω and their corresponding eigenvectors λ for each value of the wavenumber k .

4. Results and Discussion

With the implemented SBFEM method, the dispersion curves and wave structures for an elastic-viscoelastic (steel bitumen) bilayer material have been generated e ffi ciently. The numerical results obtained for the ( ω, k ) values of the uncoated metal plate are presented in green color in Fig. 2(a) to be compared with the black-colored results for the coated plate. The labels of the wave modes shown in Fig. 2 were identified from the wave structures. In Fig. 2(b), the coating material presents a frequency shifting in the di ff erent modes curves heading to higher cut-o ff frequency