PSI - Issue 52

Feifei Ren et al. / Procedia Structural Integrity 52 (2024) 730–739 Author name / Structural Integrity Procedia 00 (2023) 000–000

733

4

the global coordinate system. Through the application of matrix properties, the new equation can be derived:

2 K

2 M ) Q

( K 1 + i ξ K 2 + ξ

= 0

(8)

3 − ω

As an alternative approach, the Equation can be restructured into a linear eigenvalue form:

[ A − ξ B ] Q ξ Q

= 0

(9)

where

A =

2 M 2

, B =

− K 3

K 1 − ω

2 M 0

0

K 1 − ω

(10)

2 M K

0

K 1 − ω

2 d ς , M =

with K 1 =

1 d ς , K 2 =

1 ) d ς , K 3 =

1 − 1

1 − 1

1 − 1

1 − 1

C ( e ) B

T C ( e ) B

T C ( e ) B

T C ( e ) B

B T 1

ρ ( e ) N T N d ς .

B 2

( B 1

2 − B 2

The phase velocity can be calculated by c p = ω ξ . The determination of group velocity is essential for interpreting signals, as it provides insight into the types of modes traveling through the structure. The conventional method for computing group velocity involves calculating the derivative of angular frequency ( ω ) with respect to wavenumber ( ξ ), expressed as c g = d ω d ξ . However, this di ff erential operation may introduce numerical errors, especially when generating phase velocity dis persion curves with a large frequency increment step. An alternative approach for computing group velocity involves utilizing the ( ξ,ω ) solutions obtained through the SAFE method. This method starts with the governing equation of the SAFE approach and then evaluate its derivative with respect to the wavenumber ξ . Since ∂ω ∂ξ is a scalar, the group velocity can be calculated as Peddeti and Santhanam (2018):

=

Q L ( K 2 + 2 ξ K 3 ) Q R 2 ω Q L M Q R

∂ω ∂ξ

c g =

(11)

where Q L and Q R are the left and right eigenvectors that are computed using Equation 9. 3. Temperature-Dependent Material Properties

Given the influence of temperature on the propagation characteristics of Lamb waves, it is imperative to consider temperature variations when calculating the dispersion curve. Therefore, an investigation is carried out to analyze the impact of temperature on the group velocity of guided wave propagation using the established SAFE model as the theoretical framework. The propagation of guided wave signals is a ff ected by temperature-dependent properties of the composite structure, PZT, and thermoplastic film Giannakeas et al. (2023); Le Bourdais et al. (2019). How ever, experimentally assessing the temperature dependence of PZT and thermoplastic films poses challenges and is time-consuming. Therefore, their e ff ects are not considered in this study, with the only influencing factor taken into consideration being the temperature-dependent materials of the composite structure.