Two-dimensional DtN-FEM scattering analysis of SH guided waves by an interface debonding in a double-layered plate

TL;DR

This work addresses the forward scattering of SH guided waves by interface delamination in a bi-material plate. It develops a two-dimensional Dirichlet-to-Neumann finite element method (DtN-FEM) that imposes far-field boundary conditions via a modal expansion and mode orthogonality, avoiding absorbing layers and enabling direct, mode-resolved reflection/transmission data. The DtN-FEM yields a sparse global system and computes the scattered field with a closed-form relation, showing excellent agreement with boundary element method (BEM) benchmarks and energy-balance accuracy below 0.25%. Parametric studies reveal how material contrasts, delamination length, and interface location influence mode scattering, providing insights for damage detection and suggesting avenues for inverse reconstruction of delamination characteristics.

Abstract

In this paper, a two-dimensional Dirichlet-to-Neumann (DtN) finite element method (FEM) is developed to analyze the scattering of SH guided waves due to an interface delamination in a bi-material plate. During the finite element analysis, it is necessary to determine the far-field DtN conditions at virtual boundaries where both displacements and tractions are unknown. In this study, firstly, the scattered waves at the virtual boundaries are represented by a superposition of guided waves with unknown scattered coefficients. Secondly, utilizing the mode orthogonality, the unknown tractions at virtual boundaries are expressed in terms of the unknown scattered displacements at virtual boundaries via scattered coefficients. Thirdly, this relationship at virtual boundaries can be finally assembled into the global DtN-FEM matrix to solve the problem. This method is simple and elegant, which has advantages on dimension reduction and needs no absorption medium or perfectly matched layer to suppress the reflected waves compared to traditional FEM. Furthermore, the reflection and transmission coefficients of each guided mode can be directly obtained without post-processing. This proposed DtN-FEM will be compared with boundary element method (BEM), and finally validated for several benchmark problems.

Figures (14)

• Figure 2.1: An infinite double-layered isotropic plate is considered where the material of upper layer $A$ is steel and the material of lower layer $B$ can be steel/aluminum/titanium, $h_A+h_B=h$, the distance between virtual boundaries and debonding area is at least $2\lambda$ where $\lambda$ is the wavelength of the first SH guided mode, debonding area is located at the interface and an incident SH wave is excited in far-field, traveling as a guided wave, and scattered by an interface debonding.
• Figure 2.2: The domain mesh where displacements and tractions at the virtual boundaries $\Gamma _1$ and $\Gamma _2$ are both unknown.
• Figure 3.1: Model diagram for parametric studies
• Figure 3.2: The normalized scattered displacement on the top and bottom boundaries in a 1mm thick aluminum-steel plate: (a) frequency is 2MHz; (b) frequency is 5MHz.
• Figure 3.3: Error analysis of energy balance