Damage identification in multi-step waveguides using Lamb waves and scattering coefficients

Abstract

Damage detection in uniform structures has been studied in numerous previous researches. However, damage detection in non-uniform structures is less studied. In this paper, a damage detection algorithm for identifying rectangular notch parameters in a stepped waveguide using Lamb waves is presented. The proposed algorithm is based on mode conversion and scattering phenomena because of interaction of Lamb wave modes with defects. The analysis is divided into two steps: notch localization and notch geometry detection. The main advantage of this method is its ability to detect all of the notch parameters in a waveguide with arbitrary number of step discontinuities. The method is applied to a numerical example and the results show that it can successfully identify the notch location, depth, and width in a multi-step plate.

Appendix

In this section, the concepts and mathematical formulation of computing scattering coefficients are briefly discussed. The formulations and procedure are introduced by Kim and Roh [17], and more details of the formulations and procedure can be found in Refs. [16, 17].

The displacement and stress components of nth mode Lamb wave $w_n$ can be written as

$$\begin{aligned} w^{n}=W_n \left( {k_n ,y} \right) \mathrm{e}^{i\left( {k_n x-\omega t} \right) }, \end{aligned}$$

(A.1)

where $w_n =\left[ {u_n ,v_n ,\sigma _n ,\tau _n } \right] ^{\mathrm{T}}$ and $W_n =\left[ {U_n \left( {k_n ,y} \right) ,V_n \left( {k_n ,y} \right) ,S_n \left( {k_n ,y} \right) ,T_n \left( {k_n ,y} \right) } \right] ^{\mathrm{T}}$. Here, $u^{n}$ and $v^{n}$ denote longitudinal and transverse displacements, respectively, while $\sigma ^{n}$ and $\tau ^{n}$ denote normal and shear stresses, respectively. $U_n$, $V_n$, $S_n$, and $T_n$ are modal functions for each corresponding component. The equation form of each modal function is different for the symmetric and antisymmetric modes.

Considering the power of nth mode Lamb wave, $C_n^2$, Lamb wave modal functions are normalized by $C_n$ for direct comparison of reflection and transmission coefficients, and the normalized wave form is expressed as follows

$$\begin{aligned} {\overline{w}}^{n}={\overline{W}}_n \left( {k_n ,y} \right) \mathrm{e}^{i\left( {k_n x-\omega t} \right) }, \end{aligned}$$

(A.2)

where ${\overline{W}}_n =W_n /C_n$. The nth mode Lamb wave after reflection or transmission can be expressed with the normalized modal functions as [16]

$$\begin{aligned} {\overline{w}}^{n}=D_n {\overline{W}}_n \left( {k_n ,y} \right) \mathrm{e}^{i\left( {k_n x-\omega t} \right) }, \end{aligned}$$

(A.3)

where $D_n$ is the reflection/transmission coefficient of nth mode Lamb wave after interaction with a discontinuity provided that the incident wave has a unit power.

When an incident wave interacts with notch, the scattering mechanism can be divided into three main processes. The incident wave strikes the B1 boundary and some portion of its energy is reflected in Region 1 and the remaining portion is transmitted to Region 2. The transmitted wave strikes the boundary B2 and wave is reflected and transmitted to regions 2 and 3, respectively. The reflected wave strikes the boundary B1, and some portion is transmitted to Region 1 while the other portion is reflected to Region 2. This procedure is repeated again and again until final reflected and transmitted waves are generated (Fig. 14) [17].

Fig. 14

Scattering of Lamb wave by a rectangular notch

To solve for the scattering coefficients at each process, the displacement and stress components induced by the scattered propagating and non-propagating modes must be the same as those of the incident wave at the points on the boundaries. To find reflected and transmitted waves at each process, one can apply the boundary conditions and continuity conditions for displacements and stresses at the boundaries. For instance, in process 1 one may write (Fig. 14):

$$\begin{aligned}&\left\{ {{\begin{array}{l} {\left( \sigma \right) _{W_{i1} } +\left( \sigma \right) _{W_{r1} } =0} \\ {\left( \tau \right) _{W_{i1} } +\left( \tau \right) _{W_{r1} } =0} \\ \end{array} }} \right. ,\quad \left( {x,y} \right) \in \gamma _1\nonumber \\&\left\{ {{\begin{array}{l} {\left( u \right) _{W_{i1} } +\left( u \right) _{W_{r1} } =\left( u \right) _{W_{t1} } } \\ {\left( v \right) _{W_{i1} } +\left( v \right) _{W_{r1} } =\left( v \right) _{W_{t1} } } \\ {\left( \sigma \right) _{W_{i1} } +\left( \sigma \right) _{W_{r1} } =\left( \sigma \right) _{W_{t1} } } \\ {\left( \tau \right) _{W_{i1} } +\left( \tau \right) _{W_{r1} } =\left( \tau \right) _{W_{t1} } } \\ \end{array} }} \right. ,\quad \left( {x,y} \right) \in \gamma _2 \end{aligned}$$

(A.4)

where $w_{i1}$, $w_{r1}$ and $w_{t1}$ represent incident, reflected and transmitted Lamb wave at the left notch boundary. Normalizing Eq. (A.4) using the power, one may solve for the reflected and transmitted coefficients of each mode, i.e., $D_n$.

Note that each of the wave components $\sigma ,\tau ,u$ and v consist of several propagating and non-propagating modes. For example, $u=\sum _{i=0}^\infty u_{S,i} +\sum _{i=0}^\infty u_{A,i}$, where i is the number of considered propagating and non-propagating modes, and A and S represent symmetric and antisymmetric modes, respectively. when the product of frequency and plate thickness is low, one should only consider fundamental modes ($A_0~ \hbox {and}~S_0 )$ since higher order modes do not propagate in this range. However, sufficient number of non-propagating modes needs to be considered for acceptable accuracy. It is shown in Ref. [17] that for notch depths up to 85% of the plate thickness, considering 300 non-propagating modes, results in acceptable results.

For each scattering process, the scattering graphs are constructed in Ref. [17], and transmission and reflection coefficients of propagating modes are pre-calculated at various types of notch boundaries. All field information for non-propagating modes is included in the scattering graphs. For a Lamb wave of the same kind and a notch of the same depth, the transmission and reflection coefficients in the graphs can be referred to whenever the scattering process occurs.

The described three scattering processes occur again and again until no energy is left in Region 2. Therefore, so many iterations are needed to calculate the final transmission and reflection coefficients. To compute the final reflection and transmission coefficients, a straightforward matrix transformation method is presented by Kim and Roh [17] considering infinite repetitions of the scattering processes. This method can be used to compute the final scattering coefficients for various notch intensities (see [17] for more details). The computed scattering coefficients required for this study are shown in Fig. 4.

To verify the extracted scattering coefficients, results are compared with those obtained from finite element simulation. The results are demonstrated in Fig. 15 for selected cases which show the computed scattering coefficients from theory and simulation are in match.

Fig. 15

Comparison of transmission coefficients computed from theoretical formulations ($\circ$) and FEM simulation ($\Box$) for an 5 mm aluminum plate at 100 kHz excitation frequency. a TSA, b TSS, c TAS, and d TAA