Experimental verification and validation of nonlocal peridynamic approach for simulating guided Lamb wave propagation and damage interaction

Abstract

In this article, experimental verification and validation of a peridynamics-based simulation technique, called peri-elastodynamics, are presented while simulating the guided Lamb wave propagation and wave–damage interaction for ultrasonic nondestructive evaluation and structural health monitoring applications. Peri-elastodynamics is a recently developed elastodynamic computation tool where material particles are assumed to interact with the neighboring particles nonlocally, distributed within an influence zone. First, in this article, peri-elastodynamics was used to simulate the Lamb wave modes and their interactions with the damages in a three-dimensional plate-like structure, while the accuracy and the efficacy of the method were verified using the finite element simulation method (FEM). Next, the peri-elastodynamics results were validated with the experimental results, which showed that the newly developed method is more accurate and computationally cheaper than the FEM to be used for computational nondestructive evaluation and structural health monitoring. Specifically, in this work, peri-elastodynamics was used to accurately simulate the in-plane and out-of-plane symmetric and anti-symmetric guided Lamb wave modes in a pristine plate and was extended to investigate the wave–damage interaction with damage (e.g. a crack) in the plate. Experiments were designed keeping all the simulation parameters consistent. The accuracy of the proposed technique is confirmed by performing error analysis on symmetric and anti-symmetric Lamb wave modes compared to the experimental results for pristine and damaged plates.

Keywords

Lamb wave peridynamics peri-elastodynamics wave–damage interaction wave propagation wavenumber–frequency analysis nonlocal mechanics

Introduction

In structural health monitoring (SHM),^1–4 guided Lamb waves are the usual choice for damage detection^5,6 in metallic structures. To detect, localize, and characterize the damages in a plate, ultrasonic actuators and sensors are typically mounted on the plate structures in a strategic manner.^7,8 Symmetric (S₀) and anti-symmetric (A₀) Lamb wave modes travel through the plate, interacting with the plate boundaries, inhomogeneities, and damages.⁹ To understand the wave–damage interaction, it is neither feasible nor cost effective to perform extensive experiments in the laboratory. Thus, to perform the experiments virtually, computational nondestructive evaluation (CNDE) and computational structural health monitoring (CSHM) are proposed.¹⁰ To understand the damage type and its severity, the embedded sensor signals in the structure play an important role. In SHM, it is extremely difficult to accurately identify the damage state from the sensor signal because of infinite possibilities. Hence, an offline CNDE or CSHM tool could be extremely valuable to visualize the physics of wave–damage interactions. Although few analytical approaches are used to analyze wave propagation in simple geometries, the methods are not enough to understand the wave–damage interactions in three-dimensional (3D) complex structures. To overcome the need, several numerical techniques, such as boundary element method (BEM),¹¹ spectral finite element method (SFEM),^10,12–14 mass-spring lattice model (MSLM),¹⁵ finite element method (FEM),¹⁶ cellular automata,¹⁷ finite difference method (FDM),¹⁸ elastodynamic finite integration technique (EFIT),¹⁹and finite strip method²⁰ were developed in recent years. However, the fine spatio-temporal discretization used to improve accuracy makes these methods computationally expensive. To reduce the computational burden, a few semi-analytical techniques are also proposed, such as distributed point source method (DPSM),^21–23 semi-analytical finite element (SAFE),²⁴ and local interaction simulation approach (LISA).^25,26 One of the major disadvantages of these computation techniques is that the damage path is required to be defined ahead of time, whereas, in the practical scenario, it is almost impossible to predict a damage route. In addition, it is also essential to update the meshing of the domain alongside the damage propagation, which makes these techniques computationally expensive.

Previously, the peridynamic method was used to simulate the plane wave propagation in two-dimensional (2D) geometries,^17,27–30 but the guided wave simulations using PED was never proposed until Patra et al.¹ Hence, a new peridynamics-based^20,21 method, peri-elastodynamics (PED) for modeling guided Lamb wave modes, proposed by Patra et al.,¹ could potentially predict the damage growth and the wave propagation in bounded 3D structures. In doing so, it would not be necessary to change the parent model or the meshing and/or discretization. In PED simulations, only the damage matrix is updated to predict the damage growth. The damage matrix information will be automatically used in the wave propagation simulation. While PED has already been employed to simulate 3D guided Lamb wave propagation on a pristine plate-like structure and verified using theoretical dispersion curve from WaveFormRevealer (WFR),^31,32 there is some scope to simulate Lamb wave–damage interaction, and a detailed study on efficiency and accuracy of the proposed techniques has not yet been reported. Furthermore, PED was not verified with the experimental results.

In this article, PED was further extended to simulate Lamb wave–damage interaction with specific damage scenarios such as cracks in aluminum 6061-T6 plate. PED simulated sensor signals for pristine and damaged plate were verified with those obtained by conducting pitch-catch experiments on plates with similar dimensions and damage scenarios. Furthermore, accuracy of PED was investigated by comparing simulated symmetric and anti-symmetric Lamb wave modes to the experimental results. In addition, efficiency of the PED was investigated by comparing simulation parameters (i.e. memory requirement, simulation run time, and number of central processing unit (CPU) cores used) with FEM-based COMSOL simulation results.

Mathematical treatment with PED

Mathematical basics

PED is a nonlocal meshless simulation method developed mainly based on the bond-based peridynamics theory. In PED, a material body is discretized into a series of material points. The equation of equilibrium at the material point x after time-step t can be written as follows^33–35

ρ \overset{\cdot\cdot}{u} (x, t) = \int_{H} f_{den} (u (x', t) - u (x, t), x' - x) dV' + b (x, t)

(1)

where u, $f_{den}$ , $ρ$ , b, and $V'$ represent the displacement, pairwise force acting along the bond between x and $x'$ , density of the material body, body force per unit volume at the material point x, and the volume of the material point at $x'$ , respectively. The integral of the pairwise forces acting at x is performed over a finite region H, which is called horizon in the peridynamic theory.

Constitutive equation in peridynamics approach can be expressed as^33–35

f_{den} = b_{c} S

(2)

where $b_{c}$ is bond constant and S is stretch of the bond. Detailed discussion of mathematical formulation can be found in Patra et al.¹ and Silling et al.^33–35

Geometrical setup of the problem

In this work, to simulate the Lamb wave propagation in metal, an aluminum plate (of length = 300 mm, width = 200 mm, and thickness = 2 mm) is considered. Detailed material properties of the test specimen can be found in Table 1. To actuate the ultrasonic guided wave in the plate, a square-shaped piezoelectric (PZT) actuator with a dimension of 2.4 mm × 2.4 mm is placed 100 mm away from the left boundary of the plate. A 3.5-count 150 kHz tone-burst signal is considered to actuate the PZT patch.

Table 1.

Properties of aluminum.

Property	Values
Density	2695 kg/m³
Young’s modulus	68.9 GPa
Poisson’s ratio	0.329

It is to be noted that to represent the mode shapes of the fundamental symmetric and anti-symmetric modes of the Lamb wave, at least three layers of material points are needed. Therefore, in the PED formulation, the 3D domain of the plate is divided into three layers of material points, where L1, L2, and L3 are the top, middle or the neutral axis, and the bottom surface of the plate, respectively.

Pin-force model of the actuator

In this work, Lamb waves are generated by exciting the PZT actuator using a standard 3.5-count tone-burst signal (Figure 2(a)). Displacement in the host structure varies linearly along the length of the PZT and reaches its peak at the boundaries³⁶ (Figure 1(d)). In this study, a square PZT is modeled to attain a maximum of $1 μ m$ in-plane radial displacement at the circumferential material points. The displacement at the central material point of the PZT is maintained at zero. The equation for the variation of the displacements at the material points due to the application of the tone-burst signal can be expressed as

u (x, t) = A_{0} e^{- q t^{2} / 2} \sin (ω_{c} t)

(3)

where $A_{0}$ and $ω_{c}$ are the maximum displacement amplitude and the central frequency of the excitation signal, respectively. The parameter q is formulated as, where $A_{0}$ and $ω_{c}$ are the maximum displacement amplitude and the central-frequency of the excitation signal, respectively. The parameter q is formulated as, $q = {(kt ω_{c} / N_{cy})}^{2}$ , where k, $N_{cy}$ , and t are the signal shape factor, number of cycles of the actuation signal and the thickness of the specimen.

Figure 1.

Geometry of the aluminum plate used in the simulation: (a) pristine plate, (b) material layers (top, middle, and bottom layers; L1, L2, and L3), (c) discretization of the plate, and (d) particle displacement owing to PZT excitation.

Figure 2.

(a) 3.5-count tone-burst signal (displacement input signal) with 150 kHz central frequency, (b) dispersion curves for 2-mm thick aluminum 6061-T6 plate, (c) tuning curve of an aluminum 6061-T6 plate (2 mm thickness) with a standard 7 mm PZT.

The excitation frequency to excite the PZT is measured from the tuning curves of S₀ and A₀ modes of the base material, as shown in Figure 2(b) and (c). The tuning curves are computed using an open source software named “WaveFormRevealer(WFR),”^32,37 developed in LAMSS laboratory at the University of South Carolina. To ensure that only the S₀ and the A₀ modes are excited with different amplitudes, a central frequency of 150 kHz is selected from the tuning curve.

Numerical time integration

The equation of motion at material point i, after the time-step n, can be expressed as the finite summation over all the material points inside the horizon (equation (4))

ρ {\overset{\cdot\cdot}{u}}_{i}^{n} = \sum_{N_{f}} f_{den} (u_{f}^{n} - u_{i}^{n}, x_{f} - x_{i}) V_{f} + b_{i}^{n}

(4)

The total force on a material point can be computed by summing up all the peridynamic forces acting on that material point because of the neighboring points within its horizon. In equation (4), $N_{f}$ represents the number of material points within the horizon of the material point i. For the given boundary conditions and initial conditions, the Velocity–Verlet integration scheme is adapted to compute the time-domain displacements. Detailed steps involved in the Velocity–Verlet method can be found in Nishawala et al.¹⁷

Convergence requirements

In wave propagation modeling, it is very crucial to have an appropriate spatial and temporal discretization scheme for the convergence of the solution. According to Leckey et al.,³⁸ maximum spatial discretization $(Δ S)$ for finite element analysis must meet the criterion shown in equation (5) to have the convergence

Δ S ⩽ \frac{λ_{min}}{10}

(5)

where $λ_{min} = c_{min} / f$ , is the minimum wavelength of the Lamb wave modes, c and f represent the phase velocity and excitation frequency, respectively. $c_{min}$ is measured from the dispersion curve as the phase velocity of the $A_{0}$ mode at 150 kHz. Next, to obtain a proper time-step $(Δ t)$ , Courant–Friedrichs–Lewy condition is adopted, as presented in equation (6)³⁸

Δ t = \frac{Δ X}{c_{max} \sqrt{3}}

(6)

where $c_{max}$ is the maximum phase velocity of the propagating wave modes. Similar to $c_{min}$ , the phase velocity of the S₀ mode at 150 kHz was measured from the dispersion curve and denoted as $c_{max}$ using equations (5) and (6), to obtain a converged solution. Note that the spatial and temporal discretization mentioned in the article is valid for the finite element simulation (local approach). However, the validity of those equations for selecting spatial and temporal discretization in PED (nonlocal) simulation is not yet established. Here, we have used those equations for an initial estimate of the spatial and temporal values. We found that converged PED simulation was obtained for spatial discretization $Δ x ⩽ λ_{min} / 8$ and temporal discretization $Δ t_{PDE} ⩽ Δ t / 10$ (where $Δ t = Δ x / c_{max} \sqrt{3}$ ). We ran simulations for 1 and 1.2 mm spatial discretizations and found that results are converged for both discretizations; however, to reduce computation time and memory allocation, we choose 1.2 mm as maximum spatial distance. The temporal step is calculated as 0.01 µs. As 1.2 mm is the maximum spacing between the two material points, the distance between the two layers along the thickness direction was kept as 1 mm < 1.03 mm.

Numerical computation

Lamb wave simulation

As mentioned, a 3.5-count 150 kHz tone-burst signal is used to generate the fundamental Lamb wave modes in the plate. Figure 3 shows the in-plane ( $u_{x} (x, y, t)$ and $u_{y} (x, y, t)$ ) and the out-of-plane displacements $(u_{z} (x, y, t))$ of the Lamb wave propagation presented after three different time-steps (t = 10, 24, and 38 µs). Displacement patterns on the top surface (L1) are plotted herein (see supplemental material). From Figure 3, the fundamental physics of the high-speed S₀ mode, which travels faster than the A₀ mode, can be clearly identified. In addition, the modes travel with concentric wavefront in the isotropic material. These confirm the well-established understanding of the Lamb wave modes. In Figure 3(a1) to (a3) and (b1) to (b3), it can be observed that the S₀ and A₀ modes are visible for in-plane displacements. As shown in Figure 3(c1) to (c3), the out-of-plane displacement of the A₀ mode is strong, whereas that of the S₀ mode is barely noticeable. This is due to the dominance of the out-of-plane displacement of the particles over the in-plane displacement of the particles because of the A₀ mode.

Figure 3.

Time domain in-plane and out-of-plane displacement waveform: (a1) $u_{x} (x, y, t)$ at t = $10 μ s$ , (a2) $u_{x} (x, y, t)$ at t = $24 μ s$ , (a3) $u_{x} (x, y, t)$ at t = $38 μ s$ , (b1) $u_{y} (x, y, t)$ at t = $10 μ s$ , (b2) $u_{y} (x, y, t)$ at t = $24 μ s$ , (b3) $u_{y} (x, y, t)$ at t = $38 μ s$ , (c1) $u_{z} (x, y, t)$ at t = $10 μ s$ , (c2) $u_{z} (x, y, t)$ at t = $24 μ s$ , and (c3) $u_{z} (x, y, t)$ at t = $38 μ s$ .

Figure 4 shows the space–time representation of the in-plane $(u_{x} (x, y, t))$ displacement field at top, mid, and bottom layers (L1, L2, and L3) of the plate. The S₀ and A₀ modes are prominent in the in-plane displacement at the top and the bottom layers (Figure 4(a) and (c)). However, only the S₀ mode is prominent in the in-plane displacement plot compared to the A₀ mode in the middle layer. This is because the material points in the middle layer have no contribution from the A₀ mode in the in-plane $u_{x} (x, y, t)$ displacement field.

Figure 4.

Space–time wavefield of in-plane displacement: $u_{x} (x, t)$ at the (a) top, (b) middle, and (c) bottom layer.

Through-thickness mode shapes of the Lamb wave modes

To prove the accuracy of the PED tool, after 38 µs, when the S₀ and A₀ modes are well-separated, the through-thickness mode shapes of the plate particles are plotted in Figure 5. The out-of-plane $(u_{z} (x, y, t))$ and the in-plane $(u_{x} (x, y, t))$ displacement profiles of the A₀ mode can be found from the cross sections along the $m_{1} - m_{2}$ and $m'_{1} - m'_{2}$ lines of the plate. Similarly, the out-of-plane and the in-plane displacement profiles of the S₀ mode can be extracted from the $n_{1} - n_{2}$ and $n'_{1} - n'_{2}$ line cuts, respectively. From Figure 5(a), it can be seen that, owing to the presence of the A₀ mode, the particles move in either upward or downward directions. Given the generation of the S₀ mode, in Figure 5(c), the displacement of the particles in the middle layer is almost zero, while the top and bottom layers showing symmetric displacements. Similarly, in-plane movements of the mid-layer particles are also zero owing to the presence of the A₀ mode (Figure 5(b)), whereas in the top and the bottom layers, the particle moves in opposite directions. Vector fields and the mode shapes in Figure 5 indicate that the Lamb wave modes were accurately modeled using PED and confirmed the existing theoretical understanding.

Figure 5.

Data source: PED simulation, through-thickness mode shapes of the S₀ and A₀ mode: (a) A₀ mode for out-of-plane motion, (b) A₀ mode for in-plane motion, (c) S₀ mode for out-of-plane motion, and (d) S₀ mode for in-plane motion.

Data analysis: wavenumber–frequency analysis: verification of the simulation

Next, verification is performed using wavenumber–frequency (WF) analysis. As multidimensional Fourier transform is a popular tool to identify the different Lamb wave modes,³⁹ in this study, a 2D fast Fourier transformation is performed over the space–time data obtained from the simulation. A wavenumber–frequency representation was obtained from the 2D Fourier transform of the wavefield.³⁹

To perform the 2D transformation, first, a set of time–space data were obtained from the PED material points (#163 with 1.2 mm of resolution) along the dotted line shown in Figure 3(a1). The out-of-plane $(u_{z} (x, y, t))$ and the in-plane $(u_{x} (x, y, t))$ displacement data were obtained across the thickness. The 2D transformation was performed on the simulation data that were obtained from the top layer of the plate. To validate the wavenumber–frequency relation obtained from the PED tool, the dispersion curves from the well-established analytical tool “Disperse software” are analyzed and compared.

The wavenumber–frequency representation of the in-plane displacements $(u_{x} (x, y, t))$ is presented in Figure 6(a1). It can be observed that the S₀ and A₀ modes are identified (Figure 6(a1)). Displacements or the local energy with the mode A₀ is slightly higher compared to those in the mode S₀, which is affirmed by the tuning curve (Figure 3(b)) of the plate at 150 kHz. Figure 6(b1) represents the wavenumber–frequency relation for out-of-plane $(u_{z} (x, y, t))$ displacements of the top layer. From Figure 6(b1), it is evident that the energy distribution at the mode A₀ is much higher compared to the mode S₀. The above outcome from the PED tool agrees very well with the well-established dispersion relation of the fundamental Lamb wave modes (Figure 6(a2) and (b2)). Therefore, it can be firmly claimed that the PED is a reliable tool for Lamb wave propagation simulation and can be applied to CNDE and CSHM.

Figure 6.

Wavenumber–Frequency (WF) representation of the displacement field at the pristine state: (a1) WF of the in-plane displacement at the top surface; (b1) WF of the out-of-plane displacement at the top surface; and comparison of theoretical and numerical WF of the (a2) in-plane and (b2) out-of-plane displacements at the top surface.

Experimental design for the validationof PED

To validate the PED results, pitch-catch experiments were conducted on pristine and damaged aluminum 6061-T6 plates as shown in Figure 7(a) to (c), respectively. In this study, the plates with a crack along the centerline and with a crack offset from the center line were considered as shown in Figure 7(c) and (d). Dimensions of the plate are the same as those used in the PED simulation. Two high-frequency PZTs (type PZT 5A, purchased from STEMiNC, Florida) were attached to the plate using Hysol 9340 adhesive. The PZTs were 7 mm in diameter and 0.5 mm in thickness. The distance between the two PZTs was kept at 104 mm, measured from their center. One of the PZTs was used as an actuator, while the other was used as a sensor. A 3.5-count tone-burst, with a central frequency of ∼150 kHz and ∼20 V amplitude (∼10 V peak-to-peak) was used to excite the actuator to generate the Lamb wave propagation in the plate. Tektronix AFG3021C (25 MHz, 1-Ch arbitrary function generator, Tektronix, Inc.: Beaverton, OR) was used to generate the tone-burst actuation at the interval of 1 ms and Tektronix MDO3024 (200 MHz, 4-Ch mixed domain oscilloscope, Tektronix, Inc.: Beaverton, OR) was used to record the signals from the sensor. The sensor signal was recorded after averaging a total of 500 signals (to improve signal-to-noise ratio). The sampling rate and signal length were set to 50 MS/s and 10,000, respectively.

Figure 7.

(a) Experimental setup of pitch-catch experiments, (b) pristine plate, (c) plate with a center crack, and (d) plate with an offset crack.

Computational verification of the simulation

A commercially available finite element (FE)-based computational tool, COMSOL Multiphysics, was used to simulate the structural mechanics problem, coupled with the PZT actuator and sensor. To model the ultrasonic guided wave propagation in the aluminum plate, Solid Mechanics and Electrostatics modules, a multiphysics approach in COMSOL, was employed. COMSOL utilizes an implicit scheme to solve the transient problems. In this study, the direct solver MUMPS was chosen over an iterative solver, because of its robustness. All the direct solvers in COMSOL require significant amounts of RAM, where MUMPS can store the solution out-of-core, that is, on the hard disk. Moreover, MUMPS is substantially faster than iterative solvers. The absolute tolerance of the time-dependent solver used a global method of scaling with a specified tolerance of 0.001. The setting for time-steps was set to the generalized-alpha method with intermediate time-steps, a linear predictor, and a maximum time-step of 50 ns.

The mechanical and electrical properties of the aluminum plate and the PZT components were considered to be the same as the properties used in the experimental design. To excite the PZT of the actuator, a 20 V 3.5-count tone-burst signal was applied at the electric potential terminal. The signal response was collected from the sensing PZT for the entire duration of the simulation, which was 80 µs. Free tetrahedral (tets) meshes generated by COMSOL multiphysics were utilized to generate a mesh for the entire aluminum domain where the minimum mesh size varied between 0.1 and 1.2 mm as shown in Figure 8(b) and (c). By contrast, the minimum mesh size for the PZT actuator and sensor were varied between 0.01 and 0.2 mm as shown in Figure 8(a). A mesh convergence study was performed starting from the maximum mesh size of 2 mm to a minimum mesh size of 1.2 mm. As the mesh size was decreased, the accuracy of the simulation improved at the expense of increased computational time. A total of 24 CPU cores with a maximum memory of 80 GB were utilized to solve this problem in 45 hours.

Figure 8.

Three-dimensional FE discretization of the aluminum plate and PZT: (a) discretization of the PZT, (b) discretization of the plate and PZT, and (c) discretization of the plate.

Validation and verification of the PED simulation

In this section, the accuracy and efficiency of the PED technique to simulate Lamb wave propagation are presented. Time-domain signal (at sensor location S1) obtained from the PED was compared with the numerically (COMSOL) and analytically (WFR^31,32) obtained sensor signals and the signals acquired from the pitch-catch experiments on a pristine aluminum plate. Note that in this study a square PZT was employed for PED simulation, whereas circular PZT was used for COMSOL and the experiments. However, given the isotropic nature of the plate, at the far field, the wave fronts were circular and when the guided wave modes were fully developed, the effect of PZT size was not diagnosed. In-plane displacement $(u_{x} (x, y, t))$ from the PED was used to compare with the output voltage obtained from COMSOL, WFR, and experiment. As the sensor in the plate is located on the centerline along the X-axis, as shown in Figure 7(b) to (d), output voltage at sensor terminal is contributed primarily by $u_{x} (x, y, t)$ . To capture the sensor signal from the PED simulation, the time-domain signal was collected from a material point located 97 mm away from the PZT edge along the centerline from the actuator ( $L_{center}$ = 104 mm and $L_{eff}$ = 97 mm, shown in Figure 1(e) and (f)). The normalized amplitude of the sensor signals obtained from an experiment, COMSOL, WFR, and PED were plotted in the time domain as shown in Figure 9. Sensor signals obtained from the PED, COMSOL, and WFR were also compared with the experimental results to check the accuracy of those techniques. Good agreement between PED and the experiment was observed for symmetric and anti-symmetric modes as shown in Figure 9(b). The symmetric mode from COMSOL is in good agreement with the experiment, whereas the anti-symmetric mode slightly deviates as shown in Figure 9(c). WFR predicted the anti-symmetric mode well but overestimated the symmetric mode as shown in Figure 9(d). Mismatch of the symmetric mode may be attributed to the limitation of the WFR software; such as, being an analytical method, the material property of PZT could not be provided. To verify the accuracy of the proposed PED techniques and to compare with existing simulation techniques, error of symmetric and anti-symmetric mode is calculated by comparing them with experimental results using equation (7)

Error = \frac{\int_{t_{i}}^{t_{f}} {(A_{sim}^{env} (t) - A_{\exp}^{env} (t))}^{2} dt}{\int_{t_{i}}^{t_{f}} A_{\exp}^{env} (t) dt} \times 100

(7)

where

A_{sim}^{env} (t) = | H (A_{sim} (t) |, A_{\exp}^{env} (t) = | H (A_{\exp} (t) |

(8)

where $A_{sim}^{env} (t)$ and $A_{\exp}^{env} (t)$ are the envelope of simulated and experimental sensor signal, respectively.

Figure 9.

Time-domain comparison of sensor signal: (a) experiment, PED, COMSOL, and WFR, (b) experiment and PED, (c) experiment and COMSOL, (d) experiment and WFR, (e) error of simulated symmetric and anti-symmetric modes with respect to experimental results, and (f) memory requirement and simulation run time of PED and COMSOL simulation.

A similar approach was used to calculate the error of the COMSOL- and WFR-predicted sensor signal. Error analysis results are shown in Figure 9(e). Errors of symmetric and anti-symmetric mode of the PED simulation were 2.2% and 0.5%, respectively, which is less than the maximum permissible error tolerance of 3%. Significant error for symmetric mode of the WFR was observed. Error percentages for COMSOL simulation were 0.83% and 3.3%, respectively. PED simulation provided a better accuracy for wave propagation simulation as compared to COMSOL and WFR.

To investigate the efficiency of the PED with respect to other numerical tools in solving the proposed wave propagation problem, memory requirement, simulation run time, and CPU core used for both PED and COMSOL were plotted in Figure 9(f). Note that the element size was kept the same. While COMSOL can use multiple cores (i.e. 24 cores) for running the simulation, PED used only one core. Memory consumption and simulation run time for PED is smaller than that of COMSOL simulation. However, parallelization of the PED code can improve simulation run time significantly.

Modeling wave damage interactions by PED

Without studying the wave–damage interactions, the CNDE or CSHM may not be convincing. Hence, in this article, the PED tool is further extended to study the wave–damage interaction, where the base plate with two damage scenarios is studied. The wave–damage interaction study is performed using a through-thickness crack of dimensions 16 mm × 2.4 mm, located at a 70 mm distance from the PZT actuator (Figure 10). A crack on the centerline (red dotted lines in Figure 10) and a crack offset from the centerline are considered as shown in Figure 10(a) and (b).

Figure 10.

Geometry of aluminum 6061-T6 plate with crack: (a) center crack and (b) offset crack.

Simulation results from the center crack and offset crack are presented in Figures 11 and 12, respectively. To observe the reflection and transmission of the respective wave modes from the damage location, in-plane displacement wavefields, after three different time-steps (40, 50, and 60 µs), are shown in Figure 11(a1) to (a3). It can be seen that while the reflected S₀ mode from the crack location is observed after 40 µs, the same mode disappeared after 50 and 60 µs because of the interference of the parent mode with the reflected boundary mode. Alternatively, the reflected A₀ mode is identifiable after the 50- and 60-µs time-steps. In case of out-of-plane displacements, reflected S₀ mode is not quite visible in any of the time-steps (Figure 11(b1) to (b3)) because of its minor contribution to the out-of-plane wave propagation. However, the reflected A₀ mode is observed after the 50- and 60-µs time-steps.

Figure 11.

Time-domain displacement waveform in a plate with a center crack: (a1) $u_{x} (x, y, t)$ at t = $40 μ s$ , (a2) $u_{x} (x, y, t)$ at t = $50 μ s$ , (a3) $u_{x} (x, y, t)$ at t = $60 μ s$ ,(b1) $u_{z} (x, y, t)$ at t = $40 μ s$ , (b2) $u_{z} (x, y, t)$ at t = $50 μ s$ , and (b3) $u_{z} (x, y, t)$ at t = $60 μ s$ .

Figure 12.

Time-domain displacement waveform in a plate with an offset crack: (a1) $u_{x} (x, y, t)$ at t = $40 μ s$ , (a2) $u_{x} (x, y, t)$ at t = $50 μ s$ , (a3) $u_{x} (x, y, t)$ at t = $60 μ s$ ,(b1) $u_{z} (x, y, t)$ at t = $40 μ s$ , (b2) $u_{z} (x, y, t)$ at t = $50 μ s$ , and (b3) $u_{z} (x, y, t)$ at t = $60 μ s$ .

In-plane displacement wavefields for offset crack (Figure 10(b)) after times 40, 50, and 60 µs are shown in Figure 12(a1) to (a3). The reflection and transmission of the fundamental Lamb wave modes are observed. Similar to the center crack scenario, in the case of the offset crack, the reflected S₀ mode is observed only after 40 µs. Reflection of the S₀ mode after the other two time-steps is barely noticeable, given the interference with the boundary reflections. However, the reflected A₀ mode is visible after 50 and 60 µs. Evaluating Figure 12(b1) to (b3), in comparison to the center crack, a similar argument can also be made for the offset crack in relation to the A₀ and S₀ modes existed at different time-steps.

Next, to distinguish the amplitude of the reflected and transmitted Lamb wave modes, time–space representation of the in-plane and out-of-plane displacements for both the scenarios with center crack and the offset crack is analyzed. Displacement wavefields are computed along a selected line shown in Figure 10. In case of the center crack scenario, both the in-plane $(u_{x} (x, t))$ reflection and the transmission of the incident wave are clearly visible in Figure 13(a1). A similar phenomenon can also be noticed from Figure 13(a2) for the center crack out-of-plane displacement $(u_{z} (x, t))$ . Boundary reflection of the modes can be observed in Figure 13(a1) and (a2). Likewise, in the case of the offset crack scenario, reflection and transmission from the offset crack damage are also demonstrated in Figure 13(b1) and (b2) considering the in-plane and the out-of-plane displacement modes.

Figure 13.

Space–time wavefield representations for the top surface of the plate with a through-thickness crack: (a1) $u_{x} (x, t)$ for a plate with a center crack, (a2) $u_{z} (x, t)$ for a plate with a center crack, (b1) $u_{x} (x, t)$ for a plate with offset crack, (b2) $u_{z} (x, t)$ for a plate with offset crack.

Furthermore, the time history signals from the damage-free pristine plate at sensor location S1 marked in Figure 10 are compared with the signals from the centerline crack and offset crack scenario. Figure 14(a) compares the output response (at S1) obtained from experiments for pristine, center crack and offset crack, respectively. Figure 14(b) compares the output response (at S1) obtained from PED for the pristine, center crack and offset crack, respectively. It can be seen that the first arrival of the symmetric and anti-symmetric wave modes for the experiment and PED at the sensor location S1 are slightly delayed because of the encounter of the crack. The delay is comparatively higher at the S1 location because of the centerline crack, compared to the offset crack. Given an offset crack edge, the reflected wave energy reflects at an angle, and the sensor S1 on the centerline has less impact compared to a crack present along the centerline. In Figure 14(c), the sensor signal for the centerline crack obtained from PED and an experiment is compared. The errors of symmetric and anti-symmetric wave modes by PED prediction were 1.92% and 0.479%, respectively. In Figure 14(d), the sensor signals for offset crack obtained from PED and the experiment are compared. The errors of symmetric and anti-symmetric wave modes by PED prediction were 1.21% and 0.81%, respectively. These features were properly simulated using the PED method. Hence, it can be concluded that the PED method is a suitable wave simulation tool for CNDE and CSHM.

Figure 14.

Comparison of time-dependent signals obtained from PED and experiment at sensor location S1, in a pristine plate, plate with a crack along centerline, and a plate with an offset crack, (a) sensor signals at location S1 obtained from experiment, (b) sensor signals at location S1 obtained from PED, (c) sensor signals for centerline crack obtained from PED and experiment, and (d) sensor signals for offset crack obtained from PED and experiment.

Conclusion

In this article, a numerical tool based on bond-based peridynamic theory, named peri-elastodynamics (PED), is proposed to simulate Lamb wave propagation in an isotropic 3D plate. While several analytical and semi-analytical methods are available to computationally model the Lamb wave propagation in solids, PED can be a promising alternative tool with higher accuracy. Hence, the PED tool is introduced here, to accurately model 3D Lamb wave modes in an isotropic plate. Fundamental Lamb wave modes were generated, and their characteristics were thoroughly investigated. Particle displacements owing to the S₀ and A₀ mode propagation were visualized in vector field and across-the-thickness displacement plots. Next, to validate the PED tool, Lamb wave modes were transformed into the wavenumber–frequency domain and compared with those obtained from well-established analytical tools. PED results were also verified with experimental results for pristine, center crack and offset crack scenarios.

Footnotes

Acknowledgements

The authors acknowledge Mr Dylan Madisetti for his partial help with the PDE code management.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The study was partially supported by NASA (contract no. NNL15AA16C).

Supplemental material

Supplemental material for this article is available online.

ORCID iD

Sourav Banerjee

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