Simulation of Piezoelectric Induced Lamb Waves in Plates

C. WILLBERG, J. M. VIVAR-PEREZ, Z. AHMAD and U. GABBERT

ABSTRACT

The use of Lamb waves for damage detection and non-destructive evaluation have been increased in the last decades. For this purposes, it is strictly necessary to understand the propagation phenomena in different structures, and the influence on the geometrical and mechanical properties of the system actuator-sensors-host structure. We find in numerical simulation and computer visualization important tools to clarify this dependence under certain circumstances. This work is a compilation of some work done in regards to numerical experiments related the behavior of piezoelectric induced Lamb waves in plates. Keywords: Structural Health Monitoring (SHM), Lamb Waves, FEM, SEM. INTRODUCTION

The application of elastic guided waves to inspect structures has a long history, and its application for an online monitoring of structures is quite common nowadays. In thin-walled structures, special elastic guided waves between free stress constrained parallel boundaries occur, named after Horace Lamb in 1917 [1]. Among the advantages of Lamb wave based methods against other forms of online monitoring applications, is the relatively long range of propagation of these waves due to their relatively low rate of attenuation in comparison to other classes of ultrasonic waves. The consequence is the reduction of the number of sensors per interrogated area in the applications. The properties of Lamb waves are well discussed in the book by Victorov [2].

The recent increasing improvement on the quality of piezoelectric transducers allows the use of these materials in an easy way for exciting and sensing Lamb waves. The goal of the paper is to successfully model numerically the behavior of Lamb waves induced by piezoelectric materials and their interaction with the non healthy structure.Results are presented in the first section of the paper. Some details on the behavior of lamb waves in isotropic plates are given and their interaction with damages using a relatively simple model to study the phenomena of mode conversion.

For real applications it is necessary to understand the mechanical behavior of the coupled actuator/structure, sensor/structure system. Frequently, the electrical connection to the piezoceramic and also the coupling between the latter and the structure are assumed to be ideal and without irregularities. As consequence, the deformation over the active surface of the piezoelectric transducer in the plate is modeled as homogeneously distributed and the induced guided wave does not show a preference direction where the signal amplitude is higher. Nevertheless, Huang et al [3] showed that, in practice, circular shaped actuator can introduce a non homogeneous angle dependence on amplitude distribution in the wave front in an isotropic material and this distribution is frequency dependent. Otto-von-Guericke-University, Institute of Mechanics, Universitätsplatz 2, building 10, 39106 Magdeburg, Germany.

On this contribution, we will show that the effect of considering some

irregularities out off the standard axisymmetrical distributions of loads in the

actuator. Under these conditions, the energy distribution of amplitudes on the wave

front is shown to be directional. This directionality is frequency dependent and is

intrinsically related to the bending modes of the piezoceramic disc used as actuator.

This is the topic treated on the second section. The mentioned irregularity, on our

case, is taken as the welding point between the wire and the piezoelectric disc. The

welding area is neither uniformly nor axisymmetrically distributed on the upper

surface of the piezoactuator. Then, the deformations of the actuator under

homogeneous electric field are not axially symmetric any more, giving place to the

previously mentioned effect.

At the end of the contribution some analysis is done about the suitability of the

use of spectral methods to simulate the wave propagation phenomenon with the aim

to reduce computer memory storage and time of calculation for the mathematical

simulations. In the last section some elements of the theory behind spectral element

method is given and some comparisons to the classical finite element formulation in

simple models.

LAMB WAVES IN PLATES AND DAMAGES

Lamb waves in plates take place in more than one mode of two kinds called

symmetric (S) and anti-symmetric (A) modes. One important feature of Lamb

waves is their highly dispersive behavior. This behavior can be observed from the

Lamb wave’s dispersion curves which are the graphical representation of the phase

and the group velocities of each mode as shown for the case of an Aluminum plate

in figure 1. Once the curves are known, a compromise between shorter wave length,

less number of modes and less dispersion can be made to improve sensibility and

simplicity on the signals received on the sensors.

Figure 1. Group and phase velocity dispersion curves calculated by using analytical

formulas (asymmetric mode – dotted curve; symmetric mode – blue curve); 0A and 0S are

the left two curves

In figure 2 a two dimensional bar is shown. Under the assumption of a plane stress

state a 2D model is sufficient. With two dimensional models information regarding

mode conversion and reflection can be evaluated with less computational effort

than with comparable 3D models. By using further a pair of two coupled

piezoelectric patches as actuators on top and bottom of the plate it is possible to

send a nearly pure symmetric or a pure anti-symmetric wave with respect to the

plate mid plane. By using a further pair of two piezoelectric sensors it is possible to

interpret the different modes [7]. A Lamb wave is sent through this bar and interacts

with some different discontinuities as it travels. A “Hanning” window pulse with

central frequency 145 kHz is used. From figure 1, only the A0 and S0 modes occur

and have to be taken into account simplifying the research about mode conversion.

Figure 2. Two dimensional bar model to simulate the influence of different discontinuities to

the mode conversion of Lamb waves

Two examples from several numerically investigated experiments are presented in

relation to each other. Both of them show the changes of the sensor received signals

by a localized change of thickness (variant 1 and 2 in figure 2). The first numerical

example investigates the influence of a height change (0.6mm) at one side of a bar

only (see thickness change 1 in figure 2). In the second numerical experiment a

symmetric change of the thickness (2 times of 0.6mm) is investigated (see thickness

change 2 in figure 2). The differences of the measured response at the sensors

between the anti-symmetric and the symmetric change of thickness are discussed in

the following.

(a) (b)

Figure 3. A0 mode (a) and S0 mode (b) sensor signal

Figure 3 (a) shows the signals received at the piezoelectric sensors corresponding to

the numerical simulation of an excitation of the A0 mode. The wave travels through

the different changes of height and converts particularly into the S0 mode. The A0

mode is relatively dispersive in the chosen frequency

an evident shift of phase on the received signal due to the increase of the phase

velocity while traveling through the thickness change. This can be observed in the

differences between the results corresponding to two different thickness changes of

type 1 and type 2 (see f

respectively. In figure 3 (a)

to the transmitted S0 mode is shown, which is the result of a mode conversion after

the A0 mode has traveled through the thickness change. The dispersion effect of the

A0 mode in the signal of th

The S0 mode of the chosen numerical example shows a different behavior. In

contrast to the one site height change

for a symmetric change of thickness

of the thickness jump have influence on the amplitude maxima. Inside the mi

of Figure 3 (b) it can be seen, that the signal of the reference calculation has been

converted partially into an A0 mode. The symmetric change of t

mode conversion. Only partially reflections of the incoming signal can be seen in

the received signal. The amplitudes

reflected S0 mode in the mix zone

bonded to one side of the plate only, it is not possible to see any difference. The

investigated modes are not very sensitive against a symmetric thickness change.

Only the reflections of the traveling wave can be a sign for damage.

MODELING INTERACTIONS PIEZOACTUATORS

A circular shaped piezoceramic

perfectly bonded to an aluminum plate

in the middle of the plate

configuration is shown in

patch correspond to a zicronate lead PIC

a)

Figure 4. Model configuration: a) Piezoceramic actuator bonded on the center of the

aluminum plate. b)

Through modeling we research on

on the upper face of the piezoactuator

soldering point used to

is relatively dispersive in the chosen frequency × thickness range. This causes

an evident shift of phase on the received signal due to the increase of the phase

velocity while traveling through the thickness change. This can be observed in the

ences between the results corresponding to two different thickness changes of

figure 2) of 0.6mm height and of 2 times 0.6mm height,

(a) inside the ellipse on the left hand side the signal relative

to the transmitted S0 mode is shown, which is the result of a mode conversion after

the A0 mode has traveled through the thickness change. The dispersion effect of the

A0 mode in the signal of the sensed S0 mode can be seen as well.

The S0 mode of the chosen numerical example shows a different behavior. In

contrast to the one site height change S0 mode doesn’t convert into another mode

for a symmetric change of thickness, figure 3 (b). Only the reflections at the borders

of the thickness jump have influence on the amplitude maxima. Inside the mi

it can be seen, that the signal of the reference calculation has been

converted partially into an A0 mode. The symmetric change of thickness induces no

mode conversion. Only partially reflections of the incoming signal can be seen in

the received signal. The amplitudes of the converted A0 mode and the two times

reflected S0 mode in the mix zone are nearly the same. With a piezoelectric

e of the plate only, it is not possible to see any difference. The

investigated modes are not very sensitive against a symmetric thickness change.

Only the reflections of the traveling wave can be a sign for damage.

NG INTERACTIONS PIEZOACTUATORS -STRUCTURE

ceramic actuator (diameter = 10mm, thickness = 0.5mm)

nded to an aluminum plate (width, length = 300mm, thickness = 2mm)

in the middle of the plate is taken as model for calculation. The geometrical

shown in figure 4-a. The material properties for the piezoelectric

patch correspond to a zicronate lead PIC-181 from PI ceramics [6].

b)

Model configuration: a) Piezoceramic actuator bonded on the center of the

aluminum plate. b) Piezoceramic with soldering point on the top.

Through modeling we research on the global effects from local changes of the mass

on the upper face of the piezoactuator. Usually, piezoelectric actuators have a

used to connect them to the wires. Compared to the models on

thickness range. This causes

an evident shift of phase on the received signal due to the increase of the phase

velocity while traveling through the thickness change. This can be observed in the

ences between the results corresponding to two different thickness changes of

igure 2) of 0.6mm height and of 2 times 0.6mm height,

inside the ellipse on the left hand side the signal relative

to the transmitted S0 mode is shown, which is the result of a mode conversion after

the A0 mode has traveled through the thickness change. The dispersion effect of the

The S0 mode of the chosen numerical example shows a different behavior. In

ode doesn’t convert into another mode

eflections at the borders

of the thickness jump have influence on the amplitude maxima. Inside the mix zone

it can be seen, that the signal of the reference calculation has been

hickness induces no

mode conversion. Only partially reflections of the incoming signal can be seen in

of the converted A0 mode and the two times

ith a piezoelectric ceramic

e of the plate only, it is not possible to see any difference. The

investigated modes are not very sensitive against a symmetric thickness change.

STRUCTURE

0mm, thickness = 0.5mm) is

(width, length = 300mm, thickness = 2mm)

The geometrical

. The material properties for the piezoelectric

Model configuration: a) Piezoceramic actuator bonded on the center of the

Piezoceramic with soldering point on the top.

the global effects from local changes of the mass

piezoelectric actuators have a

Compared to the models on

which this is not considered this extra mass changes definitely the dynamic

behavior of the actuator. Therefore, two different conditions are simulated and

compared. One with a squared soldering point (2x2mm and 0.3mm height) on the

top of the piezoelectric actuator (Figure 1-b), and the other without this soldering

point. The results are shown in figure 5 - 6 in order to compare one with the other.

The contour plots of the vertical component of the displacement (u3) are visualized.

Two examples show a frequency dependency of the amplitude height of the of the

u3 component for different directions from the emitting source.

Figure 5. Contour plots of out of plane displacement U3 to show the Lamb wave behavior

100 kHz without a soldering point (left) an with a soldering point (right)

Figure 6. Contour plots of out of plane displacement U3 to show the Lamb wave behavior

300 kHz without a soldering point (left) an with a soldering point (right)

The first two contour plots for 100 kHz in Figure 5 are nearly equal in comparison

between simulations with and without a soldering point. In Figure 6 the amplitude

distributions of u3 are different in each case. The dynamical behavior of the actuator

changes on the case of extra load considered and its behavior is frequency

dependent.

SIMULATIONS USING SPECTRAL ANALYSIS

Spectral analysis was used in order to obtain better accuracy with the same level

of discretization [8]. The wave equation is considered in the way fLu = , where L

stands for a differential operator. According to our approach, the function of

displacements ),( txu is globally approximated or interpolated as,

∑=

=≈N

n

nnN xtatxutxu0

)()(),(),( φ . (1)

The functions )(xnφ are the first N members of an orthonormal function basis,

and the coefficients functions )(tan are selected in such a way that they minimize

the residual R in some sense,

),(),())(),...,(;( 0 txftxLutataxR NN −= . (2)

The result is a system of ordinary differential equations for the functions )(tan .

In our case, spectral collocation method [9] was applied for the spatial domain.

We took Chebyshev polynomial basis [8] as a function basis which is more suitable

to model waves and oscillations than the classical FEM shape functions (Figure 7).

Our criteria to minimize (2) were to make zero the residual in a Chebyshev-Lobatto

grid [9] instead of an evenly spaced grid (Figure 8). To find the values of )(tan in

every time step we used the central difference time integration scheme to solve the

system of differential equations in the time domain.

Figure 7. – Chebyshev interpolation polynomials (left) used for the complete domain and

linear shape functions (right) used for local interpolation.

Figure 8. – Example for comparison: The Chebyshev-Lobatto (left) and the classical evenly spaced grid using in FEM (right).

Figure 9. – Result of simulations 2D using shell finite elements (left) and 2D spectral

elements (right) =t 13 sµ .

Figure 10. – Result of simulations 2D using shell finite elements (left) and 2D spectral

elements (right) =t 29 sµ

Two simulations using FEM and SEM were compared in Figures 9 and 10 using

shell elements to model the 0A mode [10]. The input signal consisted in one cycle

sine burst modulated by a Hanning window. For the calculations the MATLAB

software was used. Both simulations have the same quantity of degrees of freedom

as long as the same level of discretization in each case is used. The FEM simulation

results shown on the left side of Figures 9 and 10 can be only improved by refining

the mesh side, reducing the time step length for the time integration scheme and, as

a consequence, spending more time of calculation. On the SEM results of the

calculated solution, shown in the right side of Figures 9 and 10 the propagation of

the error is evidently less. This is an effect derived from the efficiency of spectral

methods in relation of how derivable the solution is, known as spectral accuracy [8-

9].

CONCLUSIONS

This work in development shows the potentiality of available numerical methods

to efficiently model the behavior of Lamb waves in plates and the interaction

actuator structures. Accuracy of spectral analysis and Spectral Elements constitute a

potential tool to reduce computational effort to obtain the same level of accuracy. In

future work, research on the influence of the adhesive layer between piezoelectric

actuator/sensor and structure will be done, as well as the implementation of higher

order spectral elements and the use of the specific features of Lamb waves to

combine analytic-numerical methods to reduce time of calculation and memory

storage in the computer simulations.

ACKNOWLEDGEMENTS

The financial support of this work by the German Research Foundation (GA

480/13-1 and GK-828) is deeply acknowledged.

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