JSSSJournal of Sensors and Sensor SystemsJSSSJ. Sens. Sens. Syst.2194-878XCopernicus PublicationsGöttingen, Germany10.5194/jsss-5-187-2016Determination of the material properties of polymers using laser-generated broadband ultrasoundClaesLeanderclaes@emt.uni-paderborn.deMeyerThorstenBauseFabianRautenbergJensHenningBerndMeasurement Engineering Group, Paderborn University, Warburger Str. 100, 33098 Paderborn, GermanyLeander Claes (claes@emt.uni-paderborn.de)7June20165118719623September201519May201622May2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://jsss.copernicus.org/articles/5/187/2016/jsss-5-187-2016.htmlThe full text article is available as a PDF file from https://jsss.copernicus.org/articles/5/187/2016/jsss-5-187-2016.pdf
In the non-destructive determination of material properties, the utilization
of ultrasound has proven to be a viable tool. In the presented paper, a laser
is used to create broadband acoustic waves in plate-shaped specimens by
applying the photoacoustic effect. The waves are detected using a
purpose-built ultrasonic transducer that is based on piezoceramics instead of
the commonly used piezoelectric polymer films. This new transducer concept
allows for detection of ultrasonic waves up to 10 MHz with high sensitivity,
thereby allowing the characterization of highly damping materials such as
polymers. The recorded data are analysed using different methods to obtain
information on the propagation modes transmitted along the specimen. In an
inverse procedure, the gained results are compared to simulations, yielding
approximations for the specimen's material properties.
Introduction
Ultrasound transmission measurements are often used for non-destructive
testing and material characterization. In contrast to destructive testing and
characterization methods, no reference sample is required. Instead,
individual components can be tested (multiple times if required), yielding
results for the component itself and not just a certain material sample. This
results in many applications, for example in quality assurance, structural
health monitoring and the observation of material aging. The set-up for an
ultrasonic inspection typically consists of a transducer that generates the
ultrasound, the specimens through which the ultrasound is transmitted and a
receiving transducer. The received signals are then analysed to obtain
information about the specimens properties.
In this paper, plate-shaped specimens are examined. In addition to wide
availability, they also have the advantage that transducers for excitation
and detection of the acoustic wave can be placed freely on the plate's
surface. Thereby, the distance that the wave has to travel through the
specimen can be varied. Other experimental set-ups that use, for example,
cylindrical specimens are limited to a fixed distance per
specimen. Some experiments on plates are conducted using piezoelectric
transducers for both excitation and detection of the waves (,
). This has the advantage of high sensitivity; however, the size
of the active area of the transducers limits the detectable wavelength to a
certain minimum. Also piezoceramic transducers typically have a limited
bandwidth in both detecting and emitting acoustic waves. Focused, pulsed
laser radiation can be used for a broadband creation of sonic waves,
utilizing the photoacoustic effect (, ). This allows
for the excitation of many transmission modes at the same time. A multi-modal
analysis is favourable, especially for orthotropic materials, as the
components of the elastic moduli affect each mode differently and with
different sensitivities . Past studies of guided acoustic waves
in plate-shaped specimens either relied on laser Doppler vibrometers
(, ) or utilized transducers based on piezoelectric
films such as polyvinylidene fluoride for the detection.
Experiments with the piezofilm transducer used by yield no
satisfactory results on polymers, as the sensitivity of the transducer is too
low. The high cost factor of a laser Doppler vibrometer raises the question
of a more affordable alternative technique. Because of this, a transducer
based on a piezoceramic strip is developed, unifying a high detection
bandwidth with high sensitivity.
Cross section and bottom view of the purpose-build,
piezoceramics-based transducer for the detection of plate waves.
The examined specimen (a polymer plate made of polyamide 6) is used to
illustrate that the presented procedure can be applied to materials with high
damping. Because many modes are being excited, the evaluation of a single
measured signal proves difficult. Instead, the distance between excitation
and detection of the ultrasound is varied. Using traditional ultrasonic
transmitting transducers, this would be hard to realize with regards to
reproducibility. The transducers would have to be moved and the acoustic
coupling would change upon every step. In the presented measurement set-up,
the distance can be varied easily and highly reproducibly by moving the focus
of the laser radiation, using a linear actuator. The resulting data are then
transformed from the spatial and temporal domain to frequency and wavenumber,
resulting in a depiction similar to a dispersion diagram. This is done by
applying a two-dimensional Fourier transform and alternatively
via the linear prediction method . This way, the dependence on
absolute time and distance can be neglected, the data now only depend on the
increment in space and time. Those increments can be specified with higher
accuracy, as they are given by the measurement equipment (linear actuator and
oscilloscope, respectively). The dispersion diagram generated by this
procedure provides information on the modes being excited by the laser
radiation and can be analysed further to yield information about the
specimen's material properties. An inverse procedure is implemented by
comparing the result of the linear prediction to simulated dispersion
diagrams using an image-based approach.
Realization of the piezoceramic transducer
The presented transducer is a progression on the work shown in .
It features increased sensitivity and bandwidth, a better shielding against
electromagnetic interference and no longer requires a conductive sheet to be
placed between transducer and specimen. The increased sensitivity is
especially required to yield satisfactory signals when examining polymeric
specimens with high damping. Compared to transducers based on piezoelectric
polymer films, as presented for example by , the
piezoceramic-based transducer retains the drawback of a limited spatial
detection ability, similar to the Nyquist criterion. For waves to be detected
unattenuated, their wavelength has to be a least twice the width of the
piezoceramic:
λ>2⋅wcer.
For the detectable angular wavenumber k this results in an upper limit:
k<2πλ=πwcer.
For the presented transducer (wcer= 1 mm), the limit results in
k< 3142 rad m-1. However, measurements (Figs. , ) show that the limit is not as
critical as expected. Signals with an angular wavenumber larger than
3142 rad m-1 are clearly visible in the results.
The transducer is constructed using a cuboid PZT (lead zirconate titanate) piezoceramic (type PIC255,
PI Ceramics GmbH, Germany) with the dimensions lcer= 12 mm,
wcer= 1 mm and a thickness of tcer= 0.5 mm. The piezoceramic is polarized in the direction of the thickness with
the electrodes placed accordingly. Using epoxy resin mixed with silver
particles, the piezoceramic is adhered to a 50 µm thick stainless
steel sheet. Adding the silver particles to the adhesive yields an
electrically conductive coupling. A wire is connected to the back electrode
of the piezoceramic using low-temperature solder paste to avoid depolarizing
the piezoceramic. The stainless steel sheet is then folded into a trapezoid
shape, as shown in Fig. . A hole is drilled through the
sheet to mount a coaxial connector (here a SMB (SubMiniature version B) connector is used). Compared
to transducers based on piezoelectric polymers, piezoceramics normally do not
exhibit broadband sensitivity. Piezoceramics usually are sensitive mainly
around their antiresonance frequency. For the piezoceramic to be used as a
broadband sensor, the resonances need to be damped. The trapezoid shape is
therefore filled with a damping mass consisting of polyurethane mixed with
tungsten carbide . The mixing ratio of tungsten carbide and
polyurethane is 5/1 (mass tungsten carbide / mass polyurethane) to yield an
acoustic impedance of the damping mass of about 7 MRayl. In addition to
damping the resonances of the piezoceramic, the damping mass stabilizes the
transducer mechanically.
Figure shows the absolute value of the
electrical impedance of the raw piezoceramic (solid line) and the finished
transducer (dashed line). The undamped piezoceramic shows distinct
resonance–antiresonance pairs. A comparison with the impedance of the
complete transducer shows that the resonances of the ceramic are almost
completely damped. Only the resonance–antiresonance pairs at about 1.5
and 2.5 MHz remain visible as a combination of minimum and maximum in the
impedance plot. Judging from the frequency-dependent impedance, one can
assume that the transducer exhibits a uniform sensitivity over the considered
frequency band. The overall lower impedance can be attributed to a larger
static capacity compared to the bare piezoceramic resulting from stray fields
between the ceramic and the stainless steel sheet.
Absolute value of the ceramic's electrical impedance before and after embedding in the damping mass.
To further evaluate the transducer's detection capabilities, a broadband
ultrasonic source is required. For this the photoacoustic effect can be
utilized, similar to the experimental set-up described in the following
section. Using a short ultraviolet radiation pulse to heat up the active area
of the transducer, a signal that displays similarity to a step response is
recorded. The radiation is emitted by a laser (MNL
103-PDHigh Power, LTB Lasertechnik Berlin GmbH,
Germany) with a pulse width of 3 ns (full width at half maximum) and pulse
energy of 225 µJ. As the laser radiation is not focused but evenly
distributed on the transducer's active area, it can be assumed that no
ablation takes place. The transducer's active area is heated up by the
radiation abruptly. It is, however, assumed that it takes a significantly
larger amount of time to cool down. Due to thermal expansion, a similar trend
(a step followed by slow exponential decay) is expected for the displacement
in the material. The acquired, normalized step response h(t) is depicted in
Fig. (solid line). The recorded signal is
similar to the step response of a damped oscillatory system. This supports
the assumption that the laser excitation creates a step in the displacement
of the material of the transducer's active area. No unique frequency is
apparent, suggesting that multiple resonance frequencies of the piezoceramic
are excited. The initial step is followed by an exponential decay with a time
constant τ of 0.5 ms (not depicted). The decay may be caused by the
discharge of the transducer's capacitance or result from the induced
temperature gradient via the pyroelectric effect. As the decay is a low-frequency process, its influence on the analysis of the transducer's
ultrasonic detection capabilities is negligible. Deriving the step response
h(t) yields the impulse response g(t):
g(t)=∂th(t).
The normalized impulse response is depicted in Fig. as well (dashed line).
Step response and impulse response of the transducer. Recorded by heating up the transducer's active area with a short burst of laser radiation.
Assuming that the transducer is a linear and time-invariant system, its
frequency response can be derived from the impulse response using the well-known relation
G(jω)=F{g(t)},
where F is the Fourier transform operator. The absolute value of
G(jω) is depicted in Fig. .
Judging from the frequency response, signals with frequencies from 1 to
100 kHz are detected with the same sensitivity. There is a slight decrease
in sensitivity at 300 kHz followed by four peaks between 1 and 4 MHz.
The dip around 300 kHz may be caused by an antiresonance frequency, which is
visible in the impedance plot of the undamped piezoceramic. The first peak
coincides with the remaining resonance frequency of the damped piezoceramic
(Fig. , dashed line), while the second to
fourth can be accounted to resonance frequencies of the undamped piezoceramic
(Fig. , solid line). This indicates that
effects of the resonance frequencies are present even if they are not visible
in the impedance plot. At frequencies above 4 MHz the frequency response
drops sharply and becomes erratic. This suggest an upper limit in the sensors
sensitivity at about 5 MHz. The erratic trend is due to an insufficient
signal-to-noise ratio at higher frequencies. The frequency response, however,
does suggest broadband detection capabilities from about 1 kHz to 5 MHz.
This is confirmed by empirical tests, which show that the sensor is capable
of detecting sound in the audible range. The measurements presented below
further show detected signals in the megahertz regime. If a simple model for the
transducer is required, a high-order low pass with medium quality factor can
be used. A more elaborate model would be a series of second-order low-passes
with the resonance frequencies of the piezoceramic.
It should be noted that this particular method of estimating a transducer's
frequency response only yields qualitative results. This is mainly due to the
fact that the initial signal (the step) is not measured directly. Although
the recorded signal resembles a step response, we cannot be sure if the
excitation is step shaped. To evaluate this, a measurement of the
displacement of the transducer's active area, using for example a vibrometer,
is necessary. This would then enable a quantitative analysis of the
transducer's detection capabilities. As the voltage of the recorded signals
is sufficient and noise presents no problem, the described qualitative
estimation of the frequency response is considered acceptable.
Absolute value of the frequency response of the transducer.
Experimental set-up
To examine the material properties of plate-shaped specimens, the
experimental set-up shown schematically in Fig. is
used. The set-up of the measurement is similar to the ones presented for
example by and . To the authors' knowledge, all
previous set-ups applied transducers based on polymeric piezoelectrics. Pulsed
laser radiation (nitrogen laser MNL 103-PDHigh Power,
LTB Lasertechnik Berlin GmbH, Germany) is focused on a line on the
specimen's surface using a cylindrical lens. The radiation pulses locally
heat up the specimen for a short time (3 ns full duration at half maximum),
which results in the generation of acoustic waves . The
excitation on a line creates acoustic waves with straight wavefronts. The
line also distributes the electromagnetic energy introduced to the specimen's
surface to a larger area, so that no ablation of material occurs. This would
otherwise conflict with the claim of non-destructiveness. Using a line of
excitation also limits the distortion of the signal, which would be caused by
examining circular waves using a line-shaped detector. The optical elements
(the cylindrical lens and a front surface mirror) are mounted on a linear
actuator (T-LSM, Zaber Technologies Inc., Canada) to adjust the
distance between the line of excitation and detection of the acoustic waves.
This enables one to pick up measurement signals with spatial resolution and can
be used to observe the changes in the ultrasonic impulse's shape as it
travels through the specimen. The transducer is placed on the surface of the
specimen with the ceramic strip aligned in parallel to the line of
excitation. A small amount of couplant (multi-range coupling paste ZGT,
GE Measurement & Control, USA) ensures the transmission of acoustic
waves from the specimen to the transducer. Detecting the plate waves on one
surface of the specimen while exciting them on the other allows for
measurements very close to and directly above the line of excitation. A
charge amplifier (HQA-15M-10T, FEMTO Messtechnik GmbH, Germany) is
used to condition the signals for further analysis. Signals are recorded
using a USB oscilloscope. Measurements and laser pulses are triggered
simultaneously by a signal generator. This allows for repeated, identical
measurements to improve the signal-to-noise ratio by forming the ensemble
average of multiple signals.
Experimental set-up for the determination of material properties utilizing
the photoacoustic effect and the transducer shown in Fig. .
Measurements and analysis
Measurement data (normalized signal voltage) with temporal and spatial resolution.
Generated by moving the focus of the laser in equidistant steps below the
transducer and recording a signal at every step. The specimen is a polyamide
6 plate, thickness is 8.8 mm.
Time domain
For an estimation of the specimens longitudinal and transversal wave
velocity, the following measurement, similar to the one proposed by
, is conducted: the focus line of the laser is moved from one
side of the transducer to the other in equidistant steps while recording a
signal at every step. The resulting data are arranged in a two-dimensional
matrix with spatial and temporal resolution as shown in Fig. . This matrix can be used to determine the position
of the ultrasonic transducer relative to the starting position of the linear
axis. If the focus of the laser is located directly below the transducer, the
path the acoustic wave has to travel is the shortest. Therefore, the position
at which the signal arrives earliest has to be the position of the transducer
on the other side of the plate. For further measurements x is set to 0 at
this position.
Close to x= 0, there are two distinct signal groups visible. One arrives at
about 5 µs and a second at about 10 µs. These two signals can
be assigned to a longitudinal and a transversal wave, respectively
. This shows that close to the line of excitation, the observed
behaviour is more similar to bulk waves than to plate waves. The propagation
of waves as guided plate waves occurs farther from the excitation. It is to
be noted that the signal for the transversal wave (the one arriving at about
10 µs) is weaker around x=0. This is in accordance with the results
of , which also show a minimum in the transversal wave directly
below the excitation. Both the longitudinal and transversal wave velocity can
be estimated by dividing the thickness of the specimen by the delay at which
both waves are detected by the transducer. The procedure applied is
illustrated in Fig. . As it is difficult to determine
the exact point of arrival of longitudinal and transversal wave, a method of
estimation is required. Therefore, the envelope of the signal measured at x=0 is considered (solid line in Fig. ). There are two
maxima corresponding with both wave velocities. To estimate the arrival of
each respective wave, tangents are placed in the inflection points that occur
before each maximum (dashed lines in Fig. ). The point
of intersection of the tangents and the t axis is then used as an
estimation for the time it takes the longitudinal and transversal waves to
cross the specimen in thickness direction. With the thickness of the specimen
(8.8 mm) the wave velocities can be estimated as cl= 2611 m s-1 and
ct= 1117 m s-1. This arbitrary method is only used to
obtain
rough estimates that will serve as initial values in the material
characterization discussed later.
Figure shows the results of a measurement similar to the
one in Fig. . This time, the focus of the laser is
moved away from the transducer, starting at the position determined to be
x=0 in the prior measurement. This measurement allows for an observation of
the forming of propagation modes in the plate. While one can observe distinct
longitudinal and transversal waves while the distance between the laser's
focus and the transducer is small, we can detect different propagation modes
at larger distances. At a distance of x=50 mm there are at least two
different propagation modes visible. The first is detected at about
20 µs and the second at about 50 µs. It is also visible that
the faster mode is weaker and contains higher-frequency components than the
slower, dominant signal group. One might assume that these signals may still
be longitudinal and transversal bulk waves. However, the following analysis
in the frequency domain shows that there are in fact many superimposed modes
being excited by the laser.
Envelope of the signal measured with the excitation located
directly below the transducer (solid). With tangents (dashed) in the
inflection points before the maxima to estimate the arrival of the
longitudinal and transversal waves.
Measurement data (normalized signal voltage) with temporal and
spatial resolution. Generated by moving the focus of the laser away from the
transducer in equidistant steps. The specimen is a polyamide 6 plate,
thickness is 8.8 mm.
Frequency domainTwo-dimensional Fourier transform
described a method to analyse waveguides by calculating a
dispersion diagram from spatially and temporally resolved data. Applying a two-dimensional
Fourier transform to such data yields a depiction with wavenumber and
frequency resolution:
F2-D{u(x,t)}=U(k,ω).
Given that signal components from every mode are present in the measured
signals, the depiction shows ridges where curves are expected in a normal
dispersion diagram. Figure shows the result of a two-dimensional
FFT (fast Fourier transform)
applied to the measured data shown in Fig. . Compared to
the same measurement conducted on a metallic specimen (for example copper,
see Fig. ) the different ridges are not as easily
distinguishable. This is primarily due to the specimen's properties. The low-longitudinal and transversal wave velocities in polymers combined with the
ticker plate (8.8 vs. 1.53 mm) yield a substantially larger number of
modes that are capable of propagation. The copper plate, however, shows
several ridges that are easily distinguishable. A characterization of the
copper specimen was presented in , although the measurement was
conducted using a predecessor of the transducer presented above. Figure also demonstrates the broadband detection abilities of the
presented transducer. Ridges corresponding to propagation modes up to 10 MHz
are visible. Due to the low-pass characteristic of the transducer (see Fig. ), the ridges become less visible at higher
frequencies. The limited wavenumber resolution given by the transducer's
width (see Eq. ) is not visible. However, a limit is given
due to the spatial sampling given by the increment with which the linear
actuator and thereby the excitation is moved.
To allow for a possible comparison with simulation results for an inverse
procedure, the ridges of the depiction need to be detected automatically. For
the polymeric specimen, this proves to be problematic, as the ridges are too
close to each other and too thick to be distinguished. An alternative is
proposed in the following section.
Dispersion map of a polymer plate (polyamide 6, thickness 8.8 mm).
Generated by applying the two-dimensional FFT to spatially and temporally resolved data.
Dispersion map of a copper plate (thickness 1.53 mm). Generated by
applying the two-dimensional FFT to spatially and temporally resolved data.
Concept of the application of the linear prediction method for the
calculation of angular wavenumbers km for a each angular frequency
ωi.
Linear prediction method
and developed an approach based on linear
prediction that allows for a direct computation of the dominant angular
wavenumbers for a given frequency. The procedure is illustrated in Fig. . First, the matrix shown in Fig. is
Fourier transformed only in the time direction yielding a depiction with space
and frequency resolution:
F{u(x,t)}=U(x,ω).
Given that there are discrete measurement positions, x is replaced by n⋅Δx, n∈N. As Fig. shows, the
matrix can be considered as N spectra (one for each measurement position).
Assuming a fixed ωi, a column of the matrix U(n,ωi) is
expressed as a superposition of M complex oscillations:
U(n,ωi)=∑m=1MCm(ωi)⋅(e-jkmnΔx)=∑m=1MCm(ωi)⋅μmn,
where Cm(ωi)∈C and km∈R. Applying the
linear prediction method, a matrix is formed to estimate the values
U(L+1,ωi)…U(N,ωi), from the preceding L values using the
weighting factors g1…gL. For better readability, the constant
parameter ωi is omitted in the following equations.
U(L)U(L-1)…U(1)U(L+1)U(L)…U(2)⋮⋮⋮U(N-1)U(N-2)…U(N-L)g1g2⋮gL=U(L+1)U(L+2)⋮U(N).
In short, this can be expressed as
Ag=f.
The weighting vector g is estimated using the pseudoinverse of
A, A+:
g=A+f.
Inserting Eq. () into the lines of the matrix
A yields the following expression:
∑m=1MCmμmL+1-∑m=1MCmμmLg1-∑m=1MCmμmL-1g2-…-∑m=1MCmμm⋅gL=0.
Factorizing results in
∑m=1MCmμmμmL-μmL-1g1-μmL-2g2-…-gL=0.
To satisfy this equation, the expression in brackets has to be equal to zero.
As the weighting factors g1…gL are given by Eq. (),
this equation can be solved for μm, which yields L different complex
roots. These roots are contained within the unit circle and represent the
different oscillations present in the signal. Their absolute value provides a
measure for their significance. For further analysis, the P complex roots
that have the largest absolute value and are thereby closest to the unit
circle are selected. Given that μm=e-jkmΔx, the respective angular wavenumber can be calculated as
km=arg(μm)Δx.
The linear prediction method, as conducted in this paper, has two parameters.
The first is L, the number of weighting factors and thereby the number of
wavenumbers km to be calculated. In practice, setting L to one-third of
the measurement positions yields good results. The second parameter is the
number of complex roots closest to the unit circle P. They represent the
P wavenumbers that are calculated per frequency step. If P is too high,
there might be a lot of wavenumbers obtained that are not part of a mode, but
result from noise or other interferences. In waveguides the number of
propagation modes, and thereby the number of detectable wavenumbers,
increases with the frequency. As a means to resolve this issue, instead of
choosing a fixed P for all frequencies, a lower bound for the distance of
the complex roots μm to the unit circle is set. All roots that exceed
this bound are considered to represent dominant wavenumbers for the
respective frequency.
The result can be printed as a point cloud, as shown in Fig. . It is similar to a dispersion diagram, however there are
some modes that are not detected, some curves are interrupted and some points
are implausible (e.g. those that are above the A0-Mode). To enhance the
result for further analysis, it is checked if the resulting data points form
a curve by checking their distance to each other. If a detected curve
contains fewer than a given number of data points (e.g. <20), it is
discarded. In Fig. , the points that are omitted are
printed grey.
Result of the linear prediction method for dispersion analysis
applied to data measured on a polymer plate (polyamide 6, thickness 8.8 mm).
Grey points are omitted.
Material characterization
The results of the linear prediction method can be used to characterize the
material's properties by applying an inverse approach. To realize this
approach, a forward model, a cost function and initial values for the model
parameters are required.
Lamb's characteristic equations are used as the forward model, with the
transversal and longitudinal wave velocities (ct and
cl) and the thickness of the plate d as parameters
:
tan(βd/2)tan(αd/2)=-4αβk2(k2-β2)2±1,
where
α2=ω2cl2-k2andβ2=ω2ct2-k2.
Based on them, a dispersion diagram is to be simulated and compared to the
results of the linear prediction method. As two parameters of the simulation
(longitudinal and transversal wave velocity) are varied to match the
measurement result, the dispersion diagram has to be computed a number of
times. Therefore, a method that allows for a fast computation of the
dispersion diagrams is required. In this paper, the dispersion diagrams are
created by numerically solving Lamb's characteristic equations using the
Matlab® function ezplot.
This function applies a root-finding algorithm and allows for the computation
of a dispersion diagram in a few seconds. While the resulting dispersion
diagram is incomplete and shows gaps in some modes, it suffices for a
comparison with the equally incomplete result of the linear prediction
method.
Correlation coefficient ϱX,Y(cl,ct) of the measured and simulated dispersion
diagram. Maximum at cl= 2630 m s-1 and
ct= 1070 m s-1.
A cost function is required for an automatic and quantifiable comparison of
measurement and simulation. The set-up of an appropriate cost function proves
challenging as two plots with multiple incomplete lines consisting of
discrete points are to be compared. This issue is solved by applying an
image-based approach and calculating the correlation coefficient. To compare
them, both dispersion diagrams are converted to two-dimensional matrices that
contain ones where lines are found in the respective diagram and zeros
elsewhere. These matrices are regarded as images in the following steps,
where X is the image generated from the result of the linear
prediction method and Y(cl,ct,d) is the
image generated from the simulation result. To compare X and
Y(cl,ct,d), the correlation coefficient
of the two images is calculated. Comparing these two images directly,
however, yields no usable results from the correlation coefficient, as the
lines in the generated images are too sharp. Only modes with a pixel-perfect
match would contribute to the correlation coefficient
ϱX,Y in this case. To remedy this,
Gaussian blur is added to both images to allow for the correlation
coefficient to yield results if the images are similar but not identical.
Using initial values for longitudinal and transversal wave velocities obtained
from the measurement shown in Fig. , both
parameters are swept. The third parameter of Lamb's characteristic equations,
the thickness of the plate (d= 8.8 mm), is measured directly and kept
constant. Figure shows the correlation coefficient
ϱX,Y(cl,ct) of the
two blurred images as a function of the longitudinal and transversal wave
velocity parameters (cl and ct) of the simulation.
There are orthogonal ridges visible indicating that a matched transversal and
a mismatched longitudinal wave velocity (and vice versa) are registered by the
correlation coefficient. The point where those ridges intersect yields the
largest correlation coefficient and thereby an approximation for the
materials parameters. These approximation are
cl= 2630 m s-1
ct= 1070 m s-1
Comparison of simulated and measured dispersion diagram.
A comparison between the simulated dispersion diagram and the result of
linear prediction method is shown in Fig. . Even if both
dispersion diagrams are incomplete, the different modes match well. The
longitudinal and transversal wave velocities determined via ultrasound in
other works (for example by and ) are slightly
higher, which may be due to material variation or the higher frequency used.
Measuring the density ρ of the specimen
(1142 kg m-3) and using the determined wave velocities,
other material parameters can be calculated . These are for
example Young's modulus E, Poisson's ratio ν and the shear modulus G:
E= 3.66 GPa
ν= 0.40
G= 1.31 GPa
Compared to the results of tensile testing in the literature ,
Young's modulus E deviates upwards. This is to be expected, as
aforementioned tests yielded results for the quasi-static case. Polymers tend
to stiffen when subjected to higher frequencies, which is represented by an
increased Young's modulus. The estimated shear modulus and Poisson's ratio
are in good agreement with literature data . As all parameters
of the forward model (Lamb's characteristic equation) are considered
frequency independent and the results are matched over a broad frequency
range, the resulting estimations for the material parameters can be
interpreted as effective quantities for the examined frequency range. To
consider frequency dependencies in the material's parameters (for example the
frequency dependence of Young's modulus) a more complex forward model needs
to be applied. In addition to the frequency dependence of the examined
parameters, further material properties that are common in polymers, such as
damping and anisotropy, can be added to the model for a more accurate
representation of the specimen.
Summary and Outlook
Using laser-generated, broadband ultrasonic plate waves, the material
properties of a polymer specimen (polyamide 6) are determined. This is
enabled by designing a broadband, piezoelectric-based ultrasonic transducer.
Even if the results of the characterization of the transducer suggest
low-pass behaviour with a cut-off frequency of about 5 MHz, modes are
detectable at higher frequencies in metal specimens. They however become less
pronounced as the frequency increases. To further expand the bandwidth of the
transducer, an even smaller piezoceramic with higher resonance frequencies
could be applied, although this would lead to a lower overall sensitivity. To
facilitate the distinction of different modes in the dispersion map, the
examination of thinner specimens is advised. This is particularly important
when using polymer samples with low sound velocities. The examined material
(polyamide 6) represents a simple isotropic material example. For this, the
model using Lamb's characteristic equations suffices. The measurement of
material properties of more complex samples, for example inhomogeneous,
multi-layered or anisotropic specimens requires a more complex forward model.
For example, a semi-analytical finite-element (SAFE) model can be used
. More complex material models require more material
parameters. For example, a transversely isotropic material has five
independent parameters. Therefore, a simple sweep of the parameters to match
simulation and measured dispersion map would no longer be feasible. Instead,
a gradient-based optimization can be applied.
Acknowledgements
The authors would like to thank the German Research Foundation (DFG) for
financial support of the research projects HE 2897/3-1 (determination
of acoustic material properties) and HE 2897/6-1 (evaluation of
hydrothermal aging of continuous-fiber reinforced thermoplastics and
development of an ultrasonic measurement system for non-destructive
characterization of the state of aging for structural health monitoring and
remaining life-time predictions).
Edited by: A. Schütze
Reviewed by: three anonymous referees
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