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Damage Evaluation in Dimension Limestone
using Nonlinear Ultrasonics
Joshua E Love1, Megan E McGovern2 and Henrique Reis1*
1Department of Industrial and Enterprise Systems Engineering, University of Illinois, USA
2General Motors Manufacturing Systems Research Laboratory, USA
Submission: September 03, 2019; Published: September 25, 2019
*Corresponding Author: Henrique Reis, Department of Industrial and Enterprising Systems Engineering, University of Illinois, 104 S. Mathews Urbana, Illinois 61801, USA
How to cite this article: Joshua E Love, Megan E McGovern, Henrique Reis. Damage Evaluation in Dimension Limestone using Nonlinear Ultrasonics.
Civil Eng Res J. 2019; 9(1): 555755. DOI: 10.19080/CERJ.2019.09.555755
Dolomitic limestone test samples with increasing levels of damage were obtained by exposing limestone samples to temperature levels of 100°C, 200°C, 300°C, 500°C, 600°C, and 700°C for a period of 90 minutes. The samples were then nondestructively tested using nonlinear ultrasonic in the form of a non-collinear ultrasonic wave mixing approach. In addition, the test samples’ degradation of flexure strength due to damage accumulation caused by the exposure to increasing levels of temperature was also obtained using four-point bending tests. Results using the currently used non-collinear ultrasonic wave mixing approach correlate well (R2 = 92%) with the corresponding obtained reduction in strength, and with results obtained using different transducer arrangements. The approach has potential applications including quantitative evaluation of damage in stone artifacts and well as to evaluate fire-induced damage in stone infrastructure.
Limestone is frequently used as a building material due to its wide availability, comprising approximately 10% of all sedimentary rock in the world [1,2]. Typically, limestone is formed from the skeletal remains of marine organisms, such as corals, algae, and mollusks. Dolomite [CaMg(CO3)2] forms as a result of the exposure of calcium carbonate to magnesium ions. Dolomitic limestone is composed of 50-90% calcite and 10-50% dolomite [1,2]. Historically, stone has been used as structural components in buildings such as the pyramids in Giza . Since the 1800s, usage of stone as load-bearing structural component has largely stopped in favor of using steel frames [3-5]. Instead, stone is often used as non-load bearing, decorative cladding. Since the 1960s, 1-1/4 inch (30mm) thick panels anchored to masonry, concrete, or steel have been used as in façades for architectural applications. Since the late 1980s, stone-faced composite panels have also been used as decorative exteriors for buildings . These panels typically consisted of a 3/8 in (10mm) aluminum honeycomb core, sided by a 1/16-inch (1.5mm) layer of stone (facing the outside) and by a thin lightweight reinforced polymeric composite layer such as a reinforced glass fiber composite panel. A study conducted by Chin  on the most common causes of failure of stone claddings determined that 40% of failures were due to a reduction in the
strength of the stone cladding due their exposure to weather, e.g., temperature variations. Additionally, 45% of failures were due to failure of the connection between the cladding and the building and 15% of failures were due to water leakage. Both connection failures [7-9] and water leakage  can be addressed with building regulations and codes-many of which are already in place, such as the ones supported by American Society for Testing and Materials (ASTM) standards.
However, existing regulations and codes do not address the reduction in strength due to weathering [11-14]. Furthermore, some of the connection failures can also be attributed to loss of strength in the stone cladding surrounding the connection points [7-9, 15]. In addition, variability also exists in the quality of virgin quarried stone, differing from quarry to quarry and even between different locations in the same quarry. Unsurprisingly, when lower quality freshly quarried stone is used as cladding for buildings, it performs poorly. As an example of this is the Amoco Building in Chicago. The building was clad with 44,000 1-1/4 inch (30mm) thick Italian Carrera marble panels, which deteriorated quickly and were replaced for safety reasons with white Mt. Airy granite [16,17]. Laboratory testing determined that the marble cladding had experienced a 40% reduction in strength and an additional 30% reduction in strength was estimated over the next 10 years.
In a recent study on cladding panels, Schoenberg  provides a
review of 100 years’ worth of research into stone, with a particular
focus on identifying the causes of deterioration in stone paneling.
Namely, panels can warp, dish, or bow, reducing the aesthetic
appeal of a building and, in extreme cases, presenting a danger
of falling stone. For example, marble deterioration may be caused
by a variety of mechanisms. Chemical and biological attack on old
buildings and monuments have been examined by several authors
[19-21], and bowing has been studied for approximately 100
years [22,23]. While most bowing studies have focused on marble,
examples of bowing in limestone and in granite have also been observed
[24-26]. It seems that the largest factor regarding bowing
in stone is thermal hysteresis in combination with moisture, due
to the thermal anisotropy of calcite crystals within the stone .
In two previous studies by McGovern et al. [28-30], a receiving
dilatational transducer was incidentally mounted on the test specimen
surface to detect the generated shear horizontal nonlinear
wave. The longitudinal transducer was able to detect the resultant
nonlinear wave because the shear wave traveled through a
highly heterogeneous media, i.e., mesoscopic media, resulting in
a significant level of mode conversion and scattering. The scattering
and mode conversion lead to a random spatially incoherent
normal displacement at the surface, which when averaged over
the large aperture of the receiving longitudinal transducer (Panametrics
V1011, center frequency of 100kHz) lead to a temporally-
coherently but spatially incoherent output signal. In one of
these studies  the two primary longitudinal waves were two
critically refracted longitudinal waves using dilatational transducers
mounted on shear wedges. In another study , the two primary
longitudinal waves were sent by two incidentally mounted
This study seeks to address whether the type of receiving
transducer (shear instead of dilatational) and its placement to
detect the resultant nonlinear shear wave has an impact on the
observed results. In this study, the two dilatational waves are generated
by two incidentally mounted dilatational transducers, and
the receiving transducer is a shear transducer also incidentally
mounted to the test sample. However, because the dilatational and
shear velocities and corresponding attenuations depend upon the
thermal damage of the specimens (see Figures 1&2), and because
the overall dimension of the specimens are different from each
other, the resulting shear wave meets the receiving shear transducer
at a different location for each specimen, see Table 1. Please
note that the overall dimension of the specimens is irrelevant
provided that the relative position of the sending and receiving
transducers is such that the two intercepting longitudinal waves
generate the resultant shear wave, which must be captured by the
receiving shear transducer.
Acoustic techniques for structural health monitoring have
traditionally utilized principles that are valid in the linear elastic
domain. Namely, when a wave propagates in a linear elastic
medium it will maintain the same frequency in the presence of
flaws, regardless of whether or not the amplitude and/or phase
may change in the presence of flaws. Additionally, the principle
of superposition holds, where the resultant wave field associated
with the two intersecting waves is the sum of the two wave
fields associated with the two intersecting waves. On the other
hand, if a wave propagates in a nonlinear elastic medium, the frequency
will not be maintained, e.g., harmonic generation [31-34].
Furthermore, the principle of superposition does not hold true
when the propagating medium is nonlinear, mainly because of the
presence of higher order terms in the equations of motion [35,36].
As a result, in a nonlinear elastic medium, the intersection of two
waves may result in the generation of a third wave that may have
different polarization, different frequency, and may propagate in
different direction. However, for a resultant nonlinear scattered
wave to be generated the resonance and polarization conditions
need to be satisfied [35-47].
Past experiments have observed nonlinear elastic behavior
when using non-collinear wave mixing, utilizing a variety of input
frequencies [35-49]. Non-collinear wave mixing has been utilized
successfully to determine the higher-order elastic constants in
materials [41-43] and to detect degradation such as plastic deformation
and fatigue damage accumulation in metals [44,49], as
well as to evaluate aging in polymers . Johnson et al. [40,48]
observed nonlinearly generated waves in crystalline rock and developed
criteria to verify that the observed nonlinear wave was
caused by the aforementioned interaction of the two primary
waves within the material and not by possible instrumentation
Limestone has a brick and mortar type microstructure and
thus exhibits nonlinear mesoscopic elastic behavior [50,51]. In
such materials, the bricks (i.e., grains, crystals, impurities) interface
with each other across an elastic system, which behaves as
the mortar. In limestone, the mortar is a system of asperities that
holds the bricks together at the grain/crystal boundaries. Because
the majority of deformation occurs within the mortar, this system
of asperities is the source of most inherent nonlinear response
in limestone. Additionally, limestone may be subject to various
sources of damage, such as freeze-thaw cycles, acid dissolution,
frost action, and salt crystallization, which cause distributed microflaw
populations . These microflaw populations make the
material more susceptible to crack propagation as the microflaws
coalesce. Thus, the ultimate strength of the material is limited by
the presence of these microflaw populations, which act as nuclei
when the material fractures. The presence of these microflaw
populations causes nonlinear distortion in propagating mechanical
waves. Limestone has already been shown to exhibit nonlinear
behavior by Johnson et al. [39,40,48]. In this study, test specimens
with artificially induced weathering damage and different levels
of damage are investigated using a non-collinear wave mixing
approach by using two incidentally mounted dilatational transducers
as sending transducers and using an incidentally mounted
shear transducer as the receiving transducer.
In McGovern et al. [28-30] samples of Illinois dolomitic limestone
salvaged from windowsills were cut into blocks with nominal
dimensions of 155mm x 185mm x 55mm. The heating process
used and discussed by Scherer and his associates [53,54] was used
to induce controlled artificial damage. Seven samples were placed
in an oven, in which the temperature was increased at a rate of
50 °C per 20 minutes from room temperature (~25 °C) to each
sample’s individual respective desired temperature of 100 °C, 200
°C, 300 °C, 400 °C, 500 °C, 600 °C, and 700 °C. The samples were
then kept at that temperature for 90 minutes. Note that 90 minutes
was chosen, rather than 60 minutes used by Scherer and his
associates [53,54], to ensure uniform distribution of damage, as
the specimens in this experiment were larger than the specimens
used in references [53,54]. After heating, the oven was turned off
and the samples were left inside to cool overnight.
The linear acoustic properties of the test specimens, i.e., their
dilatational and shear velocities and the corresponding attenuations
were obtained and reported in McGovern and Reis .
For the benefit of the reader, Figure 2 shows the dilatation and
shear velocities and corresponding attenuations as a function of
damage, which was reproduced with permission from . For
additional discussion on specimen preparation and their linear
characterization, the reader is refereed to McGovern et al. [28-30].
To evaluate the reduction in flexural strength caused by the
temperature induced weathering process, five beam samples
were cut from each of the specimens. The bending specimens
were cut to the nominal dimensions of 180 x 55 x 15mm. These
dimensions were chosen in order to satisfy the slenderness assumption
of beam theory (h≤10 L). Before running any flexure
tests, the flexure test specimens were conditioned in an oven at
60°C for 15 hours to ensure the specimens were dry. After 13,
14, and 15 hours, the specimens were weighed to ensure their
weight remained constant, implying they were completely dried.
The prepared specimens were loaded under four-point bending
with a supporting span of 160mm and a loading span of 40 mm,
using a displacement rate for the load head of 0.05mm/min. Load
and displacement measurements were taken until the specimens
fractured. The testing was conducted in accordance with ASTM
standard . Specimen width did not meet the standard, as dimensions
were limited by original specimen geometry. Reduction
in strength was measured with respect to the average strength
of the control specimen in order to measure percentage reduction
in strength. Figure 3 shows the average percent reduction in
strength for each specimen (compared to the un-weathered specimen).
For additional information regarding the bending tests, the
reader if referred to Megan et al. [28-30].
The samples used in this study are the samples used in the
study reported by McGovern et al. [28-30] after the five thin layers
were removed from each sample to use as bending test samples.
The remaining material (with nominal dimensions of 65mm
x 185mm x 55mm) was used as the test specimens in this study,
see Figure 4, where the darker areas are the result of the heavy
vacuum grease used as couplant between the test specimens and
the interrogating transducers/wedges.
To assess damage using non-collinear wave mixing, two
monochromatic dilatational waves, 1 k and 2 k , with frequencies
1 f and 2 f , respectively, intersect at an angle, ϕ , to form a third,
scattered shear wave, 3 k , that propagates at an angle, γ , with respect
to k1 and with frequency f3 = f1 − f2 , where 1 2 f > f (see
McGovern and Reis  for a more in-depth description). Following
Kobbett et al. , For this interaction to occur, the following
equations must be satisfied,
where cL and ct are the dilatational and shear velocities respectively.
Notably, there are three interdependent parameters in
these equations,f2/f1 , γ , and ϕ . Thus, if one of these parameters is
chosen, the other two are also determined, see Figure 1.
In order to quantitatively assess the degree of damage in each
of the specimens, this study compares the nonlinear wave generation
parameter, β , of each specimen. To compare between
specimens, the normalized nonlinear wave generation parameter,
β/β0 , where β0
is the nonlinear wave generation parameter of
the undamaged specimen, is used, as was the case in McGovern
et al. [28-30]. A derivation of the normalized nonlinear wave generation
parameter can be found in McGovern et al. [28-30], and is
where is the recorded amplitude of the nonlinearly
generated wave, α (kn) is the experimentally determined attenuation
coefficient for the wave n k , and
kn D is the distance traversed
by the wave kn (D =k1 = Dk2) . Because the nonlinear wave generation
parameter, β , is a material property, the normalized nonlinear
wave generation parameter,β/β0, should be consistent for each
specimen, regardless of the frequency ratio used. This experiment
seeks to determine whether the type of the receiving transducer
has an impact on the measured normalized nonlinear wave generation
Two dilatational transducers (Panametrics V413, center frequency
500kHz) and a shear transducer (Panametrics V1548,
center frequency 100kHz) are incidentally mounted as illustrated
in Figure 1, see Table 1.
The location of the two sending dilatational transducers is
largely dictated by specimen dimensions. In this paper, the location
of the receiving shear transducer is adjusted according to the
specimen dimensions and is accumulated damage to assure the
resultant shear wave meets the receiving shear transducer at its
A schematic diagram of the experimental setup is shown in
Figure 1. A pulser-receiver (Ritec RPR-4000) was used to generate
an eight cycles sinusoidal toneburst, k1 , at f1 = 200kHz. A
function generator (Krohn-Hite Model 5920) was used to generate
an eight cycles sinusoidal toneburst, 2 k , which was amplified
by a gated amplifier (Ritec GA-2500A). This signal was iteratively
swept from f2 = 60kHz to f2 = 180kHz in 2kHz increments ( f2/f1
= 0.3 to 0.9). The sample rate was 50MHz to avoid trigger jitter
[39,48]. For each iteration, data was averaged 350 times to mitigate
the effects of noise and scatter. The received wave was filtered
and amplified by a 4-Pole Butterworth filter (Krohn-Hite Model
3945), and then sent to the computer. To measure the nonlinearly
scattered wave, three measurements were taken:
A. A measurement with both longitudinal sending transducers
B. A measurement with only one of the sending transducers
C. A measurement with only the other sending transducer
By subtracting, the two individual measurements from the
measurement obtained when the two sending transducers operate
simultaneously, the nonlinearly scattered wave (i.e., the difference
signal) can be isolated. It should be noted that this is not
a perfect subtraction to isolate the nonlinear scattered wave, as
some of the energy from the primary waves is used to generate
the nonlinear scattered wave. Therefore, the amplitude of both
transducers operating simultaneously should be slightly lower
than the sum of the amplitudes of the transducers operating
individually. As a result, the difference signal, i.e., the nonlinear
scattered wave includes a portion of the amplitude of the primary
waves. To account for this, the resultant measurement was filtered
around the theoretically predicted frequency, 3 f .
The location of the dilatational sending transducers was
largely determined by specimen geometery. The location of the
shear receiving transducer was determined by utilizing equation (2). For the other specimens, it was assumed that utilizing the
same calculated angle, γ, would be an appropriate location, similar
to the experimental method in McGovern and Reis . The
exact configuration of the transducers for each specimen along
with specimen dimensions can be seen in Table 1. Additionally,
for each independent test, all transducers were removed from
the test sample, residual couplant was cleaned from the surface
of the sample and the transducers, new couplant was applied to
the transducers, and the transducers were affixed to the sample.
Figure 5 shows the received signals in the time domain for
the un-weathered 25°C specimen. Figure 5a shows the signal recorded
when both sending transducers are operating simultaneously,
and Figure 5b shows the sum of the signals recorded when
each sending transducer operates individually. Figure 5c shows
the scattered shear wave generated by the nonlinear interaction
when both sending transducers operate simultaneously. The observed
time-of-arrival differed from the predicted time-of-arrival
within an error of 5%. In Figure 5, the amplitude of the generated
shear wave is scaled up 10 times.
When analyzing the collected data, the Johnson et al.  selection
criteria was used to ensure the observed signal was in fact
the nonlinearly generated scattered shear wave and not an artefact
of equipment nonlinearities such as nonlinearities in the used
couplant, amplifiers etc. Namely, the amplitude criteria was met
by varying the voltage of the primary waves and noting that the
received scattered wave amplitude changed in a manner proportional
to the primary scattered wave amplitudes. The directionality
criteria was also met by the nature of the experiment, i.e. placement
of the sending and receiving transducers. The time-of-flight
criteria was met by observing that the experimental arrival times
were consistent with the expected arrival times for the nonlinearly
To identify the frequency ratio that produces the maximum
amplitude of the nonlinearly generated shear wave, one transducer
was set to send a 200kHz signal for each test, whereas the signal
for the other transducer was swept from 60kHz to 180kHz in
2kHz intervals. A sample of experimental results for one frequency
sweep can be seen in Figure 6. Figure 6 shows the frequency
ratio that experimentally corresponds to the observed maximum
amplitude as compared with the theoretical predicted frequency
ratio. The discrepancy between theoretical and observed peak
frequency ratios are attributed to the mesoscopic behavior of
limestone, deviations from the theoretical assumptions used in
the non-collinear wave mixing formulation , and the transducer
placement used in this study.
Please note that under the assumptions associated with to the
traditional non-collinear wave mixing formulation , i.e., plane
longitudinal primary waves and homogeneous isotropic material,
the nonlinear wave shown in Figure 5 would have zero magnitude.
However, mainly because of the beam spread of the two primary
waves and the mesoscopic behavior of the material, the resultant
nonlinear wave has a finite amplitude, as it is illustrated in Figure
Figure 7 shows a comparison of the results obtained in this
experiment with various other transducer arrangements used
in past experiments by McGovern et al. [28-30]. Notably, there is
a discrepancy between the normalized nonlinear parameter for
the 100 °C specimen between this experiment and past experiments.
This deviation is explained by the presence of residue in
the 100 °C specimen left during the cutting process using oil to
cool the cutting saw. This residue was burned off for higher temperature
specimens, but for the 100 °C specimen the temperature
was not sufficiently high. In addition, the damage distribution for
the 100 °C specimen was not uniform throughout its volume because
of the lower temperature. Apart from the 100 °C specimen, observed normalized nonlinear wave parameter values match
closely to previous studies utilizing dilatational receiving transducers,
which validates previous experiments and implies shear
transducers and dilatational transducers can both be used as receiving
transducers to determine the normalized nonlinear wave
generation parameter, see McGovern et al. [28,30]. In Figure 7,
as it was also observed in previous experiment [28-30], the normalized
nonlinear parameter changes drastically from the 600 °C
specimen to the 700 °C specimen. This is due to the calcination
transformation that takes place at these temperatures. For additional
information regarding the calcination decomposition, the
readers are referred to [28-30, 55,56].
Figure 8 shows a correlation between the normalized nonlinear
wave generation parameter and the reduction in flexural
strength (R2 = 0.92). The Figure also shows that the percent reduction
in flexural strength error box is negative for the control
specimen. This is an artefact of the normalization process, as percent
reduction in flexural strength is measured with respect to the
mean flexural strength of the control specimen. Apart from the
results of the 100 °C specimen, the power fit seems to accurately
relate the normalized nonlinear parameter with the reduction in
flexural strength. The data point corresponding to the specimen
damaged using the 100°C is not included because of the lack of
uniformly distributed damage in the 100oC specimen; because the
specimens were cut using oil as the saw lubricant, and the 100°C
was not sufficiently high to eliminate/reduce the oil residue in the
middle of the specimen.
Finally, results of the receiving transducer placement shown
in Table 1 also indicate the maximum deviation from the position
selected for the undamaged specimen is about 15mm for the
600 °C specimen. Note that the 700°C specimen is not considered
mainly because is a much thinner specimen, i.e., it has a lower value
of the dimension c = 33mm, as compared to the other specimens,
see Table 1. Considering that the diameter of the receiving
shear transducer (Panametrics V1548, center frequency 100kHz)
is 38mm (1.5in), a conclusion can be reached that one can place
the receiving transducer assuming the material properties of the
undamaged specimen without altering the results, as it was done
in previous studies by McGovern and Reis [28-30].
Dolomitic limestone specimens were artificially weathered at
temperatures of 25°C, 100°C, 200°C, 300°C, 400°C, 500°C, 600°C,
and 700°C. Non-collinear wave mixing of two dilatational waves
was used to generate a scattered shear wave. This interaction of
the two intercepting longitudinal waves was used to characterize
the limestone specimens of varying degrees of thermal damage.
While the resultant scattered shear wave was captured using
a shear transducer mounted on the test specimens’ surface, its
location varied from specimen to specimen because the velocities
of each specimen depend of its damage accumulation. This study
shows that the use of both dilatational and shear transducers as
receiving transducers is acceptable, i.e., both lead to relatively
similar normalized nonlinear wave generation parameter values,β/β0 . It was also observed that placing the receiving transducers
are different locations did not significantly affect the results mainly
because of the relatively large diameter of the receiving sensor,
which validates the simpler approaches used by McGovern et al.
[28,30], where the relative location of the transducers’ placement
is constant. This approach has the potential to quantitative
evaluate damage in monuments and art objects for the purpose of
taking remedial actions as discussed by Scherer and his associates
[53,54], as well as to evaluate fire induced damage in stone such
as in tunnels.
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