what can we learn about the elastic properties of a material by … · 2018. 1. 9. · 1....

572非破壊検査第 66巻 12 号（2017）

非線形超音波法による非破壊検査・評価 ⅥNonlinear Ultrasonics for Non-Destructive Testing and Evaluation Ⅵ

What Can We Learn about the Elastic Properties of a Material by Monitoring the Propagation of a Finite-Amplitude Elastic Pulse?

Los Alamos National Laboratory Marcel REMILLIEUX

Review Paper

AbstractAs an elastic wave propagates in a solid, it interacts with its constituents, including its defects. It is not surprising then that elastic waves have been used for decades to characterize materials and evaluate structural integrity in various fields. One interesting fact about damaged materials is their ability to distort elastic waves under certain conditions due to an accumulation of nonlinear effects along the propagation paths of the waves. These effects may be observed even when the wavelength is much larger (by orders of magnitude) than the defects. In this paper, we will study these effects in a long thin bar of Berea sandstone by carefully monitoring the propagation of a finite-amplitude elastic pulse. The simplicity of the experimental arrangement, i.e. longitudinal wave propagating in the axial direction of a sample with a 1D-like geometry, gives us the opportunity to focus on the mechanisms at stake, namely: classical nonlinearity, hysteretic nonlinearity, and non-equilibrium dynamics.

Elastic pulse, Classical nonlinear elasticity, Non-equilibrium dynamics, Nonlinear mesoscopic elastic materialsKey Words

１.　Introduction　Dependence of the wave speed and damping parameters on strain amplitude, slow dynamics, harmonic generation, and hysteresis with end-point memory are some of the interesting properties related to nonlinear and nonequilibrium dynamics and exhibited by natural and man-made materials such as many rocks, thermally damaged concrete, and metallic components filled with micro-cracks. Such systems belong to a wider class of materials referred to as Nonlinear Mesoscopic Elastic Materials (NMEMs)１）. From a mechanical point of view, NMEMs can be thought of as a network of mesoscopic-sized

“hard” elements (e.g., grains with characteristic lengths ranging from tens of microns to a few centimeters) cemented together by a “soft” bond system. The effects of this particular structural arrangement on the propagation of elastic waves is relevant to various natural and industrial processes, ranging in scales and applications, e.g., the onset of earthquakes and avalanches in geophysics２），３）, the aging of infrastructures in civil engineering４），５）, the failure of mechanical parts in industrial settings６）−８）, bone fragility in the medical field９），10）, or the design of novel materials, including nonlinear metamaterials, for shock absorption, acoustic focusing, and energy-harvesting systems11）.　A question that is often asked is: “what creates this nonlinear elastic response?”. A material in pristine condition, i.e., free of micro-cracks, cracks, delamination, dislocations, and other defects, will most likely experience a linear elastic response, both in dynamic and quasi-static experiments, as long as the strain amplitude remains within a threshold throughout the sample. However, if a material possesses such defects, some or all of the aforementioned effects will be observed, even when the wavelengths of the elastic waves propagating in the material are much larger (by orders of magnitude) than the defects. When a closed crack is dynamically loaded with a sufficiently large amplitude, a microscopic interaction takes place within the crack that eventually results in macroscopic effects. What does this microscopic interaction consists of? It can consist of clapping from the two faces of the crack, Coulomb’s friction, the motion of some fluid trapped in the crack tips, thermal mechanisms, and possibly additional mechanisms. Unless one can devise an experiment capable of showing the full dynamic interaction within the defect without any disruption, the next best thing is to observe the effects produced by microscopic-sized defects at the macroscopic scale, which may lead us to Plato’s cave. In

that sense, flaws in the experimental setup, data analysis, or theoretical framework may lead to an erroneous interpretation of the defect dynamics.　Increasingly more elaborate techniques have emerged and evolved over the last 20 years to observe and quantify the effects of nonlinear and nonequilibrium dynamics in NMEMs. Nonlinear elasticity, in the classical sense, is usually modeled by an expansion of the elastic energy as a power series with respect to the strain tensor12）. Practically, if a sinusoidal elastic wave propagates in a medium with this sort of nonlinearity, it experiences wave distortion, eventually leading to shock formation if the wave attenuation is sufficiently small. In the context of elastic wave propagation, classical nonlinearity was studied experimentally13），14） and theoretically15）− 17） as part of the early studies on the dynamics of NMEMs. Quasi-static stress-strain measurements on rocks and other NMEMs have been made since the early 1900s 18）. These measurements show not only that rocks are nonlinear but also that they are hysteretic. Repeated looping up and down in stress-strain space results in a curved, banana-shaped path. Attempts to model this quasi-static hysteresis in NMEMs were going on simultaneously but independently as the attempts to model pulse propagation in similar samples19），20）. Practically, a sinusoidal elastic wave propagating in a medium with pure hysteretic nonlinearity progressively evolves into a triangular wave. The effects of classical and hysteretic nonlinearity on a sinusoidal waveform are summarized in Fig.1.　Besides hysteresis effects, the classical theory of nonlinearity cannot account for the other peculiar wave phenomena observed in NMEMs. The first repeatable observations showing nonclassical nonlinearity were made in resonant bar experiments 21）−23）, ultimately leading to a quantification technique known as nonlinear resonant ultrasound spectroscopy (NRUS). In these experiments, a long thin bar with free boundary conditions - which is representative of a 1D unconstrained system - is excited with a harmonic signal at one end while the elastic response is recorded at the opposite end. The source signal sweeps a frequency range around one resonance frequency of the bar with increasing source amplitudes and the variation of the resonance frequency can be tracked as a function of the maximum strain amplitude in the sample. In a linear elastic material, the resonance frequency is not strain dependent and the peak of resonance does not change with increasing drive amplitude. In NMEMs, the resonance frequency starts


解説

平成 29 年 12 月573

decreasing as a function of strain, and very noticeably so when the strain goes above a certain level, typically near 10− 6 for rocks at ambient conditions. Broadly speaking, the drop in resonance frequency indicates material softening with increasing driving strain. However, the dependence of the resonance frequency on strain is not trivial 24），25）. At very small strains (e.g., ＜10− 7 for a sandstone at ambient conditions), the material experiences an instantaneously reversible decrease of the resonance frequency, behavior dominated by classical nonlinearity. At high strains (e.g., ＞10 −6 for a sandstone at ambient conditions), the decrease of the resonance frequency is governed mostly by conditioning, a process related to the nonequilibrium dynamics of the evolving strain fields within the rock. In summary, under a continuous harmonic excitation, the elastic moduli begin to soften and eventually reach a steady state, i.e. the elasticity of the material reaches a new equilibrium state tuned to the dynamic strain amplitude induced by the excitation. Conversely, if elastic energy is no longer injected into the system, the material recovers back to its original elastic properties, with a rate generally proportional to the logarithm of time 26），27）. Notably, conditioning times are much faster than recovery times26），28） and this temporal asymmetry gives rise to all sorts of interesting observations. For instance, up and down frequency sweeps around the resonance frequency do not lead to the same resonance curves when the sweeping rates are identical but relatively fast. On the other hand, when the sweeping rates are slow enough, the upward and downward resonance curves merge. In between the two strain regimes described above, classical nonlinearity and nonequilibrium dynamics cannot be disentangled from each other in a resonance experiment because many effects are averaged and smeared over many cycles of the excitation.　Dynamic acoustoelastic testing (DAET) is a technique developed more recently, designed to delineate the mechanisms of nonlinear and nonequilibrium dynamics with high sensitivity, regardless of the strain regime. This technique is based on a pump and probe scheme in which high-frequency (HF) pulses are used to locally probe the various phases of a mechanical system set by a low-frequency (LF) pump 29）. Typically, the sample is a long thin bar and the dynamic pump is obtained by exciting the sample near a resonance mode of vibration, thus approaching the conditions used for NRUS. The probe is obtained by tracking the relative change in the arrival time of the HF pulse traveling across the shortest dimension of

the bar, with the assumption that during the short travel of this pulse, the strain field induced by the pump is varying so slowly that it can assumed to be constant. Through a careful synchronization of the HF pulse emissions with the pump signal, a stroboscopic effect can be obtained to capture the nonlinear and nonequilibrium dynamics of the material over a full cycle of the LF pump. The analysis of the relative change of the arrival times of the HF pulses as a function of the LF strain induced by the pump at the probe location gives a complete representation of classical nonlinearity, with high sensitivity typically up to at least the second order terms of nonlinearity, non-equilibrium dynamics, including conditioning and recovery, and hysteresis30）. 　An example of a DAET setup used recently by Lott et al.31） is shown in Fig.2. In this setup, the sample is held by thin wires to approach free-free boundary conditions while it is excited around its first mode of longitudinal vibration. The HF probes are positioned at the location of maximum strain, where the nonlinear elastic effects are the strongest. Ideally, the HF transducers should be mounted on a flat surface in order to launch plane waves into the sample. This is not the case for the example shown in Fig.2 because the sample has a cylindrical geometry. As a result, unwanted diffraction effects may interfere with the nonlinear elastic effects we are trying to quantify32）. In other DAET experiments29），30）, a relatively large transducer and backload are affixed to one end of the sample, resulting in fixed-free boundary conditions. In this case, the position of maximum strain is near the fixed boundary and the HF transducers located in this region may interfere with the transducer-sample interface. Another important aspect to consider is the probing direction (radial direction in Fig.2) in these experiments since it is often perpendicular to the direction of the LF pump (axial direction in Fig.2). If the tensorial nature of elasticity (e.g., Poisson effect) is ignored, the parameters of nonlinearity extracted from probing in the axial and radial directions would be different, whereas they should be identical. Going back to Plato’s cave, care should be taken at each step of these experiments to avoid or at least compensate for misleading effects and extract only those of interest (e.g., nonlinear effects).　Up to this point, the modeling of nonequilibrium dynamics and hysteresis was not addressed. Between the first theoretical description of hysteresis in NMEMs through the Preisach-Mayergoyz formalism by Guyer et al.33） and one of the most recent description developed by Pecorari34）, there has been a considerable number of assumptions and models introduced with most reported and described by Guyer and Johnson35）. Likewise, nonequilibrium dynamics has been described phenomenologically using many models, including the “soft-ratchet” model proposed originally by Vakhnenko et al.36） and recently modified by

Fig.1　Effects of pure classical nonlinearity and pure hysteretic nonlinearity on the evolution of a sinusoidal waveform in the time and frequency domains. Reproduction of a figure originally produced by Van Den Abeele et al.20）

Stress-strain

Strain-time

Strainamplitudespectrum

Hamonicamplitude dependence

No harmonics 2nd harmonic: Slope 23rd harmonic: Slope 34th harmonic: Slope 4etc.

No even harmonics3rd harmonic: Slope 25th harmonic: Slope 2etc.

Linear

1 2 3 4 5 1 3 5

A

1

t

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= M 0

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= M 0 1+7 A = M 0 1+ + f , A7 ^ h

σ

ε

σ

ε

σ

ε

ε

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ε

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Fig.2　Example of a possible DAET setup31）

Low-frequency sensor High-frequency probes

Low-frequency pump

574非破壊検査第 66巻 12 号（2017）

Berjamin et al.37）. To this date, however, a unified model capable of capturing simultaneously all the observables of the experiments described previously does not exist. The multiscale nature of the problem and lack of data on the microscopic-scale dynamics are some of the challenges to surmount in order to define the appropriate theoretical framework.　In response to this and the need for additional data at the macroscopic scale to calibrate existing models and understand the mechanisms at stake, we revisit an experiment conducted by TenCate et al.14）, this time taking a much larger amount of data over much of the surface using a scanning 3D laser vibrometer as a receiver. This experiment was originally designed to quantify classical nonlinearity. What else can we learn about the elastic properties of this sample by monitoring carefully the propagation of a finite-amplitude elastic pulse in this sample? We will demonstrate in this paper that nonequilibrium dynamics and hysteresis are additional nonlinear effects that can be evidenced and potentially quantified as part of this experiment. We use a sample of Berea sandstone for this study because it is a poster-child material for nonlinear elasticity with extensive amount of published data to which our analysis can be compared. Similar studies could be conducted in other rocks, thermally damaged concrete, damaged bones, or metallic components.

２.　Quantifying and Modeling Classical Nonlinearity

2.1　Theoretical Background　In the 1D theory of classical nonlinear elasticity, the stress (σ) can be expressed as a power series with respect to strain (ε) as,

　　　　　　　　　　　　　　　 ………………………（１）

　Where M0 is the linear elastic modulus, β and δ are, respectively, the coefficients of first- and second-order nonlinearity. 　In the absence of energy dissipation and external forces, the propagation of an elastic wave in an infinite 1D medium with first-order nonlinearity (β coefficient only) is governed by the following equation16）,

　　　　　　　　　　　　　　　　　　　　　　　 …（２）

where u is the displacement and c is the wave speed. In the frequency domain, this results in the generation of harmonics. The nonlinear coefficient β can be retrieved from a

measurement of the displacement amplitude U2 of the second harmonic generated at a distance x from a pure tone source signal as13）,

　　　　　　　　　 ………………………………………（３）

where ω is the angular frequency (ω ＝2π f ) and U1 is the displacement amplitude of the fundamental frequency. (In experiments, we often use acceleration data measured with accelerometers or particle-velocity data measured with laser vibrometers). In the frequency domain, the magnitude of the particle velocity V is related to the magnitude of the displacement U as, U ＝V/ω . Eq.(3) can then be adapted to particle velocity measurements using a factor ω ,

　　　　　　　　　 ………………………………………（４）

where V1 and V2 are now the magnitudes of the FFT of the particle velocity at the fundamental frequency and at the second harmonic, respectively.　Damping can easily be introduced in Eq.(2) by adding the term used in Burger’s equation38）,

　　　　　　　　　　　　　　　　　　　　　　 ……（５）

　In this equation, b is a damping coefficient expressed as,

　　　　　　　　　 ………………………………………（６）

where Q is the quality factor (i.e., inverse of loss factor) and proportional to the wave’s attenuation. We will demonstrate just how well Eq.(5) works with an experiment.2.2　Experimental Setup　The experimental setup is shown in Fig.3. The idea is to approach a 1D system by the appropriate choice of sample geometry and excitation frequency. The sample is a long thin bar of Berea sandstone with a length of 1794 mm and a diameter of 39.6 mm. The same sample was used 20 years earlier by

= M 0 1+ + 2 +..._ i

2 u x, t

t 2 c22 u x, t

x2 = c2

x

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- e o] g] g] g

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x2

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tc2

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-- e o] g] g] g] g

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x2

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tc2

2 u x, t

x2 = c2 u x, t

x

2

-- e o] g] g] g] g

e ox

b =c2

Q

Fig.3　Experimental setup for monitoring the propagation of an elastic pulse in the sample of Berea sandstone

＋Controller＋Signal generator＋Data management system

Voltage amplifier

Scanning heads of 3D laser Doppler vibrometer

Sample of Berea sandstone:Diameter: 39.6mm （1.56 inches）Length: 1.794m （70.63 inches）

PZT transducer＋backload

Vibration isolation table


解説

平成 29 年 12 月575

TenCate et al. in another series of experiments14）. The sample rested on foam to approach free (unconstrained) boundary conditions. A piezoelectric disk epoxied between one flat end of the sample and an inertial backload was used to generate elastic waves in the sample. Figure 4 shows the waveform and magnitude of the FFT of the normalized source signal used in the experiments. The signal consists of a sinusoidal wave with a frequency of 22.4 kHz tapered at the start and end by half Gaussian windows. The output signal from the signal generation card (internal arbitrary waveform generator in Polytec PSV-3D-500 scanning laser vibrometer) was amplified 20 times by a voltage amplifier (TEGAM 2350) before being used as input signal to the transducer. The propagation of the elastic pulse was recorded at 119 points along a line, on the surface of the sample, with a regular spacing of approximately 5 mm between the points, starting at 10 mm from the source and extending 600 mm from the source. The Cartesian components of the particle velocity were recorded on the surface of the sample with a Polytec PSV-3D-500 scanning laser vibrometer. Data were acquired with a sampling frequency of 2.56 MHz. At each scanning point, waveforms were averaged 500 times to increase the signal to noise ratio.

2.3　Experimental Results and Data Analysis　Figure 5 shows three selected waveforms and corresponding magnitudes of the FFT of the axial component of the particle velocities for a source signal with a peak amplitude of 16 Vpp (before amplification) at various distances from the source. The waveforms have been truncated to show only the direct pulse; the reflection from the far end, opposite to the source, is also recorded but does not overlap with the direct pulse and can be ignored for this analysis. The time series have been synchronized so that they can all be plotted on the same time scale and the calculation of the FFT is not affected by the arrival time of the pulse at a measurement point. The first waveform and FFT show the vibrational response measured at

just 10 mm from the source and is similar to the source signal, with some minor differences caused by the frequency response of the piezoelectric transducer – even though the transducer is operated well away from its first resonance frequency, its frequency response is not flat and modulates the source signal. More importantly, the vibrational response at this location does not exhibit a significant harmonic content, which attests to the linearity of the source and quality of the bond between the source and the sample. The next two sets of plots show how the sinusoidal waveform is distorted with propagation distance, exhibiting growth of the second harmonic (i.e., peaks observed at 44.8 kHz) and increasing higher frequency content (i.e., between 60 and 80 kHz). The evolution of the waveform with propagation distance is consistent with the features obtained by Van Den Abeele et al.19），20） who studied (analytically) classical nonlinearity and hysteresis separately: classical nonlinearity induces a steepening of the waveform (eventually leading to an N-wave in the absence of attenuation) whereas hysteresis induces a triangulation of the waveform. 　Classical nonlinearity in these data has been quantified by reducing the spectral data at all 119 measurement points with the expression given in Eq.(4). Figure 6(a) shows the variation of 2V2c

2 as a function of V12ωx, based on a wave speed of 2010m/

s. First, note that there is an offset at V12ωx＝0. This offset is not

due to the nonlinearity of the system but to the finite duration of the pulse (i.e., bandwidth). The term 2V2c

2 is calculated from the spectrum amplitude at 44.8 kHz (second harmonic amplitude) at various distances from the source. For the source signal (see Fig.4), which is a linear signal, the amplitude at the second harmonic (i.e., at 44.8 kHz) is two orders of magnitude smaller than the amplitude at the fundamental frequency but is not zero. The same applies to the signals recorded on the sample very close to the source. For this analysis, what is of interest is not the offset at V1

2ωx ＝0 but the growth of the second harmonic amplitude with propagation distance, which is quantified by the slope of the data reported in Fig.6. The slope of the linear fit through this data set is β ＝370. Experiments conducted years earlier on the same sample by TenCate et al.14） led to β ＝400. This close agreement is remarkable given that the source, source signals, and receivers were different in the two experiments. In the work of TenCate et al.14）, the elastic pulse generated by the source was centered at 12.4 kHz and the vibrational signals were recorded by a small number of accelerometers along the pulse propagation. This agreement also indicates that the material properties of the sample have not changed significantly over the long time separating the two experiments. In both studies, however, the effects of attenuation are ignored in the quantification of classical nonlinearity. It is possible to correct the measured spectra by the amount of damping in the system. The amplitude of a monochromatic wave is attenuated with distance as,

　　　　　　　　　　　　　　 …………………………（７）

where γ is the attenuation coefficient. 　This attenuation coefficient should be measured as a function of frequency, which we did not do for this sample. The alternative is to use the damping model proposed in Eqs.(5) and (6). Using this model, the attenuation coefficient can be expressed as,

　　　　　　　　　 ………………………………………（８）

　Figure 6(b) shows the variation of 2V2c2 as a function of

V12ωx for the 119 scanning points, where the amplitudes of

the fundamental frequency and second harmonic have been corrected for damping based on a quality factor Q ＝60. The quality factor was measured in resonance conditions near the fundamental frequency of the pulse. The slope of the linear fit of

Fig.4　Normalized electrical source signal used in the pulse propagation experiments: (a) waveform and (b) magnitude of the FFT. The signal consists of a sine wave with frequency of 22.4kHz, tapered at the start and end by half Gaussian windows

－1.25－1.00－0.75－0.50－0.25

00.250.500.751.001.25

（a）

（b）

Volta

ge，（

V）

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10－4

10－3

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100

Volta

ge，（

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V x = -V0 exp x] ^ hg

=b 2

2c3

576非破壊検査第 66巻 12 号（2017）

this corrected data is β ＝660.　Another important point to discuss is the high variability of the data around the linear fit for V1

2ωx＞5 in Fig.6(a) and V12ωx

＞10 in Fig.6(b). Damping is relatively large in this sample (Q ＝60). As a result, there is a strong competition between nonlinearity and attenuation throughout the propagation, which eventually results in data variability. Such variability would likely be reduced if the experiment had been conducted in a sample with much smaller damping (e.g., metallic component).　Last, it is worth mentioning that in applications related to nondestructive testing and evaluation, a similar approach is commonly used to quantify the evolution of damage in various materials, including metals and concrete 39）−41）, since the parameter of nonlinearity β is much more sensitive to the early presence of damage than the parameters of linear elasticity (e.g., relative changes in elasticity and attenuation). In the laboratory, the ideal conditions needed to measure the parameter β in an absolute sense can be approached by the appropriate choice of sample geometry, transducers, frequencies, and test configuration. In the field, however, this is rarely feasible. For instance, inspection of large concrete structures (e.g., bridges, buildings, nuclear power plants) can be achieved by using surface waves instead of compressional waves while the

effects of attenuation are not always accounted for. As a result, the parameter β will be measured in a relative sense. In the field, for most practical applications, the level of damage in the material will be quantified by monitoring the growth of the second-harmonic amplitude for a given set of source-receiver transducers separated by a fixed distance. 2.4　Simulating the 1D Wave Propagation with Classical

Nonlinearity and Damping　The simple 1D elastic wave equation with damping given in Eq.(5) is now used to model the experiments to see how well it fits the data. We use the following parameters in the model: a wave speed c ＝2010 m/s, a damping coefficient b ＝0.525 (based on a quality factor Q ＝60 and a center frequency of the impulsive source signal f ＝22.4 kHz), and a nonlinear parameter β ＝660. The wave equation is projected onto a finite-element space using the “Mathematics, General Form PDE” module of the software package COMSOL Multiphysics 5.2a. The geometry consists of a line with a length of 1794 mm. One boundary is imposed a flux condition (source) while the opposite boundary is free and so imposed a Neumann condition, i.e., ∂u(x, t)/∂x＝0. The impulsive source signal is the measured signal depicted in Fig.5 for a receiver taken at 10 mm from the source. This signal is normalized to have a peak amplitude

Fig.5　Axial component of the particle velocity measured at representative locations along the propagation path of the impulsive wave. Left column: waveforms; right column: magnitudes of the FFT; top row: 10 mm from the source; Center row: 335 mm from the source; bottom row: at 590 mm from the source

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解説

平成 29 年 12 月577

of 40.5. This approach allows us to take into account the response of the transducer in the 1D model. The computational domain is discretized into Lagrange quadratic elements with a uniform length of 3mm, which ensures at least 6 elements per wavelength up to 110kHz. The solution is integrated in time with a time step of 100ns using the Generalized-α method, which is a fully implicit, unconditionally stable, and second-order accurate method with control over the dissipation of spurious high-frequency modes. The time step is small enough to satisfy the Courant–Friedrichs–Lewy stability condition, i.e., cΔt/Δx＜1.　Figure 7(left) shows typical measured and simulated waveforms of the axial component of the particle velocity at various positions along the sample. The model captures the arrival times, amplitudes, and overall shapes of the waveforms observed in the experiments, for both the direct and reflected pulses. Note that there are two reflections: the first one from the end opposite to the source and the second one from the end where the source is attached. Eventually, these two reflections interact with each other and create more complex waveforms. The waveforms are now truncated and synchronized to retain only the direct, first arrival pulses. The measured and simulated magnitudes of the FFT of the direct pulses are shown in Fig.7(right). The FFTs again show just how well the simple model works to predict the growth of the second harmonic. In fact, after correcting the amplitudes of the fundamental frequency and second harmonic for damping, the variation of 2V2c

2 as a function of V12ωx for the 12 simulated signals

are aligned perfectly with the linear fit shown in Fig.6(b), as expected. However, it is quite clear that very soon, the model fails to capture the frequency content around the third harmonic, between 60 and 80 kHz. The spectral content in this frequency range is affected by hysteretic nonlinearity but also some dispersion effects that have not been discussed.

３.　Quantifying Nonequilibrium Dynamics　In this section, we report on a different pulse propagation

experiment where the transient features of the signals are analyzed to exhibit nonequilibrium dynamics and hysteretic nonlinearity. An alternative to the experiment reported in the previous section is to monitor the evolution of the waveform at a fixed position on the sample for various source amplitudes. The same experimental setup and source signal (Figs 3 and 4) are used for this experiment. Figure 8 shows the waveforms of the axial component of the particle velocity measured at 500 mm from the source for 20 source amplitudes ranging from 1 to 20 Vpp in steps of 1 Vpp (before amplification). The original sinusoidal waveform observed at the lowest amplitudes progressively evolves into a triangular wave, as a result of hysteretic nonlinearity, with additional distortion caused by classical nonlinearity, as discussed in the previous section. Most notably, the waveforms experience a significant delay in arrival time as the source amplitude increases. This delay is observed not only at the extrema (peaks of amplitude) but also at the zero crossings. Classical nonlinearity would not induce any delay at the zero crossings (where the strain is equal to zero) because it produces an instantaneous variation of the modulus without time constants involved. This delay is a direct consequence of nonequilibrium dynamics, specifically the strain-induced material softening (i.e., conditioning).　The data shown in Fig.8 can be further reduced to quantify nonequilibrium dynamics. For a signal measured at the source amplitude i, the signal measured at the source amplitude i −1 is taken as a reference to compute a relative time delay, Δt i/i − 1/t0. The relative time delay between two signals is estimated by cross-correlation. The time delay between the lowest source amplitude ( i ＝0) and the source amplitude i is then obtained by summation as,

　　　　　　　　　　　　　　　 ………………………（９）

　The relative time delay between the signals is also equal to the relative change in speed of the longitudinal wave, Δci/0/c0. Finally, at the perturbation level, the relative change in the Young’s modulus E (the modulus involved in the propagation of a longitudinal wave in a long thin bar) is related to the relative change in the speed of the longitudinal wave as,

　　　　　　　　　　　　　　 …………………………（10）

The relative changes in the elastic modulus over the propagation path of the waveform can be tracked as a function of the maximum strain amplitude at the measurement point. The strain component of interest is ε xx, where x is axial direction. The strain component ε xx can be expressed analytically as a function of the longitudinal component of the particle velocity vx as,

　　　　　　　　　 ………………………………………（11）

Reduced data from Fig.8 are shown in Fig.9. Three distinct regimes can be identified in the waveform: i) the earlier portion of the waveform, where the dynamic strain at the measurement point transitions from zero to steady state; ii) the center portion of the waveform, where the absolute value of the peak amplitude of the dynamic strain is quasi-constant (i.e., steady state); iii) the later portion of the waveform where the dynamic strain at the measurement point transitions from steady state back to zero, i.e., a path similar to the earlier portion of the waveform but in reverse. In the earlier portion of the waveform, the elastic modulus decreases progressively from its undisturbed equilibrium value to a value 4.3 ％ smaller. This is the portion typically referred to as conditioning. In the center potion of the waveform, the evolution of the material softening with the strain amplitude is essentially independent of the wave cycle selected for the analysis, i.e., similar results are obtained from the 9th to

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6000

8000

2V2c

22V

2c2

Raw dataLinear fit

Raw dataLinear fit

0 2 4 6 8 10

（a）

（b）

0 5 10 15 200

0.5

1.0

1.5

2.0×104

Slope of linear fit:＝370β

Slope of linear fit:＝660β

V12 xω

V12 xω

Fig.6　The variation of 2V2c2 as a function of V1

2ωx, the slope of which is β . This is a post-processing of the data set corresponding to the spectral data reported in Fig.3. (a) without any correction for damping; (b) where the amplitudes of the fundamental frequency and second harmonic have been corrected for damping, based on Q ＝ 60

!t i 0 t0 = t n n 1

n=1

i

t0-

f p

Ei 0 E0 = 2 ci 0 c0

xx =vx

c

578非破壊検査第 66巻 12 号（2017）

21st peaks (half cycles) of the waveforms. Under this condition of quasi-constant peak amplitude of the dynamic strain loading, the elasticity of the material reaches a new (but unstable)

equilibrium state where the modulus is 5.3% smaller than in the undisturbed equilibrium. In the later portion of the waveform, the evolution of the elastic modulus follows a different path from

Fig.7　Measured and simulated axial components of the particle velocity at various positions along the sample: (left) waveforms and (right) magnitudes of the FFT. In the waveforms, the time window is long enough to show the direct and reflected pulses. Only the direct pulses are used to compute the FFT

x＝15mm

－20－15－10－505

101520

Particle veloc

ity，（

mm

/s）

Particle veloc

ity，（

mm

/s）

0 0.5 1.0 1.5 2.0 2.5 3.0

Time，（ms）10 20 30 40 50 60 70 80

10－3

10－2

10－1

100

101

102


x＝415mm

－20－15－10－505

101520

Particle veloc

ity，（

mm

/s）

Particle veloc

ity，（

mm

/s）

0 0.5 1.0 1.5 2.0 2.5 3.0

Time，（ms）

ExperimentSimulation

10 20 30 40 50 60 70 8010－3

10－2

10－1

100

101

102


ExperimentSimulation

－20－15－10－505

101520

Particle veloc

ity，（

mm

/s）

Particle veloc

ity，（

mm

/s）

0 0.5 1.0 1.5 2.0 2.5 3.0

Time，（ms）10 20 30 40 50 60 70 80

10－3

10－2

10－1

100

101

102


x＝115mm

－20－15－10－505

101520

Particle veloc

ity，（

mm

/s）

Particle veloc

ity，（

mm

/s）

0 0.5 1.0 1.5 2.0 2.5 3.0

Time，（ms）10 20 30 40 50 60 70 80

10－3

10－2

10－1

100

101

102


x＝215mm

－20－15－10－505

101520

Particle veloc

ity，（

mm

/s）

Particle veloc

ity，（

mm

/s）

0 0.5 1.0 1.5 2.0 2.5 3.0

Time，（ms）10 20 30 40 50 60 70 80

10－3

10－2

10－1

100

101

102


x＝315mm

－20－15－10－505

101520

Particle veloc

ity，（

mm

/s）

Particle veloc

ity，（

mm

/s）

0 0.5 1.0 1.5 2.0 2.5 3.0

Time，（ms）10 20 30 40 50 60 70 80

10－3

10－2

10－1

100

101

102


x＝515mm


解説

平成 29 年 12 月579

that observed in the earlier portion: recovery of the undisturbed equilibrium elastic modulus occurs at a much slower rate than conditioning. This result from a pulse propagation experiment is in excellent qualitative agreement with the results obtained in resonant bar experiments by TenCate et al.27）. Also note that this experiment was repeated at other positions on the sample where similar conclusion could be reached.

４.　Conclusions　In this work, we studied the propagation of a finite-amplitude pulse in a sample of Berea sandstone, which is known to have a nonlinear elastic response starting at relatively low strain amplitudes. Such experiment contrasts with the complexity of the pump and probe scheme in a dynamic acousto-elastic experiment and with the incomplete description of nonlinearity in a resonant bar experiment. From a modeling point of view, the significance of this experiment lies in its simplicity: a 1D-like geometry with only one source frequency involved. It gave us an opportunity to observe how nonlinearity can distort an elastic wave and how such distortion can be used to characterize the material (e.g., to extract the parameters of a nonlinear equation of state). However, it is important to keep in mind that this work is just an anecdote among many others, as James TenCate (Los Alamos National Laboratory) often explains it. We have the anecdotes told by the dynamic acousto-elasticity experiments, by the resonance experiments, by the pulse propagation experiments, by the quasi-static stress-strain experiments with arbitrary stress protocols, etc. More than these anecdotes told at macroscopic scales, we need an experiment capable of imaging the microstructure dynamics in geomaterials, whether this is about fluid motion, friction, sticky contacts, or something else. This experiment is the next leap forward in nonlinear elasticity.

Acknowledgements　We wish to thank James TenCate, TJ Ulrich, Pierre-Yves Le Bas, Martin Lott, Robert Guyer, and Paul Johnson for valuable discussions and suggestions.

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Fig.8　Waveforms of the longitudinal component of the particle velocity measured at 500 mm from the source for 20 source amplitudes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7－20－15－10－505

101520

Time，（ms）

Particle veloc

ity，（

mm

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0.32 0.34 0.36－20

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Time，（ms）

Particle veloc

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mm

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ΔE/E

0，（

％）

－8

－6

－4

－2

0

xx，（με）ε

580非破壊検査第 66 巻 12 号（2017）

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Marcel REMILLIEUXLos Alamos National LaboratoryMail Stop D446, LANL, Los Alamos, New Mexico 87545, USAResearch Scientist, Ph.D.Dr. Marcel Remillieux is currently a research scientist in the Geophysics Group of Los Alamos National Laboratory where he is involved with the development, implementation, and application of novel techniques for imaging, material characterization, and nondestructive testing in industrial structures and complex media. These techniques are based on time reversal, reciprocal properties, nonlinear elasto-dynamics, resonance phenomena, multiphysics coupling, and the full integration of experiments with numerical simulations. He obtained his Ph.D. degree in mechanical engineering from Virginia Tech (USA) in 2010. He started his research work at LANL with a post-doctoral appointment in 2013.http://www.lanl.gov/expertise/profiles/view/marcel-remillieux

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