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Noncontact nonlinear resonance ultrasound spectroscopy (NRUS) for small metallic specimens
NDT and E International 98 (2018) 37–44
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Noncontact nonlinear resonance ultrasound spectroscopy (NRUS) for small metallic specimens Steffen Maier a, Jin-Yeon Kim a, Marc Forstenh€ ausler a, James J. Wall b, c, Laurence J. Jacobs a, b, * a b c
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States Electric Power Research Institute (EPRI), Charlotte, NC, 28262, United States
A R T I C L E I N F O
A B S T R A C T
Keywords: Nonlinear ultrasonic methods Hysteretic nonlinearity RUS NRUS
The objective of this research is to use nonlinear resonance ultrasound spectroscopy (NRUS) to evaluate the sensitivity of the nonclassical hysteretic nonlinearity parameter, α to microscopic material damage in metallic specimens. The proposed NRUS procedure uses both a noncontact source and receiver – an air-coupled piezoelectric transducer source and a laser vibrometer receiver – to excite and detect the axial motion of a slender rod. This fully noncontact configuration avoids mechanical contact which can adversely influence the measurements. This NRUS procedure measures αf and αQ by calculating the amount of the shift of the resonance frequency and the change of quality factor (Q) as a function of increasing excitation amplitudes. The sensitivity of this NRUS measured α parameters to small changes in microstructure is demonstrated on a series of thermally aged 17-4 PH stainless steel samples. These hysteretic nonlinearity (αf and αQ ) results are then compared to (quadratic) acoustic nonlinearity parameter (β) values previously measured on these same 17-4 PH samples to demonstrate the sensitivity of the hysteretic nonlinearity parameters to changes in microstructure due to precipitate formation.
1. Introduction Conventional linear ultrasonic techniques are capable of detecting macroscopic damage such as cracks, or in determining elastic stiffness parameters, but are less sensitive to microstructure features and damage that are orders of magnitude smaller than the wavelength of the probing ultrasonic wave. This wavelength limiting restriction means that linear ultrasonic parameters are insensitive to the microstructure changes of a number of critical damage processes like thermal or fatigue damage. In contrast, nonlinear ultrasonic methods have shown great sensitivity to microstructure features much smaller than their probing ultrasonic wavelength. Of these nonlinear ultrasonic techniques, second harmonic generation (SHG) [1] has been successfully used to measure the acoustic nonlinearity parameter, β in metallic specimens, and then to relate these measured β values to microscale changes due to thermal [2], fatigue [3], sensitization [4] and irradiation [5] damage. However, these SHG techniques are generally limited to larger size specimens and engineering components, since they require a minimum wave propagation distance to be effective. Previous research has demonstrated the effectiveness of resonant frequency acoustic methods to detect damage in small specimens. This
provides benefits in sample preparation for damage such as irradiation, because a lower radiation dose is needed for smaller sample sizes. While linear resonance ultrasound spectroscopy (RUS) has long been used to determine elastic constants and other material characteristics [6], RUS is not suitable to detect early stages of damage [7]. Nonlinear resonance ultrasound spectroscopy (NRUS) techniques exploit nonlinear elasticity effects by determining changes in specimen material properties as a function of increasing excitation amplitude. NRUS [8] is sensitive to changes in the microstructural properties of a specimen. However, NRUS measurements are also dependent on extraneous factors such as ambient temperature or bonding quality of permanently glued sensors [8]. The objective of this research is to use NRUS to evaluate the sensitivity of the nonclassical hysteretic nonlinearity parameter, α to microscopic material damage in metallic specimens. The proposed NRUS procedure uses both a noncontact source and receiver – an air-coupled piezoelectric transducer source and a laser vibrometer receiver – to excite and detect the axial motion of a slender rod. This fully noncontact configuration avoids mechanical contact points, which can adversely influence the measurements. The sensitivity of this NRUS measured αf (the α value which describes the frequency shift) to small changes in
* Corresponding author. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States. E-mail address: [email protected] (L.J. Jacobs). https://doi.org/10.1016/j.ndteint.2018.04.003 Received 12 October 2017; Received in revised form 27 March 2018; Accepted 9 April 2018 Available online 16 April 2018 0963-8695/© 2018 Elsevier Ltd. All rights reserved.
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3. Experiments
microstructure is demonstrated on a series of thermally aged 17-4 PH stainless steel samples. These αf hysteretic nonlinearity results are then compared to (quadratic) acoustic nonlinearity parameter (β) values previously measured on these same 17-4 PH samples to demonstrate the sensitivity of the αf parameter to changes in microstructure due to precipitate formation. These nonlinear acoustic results are also compared to hardness and linear resonance results. The proposed noncontact NRUS procedure provides an accurate and repeatable method to measure small changes in microstructure such as precipitate formation, that are often associated with damage evolution.
3.1. Experimental setup A schematic of the proposed NRUS experimental setup is shown in Fig. 1a, with a photo of this setup in Fig. 1b. This NRUS setup allows for the noncontact excitation and detection of a sample to measure its resonance response without forced contact or glued attachment of transducers to the sample, which could affect the nonlinear response. The slender rod (9.5 mm 100 mm) or bar samples (10 mm 10 mm x 100 mm) are hung horizontally by two thin strings, which creates an effective free-free boundary condition for longitudinal deformation with small amplitudes. For axial (longitudinal) excitation of the sample, an Agilent 33250A function generator generates continuous sinusoidal frequency sweeps, amplified by an ENI 240L power amplifier, driving an Ultran NCG50-D50 50 kHz noncontact, air-coupled transducer up to 125V peak-to-peak. While this signal amplitude is held constant for each individual sweep, the frequency increases constantly between set boundary values. The aircoupled transducer is paired with an acoustic horn, 3D-printed using thermoplastic ABS filament, with an exponentially decreasing crosssection that focuses the air pressure waves onto the cross-sectional area of the sample to increase input amplitude. Noncontact detection is accomplished with a Polytech OFV-5000 laser vibrometer that measures the out-of-plane (axial) velocity at a point on the cross-section on the opposite end of the sample. The function generator and the oscilloscope are linked to a computer via a GPIB and an Ethernet connection, respectively, and are controlled by LabVIEW. A Labview script specifies the excitation parameters for the function generator, controls the recording settings of the oscilloscope, and stores the measured velocity signal as well as the trigger signal on the computer for post-processing with MATLAB. To achieve the largest possible excitation amplitude of the sample with this setup, a comparison of multiple horn lengths is performed. A simple design procedure for exponential acoustic horns from Kim [15] (based on Webster's horn equation) is used for the design. This approach relates a gradual change of the cross-sectional area of the horn to the exponential relationship S ¼ St ekh x , where St is the cross-sectional area of the throat, the thinner end of the horn, and kh is the flare constant. The flare constant is defined by the speed of sound in air, ca and the lower cut-off frequency, fc with kh ¼ 4π fc =ca . The horn can amplify the exciting acoustic pressure as much as eðkh αa ÞLh if kh > αa , where αa is the attenuation coefficient in the air and Lh is the horn length. Below that cut-off frequency, the resistance in the throat becomes zero, as the horn cuts off. The diameter of the mouth and throat of the horn are determined by the diameters of the transducer and the sample, and the cut-off frequency is chosen to be 1800Hz for a medium sized horn with a length of hL ¼ 130 mm. Based on this geometry four more horns with lengths from 32.5 mm up to 520 mm are produced and compared to each other. Fig. 2 shows the maximum out-of-plane velocity, vmax achieved for the five different horns for a stainless steel rod sample, with an optimum value obtained for a length of Lh ¼ 130 mm averaged for five measurements. It is also observed that the highest response amplitude is not when the horn mouth is closest (almost touching) the specimen; instead, the response amplitude shows periodic up-and-downs with an overall decreasing trend with increasing gap width. By varying the gap width with the linear stage and simultaneously evaluating the amplitude of the time domain signal on the oscilloscope, the optimum gap width that provides the highest time domain signal is determined. This optimum gap width varies for the different horns between approximately 2 mm - 4 mm. The process of determining the optimum gap width can potentially lead to slightly different maximum attenuation values. In order to be able to still compare multiple measurements with each other, it is then necessary to evaluate the nonlinear behavior, as described in section 3.3, to an equal attenuation level.
2. Theory Nonlinear elastic behavior occurs when a linear stress-strain relationship (Hooke's law) no longer applies. A generalized, onedimensional nonlinear stress-strain relationship that includes nonlinear effects with different mechanisms and orders is given by the constitutive equation
σ ¼ ∫ Kðε; ε_ Þdε
(1)
with the nonlinear hysteretic modulus Kðε; ε_ Þ ¼ K0 1 βε δε2 α½Δε þ εðtÞsignð_εÞ þ … ;
(2)
where K0 is the linear modulus, β and δ are the classical quadratic (or acoustic) and cubic classical nonlinearity parameters (respectively), α is the nonclassical hysteretic nonlinearity parameter, Δε is the local strain amplitude and ε_ is the strain rate with the sign function of the strain rate, sign(_ε) [9]. Classical quadratic and cubic nonlinearity mainly depends on the crystalline structure of the material, plus local strain fields on an atomic level caused by anharmonicity of the interatomic potential, dislocations, precipitates and microcracks [1,8]. Previous research has shown that strains that arise from microstructural features such as dislocations and precipitates have a significant influence on the acoustic nonlinearity parameter, β, even more so than the lattice anharmonicity contribution [1,10,11]. Nonclassical nonlinearity, which demonstrates hysteresis and a quasistatic discrete memory involving energy dissipation, has been observed for materials of the mesoscopic-elasticity class such as rock, concrete or sandstone. Large hysteretic behavior, even for small strain levels, was measured for macro-defects and localized damage. These nonclassical effects are due to soft regions in hard materials (detectable even on a much smaller scale), caused by microcracks and pores. In metallic materials, the hysteretic motion of pinned dislocations interacting with obstacles such as precipitates can be the source of the nonlinearity [12]. Barsoum et al. [13] has shown through an NRUS experiment that the dynamic nonlinear mesoscopic response of elastic solids with dislocation-based incipient kink bands is due to the interaction of ultrasound with this microstructure. NRUS measures αf by calculating the amount of the shift of the resonance frequency as a function of increasing excitation amplitudes, eventually determining nonlinear material hysteresis [8]. The frequency normalized shift, Δf =f0 over strain amplitude, Δε specifies the hysteretic nonlinearity parameter, αf , with Δf f f0 ¼ ¼ αf Δε; f0 f0
(3)
where f0 is the resonance frequency for an infinitesimally small strain amplitude, or the linear resonance frequency. Any of the various resonance modes of a material sample, if they are of the same kind, i.e. either longitudinal or torsional, can be utilized to make these NRUS measurements [14].
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Fig. 1. a) Schematic and b) photo of noncontact NRUS experimental setup.
acoustic impedance, as well as more significant defects (like microcracks) than the precipitates and dislocations in metal samples.
3.2. Linear modal analysis The axial (longitudinal) eigenmode shapes and corresponding natural frequencies of the sample geometry are simulated with an eigenfrequency study using the finite element method (FEM) analysis software COMSOL Multiphysics 5.2, to provide an independent numerical model of the frequency response of the free-free slender rod sample and to validate the experimentally measured frequency response. Fig. 3 illustrates the displacement distribution of the first five eigenmodes with longitudinal distortion in an ideal free-free boundary condition and compares them to the experimentally measured results. In the numerical simulations, Young's modulus of the sample material was first determined such that the first numerical natural frequency matches the experimental one. It turned out that the simulations using Young's modulus determined in this way predict all natural frequencies correctly at the same time. Table 1 is a comparison of the natural frequencies determined by FEM simulation and experiment, showing very good agreement. Table 1 also includes the values of the quality factor, Q, a dimensionless parameter quantifying the degree of damping and characterizing the sharpness of a resonance peak which is determined from the experimentally gathered frequency spectrum by dividing their resonance frequency, f0 by their half-power bandwidth, Δfhp . Finally, the frequency spectrum shows that the second longitudinal mode – the mode closest to the nominal frequency of the 50 kHz piezoelectric transducer – has the largest velocity amplitude.
Fig. 2. Comparison of maximum velocity achieved vmax with the stainless steel rod for different horn lengths.
Van Den Abeele et al. [16] and Van Damme & Van Den Abeele [17] have previously used acoustic speakers to excite the bending modes of slate and composite beam samples. Van Den Abeele used a cone to increase sound pressure, and detected the subsequent motion with a laser vibrometer with the nonlinear reverberation spectroscopy (NRS) technique. Each of these approaches successfully detected nonlinear hysteretic behavior as a function of material damage, but for materials with comparably stronger hysteretic behavior than metals, much lower 39
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Fig. 3. Displacement fields (left) of the first five longitudinal eigenmodes, and corresponding natural frequencies from FEM simulation and experimentally measured frequency spectrum (right) up to 140 kHz of the stainless steel rod in free-free boundary condition. Arbitrary scaling of displacement amplitudes with dark blue indicating no displacement and dark red colored surfaces indicating large displacement. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
The frequency domain response is obtained by applying a tapered cosine window to each individual time domain signal, and performing an FFT. To reduce random noise, the peak velocity and corresponding resonance frequency is most precisely determined by filtering the frequency domain signal with a zero-phase filter, and a third order polynomial curve then approximates a small range around the peak as shown in Fig. 4b. Note that the frequency spectrum corresponds to a very small range around the peak in the time domain signal. For this reason, the peak shift (or the frequency shift) in the time domain signals cannot be observed. By repeating this process for 20 equally spaced and increasing input voltage steps, Fig. 4c is generated that shows a decreasing resonance frequency for increasing voltage amplitudes. To visualize the frequency shift for normalized resonance frequencies, Fig. 4d is generated by dividing the frequency shift, Δf ¼ f f0 for each input amplitude excitation level, by the linear resonance frequency, f0 , which is determined by the intersection of the linear fit of Fig. 4c with the x-axis. The strain amplitude in the specimen is derived from the analytical solution for the longitudinal vibration of a bar in free-free boundary condition [18].
Table 1 Comparison of natural frequencies between experiment (fexp ) and FEM numerical simulation (fsim ) of the stainless steel rod sample with the simulation parameters Young's modulus E ¼ 198.3 MPa, density ρ ¼ 7850 kg/m3 and Poisson's ratio ν ¼ 0.33. longitudinal mode 1 2 3 4 5
fexp [Hz] 25114 50157 75039 99666 123935
fsim [Hz] 25115 50137 74968 99497 123600
fsim fexp [Hz]
jfsim fexp j
1 20 71 169 335
0.003% 0.040% 0.095% 0.170% 0.270%
Qexp
fexp
12200 15600 16350 9800 5600
3.3. Nonlinear measurements An individual time domain signal is measured for each frequency sweep as shown in Fig. 4a. The frequency bandwidth selected around the linear resonance frequency is f0 Δf , where Δf ¼ 0:1%⋅f0 , with a sweep duration of 8s at a sampling rate of 200kSa/s.
Fig. 4. Data processing steps for the nonlinear measurements. 40
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Δε ¼
v c
with
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fn ¼
nc 2L
between 2 and 8s while for a sweep duration of 10s, the mean value is the same as for the 8s, but with increased error bars. The conclusion is that a sweep duration of 8 s is the best compromise. The time in between the frequency sweeps is kept to a minimum, mainly determined by the time required to save the gathered time domain signal, while the selected frequency bandwidth around the linear resonance frequency f0 Δf was chosen wide enough to ensure that the decaying amplitude can settle evenly. In order to accurately determine the material nonlinearity of the samples, the intrinsic nonlinear behavior of the experimental setup itself (including the electrical components) must be negligibly small. This is particularly true when the material nonlinearity (plus variations due to damage) of the sample itself is expected to be very small – this is the case for the metallic 17-4 PH samples examined in this research. Because NRUS resonance measurements exploit the natural amplification of the sample's motion near resonance, the electrical components involved exciting these sample resonances or other extraneous nonlinear effects such as nonlinear distortion of the air should not influence the material nonlinearity, αf measured. With this noncontact generation, the motion of the sample is not directly coupled to the generation mechanism, so these extraneous instrumentation nonlinear effects should be eliminated. To verify that this instrumentation nonlinearity is not present, consider the results from a very linear material, aluminum. Fig. 6 illustrates that an aluminum AL-6061-T6 rod with an αf -value of 0.126 is quite small compared to a 17-4 PH stainless steel rod with an αf -value of 6.43 (both averaged for five measurements of the second longitudinal mode for 2s sweeps). This also strengthens the assumption that the effect of the transducer heating up the specimen can only be minimal. Finally, Fig. 6 illustrates the nonlinear shift for the first three longitudinal modes for both the aluminum and stainless steel samples for 2s sweeps. While the overall behavior of the nonlinearity values for each of these three modes agree with each other, the measurements of the first mode result in a scattered set of data points (and an associated poor linear fit), while the results for the third mode do not excite significant strains in the samples. In contrast, the results for the second longitudinal mode provides both adequate strain excitation, and a good linear fit. As a result, all the remaining measurements on the 17-4 PH specimens are performed on this second mode. As shown by Remillieux et al. [14,19] and TenCate et al. [20,21] on geomaterials, hysteretic nonlinearity is very sensitive to the applied strain level. TenCate et al. [20] identified different strain regimes for sandstones with mostly linear elastic behavior for strains below ε 108 107 . Above a threshold of ε 106 they related the quadratic decrease in resonance frequency mostly to hystertic nonlinearity. For even higher strains, nonequilibrium effects must be considered with an often linear dependence of frequency over strain. TenCate et al. [21] show that for these higher strains, where slow dynamics superpose the
(4)
leading to Δε n ¼
nvn 2Lfn
(5)
where Δεn is the strain amplitude of the n-th mode, vn is the corresponding velocity amplitude at the specimen end (i.e. at the anti-node) measured with the laser vibrometer, c is the phase speed, fn is the resonance frequency of the n-th eigenmode, and L is the sample length. The slope of the normalized frequency shift versus the strain amplitude then defines the hysteretic nonlinearity parameter, αf [8]. A straightforward methodology of Haupert et al. [8] is used to reduce the effect of ambient environmental factors by compensating for temperature changes during the measurements. This procedure includes measuring the resonance frequency in between each increasing driving step at a base voltage amplitude that is chosen as, in this case, the lowest excitation level with narrowest resonance response. Fig. 4e shows that for this measurement setup, the reference measurements experience a negative shift (apart from possible small temperature deviations) for consistent conditions. The corrected relative frequency shift of Fig. 4f is determined by subtracting the relative frequency shift of the reference excitation from the corresponding relative frequency shift of the increasing driving steps. Even though shielding the experiment with an insulating box to reduce the temperature influence yielded more consistent results, the ambient temperature effect is still large enough to necessitate this temperature correction procedure. Because of the very low energy transmission from the air to the specimens, the influence of the driving transducer warming up the specimen can be assumed to be insignificant. The coefficient of the longitudinal acoustic wave transmission, dependent on the impedances of the media pairings, is for air to steel 0.00193% and air to aluminium 0.00521%. The difference of a factor of 2.7 between these coefficients also exactly correlates to the gain in achieved maximal strain as shown in Fig. 6. The selection of sweep parameters is a compromise between reaching a steady state at each frequency during the sweep, and reducing the influence of ambient environmental factors. A longer sweep and narrower frequency bandwidth helps in reaching a steady state, and increases the numerical accuracy of the fast Fourier transform (FFT), while extending the time span of the experimental procedure potentially increases the influence of ambient temperature variations on the measured αf results. To determine the influence of the sweep duration, αf is measured for durations from 2 to 10 s for five measurements each. Fig. 5 shows that the αf determined varies for the sweep durations with increasing values
Fig. 6. Comparison of stainless steel and aluminum rods for longitudinal modes 1-3.
Fig. 5. Normalized αf for variation of sweep duration for the stainless steel rod. 41
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nonlinearity, the applied stress isn't released instantaneously and the material properties recover logarithmic over time. While these transition strains are well-characterized for geomaterials like rocks, sandstones and concrete, the length scale of the microstructure of metals is several orders of magnitude smaller than those of the geomaterials examined by TenCate and Remillieux. Therefore, it is expected that the threshold strains would also be much smaller. However, the specific threshold strains in metals are subjected to further research. The achieved strain levels of 2 106 for stainless steel based on this air-coupled excitation method are considerably smaller than comparable measurements of Haupert et al. [8] with strain levels up to 3 105 , and while the results agree qualitatively well with aluminium behaving significantly more linear than stainless steel, a quantitative comparison of the results of hysteretic nonlinearity values is not possible. 4. Results for thermally aged 17-PH stainless steel samples Fig. 7. Evolution of linear resonance frequency for thermally aged 17-4 PH stainless steel samples. (Note: Because of the quadratic portion of the resonance frequency in the determination of the Young's modulus in Equation (6), the vertical E-tick marks of the right y-axis are slightly unequally spaced).
4.1. Material 17-4 PH stainless steel is a commonly used martensitic precipitationhardening (”PH”) alloy with a good combination of high-strength, good ductility and corrosion resistance. The steel contains about 3 wt. % Cu and is strengthened by precipitation of dispersed copper in the martensite matrix [22]. Solution annealed 17-4 PH shows initially no copper precipitates. Thermal aging at temperatures of 400 C and higher, hardens the material [23]. At first, coherent copper-rich precipitates form in the material, restraining the motion of dislocations in the matrix. The copper has a low solubility at this temperature, such that the copper atoms diffuse and form fine particles. After exceeding the peak aging time, the growing copper-rich precipitates become incoherent with the matrix, leading to lower strength and hardness [24]. Previous research [25] examined 17-4 PH stainless steel as a surrogate material for irradiation damaged reactor pressure vessel (RPV) steel, because of similarities in both microstructures with the formation of copper precipitates – irradiation damage, which leads to harmful embrittlement, is also partly caused by copper-rich precipitates of similar size and density. As a result, 17-4 PH is a surrogate sample to avoid difficulties in handling irradiated RPV steel samples. Table 2 gives an overview of the thermal aging processes and the corresponding sample designation from Ref. [25]. The samples are first solution annealed at 1040 C for 6 h followed by air cooling, and then thermally aged in increasing steps up to 6 h at 400 C. This aging temperature was selected to ensure that the dislocation density (approximately) remained constant, while simultaneously forming copper precipitates. The samples in Ref. [25] were designed to be large enough to enable the measurement of the acoustic nonlinearity parameter, β with the SHG of propagating Rayleigh waves. Four new NRUS samples were then EDM-cut (electrical discharge machining) from the original [25] samples to slender bars 10 mm 10 mm x 100 mm, with one small end face polished to provide a necessary reflective surface for the laser vibrometer.
un-aged and three thermally aged 17-4 PH samples, which indicates a monotonic increase of resonance frequency for longer thermal aging time. Note that there are very small experimental deviations between the five repeated measurements performed on each sample – the error bars for the f0 are tiny. The analytical description of the n-th natural longipffiffiffiffiffiffiffiffi tudinal frequencies fn ¼ nc=2L, where the phase speed is c ¼ E=ρ, enables the calculation of the Young's modulus from the second longitudinal mode E ¼ ðf2 LÞ2 ρ:
(6)
These Young's modulus results (also shown in Fig. 7) indicate that the formation of copper-rich precipitates stiffens the material by a small amount, leading to increased values of E with increased aging time. Rosen et al. [26] also noted an increase in stiffness for similar precipitation formation in aluminum alloy 2219. 4.3. NRUS results for the 17-4 PH bars The measured hysteretic nonlinearity parameter, αf , quantifying the hysteretic behavior and quasi-static discrete memory effects for these same four 17-4 PH bar samples is shown in Fig. 8 and Table 3. Again, the results are averaged for five repeated measurements and evaluated up to an equal maximum strain level εmax of 2 106 for all measurements. The R2 -values, which characterize the goodness of the linear fit for each
4.2. Evolution of linear resonance frequency for the 17-4 PH bars Fig. 7 shows the evolution of linear resonance frequency f0 for the one Table 2 Sample designations and heat treatment procedure for 17-4 PH samples [25]. Sample designation
Solution annealing
Cooling method
Aging time @ 400 C
AC AC-0.1 AC-1 AC-6
1040 C/6h 1040 C/6h 1040 C/6h 1040 C/6h
air cooled air cooled air cooled air cooled
– 0.1h 1h 6h
Fig. 8. Nonlinear parameters αf and αQ for thermally aged 17-4 PH stainless steel samples. 42
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Table 3 Nonlinear parameters αf and αQ for thermally aged 17-4 PH stainless steel samples. Sample condition
αf
Δαf =αf ;AC
Standard deviation
R2
AC AC-0.1 AC-1 AC-6
0.76 1.56 1.89 1.92
þ105% þ148% þ153%
9.55% 7.53% 4.46% 6.98%
0.50 0.88 0.88 0.79
Sample condition
αQ
ΔαQ =αQ;AC
Standard deviation
R2
AC AC-0.1 AC-1 AC-6
1.42 2.69 3.32 3.13
þ89% þ133% þ120%
12.5% 22.8% 10.9% 21.4%
0.37 0.46 0.72 0.62
individual measurement, indicate moderate linear approximation for less nonlinear behavior, so as the samples become more nonlinear, the accuracy of the αf measurement increases. When compared to the evolution of linear resonance frequency, the error bars for these αf -values are considerably larger. However there are clearly identifiable αf trends when comparing the results for the four different samples, where the nonlinear behavior increases with increasing aging time – becoming more negative αf -values (or higher αf -values). The trend also shows a decreasing increase in nonlinearity for each consecutive aging time step, almost leveling off between 1h and 6h. In contrast to the results from the linear resonance frequency measurements, this technique indicates that the microstructural changes effecting hysteretic material nonlinearity do not increase linearly with the thermal aging process. In addition to αf (the hysteresis parameter which describes the frequency shift), a second hysteresis parameter, αQ is introduced which describes the change of energy loss as a function of strain, or 1 1 ¼ αQ Δε: Q Q0
Fig. 9. Comparison of normalized αf , αQ and Young's modulus E (solid lines) to normalized β, Vickers hardness (HV) and TEP-measurements (dashed lines) from Matlack et al. [25]. (Note: Data points are connected for clearer visualization).
of continuous copper precipitate formation throughout the aging process. Table 4 compares the relative increase of each of these parameters from un-aged to 6h aging, and indicates that the relative change in E measured through the evolution of resonance frequency is an order of magnitude less sensitive than the Vickers hardness and TEP measurements. Finally, Table 4 gives the average standard deviation of the measurement techniques compared, which shows that while the normalized αf and αQ -results are the most sensitive, they also have, together with the β-results, comparatively the largest error bars. Nevertheless, a distinct trend is clearly seen. Now consider a comparison between the nonclassical hysteretic nonlinearity parameter, αf and the quadratic acoustic nonlinearity parameter, β. The relative β values are determined by evaluating the ratio of the amplitudes of the first (A1 ) and second (A2 ) harmonics as a func-
(7)
The parameter αQ is determined in a similar fashion to αf by evaluating the quality factor Q for each frequency sweep, and applying the data processing steps, including the correction of ambient factors, provided in section 3.3. As also shown in Fig. 8 and Table 3, αQ essentially follows the same trend as αf , but with bigger error bars and lower R2 -values.
4.4. Comparison to other monitoring techniques
tion of increasing propagation distance. The slope of the ratio of A2 =ðA1 Þ2 versus propagation distance is then proportional to β. The measured values of β show a monotonic decrease in value with increasing aging time, with a larger decline between 0.1h and 1h (at 400 C). Matlack et al. [25] related this decrease in β to the increasing number of copper precipitates, which pinned existing dislocation contributions. The β-measurements were performed on much larger specimens with dimensions of 19 mm 38 mm x 230 mm because of the inherent restrictions of this nonlinear Rayleigh wave technique due to a minimum wave propagation distance and an adequate contact surface for the exciting transducer. The hysteresis nonlinearity αf increases rapidly in the beginning, before becoming saturated at the end. This behavior of αf most closely follows the TEP-values, where they both have a slightly flattening increase between 1 and 6h aging. This correspondence could be because the TEP is also most sensitive to the formation of copper precipitates alone, independent of their pinning existing dislocations. Most importantly, the αf parameter shows the greatest sensitivity to formation of these copper precipitates – Table 4 shows that both αf , αQ and β are an order of magnitude more sensitive than the Vickers hardness and TEP values, and αf and αQ are more than twice as sensitive as β to the formation of these copper precipitates. This is important since the copperrich precipitates are the damage mechanism most directly associated
Fig. 9 compares the normalized hysteretic nonlinearities, αf and αQ , and stiffness (E) results from this research with complementary measurements of Vickers hardness (HV), thermo electric power (TEP) and quadratic acoustic nonlinearity, β, performed by Matlack et al. [25], with all normalized to their respective un-aged values. The measured quality factors increase with increasing heat treatment, exhibiting a trend quite similar to the trend of the Young's modulus, but with much bigger error bars. The Vickers hardness results show a monotonic increase in hardness with increasing aging time, corresponding to the expected increase in copper precipitate volume fraction with thermal aging. The TEP values follow the same increasing trend with thermal age – an increase in TEP corresponds to a decreasing amount of copper in the matrix material, meaning an increase in copper precipitate volume fraction [25]. For the thermo-electric power measurement technique, the voltage difference, generated by a temperature gradient in the material sample, is measured and compared to a reference material. The measurements are quantified by the Seebeck coefficient S ¼ ΔV=ΔT, the ratio between voltage difference and temperature gradient [27]. All three of these techniques show pretty much a linear relation as a function of thermal aging time, which is consistent with the assumption 43
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(DAAD) through a Graduate Research Assistantship. Additional funding was provided by the Electric Power Research Institute (EPRI). Thanks to anonymous reviewer comments on the importance of the sweep time, and to Katie Scott for additional measurements.
Table 4 Sensitivity of the measured parameters/material properties to microstructure changes due to thermal aging occurring from un-aged to 6 h. Measurement technique
Material property/ parameter
Sensitivity ΔAC 6=AC
Average standard deviation
Lin. resonance frequency NRUS
E Q*
Vickers hardness Thermo-electric power Rayleigh surface waves
HV TEP
þ2.8% þ19% þ153% þ120% þ28% þ19%
0.0049% 1.9% 7.1% 16.9% 1.9% 0.65%
β'
47%
8.2%
αf αQ
*measured at ε ¼ 2 10
6
Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi. org/10.1016/j.ndteint.2018.04.003. References [1] Matlack K, Kim J-Y, Jacobs L, Qu J. Review of second harmonic generation measurement techniques for material state determination in metals. J Nondestr Eval 2015;34(1):273. [2] Marino D, Kim J-Y, Ruiz A, Joo Y-S, Qu J, Jacobs LJ. Using nonlinear ultrasound to track microstructural changes due to thermal aging in modified 9%Cr ferritic martensitic steel. NDT E Int 2016;79:46–52. [3] Walker SV, Kim J-Y, Qu J, Jacobs LJ. Fatigue damage evaluation in A36 steel using nonlinear Rayleigh surface waves. NDT E Int 2012;48:10–5. [4] Doerr C, Kim J-Y, Singh P, Wall JJ, Jacobs LJ. Evaluation of sensitization in stainless steel 304 and 304L using nonlinear Rayleigh waves. NDT E Int 2017;88:17–23. [5] Matlack K, Kim J-Y, Wall J, Qu J, Jacobs L, Sokolov M. Sensitivity of ultrasonic nonlinearity to irradiated, annealed, and re-irradiated microstructure changes in RPV steels. J Nucl Mater 2014;448(1):26–32. [6] Migliori A, Sarrao J. Resonant ultrasound spectroscopy: applications to physics, materials measurements, and nondestructive evaluation. Wiley; 1997. [7] Windels F, Abeele KVD. The influence of localized damage in a sample on its resonance spectrum. Ultrasonics 2004;42(1):1025–9. [8] Haupert S, Renaud G, Riviere J, Talmant M, Johnson PA, Laugier P. High-accuracy acoustic detection of nonclassical component of material nonlinearity. J Acoust Soc Am 2011;130(5):2654–61. [9] Van Den Abeele KE, Johnson A, Sutin PA. Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage. Part I: nonlinear wave modulation spectroscopy (NWMS). Res Nondestr Eval 2000;12(1):17–30. [10] Hikata A, Chick BB, Elbaum C. Dislocation contribution to the second harmonic generation of ultrasonic waves. J Appl Phys 1965;36(1):229–36. [11] Cantrell JH, Zhang X-G. Nonlinear acoustic response from precipitate-matrix misfit in a dislocation network. J Appl Phys 1998;84(10):5469–72. [12] Granato A, Lücke K. Theory of mechanical damping due to dislocations. J Appl Phys 1956;27(6):583–93. [13] Barsoum MW, Radovic M, Zhen T, Finkel P, Kalidindi SR. Dynamic elastic hysteretic solids and dislocations. Phys Rev Lett 2005;94(8):085501. [14] Remillieux MC, Guyer RA, Payan C, Ulrich TJ. Decoupling nonclassical nonlinear behavior of elastic wave types. Phys Rev Lett 2016;116:115501. [15] Kim Y. Sound propagation: an impedance based approach. John Wiley and Sons, Ltd; 2010. [16] Van Damme B, Van Den Abeele K. The application of nonlinear reverberation spectroscopy for the detection of localized fatigue damage. J Nondestr Eval 2014; 33(2):263–8. [17] Van Den Abeele K, Carmeliet J, Ten Cate J, Johnson P. Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part II: single-mode nonlinear resonance acoustic spectroscopy. Res Nondestr Eval 2000;12(1):31–42. [18] Kinsler LE. Fundamentals of acoustics. Wiley; 1982. [19] Remillieux MC, Ulrich TJ, Goodman HE, Ten Cate JA. Propagation of a finiteamplitude elastic pulse in a bar of berea sandstone: a detailed look at the mechanisms of classical nonlinearity, hysteresis, and nonequilibrium dynamics. J Geophys Res Solid Earth 2017;122(11):8892–909. [20] TenCate JA, Pasqualini D, Habib S, Heitmann K, Higdon D, Johnson PA. Nonlinear and nonequilibrium dynamics in geomaterials. Phys Rev Lett 2004;93:065501. [21] TenCate JA, Smith E, Guyer RA. Universal slow dynamics in granular solids. Phys Rev Lett 2000;85:1020–3. [22] Murayama M, Hono K, Katayama Y. Microstructural evolution in a 17-4PH stainless steel after aging at 400 C. Metall Mater Trans 1999;30(2):345–53. [23] Mirzadeh H, Najafizadeh A. Aging kinetics of 17-4PH stainless steel. Mater Chem Phys 2009;116(1):119–24. [24] Hsu K-C, Lin C-K. High-temperature fatigue crack growth behavior of 17-4PH stainless steels. Metall Mater Trans 2004;35(9):3018–24. [25] Matlack KH, Bradley HA, Thiele S, Kim J-Y, Wall JJ, Jung HJ, Qu J, Jacobs LJ. Nonlinear ultrasonic characterization of precipitation in 17-4PH stainless steel. NDT E Int 2015;71:8–15. [26] Rosen M, Horowitz E, Fick S, Reno R, Mehrabian R. An investigation of the precipitation-hardening process in aluminum alloy 2219 by means of sound wave velocity and ultrasonic attenuation. Mater Sci Eng 1982;53(2):163–77. [27] Northwood DO, Sun X, Hochreiter J, Sokolowski JH, Penrod DE. Thermoelectric power (TEP) measurements as a materials characterization technique. In: AIP conference proceedings, vol. 316; 1994. p. 181–4 (1). [28] Lebedev AB. Amplitude-dependent decrement to modulus defect ratio in breakaway models of dislocation hysteresis. Philos Mag A 1996;74(1):137–50.
.
with the embrittlement of RPV steels. Finally note that the Read ratio παQ =2αf can be determined from αf and αQ , and this Read ratio is expected to be equal to 4/3 for a purely quadratic hysteretic nonlinearity [8,28]. The Read ratios calculated for these four 17-4 PH specimens are all approximately the same, with an average of 2.78 0.60, and a maximum difference ΔAC 6=AC of 15.7%. Note that these values are higher than a Read ratio of 4/3 predicted by the dislocation-hysteresis driven model of Lebedev [28]. This difference is most likely due to the fact that the microstructure of these four 17-4 PH specimens are dominated by precipitates and their interactions with dislocations, instead of the pure dislocation-hysteresis modeled in Ref. [28]. 5. Conclusion This research used the NRUS technique to evaluate the sensitivity of the nonclassical hysteretic nonlinearity parameters, αf and αQ to changes in the microstructure of thermally aged 17-4 PH stainless steel samples. These microstructural changes involved the formation of copper precipitates that are associated with embrittlement damage in RPV steels. An NRUS procedure was developed that uses both a noncontact source and receiver – an air-coupled piezoelectric transducer source and a laser vibrometer receiver – to detect the axial motion of a slender rod. This fully noncontact configuration avoids mechanical contact points which can adversely influence the measurements. The excitation of the samples with the air-coupled transducer, paired with a focusing horn excited the first five axial eigenmodes of the rod with very high quality (Q) factors. There was good agreement with the results of an FEM simulation, indicating low damping and little influence from extraneous experimental setup factors. This NRUS procedure measured the hysteretic nonlinearity parameters, αf and αQ , noting that a temperature correction was critical for repeatable measurements. Complementary linear measurements on these 17-4 PH samples determined the Young's modulus, and showed a direct relationship between thermal aging time and material stiffness, E. The measured αf (αQ ) values were two orders of magnitude more sensitive than E to changes in microstructure due to the formation of copper precipitates, and compared well to (quadratic) acoustic nonlinearity parameter (β) values previously measured on these same 17-4 PH samples. This NRUS setup provides a reliable, accurate and valid method to measure αf and αQ , demonstrating the advantages of fully noncontact generation and detection over setups with contact points. Due to the absence of operator-dependent processing, it is feasible to develop a highly automated measurement procedure and with minimal sample preparation. Acknowledgements This work is supported by the German Academic Exchange Service
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Noncontact nonlinear resonance ultrasound spectroscopy (NRUS) for small metallic specimens Steffen Maier a, Jin-Yeon Kim a, Marc Forstenh€ ausler a, James J. Wall b, c, Laurence J. Jacobs a, b, * a b c
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States Electric Power Research Institute (EPRI), Charlotte, NC, 28262, United States
A R T I C L E I N F O
A B S T R A C T
Keywords: Nonlinear ultrasonic methods Hysteretic nonlinearity RUS NRUS
The objective of this research is to use nonlinear resonance ultrasound spectroscopy (NRUS) to evaluate the sensitivity of the nonclassical hysteretic nonlinearity parameter, α to microscopic material damage in metallic specimens. The proposed NRUS procedure uses both a noncontact source and receiver – an air-coupled piezoelectric transducer source and a laser vibrometer receiver – to excite and detect the axial motion of a slender rod. This fully noncontact configuration avoids mechanical contact which can adversely influence the measurements. This NRUS procedure measures αf and αQ by calculating the amount of the shift of the resonance frequency and the change of quality factor (Q) as a function of increasing excitation amplitudes. The sensitivity of this NRUS measured α parameters to small changes in microstructure is demonstrated on a series of thermally aged 17-4 PH stainless steel samples. These hysteretic nonlinearity (αf and αQ ) results are then compared to (quadratic) acoustic nonlinearity parameter (β) values previously measured on these same 17-4 PH samples to demonstrate the sensitivity of the hysteretic nonlinearity parameters to changes in microstructure due to precipitate formation.
1. Introduction Conventional linear ultrasonic techniques are capable of detecting macroscopic damage such as cracks, or in determining elastic stiffness parameters, but are less sensitive to microstructure features and damage that are orders of magnitude smaller than the wavelength of the probing ultrasonic wave. This wavelength limiting restriction means that linear ultrasonic parameters are insensitive to the microstructure changes of a number of critical damage processes like thermal or fatigue damage. In contrast, nonlinear ultrasonic methods have shown great sensitivity to microstructure features much smaller than their probing ultrasonic wavelength. Of these nonlinear ultrasonic techniques, second harmonic generation (SHG) [1] has been successfully used to measure the acoustic nonlinearity parameter, β in metallic specimens, and then to relate these measured β values to microscale changes due to thermal [2], fatigue [3], sensitization [4] and irradiation [5] damage. However, these SHG techniques are generally limited to larger size specimens and engineering components, since they require a minimum wave propagation distance to be effective. Previous research has demonstrated the effectiveness of resonant frequency acoustic methods to detect damage in small specimens. This
provides benefits in sample preparation for damage such as irradiation, because a lower radiation dose is needed for smaller sample sizes. While linear resonance ultrasound spectroscopy (RUS) has long been used to determine elastic constants and other material characteristics [6], RUS is not suitable to detect early stages of damage [7]. Nonlinear resonance ultrasound spectroscopy (NRUS) techniques exploit nonlinear elasticity effects by determining changes in specimen material properties as a function of increasing excitation amplitude. NRUS [8] is sensitive to changes in the microstructural properties of a specimen. However, NRUS measurements are also dependent on extraneous factors such as ambient temperature or bonding quality of permanently glued sensors [8]. The objective of this research is to use NRUS to evaluate the sensitivity of the nonclassical hysteretic nonlinearity parameter, α to microscopic material damage in metallic specimens. The proposed NRUS procedure uses both a noncontact source and receiver – an air-coupled piezoelectric transducer source and a laser vibrometer receiver – to excite and detect the axial motion of a slender rod. This fully noncontact configuration avoids mechanical contact points, which can adversely influence the measurements. The sensitivity of this NRUS measured αf (the α value which describes the frequency shift) to small changes in
* Corresponding author. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States. E-mail address: [email protected] (L.J. Jacobs). https://doi.org/10.1016/j.ndteint.2018.04.003 Received 12 October 2017; Received in revised form 27 March 2018; Accepted 9 April 2018 Available online 16 April 2018 0963-8695/© 2018 Elsevier Ltd. All rights reserved.
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3. Experiments
microstructure is demonstrated on a series of thermally aged 17-4 PH stainless steel samples. These αf hysteretic nonlinearity results are then compared to (quadratic) acoustic nonlinearity parameter (β) values previously measured on these same 17-4 PH samples to demonstrate the sensitivity of the αf parameter to changes in microstructure due to precipitate formation. These nonlinear acoustic results are also compared to hardness and linear resonance results. The proposed noncontact NRUS procedure provides an accurate and repeatable method to measure small changes in microstructure such as precipitate formation, that are often associated with damage evolution.
3.1. Experimental setup A schematic of the proposed NRUS experimental setup is shown in Fig. 1a, with a photo of this setup in Fig. 1b. This NRUS setup allows for the noncontact excitation and detection of a sample to measure its resonance response without forced contact or glued attachment of transducers to the sample, which could affect the nonlinear response. The slender rod (9.5 mm 100 mm) or bar samples (10 mm 10 mm x 100 mm) are hung horizontally by two thin strings, which creates an effective free-free boundary condition for longitudinal deformation with small amplitudes. For axial (longitudinal) excitation of the sample, an Agilent 33250A function generator generates continuous sinusoidal frequency sweeps, amplified by an ENI 240L power amplifier, driving an Ultran NCG50-D50 50 kHz noncontact, air-coupled transducer up to 125V peak-to-peak. While this signal amplitude is held constant for each individual sweep, the frequency increases constantly between set boundary values. The aircoupled transducer is paired with an acoustic horn, 3D-printed using thermoplastic ABS filament, with an exponentially decreasing crosssection that focuses the air pressure waves onto the cross-sectional area of the sample to increase input amplitude. Noncontact detection is accomplished with a Polytech OFV-5000 laser vibrometer that measures the out-of-plane (axial) velocity at a point on the cross-section on the opposite end of the sample. The function generator and the oscilloscope are linked to a computer via a GPIB and an Ethernet connection, respectively, and are controlled by LabVIEW. A Labview script specifies the excitation parameters for the function generator, controls the recording settings of the oscilloscope, and stores the measured velocity signal as well as the trigger signal on the computer for post-processing with MATLAB. To achieve the largest possible excitation amplitude of the sample with this setup, a comparison of multiple horn lengths is performed. A simple design procedure for exponential acoustic horns from Kim [15] (based on Webster's horn equation) is used for the design. This approach relates a gradual change of the cross-sectional area of the horn to the exponential relationship S ¼ St ekh x , where St is the cross-sectional area of the throat, the thinner end of the horn, and kh is the flare constant. The flare constant is defined by the speed of sound in air, ca and the lower cut-off frequency, fc with kh ¼ 4π fc =ca . The horn can amplify the exciting acoustic pressure as much as eðkh αa ÞLh if kh > αa , where αa is the attenuation coefficient in the air and Lh is the horn length. Below that cut-off frequency, the resistance in the throat becomes zero, as the horn cuts off. The diameter of the mouth and throat of the horn are determined by the diameters of the transducer and the sample, and the cut-off frequency is chosen to be 1800Hz for a medium sized horn with a length of hL ¼ 130 mm. Based on this geometry four more horns with lengths from 32.5 mm up to 520 mm are produced and compared to each other. Fig. 2 shows the maximum out-of-plane velocity, vmax achieved for the five different horns for a stainless steel rod sample, with an optimum value obtained for a length of Lh ¼ 130 mm averaged for five measurements. It is also observed that the highest response amplitude is not when the horn mouth is closest (almost touching) the specimen; instead, the response amplitude shows periodic up-and-downs with an overall decreasing trend with increasing gap width. By varying the gap width with the linear stage and simultaneously evaluating the amplitude of the time domain signal on the oscilloscope, the optimum gap width that provides the highest time domain signal is determined. This optimum gap width varies for the different horns between approximately 2 mm - 4 mm. The process of determining the optimum gap width can potentially lead to slightly different maximum attenuation values. In order to be able to still compare multiple measurements with each other, it is then necessary to evaluate the nonlinear behavior, as described in section 3.3, to an equal attenuation level.
2. Theory Nonlinear elastic behavior occurs when a linear stress-strain relationship (Hooke's law) no longer applies. A generalized, onedimensional nonlinear stress-strain relationship that includes nonlinear effects with different mechanisms and orders is given by the constitutive equation
σ ¼ ∫ Kðε; ε_ Þdε
(1)
with the nonlinear hysteretic modulus Kðε; ε_ Þ ¼ K0 1 βε δε2 α½Δε þ εðtÞsignð_εÞ þ … ;
(2)
where K0 is the linear modulus, β and δ are the classical quadratic (or acoustic) and cubic classical nonlinearity parameters (respectively), α is the nonclassical hysteretic nonlinearity parameter, Δε is the local strain amplitude and ε_ is the strain rate with the sign function of the strain rate, sign(_ε) [9]. Classical quadratic and cubic nonlinearity mainly depends on the crystalline structure of the material, plus local strain fields on an atomic level caused by anharmonicity of the interatomic potential, dislocations, precipitates and microcracks [1,8]. Previous research has shown that strains that arise from microstructural features such as dislocations and precipitates have a significant influence on the acoustic nonlinearity parameter, β, even more so than the lattice anharmonicity contribution [1,10,11]. Nonclassical nonlinearity, which demonstrates hysteresis and a quasistatic discrete memory involving energy dissipation, has been observed for materials of the mesoscopic-elasticity class such as rock, concrete or sandstone. Large hysteretic behavior, even for small strain levels, was measured for macro-defects and localized damage. These nonclassical effects are due to soft regions in hard materials (detectable even on a much smaller scale), caused by microcracks and pores. In metallic materials, the hysteretic motion of pinned dislocations interacting with obstacles such as precipitates can be the source of the nonlinearity [12]. Barsoum et al. [13] has shown through an NRUS experiment that the dynamic nonlinear mesoscopic response of elastic solids with dislocation-based incipient kink bands is due to the interaction of ultrasound with this microstructure. NRUS measures αf by calculating the amount of the shift of the resonance frequency as a function of increasing excitation amplitudes, eventually determining nonlinear material hysteresis [8]. The frequency normalized shift, Δf =f0 over strain amplitude, Δε specifies the hysteretic nonlinearity parameter, αf , with Δf f f0 ¼ ¼ αf Δε; f0 f0
(3)
where f0 is the resonance frequency for an infinitesimally small strain amplitude, or the linear resonance frequency. Any of the various resonance modes of a material sample, if they are of the same kind, i.e. either longitudinal or torsional, can be utilized to make these NRUS measurements [14].
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Fig. 1. a) Schematic and b) photo of noncontact NRUS experimental setup.
acoustic impedance, as well as more significant defects (like microcracks) than the precipitates and dislocations in metal samples.
3.2. Linear modal analysis The axial (longitudinal) eigenmode shapes and corresponding natural frequencies of the sample geometry are simulated with an eigenfrequency study using the finite element method (FEM) analysis software COMSOL Multiphysics 5.2, to provide an independent numerical model of the frequency response of the free-free slender rod sample and to validate the experimentally measured frequency response. Fig. 3 illustrates the displacement distribution of the first five eigenmodes with longitudinal distortion in an ideal free-free boundary condition and compares them to the experimentally measured results. In the numerical simulations, Young's modulus of the sample material was first determined such that the first numerical natural frequency matches the experimental one. It turned out that the simulations using Young's modulus determined in this way predict all natural frequencies correctly at the same time. Table 1 is a comparison of the natural frequencies determined by FEM simulation and experiment, showing very good agreement. Table 1 also includes the values of the quality factor, Q, a dimensionless parameter quantifying the degree of damping and characterizing the sharpness of a resonance peak which is determined from the experimentally gathered frequency spectrum by dividing their resonance frequency, f0 by their half-power bandwidth, Δfhp . Finally, the frequency spectrum shows that the second longitudinal mode – the mode closest to the nominal frequency of the 50 kHz piezoelectric transducer – has the largest velocity amplitude.
Fig. 2. Comparison of maximum velocity achieved vmax with the stainless steel rod for different horn lengths.
Van Den Abeele et al. [16] and Van Damme & Van Den Abeele [17] have previously used acoustic speakers to excite the bending modes of slate and composite beam samples. Van Den Abeele used a cone to increase sound pressure, and detected the subsequent motion with a laser vibrometer with the nonlinear reverberation spectroscopy (NRS) technique. Each of these approaches successfully detected nonlinear hysteretic behavior as a function of material damage, but for materials with comparably stronger hysteretic behavior than metals, much lower 39
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Fig. 3. Displacement fields (left) of the first five longitudinal eigenmodes, and corresponding natural frequencies from FEM simulation and experimentally measured frequency spectrum (right) up to 140 kHz of the stainless steel rod in free-free boundary condition. Arbitrary scaling of displacement amplitudes with dark blue indicating no displacement and dark red colored surfaces indicating large displacement. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
The frequency domain response is obtained by applying a tapered cosine window to each individual time domain signal, and performing an FFT. To reduce random noise, the peak velocity and corresponding resonance frequency is most precisely determined by filtering the frequency domain signal with a zero-phase filter, and a third order polynomial curve then approximates a small range around the peak as shown in Fig. 4b. Note that the frequency spectrum corresponds to a very small range around the peak in the time domain signal. For this reason, the peak shift (or the frequency shift) in the time domain signals cannot be observed. By repeating this process for 20 equally spaced and increasing input voltage steps, Fig. 4c is generated that shows a decreasing resonance frequency for increasing voltage amplitudes. To visualize the frequency shift for normalized resonance frequencies, Fig. 4d is generated by dividing the frequency shift, Δf ¼ f f0 for each input amplitude excitation level, by the linear resonance frequency, f0 , which is determined by the intersection of the linear fit of Fig. 4c with the x-axis. The strain amplitude in the specimen is derived from the analytical solution for the longitudinal vibration of a bar in free-free boundary condition [18].
Table 1 Comparison of natural frequencies between experiment (fexp ) and FEM numerical simulation (fsim ) of the stainless steel rod sample with the simulation parameters Young's modulus E ¼ 198.3 MPa, density ρ ¼ 7850 kg/m3 and Poisson's ratio ν ¼ 0.33. longitudinal mode 1 2 3 4 5
fexp [Hz] 25114 50157 75039 99666 123935
fsim [Hz] 25115 50137 74968 99497 123600
fsim fexp [Hz]
jfsim fexp j
1 20 71 169 335
0.003% 0.040% 0.095% 0.170% 0.270%
Qexp
fexp
12200 15600 16350 9800 5600
3.3. Nonlinear measurements An individual time domain signal is measured for each frequency sweep as shown in Fig. 4a. The frequency bandwidth selected around the linear resonance frequency is f0 Δf , where Δf ¼ 0:1%⋅f0 , with a sweep duration of 8s at a sampling rate of 200kSa/s.
Fig. 4. Data processing steps for the nonlinear measurements. 40
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Δε ¼
v c
with
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fn ¼
nc 2L
between 2 and 8s while for a sweep duration of 10s, the mean value is the same as for the 8s, but with increased error bars. The conclusion is that a sweep duration of 8 s is the best compromise. The time in between the frequency sweeps is kept to a minimum, mainly determined by the time required to save the gathered time domain signal, while the selected frequency bandwidth around the linear resonance frequency f0 Δf was chosen wide enough to ensure that the decaying amplitude can settle evenly. In order to accurately determine the material nonlinearity of the samples, the intrinsic nonlinear behavior of the experimental setup itself (including the electrical components) must be negligibly small. This is particularly true when the material nonlinearity (plus variations due to damage) of the sample itself is expected to be very small – this is the case for the metallic 17-4 PH samples examined in this research. Because NRUS resonance measurements exploit the natural amplification of the sample's motion near resonance, the electrical components involved exciting these sample resonances or other extraneous nonlinear effects such as nonlinear distortion of the air should not influence the material nonlinearity, αf measured. With this noncontact generation, the motion of the sample is not directly coupled to the generation mechanism, so these extraneous instrumentation nonlinear effects should be eliminated. To verify that this instrumentation nonlinearity is not present, consider the results from a very linear material, aluminum. Fig. 6 illustrates that an aluminum AL-6061-T6 rod with an αf -value of 0.126 is quite small compared to a 17-4 PH stainless steel rod with an αf -value of 6.43 (both averaged for five measurements of the second longitudinal mode for 2s sweeps). This also strengthens the assumption that the effect of the transducer heating up the specimen can only be minimal. Finally, Fig. 6 illustrates the nonlinear shift for the first three longitudinal modes for both the aluminum and stainless steel samples for 2s sweeps. While the overall behavior of the nonlinearity values for each of these three modes agree with each other, the measurements of the first mode result in a scattered set of data points (and an associated poor linear fit), while the results for the third mode do not excite significant strains in the samples. In contrast, the results for the second longitudinal mode provides both adequate strain excitation, and a good linear fit. As a result, all the remaining measurements on the 17-4 PH specimens are performed on this second mode. As shown by Remillieux et al. [14,19] and TenCate et al. [20,21] on geomaterials, hysteretic nonlinearity is very sensitive to the applied strain level. TenCate et al. [20] identified different strain regimes for sandstones with mostly linear elastic behavior for strains below ε 108 107 . Above a threshold of ε 106 they related the quadratic decrease in resonance frequency mostly to hystertic nonlinearity. For even higher strains, nonequilibrium effects must be considered with an often linear dependence of frequency over strain. TenCate et al. [21] show that for these higher strains, where slow dynamics superpose the
(4)
leading to Δε n ¼
nvn 2Lfn
(5)
where Δεn is the strain amplitude of the n-th mode, vn is the corresponding velocity amplitude at the specimen end (i.e. at the anti-node) measured with the laser vibrometer, c is the phase speed, fn is the resonance frequency of the n-th eigenmode, and L is the sample length. The slope of the normalized frequency shift versus the strain amplitude then defines the hysteretic nonlinearity parameter, αf [8]. A straightforward methodology of Haupert et al. [8] is used to reduce the effect of ambient environmental factors by compensating for temperature changes during the measurements. This procedure includes measuring the resonance frequency in between each increasing driving step at a base voltage amplitude that is chosen as, in this case, the lowest excitation level with narrowest resonance response. Fig. 4e shows that for this measurement setup, the reference measurements experience a negative shift (apart from possible small temperature deviations) for consistent conditions. The corrected relative frequency shift of Fig. 4f is determined by subtracting the relative frequency shift of the reference excitation from the corresponding relative frequency shift of the increasing driving steps. Even though shielding the experiment with an insulating box to reduce the temperature influence yielded more consistent results, the ambient temperature effect is still large enough to necessitate this temperature correction procedure. Because of the very low energy transmission from the air to the specimens, the influence of the driving transducer warming up the specimen can be assumed to be insignificant. The coefficient of the longitudinal acoustic wave transmission, dependent on the impedances of the media pairings, is for air to steel 0.00193% and air to aluminium 0.00521%. The difference of a factor of 2.7 between these coefficients also exactly correlates to the gain in achieved maximal strain as shown in Fig. 6. The selection of sweep parameters is a compromise between reaching a steady state at each frequency during the sweep, and reducing the influence of ambient environmental factors. A longer sweep and narrower frequency bandwidth helps in reaching a steady state, and increases the numerical accuracy of the fast Fourier transform (FFT), while extending the time span of the experimental procedure potentially increases the influence of ambient temperature variations on the measured αf results. To determine the influence of the sweep duration, αf is measured for durations from 2 to 10 s for five measurements each. Fig. 5 shows that the αf determined varies for the sweep durations with increasing values
Fig. 6. Comparison of stainless steel and aluminum rods for longitudinal modes 1-3.
Fig. 5. Normalized αf for variation of sweep duration for the stainless steel rod. 41
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nonlinearity, the applied stress isn't released instantaneously and the material properties recover logarithmic over time. While these transition strains are well-characterized for geomaterials like rocks, sandstones and concrete, the length scale of the microstructure of metals is several orders of magnitude smaller than those of the geomaterials examined by TenCate and Remillieux. Therefore, it is expected that the threshold strains would also be much smaller. However, the specific threshold strains in metals are subjected to further research. The achieved strain levels of 2 106 for stainless steel based on this air-coupled excitation method are considerably smaller than comparable measurements of Haupert et al. [8] with strain levels up to 3 105 , and while the results agree qualitatively well with aluminium behaving significantly more linear than stainless steel, a quantitative comparison of the results of hysteretic nonlinearity values is not possible. 4. Results for thermally aged 17-PH stainless steel samples Fig. 7. Evolution of linear resonance frequency for thermally aged 17-4 PH stainless steel samples. (Note: Because of the quadratic portion of the resonance frequency in the determination of the Young's modulus in Equation (6), the vertical E-tick marks of the right y-axis are slightly unequally spaced).
4.1. Material 17-4 PH stainless steel is a commonly used martensitic precipitationhardening (”PH”) alloy with a good combination of high-strength, good ductility and corrosion resistance. The steel contains about 3 wt. % Cu and is strengthened by precipitation of dispersed copper in the martensite matrix [22]. Solution annealed 17-4 PH shows initially no copper precipitates. Thermal aging at temperatures of 400 C and higher, hardens the material [23]. At first, coherent copper-rich precipitates form in the material, restraining the motion of dislocations in the matrix. The copper has a low solubility at this temperature, such that the copper atoms diffuse and form fine particles. After exceeding the peak aging time, the growing copper-rich precipitates become incoherent with the matrix, leading to lower strength and hardness [24]. Previous research [25] examined 17-4 PH stainless steel as a surrogate material for irradiation damaged reactor pressure vessel (RPV) steel, because of similarities in both microstructures with the formation of copper precipitates – irradiation damage, which leads to harmful embrittlement, is also partly caused by copper-rich precipitates of similar size and density. As a result, 17-4 PH is a surrogate sample to avoid difficulties in handling irradiated RPV steel samples. Table 2 gives an overview of the thermal aging processes and the corresponding sample designation from Ref. [25]. The samples are first solution annealed at 1040 C for 6 h followed by air cooling, and then thermally aged in increasing steps up to 6 h at 400 C. This aging temperature was selected to ensure that the dislocation density (approximately) remained constant, while simultaneously forming copper precipitates. The samples in Ref. [25] were designed to be large enough to enable the measurement of the acoustic nonlinearity parameter, β with the SHG of propagating Rayleigh waves. Four new NRUS samples were then EDM-cut (electrical discharge machining) from the original [25] samples to slender bars 10 mm 10 mm x 100 mm, with one small end face polished to provide a necessary reflective surface for the laser vibrometer.
un-aged and three thermally aged 17-4 PH samples, which indicates a monotonic increase of resonance frequency for longer thermal aging time. Note that there are very small experimental deviations between the five repeated measurements performed on each sample – the error bars for the f0 are tiny. The analytical description of the n-th natural longipffiffiffiffiffiffiffiffi tudinal frequencies fn ¼ nc=2L, where the phase speed is c ¼ E=ρ, enables the calculation of the Young's modulus from the second longitudinal mode E ¼ ðf2 LÞ2 ρ:
(6)
These Young's modulus results (also shown in Fig. 7) indicate that the formation of copper-rich precipitates stiffens the material by a small amount, leading to increased values of E with increased aging time. Rosen et al. [26] also noted an increase in stiffness for similar precipitation formation in aluminum alloy 2219. 4.3. NRUS results for the 17-4 PH bars The measured hysteretic nonlinearity parameter, αf , quantifying the hysteretic behavior and quasi-static discrete memory effects for these same four 17-4 PH bar samples is shown in Fig. 8 and Table 3. Again, the results are averaged for five repeated measurements and evaluated up to an equal maximum strain level εmax of 2 106 for all measurements. The R2 -values, which characterize the goodness of the linear fit for each
4.2. Evolution of linear resonance frequency for the 17-4 PH bars Fig. 7 shows the evolution of linear resonance frequency f0 for the one Table 2 Sample designations and heat treatment procedure for 17-4 PH samples [25]. Sample designation
Solution annealing
Cooling method
Aging time @ 400 C
AC AC-0.1 AC-1 AC-6
1040 C/6h 1040 C/6h 1040 C/6h 1040 C/6h
air cooled air cooled air cooled air cooled
– 0.1h 1h 6h
Fig. 8. Nonlinear parameters αf and αQ for thermally aged 17-4 PH stainless steel samples. 42
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Table 3 Nonlinear parameters αf and αQ for thermally aged 17-4 PH stainless steel samples. Sample condition
αf
Δαf =αf ;AC
Standard deviation
R2
AC AC-0.1 AC-1 AC-6
0.76 1.56 1.89 1.92
þ105% þ148% þ153%
9.55% 7.53% 4.46% 6.98%
0.50 0.88 0.88 0.79
Sample condition
αQ
ΔαQ =αQ;AC
Standard deviation
R2
AC AC-0.1 AC-1 AC-6
1.42 2.69 3.32 3.13
þ89% þ133% þ120%
12.5% 22.8% 10.9% 21.4%
0.37 0.46 0.72 0.62
individual measurement, indicate moderate linear approximation for less nonlinear behavior, so as the samples become more nonlinear, the accuracy of the αf measurement increases. When compared to the evolution of linear resonance frequency, the error bars for these αf -values are considerably larger. However there are clearly identifiable αf trends when comparing the results for the four different samples, where the nonlinear behavior increases with increasing aging time – becoming more negative αf -values (or higher αf -values). The trend also shows a decreasing increase in nonlinearity for each consecutive aging time step, almost leveling off between 1h and 6h. In contrast to the results from the linear resonance frequency measurements, this technique indicates that the microstructural changes effecting hysteretic material nonlinearity do not increase linearly with the thermal aging process. In addition to αf (the hysteresis parameter which describes the frequency shift), a second hysteresis parameter, αQ is introduced which describes the change of energy loss as a function of strain, or 1 1 ¼ αQ Δε: Q Q0
Fig. 9. Comparison of normalized αf , αQ and Young's modulus E (solid lines) to normalized β, Vickers hardness (HV) and TEP-measurements (dashed lines) from Matlack et al. [25]. (Note: Data points are connected for clearer visualization).
of continuous copper precipitate formation throughout the aging process. Table 4 compares the relative increase of each of these parameters from un-aged to 6h aging, and indicates that the relative change in E measured through the evolution of resonance frequency is an order of magnitude less sensitive than the Vickers hardness and TEP measurements. Finally, Table 4 gives the average standard deviation of the measurement techniques compared, which shows that while the normalized αf and αQ -results are the most sensitive, they also have, together with the β-results, comparatively the largest error bars. Nevertheless, a distinct trend is clearly seen. Now consider a comparison between the nonclassical hysteretic nonlinearity parameter, αf and the quadratic acoustic nonlinearity parameter, β. The relative β values are determined by evaluating the ratio of the amplitudes of the first (A1 ) and second (A2 ) harmonics as a func-
(7)
The parameter αQ is determined in a similar fashion to αf by evaluating the quality factor Q for each frequency sweep, and applying the data processing steps, including the correction of ambient factors, provided in section 3.3. As also shown in Fig. 8 and Table 3, αQ essentially follows the same trend as αf , but with bigger error bars and lower R2 -values.
4.4. Comparison to other monitoring techniques
tion of increasing propagation distance. The slope of the ratio of A2 =ðA1 Þ2 versus propagation distance is then proportional to β. The measured values of β show a monotonic decrease in value with increasing aging time, with a larger decline between 0.1h and 1h (at 400 C). Matlack et al. [25] related this decrease in β to the increasing number of copper precipitates, which pinned existing dislocation contributions. The β-measurements were performed on much larger specimens with dimensions of 19 mm 38 mm x 230 mm because of the inherent restrictions of this nonlinear Rayleigh wave technique due to a minimum wave propagation distance and an adequate contact surface for the exciting transducer. The hysteresis nonlinearity αf increases rapidly in the beginning, before becoming saturated at the end. This behavior of αf most closely follows the TEP-values, where they both have a slightly flattening increase between 1 and 6h aging. This correspondence could be because the TEP is also most sensitive to the formation of copper precipitates alone, independent of their pinning existing dislocations. Most importantly, the αf parameter shows the greatest sensitivity to formation of these copper precipitates – Table 4 shows that both αf , αQ and β are an order of magnitude more sensitive than the Vickers hardness and TEP values, and αf and αQ are more than twice as sensitive as β to the formation of these copper precipitates. This is important since the copperrich precipitates are the damage mechanism most directly associated
Fig. 9 compares the normalized hysteretic nonlinearities, αf and αQ , and stiffness (E) results from this research with complementary measurements of Vickers hardness (HV), thermo electric power (TEP) and quadratic acoustic nonlinearity, β, performed by Matlack et al. [25], with all normalized to their respective un-aged values. The measured quality factors increase with increasing heat treatment, exhibiting a trend quite similar to the trend of the Young's modulus, but with much bigger error bars. The Vickers hardness results show a monotonic increase in hardness with increasing aging time, corresponding to the expected increase in copper precipitate volume fraction with thermal aging. The TEP values follow the same increasing trend with thermal age – an increase in TEP corresponds to a decreasing amount of copper in the matrix material, meaning an increase in copper precipitate volume fraction [25]. For the thermo-electric power measurement technique, the voltage difference, generated by a temperature gradient in the material sample, is measured and compared to a reference material. The measurements are quantified by the Seebeck coefficient S ¼ ΔV=ΔT, the ratio between voltage difference and temperature gradient [27]. All three of these techniques show pretty much a linear relation as a function of thermal aging time, which is consistent with the assumption 43
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(DAAD) through a Graduate Research Assistantship. Additional funding was provided by the Electric Power Research Institute (EPRI). Thanks to anonymous reviewer comments on the importance of the sweep time, and to Katie Scott for additional measurements.
Table 4 Sensitivity of the measured parameters/material properties to microstructure changes due to thermal aging occurring from un-aged to 6 h. Measurement technique
Material property/ parameter
Sensitivity ΔAC 6=AC
Average standard deviation
Lin. resonance frequency NRUS
E Q*
Vickers hardness Thermo-electric power Rayleigh surface waves
HV TEP
þ2.8% þ19% þ153% þ120% þ28% þ19%
0.0049% 1.9% 7.1% 16.9% 1.9% 0.65%
β'
47%
8.2%
αf αQ
*measured at ε ¼ 2 10
6
Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi. org/10.1016/j.ndteint.2018.04.003. References [1] Matlack K, Kim J-Y, Jacobs L, Qu J. Review of second harmonic generation measurement techniques for material state determination in metals. J Nondestr Eval 2015;34(1):273. [2] Marino D, Kim J-Y, Ruiz A, Joo Y-S, Qu J, Jacobs LJ. Using nonlinear ultrasound to track microstructural changes due to thermal aging in modified 9%Cr ferritic martensitic steel. NDT E Int 2016;79:46–52. [3] Walker SV, Kim J-Y, Qu J, Jacobs LJ. Fatigue damage evaluation in A36 steel using nonlinear Rayleigh surface waves. NDT E Int 2012;48:10–5. [4] Doerr C, Kim J-Y, Singh P, Wall JJ, Jacobs LJ. Evaluation of sensitization in stainless steel 304 and 304L using nonlinear Rayleigh waves. NDT E Int 2017;88:17–23. [5] Matlack K, Kim J-Y, Wall J, Qu J, Jacobs L, Sokolov M. Sensitivity of ultrasonic nonlinearity to irradiated, annealed, and re-irradiated microstructure changes in RPV steels. J Nucl Mater 2014;448(1):26–32. [6] Migliori A, Sarrao J. Resonant ultrasound spectroscopy: applications to physics, materials measurements, and nondestructive evaluation. Wiley; 1997. [7] Windels F, Abeele KVD. The influence of localized damage in a sample on its resonance spectrum. Ultrasonics 2004;42(1):1025–9. [8] Haupert S, Renaud G, Riviere J, Talmant M, Johnson PA, Laugier P. High-accuracy acoustic detection of nonclassical component of material nonlinearity. J Acoust Soc Am 2011;130(5):2654–61. [9] Van Den Abeele KE, Johnson A, Sutin PA. Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage. Part I: nonlinear wave modulation spectroscopy (NWMS). Res Nondestr Eval 2000;12(1):17–30. [10] Hikata A, Chick BB, Elbaum C. Dislocation contribution to the second harmonic generation of ultrasonic waves. J Appl Phys 1965;36(1):229–36. [11] Cantrell JH, Zhang X-G. Nonlinear acoustic response from precipitate-matrix misfit in a dislocation network. J Appl Phys 1998;84(10):5469–72. [12] Granato A, Lücke K. Theory of mechanical damping due to dislocations. J Appl Phys 1956;27(6):583–93. [13] Barsoum MW, Radovic M, Zhen T, Finkel P, Kalidindi SR. Dynamic elastic hysteretic solids and dislocations. Phys Rev Lett 2005;94(8):085501. [14] Remillieux MC, Guyer RA, Payan C, Ulrich TJ. Decoupling nonclassical nonlinear behavior of elastic wave types. Phys Rev Lett 2016;116:115501. [15] Kim Y. Sound propagation: an impedance based approach. John Wiley and Sons, Ltd; 2010. [16] Van Damme B, Van Den Abeele K. The application of nonlinear reverberation spectroscopy for the detection of localized fatigue damage. J Nondestr Eval 2014; 33(2):263–8. [17] Van Den Abeele K, Carmeliet J, Ten Cate J, Johnson P. Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part II: single-mode nonlinear resonance acoustic spectroscopy. Res Nondestr Eval 2000;12(1):31–42. [18] Kinsler LE. Fundamentals of acoustics. Wiley; 1982. [19] Remillieux MC, Ulrich TJ, Goodman HE, Ten Cate JA. Propagation of a finiteamplitude elastic pulse in a bar of berea sandstone: a detailed look at the mechanisms of classical nonlinearity, hysteresis, and nonequilibrium dynamics. J Geophys Res Solid Earth 2017;122(11):8892–909. [20] TenCate JA, Pasqualini D, Habib S, Heitmann K, Higdon D, Johnson PA. Nonlinear and nonequilibrium dynamics in geomaterials. Phys Rev Lett 2004;93:065501. [21] TenCate JA, Smith E, Guyer RA. Universal slow dynamics in granular solids. Phys Rev Lett 2000;85:1020–3. [22] Murayama M, Hono K, Katayama Y. Microstructural evolution in a 17-4PH stainless steel after aging at 400 C. Metall Mater Trans 1999;30(2):345–53. [23] Mirzadeh H, Najafizadeh A. Aging kinetics of 17-4PH stainless steel. Mater Chem Phys 2009;116(1):119–24. [24] Hsu K-C, Lin C-K. High-temperature fatigue crack growth behavior of 17-4PH stainless steels. Metall Mater Trans 2004;35(9):3018–24. [25] Matlack KH, Bradley HA, Thiele S, Kim J-Y, Wall JJ, Jung HJ, Qu J, Jacobs LJ. Nonlinear ultrasonic characterization of precipitation in 17-4PH stainless steel. NDT E Int 2015;71:8–15. [26] Rosen M, Horowitz E, Fick S, Reno R, Mehrabian R. An investigation of the precipitation-hardening process in aluminum alloy 2219 by means of sound wave velocity and ultrasonic attenuation. Mater Sci Eng 1982;53(2):163–77. [27] Northwood DO, Sun X, Hochreiter J, Sokolowski JH, Penrod DE. Thermoelectric power (TEP) measurements as a materials characterization technique. In: AIP conference proceedings, vol. 316; 1994. p. 181–4 (1). [28] Lebedev AB. Amplitude-dependent decrement to modulus defect ratio in breakaway models of dislocation hysteresis. Philos Mag A 1996;74(1):137–50.
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with the embrittlement of RPV steels. Finally note that the Read ratio παQ =2αf can be determined from αf and αQ , and this Read ratio is expected to be equal to 4/3 for a purely quadratic hysteretic nonlinearity [8,28]. The Read ratios calculated for these four 17-4 PH specimens are all approximately the same, with an average of 2.78 0.60, and a maximum difference ΔAC 6=AC of 15.7%. Note that these values are higher than a Read ratio of 4/3 predicted by the dislocation-hysteresis driven model of Lebedev [28]. This difference is most likely due to the fact that the microstructure of these four 17-4 PH specimens are dominated by precipitates and their interactions with dislocations, instead of the pure dislocation-hysteresis modeled in Ref. [28]. 5. Conclusion This research used the NRUS technique to evaluate the sensitivity of the nonclassical hysteretic nonlinearity parameters, αf and αQ to changes in the microstructure of thermally aged 17-4 PH stainless steel samples. These microstructural changes involved the formation of copper precipitates that are associated with embrittlement damage in RPV steels. An NRUS procedure was developed that uses both a noncontact source and receiver – an air-coupled piezoelectric transducer source and a laser vibrometer receiver – to detect the axial motion of a slender rod. This fully noncontact configuration avoids mechanical contact points which can adversely influence the measurements. The excitation of the samples with the air-coupled transducer, paired with a focusing horn excited the first five axial eigenmodes of the rod with very high quality (Q) factors. There was good agreement with the results of an FEM simulation, indicating low damping and little influence from extraneous experimental setup factors. This NRUS procedure measured the hysteretic nonlinearity parameters, αf and αQ , noting that a temperature correction was critical for repeatable measurements. Complementary linear measurements on these 17-4 PH samples determined the Young's modulus, and showed a direct relationship between thermal aging time and material stiffness, E. The measured αf (αQ ) values were two orders of magnitude more sensitive than E to changes in microstructure due to the formation of copper precipitates, and compared well to (quadratic) acoustic nonlinearity parameter (β) values previously measured on these same 17-4 PH samples. This NRUS setup provides a reliable, accurate and valid method to measure αf and αQ , demonstrating the advantages of fully noncontact generation and detection over setups with contact points. Due to the absence of operator-dependent processing, it is feasible to develop a highly automated measurement procedure and with minimal sample preparation. Acknowledgements This work is supported by the German Academic Exchange Service
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