This research studies two nonlinear ultrasound techniques: second harmonic generation and nonlinear resonant ultrasound spectroscopy, and the relationship to microstructural mechanisms in metals. The results show that there is a large change in both the classical, β, and nonclassical, α, ultrasound nonlinearity parameters in response to three specific microstructural mechanisms: precipitate growth in and along the grain boundaries, dislocations, and precipitate pinned dislocations. For example, both β and α increase with the growth of the precipitate radii (precipitate-pinned-dislocations). Additionally, both β and α increase when there is a growth of precipitates in and along the grain boundaries. As expected, β and α decrease when there is a removal of dislocations in the material. The relationship between β and α, and the microstructural mechanisms studied provide a quantitative understanding of the relationship between measured nonlinearity parameters and microstructural changes in metals, helping to demonstrate the possibility of using these two independent, but complementary, nonlinear ultrasound procedures to monitor microstructural damage.

Nonlinear ultrasound (NLU) techniques, such as second harmonic generation (SHG) (Cantrell, 2003; Cantrell and Yost, 2000, 2001; Cantrell and Zhang, 1998; Cantrell, 2004; Hikata et al., 1966; Hurley et al., 2000; Hurley and Fortunko, 1997; Kim et al., 2014; Li et al., 2019; Marino et al., 2016; Matlack et al., 2012, 2014b, 2015; Ruiz et al., 2013; Scott et al., 2017) and nonlinear resonant ultrasound spectroscopy (NRUS) (Barsoum et al., 2005; Bentahar et al., 2006; Espinoza et al., 2018; Haupert et al., 2011; Maier et al., 2018; Muller et al., 2005; Ostrovsky and Johnson, 2001; Payan et al., 2014; Remillieux et al., 2016; TenCate et al., 2004) have shown promise in detecting microstructural damage in materials before macroscopic failure occurs. However, before these nonlinear methods can be implemented outside the laboratory, a better understanding of the relationship between the microstructural mechanisms and the ultrasound nonlinearity parameters (β and α) measured by these techniques needs to be established. As a step in that direction, this research studies the relationship between multiple microstructural mechanisms, and the ultrasound nonlinearity parameters β, the classical nonlinearity parameter associated with SHG, and α, the nonclassical nonlinearity parameter for the longitudinal mode measured using NRUS (the mode number of vibration for each material is seen in Table I and was chosen because it was closest to the center frequency of the piezoelectric transducer). These ultrasound nonlinearity parameters can then be related back to material properties through the nonlinear elastic stress-strain relationship. Remillieux et al. (2017) developed a new experimental setup, where a novel 3-D scanning Doppler vibrometer is used to measure the axial displacement in a bar sample of Berea sandstone, and then extracted the classical and nonclassical nonlinearity parameters of the material. They also discussed the relative contributions of the mechanisms of classical nonlinearity, hysteresis, and nonequilibrium dynamics in such a pulse propagation experiment. This research focuses on three microstructural mechanisms found in metals: precipitate growth in and along the grain boundaries, dislocations, and precipitate pinned dislocations. These mechanisms are studied through the sensitization of 304 stainless steel (SS), the annealing of 316 L SS, and the heat treatment of Fe-Cu binary alloy, respectively. Additionally, the results from two previous studies (Fahse et al., 2020; Maier et al., 2018) are integrated with the current results to develop a deeper understanding of the relationship between measured nonlinearity parameters and microstructural changes in metals.

TABLE I.

Sample dimensions for NRUS and SHG specimens, SHG technique used for each material, and longitudinal mode of vibration.

MaterialNRUS Specimen Dimensions (mm)SHG Specimen Dimensions (mm)SHG TechniqueMode of Vibration
304 SS 10 × 10 × 100 152. × 50.8 × 12.7 Rayleigh wave 
316L SS 5 × 5 × 50 152. × 50.8 × 12.7 Rayleigh wave 
Fe-1.0% Cu alloy 5 × 5 × 50 150 × 32 × 9 Longitudinal wave 
17-4 PH 10 × 10 × 100 230 × 38 × 19 Rayleigh wave 
9% Cr ferritic steel 5 × 5 × 50 203.2 × 45.72 × 12.7 Rayleigh wave 
MaterialNRUS Specimen Dimensions (mm)SHG Specimen Dimensions (mm)SHG TechniqueMode of Vibration
304 SS 10 × 10 × 100 152. × 50.8 × 12.7 Rayleigh wave 
316L SS 5 × 5 × 50 152. × 50.8 × 12.7 Rayleigh wave 
Fe-1.0% Cu alloy 5 × 5 × 50 150 × 32 × 9 Longitudinal wave 
17-4 PH 10 × 10 × 100 230 × 38 × 19 Rayleigh wave 
9% Cr ferritic steel 5 × 5 × 50 203.2 × 45.72 × 12.7 Rayleigh wave 

Previous studies have shown the potential of SHG to detect microstructural damage in a variety of conditions, and SHG has been shown as a promising technique for metals as well as cement-based materials. These studies have demonstrated the ability of SHG to detect “material damage,” such as dislocations (Cantrell, 2003; Hikata et al., 1966; Hurley and Fortunko, 1997), dislocation dipoles (Cantrell and Yost, 2001; Cantrell, 2004), precipitates (Li et al., 2019; Scott et al., 2017), and precipitate pinned dislocations (Cantrell, 2003; Cantrell and Yost, 2000; Cantrell and Zhang, 1998; Hurley et al., 2000), which are all known precursors to macroscopic cracking and damage. Research has shown that the sensitivity of SHG to this microstructural damage is useful in a variety of applications. In one study, SHG gave promising results in the detection of radiation damage in reactor pressure vessel (RPV) steels (Matlack et al., 2012, 2014b). Another study shows how SHG can be used to determine damage in thermally aged materials (Ruiz et al., 2013). Additionally, a study looked at cement-based materials, and determined the sensitivity of SHG to microcracks and other microstructural changes (Kim et al., 2014).

Like SHG, NRUS has proven to be useful in many damage situations. So far, the majority of the research has focused on highly hysteretic, nonmetallic materials, such as rock (Remillieux et al., 2016; TenCate et al., 2004), concrete (Bentahar et al., 2006; Payan et al., 2014), and bone (Muller et al., 2005). Research on these materials has shown that the source of hysteretic nonlinearity is from “soft” inclusions found in a “hard” matrix (Ostrovsky and Johnson, 2001). One study has demonstrated the sensitivity of NRUS both theoretically and experimentally to damaged and undamaged concrete bars (Bentahar et al., 2006). Another study looked at the feasibility of NRUS to predict microcracking in bones, and saw significant changes in the hysteretic nonlinearity parameter, α, with increased damage, while wave velocity showed minimal change (Muller et al., 2005). While there has been limited research on and understanding of the mechanisms behind the hysteretic response of metals, there have been a number of studies that have demonstrated very promising experimental results (Espinoza et al., 2018; Fahse et al., 2020; Maier et al., 2018). One study looked at how dislocation densities in Al and Cu contribute to the nonlinearity (Espinoza et al., 2018). A different study demonstrated the sensitivity of NRUS to precipitate growth (Fahse et al., 2020; Maier et al., 2018). Chakrapani and Barnard (2017) used NRUS to measure the nonlinearity parameter β in aluminum and copper.

While SHG and NRUS have shown promise in detecting changes in the microstructure for a wide range of materials, the issue remains that these techniques are sensitive to multiple microstructural mechanisms, which can lead to complications when studying materials with complicated microstructures. It is important to first understand the sensitivity and relationship of the ultrasound nonlinearity parameters, β and α (SHG and NRUS, respectively), to individual mechanisms before considering more complicated situations. One objective of this research is to expand the understanding and versatility of NLU techniques by determining the degree of sensitivity of each of these separate ultrasound nonlinearity parameters to specific microstructural features. In this way, it may be possible to establish two independent, but complementary, procedures to monitor microstructural damage.

The results and discussions presented in this paper focus on understanding the relationship between β and α, and multiple microstructural mechanisms in metals. This comparison will be presented through new experimental results (β and α results for annealed 316 L SS, and α results for heat treated Fe-1.0% Cu and sensitized 304 SS), which are integrated with new analysis of previously published experimental measurements: β results for sensitized 304 SS (Doerr et al., 2017) and heat treated Fe-1.0% Cu alloy (Scott et al., 2018), and β and α results for heat treated 17–4 PH (Maier et al., 2018; Matlack et al., 2015) and 9% Cr ferritic steel (Fahse et al., 2020; Marino et al., 2016).

Nonlinear elastic behavior is observed in materials when the linear stress-strain relationships no longer hold. Considering a 1-D case, the nonlinear stress-strain relationship can be defined as

(1)

where K0 is the linear elastic modulus, β is the quadratic classical nonlinearity parameter, δ is the cubic classical nonlinearity parameter, α is the hysteretic nonlinearity parameter, Δε is the local strain amplitude, and ε̇ is the strain rate. sgn(x) is the signum function that is defined: sgn(x) =1 if x > 0, sgn(x) = 0 if x = 0 and sgn(x) = −1 if x < 0.When looking at the equation of state, K0ε refers to the linear elastic behavior, the terms containing β and δ are considered classical nonlinear elastic behavior, and the term containing α is considered nonclassical (hysteretic) nonlinear elastic behavior.

The use of SHG to measure β considers the generation of a second harmonic wave when a single frequency sinusoidal wave is introduced into a nonlinear elastic material. The quadratic classical nonlinearity parameter, β, can be determined by relating the amplitude of the fundamental wave, A1, and the amplitude of the second harmonic wave, A2,

(2)

where k is the wave number and x is the propagation distance. A full derivation of β for a material with the constitutive in Eq. (1) can be found in Levy (2019). It can be shown theoretically that the hysteretic nonlinearity (α) has no contribution to the second harmonic amplitude, and experimentally, it can be shown that the experimental setup measures a cubic dependance of third harmonic amplitude on the fundamental amplitude, demonstrating pure classical nonlinearity.

In contrast, NRUS to measure α looks at the nonlinear vibrational response through changes in the resonance frequency as a function of input amplitude (Remillieux et al., 2016). This change in resonance frequency is predominantly due to the nonclassical nonlinear hysteretic elastic behavior of the material. More specifically, the hysteretic nonlinearity (α) causes a linear dependence of the resonance frequency on the strain amplitude, and the cubic nonlinearity (δ) causes a quadratic dependence, while the quadratic nonlinearity (β) gives rise to no contribution to the resonance frequency shift (Levy, 2019; Van Den Abeele, 2007). This is important from an experiment perspective that the quadratic and hysteretic nonlinearity parameters can be measured independently from the SHG and NRUS experiments. When looking at the experimental resonance curves in this research (Fig. 1), this linear relationship between the resonance frequency shift and strain amplitude is clearly visible when plotting the change in resonance frequency as a function of the strain amplitude (Fig. 2). This linear behavior is also seen in Fahse et al. (2020) and Maier et al. (2018), which have the same experimental setup, similar input strain amplitudes, and comparable metal materials.

FIG. 1.

Sample NRUS resonance curves.

FIG. 1.

Sample NRUS resonance curves.

Close modal
FIG. 2.

Sample plot of relative frequency shift as a function of strain amplitude to determine α (a similar response is seen in Fahse et al., 2020; Maier et al., 2018).

FIG. 2.

Sample plot of relative frequency shift as a function of strain amplitude to determine α (a similar response is seen in Fahse et al., 2020; Maier et al., 2018).

Close modal

The hysteretic nonlinearity parameter, α, can be determined by relating the frequency shift, Δf, strain amplitude, Δε, and equilibrium resonance, f0:

(3)

where f is the resonance frequency of a given driving amplitude, and can be determined directly from the resonance curves. f0 can be determined through linear resonant ultrasound spectroscopy (RUS), and is the resonance frequency at a sufficiently low strain amplitude. The strain amplitude is derived by considering the longitudinal vibration of a thin 1-D bar with free-free boundary conditions at a particular resonance mode, n. A full derivation can be found in Kinsler et al. (2009), where the strain amplitude for a particular mode is derived to be

(4)

where v is the velocity amplitude at the end of the bar, and L is the length of the bar.

The materials in this study are cut into the samples of NRUS and SHG using wire electrical discharge machining (minimal introduction of stress into the material). The samples for NRUS are thin bars, and the samples for SHG are rectangular prisms. The dimensions for each of the samples are seen in Table I.

When this austenitic SS, 304 SS, is exposed to high temperatures for extended periods of time, sensitization can occur. Sensitization is known as the formation of chromium carbide precipitates. The formation of these precipitates results in a chromium depletion zone around the grain boundaries, making the steel more susceptible to intergranular stress corrosion cracking. Building on previous research that used SHG to characterize sensitization (Doerr et al., 2017), this work considers both an annealed and a sensitized 304 sample. One sample is annealed at 1080 °C for 30 min to remove any cold working effects in the form of excessive dislocations and residual stresses. The second sample is annealed and, subsequently, heat treated at 675 °C for 4 h resulting in full sensitization.

This set of specimens investigates the change in dislocation density as well as relief of stresses when exposed to annealing temperatures. This is done by taking a set of as received 316 L SS specimens and heat treating them at 1052 °C for 30 min. Heat treating SS at this temperature is known to relieve residual stresses and remove dislocations.

The set of Fe-1.0% Cu alloy specimens serves as surrogate specimens which simulate radiation damage in light-water RPV steel, and are the same set of specimens studied in Scott et al. (2017, 2018). Individual annealed specimens are thermally aged at 500 °C using the following heat treatment schedule: 5, 15, 30, 100, and 300 h, respectively. This thermal aging promotes the nucleation and growth of Cu-precipitates, and simulates varying amounts of radiation damage.

In addition to these three new sets of specimens, this research considers two additional sets of materials: 17–4 PH SS and 9% Cr ferritic steel. When 17–4 PH is thermally aged at and above 400 °C, the material hardens through the formation of Cu-precipitates (Mirzadeh and Najafizadeh, 2009). Since copper has a low solubility at 400 °C and above, the copper atoms will diffuse and form small precipitates. When the Cu-precipitates first form in the material, they will be coherent and will restrict dislocation motion (precipitate pinned dislocations). The 17–4 PH studied was solution annealed at 1040 °C for 6 h, and air cooled to ensure that the initial samples had no Cu-precipitates (Maier et al., 2018; Matlack et al., 2015). The samples were then heat treated at 400 °C for 0.1, 1, and 6 h, where an increase in the volume fraction of Cu-precipitates (corresponding to an increase in the number density of the Cu-precipitates) was seen as the heat treatment time increased (Matlack et al., 2015). The researchers chose this heat treatment temperature to ensure that the dislocation density remained constant throughout thermal aging (Matlack et al., 2015).

The research on the 9% Cr ferritic martensitic steel provides an opportunity to consider a material with multiple microstructural mechanisms (change in dislocation density and growth in radii of precipitates) (Fahse et al., 2020; Marino et al., 2016). When 9% Cr ferritic martensitic steel is heat treated, there is a formation of precipitates as well as a change in the dislocation density (Fahse et al., 2020; Marino et al., 2016). There are multiple types of precipitates that form when this material is thermally aged: M23C6, MX, and M2X particles, which form during tempering, and Laves-phase and Z-phase, which form during further thermal aging (Cipolla et al., 2009; Hald, 2008; Park et al., 2013; Sawada et al., 2005). Marino et al. (2016) heat treated the 9% Cr ferritic steel at 650 °C for 0, 200, 500, 1000, 1500, and 300 h, and found that there was an initial decrease and then levelling off of the dislocation density as well as an increase in the radii of the precipitates.

The SHG instrumentation setup (longitudinal wave and Rayleigh wave) used for this research is the same as that used in previous research (Matlack et al., 2012; Thiele et al., 2014). In these measurements, a single frequency longitudinal or Rayleigh wave is propagated through or along the surface of the material, respectively. As the wave interacts with the material, a second harmonic wave is generated which is received along with the fundamental wave by a transducer. The signal received by the transducer is then decomposed into the amplitudes of the fundamental, A1, and second harmonic, A2, waves using the signal processing techniques found in Scott et al. (2018). For the longitudinal wave measurements, the wave propagation distance remains constant, and the input amplitude of the fundamental wave is increased. β is determined by plotting A2 as a function of A12. For the Rayleigh wave measurements, the propagation distance is increased with measurements being made along the surface of the material. β is determined by plotting A2/ A12 as a function of increasing propagation distance, x. The SHG technique that is used for each of the materials is seen in Table I.

For the NRUS measurements, a completely noncontact setup is implemented, and the experimental procedure and setup follow that established in Maier et al. (2018). In the NRUS measurements, a frequency sweep around one of the vibrational modes from an air-coupled source is focused on one end of the specimen, and the response is measured at the other end of the specimen (1-D longitudinal vibration) with a laser Doppler vibrometer. The amplitude of the frequency sweep is then increased, and the response is measured again. From the received responses, the resonance curves for each input amplitude can be determined. α can then be determined by relating the shift in the resonance frequency with the strain amplitude. A sample of the resonance curves is seen in Fig. 1, and a sample plot of the relative frequency shift as a function of strain amplitude used to determine α is seen in Fig. 2. These figures demonstrate the linear shift in the resonance frequency as the input strain level is increased. This linear relationship between the shift in the resonance frequency and the strain amplitude is another demonstration that α is the parameter measured in these measurements. The approximate strain amplitudes seen in the metal specimens studied are on the order of 10−6 as is seen in Fig. 2. These curves and strain levels follow the results found in Fahse et al. (2020) and Maier et al. (2018). Additional details on the experimental procedure can be found in Levy (2019).

To provide a comparison between the nonlinear and linear ultrasound results, Young's modulus, E, is measured using RUS, where the technique follows that used in Maier et al. (2018). In this procedure, the linear resonance frequency of the material is measured when the specimen is excited at a sufficiently low stain amplitude. E can then be calculated from the material density (ρ), the linear resonance frequency (fn), length of the specimen (L), and the resonance mode (n) as seen in the following equation:

(5)

The measured α, β, and Young's modulus, E, results for the sensitized 304 SS are shown in Fig. 3, which compares the annealed and sensitized 304 SS. The complete results for β are presented in Doerr et al. (2017), and the current manuscript only shows the results for the annealed and sensitized specimens to allow for comparison with α and E. To visualize the relative change for each parameter, the measurement values are normalized to the annealed sample. The existence of sensitization of these samples is seen in the electrochemical reactivation and microscopy results presented in Doerr et al. (2017).

FIG. 3.

(Color online) Comparison of α, β (Doerr et al., 2017), and E for annealed and sensitized 304 SS.

FIG. 3.

(Color online) Comparison of α, β (Doerr et al., 2017), and E for annealed and sensitized 304 SS.

Close modal

Figure 3 shows that the NLU measurements, α and β, are much more sensitive to the formation of the chromium carbides (sensitization) than the linear parameter, E. The changes in α and β are over 20%, while the change in E is approximately 1%. This difference between the linear and nonlinear measurements demonstrates that the NLU techniques are much more sensitive to these small length scale material changes than the linear ultrasound techniques.

When comparing the two NLU results, both α and β significantly increase by 43% and 25%, respectively, between the annealed and sensitized state. The increases in both β and α between the non-sensitized and sensitized states are due to the interaction between the carbide and the grain boundaries. The mechanism is currently under investigation, but the interaction of the carbide precipitates with the two opposing grain boundary surfaces generates nonlinearity. Hysteresis nonlinearity is affected by energy dissipation, such as friction and contact. It is believed that contact and frictional motions are generated between the carbide and the grain boundaries when the ultrasonic wave propagates through the material, leading to an increase in α.

The results for α, β, and E for the 316 L SS specimens are shown in Fig. 4. First comparing the nonlinear and linear ultrasound techniques, it can be seen that both α and β are much more sensitive to the small changes in the microstructure than E, i.e., greater than 20% vs 1%. Next, comparing the NLU results, both α and β decrease between the as-received and heat treated states by 44% and 20%, respectively. This is because when the as-received sample is annealed, some dislocations are removed (decrease in dislocation density, Λ) from the material, relieving some of the residual stress in the material. These results for α are similar to the findings of Granato and Lücke (1956), who modelled the effect of a vibrating pinned dislocation on damping and hysteresis loss.

FIG. 4.

(Color online) Comparison of α, β, and E for as received and heat treated 316 L SS.

FIG. 4.

(Color online) Comparison of α, β, and E for as received and heat treated 316 L SS.

Close modal

The results for α, β, and E in the Fe-1.0% Cu specimens are shown in Fig. 5. These β results have previously been published in Scott et al. (2018), and are adopted in the present figure to allow for easier comparison to the α results. Similar to the other materials, the changes in E due to changes in the microstructure are smaller than the changes seen for both α and β, i.e., a maximum of 10% compared to 95% and 27%, respectively. The results were normalized to the 5-h specimen because of the nucleation of Cu-precipitates between the 0 and 5 h heat treatment times.

FIG. 5.

(Color online) Comparison of α, β, (Scott et al., 2018) and E results for Fe-1% Cu specimens, where inset figure clarifies the behavior from 5 h heat treatment time to 300 h heat treatment time.

FIG. 5.

(Color online) Comparison of α, β, (Scott et al., 2018) and E results for Fe-1% Cu specimens, where inset figure clarifies the behavior from 5 h heat treatment time to 300 h heat treatment time.

Close modal

Focusing on the nonlinear results, Fig. 5 shows that until the 100-h heat treatment mark, both α and β have similar trends. From the untreated specimen to the 5-h specimen, α and β decrease in value. This decrease in the nonlinearity is due to the nucleation of Cu-precipitates (Scott et al., 2018). Similar trends for β are seen by Cantrell and Yost (2000) in the nucleation of precipitates in a different material as well as in a molecular dynamics model (Setyawan et al., 2018). However, it is not fully understood why α decreases. For this uncertainty, this part is excluded from the current discussion.

From 5 to 300 h, there is growth in the radii of the Cu-precipitates which is the cause of the nonlinearity in the material. This growth can clearly be seen from the independent, ground truth, small angle neutron scattering measurements (Scott et al., 2017). While these Cu-precipitates are the source of the nonlinearities in the material, the explanations for the behaviors of α and β are discussed separately.

The behavior of β is explained by Scott et al. (2017, 2018) and is summarized here for ease of comparison. From the 5-h specimen to the 30-h specimen, the increase in β is due to the growth of the Cu-precipitate radii and a precipitate-pinned-dislocation interaction. β then continues to increase from the 30-h specimen to the 100-h specimen despite the loss in coherency between the Cu-precipitates and Fe-matrix. This increase in β may be due to some of the Cu-precipitates remaining coherent with the Fe-matrix, or some of the pinned dislocations becoming unpinned, leading to an effectively longer loop length. From the 100-h specimen to the 300-h specimen, β decreases due to the loss in coherency.

When looking at the results for α, it can be seen that after nucleation starts, α continues to increase as the radii of the Cu-precipitates grow. Since hysteresis nonlinearity is due to energy loss mechanisms, such as friction and contact, this increase in α is believed to be due to both dislocation vibration, as well as the contact between the Cu-precipitates and the Fe-matrix. The Granato and Lücke (1956) model of damping from pinned dislocations demonstrates the contribution of dislocation vibration to loss, and is a function of the dislocation density and the loop length. The hysteresis nonlinearity from the Cu-precipitates arises from the contact between the Cu-precipitates and the Fe-matrix, and acts as a function of the surface area between the two. This means that as the size of the Cu-precipitates grows, the surface area between the Cu-precipitates and the Fe-matrix will increase, resulting in an increase in α. Additionally, when there is a loss of coherency between the Cu-precipitate and the Fe-matrix (at the 100 and 300-h specimens), the contact between the Cu-precipitate and Fe-matrix may be considered even rougher, leading to even more energy loss due to contact. This rougher contact would increase α even more.

The β and α results for the 17–4 PH from Maier et al. (2018) and Matlack et al. (2015) are replotted in Fig. 6 to allow for ease of comparison for the reader. When looking at the β and α results in Fig. 6, the difference in the response of α and β can clearly be seen, where α increases as a function of heat treatment time, and β decreases as a function of heat treatment time. While a comparison of β and α is presented in Maier et al. (2018), the difference in the responses of β and α is not fully understood. This section will expand the understanding of these results by using the knowledge of hysteretic nonlinearity gained through the study of the 316 L SS, 304 SS, and Fe-1.0% Cu.

FIG. 6.

(Color online) Comparison of β and α results for 17-4 PH specimens (Maier et al., 2018; Matlack et al., 2015).

FIG. 6.

(Color online) Comparison of β and α results for 17-4 PH specimens (Maier et al., 2018; Matlack et al., 2015).

Close modal

In the 17–4 PH specimens, the Cu-precipitates pin dislocations creating similar precipitate pinned dislocations as seen in Fe-Cu specimens. However, in these specimens, the radii of the Cu-precipitates remain constant while the number density of Cu-precipitates increases. Using the Cantrell and Yost (2000) theoretical model for change in β due to precipitate pinned dislocations, Matlack et al. (2015) showed that the decrease in β was from the increase in the number density (N) of precipitates (ΔβN1/3) because the dislocation density and precipitate radii remain constant throughout the heat treatment.

When looking at the results for α from Maier et al. (2018), it can be seen that the response for α is opposite the response of β. The increase in α may be due to an additive effect, analogous to the fact that the total attenuation in a material is a simple, non-destructive sum of contributions from different physical mechanisms when they coexist. It has been shown so far that the hysteresis nonlinearity comes from dislocations, as well as friction between the precipitates and the matrix. The amount of contact between the precipitates and the matrix can be calculated as the surface area of the precipitates. In this case, when the number density of precipitates increases, the amount of surface area between the precipitates and matrix will increase, resulting in an increase in contact points. This increase in contact will result in an increase in α.

The results for α and β for the 9% Cr ferritic steel specimens from Fahse et al. (2020) and Marino et al. (2016) have been replotted in Fig. 7 to allow for an easier comparison of α and β for the reader. The changes in the microstructure of 9% Cr ferritic steel from heat treatment are more complicated than the other four materials presented above because there are multiple microstructural mechanisms, such as decrease in dislocation density and growth in precipitates. There is initially a rapid decrease in the dislocation density, which then begins to level off as heat treatment time increases. As the dislocation density begins to level off, the precipitates begin to nucleate and grow as the heat treatment times continue to increase. Marino et al. (2016) and Fahse et al. (2020) suggest that the change in nonlinearity in the material can be broken into two phases: the first phase where the decrease in β and α is dominated by the decrease in dislocation density and precipitate nucleation; and the second phase where the increase in β and α is dominated by the growth of the precipitates. These results show that while there are two competing mechanisms, the ultrasound nonlinearity parameters still show a significant response to the changes. These results suggest that there may be a critical precipitate radius where the nonlinear contribution from precipitates dominates the nonlinear contribution from dislocations. While this material is very complicated and still not yet fully understood, these results show that it is possible to make separate, independent ultrasound nonlinearity parameter measurements on a material with different mechanisms in each stage, thus gaining a better understanding of the microstructural changes in the material.

FIG. 7.

(Color online) Comparison of β and α results for 9% Cr ferritic steel specimens (Fahse et al., 2020; Marino et al., 2016).

FIG. 7.

(Color online) Comparison of β and α results for 9% Cr ferritic steel specimens (Fahse et al., 2020; Marino et al., 2016).

Close modal

This paper has shown the relationship between α and β, and three prominent microstructural mechanisms: dislocations, precipitate pinned dislocations, and precipitate growth in and along the grain boundaries, where a summary of these relationships can be seen in Table II. Understanding how these microstructural mechanisms relate to the ultrasound nonlinearity parameters α and β allows for better material damage characterization. Additionally, understanding the response to individual mechanisms builds the foundation for the understanding of more complicated materials in the future.

TABLE II.

Summary of β and α response to microstructural mechanisms.

MaterialMicrostructural Mechanismβα
Response/DescriptionResponse/Description
304 SS Growth of precipitates along grain boundaries ↑ Interaction of carbide with the grain boundary ↑ Contact/friction between carbide and grain boundary 
25% 43% 
316L SS Removal dislocations from cold working ↓ ΔβΛ ↓ Reduced contact from dislocations 
20% 44% 
Fe-1.0% Cu Nucleation of precipitates ↓ Cu atoms to Cu precipitates ↓ Precipitate nucleation 
25% 400% 
Precipitate pinned dislocations ↑ then ↓ Growth of radii then loss of coherency Δβr4 ↑ Growth in contact area between precipitate and matrix 
27% 95% 
17-4 PH Precipitate pinned dislocations ↓ Increase # density ΔβN1/3 ↑ Growth in contact area as # increases 
50% 150% 
9% Cr ferritic steel Dislocation Density decrease ↓ Competing effects ΔβΛ ↓ Reduced contact from dislocations 
30% 55% 
Growth of precipitate radius ↑ Δβr4 ↑ Growth in contact area 
60% 80% 
MaterialMicrostructural Mechanismβα
Response/DescriptionResponse/Description
304 SS Growth of precipitates along grain boundaries ↑ Interaction of carbide with the grain boundary ↑ Contact/friction between carbide and grain boundary 
25% 43% 
316L SS Removal dislocations from cold working ↓ ΔβΛ ↓ Reduced contact from dislocations 
20% 44% 
Fe-1.0% Cu Nucleation of precipitates ↓ Cu atoms to Cu precipitates ↓ Precipitate nucleation 
25% 400% 
Precipitate pinned dislocations ↑ then ↓ Growth of radii then loss of coherency Δβr4 ↑ Growth in contact area between precipitate and matrix 
27% 95% 
17-4 PH Precipitate pinned dislocations ↓ Increase # density ΔβN1/3 ↑ Growth in contact area as # increases 
50% 150% 
9% Cr ferritic steel Dislocation Density decrease ↓ Competing effects ΔβΛ ↓ Reduced contact from dislocations 
30% 55% 
Growth of precipitate radius ↑ Δβr4 ↑ Growth in contact area 
60% 80% 

These results in Table II show that depending on the specific microstructural mechanism, α and β may respond similarly or differently in response to these changes. This lends itself to the conclusion that the physics behind their response is different. α is sensitive to energy loss mechanisms, such as friction and contact, and seems to be an additive response, while β is sensitive to specific interactions between microstructural components.

It is of interest to note that in all the results, the percent change in α is generally much higher than the change in β. This lends itself to the conjecture that α is more sensitive than β to the changes in the microstructures considered in this paper. However, other possibilities, such as ease of measurement technique, may lend a hand in the exact sensitivities. Leading with the notion that α is more sensitive than β, the question remains: why not always make NRUS measurements instead of SHG measurements? The answer to this is that while NRUS may provide important information about the material, it is not a practical technique to use in situ and on complicated geometries. In contrast, SHG, especially Rayleigh waves, has the potential for easy field measurements where it is not practical to assume a uniform geometry. Finally, these results demonstrate that it is possible to make two independent, but complementary, measures of ultrasound nonlinearity parameters in metal components. Having two different measures (albeit with similar physics) that are each sensitive to microstructural changes with different responses should help in the development of a procedure to determine the specific damage state in a component.

This research investigates the sensitivity of the ultrasound nonlinearity parameters, α and β, to three prominent microstructural mechanisms: dislocations, precipitate pinned dislocations, and precipitate growth in and along the boundaries of grains. Understanding the sensitivity and relationship of β and α to each of these individual mechanisms provides a critical foundation for considering more complicated material microstructures in the future. One important conclusion of this research is an integrated approach that provides a more fundamental understanding of the versatility of these NLU techniques. These results show that the response of α and β may differ, depending on the microstructural mechanisms, suggesting that the physics behind their response are different. α is sensitive to energy loss mechanisms, such as friction and contact, while β is sensitive to specific interactions between microstructural components. The higher percent change in α compared to the percent change in β leads to the conjecture that α is more sensitive than β to the changes in the microstructures considered in this paper. However, other possibilities, such as ease of measurement technique, may lend a hand in the exact sensitivities. The understanding of these parameters, α and β, which are each sensitive to microstructural changes in independent fashions is an important step towards quantitatively tracking the material damage state in a component.

This research was partially funded through Fellowships for KML from the Nuclear Energy University Program (NEUP) and American Society for Nondestructive Testing (ASNT).

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