Attenuation of ultrasound in severely plastically deformed nickel

Abstract

Ultrasound attenuation was measured in nickel specimens of about 30 mm diameter prepared using the high pressure torsion technique. The cold working process produced an equivalent shear strain increasing from zero at the center up to 1000% at the edge of the specimen. The fragmentation of the grains due to multiple dislocations led to an ultrafine microstructure with large angle grain boundaries. The mean value of the grain size distribution gradually decreased from ∼50 μm at the center to 0.2 μm at the edge. Laser pulses of 5 ns were employed for the excitation of broadband ultrasound pulses covering the spectral range of 0.1–150 MHz. The ultrasound pulses were measured from the opposite side of the specimen by means of an optical interferometer and a piezoelectric foil transducer in two experimental setups. The features of the detected signal forms are discussed. The absolute value of the attenuation decreases from the center to the edge of the specimen showing nearly linear frequency dependence. The variation of the phase velocity was measured in a 6 mm-thick high pressure torsion nickel sample, revealing a velocity increase from the center to the edge.

1. Introduction

The mechanical properties of polycrystalline materials strongly depend on their microstructure. High pressure torsion (HPT) is a relatively new technique employed for severe plastic deformation of metals and composites [1]. It produces huge equivalent strain ε_v, which is proportional to the number of rotations n exerted on a disk-like sample of a thickness d under high hydrostatic pressure. The equivalent strain is calculated as ${\epsilon }_v=2\pi nr/(\sqrt{3}d)$, where r is the distance from the center of the disk [2]. The shear strain increases drastically the dislocation density according to the dislocation multiplication mechanism and induces a grain fragmentation increasing gradually with radius. The reduction in the grain size depends on the material composition and approaches few hundreds of nanometers for metals, after which a saturation of refinement is reached [3]. The refinement of the grain structure increases the yield strength of the materials often without loss of ductility, which is very promising for applications. The nondestructive evaluation of the mechanical properties of such materials is an important task. Different techniques of ultrasound excitation and detection are employed in material sciences for the correlation of the changes of elastic properties and attenuation with microstructure [4–9]. Laser induced ultrasound provides essentially broadband probe pulses from few nanosecond down to tens of picoseconds, allowing a diversity of applications in material science and nondestructive methods [10–14]. The achievable level of elastic strain may involve nonlinear elastic deformation and can be used as the noncontact method for the material strength estimation [15–17]. This study reports on the broadband measurements of the frequency-dependent attenuation of the time-resolved laser induced ultrasound pulses in an HPT nickel sample.

2. HPT nickel sample

The considered HPT-specimen of nickel has a diameter of 33 mm and a thickness of 2.5 mm. The equivalent strain of our specimens linearly increases from zero at the geometrical center up to thousand percent (ε_v=10) near the circumference. Typical electron back scatter diffraction micrographs taken at zones with defined strain are presented in Fig. 1. The center of the specimen does not demonstrate significant changes of the microstructure and can be considered as polycrystalline material. At the beginning of severe plastic deformation (ε_v=1, about 1.5 mm from the center for our specimen) the microstructure consists of big grains with large disorientations in their interior. Due to the plastic deformation the dislocation density rises immediately. The dislocations do not arrange randomly. By contrast, they form dislocation cells with very small disorientations. The proceeding deformation increases the disorientations between neighboring cells leading to the occurrence of high angle grain boundaries, more than 15°. At this point new smaller grains appear in the initially coarse structure [1]. The further increase in the shear strain, ε_v>2, induces fragmentation of the small grains that were just developed before. This fragmentation process scales the grain size of HPT deformed nickel down to the hundred nanometers region. The saturation of the grain fragmentation process happens at a strain ε_v of about 8, approximately 12 mm from the center. Grains are elongated in tangential direction with a typical length of 200–300 nm and have a lateral size of 150–200 nm in radial direction. The saturation is given by the balance of the restoration and grain fragmentation processes. Generally, the HPT technique produces an ultrafine microstructure with a typical ultimate grain size of more than 100 nm. This microstructure is intermediate between nanocrystals and polycrystals. A high chemical purity and multiplied dislocation density distinguish the HPT method from others approaching nanocrystalline and ultrafine microstructure.

Fig. 1.

Electron back scattering diffraction micrographs of the HPT deformed nickel specimen at spots with equivalent strain ε_v. Small angle disorientation within the interior of the grains is observed for the coarse grain structure and a large angle grain boundary is typical for the ultrafine microstructure. The micrographs are obtained for the axial direction, the radial axis is horizontal and the tangential axis is vertical.

3. Attenuation of ultrasound pulses

An estimation of the scattering constants was carried out on the base of the unified scattering theory [18] and can be found in Ref. [19] for nickel. This estimation showed that the contribution of dislocation damping to attenuation is the most important effect for the cold working metal and that the phenomenological Granato–Lücke model can be employed [20]. A further development of the theory includes averaging of the dislocation orientation and Burgers vectors but gives similar results [21]. Practically the most important result is that the decrease in the dislocation loop length shifts the maximum of the ultrasonic attenuation to higher frequencies. The variation in the phase velocity depends on the dislocation drag force, which in general is specific for the material and dislocations arrangement. It is noteworthy that there is no comprehensive theory about dislocation dynamics in polycrystals. The said models used limitations of the stress amplitude of elastic waves, as well as low dislocation density, and therefore the models neglect the possible interaction of the dislocations as well as the formation of dislocations as consequence of the interaction.

4. Experimental

The surfaces of HPT nickel samples were mirror-like polished and the planes were parallel within 0.01 mm/cm over the entire specimen. The specimen was mounted onto the side wall window of a water filled bath. A sketch of the experimental setup is presented in Fig. 2. The short transient disturbance is one of the main advantages of the laser ultrasound technique for the evaluation of mechanical properties of materials [13,22]. A pumping pulse of a Nd:YAG laser of 5 ns duration, 1064 nm wavelength and with a pulse energy of<20 mJ illuminated the water–metal interface. To increase the efficiency of ultrasound generation the surface of the sample was covered by a<20 μm-thick strongly absorbing layer of black acrylic paint, leading to absorption of most of the laser radiation. The power density on the sample surface was below 15 MW/cm² for a diameter of the pumping laser spot of about 6 mm, therefore a maximum pressure pulse magnitude of 15 MPa can be expected [23]. The transient form of the induced pressure pulse is a replica of the envelope of the laser pulse intensity [24]. Thus, a broadband longitudinal pulse with a plane wave front and a pronounced compression phase is launched into the sample.

Fig. 2.

Sketch of the experimental setup that employs an actively stabilized Michelson interferometer.

The optical detection scheme is based on a Michelson interferometer with active stabilization that keeps the operating point at maximum sensitivity for a sufficiently long time for signal averaging [25]. Optical noncontact methods detect frequencies above 100 MHz and possess high lateral resolution on the order of the size of the probe beam spot [10,22]. In the scanning experiment the size of the probe beam spot was estimated as $\delta x=4{\lambda }_{p}F/\pi d_b\approx 13\mathrm{\mu }m$, where λ_p is the wavelength of the probe He–Ne laser, F is the focal length of the focusing lens and d_b is the diameter of the beam. The vertical particle displacement at the surface induces transient changes in the interferometer output beam intensity, which is detected by a photodiode (New Focus) with a bandwidth of 0.1–125 MHz. The power of the probe laser beam of 5 mW and the noise of the detector give a sensitivity of ∼0.4 nm to the setup [10].

A series of transient wave signals obtained at different radii from the center of the HPT-sample are presented in Fig. 3. The signals were 256 times averaged at each detection point to reduce the noise of electronics. The primary longitudinal pulse is detected on the surface opposite to the excitation spot at about 0.46 μs after the pumping laser pulse and produces the steeply ascending front of the signal. The ultrasound pulse is completely reflected at the borders due to the large difference of acoustical impedance of air and nickel. The next reflection of the probe pulse from the nickel–water interface reduces the pressure amplitude by less than 1 dB. The first reverberation arrives at the detection spot at time ∼1.34 μs and induces a second rising front in the measured signal. The following reverberation arrives with a time interval equal to a delay of double sample thickness, giving the calculated value of the longitudinal velocity as 5.63±0.02 km/s.

Fig. 3.

Transient signals measured at different radial distances of the HPT nickel specimen with different equivalent strain ε_v (a). Time derivatives combining with a low pass filter evidently show decreasing attenuation and the scattered ultrasound from the center to the edge of the specimen (b). The lines are presented with vertical offset.

The Michelson interferometer is sensitive to the vertical surface displacement. Scattering reduces the coherent part of the longitudinal beam mainly due to conversion to shear waves. The noise between the rising fronts of the signals combines converted pulses of the shear and even surface acoustic waves with longitudinal waves coming by complex trajectories. Under the assumption of omni-directional scattering the primary longitudinal pulse contains a negligibly small part of the forward scattered ultrasound or structure noise. After the reflection of the primary pulse deviated waves and the backscattered part of ultrasound approach the detection spot. The mean displacement after the first rising slope has a very similar form independent of microstructure whereas the signal obtained in the coarse grain region near the center shows additionally small amplitude pulses that gradually decay with radial distance (see Fig. 3a). The time derivatives of the measured signal yield the pressure time dependences that show strong scattering and attenuation typical for the coarse structure at the center, see Fig. 3b). The scattered field is reduced with the refinement of the microstructure.

The discussed dependences were demonstrated by Scruby et al. [26]. There it was shown that the elastic disturbance relaxes in 8 mm steel specimens and the displacement induced by the reverberations starts nearly from the base line; the specimen with coarse grains showed stronger noise on a very similar signal. The piezoelectric transducers measure the averaged coherent part of the ultrasound pulse over the volume of the specimen while the optical method employed by us gives additionally locally varying scattering signals that can be used for the estimation of the structural noise and measuring of the backscattered ultrasound. The influences of the experimental arrangement and the backscattered structure noise on the attenuation measurements in pulse echo experiments were discussed in Ref. [4]. Generally, the interference of scattered ultrasound with the coherent part yields an error of ultrasound attenuation.

During propagation in the sample the pulse is subjected to diffraction losses, leading to a lowest measured frequency of ≈5 MHz. This can be determined from the dimensionless factor S=λx/r², where λ is the wavelength, x is the traveling distance and r is the radius of the pumping laser spot. The influence of the diffraction is negligible if S<1. Due to diffraction the initially induced pulse with a pronounced compression phase shows an additional expansion phase in the far field (see for instance Ref. [13]). Thus, spectra of the reverberations become narrower from the low and high frequency sides due to the simultaneous influence of diffraction and attenuation, respectively. Consequently, for short transient pulses a thinner sample allows estimation of the attenuation in a broader frequency range.

5. Procedure of the signal treatment

The rise time of the front of the signal was used to demonstrate the influence of the grain size on the scattering of ultrasound [26]. It is seen that the rise time is shorter if the grain structure is finer and longer for the coarse grains, which is more pronounced for the third detected fronts, as it is seen in Fig. 3a. The reduction of the steepness of the front is related to the decay of the high frequencies of the spectrum. A method for quantitative estimation of the attenuation in frequency domain from the signal fronts is suggested here. It uses the decay of the coherent part of the probe pulse spectrum contained in the rising front after filtering out the noise of the structure.

The details of the procedure are presented in Fig. 4. The signal detected at a distance of 5 mm from the center is taken as an example. It is assumed that only the fronts of the signal contain the coherent part of the induced longitudinal pulse. The separated front is cut out from the signal and continued by horizontal lines on either side, providing a step-like function. A base line is defined either at the mid value or at the starting value of the front as it is indicated by the arrows in Fig. 4a. A time window of the form exp(−t²/t_w²) is multiplied with the step-like function, where t_w=100 ns is determined by the lowest detectable frequency of 5 MHz, which gives a width of 200 ns at 1/e level. The resulting signals for the base line at the bottom or at the mid of the fronts are presented in Fig. 4b and 4c, respectively. A weak point of the procedure is the extraction of the signal front in presence of the strong scattering field, since the second front may contain displacements due to the non-coherent field.

Fig. 4.

Measured signal (dashed line), the fronts of the primary pulse and first reverberation (thick lines) and the prolongation of the fronts, which build the step-like function (thin lines). Arrows indicate the level of the base lines for further calculations presented in (b) and (c). Inset shows the form of the time filtering window. Two polar signals are obtained for the case of the base line at mid level (b) and uni-polar signals for the bottom shift of the base line (c).

The ratio of the Fourier spectra of the treated signals was used for the estimation of the ultrasound attenuation in dB/cm according to the formula:

$$\alpha (f)=20\log (R{U}_p(f)/U_1(f))/2d$$ (1)

where U_p(f) and U₁(f) are the spectral amplitudes of the primary pulse and its first reverberation, respectively, R is a reflection coefficient that compensates for the decrease of the first reverberation due to the reflection at the metal–water interface and 2d is twice the thickness. Due to the employed procedure of front extraction the separate signals give smooth spectra with a deficiency at low frequencies in the case of bipolar signals. The difference of the ultrasound attenuation estimation is less than 0.5 dB for the frequency range <8 MHz and is negligible for higher frequencies since the applied filters are canceled in frequency domain. A diffraction correction procedure was employed according to Ref. [27], decreasing the attenuation in the frequency range <10 MHz.

Additional measurements were done by an immersion technique using a home-made piezoelectric foil transducer. A polarized polyvinylidenefluoride (PVDF) foil of 25 μm thickness and 2 mm diameter was used in the construction of the transducer, allowing a broadband detection in the frequency range from 5 up to 90 MHz [19]. The sensitive area of the transducer is small enough for a spatially resolved scanning experiment. The attenuation was measured at different radii of the specimen. The absolute value of the attenuation decreased from the center to the edge. Measurements with an HPT nickel sample of ∼30 mm diameter and 6.00±0.02 mm thickness with an equivalent strain at the edge of 5000% were carried out to estimate the variation of the phase velocity due to the changes of the microstructure. An equivalent strain of 1000% and the saturation of the grain structure refinement were achieved at a radius of ∼3 mm in contrast to the 2.5 mm-thick HPT nickel sample, where this was achieved at the edge. Therefore, the measurements were carried out at the center; 4 mm from the center and 8 mm from the center. The measured signals are presented in Fig. 5. The primary pulse has bipolar shape since the piezoelectric transducer functioned in the current detection mode and measured the derivative of the pressure pulse. Fig. 5a shows the signal measured at the center where the first reverberation has a three times smaller peak-to-peak amplitude in comparison with the primary pulse. The peak-to-peak signal amplitudes of the primary pulses and the first reverberations are larger with steeper slopes for the detection points at distances of 4 and 8 mm from the center in Fig. 5b.

Fig. 5.

Signals detected by PVDF foil transducer in 6 mm-thick HPT nickel with equivalent strain of 5000% at the edge. (a) Signal at the center of the specimen and (b) signals at a distance of 4 mm from the center (dashed line) and at a distance of 8 mm from the center (solid line).

It is noteworthy that the signal measured by the piezoelectric PVDF foil detector is proportional to the derivative of the pressure pulse in comparison to the used optical detection scheme where the signal is proportional to the particle displacement or integral of the pressure. The laser induced particle displacement can be numerically calculated for the considered arrangement [10]. The second derivative of the optically measured signal yields a transient form similar to the signal detected by the piezoelectric transducer. The noise observed by the optical method between the reverberations of the pulse in Fig. 3a due the scattering in the coarse microstructure is transformed to small ripples of the signal detected by the piezoelectric transducer (Fig. 5a).

6. Results

The estimations of the phase velocity were carried out for 6 mm-thick HPT nickel using the time interval between zero crossing points (see Fig. 5). The lowest value of phase velocity, 5.63±0.02 km/s was obtained for the center. The velocity increases to 5.67±0.02 and 5.69±0.02 km/s at 4 and 8 mm from the center, respectively. The corresponding time-delay between the arrivals of the first reverberations can be seen in Fig. 5b. The measured velocity of polycrystalline nickel is 5.82±0.02 km/s.

Fig. 6 shows the ultrasound attenuation estimated from signals obtained by both methods at the center and at the edge of the HPT nickel specimen. The results show a decrease in the attenuation from the center to the edge of the specimen. The attenuation obtained from PVDF foil measurement at the center of the specimen has a higher value within the range from 10 to 50 MHz in comparison with a mean value obtained by the optical method. The variation of the microstructure from the coarse microstructure at the center to ultrafine grain structure is significant, leading to the nonmonotonic decrease of attenuation with distance. Approaching the ultrafine microstructure the results obtained by both methods are in good agreement. The standard deviation increases with the frequency since the spectral amplitude decreases.

Fig. 6.

Attenuation spectra of ultrasound obtained at different points of the HPT nickel sample by two methods. The mean value of the attenuation (dash line) and standard deviation calculated for 15 points at the center of HPT nickel within the region limited by the radius 1.5 mm. The mean value of the attenuation (solid line) and calculated standard deviation obtained for 20 points at a distance>9 mm. The results obtained by PVDF transducer include the measurement in single point at the center of the specimen (circles) and the mean value (triangles) with standard deviation calculated at 5 different points at a distance>9 mm.

The dynamic range can be estimated according to the formula $20\log (2^{N_R}\sqrt{N})$, where N_R is the number of effective bits equal to 7 for an oscilloscope with 8 bit vertical resolution and N is the number of the averaged signals, which is 255 in our experiments. The dynamic range of the signal spectral amplitude measured by the PVDF foil transducer shows a range of more than 60 dB, whereas optical detection is limited to 40 dB. The signal to noise ratio of optical detector is ∼4, which is about five times lower in comparison with PVDF detector. The calculated spectrum has an error of at least 1 dB for both methods. The ratio of the spectra used in the calculation of the attenuation doubles this value thus the instrumental error is 2 dB/cm.

7. Discussion

The presented optoacoustic method of probe pulse excitation possesses several advantages for the measurement of acoustic attenuation and sound velocity. The transient pressure pulse has a pronounced compression phase. The spectrum is essentially broadband and covers the range from 0.1 up to 150 MHz. The lower frequency border is defined by the size of the pumping laser spot while the upper border depends on the laser pulse duration. This provides better temporal resolution for the measurements of the time of flight and a broader bandwidth for the estimation of the attenuation in comparison with conventional ultrasound methods.

Regarding the measurements of attenuation the known models do not require specific detection methods. However, there are some features of the optical and piezoelectric detection methods considered in this study. The optical methods possess the potential to operate in a much higher frequency range but suffer from low sensitivity. Due to the higher temporal and spatial resolution the contribution of the locally scattered ultrasound can be compared with the coherent part of the pulse. In comparison, piezoelectric transducers have a larger size of the detection area and therefore their sensitivity is higher but the contribution of the locally scattered field is diminished due to the sign variation. The detected amplitude of the scattered ultrasound induces a larger error for optical methods. Despite the noncontact advantage of the optical methods the results of their application for polycrystalline materials show limited dynamic range and frequency bandwidth, mainly due to the scattering of the probe pulse. With the exception of the results obtained for aluminum, which is a good example for a weakly scattering polycrystal, the studies considering attenuation in polycrystals reported measurements in the range below 50 MHz [8,9,28]. Nevertheless, the results obtained by both methods show that the attenuation decreases from the center to the edge of the specimen.

Estimation of the scattering contribution showed that attenuation of our specimen is mainly due to dislocation damping [19]. The cold working process is responsible for the dislocation multiplication as well as for the changes of the dislocation loop distribution. The fragmentation of the grains starts at a distance of few millimeters from the center of the HPT nickel, reducing the number of dislocations as well as the length of dislocation loops. The dislocations are pinned by grain boundaries and the absolute attenuation becomes lower as it is seen from our experimental results. The recent revision of the Granato–Lücke model reveals quantitative difference due to the averaging of the stress field with random distribution of Burger's vector and dislocation lines [21]. The HPT specimens have a dislocation density of ∼10¹⁶ m⁻², exceeding the range of the used approach, which assumes a small value of the product ΛL², where Λ is the dislocation density and L is dislocation loop length. Experimental works considered mainly the low frequency range, whereas in the frequency range of about 100 MHz attenuation should reach a maximum value for polycrystals. According to the calculations, for an exponential distribution of loop lengths attenuation linearly increases within a broad frequency range around the maximum [20]. The measured almost linear frequency dependence of attenuation can be also due to the nonlinear interaction of dislocations.

8. Conclusion

Two different detection methods were employed in our experiments. The combination of laser induced ultrasound and a noncontact optical detection method possesses the potential of a broad operating bandwidth but the signal to noise ratio is five times lower in comparison with piezoelectric detector. The method suffers from high sensitivity to the locally scattered field, causing a higher deviation of the estimated attenuation in coarse microstructure. The combination of laser excitation with the immersion technique extends the dynamic range of the measurements to more than 60 dB and approaches a frequency of 90 MHz showing very good possibilities for nondestructive applications and material properties evaluation.

The obtained results showing variation of phase velocity and attenuation in HPT nickel are in general consistent with a model developed by Granano and Lücke considering an elastic motion of dislocation. The measured attenuation dependences show that nonlinear effects should be observable both in highly dense dislocation nickel and in the nickel with ultrafine microstructures but the discussion of these results is beyond of the presented paper topic.