# Linear Versus Nonlinear Acoustic Probing of Plasticity in Metals: A Quantitative Assessment

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RUS | Resonant ultrasound spectroscopy |

TEM | Transmission electron microscopy |

XRD | X-ray diffraction |

SHG | Second harmonic generation |

NRUS | Nonlinear resonant ultrasound spectroscopy |

## References

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**Figure 1.**Schematic illustration of samples. For each thermo-mechanical treatment and for both groups, three pieces were cut for the application of acoustic methods and XRD. The longest dimension corresponds to the original’s bar axis and the cold-rolling direction. Consequently, the XRD samples are named longitudinal and transversal.

**Figure 2.**Schematic illustration of the experimental setup used for both RUS and NRUS. The sample is positioned between a contact ultrasonic transducer and a high frequency response pressure sensor. The set of springs and the air bearing ensure that the contact force applied to the sample is very small, which enables to compare the measured resonant frequencies with those of a free-stress parallelepiped. For resonances below 102.4 kHz a spectrum analyzer is used for the frequency sweep. Above this frequency limit, this apparatus is replaced by a National Instruments digital-to-analog acquisition card, model PCI-6251.

**Figure 3.**Schematic illustration of the experimental setup used for SHG. A sinusoidal voltage waveform is amplified and used to generate an ultrasonic signal, emited by one of the transducers. The second one receives the transmitted signal and its FFT spectrum is computed by an oscilloscope. Both the fundamental and first harmonic amplitudes are recorded on a personal desktop computer.

**Figure 4.**Example of (

**a**) Aluminum and (

**b**) Copper XRD pattern. (

**a**) Five peaks are observed for Al, corresponding to different lattice planes: (111) ($2\theta ={38.55}^{\circ}$), (200) ($2\theta ={44.81}^{\circ}$), (220) ($2\theta ={65.21}^{\circ}$), (311) ($2\theta ={78.35}^{\circ}$) and (400) ($2\theta ={99.22}^{\circ}$). Inset: A distribution of crystallite sizes (Al${}_{i},i=1,2,3,4$) contribute to the (200) diffraction peak (shown) as well as to the others (not shown). (

**b**) For Cu samples, five peaks are observed in the angular range measured, corresponding to the following lattice planes: (111) ($2\theta ={43.37}^{\circ}$), (200) ($2\theta ={5051}^{\circ}$), (220) ($2\theta ={74.2}^{\circ}$), (311) ($2\theta ={90.01}^{\circ}$) and (222) ($2\theta ={95.23}^{\circ}$).

**Figure 5.**Normalized variations of nonlinear acoustic parameters ${\alpha}^{\prime}$ and ${\beta}^{\prime}$ of each sample respect to the purely rolled one as functions of the variations of dislocation density, obtained with the linear measurements. For both groups Al and Cu, $\Delta {\Lambda}_{RUS}$ are similar, which are obtained from changes in the transverse elastic wave speed ${v}_{T}$, which are of the order of a few percent. (

**a**) For the Al group, ${\alpha}^{\prime}$ shows changes of $39\%$ to $125\%$ and ${\beta}^{\prime}$ of $14\%$ to $20\%$; (

**b**) For the Cu group, ${\alpha}^{\prime}$ shows changes of $320\%$ to $510\%$, and ${\beta}^{\prime}$ of $19\%$ to $62\%$.

**Table 1.**Aluminum and Copper group characteristics: rectangular parallelepiped dimensions, mass density and treatments for the four samples of each group. Columns are ordered for decreasing expected dislocation density.

Aluminum | ||||

Parameter | Al Roll | Al Roll-A15 | Al Roll-A30 | Al Roll-A60 |

${d}_{1}$ (cm) | $0.501\pm 0.001$ | $0.503\pm 0.001$ | $0.499\pm 0.001$ | $0.497\pm 0.001$ |

${d}_{2}$ (cm) | $1.702\pm 0.001$ | $1.704\pm 0.001$ | $1.702\pm 0.001$ | $1.709\pm 0.001$ |

${d}_{3}$ (cm) | $5.002\pm 0.001$ | $5.004\pm 0.001$ | $5.004\pm 0.001$ | $5.008\pm 0.001$ |

$\rho $ (g/cm${}^{3}$) | $2.670\pm 0.006$ | $2.661\pm 0.006$ | $2.672\pm 0.006$ | $2.665\pm 0.006$ |

Treatments | ||||

Rolled | $82.8\%$ | $82.8\%$ | $82.8\%$ | $82.8\%$ |

Annealed | - | $450{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C $\times \phantom{\rule{3.33333pt}{0ex}}15$ min | $450{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C $\times \phantom{\rule{3.33333pt}{0ex}}30$ min | $450{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C $\times \phantom{\rule{3.33333pt}{0ex}}60$ min |

Copper | ||||

Parameter | Cu Roll 2 | Cu Roll-A15 | Cu Roll-A30 | Cu Roll-A60 |

${d}_{1}$ (cm) | $0.401\pm 0.001$ | $0.401\pm 0.001$ | $0.401\pm 0.001$ | $0.392\pm 0.001$ |

${d}_{2}$ (cm) | $1.700\pm 0.001$ | $1.700\pm 0.001$ | $1.702\pm 0.001$ | $1.701\pm 0.001$ |

${d}_{3}$ (cm) | $5.006\pm 0.001$ | $4.999\pm 0.001$ | $5.000\pm 0.001$ | $4.999\pm 0.001$ |

$\rho $ (g/cm${}^{3}$) | $8.882\pm 0.023$ | $8.883\pm 0.023$ | $8.901\pm 0.023$ | $8.898\pm 0.023$ |

Treatments | ||||

Rolled | $88.3\%$ | $88.3\%$ | $88.3\%$ | $88.3\%$ |

Annealed | - | $850{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C $\times \phantom{\rule{3.33333pt}{0ex}}15$ min | $850{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C $\times \phantom{\rule{3.33333pt}{0ex}}30$ min | $850{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C $\times \phantom{\rule{3.33333pt}{0ex}}60$ min |

**Table 2.**Acoustic parameters, both linear and nonlinear, obtained for each group of samples compared and contrasted. Nonlinear parameters ${\alpha}^{\prime}$ and ${\beta}^{\prime}$ exhibit a considerably higher change from sample to sample than the linear parameter ${v}_{T}$. Errors are obtained by standard deviation of ten measurements with each method. See text for symbol definition.

Aluminum | |||

Treatment | ${\mathit{v}}_{\mathit{T}}$(m/s) | $\frac{{\mathit{\alpha}}^{\mathbf{\prime}}}{{\mathbf{10}}^{\mathbf{-}\mathbf{4}}}$(V${}^{\mathbf{-}\mathbf{1}}$) | ${\mathit{\beta}}^{\mathbf{\prime}}$(V${}^{\mathbf{-}\mathbf{1}}$) |

Roll A60 | $3116\pm 4$ | $-39\pm 8$ | $0.42\pm 0.02$ |

Roll A30 | $3130\pm 7$ | $-44\pm 7$ | $0.39\pm 0.02$ |

Roll A15 | $3146\pm 4$ | $-63\pm 5$ | $0.39\pm 0.02$ |

Roll | $3065\pm 4$ | $-28\pm 5$ | $0.49\pm 0.01$ |

Copper | |||

Treatment | ${\mathit{v}}_{\mathit{T}}$(m/s) | $\frac{{\mathit{\alpha}}^{\mathbf{\prime}}}{{\mathbf{10}}^{\mathbf{-}\mathbf{4}}}$(V${}^{\mathbf{-}\mathbf{1}}$) | ${\mathit{\beta}}^{\mathbf{\prime}}$(V${}^{\mathbf{-}\mathbf{1}}$) |

Roll A60 | $2294\pm 6$ | $-168\pm 21$ | $0.90\pm 0.10$ |

Roll A30 | $2304\pm 4$ | $-244\pm 31$ | $0.35\pm 0.01$ |

Roll A15 | $2326\pm 3$ | $-176\pm 18$ | $0.42\pm 0.01$ |

Roll | $2229\pm 4$ | $-40\pm 10$ | $1.11\pm 0.03$ |

**Table 3.**Comparison of XRD and RUS measurements of relative dislocation density for the Al and Cu samples. Errors for XRD measurements are calculated with the contribution of the Rietveld refinement results and the statistical error from the repetition of the experiment in two pieces of the same sample (longitudinal and transversal). These errors are large and preclude a sample-to-sample comparison. By contrast, the errors associated with the acoustic measurements are sufficiently small that a quantitative comparison can be confidently provided.

Aluminum | ||

Compared Samples | $\frac{\mathbf{\Delta}{\mathbf{\Lambda}}_{\mathit{XRD}}}{{\mathbf{10}}^{\mathbf{7}}}$(mm${}^{\mathbf{-}\mathbf{2}}$) | $\frac{\mathbf{\Delta}{\mathbf{\Lambda}}_{\mathit{RUS}}}{{\mathbf{10}}^{\mathbf{7}}}$(mm${}^{\mathbf{-}\mathbf{2}}$) |

Roll & Roll A60 | $1.24\pm 1.47$ | $4.47\pm 0.70$ |

Roll & Roll A30 | $0.87\pm 1.35$ | $5.68\pm 0.96$ |

Roll & Roll A15 | $0.42\pm 7.12$ | $7.07\pm 0.69$ |

Copper | ||

Compared Samples | $\frac{\mathbf{\Delta}{\mathbf{\Lambda}}_{\mathit{XRD}}}{{\mathbf{10}}^{\mathbf{7}}}$(mm${}^{\mathbf{-}\mathbf{2}}$) | $\frac{\mathbf{\Delta}{\mathbf{\Lambda}}_{\mathit{RUS}}}{{\mathbf{10}}^{\mathbf{7}}}$(mm${}^{\mathbf{-}\mathbf{2}}$) |

Roll & Roll A60 | $2.34\pm 21.74$ | $3.31\pm 0.51$ |

Roll & Roll A30 | $4.73\pm 19.35$ | $3.81\pm 0.41$ |

Roll & Roll A15 | $5.04\pm 19.0$ | $4.90\pm 0.35$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Espinoza, C.; Feliú, D.; Aguilar, C.; Espinoza-González, R.; Lund, F.; Salinas, V.; Mujica, N.
Linear Versus Nonlinear Acoustic Probing of Plasticity in Metals: A Quantitative Assessment. *Materials* **2018**, *11*, 2217.
https://doi.org/10.3390/ma11112217

**AMA Style**

Espinoza C, Feliú D, Aguilar C, Espinoza-González R, Lund F, Salinas V, Mujica N.
Linear Versus Nonlinear Acoustic Probing of Plasticity in Metals: A Quantitative Assessment. *Materials*. 2018; 11(11):2217.
https://doi.org/10.3390/ma11112217

**Chicago/Turabian Style**

Espinoza, Carolina, Daniel Feliú, Claudio Aguilar, Rodrigo Espinoza-González, Fernando Lund, Vicente Salinas, and Nicolás Mujica.
2018. "Linear Versus Nonlinear Acoustic Probing of Plasticity in Metals: A Quantitative Assessment" *Materials* 11, no. 11: 2217.
https://doi.org/10.3390/ma11112217