# Nonlinear acoustic characterization of heterogeneous plasticity in bent
aluminium samples

Carolina Espinoza<EMAIL_ADDRESS>Departamento de Sonido,
Facultad de Artes, Universidad de Chile, Compañía de Jesús 1264, Santiago,
Chile.  Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile, Avenida Blanco Encalada 2008, Santiago, Chile.  Vicente
Salinas<EMAIL_ADDRESS>Grupo de Investigación en Física
Aplicada, Instituto de Ciencias Químicas Aplicadas, Facultad de Ingeniería,
Universidad Autónoma de Chile, Avenida Pedro de Valdivia 641, Santiago, Chile.
Makarena Osorio<EMAIL_ADDRESS>Departamento de Física, Facultad
de Ciencias Físicas y Matemáticas, Universidad de Chile, Avenida Blanco
Encalada 2008, Santiago, Chile.  Edgar Pío<EMAIL_ADDRESS>Departamento de Ingeniería Metalúrgica y Materiales, Universidad Técnica
Federico Santa Maria, Av. España 1680, Valparaíso, Chile.  Claudio Aguilar
<EMAIL_ADDRESS>Departamento de Ingeniería Metalúrgica y Materiales,
Universidad Técnica Federico Santa Maria, Av. España 1680, Valparaíso, Chile.
Fernando Lund<EMAIL_ADDRESS>Departamento de Física, Facultad de
Ciencias Físicas y Matemáticas, Universidad de Chile, Avenida Blanco Encalada
2008, Santiago, Chile.  Nicolás Mujica<EMAIL_ADDRESS>Departamento de
Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,
Avenida Blanco Encalada 2008, Santiago, Chile.

###### Abstract

Knowledge of the state of plastic deformation in metallic structures is vital
to prevent failure. This is why non-destructive acoustic tests based on the
measurement of first order elastic constants have been developed and
intensively used. However, plastic deformations, which are usually
heterogeneous in space, may be invisible to these methods if the variation of
the elastic constants is too small. In recent years, digital image correlation
techniques, based on measurements carried out at the surface of a sample, have
been successfully used in conjunction with finite element modeling to gain
information about plastic deformation in the sample interior. Acoustic waves
can penetrate deep into a sample and offer the possibility of probing into the
bulk of a plastically deformed material. Previously, we have demonstrated that
nonlinear acoustic methods are far more sensitive to changes in dislocation
density than linear ones. Here, we show that the nonlinear Second Harmonic
Generation method (SHG) is sensitive enough to detect different zones of von
Mises stress as well as effective plastic strain in centimeter-size aluminium
pieces. This is achieved by way of ultrasonic measurements on a sample that
has undergone a three-point bending test. Because of the relatively low stress
and small deformations, the sample undergoes plastic deformation by
dislocation proliferation. Thus, we conclude that the nonlinear parameter
measured by SHG is also sensitive to dislocation density. Our experimental
results agree with numerical results obtained by Finite Element Method (FEM)
modeling. We also support the acoustic results by X-Ray Diffraction
measurements (XRD). Although intrusive and less accurate, they also agree with
the acoustic measurements and plastic deformations in finite element
simulations.

##  I Introduction

Plastic strain in metals and alloys usually exhibits a non-uniform
distribution. This is true under controlled laboratory conditions due to the
anisotropy of the various mechanisms responsible for plastic deformation, such
as dislocation slip, twinning, and phase transformation [1] and, most
certainly, for pieces in service due to the actual heterogeneous boundary
conditions at play [2]. Recently, the advent of additive manufacturing has
raised awareness of the fact that the fabrication process itself can introduce
heterogeneous microstructures [3, 4].

The behavior of aluminium, of particular interest to the automobile and
aerospace industries, under plastic strain, has been a subject of special
interest. Sachtleber et al. [1] determined the spatial distribution of plastic
surface strains of aluminium polycrystals compressed in a channel die, using
an image analysis method to determine the change in surface patterns of the
sample under consideration. This work spawned a large body of subsequent
research, particularly using the optical correlations obtained at the surface
of a specimen in tandem with finite element modeling to obtain information
about the behavior of the innards of a sample for a variety of materials. The
same principle can be used substituting light by back-scattered electrons, and
falls under the general label of Digital Image Correlation (DIC) [5]. In
addition to aluminium alloys [6] examples include $\alpha$-iron [7], Zr and Ti
alloys [8], stainless steel [9] and Inconel 718 [10]. The field has been
reviewed recently by Weidner and Biermann [11].

While the combination of experimental surface measurements with three
dimensional finite element numerical modeling has met with undoubted success,
the fact remains that it would be desirable to have an experimental tool
capable of penetrating a plastically deformed sample without destroying it,
and capable of exploring space-dependent plasticity. In this work we show that
such a tool is provided by nonlinear ultrasound. Although the length scales
available for exploration compare with wavelength, say in the millimiter to
hundreds of microns range, ultrasonic waves penetrate with small, and, in any
case controlled, attenuation, in metals and alloys [12]. It is thus reasonable
to propose ultrasound as a non-intrusive probe to characterize heterogeneous
plastic deformation.

The mechanism we have chosen to obtain heterogeneous plastic deformation is a
three-point bending test. This generates sufficient heterogeneity in the
sample (as compared for example with a tension test) and is standardized so
our results should be easy to replicate. In order to study these
heterogeneities, bending tests can provide more information about the expected
behavior of a piece in service than, say, uniaxial testing [13, 14]. More
broadly, bending tests are often used to assess the mechanical performance of
aluminium alloy sheets [15, 16, 17, 18].

Acoustic methods have been used in crack detection [19, 20, 21, 22] and
nondestructive evaluation of materials in general [23, 24, 25, 26]. In recent
years, their application to measure dislocation density in metallic materials
has been an active area of research. Advances in theoretical modeling and the
development of instrumentation have allowed acoustic measurements to emerge as
a quantitative tool for measuring dislocation density in the study of the
plastic behavior of metals and alloys [27, 28, 29, 30]. In previous work, we
have demonstrated that, in addition, different plastic deformation mechanisms,
such as dislocation slip and twinning, can be thereby characterized and
identified [31]. A correlation between changes in nonlinear acoustic
parameters and the dislocation density in aluminium and copper samples
subjected to different thermomechanical treatments has been observed [32]. We
also demonstrated that the nonlinear acoustic parameter obtained by Second
Harmonic Generation (SHG) is more sensitive to changes in dislocation density
than the one obtained by Nonlinear Resonance Ultrasound Spectroscopy (NRUS)
[32], and is thus preferable to probe the plastic behavior of metals and
alloys. In this work we show that SHG, in combination with finite element
modeling, allows us to obtain local measurements of plastic deformation within
aluminium specimens subjected to three-point bending tests, obtaining
information about the plastic state of the samples at different spatial
locations.

##  II Experimental and numerical methods

###  II.1 Bending test

It is well known that a bending test will induce larger stresses around the
pushing points, so it is expected that different levels of plastic deformation
will occur in the longitudinal direction of a plastically deformed specimen.
In order to demonstrate the sensitivity of SHG in identifying zones with
different plastic deformations, we decided to apply a bending test to a
metallic sample. The test specimen was fabricated out of commercially 1100
pure aluminium (99.0% pure) with a geometry defined in the standard ASTM
E290-14 for bending test [33] with dimensions $16\times 32\times 200$ mm3. The
as received sample without any mechanical deformation nor thermal treatment
was labeled as Original sample (OS). Before the bending test, the OS was
annealed at $400^{\circ}$C for $125$ hours to remove dislocations introduced
during the manufacturing processes. The annealed specimen before the
deformation was labeled as Annealed sample (AS). Then, a quasi-static three-
point bending test was performed using an Instron 3369 machine, with a maximum
load capacity of $50$ kN. The applied load was $5$ kN with a velocity of
$0.01$ mm/min. During the test, the mandrel was positioned above and at the
center of the specimen, and the bottom supports were at $1.0$ cm from each
edge. The annealed sample after the deformation was labeled as Post Bending
sample (PBS).

###  II.2  Acoustic measurements: speed of shear and longitudinal waves

In this study, the speed of elastic waves were measured with the setup
presented in Fig. 1(a) using the time of flight (TOF) method: an ultrasonic
(US) pulse of carrier frequency $f=3$ MHz that is modulated by a Gaussian
function, with a final width of about 5 oscillations, is transmitted into a
probe, as in [30]. A pair of ultrasonic transducers are used to emit and
receive the ultrasonic signal (Panametrics - V110 for longitudinal waves and
V156 for transverse waves, both resonant at $5$ MHz, with element diameter
$8.8$ mm). The measurements are performed at points along the top-bottom line
as shown in Fig. 1(b). The wave speeds are computed by measuring the US pulse
time of flight through the sample by cross-correlation between the electrical
signal used for US emission and the measured US signal. The distance traveled
by the wave, $l$, is obtained at each ultrasonic measurement point using a
Mitutoyo micrometer screw, with $0.001$ mm precision.

Figure 1: Schematic illustration of the experimental setup for SHG
measurements. (a) For the TOF measurement, a Gaussian modulated US pulse is
generated and amplified. For the SHG measurement, a sinusoidal voltage
waveform is generated and amplified. In both cases, the US signal is emitted
by a contact transducer, and a second one receives the transmitted signal. For
the TOF measurement, both the electrical emission signal and the received US
one are acquired by an oscilloscope and these both signals are transferred to
a computer. For the SHG measurement, the FFT is computed by an oscilloscope
and the first and second harmonic amplitudes are recorded on a computer.
Measurements are performed at four different lines: (b) Top-Bottom Line (TBL),
(c) Top Line (TL), Middle Line (ML) and Bottom Line (BL). The letters in (b)
refer to points where X-ray diffraction measurements were performed.

###  II.3 Acoustic measurements: Second Harmonic Generation

In SHG a second harmonic wave is generated from a propagating monochromatic
elastic wave. This is due to the anharmonicity of the elastic material and the
presence of micro-structural features such as dislocations. The second
harmonic nonlinear response is quantified by the nonlinear parameter [34]

$\beta=\frac{8}{lk^{2}}\frac{A_{2\omega}}{A_{\omega}^{2}},$ (1)

where $k$ is the wave number and $l$ is the elastic wave propagation distance.
$A_{\omega}$ and $A_{2\omega}$ are the absolute physical displacements of the
fundamental and second harmonic waves. The nonlinear parameter $\beta$ has a
linear relationship with dislocation density, to a first approximation [32].
In this study, the SHG method was applied using the setup presented in Fig.
1(a): in this case a continuous ultrasonic longitudinal sine wave of frequency
$f=3$ MHz is transmitted into a probe. Through Fourier analysis of the
received signal, the fundamental ($\bar{A}_{\omega}$) and the second harmonic
($\bar{A}_{2\omega}$) amplitudes, in volts units, are measured. Therefore,
using (1) and $k=2\pi f/v_{L}$ the nonlinear parameter $\beta^{\prime}$ is
obtained in ${m/V}$ units as

$\beta^{\prime}=\frac{2v_{L}^{2}}{l\pi^{2}f^{2}}\frac{\bar{A}_{2\omega}}{\bar{A}_{\omega}^{2}},$
(2)

where $f$ is the US wave frequency and $v_{L}$ is the longitudinal speed of
wave propagation in aluminium. Finally, in order to identify heterogeneities
in the dislocation density along the samples, four measurement lines were
defined: Top-Bottom Line (TBL, Fig. 1(b)), Top Line, Middle Line and Bottom
Line (TL, ML and BL, Fig. 1(c)). The reason for this choice is to explore the
different plastic regimes in a three-point bending test: The “Top” (concave)
region bends in compression, the “Bottom” (convex) region bends in tension,
and the “Middle” region is in between. In this way we explore the fairly
complex space dependence of stress that is associated with the plastic bending
regime.

###  II.4 X-Ray Diffraction

In order to support the SHG results, we applied X-Ray Diffraction technique
(XRD) to obtain the dislocation density $\Lambda_{XRD}$ at different points of
the Post Bending sample, and compare them with the same measurements of the
annealed sample. For the XRD measurements, we have used the same procedure and
equipment reported by Espinoza et al. [32]. Lattice parameter $a$ and
microstrain $\langle\epsilon^{2}\rangle^{1/2}$, are obtained by Rietveld
refinements of the X-ray patterns with the Materials Analysis Using
Diffraction (MAUD) software. We use $LaB_{6}$ ($a=4.1565915(1)$ Å) as external
standard for the determination of instrumental broadening. The calculation of
dislocation density $\Lambda_{XRD}$ from the microstructural parameters $a$
and $\langle\epsilon^{2}\rangle^{1/2}$ is obtained through

$\Lambda_{XRD}=\frac{24\pi E}{GF}\frac{\langle\epsilon^{2}\rangle}{a^{2}},$
(3)

where $F\approx 5$ for FCC materials, $E=74.4\pm 1.9$ GPa is the Young’s
modulus and $G=28.1\pm 0.8$ GPa the shear modulus for Al. These values are
calculated as the averages of those reported in [36, 37, 38], with error bars
given by the computed standard deviations.

###  II.5 Finite Element Method Simulations

Finite Element Method (FEM) simulations of the bending test have been
developed to complement the results obtained experimentally, as well as to
obtain parameters that are not possible to measure experimentally. The
simulations have been carried out with the finite element technique using the
Solid Mechanic module of the COMSOL 5.6 [39] program. The simulation
reproduces the bending test performed experimentally, with the same probe
dimensions (see Sec.II.1). The simulated material properties are obtained from
the COMSOL material library and are complemented by the experimentally
measured behavior.

The simulation is performed up to the experimentally measured loading limits,
which implies that the material, in certain regions, exceeds the yield stress.
When this occurs, an isotropic bilinear plasticity model is used, in which
Young’s modulus, Yield stress and isotopic tangent modulus are required as
input variables. The bilinear method consists of representing the stress-
strain curve as two straight lines intersecting at a point corresponding to
creep. The first line, which has zero intercept and ends at creep, corresponds
to the elastic zone of the material. The slope of this first line corresponds
to its Young’s modulus. The second straight line indicates the creep point and
ends at the maximum deformation point. The slope of this line, which
corresponds to the plastic zone of the material, is called the isotopic
tangent modulus. The values used for these parameters are: Elastic modulus
$E=74.0$ GPa ; Yield stress $\sigma_{Y}=30.0$ MPa; Isotropic tangent modulus
$E_{t}=2.0$ GPa. These values are obtained from previous tests performed on
the same material [30].

The boundary conditions of the system are presented in Figure 2. On top, at
the center of the specimen, a force is exerted on the surface

$S\cdot\boldsymbol{n}=\boldsymbol{F_{A}}=\frac{\boldsymbol{F_{Tot}}}{A},$ (4)

where $S$ is the second Piola-Kirchhoff stress tensor, $\boldsymbol{n}$ is the
normal vector of the contact surface, $\boldsymbol{F_{A}}$ is the force per
unit area, $\boldsymbol{F_{Tot}}$ is the total applied force and $A$ is the
contact surface. The contact surface used in the simulation corresponds to the
contact area between the upper cylinder of the tensile machine and the upper
face of the specimen. This area is obtained from the surface deformation
experienced by the sample after the experimental test. The latter varies
between $0$ and $5$ kN, with $500$ N steps. The simulation is performed quasi-
statically, in agreement with experimental conditions. At the bottom surface,
two stainless steel cylinders are configured as pivots and no displacement is
allowed at these points ($\boldsymbol{\bar{u}=0}$, where
$\boldsymbol{\bar{u}}$ is the displacement vector in the pivot). The lower
pivots are $180$ mm apart.

Figure 2: Schematic drawing of the boundary conditions used in the 3-point
bending test simulation process. The lower pivots have the “fixed volume
constraint” ($\boldsymbol{\bar{u}=0}$) which does not allow deformation or
movement of these pivots. The contact edge condition is continuity, so the
deformation of the sample at the contact edge ($\boldsymbol{u}$ is the
displacement vector in the sample) with the pivots is zero. The upper cylinder
imposes the force on the specimen with the “boundary load condition” on the
contact surface between the cylinder and the specimen.

From the simulation, the von Mises stress ($\sigma_{VM}$), the plastic strain
($\epsilon_{pe}$) and the strain in the whole specimen are obtained, i.e. as
function of position $(x,y,z)$. With this information, and by proper space
integration, the averages $\sigma_{VM}(x_{i})$ are obtained at the same
positions $x_{i}$ where the nonlinear parameter $\beta^{\prime}$ is
experimentally measured (TBL, TL, ML and BL measurements).

##  III Results

###  III.1 XRD results

As in previous work with polycrystalline aluminium, the XRD diffraction
patterns showed a distribution of crystallite sizes (phases) contributing to
each diffraction peak. Each phase has an associated microstrain
$\langle\epsilon^{2}\rangle^{1/2}$. To obtain the dislocation density
$\Lambda_{XRD}$ using Eqn. (3), we use the volume fraction of each phase
provided by MAUD as a weight factor. Table 1 shows the XRD results of the
dislocation density at different points of the measurement lines, see Fig. 1
(b), calculated as a weighted average of the results for different crystallite
sizes. This is to be compared with the average dislocation density of the
original sample, $\Lambda_{XRD}^{ORI}=8.12\pm 2.89\times 10^{7}$ mm-2, as
reported in [30]

Table 1: XRD measurements of dislocation density $\Lambda_{XRD}$ of the Post
Bending sample. Errors for XRD measurements are calculated with the Rietveld
refinement results. Post Bending sample
---
Measurement point | TL $\frac{\Lambda_{XRD}}{10^{8}}$ (mm-2) | ML $\frac{\Lambda_{XRD}}{10^{8}}$ (mm-2) | BL $\frac{\Lambda_{XRD}}{10^{8}}$ (mm-2)
$A$ | $1.37\pm 0.46$ | $1.95\pm 0.84$ | $2.20\pm 1.99$
$B$ | $0.57\pm 0.24$ | $1.27\pm 0.46$ | $0.60\pm 0.27$
$C$ | $0.57\pm 0.24$ | $1.60\pm 0.55$ | $0.59\pm 0.69$
$D$ | $0.22\pm 0.08$ | $0.64\pm 0.18$ | $0.11\pm 0.25$
$E$ | $0.43\pm 0.10$ | $1.79\pm 0.92$ | $0.25\pm 0.08$

###  III.2 Simulations

With the numerical simulation, carried out with the same experimental
conditions, the values of deformation and stress at each of the points of the
mesh used are obtained. For this simulation the maximum finite element size
used is $1$ mm. Figure 3 shows an isometric projection view of the specimen in
its final condition, i.e. when subjected to a deformation of $50$ mm in the
direction of the bending test, for which it was subjected to $5$ kN of force
applied in the same direction. Figure 3 shows that the stress field is
heterogeneous in the complete volume, maintaining the symmetry of the bending
test. It is observed that along the specimen thickness the stress is not
constant, so that the imposed deformation is a function of the position where
it is measured.

Figure 3: Final condition of the specimens subjected to simulated three-point
bending test for a force of 5 kN. The color scale shows the Von Misses stress
($\sigma_{VM}$) at each node of the mesh simulation. The points of highest
stress are in the contact zone of the lower pivots as well as in the zone
where the force is imposed. The spatial distribution of the effective plastic
strain ($\varepsilon_{pe}$, not shown) is qualitatively the same.

To obtain the stress to which the specimen is subjected at the acoustic
measurement points described in Fig 1(c), lines are created in the simulation
that cross the specimen starting at the point of contact of the transmitter
transducer axis and ending at the point of contact of the receiver transducer.
These lines are created with a separation of $2$ mm between them. Along each
of these lines, the average von Mises stress is calculated, thus obtaining a
quantity that reflects the average deformation along the acoustic wave path in
the experimental measurement.

The average von Mises stress of each acoustic wave path is presented in Fig
4(a). As expected, when plotting this average stress as a function of the x
coordinate, it can be seen that its maximum is achieved in the central zone,
at the point of force application by the mechanical testing machine, for all
the analyzed zones (TL, ML, BL and TBL). Between the coordinates $x=90$ mm and
$x=80$ mm (also at $-80$ mm and $-90$ mm) there is a local maximum which
correlates with the position of the lower pivots of the bending test. This
local maximum is highly attenuated at the TL line, showing that at that height
the effect of the pivots is highly reduced. On the ML line, the von Mises
stress values are lower than for the other lines, but not zero. This is in
agreement with what one would expect of not-too-large deviations from elastic
displacements in plane strain, which indicate that the upper zone of the
specimen is subjected to compression, while the lower zone is subjected to
tension, and that there is a line, near the center of the specimen, in which
the deformation is zero [40].

For the reader’s reference, the black segmented lines in Fig 4(a) show the
position in which the ultrasonic transducers were positioned to obtain the
$\beta^{\prime}$ parameter.

In the FEM simulation, it is possible to obtain a quantity representing the
plastic deformation, i.e. the non-reversible deformation that occurs for the
different stresses applied at each finite element node. This quantity is known
as ”effective plastic strain”. In the process of finite element simulation
with the COMSOL structural mechanics module, the stress is calculated:

$\sigma=\sigma_{ext}+C:\varepsilon_{el},$ (5)

where $\sigma$ is the stress at each node, $\sigma_{ext}$ is the applied
external stress, $C$ is the elastic constants tensor and $\varepsilon_{el}$
the elastic deformation. The total deformation ($\varepsilon$) can be
decomposed into the elastic deformation ($\varepsilon_{el}$) and the plastic
deformation ($\varepsilon_{pe}$), i.e.,
$\varepsilon_{el}=\varepsilon-\varepsilon_{pe}$. In this way, a quantity is
obtained representing the material’s permanent changes, which are associated
with the microstructural changes [35] of the sample subjected to the bending
test.

Fig 4(b) shows the average effective plastic strain along each of the acoustic
wave paths as a function of the x-coordinate. It can be seen that for the TL
line the effective plastic strain is zero for values of $|x|>70$ mm, thus
showing that the pivots do not generate plastic strain at that position. For
the ML line, the effective plastic strain values are lower than $0.5\%$,
showing that along this line the plastic deformation is small and is close to
the zero plastic deformation zone of the bending test. For the TBL and BL
lines the behavior is quite similar, presenting local maximum strain in the
zone of the lower pivots and a maximum plastic strain in the zone where the
force is applied by the testing machine.

Figure 4: Results of simulations of three-point bending test with $5$ kN
applied force. (a) Spatial average of von Mises stress along the acoustic wave
path as a function of the x-coordinate. The dashed black lines show the
locations where the $\beta^{\prime}$ parameter was experimentally measured.
(b) Spatial average of effective plastic strain along the acoustic wave path
as a function of the x-coordinate.

###  III.3 Velocity of shear waves for Annealed and Post Bending samples

Figure 5 shows the speed of shear waves measured at different locations of the
Original and Post Bending samples, along the top-bottom measurement line.
Using the formula that relates change in the speed of shear wave $\Delta
v_{T}$ with change in dislocation density $\Delta\Lambda$ and dislocation
length $L$ [28]

$\frac{\Delta v_{T}}{v_{T}}=-\frac{8\Delta(\Lambda L^{2})}{5\pi^{4}}$ (6)

we obtain, using $L\sim 80$ nm [30], $\Delta\Lambda\sim 0.6\times 10^{8}$
mm-2. This is similar to the values obtained using XRD, as reported above. The
scatter in the data, as well as the experimental uncertainties, preclude a
more refined determination of the plasticity behavior of the bent sample as a
function of position. As we now show, these shortcomings can be overcome with
the measurements of the nonlinear acoustic parameter, in combination with
finite element simulations.

Figure 5:  Shear wave velocity. Blue circles, annealed sample; orange
diamonds, post bending sample. The solid blue line represents the shear speed
average for the annealed sample, and the dashed blue lines to the standard
deviation around this average. Notice that this standard deviation of $2.8$
m/s corresponds to $0.1\%$ of the average value. The red solid lines are
guides-to-the-eye showing the linear behavior observed for the shear speed for
the post bending sample. In this case, the variation between the maximum and
minimum speeds is $\sim 20$ m/s, a difference of about $0.6\%$ respect to the
average.

###  III.4 Nonlinear ultrasonics of the original and annealed samples

Figure 6: Nonlinear acoustic parameter $\beta^{\prime}$ as function of
longitudinal coordinate $x$ for the original (a) and annealed (b) samples for
the four measurement lines. Spatial averages for the different lines are shown
before and after the annealing procedure in (c) and (d), respectively. A
larger dispersion of values is observed for the original sample (a) than for
the annealed one (b), as expected from the latter’s lower dislocation density.
In fact, in each measurement line of the original sample the nonlinear
parameter covered clearly separated ranges of values, as shown in (c). The
mean value of all the measurements over this sample is
$\langle\beta^{\prime}\rangle_{\rm OS}=1.0\times 10^{-5}$ m V-1 with a
dispersion $\delta\beta^{\prime}_{\rm OS}=0.3\times 10^{-5}$ m V-1. Panel (d)
shows that for the annealed sample the nonlinear parameters tend to collapse
to a mean value $\langle\beta^{\prime}\rangle_{\rm AS}=1.0\times 10^{-5}$ m
V-1 with a dispersion $\delta\beta^{\prime}_{\rm AS}=0.1\times 10^{-5}$ m V-1.
Although the mean values of the nonlinear parameter $\beta^{\prime}$ are the
same in the original and annealed samples, the dispersion is three times
smaller for the annealed sample, which is evidence of the homogenization
obtained after thermal treatment.

First, using the TOF method we measured an average value of the longitudinal
wave speed in the annealed sample, $\langle v_{L}\rangle=6071$ m/s, with a
dispersion $\delta{v_{L}}=15$ m/s given by its standard deviation. To
calculate the value of $\beta^{\prime}$ using the expression (2) we consider
this wave speed as a constant before and after the thermomechanical processes.
This is justified due to the very small changes it suffers with variations of
dislocation density, of the order of $1\%$, much smaller than the variations
in $\beta^{\prime}$ [32].

We have characterized $\beta^{\prime}$ in both the original and annealed
sample. The summary of these measurements is presented in Fig. 6. Before the
thermal treatment there is an initial distribution of dislocations that is
reflected in an inhomogeneous distribution of $\beta^{\prime}$. This is
clearly demonstrated with the spatial averages shown in panels (c) and (d). In
the former, the top and bottom line $x$-coordinate average values are, within
experimental errors, the same, and larger than those of the middle line and
the top-bottom one. This is most likely due to the manufacturing processes of
the probes, where by stress induced machining the surfaces end up with more
dislocations, resulting in larger values of $\beta^{\prime}$ [32]. On the
other hand, in (b) and (d) we show that after the thermal treatment, the
nonlinear parameters collapse to a constant value, with a mean value
$\langle\beta^{\prime}\rangle_{\rm AS}=(1.0\pm 0.1)\times 10^{-5}$ m V-1, and
a dispersion that is three times smaller compared with the original, pre-
annealed sample. This is in good agreement with previous results [32].

###  III.5 Post bending ultrasonics and numerical results

In order to measure the effect of plastic deformation on the nonlinear
acoustic parameter, the SHG method was applied before and after the bending
test, on the Annealed and Post Bending samples. In Fig. 7 we show the spatial
dependence of the nonlinear parameter $\beta^{\prime}$ as function of the
longitudinal coordinate $x$ for the PB sample. For all measurement lines
except the middle one (ML), it clearly increases above the base value $1\times
10^{-5}$ m V-1 as it approaches the central bending point. Concomitantly, it
tends to this value as $|x|$ increases. The sample global average, for all
lines and all $x_{i}$ measurement positions, does increase slightly above this
value too, as indicated in the figure caption.

##  IV Discussion

In order to quantify the changes in the acoustic behavior of the material as a
consequence of plastic deformation we consider the change in the nonlinear
parameter $\beta^{\prime}$, which is calculated as

$\Delta\beta^{\prime}=\beta^{\prime}_{\rm PBS}-\beta^{\prime}_{\rm AS},$ (7)

where the subfix ’PBS’ stands for the Post Bending sample, and ’AS’, for the
Annealed one. Since we have $\Delta\beta^{\prime}$ at various positions of the
sample, and, as a result of the FEM simulation, we have both the von Mises
stress $\sigma_{VM}$ and the effective plastic strain $\epsilon_{pe}$ at those
same locations, it is possible to consider $\Delta\beta^{\prime}$ as a
function of either of the two latter quantities. This we now do.

In Fig. 8 we present $\Delta\beta^{\prime}$ obtained experimentally as
function of the von Mises stresses $\sigma_{VM}(x_{i})$ obtained numerically,
calculated for each measurement line, i.e. averaging, along the thickness of
the sample, at each measurement position $x_{i}$. At low stress, the data is
consistent with a constant $\Delta\beta^{\prime}$, very close to zero; a clear
deviation is observed above a given stress value, indicated by a vertical
dashed line in each panel. This threshold value is more or less consistent
with the aluminium’s yield stress, although a bit larger than the one of an
annealed sample, which is of the order of $30$ MPa. Indeed, for the same
material we measured $\sigma_{Y}\approx 30$, $44$ and $51$ MPa for three
consecutive tensile tests on a single probe [30], where the successive
increments are due to hardening by dislocation proliferation. In Fig. 8(a),
each measurement line data set is identified with a different symbol and
color. Most of the middle line data is below the threshold stress, with
$\Delta\beta^{\prime}\approx 0$. This is expected because this line is between
a tensile zone and a compression one, so the local stresses are expected to be
very low. The top and bottom line data are more or less symmetric and deviate
clearly from $0$ above a threshold. The top-bottom line data also shows this
behavior but with a stronger slope, although errors are much larger for this
data set. The continuous red line is the data fit to all measurements merged
in one data set, of the form $\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}$
for $\sigma_{\rm VM}<\sigma_{o}$ and
$\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}+A(\sigma_{\rm VM}-\sigma_{o})$
for $\sigma_{\rm VM}>\sigma_{o}$. Here, for the fitting procedure we have used
weight factors $w=1/e^{2}$, where $e$ are the errors for each
$\Delta\beta^{\prime}$. The black dashed vertical line corresponds to the
fitted parameter $\sigma_{o}=35.8$ MPa. The gray shaded region corresponds to
the $95\%$ confidence bounds of this fitted parameter, which is interpreted as
the parameter’s error bar; in this case $\pm 7.6$ MPa. The other parameters
are given in the figure caption. In Fig. 8(b), grey solid circles show the
merged data set and the solid red squares are window averages: the complete
$\sigma_{VM}$ range is divided in 10 bins and all the data that fall into each
bin is averaged. In this case, the solid red line is the same functional fit
but to this window averaged data set. The fit is also done using weight
factors, as defined previously but with the windowed data standard deviation
as error. In this case, $\sigma_{o}=37.3\pm 15.8$ MPa, with its error bar also
given by the fitted $95\%$ confidence bounds. It is larger in this case, by a
factor 2.

Figure 7: (a) Nonlinear acoustic parameter $\beta^{\prime}$ as function of
longitudinal coordinate $x$ of the post bending sample for the four
measurement lines. Spatial averages for the different lines are shown in (b).
Note the difference in the vertical scales compared to Figure 6. The parameter
$\beta^{\prime}$ increases for smaller distances to the central bending point,
for all lines except the middle one, for which it is almost constant. The mean
value of all the measurements over this sample is
$\langle\beta^{\prime}\rangle_{\rm PBS}=1.1\times 10^{-5}$ m V-1 with a
dispersion $\delta\beta^{\prime}_{\rm PBS}=0.2\times 10^{-5}$ m V-1, which
reflects a small increment with respect to the annealed and original samples.
However, the importance here is the spatial sensitivity of $\beta^{\prime}$,
as it clearly increases above the base values in regions of larger stress.

Figure 8: Nonlinear parameter difference $\Delta\beta^{\prime}$ as function of
von Mises stress $\sigma_{\rm VM}$ for all measurements. The continuous lines
show the linear fits of the form
$\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}$ for $\sigma_{\rm
VM}<\sigma_{o}$ and
$\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}+A(\sigma_{\rm VM}-\sigma_{o})$
for $\sigma_{\rm VM}>\sigma_{o}$. In (a) the fit is done merging data from all
measurement lines. In (b), it is done for the window averaged data presented
with solid red square symbols. The adjusted values are: (a)
$\Delta\beta^{\prime}_{o}=(-1.5\pm 0.8)\times 10^{-6}$ m V-1, $A=(2.1\pm
1.2)\times 10^{-7}$ m V-1MPa-1 and $\sigma_{o}=35.8.0\pm 7.6$ MPa; (b)
$\Delta\beta^{\prime}_{o}=(-1.9\pm 1.2)\times 10^{-6}$ m V-1, $A=(2.4\pm
2.5)\times 10^{-7}$ m V-1MPa-1 and $\sigma_{o}=37.3\pm 15.8$ MPa. The vertical
dashed lines are the fitted threshold stresses $\sigma_{o}$. The gray shaded
regions correspond to the $95\%$ confidence bounds of these fitted parameters.

Here, a natural question is if the von Mises stress is the best quantifier for
the local material state. Is it the best correlator to the
$\Delta\beta^{\prime}$ measurements? In other words, if a
transition/bifurcation is to be expected/evidenced, is $\sigma_{VM}$ the
appropriate diagnose parameter. There are two facts that have to be addressed:
(1) The threshold value $\sigma_{o}$ is about $20\%$ larger than the yield
stress of an annealed Al sample; (2) Its associated error bar for the window
average procedure is about 2 times the one of the fit using all the
measurements merged into one data set.

Thus, we propose as an alternative control parameter the plastic strain
$\epsilon_{pe}$. In Fig. 9 we show $\Delta\beta^{\prime}$ obtained
experimentally as function of $\epsilon_{pe}(x_{i})$ obtained numerically,
calculated for each measurement line at each ultrasonic measurement point
$x_{i}$. A similar behaviour is observed: a constant
$\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}\approx 0$ is observed for
$\epsilon_{pe}<\epsilon_{o}$ and
$\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}+B(\epsilon_{pe}-\epsilon_{o})$
for $\epsilon_{pe}>\epsilon_{o}$. However, an importance difference exists:
for $\epsilon_{pe}<\epsilon_{o}$, most plastic strain values are concentrated
near 0, whereas the von Mises stress values are more evenly distributed
between 0 and $\sigma_{o}$. Here, again all of the middle-line data is
concentrated well below the threshold value (at less than one-third), in
contrast to the von Mises stress, and the top and bottom line data deviate at
a given threshold with a similar slope. Also, again the top-bottom line data
deviates with a larger slope but has larger errors. The fitting procedure also
uses weight factors given by the errors in each case. The threshold plastic
strain $\epsilon_{o}$ is small, although not $0$, as could be naively
expected. Its associated error bars are small, and differ only by a factor of
1.25 between the two procedures in this case.

Figure 9: Nonlinear parameter difference $\Delta\beta^{\prime}$ as function of
the effective plastic strain $\epsilon_{pe}$. The continuous lines show the
linear fits of the form $\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}$ for
$\epsilon_{pe}<\epsilon_{o}$ and
$\Delta\beta^{\prime}=\Delta\beta^{\prime}_{o}+B(\epsilon_{pe}-\epsilon_{o})$
for $\epsilon_{pe}>\epsilon_{o}$. In (a) the fit is done merging data from all
measurement lines. In (b), it is done for the window averaged data presented
with solid red square symbols. The adjusted values are: (a)
$\Delta\beta^{\prime}_{o}=(-1.1\pm 0.6)\times 10^{-6}$ m V-1, $B=(6.9\pm
4.2)\times 10^{-4}$ m V-1 and $\epsilon_{o}=(1.13\pm 0.36)\times 10^{-2}$; (b)
$\Delta\beta^{\prime}_{o}=(-1.0\pm 1.1)\times 10^{-6}$ m V-1, $B=(6.2\pm
4.3)\times 10^{-4}$ m V-1 and $\epsilon_{o}=(1.19\pm 0.45)\times 10^{-2}$. The
vertical dashed lines are the fitted threshold plastic strains $\epsilon_{o}$.
The gray shaded regions correspond to the $95\%$ confidence bounds of these
fitted parameters.

##  V Conclusions

The results discussed in the previous sections show that the measurement of
the acoustic nonlinear parameter $\beta^{\prime}$ is a reliable measure of
plastic deformation within aluminium, and that this measure can be realized
with space resolution limited only by the transducers size. First of all, we
note that changes in $\beta^{\prime}$ between annealed and plastically
deformed regions are, broadly speaking, of order 50%. This is a significant
change, not difficult to measure. We also note that a universal indicator,
long used of plastic behavior is the von Mises stress $\sigma_{\rm VM}$
increasing beyond a given value. Our measurements indicate that the onset of
plastic behavior, as diagnosed by the behavior of $\sigma_{\rm VM}$ is
accurately captured by the behavior of the nonlinear acoustic parameter
$\Delta\beta^{\prime}$. Beyond onset, both quantities are, to a good
approximation, linearly related. We have also found that a more reliable
experimental indicator to relate plastic behavior to nonlinear acoustics is
the effective plastic strain $\epsilon_{pe}$. This is not surprising, since
$\epsilon_{pe}$ is non-vanishing only within plastically deformed regions. A
comparison of the ML values in Figures 8(a) and 9(a) illustrates this point.
The ML points fall within the region that separates the “upper” from the
“lower” region of the deformed sample where, because of symmetry (which is,
admittedly, only approximate in the heavily deformed case) the plastic
deformation is smallest. In figure 8(a) the ML points are broadly distributed
for non-vanishing values of $\sigma_{\rm VM}$, whereas in figure 9(a) they are
clustered near the origin. This is also reflected in the fact that it is
possible to ascertain fairly unambiguously that a value of
$\Delta\beta^{\prime}>0.1$, say, is indicative of $\epsilon_{pe}>0.01$. A
similar relation with $\sigma_{\rm VM}$ is more uncertain. Also, the onset of
plasticity as determined by $\epsilon_{pe}>0.01$ is independent of the yield
stress.

We have also measured the speed of linear shear waves. As noted elsewhere
[32], this indicator is less sensitive than changes in nonlinear parameters to
changes in microstructure and, indeed, our results show changes (see figure 5)
that are less accurate. Something similar can be said about dislocation
measurements using XRD. In addition to its being an intrusive technique, its
accuracy (see Table 1) is also inferior to nonlinear acoustics.

##  VI Acknowledgements

This work was supported by Fondecyt, Chile Grants #11190900, #1191179,
Fondecyt Postdoctoral Grant #3200239 and by U. de Chile VID Grant ENL12/22.

##  VII Data Availability

The raw/processed data required to reproduce these findings cannot be shared
at this time as the data also forms part of an ongoing study.

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