metals
ArticleA New and Direct R-Value Measurement Method of Sheet
Metal Based on Multi-Camera DIC System
Siyuan Fang 1 , Xiaowan Zheng 1,2 , Gang Zheng 1 , Boyang Zhang 1 , Bicheng Guo 1 and Lianxiang Yang 1, *12*

Citation: Fang, S.; Zheng, X.; Zheng,G.; Zhang, B.; Guo, B.; Yang, L. A
New and Direct R-Value Measurement
Method of Sheet Metal Based on

Department of Mechanical Engineering, School of Engineering and Computer Science, Oakland University,
Rochester, MI 48309, USA; siyuanfang@oakland.edu (S.F.); xiaowanzheng@oakland.edu (X.Z.);
gangzheng@oakland.edu (G.Z.); boyangzhang@oakland.edu (B.Z.); bichengguo@oakland.edu (B.G.)
College of Electrical and Information Engineering, Zhengzhou University of Light Industry,
Zhengzhou 450002, China
Correspondence: yang2@oakland.edu

Abstract: More and more attention has been given in the field of mechanical engineering to a material’s R-value, a parameter that characterizes the ability of sheet metal to resist thickness strain.
Conventional methods used to determine R-value are based on experiments and an assumptionof constant volume. Because the thickness strain cannot be directly measured, the R-value is currently determined using experimentally measured strains in the width, and loading directions in
combination with the constant volume assumption, to determine the thickness strain indirectly. This
paper provides an alternative method for determining the R-value without any assumptions. Thismethod is based on the use of a multi-camera DIC system to measure strains in three directions
simultaneously. Two sets of stereo-vision DIC measurement systems, each comprised of two GigE
cameras, are placed on the front and back sides of the sample. Use of the double-sided calibration
strategy unifies the world coordinate system of the front and back DIC measurement systems to one
coordinate system, allowing for the measurement of thickness strain and direct calculation of R-value.
The Random Sample Consensus (RANSAC) algorithm is used to eliminate noise in the thickness
strain data, resulting in a more accurate R-value measurement.

Multi-Camera DIC System. Metals
2021, 11, 1401. https://doi.org/

Keywords: R-value; thickness strain; digital image correlation; multi-camera DIC; non-destructive testing

10.3390/met11091401
Academic Editors: Gabriel Centeno
and Ricardo Jose Alves de Sousa
Received: 10 July 2021
Accepted: 2 September 2021
Published: 4 September 2021

Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims inpublished maps and institutional affiliations.

Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms andconditions of the Creative Commons
Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

1. Introduction
The plastic strain ratio r is a parameter that indicates the ability of a sheet metal to resist
thinning or thickening when subjected to either tensile or compressive forces in the planeof the sheet [1,2]. It has been used as an important parameter to evaluate the formabilityof automotive sheet metal. It is typically advantageous for the material to experience aminimal reduction in the area when subjected to a force; this means that it is a good drawing
material with a high R-value. Therefore, an accurate measurement of R-value is of greatsignificance to the study of the tensile and compressive properties of materials. Stickels [3]predicted the plastic-strain ratio of low-carbon steel sheets using the Young’s Modulus,
while Ghosh [4] calculated the R-value of Al-Mg-Si alloy sheets based on crystal plasticity
models. However, since the crystal structure of polycrystalline materials is often difficult
to model accurately, the accuracy of the current theoretical estimates is not high [5,6].
Some scholars [7–9] employed laser-ultrasound resonance spectroscopy to measure thetexture, thickness, and plastic strain ratio on-line. Although the laser-ultrasonic method
has the advantages of being fast and non-destructive, the detection accuracy is greatly
affected by the vibration of the tested strip and changes in the environment temperature. In
some traditional R-value measurement methods [10,11], the thickness strain is calculatedusing length strain and width strain. The stated measurement methods are based on theassumption of constant volume, which states that the volume of the material will notchange during the deformation process. However, some materials do not always follow

Metals 2021, 11, 1401. https://doi.org/10.3390/met11091401

https://www.mdpi.com/journal/metals

Metals 2021, 11, 1401

2 of 16

the constant volume assumption, especially inside the necking band. This makes theexact determination of the R-value challenging [12]. Because of this, development of anexperimental method for direct measurement of R-value becomes meaningful.
Digital Image Correlation (DIC), a non-contact measurement method, has achieved
great success in the field of optical measurement and is still developing rapidly [13–17]. A
first attempt to experimentally measure R-value using DIC was reported by Xie in 2017 [18].
Instead of observing the front surface of the sheet metal, this method observed the sheet
metal at a 45-degree angle. As a result, both the front surface and the depth side of the sheet
metal can be observed at the same time by two DIC cameras, and the three strains ε 1 , ε 2and ε 3 can be measured simultaneously. A shortcoming of this method is that the thickness
strain can only be measured on the edge. Additionally, the edge needs to be cut, which will
introduce a residual strain and ultimately affect the measurement results.
Thickness strain measurement at any point on the sheet metal surface using DIC was
first reported by our lab at Oakland University together with Dantec-Dynamics GmbH,Germany [19]. This method was based on using multiple multi-camera DIC systems
to measure strains in three directions simultaneously. Two sets of stereo vision DIC
measurement systems, each consisting of two GigE cameras, were used in this method.
One stereo vision DIC system was placed on the front side of the test sample, while the othersystem was placed on the back side of the sample. The use of the double-sided calibration
strategy unified the world coordinate system of the front and back DIC measurement
systems to one coordinate system, thus enabling the measurement of the thickness strain.
This paper describes the use of this technology to measure the R-value directly without any
assumptions. A tensile test of DP980 was conducted to directly measure the R-value. The
theory of multi-camera DIC and the experimental setup will briefly be explained. Section 2of this paper explains the DIC related theories, the two-sided calibration strategy, and thealgorithms to smooth the thickness strain, while Section 3 presents the experimental setup
and experimental process. The two R-value calculation methods will be presented, as wellas the experimental results of DP980 and a comparison between the R-value determinedthrough experimental results determined using the constant volume assumption. The
potential and limitation of the method, as well as conclusions, will be discussed andpresented in Section 4.
2. Fundamental of DIC
2.1. Basic Theory
The stereo vision system shown in Figure 1, or a two-camera system with different
perspectives, plays an important role in the DIC measurement system. A space point p isrecorded simultaneously by two cameras on the image plane with a different perspective.
The points p1 and p2 on the image plane of the two cameras are then matched using the
DIC matching algorithm, and the 3D coordinates of point p can be recovered based on thegeometry of the stereo-vision system.
The core idea of the DIC method is to measure displacement and strain information
on an object’s surface by matching and tracking feature points in the natural texture or
artificial speckle pattern before and after deformation. The DIC matching process obtains
the coordinate on the target image after deformation corresponding to a known coordinate
position on the reference image taken before deformation. It is usually necessary to divide
the image into multiple grids, or subsets, and the deformation of the subset is used torepresent the deformation of the local area of the object. Figure 2 shows this principle. On
the reference image, a rectangle with a size of (2M + 1) pixel × (2M + 1) pixel and pointp( x0 , y0 ) at the center is taken as the reference subset. A correlation function is then used
to find a subset on the target image with a similar distribution to that of the reference
subset. The subset with the largest correlation coefficient is considered the deformed subset.
The sum squared difference correlation criterion (SSD) is commonly used to evaluate thesimilarity between subsets [20]. Assuming that the intensity distribution of the two facetsis represented as F ( x, y) and G ( x 0 , y0 ), the SSD function has the following form:

Metals 2021, 11, 1401

subset. The subset with the largest correlation coefficient is considered the deformed subsubset.
subsetwithdifference
the largest
correlation
coefficientis consideredtheuseddeformed
subset. TheThesumsquaredcorrelation
criterion(SSD)
is commonly
to evaluateset.
sum squared
difference
correlation
criterion
is commonlyused toofevaluate
the Thesimilarity
betweensubsets [20].
Assumingthat (SSD)
the intensity
distribution
the twothesimilarity
betweenthat
thefunction
intensity
distributionof thefacetsis represented
assubsets
𝐹(𝑥, 𝑦) [20].
and Assuming𝐺(𝑥 ′ , 𝑦 ′ ), theSSD
has
the following
form:
3 two
of 16
facets is represented as 𝐹(𝑥, 𝑦) and 𝐺(𝑥 ′ , 𝑦 ′ ), the SSD function has the following form:
𝐶𝑆𝑆𝐷 = ∑ 𝐹(𝑥, 𝑦) × 𝐺(𝑥 ′ , 𝑦 ′ )(1)
𝐶𝑆𝑆𝐷 = ∑ 𝐹(𝑥, 𝑦) × 𝐺(𝑥 ′ , 𝑦 ′ )(1)

CSSD = ∑ F ( x, y) × G x 0 , y0(1)Figure1.1. Schematic
Schematicofof stereo-visionstereo-visionsystem.
system.
FigureFigure 1. Schematic of stereo-vision system.

FigureFigure2.2. Basic
Basic principle
principleof
ofdigital
digitalimage
imagecorrelation.
correlation.
Figure 2. Basic principle of digital image correlation.

2.2. Double-Sided Calibration
2.2. Double-Sided Calibration
Camera calibration
plays an important role in DIC measurement, as its quality directly
2.2. Double-Sided
Calibration
Camera calibration
plays an important role in DIC measurement, as its quality diaffects measurement accuracy. The principle of camera calibration is to establish the
Cameraplays
an importantrole inofDIC
measurement,
itsestablish
quality directly
affects calibration
measurement
accuracy.
The principle
cameracalibration as
is to
the
relationship between the position of the camera image pixel and the scene point position.
rectlyaffects measurement
accuracy.
principleof camera
is topoint
establishtherelationshipbetween the positionof Thethe camera
image
pixel calibrationand the scene
position.
According to the camera imaging model, the relationship between the image coordinate
relationshipbetweenthe position
the camera
image pixelbetween
and the scene
pointcoordinate
position.
According to
the camera
imagingofmodel,
the relationshipthe imagesystem and the world coordinate system is as follows [21]:Accordingtothecameraimagingmodel,
therelationshipbetweentheimage
coordinate
system and the world coordinate system is as follows [21]:* [21]:
system and the world coordinate system* is as follows
⃑⃑
(2)
⃑⃑ =
sm𝑠𝑚
=
A𝐴
t] 𝑡]
M𝑀
(2)
[r [𝑟
⃑⃑
(2)
𝑠𝑚
⃑⃑ = 𝐴 [𝑟 𝑡] 𝑀
̅ = [𝑋, 𝑌, 𝑍, 1]𝑇 is
where 𝑚̅ = [𝑢, 𝑣, 1]𝑇 is the image homogeneous coordinate system, 𝑀
where m = [u, v, 1] T𝑇 is the image homogeneous coordinate system, M̅= [ X, Y, Z, 1]𝑇T is
where𝑚̅ =homogeneous[𝑢, 𝑣, 1] is theimage homogeneous
coordinate
system,𝑀 =camera[𝑋, 𝑌, 𝑍,internal
1] is
the worldcoordinate
system, 𝑠 is thescale factor,
𝐴 is thethe world homogeneous coordinate system, s is the scale factor, A is the camera internal
the
world
homogeneous
coordinate
system,
𝑠
is
the
scale
factor,𝐴
isthecamerainternal[𝑟
parameter matrix, and𝑡] is the camera external parameter matrix. Both the internalparameter matrix, and [r t] is the camera external parameter matrix. Both the internal andparametermatrix,and [𝑟of𝑡] theis thecameraparametermatrix.calibration
Both the internaland externalparameters
cameracanexternalbe calculatedby Zhang’s
methodexternal parameters of the camera can be calculated by Zhang’s calibration method [21].
and[21].external parameters of the camera can be calculated by Zhang’s calibration method
The double-sided calibration strategy uses a double-sided calibration plate to calibrate[21]. The double-sided calibration strategy uses a double-sided calibration plate to calithe front and back dual-camera DIC systems simultaneously. The schematic of the doublecalibration
strategy systems
uses a double-sided
calibration
plate toofcalibrateThethedouble-sided
front andback
dual-camera
simultaneously.
schematicthesided
calibration
strategy
is shown in DIC
Figure 3. Figure
3 shows two The
typicaltwo-camera
brate
the
front
and
back
dual-camera
DIC
systems
simultaneously.
The
schematicofthedouble-sided
calibrationand
strategy
shown
in Figure
3. Figure3 shows
two typicaltwostereo
vision subsystems,
these istwo
subsystems
are linked
throughcalibration.
Figure4
double-sided
calibration
strategy
is
shown
in
Figure3.
Figure3shows
two
typicaltwoshows the design of the calibration plate. The role of this calibration is to connect these
two subsystems into one single global system. The world coordinate system of the front
camera is taken as the reference global coordinate, and the back camera coordinate system
is transformed to the reference global system. This transformation can be expressed
as follows:Metals 2021, 11, 1401

camera stereo vision subsystems, and these two subsystems are linked through calibracamera
stereovisionsubsystems,
andofthese
two subsystems
calibra- is totion.
Figure4 shows
the design
the calibration
plate. are
Thelinked
role ofthroughthis calibration
tion. connect
Figure 4these
shows
the
design of into
the calibration
plate. system.
The roleTheof thiscalibration
is system
totwosubsystems
one single global
world
coordinate
connect
thesetwosubsystems
intoone
singleglobalsystem.
The
world
coordinate
systemof the front camera is taken as the reference global coordinate, and the back camera coorof thedinate
front camera
as the reference
global coordinate,
and Thisthe back
camera coorsystem isis takentransformed
to the reference
global system.
transformationcanbe
4 of 16
dinatesystemistransformedtothereference
globalsystem.
Thistransformation
can beexpressed as follows:
expressed as follows:−1 0 0 0−1 00 01 00 0
𝑓𝑟𝑜𝑛𝑡
𝑏𝑎𝑐𝑘
⃑⃑
𝑀=
𝑚
⃑⃑
= 𝑚
⃑⃑
∙ 𝑇𝜔
(3)
 𝑇𝜔0= [ 1 0 0  𝑑 ]𝑓𝑟𝑜𝑛𝑡 𝑤𝑜𝑟𝑙𝑑 𝑏𝑎𝑐𝑘 𝑤𝑜𝑟𝑙𝑑⃑⃑ = 𝑚𝑀⃑⃑ 𝑤𝑜𝑟𝑙𝑑 = 𝑚⃑⃑ 𝑤𝑜𝑟𝑙𝑑 ∙ 𝑇𝜔𝑇𝜔 = [−1 00 00 0−1](3)
0 0 −1 𝑑*
 0 10 00 00  1*back* f ront
0 0 0 1(3)
M =mworld =mworld · Tω Tω = 
 0 0 calibration
In this equation, 𝑑 is the thickness of the double-sided−1 d  plate.
In this equation, 𝑑 is the thickness of the double-sided
0 calibration
0 0 1plate.

Figure 3. Schematic of double-sided calibration strategy.
Figure 3. Schematic of double-sided calibration strategy.
Figure 3. Schematic of double-sided calibration strategy.

Figure4.4.Double
Doublesidesidecalibration
calibrationplate
plate(front(frontandandback
backside).
side).
FigureFigure 4. Double side calibration plate (front and back side).

2.3. Optimized
Thickness
Measurement
In this equation,d isStrain
the thicknessof the double-sided calibration plate.
2.3. Optimized Thickness Strain Measurement
The measurement of the thickness strain is the core part of calculating the R-value.
2.3.
Optimized
Thickness
StrainMeasurementis the core part of calculating the R-value.
Themeasurementof thethickness
Li built
a dual-camera
DIC
systemstrain
on the front and back of the test piece to realize the
Li builtaThedual-camera
DIC ofsystemona thefront
andbackcoreof thetest
piece
to realize
the inmeasurement
the thickness
strainis thepartof calculatingthenon-destructive
measurementofsample’s
thicknessstrain[19].
However,as
Li R-value.
wrote
non-destructive
measurementofasample’s
thicknessstrain[19].
However,
as
Li
wrote
Li
built
a
dual-camera
DIC
system
on
the
front
and
back
of
the
test
piece
to
realizethe[19], the strain measurement results contain noise, so it is necessary to remove theinnoise[19],non-destructive
thestrainmeasurement
results
containnoise,
soitisnecessarytoremovethenoisemeasurementofasample’s
thicknessstrain[19].
However,
as
Li
wrote
from the strain data. Since the thickness of the test sample is usually a few millimeters,
frominthestraindata.
Since
the thickness
ofcontain
thestrain
testnoise,samplea few
millimeters,[19],
theisstrain
measurement
results
so itisisusuallynecessary
to5removetheDP980
noise
the
noise
amplifiedduringthethicknessmeasurement.
Figureshows
thethe noiseamplifiedduringthethethicknessstrain5 shows
themillimeters,
DP980
from isthestrain data.
Since
thicknessof measurement.
the test sampleFigure
is usuallya few
the noise is amplified during the thickness strain measurement. Figure 5 shows the DP980
thickness strain history, where the x-axis represents the number of photos taken and theyellow circle indicates the noise data. Two smoothing algorithms, namely the Least squares
algorithm and Random sample consensus (RANSAC) algorithm, were used to fit the curveand reduce noise. A comparison of the results using these two algorithms follows.

Metals 2021, 11, 1401

thickness strain history, where the x-axis represents the number of photos taken and theyellow circle indicates the noise data. Two smoothing algorithms, namely the Leastsquares algorithm and Random sample consensus (RANSAC) algorithm, were used5 to
fit
of 16
the curve and reduce noise. A comparison of the results using these two algorithms follows.

Figure5.5.DP980
DP980thickness
thicknessstrainstrainfitted
fittedby
byleastleastsquares
squaresalgorithm.
algorithm.
Figure2.3.1.
2.3.1. Least
Least Squares
SquaresAlgorithm
Algorithm
The
The least
least squares
squares method
method isis thethe most
most common
common method
method used
used totosolvesolve curve
curve fitting
fittingproblems.
The
basic
idea
of
the
methodistominimizethesumofsquares
of
the
error
problems. The basic idea of the method is to minimize the sum of squares of the errorto
to
find
can
model
thetheexpected
value
of of
y
findthethebestbestfunction
functionmatch
matchforforthethedata.
data.InIngeneral,
general,we
we
can
model
expected
value
as
an
nth
degreepolynomial,yieldingthegeneralpolynomial
regression
model:y as an nth degree polynomial, yielding the general polynomial regression model:𝑦=+1 𝛼𝑥 α+2 x𝛼22 𝑥+2 α+3 x𝛼33 +𝑥 3 ·+y=f (𝑓(𝑥,x, α)𝛼)==α0 𝛼+0 αx 1+· ·⋯αn𝛼x𝑛n𝑥 𝑛(4)(4)

where 𝛼 = [𝛼 , 𝛼 , ⋯ , 𝛼 ] is a parameter to be determined.
where α = [α1 ,1α2 ,2· · · , α𝑛n ] is a parameter to be determined.
Theobjective
objectivefunction
function L𝐿(𝑦,minimizedtotofindfindthetheoptimal
optimalestimatedestimatedvalue
value
The
(y, f𝑓(𝑥,( x, α𝛼)))) isisminimized
(𝑥
)
of
the
parameter𝛼inthefunction
𝑓(𝑥,
𝛼)
for
m
given
sets
of
data
,
𝑦
(𝑖
=
1,2,
⋯
𝑚).
𝑖
𝑖
of the parameter α in the function f ( x, α) for m given sets of data ( xi , yi ) (i = 1, 2, · · · , ,m).
𝑚

𝐿(𝑦, 𝑓(𝑥, 𝛼)) =m∑[𝑦 − 𝑓(𝑥 , 𝑦 )]2

L(y, f ( x, α)) =∑𝑖=1[yi − f (xi , yi )]2
𝑖

i =1

𝑖

𝑖

(5)
(5)

The result using least squares fitting is shown in Figure 5 as a red curve. The red
Theresult usingsquaresfitting ispolynomial
shown in Figure
as a red curve.
red 5curve
curve represents
the least
resultsof quadratic
fitting.5 ObservationofThe
Figureindirepresentsof quadratic
polynomial
Observationof Figure
5 indicates
cates that the results
fitted curve
shifts relative
to thefitting.
expected
data. Becauseof this,the algothat
the fitted
shiftsrelativeto the expected
data.
Becauseof this,the algorithm’s
rithm’sfittingcurve
results
becomeinaccurate
if the datacontainsa largeamountof error or
fitting
inaccurate
if least
the data
containsa large amount of error or noise. This
noise. results
This is become
the limitationof thesquaresalgorithm.
is the limitation of the least squares algorithm.
2.3.2. Random Sample Consensus (RANSAC) Algorithm
2.3.2. Random Sample Consensus (RANSAC) Algorithm
The RANSAC algorithm based on the iterative method was proposed by Fischler [22]TheRANSAC
algorithm
basedsolutions
on the iterative
methodby Fischler[22]
to solve theproblemof inaccurate
when using
the wasleastproposedsquares modelon sample
to
solve
the
problemofinaccurate
solutions
when
using
the
least
squaresmodel
onsampledata with a large proportion of outliers. Song used the improved RANSAC algorithm todatawith errors
a largecaused
proportionof outliers. Songusedthedisturbances
improved RANSAC
algorithm
eliminate
by high-temperature
heat
wave
in DIC measurement
to eliminate errors caused by high-temperature heat wave disturbances in DIC measurement [23]. This algorithm is able to classify data points as outliers or inliers and fit themathematical model through inliers while ignoring outliers. The basic algorithm is summarized as follows:Metals 2021, 11, x FOR PEER REVIEW

Metals 2021, 11, 1401

6 of 16[23]. This algorithm is able to classify data points as outliers or inliers and fit the mathe6 of 16
matical model through inliers while ignoring outliers. The basic algorithm is summarized
as follows:1.
1.

Randomly select several points in the thickness-strain data to calculate the polynoRandomly
select severalhistory
points in
the thickness-strain data to calculate the polynomial
mial thickness-strainmodel.
thickness-strain history model.
2. Set a preset tolerance value 𝜀 and calculate the number of data points that match the2.
Set a preset tolerance value ε and calculate the number of data points that match themathematical model in Step 1.
mathematical model in Step 1.
3.
If the proportion of data points that meet the mathematical model in Step 2 exceeds
3.
If the proportion of data points that meet the mathematical model in Step 2 exceeds
the preset
preset threshold
thresholdτ,𝜏,recalculate
recalculatethe
thethickness-strain
thickness-strainhistory
historymodel
modeltotouse
usethese
these inthe
liers andinliers
andterminate
terminatethe
the algorithm.
algorithm.
4.
Otherwise,
repeat
Steps
1 to
3K
times.
4.
Otherwise, repeat Steps 1 to
3Ktimes.
Thenumbernumberofofiterations
iterations
K should
large
enoughto ensurethe probabilityThe
K should
be be
large
enoughto ensurethat thatthe probability
p of p of
atleastleastoneonesetsetofofrandom
randomsamples
includingoutliers
is greater
Assumingatsamples
notnotincludingoutliers
is greater
thanthan0.99.0.99.
Assuming
that
that
thethedata
is selected
as anis a, isthenb = 1 b−= a1is− thethatthetheprobability
probability
that
data
is selectedasinlier
an inliera, thena isprobability
the probabilityofnumberof iterationsof thenumberof ofofobserving
observingan
anoutlier.
outlier.The
Therequired
requirednumberof iterationsof minimum
the minimum
numberpoints,denotedk,iscalculated
using
the
following
function:
points, denoted k, is calculated using the following function:1−1p=ak )−K 𝑎𝑘 )𝐾− 𝑝(1=−(1

(6)

(6)

log(1log(1− p) − 𝑝)
K = 𝐾 = log(1 − (1 k− 𝑏)𝑘 )(7) (7)log(1 − (1 − b) )Figure6 6shows
showsthat
that
the
RANSAC
algorithm
classifies
thicknessstrainFigurethe
RANSAC
algorithm
classifies
the thethicknessstraindata data
into into
inlierpointsandoutlierpoints.
The
blue
point
set
represent
theinlierpoints,andthe redinlier points and outlier points. The blue point set represent the inlier points, and the redcirclesrepresent
representthetheoutlier
outlierpoint
Usingmethod,outlierpoints,or noisy
circlespoint
set.set.
Using
thisthis
method,the theoutlierpoints,or noisydata,data,
canbe
beignored
ignoredduring
duringfitting.
fitting.The
The
resultRANSAC
algorithm
fitting
is shown
canresultof ofthetheRANSAC
algorithm
fitting
is shown
in in
Figurered
fitting
curve
is fitted
withinlierpointsas theset. set.
Figure7.7.ItItcan
canbebeseen
seenthat
thatthetheredfitting
curve
is fitted
withinlierpointsasdatathe data
Comparedsquaresalgorithm,
it isithardly
affectedby noisein theThis This
Comparedwith
withthetheleast
least
squaresalgorithm,
is hardly
affectedby noiseindata.
the data.
indicates
algorithm
is effective
andandhashasimportantsignificancefor theindicatesthat
thatthetheRANSACRANSAC
algorithm
is effective
importantsignificancefor thesubsequent
R-valuecalculation.
subsequent R-value calculation.

Figure6.6.Thickness
Thicknessstrainstraindata
is classified
inlieroutlier.
Figuredata
is classified
inlierandandoutlier.

Metals 2021, 11, 1401
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Figure7.7.Fitting
Fittingcurve
curvebyby
RANSAC
algorithm.
FigureRANSAC
algorithm.

3.3.DP980
DP980R-Value
R-ValueDetermination
DeterminationIn
this
section,the
R-valuecalculation
methodis reviewed
first, followed
by a discusIn this section, theR-valuecalculation
methodis reviewed
first, followed
by a dission of the experimental setup and process. The composite material DP980 was used as thecussion of the experimental setup and process. The composite material DP980 was used
experimental sample. Finally, the experimental results are analyzed and discussed.
as the experimental sample. Finally, the experimental results are analyzed and discussed.
3.1. R-Value Calculation
3.1. R-Value Calculation
The equation to calculate R-value is shown as Equation (8).
The equation to calculate R-value is shown as Equation (8).
ε2
R=
(8)
𝜀
ε3 = 2𝑅(8)
𝜀3where ε 2 , ε 3 represent the true width strain and true thickness strain, respectively.
where 𝜀2 , 𝜀3 represent the true width strain and true thickness strain, respectively.
For sheet metals, the R-value is usually determined from three different directions of
Forloading
sheet metals,
is usually
determinedthree
different
directions ofin-plane(0◦ , 45◦the, 90◦R-value
in the rolling
direction)and thefrom
normalR-valueis calculated
in-planeloading
(0°,
45°,
90°
in
the
rolling
direction)andthenormalR-valueis calculatedusing the following equation:using the following equation:
1
R = ( R01 + 2R45 + R90 )(9)(9)
𝑅4 = (𝑅0 + 2𝑅45 + 𝑅90 )4
3.2. Experiment System
The experimental
3.2. Experiment
Systemsystem is shown in Figures 8 and 9. Four GigE cameras (maximum
resolution 2752 × 2200) were used to form two independent DIC binocular measurement
The experimental system is shown in Figures 8 and 9. Four GigE cameras (maximum
systems. One DIC system was placed in front of the test sample, while the other DIC system
resolution 2752 × 2200) were used to form two independent DIC binocular measurement
was placed behind the test sample. Two pairs of 50 mm fixed focus lenses were used tosystems. One DIC system was placed in front of the test sample, while the other DIC sysachieve a 20 mm × 20 mm field of view. The spatial resolution was 0.08 mm/pixel and thetem was placed
behindtheTwotestpowerfulsample. LED
Two lights
pairs providedof 50 mmsufficient
fixed focus
lenses wereacquisition
rate was
10 Hz.
illumination
for usedto achieve
20 mm
20 mm field systems.
of view. A
The
resolution
0.08 mm/pixel
thefront anda backDIC×measurement
50 spatial
KN MTStensile testwas
machinewas used andtheacquisition
rate
was
10
Hz.
Two
powerfulLED
lightsprovidedsufficient
illumination
to do the tensile test, and a 1 mm thick dog bone shape sample made from DP980
sheet
for
the
front
and
back
DIC
measurement
systems.
A
50
KN
MTS
tensiletestmachinemetal was prepared for the test. The tensile speed was set at 3 mm/min. The slow tensile wasusedused
to do
test, anda 1images
mm thick
bone
shape
from
DP980
rate
forthethetensile
test enabledmore
to bedog
taken
beforethesample
sample madecracked,thus
providingmore
data
to analyze.
Figureis a tensile
schematic
diagramof thisCurrent
sheet metal
was
prepared
for thetest.10The
speed
was setat 3sample.
mm/min.
The slowtensile rate used for the test enabled more images to be taken before the sample cracked,
thus providing more data to analyze. Figure 10 is a schematic diagram of this sample.

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Current limitations with this system include the cost of the cameras used for the DIC systems and the large amount of space required to accommodate the front and rear camera
8 of 16
Currentgroups.
limitations with this system include the cost of the cameras used for the DIC systems and the large amount of space required to accommodate the front and rear cameragroups.
limitations with this system include the cost of the cameras used for the DIC systems andthe large amount of space required to accommodate the front and rear camera groups.

FigureFigure8.8.Schematic
Schematicdiagram
diagramof
ofthe
theexperimental
experimentalsystem.
system.

Figure 8. Schematic diagram of the experimental system.

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Figure9.9.Experimental
Experimentalset-up
set-upof
ofthe
theR-value
R-valuemeasurement.
measurement.
FigureFigure 9. Experimental set-up of the R-value measurement.

Figure10.
10.TheTheschematic
schematicdiagram
diagramof
ofDP
DP980.
980.
Figure3.3. Experimental Results Analysis and Discussion
The 3D contour and strain distribution on the surfaces of the DP980 was obtained
through DIC measurement. Figure 11 illustrates the simultaneous measurement results oflength strain (𝜀1 ) map, width strain (𝜀2 ) map and thickness strain (𝜀3 ) map before fracture

Metals 2021, 11, 1401

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Figure 10. The schematic diagram of DP 980.

3.3. Experimental Results Analysis and Discussion
3.3. Experimental
Results Analysis
anddistribution
Discussion on the surfaces of the DP980 was obtained
The 3D contour
and strain
The 3DDIC
contourand strainFigure
distribution
on thethesurfacesof the DP980
was obtained
throughmeasurement.
11 illustrates
simultaneous
measurement
results ofthroughDIC
measurement.
Figure11 illustrates
measurement
resultsoflengthstrain(ε 1 ) map, width
strain(ε 2 ) map theandsimultaneous
thickness strain(ε 3 ) map before
fracturelengthstrain(𝜀1 ) map,strain (𝜀area,
and thickness
strainof(𝜀a3 )strain
map before
fracture isin thetensiletest. width
The necking
whichis indicative
concentration,2 ) map
in thetensiletest. The necking area, which is indicative of a strain concentration, is clearlyclearlyvisible.
visible. R-value measurements were taken using two different methods, and these results
R-valuemeasurements
were taken
using
two different
methods,measurement,
and these results
werecomparedand analyzed.
The first
methodis the traditional
or indirect
measurement,
methodThebased
the constantvolume assumption.
Accordingto thewerecomparedand analyzed.
firstonmethod
is the traditional
measurement,
or indirect
assumptionmethodof constantvolume,= 0; thus,
the thickness
strain canmeasurement,
basedon theεconstantassumption.
Accordingtobe
thederived
as1 + ε 2 + ε 3 volume
from measurementsof the lengthstrain:sumptionof constant volume,
𝜀1 + 𝜀2 +and
𝜀3 =width
0; thus,the thickness strain can be derived frommeasurements of the length and width strain:
ε 3 = −ε 1 + ε 2
(10)
(10)
𝜀3 = −(𝜀1 + 𝜀2 )Based on the definition of R-value:
Based on the definition of R-value:ε2
R = 𝜀2(11)
𝑅 = −ε 1 + ε 2(11)−(𝜀1 + 𝜀2 )othermethodusedis thedirectmeasurement
method,or thedirectthickness
TheTheothermethodusedis thedirectmeasurement
method,or thedirectthicknessstrainmeasurement
methodproposedin
thispaper.
The
calculation
methodisshownstrain measurement method proposed in this paper. The calculation method is shown
in inEquation
(8).
Data
in
both
the
non-necking
andneckingarea
wasselectedfor
analysis
Equation (8). Data in both the non-necking and necking area was selected for analysis to tobettershow
difference
betweenmeasurement
methods.
bettershow
the thedifference
betweenthe thetwotwomeasurement
methods.

Figure11. Fullstraindistributionof sample
surfacemeasuredby DICsystem:(a) distribution
Figure11. Fullfieldfieldstraindistributionof sample
surfacemeasuredby DICsystem:(a) distributionof length
strain;(b) distributionof widthstrain;(c) distributionof thickness
strain.
of length
strain;(b) distributionof widthstrain;(c) distributionof thickness
strain.

3.3.1. Non-Necking Area
A square area of 8 × 8 pixels, as shown in Figure 12a, was selected instead of a single
point in the non-necking area to reduce the influence of noise on the data. Each pixel
represents 0.2 mm. The calculation results for the traditional and proposed methods areshown in Figure 13. According to the definition of R-value, the R-value for DP980 is equal
to the slope of these two curves. It can be seen from the figure that the relationship between
thickness strain and width strain is approximately linear, and the amount of noise in thedata is low. Therefore, the least squares principle can be used to calculate the slope ofthe two curves. The directly measured R-value is 0.8546, while the indirectly measured
R-value is 0.8656. Based on previous research of the properties of DP980 material, its

resents 0.2mm.
The calculation
results
for onthethetraditional
methods arepoint in the non-necking
area
to reduce
the influenceof noisedata. Each andpixelproposed
repshown
in Figure 13.
Accordingthe definition
R-value,methods
the R-value
resents 0.2 mm.
The calculation
results
for thetotraditionaland ofproposed
are for DP980 is equal
to13.
theAccording
slope of tothese
two curves.
It can bethe
seen
fromfortheDP980
figureisthat
the relationship beshown in Figure
the definitionof R-value,
R-valueequal
tween
thicknessstrainandwidthstrainisapproximately
linear,andtheto the slope of these two curves. It can be seen from the figure that the relationship be- amount of noise
Metals 2021, 11, 1401
10theof 16slope
the data
low. strain
Therefore,
the least squares
can be used
to calculate
tween thicknessinstrain
andis
width
is approximately
linear,principle
and the amountof noisein the data is low.
Therefore,
the least
canR-value
be usedisto0.8546,
calculate
the theslopeof thetwo curves.
Thesquares
directlyprinciple
measuredwhile
indirectly measuredof the two curves.
The
directly
measured
R-valueis0.8546,while
theindirectly
measured
R-value is 0.8656. Based on previous research of the properties of DP980 material, its RR-value is 0.8656.
Basedon previous researchof thepropertiesof DP980
material,
its R- composition [24].
value
generally
between
0.7–1.0,
dependingmaterial’s
R-value
generally ranges
ranges between
0.7–1.0,
dependingonon
thethematerial’s
composition [24].
value generallyTherangesbetween
0.7–1.0,
dependingon thematerial’s
composition[24].
results
measuredbothmethods
canbe consideredreasonablethey
The results
measuredby byboth
methods
can beconsideredreasonablebecausebecause
they bothfall both
The results measuredbybothmethods
canbeconsideredreasonablebecausethey
bothfall within
this range,andthe differencein thebetween
results between
the twoismethods
is very
withinthis range,
and thedifferencein the results
the two methods
very small.
fall within thisTherefore,
range, Therefore,
andboththe methods
difference
the results
between the two methods is verysmall.
both methods
are acceptable.
areinacceptable.
small. Therefore, both methods are acceptable.

Figure12.
of ofR-value:(a)Non-necking
area;(b) Necking
area.area.
Figure 12. Calculationof
R-value: area(a)areaNon-necking
area;(b) Necking
area.
Figurearea
12.Calculation
Calculation
R-value:(a)Non-necking
area;(b) Necking

Figure 13. The R-value curve by two different measurement methods in the non-necking area.

Figure13.
13.The
TheR-value
R-valuecurve
twodifferent
measurement
methods
innon-necking
the non-necking
Figure
curve
byby
two
different
measurement
methods
in thearea. area.

3.3.2. Necking Area
When necking occurs during a tensile test, a large amount of strain is disproportionately distributed in a central location of the material. As the deformation continues, thestrain in the necking area continues to increase until the material ruptures. In the necking
stage, the cross-sectional area of the material decreases drastically, while the strain in the
thickness and width directions increases sharply. Figure 14 displays an example of the
sharp strain increase that can occur during the necking stage.

Metals 2021, 11, 1401

When necking occurs during a tensile test, a large amount of strain is disproportionately distributed in a central location of the material. As the deformation continues, thestrain in the necking area continues to increase until the material ruptures. In the necking
stage, the cross-sectional area of the material decreases drastically, while the strain
in the11 of 16
thickness and width directions increases sharply. Figure 14 displays an example of the
sharp strain increase that can occur during the necking stage.

Figure14.
14.Width
Widthandand
Thicknessstrainhistory.
FigureThicknessstrainhistory.

An
wasselectedin thenecking
areaarea
to obtain
straindata,data,
as as
An88××88pixel
pixelsquaresquarearea
area
wasselectedin thenecking
to obtain
strainshown
inFigure12b.
Figure15
illustrates
theR-valuemeasuredbythetwo
methods
intheshown in Figure 12b. Figure 15 illustrates the R-value measured by the two methods inneckingarea. Inthe Inlinearstage, thestage,
thicknessis proportionaltothe necking
area.
the elastic
linear deformation
elastic deformation
the strain
thicknessstrain is proporthe
widthstrain.
The
R-value
at
this
time
is
a
constant,
andtheresults
measuredbythetwotional to the width strain. The R-value at this time is a constant, and the results measured
methods
are methods
0.8471 andare
0.8526,
respectively;
theseresults arethesesimilarto thoseby the two
0.8471and 0.8526,
respectively;
results
aremeasured
similar tointhose
the non-necked area. This shows that the material is uniformly distributed, stable in nature,measured in the non-necked area. This shows that the material is uniformly distributed,
and isotropic. When the width strain exceeds 60 mm/m, the strain value increases rapidly,
stable in nature, and isotropic. When the width strain exceeds 60 mm/m, the strain value
and the material deformation enters the necking stage. It is worth noting that the direct
increases rapidly, and the material deformation enters the necking stage. It is worth noting
method measurement curve no longer increases linearly; the growth rate reduces, and thethat thecorrespondingly
direct method measurement
curve
noshows
longerthat
increases
linearly;
growthR-valuebecomes smaller.
This
the abilityof the thematerial
to ratereduces,andcorrespondingly
becomessmaller.toThis
thatthethedirect
ability ofresiststraininthe
the R-value
thicknessdirection is reduced.
Comparedthe showscurve forthematerial
toresiststraininthethicknessdirectionisreduced.
Comparedtothemeasurement method, the curve for the indirect measurement method remains linear in thecurvefor thedirectstage,
measurement
method,thefor theindirect until
measurement
method reearlynecking
and the R-value
does
notcurve
decreasesignificantly
the later necking
mains This
linearresultin theearlyconfirms
necking somestage,researcher
and the R-value
decreasedoes
significantly
stages.
further
opinionsdoes
that notthe material
notnecessarily
follow
the constantassumption
in the necking[12]. Therefore,
until the laterneckingstages. volume
This result
further confirms
somestage
researcher
opinions thatthethe direct follow
measurement
methodis moreassumption
reliable andinaccurate
theR-valuematerialmeasured
does notby
necessarily
the constantvolumethe necking
intheneckingstage.
It
should
be
noted
that
some
materials
dofollowtheassumption
stage [12]. Therefore, the R-value measured by the direct measurement method isofmore
constantin the inneckingstage. The
R-values
for DP980
in therolling
directions
reliable volume
and accurate
the necking
stage.
It should
be noted
thatthreesomematerials
do follow
arein Table
1. Thevolume
R-values
three rollingare verysimilar,thedisplayed
assumptionof constant
in inthetheneckingstage. directions
The R-values
for DP980
in theindicatingthatthematerialhas
similarresistancetothicknessdeformationindifferent
three rolling directions are displayed in Table 1. The R-values in the three rolling direcrolling directions.
tions are very similar, indicating that the material has similar resistance to thickness deformationin different
3.4.
Additional
Verificationrolling
Tests directions.
Two additional materials were used for uniaxial tensile tests to verify the repeatabilityof the method proposed in this paper: aluminum alloy 6061 and a polymer material.

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12 of 1Figure15.
15.TheThe
R-value
curve
by twodifferent
measurement
methods
in the area.
necking area.
FigureR-value
curve
by twodifferent
measurement
methods
in the necking
Table1. 1.
R-valueof DP980
in different
rolling
direction.
Table
R-valueof DP980
in different
rolling
direction.

Rolling
Direction
Rolling
Direction0◦

0°

45◦ 45°90◦ 90°

Sheet
Metal
Sheet
Metal

R-ValueR-Value

0.86560.8734 0.8734
0.8853 0.8853
0.8749 0.8749
0.86563.4. Additional
3.4.1.
AluminumVerification
Alloy 6061 Tests

ATwo
3 mm
thick dog-bone
shapedalloy 6061
samplewasforrepeatabiaadditional
materials
werealuminum
used for uniaxial
tensiletests
to utilized
verify thetensiletest,method
and the test
conditions
wereconsistent
with those
DP980.
The nonity of the
proposedin thispaper:
aluminumalloyused
6061foranda polymer
material.
necking and necking areas were used to analyze the R-value of aluminum alloy 6061.
Figures
16 and 17 show
the6061
R-value curves of the non-necking area and the necking area
3.4.1. AluminumAlloyusing the direct measurement method and the indirect measurement method, respectively.
A 3 mmthickdog-boneshapedalloy 6061
sample before
was utilized
All thickness
straindata
have been
filteredaluminumusing the RANSAC
algorithm
plottingfor a ten
sileR-value
test, andthe test
conditions
weretheconsistent
with
those
used for DP980.
Theofnon-neckthecurves.
It canbe seen fromfigures thatthetwo measurement
curvesingnon-necking
and necking
areas
wereclose.
usedTheto analyze
R-valueof aluminumalloy 6061.
thearea
are veryR-values theobtainedby fitting
the two curves
with Figure
leastsquaresare 0.4895
and 0.4925,
respectively.
However, thedatain the necking
16 and17 show
the R-value
curvesof the non-neckingareaandtaken
the necking
area using th
area
shows
the curves ofmethod
the two measurement
methods
are different in
the necking
stage
directmeasurement
and the indirect
measurement
method,respectively.
A
beforefracture.
The
thicknessstrainofthedirectmeasurement
methodincreases
more
thickness strain data have been filtered using the RANSAC algorithm before plotting th
rapidly than that resulting from the indirect measurement method; thus, the R-value
R-value curves. It can be seen from the figures that the two measurement curves of th
becomes smaller. The R-value curve change of this aluminum alloy 6061 based on direct
non-neckingisarea
are veryclose.
The R-values
obtainedby assumption
fitting the two
curvesmeasurement
similarto thatof DP980.
The constantvolumeis nolongerwith leassquares inarethe0.4895
and
0.4925,
However,taken structurein the necking
arefollowedneckingstage.
Therespectively.
possible reason
for this isthe
thatdata
the crystal
of
shows
the
curvesofthetwomeasurement
methods
aredifferent
intheneckingstage
bethe metal material is dislocated during the necking process.

fore fracture. The thickness strain of the direct measurement method increases more rap
idly than that resulting from the indirect measurement method; thus, the R-value become
smaller. The R-value curve change of this aluminum alloy 6061 based on direct measure
ment is similar to that of DP980. The constant volume assumption is no longer followein the necking stage. The possible reason for this is that the crystal structure of the metamaterial is dislocated during the necking process.

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13 of 16

Figure16.
16.R-value
R-valuecurves
curvesof
ofaluminum
aluminumalloy
alloy6061
6061ininthe
thenon-necking
non-neckingarea.
area.
FigureFigure 16. R-value curves of aluminum alloy 6061 in the non-necking area.

Figure 17. R-value curves of aluminum alloy 6061 in the necking area.
Figure17.
17.R-value
R-valuecurves
curvesof
ofaluminum
aluminumalloy
alloy6061
6061ininthe
thenecking
neckingarea.
area.
Figure3.4.2. Polymer
3.4.2.
3.4.2. Polymer
Polymer
TheR-value
R-value ofthe
the polymermaterialmaterial is measuredinin thissection.
section. A2.5
2.5 mm thick
thick dogThe
The R-valueofof thepolymer
polymer materialisismeasured
measured inthis
this section. A
A 2.5 mm
mm thick dogdogbone specimenwas
was usedfor
for uniaxialtensile
tensile experiments.Since
Since thepolymer
polymer materialisisaa
bonebonespecimen
specimen wasusedused foruniaxial
uniaxial tensileexperiments.
experiments. Sincethe
the polymermaterialmaterial is atypical brittlematerial,material, thematerialmaterial doesnot
not neckduring
during thetensile
tensile test.Any
Any rectangulartypicaltypicalbrittle
brittle material,thethe materialdoes
does notneck
neck duringthe
the tensiletest.
test. Anyrectangular
rectangular
area
of
8
×
8
pixelsonthematerial
surfacecanbeselectedfor
R-valuecalculation.
Figure
area
of
8
×
8
pixelsonthematerial
surfacecanbeselectedfor
R-valuecalculation.
Figure18
area of 8 × 8 pixels on the material surface can be selected for R-value calculation.
Figure18shows
the
R-valuecurvesofthetwomeasurement
methodsforthismaterial.
It
canbe
be
shows
the
R-valuecurvesofthetwomeasurement
methodsforthismaterial.
Itcan
18 shows the R-value curves of the two measurement methods for this material. It can be
found
that
the
measurement
resultsofthetwo
methods
arelinearly
increasing,
but
the
found
found thatthat thethe measurement
measurement results
results ofof thethetwotwomethods
methodsare
arelinearlylinearlyincreasing,increasing,butbutthethegrowth rate is quite different. The R-value of the direct measurement is 0.3101, while thegrowth rate is quite different. The R-value of the direct measurement is 0.3101, while theMetals 2021, 11, 1401

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growth rate is quite different. The R-value of the direct measurement is 0.3101, whiletheindirectmeasurement
resultis 0.2499.
This
statespolymermaterialdoes
indirectmeasurement
resultis 0.2499.
This
statesthatthat
thethepolymermaterial
does
notnotenentirely
followtheconstant
volumeassumption
inthetensiletest.
Onepossible
explanation
tirely follow the constant volume assumption in the tensile test. One possible explanation
isisthetheconstantconstant volume
volume assumption
assumption isis based
based onon thethe crystal
crystal structure
structure ofof metal
metal materials,
materials,
whilepolymers
arecomposedofpolymercompounds
withcomplexstructures
while polymers are composed of polymer compounds with complex structuresandandno
nolongerlongerfollow
followthe
thedeformation
deformationconditions
conditionsof
ofmetalmetalmaterials.
materials.

Figure18.
18.R-value
R-valuecurves
curvesof
ofpolymer.
polymer.
Figure4.4.Conclusions
Conclusions
A
Anew
newR-value
R-valuemeasurement
measurement method
method based
based ononthe
theuseuseof
oftwo
twostereo-DIC
stereo-DICsystems
systemshas
beenpresented
in
this
paper.
This
methodprovidesanewpossibility
has been presented in this paper. This method provides a new possibilityfor
fordirect
directandandsimultaneous
simultaneousmeasurement
measurementof
ofstrains
strainsin
inthe
thelength,
length,width,
width,and
andthickness
thicknessdirections,directions,thus
thus
allowing
allowingfor
fordirect
directmeasurement
measurementof
ofthe
theR-value.
R-value.During
Duringthethemeasurement
measurementprocess,process,ititwas
was
found
that
DP980
and
aluminum
alloy
6061
do
not
followtheconstant
volumeassumption
found that DP980 and aluminum alloy 6061 do not follow the constant volume assumpin
theinnecking
stage.
It wasalsoalso
foundthata polymer
materialtion
the necking
stage.
It wasfoundthata polymer
materialdoes
doesnot
notfollow
followthis
thisassumption
duringtheentiretensile
test.
The
test
results
havecertainsignificanceassumption during the entire tensile test. The test results have certain significance forforthetheresearchdifferentmaterials.
materials.Compared
Comparedwith
with thethe traditional
research
of ofdifferent
traditional R-value
R-value measurement
measurement
method,method,thisthismethod
methodhas
hasthetheadvantages
advantagesof
ofdirectdirectmeasurement
measurementwithout
withoutassumption.
assumption.The
The
constant
assumption has
hascertain
certainlimitations,limitations,especially
especiallynecking
stage
of
constant volume
volume assumption
in inthethenecking
stage
of mamaterial
tensiledeformation
andpolymertensiledeformation.
Thus,
the
R-valueobtained
terial tensile deformation and polymer tensile deformation. Thus, the R-value obtained byby the direct measurement method has more research value.
the direct measurement method has more research value.
A main challenge to measuring strains in the third direction is that the strain data
A main challenge to measuring strains in the third direction is that the strain data inin this direction has more noise than other two strains due to the small value of thethis direction has more noise than other two strains due to the small value of the thickness.
thickness. A standard DIC algorithm cannot be applied for such a measurement without
A standard DIC algorithm cannot be applied for such a measurement without using ausing a special noise-removing algorithm, such as RANSAC presented in this paper. Some
special noise-removing algorithm, such as RANSAC presented in this paper. Some appliapplication limitations include the cost of four cameras and the large space requirement of
cation limitations include the cost of four cameras and the large space requirement of thethe front and rear camera groups. Nevertheless, this paper provides an alternative method
front and rear camera groups. Nevertheless, this paper provides an alternative methodthat directly measures R-value. The results for direct R-value measurement, as well as thethat directly measures R-value. The results for direct R-value measurement, as well as thecomparison with the indirect R-value method, show that the method and the algorithm
comparison with the indirect R-value method, show that the method and the algorithm
proposed is feasible and effective.
proposed is feasible and effective.
This paper has directly and indirectly measured the R-value of different materials,
This paper
has directly
andmeasurement
indirectly measured
theandR-value
ofthe
differentmaterials,
compared
the resultsof the two
methods,
shown
accuracyof thecompared the results of the two measurement methods, and shown the accuracy of thedirect measurement method. However, these measurement results are based on uniaxial

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direct measurement method. However, these measurement results are based on uniaxial
tensile tests. Thus, it cannot completely reveal the reasons and limitations of the constantvolume assumption. Future work may be necessary to study the microstructure of different
materials before and after uniaxial tensile tests and determine the scope of application of
the constant volume assumption through the analysis of the material microstructure.
Author Contributions: Conceptualization, S.F. and L.Y.; methodology, S.F., X.Z., B.Z.; validation, S.F.,
G.Z. and B.Z.; formal analysis, B.G.; investigation, G.Z.; resources, S.F., X.Z., B.Z.; data curation, S.F.,
G.Z.; writing—original draft preparation, S.F.; writing—review and editing, L.Y.; visualization, S.F.,
G.Z.; supervision, L.Y.; project administration, L.Y. All authors have read and agreed to the publishedversion of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data sharing not applicable.
Acknowledgments: The authors would like to express their sincere thanks to Bernard Sia, who madecareful correction of the manuscript as well as the English.
Conflicts of Interest: The authors declare no conflict of interest.

References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

ASTM. E517–00. Standard Test Method for Plastic Strain Ratio R for Sheet Metal; American Society for Testing and Materials: WestConshohocken, PA, USA, 2006.
Lankford, W.T.; Snyder, S.C.; Bausher, J.A. New criteria for predicting the press performance of deep drawing sheets. Trans. Am.
Soc. Met. 1950, 42, 1197–1232.
Stickels, C.; Mould, P. The use of Young’s modulus for predicting the plastic-strain ratio of low-carbon steel sheets. Metall. Mater.
Trans. B 1970, 1, 1303–1312. [CrossRef]
Ghosh, M.; Miroux, A.; Kestens, L. Correlating r-value and through thickness texture in Al–Mg–Si alloy sheets. J. Alloys Compd.
2015, 619, 585–591. [CrossRef]
Moreau, A.; Lévesque, D.; Lord, M.; Dubois, M.; Monchalin, J.-P.; Padioleau, C.; Bussière, J. On-line measurement of texture,
thickness and plastic strain ratio using laser-ultrasound resonance spectroscopy. Ultrasonics 2002, 40, 1047–1056. [CrossRef]
Huang, M.; Cheng, H. Determination of all elastic and plastic parameters for sheets of cubic metals only by uniaxial tension tests.
Eur. J. Mech. A/Solids 2015, 49, 539–547. [CrossRef]
Bo, W.; Chen, X.-H.; Pan, F.-S.; Mao, J.-J.; Yong, F. Effects of cold rolling and heat treatment on microstructure and mechanicalproperties of AA 5052 aluminum alloy. Trans. Nonferrous Met. Soc. China 2015, 25, 2481–2489.
Seiner, H.; Bodnárová, L.; Sedlák, P.; Janeček, M.; Srba, O.; Král, R.; Landa, M. Application of ultrasonic methods to determine
elastic anisotropy of polycrystalline copper processed by equal-channel angular pressing. Acta Mater. 2010, 58, 235–247. [CrossRef]
Mao, W.; Ma, Q.; Feng, H.; Chen, C.; Li, J. On-line determination technology on Lankford parameter of deep drawing aluminum
sheets. Chin. J. Nonferrous Met. 2006, 16, 1149.
Huh, J.; Kim, Y.; Huh, H. Measurement of R-values at intermediate strain rates using a digital speckle extensometry. In Dynamic
Behavior of Materials; Springer: Cham, Switzerland, 2011; Volume 1, pp. 317–322.
Esmaeilizadeh, R.; Khalili, K.; Mohammadsadeghi, B.; Arabi, H. Simulated and experimental investigation of stretch sheet
forming of commercial AA1200 aluminum alloy. Trans. Nonferrous Met. Soc. China 2014, 24, 484–490. [CrossRef]
Xie, X.; Li, J.; Sia, B.; Bai, T.; Siebert, T.; Yang, L. An experimental validation of volume conservation assumption for aluminumalloy sheet metal using digital image correlation method. J. Strain Anal. Eng. Des. 2017, 52, 24–29. [CrossRef]
Chu, T.; Ranson, W.; Sutton, M.A. Applications of digital-image-correlation techniques to experimental mechanics. Exp. Mech.
1985, 25, 232–244. [CrossRef]
Chen, X.; Xu, N.; Yang, L.; Xiang, D. High temperature displacement and strain measurement using a monochromatic lightilluminated stereo digital image correlation system. Meas. Sci. Technol. 2012, 23, 125603. [CrossRef]
Helm, J.D.; McNeill, S.R.; Sutton, M.A. Improved three-dimensional image correlation for surface displacement measurement.
Opt. Eng. 1996, 35, 1911–1920. [CrossRef]
Chen, F.; Chen, X.; Xie, X.; Feng, X.; Yang, L. Full-field 3D measurement using multi-camera digital image correlation system. Opt.
Lasers Eng. 2013, 51, 1044–1052. [CrossRef]
Li, J.; Dan, X.; Xu, W.; Wang, Y.; Yang, G.; Yang, L. 3D digital image correlation using single color camera pseudo-stereo system.
Opt. Laser Technol. 2017, 95, 1–7. [CrossRef]

Metals 2021, 11, 1401

18.
19.
20.
21.
22.
23.
24.

16 of 16

Xie, X.; Li, J.; Zhang, B.; Sia, B.; Yang, L. Direct measurement of R value for aluminum alloy sheet metal using digital image
correlation. In International Digital Imaging Correlation Society; Springer: Cham, Switzerland, 2017; pp. 89–94.
Li, J.; Xie, X.; Yang, G.; Zhang, B.; Siebert, T.; Yang, L. Whole-field thickness strain measurement using multiple camera digital
image correlation system. Opt. Lasers Eng. 2017, 90, 19–25. [CrossRef]
Vendroux, G.; Knauss, W. Submicron deformation field measurements: Part 2. Improved digital image correlation. Exp. Mech.
1998, 38, 86–92. [CrossRef]
Zhang, Z. A flexible new technique for camera calibration. IEEE Trans. Pattern Anal. Mach. Intell. 2000, 22, 1330–1334. [CrossRef]
Fischler, M.A.; Bolles, R.C. Random sample consensus: A paradigm for model fitting with applications to image analysis andautomated cartography. Commun. ACM 1981, 24, 381–395. [CrossRef]
Song, J.; Yang, J.; Liu, F.; Lu, K. High temperature strain measurement method by combining digital image correlation of laser
speckle and improved RANSAC smoothing algorithm. Opt. Lasers Eng. 2018, 111, 8–18. [CrossRef]
Li, Q.; Zhang, H.; Chen, F.; Xu, D.; Sui, D.; Cui, Z. Study on the plastic anisotropy of advanced high strength steel sheet:
Experiments and microstructure-based crystal plasticity modeling. Int. J. Mech. Sci. 2020, 176, 105569. [CrossRef]