Optics and Lasers in Engineering 48 (2010) 1285–1290

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Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Necking progression in tensile specimens monitored in real-timeby using fringe projection

Raul R. Cordero a,n, Amalia Martinez b, Juan A. Rayas b, Fernando Labbe c

a Universidad de Santiago de Chile, Avenida Libertador Bernardo O’Higgins 3363, Santiago, Chileb Centro de Investigaciones en Optica, A.C. Apartado Postal 1-948, C.P. 37000, Leon, Gto., Mexicoc Universidad Tecnica Federico Santa Marıa, Ave. Espana 1680, Valparaıso, Chile

a r t i c l e i n f o

Article history:

Received 20 April 2010

Received in revised form

2 June 2010

Accepted 2 June 2010Available online 19 June 2010

Keywords:

Fringe projection

Necking

Tensile test

66/$ - see front matter & 2010 Elsevier Ltd. A

016/j.optlaseng.2010.06.001

esponding author.

ail address: raul.cordero@usach.cl (R.R. Corde

a b s t r a c t

A specimen subjected to uniaxial tensile tests undergoes a thickness reduction that leads to out-of-

plane displacements. These can be approximated by monitoring the progression of the changes in shape

induced on the specimen. The combination of the fringe projection technique and the Fourier transform

method (FTM) allowed us to monitor in real-time the out-of-plane displacement fields, induced on a

brass sheet specimen in different regions of the tensile test. These maps enabled us to detect different

trends in the deformation process and nonlinear effects linked to the progression of the thickness

necking. Methodological details are provided below.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

A specimen subjected to a tensile test initially undergoeselastic deformation that is followed by a transition zone to theplastic deformation stage. Although during the transition zone thedeformation is still stable, the strains begin to localize within arelatively broad zone known as diffuse necking. The stabledeformation, with continuously rising load, is followed, afterreaching the maximum tensile force, by the instability zonewhereby a local neck is produced due to strong localization ofdeformation. Because in-plane strains are associated with out-of-plane deformations, the detection of the thickness reduction oftensile specimens can be used to monitor the progress of both thediffuse necking and the localized necking.

The whole-field deformation progression on tensile specimenscan be monitored by using interferometric techniques such aselectronic speckle-pattern shearing interferometry (ESPSI) [1–3],electronic speckle-pattern interferometry (ESPI) [4,5] and whole-field subtractive Moire (WSM) [6,7]. By using interferometerswith in-plane sensitivity [8,9], ESPI and WSM have been used tovisualize the onset of strain localization. However, due to theirhigh-sensitivity, these interferometric techniques cannot beused to efficiently measure the relatively large deformations thata sample undergoes during the plastic deformation. Therefore,regarding tensile specimens, interferometric techniquesare nowadays mostly applied to monitor the strain rates by

ll rights reserved.

ro).

measuring small changes in the strains induced at known timeintervals [10–13].

By using interferometers with out-of-plane sensitivity, ESPIand WSM have been also used to follow the thickness reduction,associated with the strain localization onset [14,15]. However,the high-sensitivity of the applied inteferometric techniquesrestricted those measurements such that thickness reductionsonly up to 10 mm could be measured. Nevertheless, by analyzingthe out-of-plane displacements between close load stages, it wasobserved in [14,15] that, during the transition zone, sheet metalspecimens were thinned mainly along a relatively broadband,which seemed to be linked with the diffuse necking progress.

Monitoring greater thickness reductions as those expected atthe localized necking requires applying whole-field opticaltechniques of greater dynamical range than the interferometry-based methods, such as those based on the fringe projectiontechnique [16–21]. As pointed out in [16], these techniques areoften referred to as the particular fringe analysis method used inthe measurement: phase stepping profilometry (PSP) [17,18],Fourier transform profilometry (FTP) [19], wavelet transformprofilometry (WTP) [20], etc. In [21], a comparison is presented.

The fringe projection technique for contouring involvesprojecting gray-code fringes onto the surface of the specimenthat are captured by using a camera. The application of load leadsto changes in the fringe pattern, which are related to the out-of-plane displacements induced on the surface by the load applica-tion. The technique has been successfully applied to monitorchanges in shape that involved out-of-plane deformations greaterthan 10 mm [22–25].

We used a combination (often referred to as FTP) of the fringeprojection technique and the Fourier transform method (FTM) in

R.R. Cordero et al. / Optics and Lasers in Engineering 48 (2010) 1285–12901286

order to follow the progression of the changes induced on thetopography of a brass sheet sample subjected to uniaxial tensiletests. These techniques allowed us to estimate the real-time out-of-plane displacement fields, induced on the sample in differentregions of the test. Our attention was mostly paid to thicknessreductions undergone by the specimen during the transition zone(within which the diffuse necking begins to form) as well asimmediately after reaching the maximum tensile force (when thelocalized necking appears). During the transition zone, weobserved that the tested sample was thinned along a relativelybroadband that we identified as the diffuse necking. Moreover,after reaching the maximum tensile force, the thickness reduc-tions undergone by the specimen exhibited the remarkabledegree of localization normally associated with the localizednecking.

2. Fringe projection

This technique for contouring involves projecting gray-codefringes onto the surface of the specimen and then viewing fromanother direction. Fig. 1 shows the required setup; a computerprojector can be used as illuminating source. If the illuminationdistance is adequately large, the CCD camera in Fig. 1 captures,from the initially flat surface of the specimen, an intensity patternI of nearly straight equally spaced fringes of period p

Iðx,yÞ ¼ Aðx,yÞþBðx,yÞcos2pp

xþyðx,yÞ

� �, ð1Þ

where A and B are functions of position and y is the modulationterm related to perspective and optical aberration effects (see [26]for further discussions on the aberration dependence on thegeometry of the object). As the specimen undergoes a tensile test,the out-of-plane deformation leads to the departure of the viewedfringes from straight lines such that a new intensity pattern I canbe captured by the camera

Iðx,yÞ ¼ Aðx,yÞþBðx,yÞcos2pp

xþyðx,yÞþfðx,yÞ

� �, ð2Þ

Fig. 1. Optica

where f is the phase modulation due to the thickness reductioninduced by the deformation.

Note that the arguments 2pp xþy x,yð Þ

� �and 2p

p xþy x,yð Þþ�

f x,yð ÞÞ in Eqs. (1) and (2), respectively, can be efficiently retrievedby applying the Fourier transform method (FTM) [27]. The FTMinvolves the Fourier transform application to the fringe patterns,the isolation of the term carrying the phase information byapplying a band-pass filter in the spatial frequency domain andthen the inverse Fourier transform application.

The subtraction of arguments in Eqs. (1) and (2) allowsisolating the phase modulation f and cancelling perspective andaberration effects; therefore, no absolute calibration of the systemis required. If the in-plane displacement adequately small, fencodes information only on the out-of-plane displacement W

(the departure of the surface from the initially surface) induced onthe illuminated surface by the application of load. According to[28,29], the phase modulation f is related to the correspondingout-of-plane displacement W by

Wðx,yÞ ¼fðx,yÞ

2pp

tana

� �, ð3Þ

where a is the angle between the observation direction and theillumination direction.

Note that in general, f stands for the phase modulationinduced between any pair of load stages. When the FTM is used toretrieve the phases at two load stages, the corresponding phasedifference f can be computed and then used (by applying Eq. (3))to evaluate the out-of-plane relative displacement W betweenthese two load stages. The fringe projection technique has beensuccessfully applied to monitor changes in the shape thatinvolved out-of-plane deformations greater than 10 mm [22–25].

3. Experimental details and data exploitation

We applied the fringe projection technique to monitor thenecking progression of a brass sheet sample (see Fig. 2) subjectedto a tensile test. The initial thickness of the samples was 0.6 mm.It was tested on an Instrons machine working in tension at

l Setup.

R.R. Cordero et al. / Optics and Lasers in Engineering 48 (2010) 1285–1290 1287

0.5 mm min�1 speed. The pulling direction was the y axis. Fig. 3shows a plot of the sequence of the brass sheet tensile test; it wasconstructed from the values of force and specimen elongationindicated by the display device of the testing machine.

We used the optical setup shown in Fig. 1. A projector wasutilized for projecting gray-code fringes onto the surface of thespecimen. a was equal to 351. The illumination distance wasselected adequately large, such that the CCD camera in Fig. 1captured, at the beginning of the test, nearly straight equallyspaced fringes of period p¼1 mm. The out-of-plane deformationinduced by the load application led to the departure of the viewedfringes from straight lines.

As the test progressed, we retrieved the whole-field phase mapby using FTM from the fringe patterns captured by the CCD. Wedecided to retrieve only every 0.5 s the whole-field phase map byusing FTM from the patterns captured by the CCD. This meansthat, for example, during the transition zone of the test (thatinvolved a total elongation of about 0.5 mm, see Fig. 3), 120whole-field phase maps were computed; during the whole testmore than 1500 3D measurements were carried out.

The subtraction of phase maps corresponding to two loadstages allowed us to calculate the phase modulation f due to thethickness reduction between these two load stages, and then tocompute the corresponding out-of-plane relative displacement W

by using Eq. (3).Because out-of-plane relative displacements are due to the

thickness reduction induced by the test, the detection of boththe diffuse necking and the localized necking implied monitoringthe W maps. Our attention was paid to monitoring the thicknessreductions undergone by the specimen during the transition zone(within which the diffuse necking begins to form) as well asduring the instability zone (immediately after reaching themaximum tensile force, when the localized necking appears).Both the transition zone and the instability zone are indicated byrectangles in Fig. 3. Besides the significance that the localizationstudies assign to these zones, we decide to monitor the changes inshape in relatively short stages of the tensile test in order to avoidsignificant effects on W due to the in-plane deformation of thesample.

Fig. 3. Force–displacement sequence of the brass sheet tensile test. Th

LoadLoad 25 mm 20 mmy

x

30 mm

r = 30 mm

Fig. 2. Specimen (Brass). The pulling direction was y. The illuminated area was

close to the center of the sample.

4. Results

Fig. 4 shows a sequence the whole-field maps of the relativedisplacement W induced by the deformation that the sampleunderwent during the transition zone (between d¼2.9 and3.4 mm, see Fig. 3). Note that plots in Fig. 4 do not stand forthe actual shape of the illuminated sample but they showthe progression of the displacements relative to the shape of theilluminated surface at d¼2.9 mm (when the transition began).This means that the phase map, retrieved by using FTM from thecorresponding fringe pattern at d¼2.9 mm, was taken asreference.

Although the whole-field W maps shown in Fig. 4 werecomputed every time by applying Eq. (3), f was calculated in thecase of Fig. 4a by subtracting the phase map at d¼2.9 mmfrom the reference; in the case of Fig. 4b, f was calculated bysubtracting the phase map at d¼3.0 mm from the reference; inthe case of Fig. 4c, f was calculated by subtracting the phase mapat d¼3.1 mm from the reference; in the case of Fig. 4d, f wascalculated by subtracting the phase map at d¼3.2 mm from thereference; in the case of Fig. 4e, f was calculated by subtractingthe phase map at d¼3.3 mm from the reference; and in the caseof Fig. 4f, f was calculated by subtracting the phase map atd¼3.4 mm from the reference.

The three-dimensional sequence depicted in Fig. 4 shows that,although during the transition stage of the test, the shear bandwas not formed yet and hence the specimen was still deformingeverywhere, the necking process began already to localize withina relatively broadband (of about 15 mm width) that we identifiedas diffuse necking.

The whole-field W maps shown in Fig. 4 were in sharp contrastto those obtained between any pair of load stages within theelastic zone of the test; before the elongation reached d¼2.9 mm,the sample did not exhibit any spatial variation in its out-of-planedeformation. This latter result was anyway expected because theconstant deformation that characterizes the elastic zone ofmechanical tests.

Fig. 5 shows a sequence the whole-field maps of the relativedisplacement W induced by the deformation that the sampleunderwent immediately after reaching the maximum tensileforce and just before the fracture (between d¼5.8 and 6.3 mm,see Fig. 3). Again note that plots in Fig. 5 do not stand forthe actual shape of the illuminated sample but they showthe progression of the displacements relative to the shape of theilluminated surface at d¼5.8 mm. Therefore, this time we took asreference the phase map, retrieved by using FTM, from thecorresponding fringe pattern at d¼5.8 mm.

In order to compute the whole-field W maps shown in Fig. 5 byapplying Eq. (3), f was calculated in the case of Fig. 5a bysubtracting the phase map at d¼5.8 mm from the reference; in

e rectangles indicate the transition zone and the instability zone.

Fig. 4. Relative displacement W induced on the sample during the transition zone. These displacements are relative to the shape of the illuminated surface at d¼2.9 mm,

when the transition began.

Fig. 5. Relative displacement W induced on the sample during the instability zone. These displacements are relative to the shape of the illuminated surface at d¼5.8 mm,

immediately after reaching the maximum tensile force.

R.R. Cordero et al. / Optics and Lasers in Engineering 48 (2010) 1285–12901288

the case of Fig. 5b, f was calculated by subtracting the phase mapat d¼5.9 mm from the reference; in the case of Fig. 5c, f wascalculated by subtracting the phase map at d¼6.0 mm from thereference; in the case of Fig. 5d, f was calculated by subtracting thephase map at d¼6.1 mm from the reference; in the case of Fig. 5e,f was calculated by subtracting the phase map at d¼6.2 mm fromthe reference; and in the case of Fig. 5f, f was calculated bysubtracting the phase map at d¼6.3 mm from the reference.

The sequence depicted in Fig. 5 shows that, immediately afterreaching the maximum tensile force, the thickness reductionsundergone by the specimen exhibited a remarkable degree oflocalization. In the instability zone of the test, the necking leads toa thinner band (of about 8 mm width) that we identified aslocalized necking.

Fig. 6 shows the out-of-plane displacements along x¼0 (onthe center of the observed area) induced by the deformation

Diffuse necking15 mm

Localized necking8 mm

0

-5

-10

-15

-20

-25 d=6.3mm

d=6.2mm

d=6.1mm

d=6.0mm

d=5.9mm

d=3.4mm

d=3.3mm

d=3.2mm

d=3.1mm

d=3.0mm

0

-10

-20

-30

-40

-10 -5 0 5 10 -10 -5 0 5

y (mm) y (mm)

w (µ

m)

w (µ

m)

Fig. 6. (a) Diffuse necking progression within the transition zone and (b) Localized necking progression after the maximum tensile force was reached.

x

y

15 mm

Fig. 7. Picture taken just after the fracture of the specimen.

R.R. Cordero et al. / Optics and Lasers in Engineering 48 (2010) 1285–1290 1289

(a) during the transition zone (between d¼2.9 and 3.4 mm) and(b) after reaching the maximum tensile force and just before thefracture (between d¼5.8 and 6.3 mm). It can be observed inFig. 6b that at the location where the greatest deformations tookplace, the sample underwent out-of-plane displacements of about40 mm. This necking represented a thickness reduction of about12% on the illuminated side of the specimen. As shown in Fig. 7,the fracture passed through the zone where we measured thegreatest out-of-plane displacement during the instability zone ofthe test.

Note that both Fig. 6a and b report on the out-of-planedisplacements that the sample underwent at comparable time

intervals. However, when compared with those shown in Fig. 6a,the W values in Fig. 6b show a faster thickness reduction. Thisresult agrees those reported in [12,13]; those prior efforts haveshown that the strain rates in tensile specimens augment with theplastic activity and this increment is linked with both the strainlocalization and the necking progression.

5. Summary and conclusions

We monitored the progression of out-of-plane displacementsinduced on a brass sheet sample subjected to uniaxial tensiletests. These out-of-plane displacements are due to the thicknessreduction induced by the test.

The out-of-plane relative displacements between any two loadstages can be measured by applying the fringe projectiontechnique. It implies projecting gray-code fringes onto the surfacethat are captured by using a camera. The fringe projectiontechnique involves calculating the difference between the phasescorresponding to fringe patterns captured at different load stages.In our case, these phases were retrieved by applying the Fouriertransform method (FTM). As shown above, the phase differenceallows calculating the out-of-plane relative displacements, whichare in turn related to the thickness reductions induced on thesurface by the application of load.

We paid special attention to the out-of-plane displacementsundergone by the specimen during the transition zone (withinwhich the diffuse necking begins to form) as well as immediatelyafter reaching the maximum tensile force (when the localizednecking appears).

We observed that during the transition zone, the sheet metalspecimen was thinned mainly along a relatively broadband.Although this thickness necking was expected, it exhibited certaindegree of localization The whole-field out-of-plane displacementmaps obtained during the transition zone were in sharp contrastto those obtained within the elastic zone of the test; before theelongation reached d¼2.9 mm, the sample did not exhibit anyspatial variation in its out-of-plane deformation. This was inagreement with the constant deformation that characterizes theelastic zone of mechanical tests. We identified the detectedbroadband as the diffuse necking because, during the transition tothe plastic deformation, the strains were not zero at any place ofthe field.

R.R. Cordero et al. / Optics and Lasers in Engineering 48 (2010) 1285–12901290

After reaching the maximum tensile force, the out-of-planedeformation of the sample exhibited a remarkable degree oflocalization. The sheet metal specimen was mostly thinned alonga thinner band that we identified as localized necking. Thelocalized necking preceded the development of the fracture and itis associated with the so-called strain localization. We were alsoable to note a significant growth in the thickness reduction rate,which was in agreement with the expected increment of theplastic activity within the instability zone, whereby the local neckdevelops.

We conclude that, in combination with FTM, the fringeprojection technique is a valuable tool for monitoring dynamicalchanges in shape that involve out-of-plane displacements of theorder of tens of micrometers.

Acknowledgements

The support of CONICYT (FONDECYT Preis 1090471, ANILLOPreis ACT98 and ANILLO Preis ACT95), UTFSM (DGIP Preis 250915)and USACH (DICYT, academic exchange program) is gratefullyacknowledged.

References

[1] Patorski K. Digital in-plane electronic speckle pattern shearing interferome-try. Opt Eng 1997;36:2010–5.

[2] Aebischer HA, Waldner S. Strain distributions made visible with image-shearing speckle pattern interferometry. Opt Laser Eng 1997;26:407–20.

[3] Labbe F, Cordero RR, Martınez A, Rodriguez-Vera R. Measuring displacementderivatives by electronic speckle pattern shearing interferometry (ESPSI).Meas Sci Technol 2005;16:1677–83.

[4] Rastogi PK, Jacquot P, Pflug L. Holographic interferometry at LSA: someselected examples of its application. Opt Laser Eng 1990;13:67–77.

[5] Martinez A, Rodriguez-Vera R, Rayas J, Puga H. Fracture detection by gratingmoire and in plane ESPI techniques. Opt Laser Eng 2003;39:525–36.

[6] Han B, Post D, Ifju P. Moire interferometry for engineering mechanics: currentpractices and future developments. J Strain Anal 2001;36:101–17.

[7] Post D, Han B, Ifju P. High sensitivity moire: experimental analysis formechanics and materials. New York: Springer-Verlag; 1994.

[8] Vial-Edwards C, Lira I, Martınez A, Munzenmayer M. Electronic specklepattern interferometry analysis of tensile tests of semihard copper sheets.Exp Mech 2001;41:58–62.

[9] Cordero RR, Franc-ois M, Lira I, Vial-Edwards C. Whole-field analysis of uniaxialtensile tests by moire interferometry. Opt Laser Eng 2005;43:919–36.

[10] Labbe F. Strain-rate measurements by electronic speckle-pattern interfero-metry (ESPI). Opt Laser Eng 2007;45:827–33.

[11] Guelorget B, Francois M, Vial-Edwards C, Montay G, Daniel L, Lu J. Strain ratemeasurement by electronic speckle pattern interferometry: a new look at thestrain localization onset. Mater Sci Eng A 2006;415:234–41.

[12] Cordero RR, Labbe F. Monitoring the strain-rate progression of an aluminumsample undergoing tensile deformation by electronic speckle-pattern inter-ferometry (ESPI). J Phys D: Appl Phys 2006;39:2419–26.

[13] Labbe F, Cordero RR. Monitoring the plastic deformation progression of aspecimen undergoing tensile deformation by moire interferometry. Meas SciTechnol 2005;16:1469–76.

[14] Cordero RR, Labbe F. Measuring out-of-plane displacements by electronicspeckle-pattern interferometry (ESPI) and whole-field subtractive moire.Meas Sci Technol 2006;17:825–30.

[15] Labbe F, Cordero RR. Detecting the beginning of the shear band formation inuniaxial tensile tests by out-of-plane displacement measurements. Opt LaserEng 2007;45:153–9.

[16] Gorthi SS, Rastogi P. Fringe projection techniques: Whither we are?Opt LasersEng 2010;48(2):133–40.

[17] Chen LC, Liao CC. Calibration of 3D surface profilometry using digital fringeprojection. Meas Sci Technol 2005;16:1554–66.

[18] Zhang S. Recent progresses on real-time 3D shape measurement using digitalfringe projection techniques. Opt Lasers Eng 2010;48(2):149–58.

[19] Su X, Zhang Q. Dynamic 3-D shape measurement method: A review. OptLasers Eng 2010;48(2):191–204.

[20] Zhong J, Weng J. Spatial carrier-fringe pattern analysis by means of wavelettransform: Wavelet transform profilometry. Appl Opt 2004;43(26):4993–8.comprsion de algunas.

[21] Quan C, Chen W, Tay CJ. Phase-retrieval techniques in fringe-projectionprofilometry. Opt Lasers Eng 2010;48(2):235–43.

[22] Barrientos B, Cywiak M, Lee WK, Bryanston-Cross P. Measurement ofdynamic deformation using a superimposed grating. Rev Mex Fıs 2004;50:12–8.

[23] Tay CJ, Quan C, Wu T, Huang YH. Integrated method for 3-D rigid-body displacement measurement using fringe projection. Opt Eng2004;43:1152–9.

[24] Quan C, Tay CJ, Huang YH. 3-D deformation measurement using fringeprojection and digital image correlation. Optik 2004;115(4):164–8.

[25] Barrientos B, Moore AJ, Perez-Lopez C, Wang L, Schudi T. Three-dimensionaldisplacement fields measured in a deforming granular-media surface bycombined fringe projection and speckle photography. Opt Eng 1999;38:2069–74.

[26] Fagg AH, Hales BS, Stahl HP. Systematic errors of a projection-moirecontouring system proc. SPIE 1992;vol. 1776:120.

[27] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringe-patternanalysis for computer-based topography and interferometry. J Opt Soc Am A1982;72:156–60.

[28] Barrientos B, Cerca M, Garcıa-Marquez J, Hernandez-Bernal C. Three-dimensional displacement fields measured in a deforming granular-mediasurface by combined fringe projection and speckle photography. J Opt A: PureAppl Opt 2008;10. 104027 (10).

[29] Gasvik KJ. Optical metrology. 3rd Edition. Chichester: Wiley; 2003.