Experimental Mechanics (2015) 55:1575–1590DOI 10.1007/s11340-015-0054-9

Effects of Various Shape Functions and Subset Size in LocalDeformation Measurements Using DIC

Xiaohai Xu1 ·Yong Su1 ·Yulong Cai1 ·Teng Cheng1 ·Qingchuan Zhang1

Received: 6 March 2015 / Accepted: 9 June 2015 / Published online: 30 June 2015© Society for Experimental Mechanics 2015

Abstract The digital image correlation (DIC) methodobtains comparable results with strain gauges and its reli-ability and accuracy are commonly accepted in the mea-surement of affine deformations. However, in engineeringmeasurements, there are always substantial local deforma-tions with high strain gradients, such as the Portevin-LeChatelier (PLC) shear bands, deformations near gaps, andcrack tips. In these situations, strain gauges are restrictedbecause the results within the contact areas are smoothed.Although the DIC method can be employed to measurethese local deformations, the calculation parameters (e.g.,the order of the shape functions, and template size) seri-ously impact the results. By analyzing PLC shear bandswith different gradients in tensile tests and simulated bands,the deep mechanism on how shape functions and tem-plates impact on the accuracy of DIC results is estab-lished. This study also demonstrates that second-order shapefunctions are more suitable than first-order shape func-tions to describe local deformations. The theory that theresults of second-order shape functions are reliable andaccurate when the relative error between first- and second-order shape functions is less than 10 %, is proposed. Inaddition, improving the spatial resolution and the acquisi-tion frequency is proposed, and proved to achieve reliableresults.

� Qingchuan Zhangzhangqc@ustc.edu.cn

1 CAS Key Laboratory of Mechanical Behavior and Designof Materials, University of Science and Technology of China,Hefei 230027, China

Keywords Digital image correlation · Localdeformations · Reliability · Shape functions · Templates

Introduction

The digital image correlation (DIC) method has been widelyused in non-contact measurements of displacements anddeformations. This method is the best option for two-and three-dimensional curved surface non-contact measure-ments in engineering because it has advantages such assimple experimental setup, minimal requirements in mea-surement environments, a wide range of measurements, andhigh accuracy [1–5].

Traditional measuring methods such as strain gauges canemployed, but the results are smoothed in the contact areaswith local deformations, such as the Portevin-Le Chatelier(PLC) shear bands [6, 7], deformations near gaps [4, 8], andcrack tips [9, 10]. The DIC method is able to solve this issue[11–15]. Zhang first employed the DIC method to measuredeformations in the PLC bands [11]. Other researchers theninvestigated PLC bands using the DIC method and ana-lyzed their mechanism [12–15]. Rethore et al. also discussedthe discontinuities of displacement fields using both sim-ulations and physical PLC bands [14]. However, there arestill doubts about the reliability [16] and accuracy [17, 18]of the results because there are neither accepted standardsnor comparisons of the measurements of local deformations.Changes in DIC calculation parameters such as the orderof the shape functions [4, 17, 18], template size [18, 19],grid step, the type and order of the interpolation functions[18, 20], and size of the strain window may lead to quitedifferent results, especially in local deformations with highgradients. In view of this, researchers often merely report

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qualitative results [8–15]. However, the reliability and accu-racy of the DIC results have always been a major concernof researchers and users. Questions have been raised, suchas how DIC calculation parameters affect the results in themeasurements of local deformations, how the results can bejudged as correct, whether there are any techniques for set-ting proper parameters to ensure correct results and, if so,how they are optimized, and whether there are any othertechniques for improving the results, for example, whetherchanges in the experimental setup or the specimen ensurecorrect results. All these issues need solutions so that DICusers can judge if the results are accurate.

This study takes the PLC shear bands occurring in tensiletests of alloy material specimens as examples in studying theabove problems. Section Shape Functions introduces shapefunctions with different orders. Section Tensile Test obtainsa PLC shear band occurrence by experiment and analysesof the DIC results. Section Simulations assumes commondisplacement and strain functions of the PLC shear bandsand generates speckle images with different gradient defor-mations using a computer program. Section Discussionsconcentrates on how shape functions and templates affectDIC results in different conditions. Section Improving theExperiment proposes an effective technique for achievingreliable results. Section Conclusions summarizes the study.

Shape Functions

The DIC method is a full-field non-contact optical measure-ment using tracking speckle markers within the image tem-plate region (subset). In this process, correlation functionswere used to evaluate the degree of similarity between thereference and deformed images. The inverse compositionalGauss–Newton algorithm [4, 21] with the robust zero-meannormalized sum of squared differences (ZNSSD) criterionwas used in this work to optimize the correlation functions.Sub-pixel displacements were obtained and the full strainfield was calculated by post-processing. In the optimiza-tion, the deformation mode of the subset, called the shapefunction, was assumed. In the case of two-dimensional(2D)-DIC, zero-order shape functions are commonly used(a pure translation):

[x′y′

]=

[1 0 u

0 1 v

] ⎡⎣ x

y

1

⎤⎦ (1)

and first-order shape functions (translation, rotation, shearand normal strains):

[x′y′

]=

[1 + ux uy u

vx 1 + vy v

] ⎡⎣ x

y

1

⎤⎦ (2)

In addition, second-order shape functions (concerningstrain gradients) were also used:

[x′y′

]=

[ 12uxx uxy

12uyy 1 + ux uy u

12vxx vxy

12vyy vx 1 + vy v

]⎡⎢⎢⎢⎢⎢⎢⎣

x2

xy

y2

x

y

1

⎤⎥⎥⎥⎥⎥⎥⎦

(3)

In Eqs. 1–3, (x, y) is the local coordinate whose originis the reference subset center in the reference image and(x′, y′) is the subset after deformation, u and v are the dis-placements of the subset center, ux , uy , vx and vy are thecorresponding displacement gradients, uxx , uxy , uyy , vxx ,vxy and vyy are the corresponding second-order displace-ment gradients. Second-order shape functions are supposedto achieve more accurate results in complex deformations[22].

Three orders of shape functions have been embedded intothe correlation module as an option in some commercialDIC software. The work below was carried out with the helpof PMLAB’s DIC-3D software.

Tensile Test

Experiment

To discuss the reliability of local deformation measure-ments using DIC method, a tensile test with alloy specimenwas performed. Figure 1(a) presents the hardware systembuilt. The specimen was dumbbell-shaped with a gauge60 mm long, 12 mm wide and 1 mm thick. The materialwas a commercial Al-based alloy (type 5456). The loadingspeed was 3 mm/min, the corresponding nominal strain rate8.4×10−4 /s, and the load sampling frequency 12.5 Hz. Thedetailed relationship between the stress and strain is shownin Fig. 1(b). The three-dimensional (3D)-DIC method wasused to avoid errors caused by out-of-plane displacements[23, 24]. The cameras’ sampling frequency was 5 Hz, andthe image resolution was 2448× 2048 pixels. The referenceimage captured by the left camera is shown in Fig. 1(c).

Results

Shape functions and templates are the main focus in thispaper, so the other DIC calculation parameters are all fixed.For example, the grid step is set at 1 pixel, the strain windowcontains 17 × 17 calculation points, and the interpolationfunction is the bi-septic B-spline function [21].

Figure 2 indicates strain results for different coordi-nates using first-order shape functions with a 29-pixel tem-plate and the DIC results calculated using different shape

Exp Mech (2015) 55:1575–1590 1577

Fig. 1 Experimental setup and data. a Hardware system. b Stress–strain curve with an inset indicating the capture time A and B during one stressserrated drop. c Reference speckle image cut into a small map

functions with different template sizes. The correspond-ing speckle images were captured at times A and B (asshown in Fig. 1(b) during one stress serrated drop, and thearea of interest (AOI) was 180 × 700 pixels. Figure 2(a1-a3,b1-b3) show all the strain components (in plane) in twocoordinate systems. Under the first coordinate system, the x-direction equals the tensile direction. Strain componets exx(Fig. 2(a1)) and eyy (Fig. 2(a2)) have the same distributionexcept for their signs and exy (Fig. 2(a3)) is nearly uniform.Under the second coordinate system, the t-direction is par-allel to the direction of the PLC band. Strain componetsenn (Fig. 2(b2)) and etn (Fig. 2(b3)) have the same distri-bution except for their signs and ett (Fig. 2(b1)) is nearlyuniform. The comparison between the two systems indi-cates that coordinate transforms do not change the natureof the strain. We conclude that the strain component in thetensile direction can represent the deformation occurred inthe PLC bands. At the bottom of Fig. 2(d1), the directionof the PLC shear band is given and the coordinate sys-tem is defined. Full-field u-displacements calculated usingfirst- and second-order shape functions with a 29-pixel tem-plate size are shown in Fig. 2(d1) and (d2). The obliquesteps correspond to the PLC shear bands in both displace-ment fields. The step in Fig. 2(d1) is wider than the step inFig. 2(d2). Because the full-field displacements and strainare difficult to assess because of random variations, anequivalent template shown in Fig. 2(c2) was used to smooththe displacements along the direction of the band. Theactual width of the template was the horizontal grid count,and the height-width ratio of the template was the tangentvalue of the oblique angle. Smaller displacement fields thesame size as the equivalent template were taken out alongthe x-direction. Each sub field multiplied by the equiva-lent template had a new displacement value. All the newdisplacements along the x-direction made up the smootheddisplacement curve. Figure 2(e1) shows the displacementcurves in the x-direction. The upside of the band shifts up

about 20 μm to the under-part, and the negative slope out-side the band represents the shrinkage deformation [11, 15].Full strain fields and smoothed strain curves are shown inFig. 2(e1-e3). There were also non-uniform strain fields cor-responding to the PLC shear bands. There were negligibledifferences in the strain fields and curves outside the band,however, the strain results inside the band were quite differ-ent from each other when different shape functions (first-and second-order) and templates (29 and 39 pixels) wereused. The relative errors reached 47 %, where the corre-sponding maximum strain values using second-order shapefunctions with a 29-pixel template and using first-ordershape functions with a 39-pixel template were respectively14300 and 9700 με.

Simulations

Non-linear Fitting

The above indicates that different shape functions and tem-plates may lead to quite different results even when otherDIC calculation parameters are fixed, especially in themeasurements of local deformations. This raises severalquestions. How do shape functions and templates impact onthe DIC results? Which of the above results is more accu-rate? Are there any better results? We use further analysis toanswer these questions.

exx(x) = a × e− (x−b)2

2c2 + d (4)

FWHM = 2√2 ln 2c (5)

We take the results of second-order shape functions witha 29-pixel template size as an example. The gauss functionequation (4) in Origin 9.1 was employed to fit the straincurve as shown in Fig. 3(a). The fitting parameters obtainedwere: a = 0.015, b = 192 pixels, c = 12 pixels, and d = -59

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Fig. 2 Strain maps in differentcoordinate systems anddisplacement u and strain exxcalculated by different shapefunctions with different templatesizes. (a1-a3, b1-b3) Full fieldstrain exx, eyy, exy, enn, ett andetn by first-order shape functionswith a 29 pixel template. (c1)Traditional average template and(c2) equivalent template. (d1,d2) Full filed displacement uand (e1, e2) strain exx by first-(left) and second-order (right)shape functions with a 29 pixeltemplate size, the direction line(the red slash in the bottom) ofthe PLC band and the coordinatesystem definition. (d3)Smoothed displacement curvesand (e3) smoothed strain curvesby first- (blue) and second-order(red) shape functions with 29(solid) and 39 pixel (dash)template sizes

Exp Mech (2015) 55:1575–1590 1579

Fig. 3 a Smoothed strain curveand its Gaussian fit curve and bsmoothed displacement curveand the integral curve whosederivative is the fit curve justmentioned

με. Parameter a is the intensity of the Gauss function andcorresponds to the maximum strain value within the band(MSV B), b represents the location of the maximum strain,c is the standard error of the Gauss function and is relative tothe full width at half the maximum strain (FWHM , equa-tion (5) used to describe the band width, and d correspondsto the shrinkage strain outside the band caused by the shear-ing deformation within the band. The integral of the strainshown in equation (4) is the displacement in equation (6).The displacement and integral curves match and satisfy anapproximate linear relationship (Fig. 3(b)).

u(x)=a × c√

π/2

[erf

(x − b√

2c

)+ erf

(b√2c

)]+d ×x

(6)

Simulated Speckle Images

Because the location of the maximum depends on the coor-dinate system definition, and the shrinkage strain outside theband has little effect on the high gradient local deformationwithin the band, parameter b and d can be ignored, reducingEqs. 4 and 5 to Eqs. 7 and 8.

exx(x) = a × e− x2

2c2 (7)

u(x) = a × c√

π/2erf

(x√2c

)(8)

In this way, the maximum strain value within the band a,and band width related parameter c, entirely determine the

displacement and strain fields. To separate the errors in andoutside the band, the dividing lines locating at ±4c wereselected (Fig. 4(a)).

FFDE =

√√√√√√M∑i=1

N∑j=1

(ui − utij )

2

M × N(9)

SDE =

√√√√√√√M∑i=1

(

N∑j=1

uij

Nut

i)2

M(10)

Two types of displacement error statistics wereemployed: equation (9) illustrates the full-field displace-ment error (FFDE); and equation (10) illustrates thesmoothed displacement error (SDE). In the two equa-tions, ut

ij represents the theoretical displacement field, uti

is theoretical displacement curve, uij is the calculated dis-placement, and M and N are the sample size in the x- andy-directions.

A computer program [25] was used for simulations ignor-ing the oblique angle of the PLC shear bands. The computerprogram used 80000 random speckles with 600 (gray scale)intensity and 2.5-pixel radius (How the variable sizes ofthe speckles influence the results is discussed later, seeAppendix). The speckles were randomly distributed in asquare area of 1201 × 1201 pixels and the square sampling

Fig. 4 a Definition of the areawithin and outside the band withthe dividing lines located wherePosition equals ±4c (c is relatedto the width of the band) and breference speckle image (80000random speckles, 2.5-pixelradius) generated by a computerprogram

1580 Exp Mech (2015) 55:1575–1590

area was 1000 × 1000 pixels. The center of the distribu-tion area, the center of the sampling area, and the originpoint of the defined coordinates all coincided. The displace-ments of the deformed images were the same as shown inequation (8), with a or c varying. Figure 4(b) presents thereference speckle image. Two groups of deformed speckleimages were separately generated: (1) c fixed, a varied from0.0025 to 0.08 and (2) a fixed, c varied from 4 to 128pixels (with exponential growth and an index of 2). TheDIC results of these images are calculated and analyzed inSection Discussions.

Discussions

When the Deformation is Fixed

According to the simplification and simulation inSection Non-linear Fitting, the deformed speckle imagewhen a = 0.01 and c = 11 pixels was analyzed, using inthis section the DIC method with different shape functions(zero-, first- and second-order) and templates (9, 19, 29 and39 pixels). Figure 5 shows some of the DIC results. Full

displacement fields for different orders of shape functions,with a fixed 29-pixel template, are shown in Fig. 5(a-c). Figure 5(d) shows the theoretical displacement curve(black) and displacement error curves by first- (blue) andsecond-order (red) with 19- (solid), 29- (dash) and 39-pixel(dot) template sizes. The displacement error curve wasobtained by smoothing the calculated results minus thetheoretical value. The errors of the first- and second-ordershape functions were approximately the same outside theband, however, the first-order shape functions led to largererrors within the band. The inset of Fig. 5(d) indicates thata larger template gains larger errors when the shape func-tions under-match the deformations. Full strain fields andstrain curves are shown in Fig. 5(e-h). Second-order shapefunctions acted better than first-order shape functions asshown in the inset of Fig. 5(h).

Full-field displacement errors (FFDEs) and smootheddisplacement errors (SDEs) within and outside the bandare shown in Fig. 6. FFDEs outside the band as shown inFig. 6(a) and SDEs outside the band as shown in Fig. 6(c)both decrease with an increase in the template size. This isbecause a larger template leads to more stable results out-side the band where the affine deformations are completely

Fig. 5 Displacement u andstrain exx calculated usingdifferent shape functions andtemplate sizes. (a-c) Full-fielddisplacement u and (e-g) strainexx by zero- (top), first- (center)and second-order (bottom) shapefunctions with a 29-pixeltemplate. (d) A theoreticaldisplacement curve (black) andsmoothed displacement errorcurves using first- (blue) andsecond-order (red) shapefunctions with templates of 19(solid), 29 (dash) and 39 pixels(dot). (h) Theoretical straincurve and smoothed straincurves using first- and second-order shape functions withtemplates of 19, 29 and 39 pixels

Exp Mech (2015) 55:1575–1590 1581

described by the employed shape functions. The FFDEsoutside the band calculated using different orders of shapefunctions share the same order of magnitude as do theSDEs outside the band. However, the FFDEs ( 0.01 pix-els) outside the band are an order of magnitude larger thanthe SDEs outside the band ( 0.001 pixels) which indicatesthat random errors are greater outside the band.

Figure 6(b) and (d) display the FFDEs and the SDEswithin the band. As shown in Fig. 6(d), the SDEs within theband calculated using zero- and first-order shape functionstend to increase with increasing template size. In contrast,the SDEs within the band calculated using second-ordershape functions, first increase then decrease with increas-ing template size. In addition, the SDEs within the bandof second-order shape functions are an order of magnitudesmaller and vary much slower than those calculated usingzero- and first-order shape functions. This is because thesecond-order shape functions are more suitable for describ-ing deformations within the band. Assuming that randomerrors within and outside the band are the same, randomerrors within the band can be replaced by FFDEs out-side the band because, as mentioned above, random errorsdominate outside the band. In this way, FFDEs withinthe band shown in Fig. 6(b) are approximately the sum ofthe FFDEs outside the band shown in Fig. 6(a) and theSDEs within the band shown in Fig. 6(d). As shown inFig. 6(b), the FFDEs within the band calculated usingzero- and first-order shape functions tend to first decreasethen increase when the template size increases, however, theFFDEs within the band of second-order shape functions

simply decrease. In addition, the FFDEs within the bandcalculated using second-order shape functions are smallerthan those calculated using zero- and first-order shapefunctions when the template size is large enough.

In general, a larger template leads to smaller randomerrors, however, larger systemic errors occur when the shapefunctions used under-match the deformations to be calcu-lated. None of the three orders of shape functions employedhere can completely match the high gradient deformationswithin the band, so the FFDEs within the band, and thesize of the template, are positively related when the sizeof the template grows larger than the critical value (criti-cal template size). The critical template size of zero- andfirst-order shape functions is approximately 19 pixels. TheFFDEs within the band of the second-order shape func-tions shown in Fig. 6(b) do not increase when the template isno larger than 39 pixels. However, there is certainly a criticaltemplate size larger than 39 pixels, because the correspond-ing SDEs shown in Fig. 6(d) have a positive relationshipwith the template size, and the corresponding random errorsare relative small. At this point, the mechanism has beenestablished, in which second-order shape functions havea larger critical template and lead to better displacementresults within high gradient local deformations when thetemplate is large enough.

We now focus on the strain results. The non-linear fit-tings mentioned in Section Non-linear Fitting were com-pleted with strain curves calculated using different shapefunctions and templates. Figure 7 shows the theoreticalvalue and how the calculated MSV B and FWHM for

Fig. 6 Displacement errorstatistics calculated usingdifferent shape functions andtemplate sizes. a Full-fielddisplacement errors outside theband. b Full-field displacementerrors within the band. cSmoothed displacement errorsoutside the band. d Smootheddisplacement errors within theband

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different shape functions varied with the template size.MSV B calculated using different shape functions had anegative relationship with the template size, and FWHM

had a positive relationship, as shown in Fig. 7(a) and (b),respectively. MSV B and FWHM were far from the the-oretical value when the template size increased, whichcoincided with the tendency of SDEs to be within the bandshown in Fig. 6(d). MSV B and FWHM calculated usingsecond-order shape functions were nearest to the theoreticalvalue. In other words, the second-order shape functions hadbetter strain results within the band.

To sum up, shape functions and templates affected theDIC results jointly, not individually, and the effects werequite different between affine and inhomogeneous defor-mations. However, optimal shape functions and templatescan be found for given deformations as discussed above.Deformations with different gradients are discussed belowto further discover how shape functions and templates takeeffect in the DIC method.

When the Maximum Strain Varies Only Within theBand

Section When the Deformation is Fixed discussed a fixeddeformation with different shape functions and templates.The first group (with a varying) of speckle images withdifferent deformations mentioned in Section SimulatedSpeckle Images will be the focus of this sub section.

To simplify the discussion, the DIC results for differ-ent deformations calculated using different shape func-tions with a fixed 29-pixel template are discussed first.Full displacement error fields calculated using zero- (left),first- (center) and second-order (right) shape functions areshown in Fig. 8(a-c). The theoretical displacement curve(black) and displacement error curves calculated using zero-(green), first- (blue) and second-order (red) shape functionsare displayed in Fig. 8(d) with a = 0.01. The same dis-placement results, with a = 0.02, are shown in Fig. 8(e-h).The displacement errors changed little outside the band andapproximately doubled within the band when a changedfrom 0.01 to 0.02.

FFDEs and SDEs within and outside the band whilea varies are shown in Fig. 9. The negligible changes inFig. 9(a) and (c) indicate that the FFDEs and SDEsoutside the band were almost unaffected by the maxi-mum strain within the band. The FFDEs and SDEscalculated using second-order shape functions were a lit-tle larger than, and had the same order of magnitude as,those calculated using zero- and first-order shape func-tions. The FFDEs within the band had direct propor-tional relationships with the maximum strain within theband as shown in Fig. 9(b) as do the SDEs withinthe band shown in Fig. 9(d). In summary, displace-ment errors outside the band remained unchanged anddisplacement errors within the band increased in proportionto the maximum strain within the band.

Theoretical strain curves (black) and strain curves cal-culated using different shape functions while a varies, areshown in Fig. 10(a-f). They have similar shapes althoughthe scales are not the same. The calculated strain errorschanged little outside the band and kept increasing withinthe band when both the calculated and theoretical strain val-ues grew in a similar manner. The curves for second-ordershape functions were located closer to the theoretical curves,while the curves for the zero- and first-order shape functionsalmost completely coincided. Figure 10(g) shows the ratiobetween MSV B and a changed little and Fig. 10(h) showsFWHM remained approximately unchanged while a var-ied. Therefore, it was established that the mechanism, inwhich the maximum strain within the band was proportionalto the strain errors, made no difference to the band width.

All the above results in this section were obtained whenthe template size was fixed at 29 pixels. Figure 11 givesmore complex results when the change in the templatesize was obvious. Figure 11(b) indicates the SDEs withinthe band calculated using different shape functions whenthe maximum strain within the band, and the templatesize, were both changing. The SDEs within the band ofsecond-order shape functions were smaller, especially whenthe template size and the maximum strain of the bandwere adequately large. The FFDEs within the band areshown in Fig. 11(a). The FFDEs within the band for

Fig. 7 Parameters fitted usingthe smoothed strain curvescalculated using different shapefunctions and template sizes. aMSV B (maximum strain valuewithin the band) and b FWHM

(full width at half maximum)calculated using zero- (green),first- (blue) and second-order(red) shape functions withdifferent template sizes and theirtheoretical values (black)

Exp Mech (2015) 55:1575–1590 1583

Fig. 8 Displacement errors calculated using different shape functions with a fixed template size (29 pixels) while a varied. (a-c) a = 0.01 and(e-g) a = 0.02, full-field displacement errors calculated using zero- (left), first- (center) and second-order (right) shape functions. (d) a = 0.01 and(h) a = 0.02, theoretical displacement curve (black) and smoothed displacement error curves using zero- (green), first- (blue) and second-order(red) shape functions

second-order shape functions were larger than those forfirst-order shape functions when the template was 9 pix-els, which was caused by larger random errors. The crit-ical template sizes for all three types of shape functionsall decreased when the maximum strain within the bandincreased. All the results were in agreement with the state-ments from previous papers [17, 22].

In summary, second-order shape functions obtained bet-ter results within the band and marginally worse resultsoutside the band when the band related parameter c wasfixed (11 pixels) and the maximum strain within theband a varied from 0.0025 to 0.08. In the next subsection, bands with different widths will be calculated andanalyzed.

Fig. 9 Displacement errorstatistics calculated usingdifferent shape functions with afixed template size (29 pixels)while a varies from 0.0025 to0.08. a Full-field displacementerrors outside the band. bFull-field displacement errorswithin the band. c Smootheddisplacement errors outside theband. d Smoothed displacementerrors within the band

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When Only the Band Width Varies

The previous sub section studied bands with different maxi-mum strains within them, while the second group of speckleimages with different band widths will be discussed in thissub section.

As before, the template size was fixed initially at29 pixels. Full displacement error fields calculated usingzero- (left), first- (center) and second-order (right) shapefunctions are shown in Fig. 12(a-c) and the theoreticaldisplacement curve (black) and displacement error curves

calculated using zero- (green), first- (blue) and second-order(red) shape functions are displayed in Fig. 12(d) with c = 8pixels. The same displacement results with c = 16 pixels areshown in Fig. 12(e-h). The displacement errors changed lit-tle outside the band and reduced by nearly half within theband, when c changed from 8 to 16 pixels.

FFDEs and SDEs within the band while c variedare shown in Fig. 13. Figure 13(b) indicates that theSDEs within the band decreased sharply at first and thenslowly with the increase in the band width. The reasonfor the decrease was that the increasing band width led to

Fig. 10 Smoothed strain curvescalculated using different shapefunctions with a fixed templatesize (29 pixels) while a variedfrom 0.0025 to 0.08 and theirfitting parameters. a a = 0.0025,b a = 0.005, c a = 0.01, d a =0.02, e a = 0.04 and f a = 0.08. gThe ratio between MSV B and a

and h FWHM calculated usingzero- (green), first- blue andsecond-order red shapefunctions while a varied, andtheir theoretical values black

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Fig. 11 Displacement errorstatistics calculated usingdifferent shape functions withdifferent template sizes while a

varied from 0.0025 to 0.08. aFull-field displacement errorswithin the band. b Smootheddisplacement errors within theband

decreasing gradients. In Section When the Maximum StrainVaries Only Within the Band, we learnt that the displace-ment errors were linearly related to the gradients when thetemplate size and the band width were both fixed. Thesenon-linear decreases were caused by the increase in the bandwidth when the template size was fixed. The SDEs withinthe band for the second-order shape functions were obvi-ously much smaller than the corresponding errors for thezero- and first-order shape functions when the template sizewas small. However, the differences in the SDEs calculatedusing the three orders of shape functions were noticeablysmall when the template was more than 64 pixels despite thesecond-order shape functions leading to the smallest SDEs.FFDEs within the band also initially decreased sharply andthen slowly as shown in Fig. 13(a). Assuming the random

errors for the different band widths remained the same, theFFDEs within the band for the second-order shape func-tions were larger than the errors for the first-order shapefunctions when c was larger than 64 pixels. This is explainedby the fact that the random errors of second-order shapefunctions were larger and random errors take hold when theband width is large enough.

Theoretical strain curves (black), and strain curves cal-culated using different shape functions while c varied, areshown in Fig. 14(a-f). They are quite different from eachother although the scale was the same. The strain curvesfor zero- and first-order shape functions nearly coincided,however, the fluctuations in the strain curves using zero-order shape functions were larger, especially when c wasgreater than 64 pixels. Strain errors changed little outside

Fig. 12 Displacement errors calculated using different shape functions with a fixed template size (29 pixels) while c varied. (a-c) c = 8 pixelsand (e-g) c = 16 pixels, full-field displacement errors calculated using zero- (left), first- (center) and second-order (right) shape functions. (g) c =8 pixels and (h) c = 16 pixels, the theoretical displacement curve (black) and smoothed displacement error curves using zero- (green), first- (blue)and second-order (red) shape functions

1586 Exp Mech (2015) 55:1575–1590

Fig. 13 Displacement errorstatistics calculated usingdifferent shape functions with afixed template size (29 pixels)while c varied from 4 to 128pixels. a Full-field displacementerrors within the band. bSmoothed displacement errorswithin the band

Fig. 14 Smoothed strain curvescalculated using different shapefunctions with a fixed templatesize (29 pixels) while c variedfrom 4 to 128 pixels and theirfitting parameters: a c = 4pixels, b c = 8 pixels, c c = 16pixels, d c = 32 pixels, e c = 64pixels and f c = 128 pixels. gThe ratio between FWHM andc and h MSV B calculated usingzero- (green), first- (blue) andsecond-order (red) shapefunctions while c varied, andtheir theoretical values (black)

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Table 1 Errors for the maximum strain within the band

c/pixel Relative First-order Second-order

4 63.8 % 62.3 % 38.3 %

8 35.7 % 35.9 % 13.0 %

16 12.1 % 13.0 % 2.5 %

32 3.3 % 3.3 % 0.2 %

64 0.8 % 0.6 % 0.3 %

128 0.2 % 0.2 % 0.4 %

the band and decreased within the band when the band widthincreased. Figure 14(g) shows the ratio between FWHM

and c first decreased sharply then remained almost thesame. Figure 14(h) shows theMSV B first increased sharplythen remained almost the same. Both the ratio and MSV B

for second-order shape functions were closest to the the-oretical value. The small strain errors when c was greaterthan 32 pixels indicated that the first- and second-ordershape functions behave similarly when the gradients weresmall enough. Hence, second-order shape functions may actno better than first-order shape functions in low gradientdeformations.

Table 1 illustrates the errors for the maximum strainwithin the band calculated in different ways. The rela-tive errors shown in the second column present the ratiobetween the absolute difference and the MSV B of first-order shape functions. The absolute difference was thedifference between the MSV Bs calculated using first- andsecond-order shape functions. The errors shown in the thirdand last column represent the relative errors of the first- andsecond-order shape functions. When these relative errorswere less than 10 %, the second-order shape functionsobtained results with errors of less than 5 %.We suggest thatthe DIC results for second-order shape functions are reli-able when the relative error between first- and second-ordershape functions is less than 10 % because the error resultsof less than 5 % are generally believed to be reliable.

The above results were obtained when the template sizewas fixed at 29 pixels. Figure 15 displays more com-plex results when the change in template size is obvious.

Figure 15(b) indicates the SDEs within the band calculatedusing different shape functions when the band width and thetemplate size both varied. The SDEs within the band forsecond-order shape functions were smaller, especially whenthe template size was large enough and the band width wassmall enough. The FFDEs within the band are shown inFig. 15(a). The FFDEs within the band for second-ordershape functions were larger than those for first-order shapefunctions when the template size was 9 pixels which wascaused by larger random errors. The critical template sizefor all three orders of shape functions increased when theband width increased.

In this section, we establish that second-order shape func-tions obtain better results when the band width is small andno better results when the band width is large.

All the studies in the three Sections When the Defor-mation is Fixed–When Only the Band Width Varies werebased on the deformation gradients in speckle images ratherthan the forms of deformations, so the established mecha-nism and proposed theory were suitable for general localdeformations such as deformations near gaps and cracktips.

So far, the detailed mechanism on how shape functionsand templates impacted on the DIC results has been estab-lished and a method for judging the DIC results has beenproposed. For the unacceptable DIC results, an effectiveway to obtain reliable results will be proposed and discussedin the next section.

Improving the Experiment

The relative error for the experiment results discussed inSection Results was 25 % when the template size was 29pixels. According to the theory proposed in Section WhenOnly the Band Width Varies, an error of more than10 % (25 %) means even second-order shape functionsmay not obtain reliable results. In spite of this, Sec-tions When the Maximum Strain Varies Only Within theBand and When Only the Band Width Varies indicated that

Fig. 15 Displacement errorstatistics calculated usingdifferent shape functions withdifferent template sizes, while c

varied from 4 to 128 pixels. aTotal error within the band. bSmoothed error within the band

1588 Exp Mech (2015) 55:1575–1590

Fig. 16 a Smootheddisplacement curves and bsmoothed strain curvescalculated using first- (blue) andsecond-order (red) shapefunctions with a fixed templatesize (29 pixels) in an improvedexperiment

smaller maximum strains within the band and larger bandwidth led to smaller errors, so effective ways for improv-ing the spatial resolution of the image and the frequency ofthe acquiring system may be used to improve the experi-ments. An improved experiment with a larger band widthwas performed to verify this method by increasing only thethickness of the tensile specimen from 1 mm to 3 mm [11].In this manner, the spatial resolution of the band increasedso that reliable results could be obtained. Figure 16 showsthe improved displacement and strain curves calculatedusing first- and second-order shape functions with a fixed29-pixel template size. Comparing this with the results fromthe first experiment in Fig. 2, we see the difference causedby different shape functions becomes obviously smaller.The relative error between the first- and second-order shapefunctions reduced from 25 % to 7 %. The error of lessthan 10 % (7 %) demonstrated the improved method andproved that second-order shape functions achieved reliableand accurate DIC results.

Conclusions

Inhomogeneous deformations such as PLC shear bandsoccurring in tensile tests were illustrated to study how shapefunctions and templates affected the DIC results for localdeformations. Experiments and simulations both demon-strated that shape functions with different orders obtainedconstant results in affine deformations, and that second-order shape functions were more suitable in local defor-mations. By simulating and analyzing bands with differentwidths and heights, the detailed mechanisms on how shapefunctions and templates impact on DIC results in localdeformations, were established:

1) Higher order shape functions have a larger critical tem-plate size and lead to better results when the templateis large enough;

2) The maximum strain within the band is proportional toboth displacement and strain errors and does not affectthe band width;

3) Second-order shape functions can obtain better resultswhen the deformations have a high gradient but worseresults in low gradient deformations [17, 22].

To judge the reliability of DIC results, the theory thatsecond-order shape functions produce reliable results (lessthan 5 % error) if the relative error between the first- andsecond-order shape functions appears to be less than 10 %,has been proposed. In the measurements using the DICmethod, we suggested employing first-order shape func-tions with a large template size in uniform deformationfields, and second-order shape functions with a moderatetemplate size in high gradient inhomogeneous deforma-tions. When the relative errors are larger than 10 %, aneffective way of improving the spatial resolution and theacquisition frequency was suggested for obtaining relativeerrors of less than 10.

Acknowledgments This work was supported by the National BasicResearch Program of China (2011CB302105), and the National Nat-ural Science Foundation of China (Grant Nos. 11372300, 11332010,51271174, 11127201, 11472266 and 11428206).

Appendix: Influence of Variable Sizes of Speckles

To discover how they influence the results, speckles withvariable sizes were employed and a typical deformation (a =0.01 and c = 11 pixels) was applied. The sizes of the speck-les satisfied a normal distribution with μ = 2.5 and σ = 1/3so the sizes of 99.7 % speckles were located between 1.5and 3.5 pixels.

Figure 17 shows that variable sizes of speckles scarcelycause any difference compared with speckles with a fixed2.5-pixel radius. The two types of speckles produced nearlythe same results regardless of the order of the shape func-tions.

The simulations are highly dependent on the speckles,and simulated speckles are quite different from spray spots.

Exp Mech (2015) 55:1575–1590 1589

Fig. 17 Influence of variable sizes of speckles

Results with simulated speckles and spray spots employedin the simulations are presented in Fig. 18 to illustrate thedifference. Figure 18(a) indicates the reference speckleimage with simulated speckles and with spray spots shownin Fig. 18(b). Figure 18(c) shows the displacement errorscalculated using different shape functions with the simu-lated speckles, and spray spots, respectively (The displace-ments were imposed by equation (8) while a = 0.01 and c =11 pixels). There were interpolation errors (of about 0.004pixels) outside the band when spray spots were employed.However, the tendency was the same regardless of the typeof speckles used and the strain curves were little different(shown in Fig. 18(d)).

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