Journal of Mechancs Engneerng and Automaton 7 (2017) 26-38 do: 10.17265/2159-5275/2017.01.004 D DAVID PUBLISHING Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm Gancarlo L. G. Gonzáles 1, Leonardo D. Rodrgues 2 and José L. F. Frere 1 1. Department of Mechancal Engneerng, Pontfcal Catholc Unversty of Ro de Janero, Rua Marquês de São Vcente 225, Gávea, Ro de Janero RJ 22543-900, Brazl 2. Department of Mechancal Engneerng, UFPA, Rua Augusto Corrêa, 1 - Guamá, Belém-PA 66075-110, Brazl Abstract: In ths work, the effcency of a feature-based DIC (dgtal mage correlaton) algorthm for measurng hgh stran gradents was nvestgated by means of numercal and actual experments. The so-called SIFT-Meshless method conssted of a novel formulaton nvolvng the SIFT (scale-nvarant feature transform) feature detector wth a self-adaptve meshless formulaton. Whereas the numercal experments amed to evaluate the accuracy and the spatal resoluton, the actual experments amed to demonstrate n practce the above fndngs. A stereoscopc system and a mcro-stereoscopc system were used to perform hgh stran gradent measurements n notched specmens of dfferent materals and notch szes. Ths paper concludes that the feature-based algorthm s able to provde accurate stran measurements at hgh stran gradent regons, even under condtons of plastcty. Moreover, the algorthm showed ts effcency to capture the peak stran near the notch boundary. Lastly, a spatal resoluton study proposes a lnk between the desred accuracy and the pxel resoluton requred to perform accurate measurements of hgh stran gradents. Key words: Hgh stran gradents, dgtal mage correlaton, SIFT, meshless. 1. Introducton Regons of hgh stran gradents (.e., large change n stran values wthn a small dstance) are commonly observed surroundng stress concentrators, such as holes, shoulder fllets, and regons generally called shallow or deep notches. Therefore, ther characterzaton requres of expermental technques wth hgh spatal resoluton to capture the peak stran expected near the notch boundary, whch s an area of great nterest n fatgue analyss. In that feld, the DIC (dgtal mage correlaton) [1-4] s presently the most employed technque by researchers. If the standard subset-based DIC s used [5-8], accurate stran measurements near the notch boundary are dffcult to obtan because the pont of measurement s consdered by default at the center of the subset, therefore, the nformaton at the boundary of the regon of nterest Correspondng author: José L. F. Frere, assocate professor, Ph.D., research felds: structural ntegrty and expermental stress analyss. (.e., the notch boundary) s absent. Moreover, the measured gradent can be more flattened than the real one, dependng on the DIC parameters used n the analyss. In recent years, alternatve algorthms [9-12] have been proposed to optmze the standard DIC n order to ensure more accurate solutons when non-homogenous stran feld mesurements are requred. Another recently proposed algorthm s the SIFT-Meshless method [13] whch uses a feature-based matchng approach for correlatng mages. Features are ponts or small patches on the mage that dffer from ther mmedate surroundng regon and can easly be extracted by means of effcent algorthms. The SIFT-Meshless algorthm adopts the SIFT [14, 15] whch s a robust local feature extractor wth outstandng performance. By usng SIFT, dsplacement measurements are obtaned by trackng the successfully matched features from the reference to the deformed mage. Then, a meshless
Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm 27 formulaton wth self-adaptve doman sze s appled for solvng the dsplacement feld and ts dervatves (more nformaton about the constructon of meshless shape functons and ther propertes can be found n Ref. [16]). The self-adaptve meshless formulaton mproves the accuracy of the measurements when low and hgh stran gradent felds co-exst n the same analyss. Moreover, the method s capable to obtan stran data at the notch boundary, snce feature ponts are extracted from ths regon. In ths paper, the SIFT-Meshless method s used to perform hgh stran gradent measurements around shallow and deep U-notches. Two applcatons usng dfferent materals and dfferent types of loadng are presented: the elastc problem of deep U-notches n two polycarbonate specmens under unaxal tensle load; and the elastc-plastc problem of a semcrcular notch n a stanless steel specmen under bendng load. In addton, a subset-based DIC analyss was performed by usng the commercally avalable VIC-2D and VIC-3D systems from Correlated Soluton Inc [17]. Moreover, the experments were smulated usng the fnte element ANSYS software to provde an addtonal source of verfcaton. Lastly, the spatal resoluton and accuracy of the two DIC algorthms were nvestgated by means of numercal experments. The spatal resoluton s defned n Refs. [11,18] as the mnmum spatal unt between two ndependent data ponts requred to perform a measurement. Based on ths concept, the spatal resoluton for the subset-based DIC method s determned by the subset sze, whch defnes the dstance between two adjacent measurement ponts n the correlaton process. However, ths defnton s not applcable to non-tradtonal DIC algorthms, such as the SIFT-Meshless method. From the lterature revew [11], one way to evaluate the spatal resoluton for non-tradtonal DIC algorthms s by usng smulated mages undergong snusodal dsplacements wth varous ampltudes and spatal frequences. In the present study, not only a snusodal functon but also an analytcal stress-concentraton dsplacement felds were consdered to smulate dfferent cases of stran gradents. The numercal experments amed to establsh a methodology to evaluate the accuracy and spatal resoluton of tradtonal and non-tradtonal DIC algorthms for hgh stran gradent measurements. Moreover, the fnal objectve s to present a smple procedure to obtan the expected error when measurng the actual maxmum stran at the notch boundary for a gven optcal confguraton, where the feld magnfcaton and consequently pxel resoluton are known beforehand. 2. Fundamentals of the Local-based SIFT-Meshless Method 2.1 Dsplacement Measurements Usng SIFT The SIFT technque s used to measure relatve dsplacement of features successfully matched n two mages of the specmen s surface one captured before and the other after deformaton (Fg. 1). The trackng feature procedure nvolves three steps: feature detecton, feature descrpton, and feature matchng. These are brefly descrbed as follows (more detaled nformaton about the SIFT algorthm can be found n Refs. [14, 15]). The feature detecton step conssts n searchng for dstnctve and representatve feature ponts. The cost of extractng the SIFT features s mnmzng by usng a cascade flterng approach n whch the more expensve operatons are appled only at locatons that pass all pror tests. Durng ths stage, features senstve to nose and poorly localzed features are dscarded. In the feature descrpton step, a numercal vector called feature descrptor s bult based on the mage propertes of a support regon around each feature pont, as shown n Fg. 2. The mage propertes nclude gradent magntude, gradent drecton, and pxel ntensty. The fnal feature descrptor s a vector wth 128 elements descrbng the unque propertes of the correspondng feature pont and ts support regon.
28 Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm undeformed-deformed par of mages, a meshless formulaton based on the Movng Least Squares approxmaton [19] s used for solvng the dsplacement and stran feld. Thus, the meshless approxmaton for the u-component of the dsplacement feld can be wrtten, Fg. 1 Dsplacement measurements obtaned by matchng SIFT features. Fg. 2 Example of SIFT features detected on a dotted mage test (left), and feature dentfcaton gven by ts descrptor (rght). In the feature matchng step, correspondence features between mages are computed based on the smlarty of ther descrptors. To do so, the Eucldean dstance s the commonly used method for matchng features n the SIFT algorthm. The best match for each descrptor s found by mnmzng the Eucldean dstance. In order to dscard poorly or overly ambguous matched features, the dstance rato between the closest dstance and the second-closest dstance s evaluated. The SIFT-Meshless method uses a dstance rato of 0.3-0.4 n order to reduce the possblty of false matches, due to the large number of ponts extracted from mages. Moreover, snce a stereovson system s used, the SIFT features are also used to determne correspondng features n the stereo mages for recoverng the 3-D postons of the mages captured by the cameras usng a stereo trangulaton algorthm. 2.2 Meshless Formulaton Once the SIFT features are located n the n h u ( x) ( x) u ( x) U (1) where n s the number of nodes (SIFT features) n the support doman of any pont x = [x, y] T, Φ(x) s the matrx of the shape functon values of the -th node, and U s the vector of the nodal dsplacements provded by trackng the SIFT features. The support doman determnes the number of nodes to be used to support or approxmate the functon value at a specfc pont of evaluaton. In addton, the support doman can be weghted usng a weght functon, whch gves more nfluence to ponts close to the evaluaton pont. The authors tested several weght functons avalable n the lterature, and smlar results were obtaned. The SIFT-meshless formulaton adopted the exponental weght functon. r 0.4 2 wr () e, r 1 0, r 1 (2) In Eq. (2), the value of the weght functon decreases wth the dstance r from the center of the functon,.e., ponts near to the evaluaton pont have more nfluence on the value of the functon than those ponts far away. The dmenson of r s calculated usng the equaton: x x r dm (3) In ths equaton, dm s the dmenson of the nfluence doman determned by dm d (4) s where, d s the radus of the doman, and α s s a scalng parameter, both ntally prescrbed by the analyst. As shown n Fg. 3, the support doman never extends beyond the perphery of the problem doman.
Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm 29 h x x 0 h, 0 n x 0 u u y y 0, y v, v xy y x y, x (9) Fg. 3 Example of the support doman for the node x. For domans wth rregularly dstrbuted nodes (such as the features ponts provded by SIFT), the use of an nfluence doman works well to select nodes for constructng shape functons. 2.2.1 Movng Least Squares Approxmaton The MLS shape functon assocated wth node x can be calculated as 1 T 1 ( ) p j( ) ( ) ( ) j j m x x A x B x = P A B (5) where, P T s a polynomal bass of order m (e.g., lnear, quadratc). The matrces A and B are defned by T n w 1 Ax ( ) ( x x) Px ( ) P ( x ) (6) T T w Bx ( ) w xx 1 P x1 xxn P x n (7) From Eq. (6), one can see that the mnmum number of nodes needed for matrx A to be regular depends on the polynomal bass functon used. In ths work, a quadratc bass functon was adopted. The expresson n Eq. (1) can also be used for the dsplacement component v(x). Thus, the dsplacement feld n the deformed surface s defned as h n u 0 u 0 v v (8) The strans at any pont n the problem doman are obtaned n terms of the nodal dsplacement components by usng the stran-dsplacement relaton defned by In ths way, the frst order partal dervatves of the MLS shape wth respect to x and y drecton are requred, whch can be easly computed by dfferentaton of Eq. (5). It s worth mentonng that the small-stran approxmaton defned n Eq. (9) s used for strans less than 1%. When needed, other stran defntons (e.g. large strans) can be used. 2.2.2 Self-adaptve Doman Process It was verfed that hgh stran gradents requre small nfluence domans to perform accurate approxmatons. On the other hand, large nfluence domans provde a smoothed soluton, whch s most approprate for characterzng unform stran dstrbutons. Moreover, accordng to Eq. (4), the dmenson of the nfluence doman s determned by the scalng parameter α s. In ths way, the self-adaptve process conssts n determnng a sutable value for α s for each evaluaton pont accordng to the stran-gradent behavor. For that, the proposed test functon shown n Fg. 4b s used. Ths functon relates the stran-gradent behavor wth an approprate value for α s. To obtan an approxmaton of the stran-gradent behavor n the analyzed regon, a frst meshless approxmaton s calculated wth a hgh value for α s, such as 3 or 4. Then, the gradent of ths frst approxmaton s computed n the drecton of the notch tp (e.g., along the x-drecton of the notch shown n Fg. 4a). The magntude of ths gradent shows how fast the stran rses n that drecton, ndcatng areas of hgh stran concentraton. Lastly, by usng the test functon n Fg. 4b, the approxmatons are re-computed wth the new values for α s, and the correspondng stran components on the surface are obtaned. 3. Expermental Detals Two data acquston systems were used: a
30 Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm Fg. 4 (a) Example of gradent behavor calculated for the regon ahead of the notch (b) test functon used for calculatng sutable values for α s. Table 1 Expermental matrx. Specmen Notch radus Materal Analyss Test Optcs System PC-1 2.4 mm Poly-carbonate Elastc Tensle Mcro PC-2 1.0 mm Poly-carbonate Elastc Tensle Mcro SS-3 12.5 mm Stanless steel Elastc-plastc Bendng HM lenses stereoscopc system wth HM (hgh magnfcaton) lenses for measurng areas wth sde-lengths rangng from 5 mm to 50 mm, and a mcro-stereoscopc (mcro) system for measurng areas wth sde-lengths rangng from 1 mm to 7 mm. The dgtal CCD (charge-coupled devce) cameras had resolutons of 2,448 2,048 pxels. Table 1 gves the expermental matrx. Calbratng stereovson systems was completed n the calbraton module of the VIC-3D software. The same calbraton parameters were used for both DIC algorthms to exclude possble sources of errors. The sample surfaces were covered wth a unform coat of whte pant, after whch black dots were added. The sze and dstrbuton of such dots s related to the optcal system used. For the macroscopc length scales, the random black-and-whte pattern was performed wth spray pantng. For the mcroscopc length scales, small szes of dots were obtaned wth an arbrush that provded a smooth transton from fne to medum sprayng. The same speckle pattern was used for both algorthms. In the tradtonal DIC approach, the surface preparaton ams to create a random hgh contrast pattern on the specmen s surface. In the SIFT-Meshless method, the objectve s to provde a large number of potental canddates to be feature ponts that can be detected by the SIFT algorthm. 4. Stran Concentraton n Deep U-Notch Specmens under Lnear-Elastc Tensle Load 4.1 Expermental Condtons The expermental test performed n ths secton amed to obtan the hgh stran gradent dstrbuton ahead of a sharp notch subjected to tensle stresses. Two specmens made of polycarbonate wth 3 mm thckness contanng central deep U-notches wth rad of 2.4 mm (PC-1) and 1 mm (PC-2) were used. The geometry of the samples and the expermental set-up are shown n Fg. 5. The mechancal propertes of the polycarbonate, namely Modulus of Elastcty and Posson s rato are respectvely 2.3 MPa and 0.4. The expermental setup used the mcro-stereoscopc system. The feld of vew was about 6.1 5.1 mm 2 surroundng the notch tp wth pxel resoluton of approxmately 2.5 μm on the object plane for both PC-1 and PC-2 specmens. The loads were appled to the specmens by means of a force applcaton system nto whch a load cell devce was nstalled. Thus, successve tensle loadngs of 25, 50, 100 N were
Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm 31 Fg. 5 Geometry of the polycarbonate specmens. appled to PC-1, and of 15, 30, 60 N to PC-2. One mage was captured at each load ncrement. 4.2 Results Surface strans around the notch were assessed wth the VIC-3D software usng subset szes of 43 pxels and 49 pxels for PC-1 and PC-2, respectvely. For both analyses, the step sze was set to 11 pxels and stran wndows of 15. In the SIFT-Meshless analyss, approxmately 6,500 and 5,900 feature ponts were successfully matched n the PC-1 and PC-2 captured mages, respectvely. The mnmum doman (d ) was set to 200 pxels for both analyses. Fg. 6 depcts the stran dstrbuton determned by the VIC-3D and SIFT-Meshless algorthms along the x-axs (y = 0) at the dfferent loadng stages. These stran values were compared wth results determned by the FEM smulaton usng Ansys (3-D FEM analyss, tetrahedral elements, Sold-187). From these test results, t can be seen that the expermental measurements and the numercal results showed good agreement. Moreover, the hgh stran gradent regons can be clearly dentfed beng that t was more severe n PC-2, where, for the maxmum loadng of 60 N, the stran value decreased drastcally from 0.2% (2,000 με) to 0.7% (7,000 με) n a dstance of 1 mm ahead of the notch tp. Notce that the maxmum strans at the notch tp were successfully quantfed by the SIFT-Meshless method. In the Fg. 6 Comparson between numercal and expermental stran dstrbutons extracted from the stran component ε yy along the x-axs (a) PC-1 specmen (b) PC-2 specmen. VIC-3D analyss, the nearest pont of measurement from the notch was located at about 75 μm and 70 μm for PC-1 and PC-2, respectvely. In addton, Fg. 7 depcts the stran felds for the deformed state measured n PC-1 at 100 N by the two DIC algorthms. 5. Stran Concentraton n Crcumferental Notch Specmen under Elastc-Plastc Bendng 5.1 Expermental Condtons The experment performed n ths secton amed to nduce a plastc stran-gradent near the notch tp by four-pont bendng test. Therefore, t s requred that the algorthm be able to quantfy the stran-gradent and the nonlnear behavor of the materal ahead of the notch tp after the yeld stress (or elastc lmt) s reached, generally at 0.2% stran (.e., 2,000 με). The bendng test was performed n a servo-hydraulc Instron machne equpped wth a four-pont bendng
32 Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm resultng n a feld of vew of about 35 30 mm 2 wth a pxel resoluton of approxmately 14 μm on the object plane. Images were taken from the specmen wth loadngs of 4, 6, 8 and 10 kn, whereby the elastc lmt was attaned and plastc strans at the notch tp were nduced. 5.2 Results Fg. 7 Full-feld dstrbuton for the stran component ε yy at 100 N (a) VIC-3D (b) SIFT-Meshless method. Fg. 8 Sketch of the expermental setup of the 4-pont bendng. fxture. The specmen was a 304 stanless-steel plate wth thckness of 5 mm and a semcrcular notch wth radus of 12.5 mm. The specmen geometry and the four-pont bendng setup are shown schematcally n Fg. 8. The mechancal propertes of the stanless-steel were determned expermentally: Modulus of Elastcty (E) = 195 MPa and the yeld stress S y, 0.2% = 250 MPa. In ths experment, the stereoscopc system equpped wth hgh magnfcaton lenses was used. The cameras were placed n front the specmen After the mages were captured, they were processed usng the VIC-3D software wth a subset sze of 25 pxels, step sze of 8 pxels, and stran wndows of 15. In the SIFT-Meshless analyss, approxmately 8,500 feature ponts were successfully matched between mages, and the mnmum doman used was set to approxmately 180 pxels. The elastc-plastc analyss for the 4-pont bendng problem was smulated usng Ansys software (3-D FEM analyss, tetrahedral elements, Sold-187). The numercal soluton used an sotropc multlnear model, descrbed by the plastc stress-stran curve of the 304 stanless-steel. The maxmum stran values obtaned by the FEM smulaton and measured by the two expermental technques are plotted n Fg. 9. The nclnaton of the curve ndcates the ncrement of the plastcty at the notch tp. It can be seen that sgnfcant plastc stran was nduced after the load of 6 kn. Fg. 10 shows the axal stran values determned by the VIC-3D and SIFT-Meshless methods along the y-axs (x = 0) for the loadngs of 8 kn and 10 kn. The stran dstrbutons obtaned by the two DIC algorthms were n good agreement wth the fnte Fg. 9 Maxmum stran at notch tp n SS-3 specmen.
Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm 33 Fg. 10 Comparson between numercal and expermental stran dstrbutons extracted from the stran component ε xx along the selected path (a) 8 kn (b) 10 kn. element smulatons usng the Ansys software, even when sgnfcant plastc stran has been reached n the area surroundng the notch. In addton, stran measurements at the notch boundary were successfully obtaned usng the SIFT-Meshless method, where the maxmum plastc stran was approxmately 0.5% when the test load was 10 kn. In the VIC-3D analyss, the pont nearest to the notch was located at 250 μm from the notch boundary. It can be also observed n Fg. 11, where the stran felds for the deformed state measured n SS-3 at 10 kn by the two DIC algorthms are shown. Moreover, t was notced that the SIFT-Meshless algorthm adapts the calculaton of the full-feld soluton to the presence of hgh and low stran gradents by decreasng or ncreasng the nfluence doman at each evaluated feature pont. 6. Accuracy and Spatal Resoluton n Hgh Stran Gradent Measurements The spatal resoluton study was focused on the Fg. 11 Full-feld dstrbuton for the stran component ε xx at 10 kn (a) VIC-3D (b) SIFT-Meshless method. Fg. 12 Speckle pattern used for numercal test. assessment of the stran error, whereas that, stran s manly the fnal desred quantty to be measured. For that, dfferent numercal tests were carred out usng numercally deformed mages by mposng undrectonal n-plane deformaton felds wth hghly heterogeneous behavor. Fg. 12 shows the speckle pattern used as reference mage (1,200 800 pxels) whch was cropped from an actual expermental mage (8-bt, 2,248 2,048 pxels). The deformed mages were generated usng the Matlab and ts Image Processng Toolbox TM.
34 Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm The present analyss assumed dsplacement feld functons mposed along the horzontal drecton wth zero dsplacement n the vertcal drecton. The error assessment was based on the measurement accuracy to capture the maxmum peak stran for the dfferent confguratons of mposed stran gradents. Snce the synthetc dsplacement feld s undrectonal (x-drecton), the correspondng mposed stran gradent s the same for all vertcal lne (column) of the mage. Therefore, the strans measured for all ponts located over any column can be statstcally analyzed and compared wth the mposed stran value to the consdered column. Thus, the dfferences between the measured and mposed strans can be used n order to predct the expected AL (ampltude loss) for each DIC algorthm evaluated. For that, the followng model was used: max mposed max 3max AL 100 (10) max mposed In ths model, the 3-sgma ndcates a 99.8% of certanty on ampltude determnaton. The arthmetc mean (μ) and the standard devaton (σ) are defned as: max measured max n 2 1 1 measured n max max max (11) (12) Fnally, the error assocated wth the whole feld stran measurement n terms of the RMSE (root mean square error) s calculated by mposed RMSE 1 ( x, y ) ( x, y n ) 2 measured where, n s the number of evaluaton ponts. (13) reproduced wthout sgnfcant loss of ampltude, whch s defned by the analyst accordng to the measurement requrements. The snusodal dsplacement appled n the x-drecton to generated deformed mages satsfes the followng equaton: (14) uxy (, ) Asn 2 x/ P where, A s the ampltude, and P s the perod. Thus, a set of eght snusodal deformed mages were generated wth perods varyng from 120 to 400 pxels. For each smulaton, the ampltude A n Eq. (14) was adjusted n order to obtan a maxmum peak of 1.2% n stran response. Fg. 13 shows an example of a pre-assgned snusodal stran feld. The deformed mages were processed usng both DIC algorthms. The VIC-2D software was tested usng two dfferent parameter settngs. The frst confguraton (VIC-2D #1) uses a subset sze of 21 pxels; step sze of 2 pxels; and stran wndow of 11. The second confguraton (VIC-2D #2) uses a subset sze of 41 pxels; step sze of 2 pxels; and stran wndow of 21. In the SIFT-Meshless method, the 2-D analyss adopted two confguratons correspondng to two dfferent values of the radus of doman: d = 30 pxels (SIFT-M #1) and d = 40 pxels (SIFT-M #2). In ths case, snce t s a numercal experment and nose was not nvolved, the test functon showed n Fg. 4b was modfed for values of α s varyng from 2 to 3. Thus, by usng Eq. (4), the mnmum nfluence doman szes are 60 pxels and 80 pxels for the frst confguraton (d = 30) and the second confguraton (d = 40), respectvely. 6.1 Synthetc Snusodal Stran Feld Ths secton nvolves the use of synthetc mages undergong snusodal dsplacements wth dfferent values for the perod, smlar to what was reported n recent papers [11, 20-21]. Based on these studes, the spatal resoluton s defned as the lowest perod over whch the dsplacement or stran gradents can be Fg. 13 Example of mposed snusodal stran feld (P = 400 pxels) used n the numercal tests.
Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm 35 Fg. 14 Results obtaned from synthetc snusodal gradent analyss. Fgs. 14a and 14b show the percentage of ampltude loss to reach the stran peak (maxmum and mnmum values) for the mposed snusodal stran feld and the RMSE of the computed stran feld usng the two DIC technques. From these fgures, t can be seen that the SIFT-Meshless method was able to reproduce snusodal strans of perods up to 200 pxels wth values of ampltude loss less than 10%. As expected, more accurate results are obtaned wth the mnmum radus of doman when the values of the perod decreased,.e. more heterogeneous feld. In the same manner, the selecton of a small vrtual stran gage [22] sze n the DIC analyss clearly mproved the accuracy of the soluton. 6.2 Synthetc Gradent from a Closed-form Soluton for the Stress Concentraton around a Notch Ths secton amed to use a closed-form soluton for stress concentraton around notches to generate synthetc mages n order to evaluate the accuracy and the spatal resoluton of tradtonal and non-tradtonal DIC algorthms. For that, a dsplacement functon derved from the Krsch s soluton [23] was formulated. The Krsch s soluton s a well-know analytcal soluton used to descrbe the stran dstrbuton around a small crcular hole n an nfnte plate subjected to unaxal tensle loads. Therefore, a more sutable error assessment can be performed n order to compare the performance of the DIC algorthms by smulatng the actual stran gradent that develops around a notch. The stran dstrbuton from the Krsh s soluton plotted n Fg. 15 shows that the maxmum stran at the notch boundary s three tmes hgher than the nomnal stran far away from the hole. Moreover, t can be seen that at a dstance of half the notch radus, the stran decays to approxmately half of ts maxmum value, named here as half-radus axal decay. Ths behavor s also observed n stran dstrbutons around notches subjected to unaxal tensle loadng (see expermental results of the Secton 4). Therefore, ths characterstc can be used as control parameter to obtan dfferent confguratons of stran gradents to be smulated. For smplcty, the Krsch s equatons were frst rewrtten to apply a dsplacement only along the x-drecton, as shown n Fg. 16. Hence, the horzontal dsplacement was defned as 2 4 R R uxy (, ) 2x 3 (15) 6 x x where, ε s the maxmum stran at the notch boundary, and R s the notch radus n pxels. In ths analyss, a set of seven deformed mages were smulated wth maxmum stran value of 0.7% (ε = 0.007) and values of radus (R) varyng from 200 to 500 pxels, correspondng to a mnmum half-radus axal decay of 100 pxels and maxmum of 250 pxels, respectvely. The VIC-2D analyss adopted two dfferent parameter settngs. The frst confguraton uses a subset sze of 21 pxels; step sze of 5 pxels; and stran wndow of 11. Fg. 15 Stran dstrbuton ε xx along the horzontal axs from the Krsch s soluton.
36 Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm results confrm the good performance of the method to obtan relable stran measurements near the notch boundary. In the other hand, the VIC-2D algorthm presented values of ampltude loss above of 10% for all the smulatons. It s worth mentonng that the maxmum values were extracted drectly from the soluton, and no extrapolaton was used to estmate the maxmum value at boundary of the smulated notch. 6.3 Dscusson Fg. 16 (a) Area of nterest used for valdaton (b) example of mposed stran feld (ε = 0.007, R = 200 pxels, half-radus axal decay = 100 pxels) used n the numercal tests. Fg. 17 Results obtaned from synthetc gradent analyss wth closed-form soluton. The second confguraton uses a subset sze of 41 pxels; step sze of 5 pxels; and stran wndow of 21. The SIFT-Meshless analyss adopted two dfferent values for the radus of doman: d = 80 pxels and d = 100 pxels. Smlar to the subsecton 6.1 the values of α s vary from 2 to 3. Thus, by usng Eq. (4), the mnmum nfluence doman szes are 160 pxels and 200 pxels for the frst confguraton (d = 80) and the second confguraton (d = 100), respectvely. The regon outsde of the area of nterest shown n Fg. 16.a smulates the presence of a notch; therefore, the results obtaned from ths analyss shown n Fg. 17 take nto consderaton the error to capture the stran value at the notch boundary. For the SIFT-Meshless method, t can be seen that values of ampltude loss below of 10% are expected for values of half-radus axal decay above 150 pxels. These In ths study, two dfferent metrcs for evaluatng the spatal resoluton of the non-tradtonal DIC algorthm have been tested: synthetc snusodal stran feld and synthetc stran gradent from a closed-form soluton. The frst metrc, proposed n Ref. [11], was focused on the ampltude loss to capture the maxmum stran n a hghly heterogeneous stran feld. The control parameter s the perod, P, of the snusodal functon. The numercal results shown n Fg. 14 confrmed that the SIFT-Meshless algorthm s able to perform hgh stran gradent measurements usng an adequate expermental confguraton. The second metrc, proposed n ths paper, was focused on the ampltude loss to capture the maxmum stran n the regon surroundng a notch tp. The control parameter for ths approach s the half-radus axal decay, whch establshes that at a dstance from the notch root equal to half the notch radus, the stran decreases to approxmately half the maxmum value reached. The numercal results shown n Fg. 17 confrmed the capablty of the SIFT-Meshless algorthm to obtan accurate measurements along to the notch boundary. As shown n Fg. 18, snce features ponts are detected over the notch boundary, stran measurements are possble to be determned along ths crtcal area usng the SIFT-Meshless method. Ths s n contrast to the tradtonal subset-based DIC algorthm that showed a lmtaton n obtanng the stran nformaton at ths crtcal area. Ths lmtaton happens because the pont of measurement n subset-based
Accurate Measurements of Hgh Stran Gradents near Notches Usng a Feature-Based DIC Algorthm 37 Fg. 18 Data ponts at ROI boundary n SIFT-Meshless method and tradtonal subset-based DIC. Fg. 19 Closed form solutons ftted by a snusodal functon. methods s located at the center of the subset; therefore, a data pont s obtaned at least half the subset away from the ROI boundary. It reflects n the error value to fnd the maxmum stran values, as shown n the numercal and expermental results. In addton, the half-radus axal decay can be also related to the pxel resoluton needed to perform a measurement wth a prescrbed accuracy. For example, based on the results shown n Fg. 17a, values of ampltude loss of about 5% and 15% are expected for the SIFT-Meshless method and the VIC-2D software, respectvely, when the half-radus axal decay s 200 pxels and the maxmum expected stran s 7,000 µε. If the radus of the notch to be analyzed s equal to 1 mm (PC-2 specmen),.e., R = 2 half-radus axal decay = 400 pxels, then the pxel resoluton requred s 2.5 μm/pxel. Ths example can be compared wth the PC-2 specmen case (Secton 4): a plate subjected to tensle stress wth notch radus of 1 m, at whch a pxel resoluton of approxmately 2.5 μm/pxel was used. From the expermental results shown n Fg. 6b, the values of loss ampltude obtaned for the maxmum load appled to the specmen (.e., P = 60 N wth maxmum stran of approxmately 7,000 µε at notch boundary) were 2% and 12% for SIFT-Meshless method and VIC-3D software, respectvely. These results are n agreement wth the predcted values n Fg. 17a. Therefore, t can be concluded that ths approach can be used to evaluate the expected error n obtanng the maxmum stran value usng tradtonal and non-tradtonal DIC algorthms wth a certan expermental confguraton. Fnally, the half-radus axal decay can be also related to the perod of the snusodal functon as shown n Fg. 19. It s shown that, the stran gradent behavor from the Krsch s soluton (for descrbng stresses around a hole n an nfnte plate) and from the Creager & Pars equatons [24] (for descrbng stresses n parabolc, ellptc, U and narrow V-shaped notches) can be approxmated by a sne functon,.e., half-radus axal decay = P/4. Thus, a pror error estmaton of an algorthm can be obtaned f numercal experments are smulated beforehand usng a dsplacement snusodal functon wth perod P = 2 R. 7. Conclusons The present work shows and compares by means of expermental and numercal tests, the ablty of a recently developed method, the SIFT-Meshless method, to perform hgh stran gradent measurements around shallow and deep notches, even under condtons of sgnfcant plastcty. The expermentally determned full-feld stran responses are n agreement wth results determned from numercal fnte element smulatons and from a commercally avalable DIC software. The capablty of the SIFT-Meshless method for obtanng relable stran measurements at the notch boundary zone, where the maxmum stran values were expected, was
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