EVALUATION OF ALGORITHMS FOR RESIDUAL STRESS CALCULATION FROM RADIAL IN-PLANE DSPI DATA

Transcription

1 EVALUATION OF ALGORITHMS FOR RESIDUAL STRESS CALCULATION FROM RADIAL IN-PLANE DSPI DATA M. R. Viotti, nd A. V. Fntin, A. Albertzzi Lbortório de Metrologi e Automtizção, Universidde Federl de Snt Ctrin, CEP , Florinópolis, SC, Brzil ABSTRACT Residul stress mesurements cn be performed by using the combintion of rdil in-plne digitl speckle pttern interferometer nd the hole drilling technique. In prcticl situtions it is very usul tht displcements due to relieved residul stresses produce opticl phse distribution corrupted by severl noise levels. In consequence, lgorithms used to compute these stresses cn be influenced by the level noise introducing errors during the computtion. This pper reports on the evlution nd comprison of four methods: () displcement-lest squres pproch, (b) strin-lest squres pproch A, (c) strin-lest squre pproch B nd (d) strin-lest bsolute error minimiztion one. These methods determine residul stress sttes from only one opticl phse distribution produced by the hole drilling technique. The former pproch mkes the computtion from the mesured displcement field round the hole. On the other hnd, the other pproches use the computed rdil strin field from the mesured rdil in-plne displcement field. To evlute the performnce of these methods, phse mps for severl levels of residul stresses nd different levels of opticl noise were simulted. Keywords: Digitl speckle pttern interferometry; Residul stresses; Rdil in-plne displcements. 1. Introduction In the lst decdes, severl opticl techniques hve been developed hving the bility to generte fringe ptterns from which the displcement field cn be evluted. Among them, digitl speckle pttern interferometry (DSPI) is the most verstile one [1]. As it is well known, DSPI is bsed on the genertion of speckle distributions by lser source. In contrst to hologrphic interferometry, which needs photogrphic process, DSPI uses speckles sufficiently lrge to be resolved by video system. The ppliction of digitl techniques in DSPI llows the utomtion of the dt nlysis process, which is usully bsed on the extrction of the opticl phse distribution encoded by the generted correltion fringes [2]. From these dt, in-plne nd outof-plne whole-field displcement fields cn be mesured over the surfce of ny rough object without mking contct with it. A novel nd importnt ppliction of DSPI is its combintion with the hole-drilling technique to mesure residul stress fields [3, 4]. Severl steps tke plce during the mesurement. First, set of speckle interferogrms is cquired nd next the hole is drilled to remove stressed mteril. After this, second set of interferogrms is cquired nd the opticl phse difference distribution is clculted. Finlly, the displcement field due to the relieved stresses is computed. To perform the drilling process between both imge cquisitions, kinemtic mountings re used [5]. They llow cquiring the imge sets on the opticl bench nd to mke the hole in n externl milling mchine. These repositioning systems present some drwbcks, nmely: () they cn introduce rigid body displcements, (b) they cn be used to evlute only smll prts, nd (c) they cn not be pplied to perform in-situ mesurements. In order to void these problems, Albertzzi et l. [6-8] hve developed portble double illumintion DSPI device tht enbles mesuring of residul stresses outside of the opticl bench nd the lbortory. This system ws designed tking into ccount the fulfilment of some requirements such s: () high stiffness to keep the reltive motion between criticl prts of the interferometer below on n cceptble level; (b) strong clmping system to keep negligible the reltive motion between the device nd the specimen to be mesured; (c) fst nd esy positioning in order to plce the mesuring device precisely in given point of interest, fst processing to obtin the rdil in-plne displcement field from the nlysis of only one correltion fringe pttern. Despite the fct tht the good performnce of this portble device to mesure residul stresses hs been firmly estblished nd its use for in-situ mesurements, its compct configurtion cn introduce opticl noise in the mesured opticl phse distribution. This noise cn be increse by the drilling process becuse it produces locl heting nd smll locl rigid body

2 displcements which cn decrese interferogrms correltion consequently incresing the signl to noise rtio. This opticl noise ffects the mesured opticl phse distribution nd, depending on their mgnitude, they could introduce considerble errors during the residul stress determintion. In consequence, it is importnt to hve processing method cpble to del with opticl phse distributions corrupted with opticl noise in order to compute the residul stress field. This pper reports on the evlution nd comprison of four methods: () displcement-lest squres pproch, (b) strinlest squres pproch A, (c) strin-lest squres pproch B nd (d) strin-lest bsolute error minimiztion one. These methods determine residul stress sttes from only one opticl phse distribution produced by the hole drilling technique. The former method uses Kirsch s displcement equtions to pply the minimizing technique nd to perform the computtion of residul stress mgnitudes nd rigid body trnsltions. On the other hnd, the ltter methods use the computed rdil strin field from the mesured rdil in-plne displcement field. To evlute the performnce of these methods, phse mps for severl levels of residul stresses nd different levels of opticl noise were simulted. As in prctise, it is very difficult to mke specimen with reference vlue of residul stress tht llows the vilbility of set of phse mps for different residul stress sttes. In the present work, simulted ones were used s reference. These phse mps were corrupted for five levels of rndom opticl noise. These new set mps were processed in order to evlute the performnce nd the error of the methods to quntify the residul stress fields. 2. Residul stress mesurement 2.1. Displcement-lest squres pproch. The polr rdil displcement field mesured in circulr region provides sufficient informtion for chrcteriztion of displcements, stresses or residul stresses tht occurs in the interrogted region. If uniform in-plne trnsltion is pplied on the specimen surfce, the following rdil displcement field is developed [9] u (, r θ) = u cos( θ α) (1) r where ur (, r θ ) is the polr in-plne rdil displcement component, u t is the mount of uniform trnsltion, α is the ngle tht defines the trnsltion direction, r nd θ re polr coordintes. When the hole-drilling technique is used, it is possible to obtin quntittive vlue of the residul stress from the mesurement of the displcement field generted round through hole by using the model developed by Kirsch [10] nd round blind hole by pplying the numericl model developed by Schjer [11]. The Kirsch s model is bsed on the elstic solution for n infinite plte subjected to uniform stress stte, when cylindricl hole is drilled through the plte thickness. The rdil component of the rdil in-plne displcement field developed by the introduction of the hole cn be written in polr coordintes s [10]: where the functions A(r) nd B(r) re given by [10]: t u (, r θ) = A()( r σ + σ ) + B()( r σ σ )cos(2θ 2 β) (2) r υ Ar () = (3) r 2 1+ υ Br () = 3 r 1+ υ r being u r ( r, θ ) the rdil component of the rdil in-plne displcement field, σ1 nd σ 2 re the principl residul stresses, β is the principl direction, E is the elstic modulus, υ is the Poisson s rtio, nd is the rdius of the hole. Replcing the vlues of A(r) nd B (r) from Eqs (3) nd (4) into Eq. (2): (4) 3 ur (, r θ ) = (1 + ) ( 1 2) 4 -(1 ) ( 1 2)cos( ) υ ρ σ + σ + ρ υ ρ + σ σ θ β where ρ = / r is the rtio of the hole rdius to the polr coordinte r. (5) The rdil component of the rdil in-plne displcement field developed by the combintion of displcements produced by rigid body motions nd relieve residul stresses cn be obtined by dding Eqs. (1) nd (5) giving

3 3 ur( r, θ ) = u cos( θ t α ) + (1 + ) ( 1 2) 4 -(1 ) ( 1 2) cos( ) ν ρ σ + σ + ρ + ν ρ σ σ θ β (6) When lest squres pproch is used, set of experimentl dt is smpled from the mesured displcement field nd fitted to mthemticl model by lest squres. No prticulr smpling strtegy is required, but it is good prctice to select smpling points regulrly distributed over ll mesured region. An pproprite mthemticl model cn be obtined by dding nd rewriting Eqs (1) nd (5): (, ) 0. θ 1 cos( θ ) 1 sin( θ ) 2. r u R u C u S u C cos(2 θ ) u2s. = sin(2 θ ) + u0 u r K K K K K K r (1 + υ) r r (1 + υ) r r The terms Ku 0 R, Ku 1 C, Ku 1 S, Ku 2 C nd Ku 2 S re esily identified by comprison with equtions (1) nd (5). The term K u0 is n dditionl one used to bsorb constnt bis in the displcement field, which cn be occsionlly cused by therml drift. At lest six mesured points re necessry to determine the six coefficients. With DSPI few tens of thousnds mesured points re vilble being used to compute the coefficients by mens of the lest squres method. Since the coefficients re ll liner, the lest squres cn be crried out by stright forwrd wy using multi-liner fitting procedure. The displcement nd stresses components cn be computed from the fitted coefficients of Eq. (7) by (7) u = K + K t u1c u1s α tn 1 u1s = K uc 1 ( Ku R Ku C Ku S) 1 0 ( Ku R Ku C Ku S) tn K E σ = + + (1 + υ). E σ = + (1 + υ). β K 1 u2s = 2 K u2c (8) In prcticl situtions it is very usul tht both residul stresses nd rigid body displcements pper mixed up in the sme opticl phse difference distribution. They cn be mesured simultneous nd independently since different orthogonl lest squres terms re involved in their computtion Strin-lest squres pproch A Tking into ccount tht ρ = / r, it cn be replced in Eq. (5). Thus, this eqution cn be written in the following wy: 4 (1 + ν) 1 ur (, r θ) = ( σ1+ σ2) + 4 -(1 + ν) ( σ1 σ2)cos(2θ 2 β) r r r In order to obtin the rdil component of the rdil in-plne strin ε r, Eq. (8) cn be derivte in r (9) 4 (1 + ν) 1 (, ) ( 1 2) εr r θ = σ σ -4 3(1 ν) ( σ1 σ2) cos(2θ 2 β) r r r As Eq. (1) does not depend on the rdil position, the rdil in-plne strin produced by rigid body displcements is zero. Thus, by rerrnging the terms nd replcing gin the reltionship between the rdius of the hole nd the rdil position, Eq. 10 cn be written in more dequte form (1 + ν) εr (, r θ) = ρ ( σ1+ σ2) + -4ρ + 3(1 + ν) ρ ( σ1 σ2)cos(2θ 2 β) Since the rdil in-plne displcement produced by rigid body trnsltions is independent of the rdil coordinte, its derivte is zero for every point to be nlyzed. In consequence, the use of rdil strins to compute residul stress from phse mps (10) (11)

4 where displcements produced by relieved residuls stresses nd unwnted rigid body trnsltions re mixed up is very ttrctive method becuse this method is independent of rigid body motions. The reltion between the displcement ur (, r θ ) nd the mesured phse difference φr (, r θ) is given by [2] 4π φr(, r θ) = ur(, r θ)sinγ (12) λ where λ is the wvelength of the light nd γ is the ngle between the direction of illumintion nd the norml to the specimen surfce. Thus, to compute the residul stress field, the displcement field is obtined from the continuous opticl phse difference distribution nd then the relted strin field is clculted performing the numericl derivte of these displcements. As it is known, DSPI techniques llows obtining phse difference mps whose vlues re between π nd + π, the continuous phse difference distribution should be computed using phse unwrpping methods which dd or subtrct n dequte multiple of 2π to ll pixels to remove phse jumps [14]. As before, n pproprite mthemticl model cn be obtined by rewriting Eq. (11) s follows r( r, ) = Kε0R. + K 2C. ε 3 cos(2 ) Kε2S. 3 + sin(2 ) + Kε0 ε θ r θ θ (1 + υ) r r (1 + υ) r r The stress components cn be computed from the fitted coefficients of Eq. (12) by E σ1 = Kε0 + Kε2 + Kε2 (1 + υ) E σ2 = Kε0 Kε2 + Kε2 (1 + υ) 1 tn 1 K ε2s β = 2 K ( R C S ) ( R C S ) 2.3. Strin-lest squres pproch B The previous pproch evlutes the residul stress field by strin pproch using the strin field computed from the rdil in-plne displcement fields. Previously, n imge processing hs to be crried out, which includes the ppliction of phse unwrpping techniques. Tking into ccount Eq. (12) nd the reltionship between strin nd displcement fields λ ur(, r θ) =. φr(, r θ) 4πsinγ nd ur (, r θ) εr (, r θ) = (15) r it is possible to obtin the following reltionship between the opticl phse distribution nd the rdil strin component ε2c (13) (14) λ φr (, r θ) εr (, r θ) =. 4πsinγ (16) Consider tht Squring nd dding member to member cos φr( r, θ) φr( r, θ) = sin φr ( r, θ). sin φr( r, θ) φr( r, θ) = cos φr ( r, θ). (17) r(, r ) cos r(, r ) sin r(, r ) φ θ φ θ φ θ =± + (18) Thus, it cn be noted tht Eq. (18) nd (16) llows performing the direct computtion of the strin field from the opticl phse distribution being unnecessry the use of phse unwrpping lgorithms becuse Eq. (18) uses sin() nd cos() functions which re continuous functions. The sign of the derivtive is determined, for exmple, compring the sign of cos φr ( r, θ ) with the

5 sign of sin φr ( r, θ ) / r. The sign of eqution (18) is positive if both signs re the sme nd negtive if otherwise. Once the strin field is computed, Eqs. (13) nd (14) cn be used in order to compute the ssocited residul stress field Strin-lest bsolute error minimiztion pproch In this pproch, n error function is built nd minimized for the complete set of points of the processed phse difference mp by using lest bsolute minimiztion method in order to compute the principl residul stresses nd their principl direction. This error function is shown in Eq. 19. Npoint s Fe1( r, θ) = ε( r, θ) u( r, θ) (19) n= 1 where Fe1( r, θ) is the error function, urθ (, ) is the numericl derivte of the mesured displcements computed by equtions (16) nd (18), nd Npoints re the number of vlid points on the imge. As pproch 2.3, the use of the direct computed strin field presents the sme dvntge voiding the implementtion of phse unwrpping lgorithm. The lest bsolute method hs some dvntges over the lest squres ones, nmely it is insensitive to lrge differences produced by the presence of outlier phse vlues into the opticl phse distribution. The min drwbck is tht it is mthemticlly less stble nd computtionlly hevier to be implemented. 3. Evlution procedure nd results discussion As it ws previously mentioned, computer simultion ws used in order to comprtively evlute the performnce of the proposl methods. The computer simultion considered tht perfectly relieved displcement fields, s in eqution (2), were mesured by rdil in-plne interferometer with symmetricl dul-bem illumintion hving sensitivity vector in the rdil direction. () (b) (c) (d) (e) Figure 1. Simulted residul stress phse difference mps for σ 1 =150 MP, σ 2 = 30 MP nd β = 45º. Opticl Gussin noise distributions were dded to the phse vlues with: () stndrd devition of 3 %, (b) stndrd devition of 6 %,(c) stndrd devition of 9 %, (d) stndrd devition of 12 % nd (e) stndrd devition of 15 % of one fringe order.

6 The wvelength used in the simultion ws chosen s λ = 785 nm nd the ngle between the direction of illumintion nd the norml to the specimen surfce ws γ = 60 o. The specimen ws simulted over circulr region of 10 mm in dimeter with resolution of pixels centred t the hole centre. Steel ws selected s the specimen mteril, with n elsticity module E = 210 GP nd Poisson s rtio υ = 0.3. Rdil in-plne displcement fields for severl residul stress sttes were simulted using Eq. (2). Then, they were trnsformed in phse vlues using Eq. [12] being finlly wrpped in order to obtin discontinuous opticl phse distributions. On the other hnd, severl sets of imges contining opticl noise with Gussin distributions were obtined by using NORM ( µ, δ ) = µ + δ 2ln( rdn).cos(2 π. rdn) (20) where the function NORM ( µ, δ) in eqution (20) genertes Gussin (norml) distribution, where µ nd δ re the men vlue nd the stndrd devition of the Gussin distribution, nd rdn is pseudo-rndom numbers genertor with rectngulr distribution between 0 nd 1. Five levels of Gussin noise were tking into ccount by using five mgnitudes of stndrd devitions, nmely, 3, 6, 9, 12, nd 15 % of one fringe order dded to the imges of the phse difference mps. As these imges hd 256 grey levels (8 bits) the used stndrd devitions were 7.7, 15.4, 23.0, 30.8 nd 38.4 grey levels. To ccomplish sttisticl nlysis, set of 10 imges were finlly obtined for ech level of opticl noise nd for ech residul stress stte simulted. At lst, ech combined phse mp (residul stress dded to opticl noise) ws processed with the proposl pproches nd the residul stress fields were computed. As n exmple, Figs. 1 (), (b), (c), (d) nd (e) show simulted residul stress phse mps for σ 1 =150 MP, σ 2 = 30 MP nd β = 45º nd they correspond to 3, 6, 9, 12, nd 15 % of fringe order of stndrd devition of the opticl noise respectively. A summry of the computed results is shown in Tbles 1, 2 3 nd 4. These tbles list the stress men vlue nd its stndrd devition for ech set of 10 imges. In these tbles, the identifiction number of every method is corresponded to the pproch presenttion order in heder 2. Residul stress field: σ 1 = 150 MP, σ 2 = - 50 MP, nd β=45º Noise stndrd Approch 2.1 Approch 2.2 Approch 2.3 Approch 2.4 devition [%] Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. σ 1 [MP] 3 149,98 0,10 150,44 0,20 145,23 0,17 145,26 0,23 σ 1 [MP] 6 149,95 0,33 150,35 0,43 144,58 0,77 144,69 0,50 σ 1 [MP] 9 150,1 0,37 150,13 1,53 143,37 0,77 143,66 0,67 σ 1 [MP] ,21 0,50 150,77 1,73 141,42 0,93 142,12 1,03 σ 1 [MP] ,79 0,67 151,14 2,50 138,37 0,87 140,72 0,93 σ 2 [MP] ,13-46,46 0,17-49,73 0,17-49,7 0,23 σ 2 [MP] 6-50,05 0,33-46,61 0,40-49,29 0,47-49,26 0,60 σ 2 [MP] 9-49,89 0,37-46,52 1,60-48,79 0,90-48,93 0,80 σ 2 [MP] 12-49,84 0,50-46,25 1,87-49,06 0,90-48,82 0,87 σ 2 [MP] 15-50,27 0,63-46,39 2,20-48,08 1,33-48,17 1,40 β [º] ,00 45,01 0,03 44,99 0,03 44,99 0,03 β [º] ,00 44,99 0,07 44,99 0,07 44,99 0,07 β [º] 9 44,99 0,00 44,99 0,07 45,02 0, ,10 β [º] ,03 45,01 0,10 45,07 0,07 45,01 0,13 β [º] 15 45,01 0,03 45,1 0,23 45,05 0,17 44,93 0,13 Tble 1. Computed residul stress fields from simulted phse mp with five levels of opticl noise nd for σ 1=150 MP, σ 2 = -50 MP nd β = 45º. All results re expressed in MP. According to results list in Tbles 1, 2, 3 nd 4, it is possible to see tht the displcement-lest squres pproch (in these tbles it is clled s Approch 2.1) hs mnged to correctly compute the residul stress from ll the phse mps for ech level of opticl noise. In ddition, this lgorithm presented n bsolute error between the simulted residul stress field nd the computed one lwys inside MP.

7 These tbles lso show tht the strin-lest squres pproch A (clled s Approch 2.2) hd good performnce. This pproch only filed during the computtion of one of the principl residul stress, nmely the smller one σ 2. For this stress, the bsolute error rnged from 1 to 3 MP. This method presented n bsolute error in the lrger stress σ 1 which oscillted between 0.1 MP nd 1.0 MP. The other strin pproches presented bsolute errors which increse considerbly s the opticl noise rises. Only for n opticl noise with stndrd devition up to 7.7 grey levels (3%), they showed resonble performnce hving n bsolute error of bout 5 MP. For the other level noises, they presented poor performnce rnging the bsolute error between 5 nd 10 MP. Residul stress field: σ 1 = 150 MP, σ 2 = 0 MP, nd β=45º Noise stndrd Approch 2.1 Approch 2.2 Approch 2.3 Approch 2.4 devition [%] Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. σ 1 [MP] 3 149,96 0,23 150,49 0,33 146,62 0,23 146,58 0,30 σ 1 [MP] 6 149,98 0,27 150,49 0,63 145,57 0,43 145,98 0,70 σ 1 [MP] 9 149,69 0,43 150,21 0,57 144,51 0,47 144,61 0,80 σ 1 [MP] ,98 0,43 150,14 1,43 142,59 0,90 142,76 1,03 σ 1 [MP] ,59 0,60 149,58 1,40 139,25 1,03 140,55 1,53 σ 2 [MP] 3-0,03 0,23 3,22 0,33-0,58 0,20-0,57 0,30 σ 2 [MP] 6 0,04 0,30 3,27 0,57-0,69 0,40-0,32 0,40 σ 2 [MP] 9-0,3 0,37 2,69 0,83-0,84 0,60-0,82 0,57 σ 2 [MP] 12-0,03 0,43 2,67 1,33-1,07 0,43-1,12 1,07 σ 2 [MP] 15-0,27 0,60 2,38 2,23-1,79 0,90-2,35 1,20 β [º] ,00 44,98 0, , ,03 β [º] 6 44,99 0,00 44,99 0,07 44,98 0,03 45,06 0,07 β [º] ,00 44,98 0,13 45,02 0,10 45,03 0,13 β [º] ,03 45,07 0,13 45,04 0,20 44,93 0,23 β [º] 15 45,01 0,03 45,05 0,17 45,04 0,13 45,1 0,23 Tble 2. Computed residul stress fields from simulted phse mp with five levels of opticl noise nd for σ 1 =150 MP, σ 2= 0 MP nd β = 45º. All results re expressed in MP. Residul stress field: σ 1 = 150 MP, σ 2 = 30 MP, nd β=45º Noise stndrd Approch 2.1 Approch 2.2 Approch 2.3 Approch 2.4 devition [%] Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. σ 1 [MP] 3 149,99 0, ,17 147,44 0,20 147,25 0,37 σ 1 [MP] 6 149,93 0,23 150,95 0,40 146,74 0,37 146,58 0,87 σ 1 [MP] 9 149,78 0,27 150,64 0, ,80 145,74 0,93 σ 1 [MP] ,86 0,27 150,17 1,37 143,22 0,70 143,67 1,27 σ 1 [MP] ,09 0,73 149,85 1,63 140,66 1,23 141,57 1,10 σ 2 [MP] ,10 32,22 0,30 29,37 0,23 29,36 0,43 σ 2 [MP] 6 29,95 0,20 32,24 0,47 29,67 0,33 29,45 0,77 σ 2 [MP] 9 29,79 0,27 32,27 0,80 29,12 1,03 29,46 0,87 σ 2 [MP] 19,84 0,27 31,8 1,30 28,4 0,77 28,65 1,00 σ 2 [MP] 15 30,17 0,73 31,05 1,23 27,63 0,70 27,32 1,40 β [º] , ,07 44,99 0,07 44,98 0,07 β [º] 6 45,01 0,00 45,01 0,07 44,95 0,13 45,03 0,10 β [º] ,03 45,02 0,13 45,06 0,17 45,01 0,30 β [º] ,03 45,02 0,20 45,04 0,23 44,98 0,17 β [º] 15 44,99 0,03 44,92 0,37 44,92 0,23 45,01 0,17 Tble 3. Computed residul stress fields from simulted phse mp with five levels of opticl noise nd for σ 1 =150 MP, σ 2 = 30 MP nd β = 45º. All results re expressed in MP. As it ws previously shown, strin-lest squres pproch B (2.3) nd strin-lest bsolute error minimiztion one (2.4) dispense the ppliction of phse unwrpping techniques to compute the rdil strin field from the phse mp, gretly

8 reducing the numericl effort required. According to previous tbles, the introduced opticl noise influence over the residul stress bsolute error ws considerble. This conclusion highlighted the dvntge of the ppliction of robust phse unwrpping lgorithms tht use weighting mtrix to msk inconsistent pixels voiding their influence during the residul stress computtion. In consequence, the displcement-lest squres pproch (2.1) nd the strin-lest squres pproch A (2.2) re more robust thn the strin-lest squres pproch B (2.3) nd the strin-lest bsolute error minimiztion pproch (2.4). The men vlues of the residuls stresses re lwys closer to the reference vlues for the first two pproches, wht lso mke them more dequte to processing rel hole drilling phse mps. By compring the displcement-lest squres pproch (2.1) nd the strin-lest squres pproch A (2.2) nd by nlysing the stndrd devition of the obtined results, it is possible to see tht the stndrd devition of both methods increses s the opticl noise level increse. For the former method the stndrd devition ws smller thn 1 MP whtever the noise level. On the other hnd, the ltter method showed stndrd devition up to 2 MP. Thus, the displcement-lest squres pproch cn be considered s the best of these four methods to compute residuls stresses. Residul stress field: σ 1 = 150 MP, σ 2 = 75 MP, nd β=45º Noise stndrd Approch 2.1 Approch 2.2 Approch 2.3 Approch 2.4 devition [%] Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. Men vlue Stndrd dev. σ 1 [MP] 3 149,97 0,10 151,09 0,23 148,16 0,23 148,26 0,30 σ 1 [MP] 6 149,95 0,27 150,81 0,53 147,18 0,37 147,5 0,57 σ 1 [MP] 9 149,99 0,40 150,62 0,93 146,27 0,73 146,63 0,87 σ 1 [MP] ,99 0,63 150,39 1,23 143,43 0,73 144,73 0,70 σ 1 [MP] ,11 0,73 150,45 1,37 141,53 0,83 143,58 1,13 σ 2 [MP] 3 74,97 0,10 77,29 0,20 74,43 0,30 74,65 0,37 σ 2 [MP] 6 74,96 0,23 77,02 0,57 74,01 0,53 74,18 0,77 σ 2 [MP] 9 74,97 0,40 76,72 1,23 73,51 0,67 74,44 1,07 σ 2 [MP] 12 74,9 0,63 76,81 1,10 71,7 0,77 74,64 1,00 σ 2 [MP] 15 75,04 0,77 76,33 1,07 70,35 0,73 74,48 0,77 β [º] ,00 44,96 0,07 44,99 0,07 44,98 0,10 β [º] 6 45,01 0,03 44,92 0,17 44,96 0,17 45,07 0,20 β [º] 9 44,99 0,03 45,04 0,20 44,9 0,20 44,91 0,23 β [º] 12 44,98 0,03 45,1 0,20 45,08 0,27 44,94 0,47 β [º] 15 44,99 0,03 44,92 0,47 44,95 0,37 45,09 0,43 Tble 4. Computed residul stress fields from simulted phse mp with five levels of opticl noise nd for σ 1=150 MP, σ 2 = 75 MP nd β = 45º. All results re expressed in MP. 4. Conclusion The ttrctiveness of DSPI to the opticl metrology community rises not only form its non contcting nture but lso from the reltive speed of the inspection procedure, minly due to the use of video detection nd digitl imge processing. A novel nd importnt ppliction of this technique is its combintion with the hole-drilling technique to mesure residul stress fields. However, phse mps tht re generted during the hole drilling process present sever dt contmintion by noise nd lso include the edge of the hole nd decorreltion due to rigid body displcements of the specimen to be mesured. This opticl noise ffects the mesured opticl phse distribution nd, depending on their mgnitude, they could introduce considerble errors during the residul stress determintion. In consequence, it is obvious the need of robust processing lgorithms which should be cpble to del with these rel phse mps. This pper presents some lgorithms to compute residul stress fields from the phse difference mps. According to the preceding results it is possible to see tht the displcement-lest squres pproch (2.1) hs presented the better overll performnce. In ddition, it hs mnged to compute residul stress fields with n bsolute error rnging between MP. According evlution results, it cn be noted tht this lgorithm is more robust becuse it is slightly ffected by opticl noise level present in the residul stress simulted phse mps nd no significnt bis error ws detected in ny simulted mp. However, robust phse unwrpping lgorithm is required. Further reserch will be done in order to evlute the displcement-lest squres pproch nd the strin-lest squres pproch A with phse mps generted by combining displcements due to rigid body motions nd residul stress fields. The results will be presented in future works.

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