Copyright JCPDS-International Centre for Diffraction Data 2009 ISSN Advances in X-ray Analysis, Volume 52

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1 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume A NEW PROCEDURE FOR THE EVALUATION OF RESIDUAL STRESSES BY THE HOLE DRILLING METHOD BASED ON NEWTON-RAPHSON TECHNIQUE ABSTRACT G. Petrucc, M. Scafd Dpartmento d Meccanca, Unverstà degl Stud d Palermo, Vale delle Scenze 9128 Palermo, The hole drllng method s one of the most used sem-destructve technques for resdual stress analyss of mechancal parts. In the presence of non-unform resdual stresses, the stress feld can be determned from the measured relaxed strans usng several methods, but the most used s the so called ntegral method. Ths method s characterzed by some smplfcatons that lead to approxmate results especally when the resdual stress vares abruptly. In ths paper a new calculaton procedure based on the Newton-Raphson method for the determnaton of zeroes of functons s presented. THE HOLE DRILLING METHOD The hole drllng method s one of the more used sem-destructve technques for the analyss of the resdual stresses n mechancal components [1-3]. In the case of stresses not unform n the thckness, a hole of radus equal to R s drlled n varous steps, up to the maxmum depth z M and the relaxed deformatons are recorded and subsequently elaborated to calculate the resdual stresses. The relatonshp between stresses and strans s the followng z p z = F( Z, z) P Z dz, z zm, Z z, (1) beng p ( z) = ( ε + )2, ( z) = ( σ + )2 1 ε 3 P, (2,3) 1 σ 3 where ε 1 and ε 3 are the deformatons measured by the grds 1 and 3 of the rosette, σ 1 and σ 3 are the stresses actng n the same drectons, z s the current depth of the hole, Z s the abscssa measured from the surface of the component, F(Z,z), defned n the feld Z z, s the nfluence functon that depends on the geometry of the analyzed component, the hole-rosette assembly and can be determned through numercal smulatons by the fnte element method or the boundary method [4]. Typcally, the stresses are evaluated up to a maxmum depth z M equal to half the value of the mddle radus of the rosette R m,.e. z M =R m /2. The determnaton of the functon P(z) through eq.(1) consttutes an nverse problem for whose resoluton dfferent approxmate methods have been proposed, lke ntegral method [1], the power seres method [3], the splne method [5], procedures based on the applcaton of methods of soluton of the nverse problem [6-7]. In each method, the P(Z) stresses are approxmated by a functon P (z) whose parameters are determned n such a way that the dfferences between the measured deformatons p(z) and the deformaton calculated by eq.(1) ntroducng the P (z) stresses p (z)

2 Advances n X-ray Analyss, Volume 52 Ths document was presented at the Denver X-ray Conference (DXC) on Applcatons of X-ray Analyss. Sponsored by the Internatonal Centre for Dffracton Data (ICDD). Ths document s provded by ICDD n cooperaton wth the authors and presenters of the DXC for the express purpose of educatng the scentfc communty. All copyrghts for the document are retaned by ICDD. Usage s restrcted for the purposes of educaton and scentfc research. DXC Webste ICDD Webste -

3 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume z (, ) p z = F Z z P Z dz (4) are mnmzed n correspondence to a dscreet number of properly chosen depth z (that, n general don't concde wth all the steps n whch the drllng of the hole s dvded). The ntegral method [1-4] s based on the assumpton that the stress actng between two levels z -1 and z s equvalent to a unform stress,.e. a stepped functon P (z) s consdered. In the power seres method [3] the approxmatng functon P (z) s a polynomal of proper degree, usually not hgher than 3. In the splne method [5] the P (z) functon s a splne whose number of polynomals and degree s assgned. The coeffcents of the splne are determned by solvng a lnear system gven by the ntegral equaton n correspondence of the extremes of the polynomals of the splne. As also shown n [5], n each of the proposed methods, the p (z) deformatons evaluated by eq.(4) do not concde wth the expermental p(z) deformatons, apart from the levels z n correspondence of whch the parameters of the approxmatng functons have been evaluated, and a dfference between evaluated deformaton and the expermental ones defned as e z = p z p z (5) can be evaluated at each quote z. Recently methods of soluton of the nverse problem that, usng eq.(4), try to correct the stresses P (z) n such a way that the dfferences (5) are mnmzed, have been proposed [6-7]. As an alternatve, n ths paper, the teratve Newton-Raphson method s used to correct the P (z) functon so that the dfference between the p (z) and p(z) deformatons n the whole feld (5) s mnmzed n all the feld. THE PROPOSED METHOD In the proposed method, the P(Z) functon s approxmated by a P (Z) functon gven by n polynomals P (Z) of M degree, each representng the functon n the feld z 1 Z z, wth =1 n: M P( Z) = a, Z, z 1 Z z, = 1 n. (6) = In ths case, the a, coeffcents n (6) are the unknown of the problem represented by eq.(1); n the proposed method ther determnaton s effected by teratve approxmatons through the Newton-Raphson (NR) method [8] n the way descrbed n the followng. In practce the proposed method requres the ntal choce of the followng parameters: the number of polynomals n, the maxmum degree of the polynomals M, the coordnates of the extremes of the feld of defnton of every polynomal z, wth = n. In the followng, some practcal suggestons for effectng such choces wll be furnshed. Introducng n (4) the polynomals defned n (6) n place of P (Z), the followng expresson for the deformatons p (z) can be obtaned: z1 z2 zn M M M (, ) 1, (, ) 2,... (, ) n, = z = 1 z = n 1 (7) p z = F Z z a Z dz + F Z z a Z dz + F Z z a Z dz

4 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume that can be syntheszed as n M p z = a I z, (8) where k= 1 = zk zk 1 k, k, Ik, z = F( Z, z) Z dz, (9) beng FZz (, ) =, Z> z. (1) In general, the abscssas z at whch the p (z) functon s evaluated le nsde the nterval of exstence of one of the polynomals (6),.e. t s z k 1 z z k, wth z k z M. Snce the ntegral (4) s extended from Z= to Z=z, the ndex of the frst summaton starts, obvously, from the value k=1, nvolvng n the calculaton of p (z) all the polynomals P k (Z) for whch the condton z k z s verfed. The summaton (8) s extended up to k=n, for smplcty of notaton and mplementaton. Ths would mply that the ntegraton (4) s always extended up to the maxmum depth z M >z, so that, n order to avod to ntroduce errors, the functon F(Z,z) has to be set to zero for Z>z, although t has no physcal meanng n such feld. Consderng eq.(8), the functon e(z) defned n (5), can be rewrtten as, n M k, k,. (11) k= 1 = = e z a I z p z For each of the n polynomals that consttute the P (Z) functon, the followng error functon, whose varables are the a, coeffcents, can be defned z z 1 2 E = e z dz. (12) The optmal values of the a, coeffcents are those that mnmze the E functons and, as antcpated, they can be obtaned by the NR method [8]. In general, the NR method allows the determnaton of the zeros of a functon through teratve approxmatons. In partcular, n ths case, the searched values of the a, coeffcents are those that make null the dervatve of the E functon tself wth respect to the a, varables, n correspondence of whch the E functon assumes ts mnmum value. The NR method allows to obtan the values of the zeros of the functon n teratve way, startng from frst attempt values and brngng opportune correctons at every teraton. Introducng the a vectors that contan the a, coeffcents related to the -th polynomal, the recursve relatonshp for the determnaton of the zeros s the followng: a = a + Δa, (13) q q 1 n whch q s the number of the teraton and Δa s the vector of the correctons to be assgned to the values of the zeros that can be obtaned by the followng relatonshp:

5 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume Δa = J H, (14) where J and H are respectvely the Jacoban vector and the Hessan matrx of the E functons, whose components are gven by the partal dervatves of the E functons wth respect to the a, coeffcents as follows: J E a =, =1,2,.. M,, H s, a E a 2 =, =1, 2,.. M ; s=1, 2,.. M. (15,16),, s Rememberng eq.(12), eq.(15) can be rewrtten as z e z J = 2 e( z) dz. (17) a z 1, Referrng to eq.(11), the dervatves of the e(z) functon can be wrtten as e z n M = = ak, Ik, ( z) p ( z) I, ( z) (18) a, a, k= 1 = and eq.(17) becomes z z 1 J = 2 e z I z dz, (19), beng the I, (z) functons defned n eq.(9). Operatng as n the prevous case, the elements of the Hessan matrx can be smply obtaned as z s, 2, s, z 1 H = I z I z dz. (2) It s easy to observe that, once the abscssas of the extremes of the felds of defnton of the polynomals z are fxed, the elements of the Hessan matrx are constants. In concluson, fxed the number of the polynomals n, the degree of the polynomals M and the coordnates of the extremes of the felds of defnton of the polynomals z, the procedure for the calculaton of the a, coeffcents conssts n the followngs steps: 1. the I, (z) functons are evaluated by eq.(9), 2. the elements of the Hessan matrx are evaluated by eq.(2), 3. frst attempt values are assgned to the a, coeffcents (they can be smply set to ), 4. the teratve procedure for the determnaton of the a, coeffcents for each polynomal P, s effected; n partcular t conssts n the followng steps: 4.1. the e(z) and E functons are evaluated by eq.(11) and (12); f the value of E or the varaton of E wth respect to the prevous step s lower than a pre-defned low value the teraton s stopped, 4.2. the elements of the Jacoban vector are evaluated by eq.(19), 4.3. the correcton to be assgned to the a, coeffcents are evaluated by eq.(14), 4.4.the new values of the a, coeffcents are obtaned ncreasng those of the prevous step by eq.(13).

6 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume NUMERICAL ANALYSIS AND DISCUSSION The effectveness of the proposed method has been verfed consderng smulated resdual stress dstrbutons. In partcular, n ths work: 1. shot peenng [6] and lnear resdual stress felds P(z) on a mechancal component made of steel (wth E=2 GPa, ν=.3) have been smulated, consderng a ASTM Type A rosette wth mean dameter D m =5.13 mm [1] and hole radus D=.4 D m ; 2. by eq.(1), the correspondng theoretcal stran dstrbuton p(z) has been evaluated usng the nfluence functon reported n [9]; 3. the stran dstrbutons p(z) have been used as expermental stran functons to evaluate the resdual stresses by the proposed method and the ntegral method; 4. smulated stran errors (havng standard devaton smlar to that of typcal expermental errors on strans) have been added to the theoretcal strans p(z); these nosy strans have been used to evaluate the resdual stresses operatng n the same way of pont 3; 5. the nosy strans have been ftted by a smoothng splne dstrbuton n order to smulate a typcal elaboraton of expermental strans; these smoothed strans have been used to evaluate the resdual stresses operatng n the same way of pont 3. Fg.1 p(z) theoretcal strans relatve to the resdual stresses of fg.2 Fg.2 Shot peenng resdual stresses: theoretcal P(z) dstrbuton and P (z) dstrbutons obtaned by the proposed method and the ntegral method, usng theoretcal strans In the fgures the non-dmensonal varable h=z/r m, defned n the feld h=..5, has been used n place of depth z. In fg.1, the theoretcal strans p(z) obtaned from the shot peenng resdual stresses dstrbuton shown n fg.2 are shown. In fg.2 the theoretcal stress P(z) and the resdual stress P (z) obtaned by the proposed method and the ntegral method are shown. Three polynomals (n=3) of thrd degree ( M =3) have been used for the proposed method; the three polynomals have been defned n the felds whose extremes were z /R m =, z 1 /R m =.75, z 2 /R m =.15, z 3 /R m =.5. Sx optmzed steps [11] have been used for the ntegral method. In ths Fg.3 Nosy strans and smoothng splne stran dstrbuton case, the resdual stresses evaluated wth the

7 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume proposed method concde wth the theoretcal stresses, whle the results obtaned by the ntegral method, although more than acceptable, present the typcal steps. The nosy strans obtaned by addng random stran errors to the theoretcal strans and the smoothng splne are shown n fg.3. To smulate the effect of the measurement procedure, the random errors were smulated usng a 3% standard devaton from the theoretcal values and a roundng of.5 με. The resdual stresses P (z) evaluated by the proposed method and the ntegral method ntroducng drectly the nosy strans are shown n fg.4a. Practcally, the central polynomal of the proposed method presents varous undesred oscllatons. In fg.4b the results obtaned usng a P(z) functon consttuted by 3 straght lnes, that s 3 polynomals of frst degree, s shown: n ths case, havng reduced the degree of the polynomals, the results appear satsfactory. As shown n fg.5, fttng the nosy strans by the smoothng splne shown n fg.3, the results become agan good and the P (Z) stress functon compares very well wth the theoretcal values. a) b) Fg.4 P (z) functon for the shot peenng case obtaned usng nosy strans. a) n=3, M =3; b) n=3, M =1 Comparng the results obtaned usng theoretcal and nosy strans t s possble to observe that, smlarly to other methods, the proposed method presents a senstvty to the expermental errors. The senstvty ncreases as the n and M parameters ncreases and t s dfferent n dfferent zones of the h feld. In partcular, the senstvty s more elevated n the feld.3<h<.5. Consequently, t s opportune: a) to lmt the number of polynomals,.e. consderng n 4, usually n=3, b) to use a maxmum degree of polynomals M =3, c) to use a properly extended feld for the last polynomal, operatng n such a way that t s z n 1 <.4z M, usually.25z M z n 1.35z M. As an example, n the followng, results obtaned by the proposed method properly reducng the n and M parameters are shown. Lettng n=1, the P(z) functon s approxmated by a sngle polynomal. The results obtaned usng a sngle 2 nd degree polynomal are shown n fg.6; n partcular, the results relatve to the shot peenng dstrbuton are shown n fg.6a and that relatve to the lnear behavor are shown n fg.6b. In both cases, the obtaned approxmaton s good enough, a part from the zone near h= for the case of the shot peenng. Fg.5 P (z) functon for the shot peenng case obtaned usng smoothed strans

8 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume a) b) Fg.6 P (z) functon obtaned by a sngle 2 nd degree polynomal (n=1 e M =2): a) shot peenng case, b) lnear stresses dstrbuton A further possblty of the proposed method s to use a stepped P(z) functon as n the case of the ntegral method, by usng n polynomals of M = degree. In ths case, the use of z values correspondng to the optmzed steps for the ntegral method [11] s opportune. The results obtaned are shown n fg.7a (shot peenng case) and n fg.7b (lnear case). It s easy to see that the results obtaned by proposed method are better than those obtaned by the ntegral method. CONCLUSIONS a) b) Fg.7 P(z) functon obtaned by degree polynomals: a) shot peenng case, b) lnear stresses dstrbuton In ths paper a new method for the elaboraton of the expermental strans for the evaluaton of the non-unform resdual stresses by the hole drllng method has been ntroduced. It s based on the approxmaton of the resdual stress dstrbuton by a proper amount of polynomals of proper degree, whose unknown coeffcents are determned by the Newton-Raphson teratve technque, mnmzng the squared dfference between the expermental strans the strans evaluated ntroducng the measured stress n the ntegral equaton. The proposed method s of smpler mplementaton than the splne method [5] and t s more precse than the ntegral method. Varous numercal smulatons have been performed to verfy the method n comparson wth the well known ntegral method and n ths paper results about the shot peenng [6] and the lnear stress felds are presented. The comparsons have been performed usng both the theoretcal stran

9 Copyrght JCPDS-Internatonal Centre for Dffracton Data 29 ISSN Advances n X-ray Analyss, Volume dstrbutons, nosy stran dstrbutons obtaned addng typcal random errors to the theoretcal strans and smoothed splne approxmatons of the nosy strans. Usng theoretcal strans, the proposed method gves practcally exact results, n the ntrnsc lmt of the ntegral equaton. Usng nosy strans, t s possble to obtan better results than the ntegral method and than the other exstng methods, prncpally usng a stepped functon or a sngle 2 nd degree polynomal. Usng the smoothng splne approxmaton of the nosy strans, the results obtaned are comparable wth the ones obtaned usng theoretcal strans. Further developments, not descrbed n ths paper, are actually n study n order to mprove the proposed method n the case of nosy strans; n partcular they regard the ntroducton of a condton of contnuty between the adacent polynomals that consttutes the P(z) stress functon. REFERENCES [1] Bak-Zochosk, M., A sem-destructve method of measurng resdual stresses, VDI- Berchte, 1978, 313, [2] Flaman M.T., Mannng B. H., Determnaton of resdual-stress varaton wth depth by the hole-drllng method, Expermental mechancs, 1985, 25 (3). [3] Schaer, G.S., Measurement of non-unform resdual stresses usng the hole-drllng method. Part. I and Part.II, J. of Eng. Mat. and Tech., 1988, 11 (4), [4] Schaer G.S., Applcaton of Fnte Element Calculatons to Resdual Stress Measurement, J. of Eng. Mat. and Tech., 1981, 13 (2), pp [5] Petrucc G., Zuccarello B., A New Method for Non-unform Resdual Stresses Analyss Usng the Hole Drllng Technque, J. of Stran Analyss, 1997, vol.33, n.1, pp [6] Beghn M., Bertn L., Roselln W., Applcazone d algortm genetc nella msura d autotenson varabl nello spessore, 1999, XXVIII Convegno nazonale AIAS (n talan). [7] Schaer G. S., Prme M. B., Use of nverse solutons for resdual stress measurements, Journal of engneerng materals and technology, 26, 128. [8] Hoffman J. D., Numercal methods for engneers and scentsts, Marcel Dekker Inc, New York, 21. [9] Beghn M., Bertn L, Analytcal expresson of the nfluence functon for accuracy and versatlty mprovement n the hole-drllng method, J. of Stran Anal., 2, 35. [1] ASTM E837-1 Standard test method for determnng resdual stresses by the hole-drllng stran-gage method, 21, ASTM Internatonal. [11] Zuccarello B., Optmal calculaton steps for the evaluaton of resdual stress by the ncremental hole-drllng method, Expermental mechancs, 1999, 39 (2).