Guidelines on the Estimation of Uncertainty in Hardness Measurements

Transcription

1 Eropean Assocaton of Natonal Metrology Instttes Gdelnes on the Estmaton of Uncertanty n ardness Measrements EURAMET cg-16 Verson.0 03/011 Prevosly EA-10/16

2 Calbraton Gde EURAMET cg-16 Verson.0 03/011 GUIDELINES ON TE ESTIMATION OF UNCERTAINTY IN ARDNESS MEASUREMENTS Prpose Ths docment has been prodced to enhance the eqvalence and mtal recognton of calbraton reslts obtaned by laboratores performng calbratons n the feld of hardness. 1

3 Athorshp and Imprnt Ths docment was developed by the EURAMET e.v., Techncal Commttee for Mass and. nd verson March 011 1st verson Jly 007 EURAMET e.v. Bndesallee 100 D Branschweg Germany e-mal: phone: Offcal langage The Englsh langage verson of ths docment s the defntve verson. The EURAMET Secretarat can gve permsson to translate ths text nto other langages, sbject to certan condtons avalable on applcaton. In case of any nconsstency between the terms of the translaton and the terms of ths docment, ths docment shall preval. Copyrght The copyrght of ths docment EURAMET cg-16, verson.0 Englsh verson s held by EURAMET e.v The text may not be coped for sale and may not be reprodced other than n fll. Extracts may be taken only wth the permsson of the EURAMET Secretarat. ISBN Gdance Pblcatons Ths docment gves gdance on measrement practces n the specfed felds of measrements. By applyng the recommendatons presented n ths docment laboratores can prodce calbraton reslts that can be recognzed and accepted throghot Erope. The approaches taken are not mandatory and are for the gdance of calbraton laboratores. The docment has been prodced as a means of promotng a consstent approach to good measrement practce leadng to and spportng laboratory accredtaton. The gde may be sed by thrd partes e.g. Natonal Accredtaton Bodes, peer revewers wtnesses to measrements etc., as a reference only. Shold the gde be adopted as part of a reqrement of any sch party, ths shall be for that applcaton only and EURAMET secretarat shold be nformed of any sch adopton. On reqest EURAMET may nvolve thrd partes n a stakeholder consltatons when a revew of the gde s planned. Please regster for ths prpose at the EURAMET Secretarat. No representaton s made nor warranty gven that ths docment or the nformaton contaned n t wll be stable for any partclar prpose. In no event shall EURAMET, the athors or anyone else nvolved n the creaton of the docment be lable for any damages whatsoever arsng ot of the se of the nformaton contaned heren. The partes sng the gde shall ndemnfy EURAMET accordngly. Frther nformaton For frther nformaton abot ths docment, please contact yor natonal contact person of the EURAMET Techncal Commttee for Mass and Related Qanttes see

4 Calbraton Gde EURAMET cg-16 Verson.0 03/011 GUIDELINES ON TE ESTIMATION OF UNCERTAINTY IN ARDNESS MEASUREMENTS Contents 1 INTRODUCTION... 1 PARAMETERS TAT AFFECT TE UNCERTAINTY OF INDENTATION ARDNESS MEASUREMENT Reference/test materal 4. ardness machne 6.3 Envronment 6.4 Operator 7 3 GENERAL PROCEDURE FOR CALCULATING TE UNCERTAINTY OF ARDNESS MEASUREMENT APPLICATION TO TE ROCKWELL C SCALE: EVALUATION AND PROPAGATION OF UNCERTAINTY Calbraton ncertanty of hardness testng machnes drect calbraton method 9 4. Calbraton ncertanty of the ndrect calbraton method 13 5 REFERENCES... 19

5 Calbraton Gde EURAMET cg-16 Verson.0 03/011 Gdelnes on the Estmaton of Uncertanty n ardness Measrements 1 INTRODUCTION 1.1 In the feld of hardness measrement a wde varety of methods and eqpment s appled whch may dffer accordng to the materal. A hardness measrement s sefl when the reslts obtaned at dfferent stes are compatble to wthn a determned nterval of measrement ncertanty. The gde ams to demonstrate the concepts of measrement ncertanty appled to ths specal feld. Only ncertantes of the commonly sed ndentaton hardness measrng methods for metals Brnell, Rockwell, Vckers are dscssed, for the ranges generally employed n engneerng practce where nversal metrologcal methods have already been mplemented n ndstral contres. 1. A hardness vale s the reslt of a measrement performed on a test pece nder standard condtons, and t s based on an agreed conventon. The hardness determnaton s essentally performed n two steps: 1. An ndentaton s made nder prescrbed condtons,. The determnaton of a characterstc dmenson of the ndentaton mean dameter, mean dagonal or ndentaton depth. 1.3 The dssemnaton of hardness scales s based on three man elements: a the hardness scale defnton: descrpton of the measrement method, the relevant tolerances of the qanttes nvolved and the lmtng ambent condtons. b the hardness reference machne: metrologcal devces that materalse the hardness scale defntons. Dstncton shold be made between prmary standard machnes, whch consttte the best possble realsaton of the hardness scale defntons, and calbraton machnes, sed for the ndstral prodcton of hardness reference blocks. c the hardness reference block: One may dstngsh between prmary hardness reference blocks, calbrated by prmary hardness standard machnes and sed when the hghest accracy s reqred, e.g. for verfcaton and calbraton of hardness calbraton machnes, and hardness reference blocks ntended manly for the verfcaton and calbraton of ndstral hardness testng machnes. 1.4 Fgre 1.1 shows the for-level strctre of the metrologcal chan necessary to defne and dssemnate hardness scales. Note that at each level both drect calbraton and ndrect calbraton are reqred. Drect calbraton gves any possble reference to mass, length and EURAMET cg-16, Verson.0 03/011 Page 1

6 tme natonal standards, and checks the conformty to tolerances reqred by the scale defnton. Indrect calbraton s reqred becase a nmber of factors, not yet completely defned e.g. dsplacement-tme pattern drng the ndentaton, shape rreglartes and mechancal performances of the ndenter cannot be evalated by drect measrement. Comparsons lke nternatonal comparsons for the Prmary ardness Standard Machnes, comparsons wth Prmary ardness Standard Blocks for the ardness Calbraton Machnes and fnally comparsons wth ardness Reference Blocks for ardness Testng Machnes are consdered, therefore, as ndrect measrements. Drect calbraton and ndrect calbraton cover, as shown before, dfferent contrbtons to the ncertanty, so that dfferent expressons of the ncertanty, wth dfferent meanng, can be obtaned: a ncertanty of the scale defnton, prodced by the tolerances adopted and by the lack of defnton of some nflence factors; b ncertanty of the nomnal materalsaton of the scale defnton, prodced by the ncertanty of the factors defned by the scale defntons taken nto accont by the drect calbraton; c ncertanty of the effectve materalsaton of the scale defnton, prodced by the factors not defned by the scale defntons taken nto accont by the ndrect calbraton. Notce that contrbton a s nherent to the defnton tself and therefore shall always be combned wth contrbtons b and c that are, at least partally, overlappng, so that one can take the maxmm vale of the two separate evalatons. 1.5 The metrologcal chan starts at the nternatonal level sng nternatonal defntons of the varos hardness scales to carry ot nternatonal ntercomparsons. 1.6 A nmber of prmary hardness standard machnes at the natonal level "prodce" prmary hardness reference blocks for the calbraton laboratory level. Natrally, drect calbraton and the verfcaton of these machnes shold be at the hghest possble accracy. 1.7 No nternatonal standards are avalable for ths frst step n the materalsaton of hardness scales. De to the small nmber of laboratores at the natonal level, ther work s reglated by nternal operaton procedres for the prmary machnes only and, of corse, by the reglatons for nternatonal ntercomparsons. 1.8 At the calbraton laboratory level, the prmary hardness reference blocks are sed to qalfy the hardness calbraton machnes, whch also have to be calbrated drectly and ndrectly. These machnes are then sed to calbrate the hardness reference blocks for the ser level. 1.9 At the ser level, hardness reference blocks are sed to calbrate the ndstral hardness testng machnes n an ndrect way, after they have been drectly calbrated The stablty of hardness scales s essentally nderpnned by ths two-step calbraton procedre for hardness machnes: I Drect calbraton ensres that the machne s fnctonng correctly n accordance wth the hardness defntons and regardng the approprate parameters; II Indrect calbraton wth hardness reference blocks covers the performance of the machne as a whole The man reqrements for the hardness of reference blocks are stablty wth tme and nformty over the block srface. EURAMET cg-16, Verson.0 03/011 Page

7 1.1 In some cases hardness blocks calbrated by prmary standard machnes are sed drectly for the verfcaton and calbraton of ndstral hardness testng machnes. Ths s not n lne wth the for-level strctre of fgre 1.1, bt there are good reasons for t. In hardness metrology the classcal rle of thmb - namely that the reference nstrment shold be an order of magntde or at least a factor of three better than the controlled devce - n many cases cannot be appled. The ncertanty gap between the natonal level and the ser level s farly small and each step from one level to the next adds an addtonal contrbton to the total ncertanty; so the for-level herarchy may lead to ncertantes too large for relable hardness vales at the ser level. Most metrologcal problems of hardness comparson, of error propagaton and traceablty to standards have ther orgns n ths fact. The calclatons n secton 4 llstrate ths problem. Internatonal level Internatonal Internatonal comparsons defntons Natonal level Prmary hardness Drect standard machnes calbraton Calbraton Prmary hardness ardness calbraton Drect laboratory level reference blocks machnes calbraton User level ardness reference ardness Drect blocks testng machnes calbraton Relable hardness vales Fg. 1.1: The strctre of the metrologcal chan for the defnton and dssemnaton of hardness scales EURAMET cg-16, Verson.0 03/011 Page 3

8 PARAMETERS TAT AFFECT TE UNCERTAINTY OF INDENTATION ARDNESS MEASUREMENT.0.1 Indentaton hardness measrement can often be rghtly consdered non-destrctve snce the tested part s stll sable afterwards. owever, destrcton at the actal pont of test makes t mpossble to verfy the ncertanty of the process by a repeated measrement at that same pont. It s therefore mportant that every sngle measrement be performed to a hgh degree of accracy see secton There are several nflencng parameters that affect the ncertanty of hardness measrements more or less serosly; they are lsted n table.1 and dvded nto grops accordng to ther orgns: 1. Test pece. ardness testng machne 3. Envronment 4. Operator.0.3 The table lsts more than 0 sorces of ncertanty whch may all contrbte sgnfcantly to the total ncertanty of a hardness measrement. These sorces of ncertanty may not always contrbte to every measrement at every level of the metrologcal chan llstrated n fgre Reference/test materal.1.1 Table.1 shows that the test pece materal ntrodces a sgnfcant nmber of ncertantes. For example, the test pece thckness may affect the measred hardness f the wrong method s selected. The deeper the ndent, the thcker the test pece needs to be. Materal whch s too thn wll yeld harder reslts becase of the anvllng effect. In addton, f the materal s too thn to spport the test force drng measrement, the ndenter tself cold be damaged and ths wll ndermne the relablty of any frther measrement performed wth that ndenter..1. The srface qalty of the test pece may also consderably nflence the reslts of hardness measrements. A rogher srface wold reqre a greater force and/or a larger ndenter to prodce a larger ndentaton. The Brnell method may be the most approprate snce t s less affected by a rogh srface than Rockwell or Vckers. Althogh Brnell measrements are more tolerant of varyng fnsh, there are lmts to the permssble srface roghness for ths method too. In general, nformty of srface fnsh s mportant for accrate and reprodcble reslts..1.3 Srface cleanlness s also crtcal for precse and reprodcble hardness measrement. Srface solng wth grease, oxdes or dst may case consderable devatons n the reslts; moreover, the test materal or reference block may even be rreversbly damaged. EURAMET cg-16, Verson.0 03/011 Page 4

9 Table.1: Parameters that affect the ncertanty of ndentaton hardness measrement Inflencng factor 1. Test pece Test pece thckness too low Stffness of the spport Gran strctre too coarse Sorce of ncertanty Remarks Parameters consdered for calclaton Only relevant, f the chosen test method s not approprate. Srface roghness Inhomogeneos dstrbton of hardness Srface cleanlness. ardness testng machne a Machne frame Frcton loss Elastc deflecton Msalgnment of the ndenter holder b1 Depth Indcatng error Only relevant for Rockwell ndentaton depth h measrng system Poor resolton Nonlnearty ysteress b Lateral measrng system c Force applcaton system Indcatng error Poor resolton Nmercal apertre of lens or llmnator Inhomogeneos llmnaton of the ndentaton Devaton from nomnal forces Devaton from tme ntervals of the testng cycle Force ntrodcton Overrn of test forces d Indenters Devaton from the deal shape Damage Deformaton nder force 3. Envronment Temperatre devaton or drft Vbraton and shocks 4. Operator Wrong selecton of test method andlng, readng, evalaton errors Only relevant for Brnell, Vckers, Knoop prelmnary/total test force prelmnary/total test force dwell tme ndentaton velocty ndenter rads and angle F0, F t0, t v r, α EURAMET cg-16, Verson.0 03/011 Page 5

10 . ardness machne..1 The desgn, assembly and condton of the hardness testng machne are all crtcal to accrate reslts. Excessve frcton can case bas and non-repeatablty. Even nstrments that are operated properly can gve poor reslts de to excessve frcton n the force applyng system. Smlar ncertanty contrbtons de to small amonts of frcton can be expected from the depth measrng system... Excessve deflectons of the spportng frame of the testng machne and the test pece spport system can case problems too. Devatons of 1 to 3 hardness nts are not ncommon de to mproper spport of the test pece and excessve deflecton of the nstrment s frame...3 De to the very small dmensons that are measred, the measrng system s crtcal. For example, one reglar Rockwell scale nt s eqvalent to only µm ndentaton depth and the sperfcal scale s half of that, so measrng system ncertanty s very mportant...4 The force applcaton system mst constantly apply accrate forces. gh-qalty measrng eqpment shold be able to apply forces well wthn the lmts of ±1.0% for the ser level, and even wthn 0.1% of the nomnal force for calbraton machnes...5 Applcaton of the forces reqres that both the velocty and the dwell tme of the forces be defned. Varatons of testng cycle parameters that may occr wth some manally controlled machnes can prodce varatons n the reslt of p to 1 RC at 60 RC. Softer materals and materals sbject to work hardenng cold gve sgnfcantly hgher ncertantes. In these cases contrbtons of dwell tme ncertanty and ndentaton velocty shall be evalated specfcally for the materal tested...6 The propertes of the ndenter also nflence the ncertanty of hardness measrements. It s relatvely easy to manfactre a ball to the reqred shape. owever, the ball holder s the man sorce of ncertanty...7 Damond ndenters are more dffclt to manfactre to the reqred shape. The potental sorces of ncertanty are sgnfcant, bt n ths context t s not necessary to categorse the effect of each n detal. It s mportant to note here that the best Rockwell damond ndenters manfactred today wll exhbt varatons p to ±0.5 RC when compared on the same testng machne. Lower qalty ndenters wll gve sgnfcantly larger varatons..3 Envronment.3.1 Ambent temperatre may have consderable nflence on the reslts of hardness measrements, especally f small lengths have to be determned. The lower lmt for Vckers ndentatons s 0 µm, and the mnmm depth for Rockwell scales N and T s only 6 µm to 7 µm. Accordng to the relevant standards, the temperatre ranges are 10 C to 35 C for the test methods and 3±5 C for the calbraton of reference blocks. These ranges are too wde for some hardness scales, bt operaton otsde these ranges shold n any case be case of concern. If ths s navodable, comparatve measrements shold be performed to assess the nflence of temperatre..3. Vbratons, electrcal nterference and lack of cleanlness, can case sgnfcant problems that are dffclt to qantfy. Ultra-low force mcrohardness measrements of corse reqre an absoltely vbraton-free envronment, whereas vbraton reqrements for test forces above 00 mn are not so crtcal. EURAMET cg-16, Verson.0 03/011 Page 6

11 .4 Operator Measrement postons on the srface of the sample become mportant n many cases. Measrements, for nstance, near the edge of a pece or at ponts close to each other mst be properly located to ensre accrate reslts. Uncertantes of p to RC are not ncommon here. Overall montorng of the operaton s very mportant. Some modern testng machnes have featres that mnmse operator nflence; nevertheless, the latter s stll essental for a sccessfl hardness measrement. 3 GENERAL PROCEDURE FOR CALCULATING TE UNCERTAINTY OF ARDNESS MEASUREMENT The followng procedre s based on EA/4-0 [1] cf. worked examples n secton 4. a Express the relatonshp between the measred hardness otpt qantty and the npt qanttes X model fncton n mathematcal terms: = f X 1,X,...,X N 1 Notce that n the case of ardness a mathematcal relatonshp connectng npt qanttes X wth the otpt qantty s not known at the state of the art. The connecton s gven by the scale defntons that are emprcal procedres. The model fncton, therefore, does not gve mch more than a lst of factors affectng the measrement reslts. In practce ths s sffcent for establshng a procedre based on EA/4-0, provdng that specal care s adopted for evalatng standard ncertantes of the npt qanttes and senstvty coeffcents, as shown here after. b Identfy and apply all sgnfcant correctons. c Lst all sorces of ncertanty n the form of an ncertanty analyss n accordance wth the followng table: Table 3.1: Schematc of an ordered arrangement of the qanttes, estmates, standard ncertantes, senstvty coeffcents and ncertanty contrbtons sed n the ncertanty analyss of a hardness measrement qantty X estmate x standard ncertanty x senstvty coeffcent c contrbton to the standard ncertanty X 1 x 1 x 1 c X n x n x n c n n ardness The qanttes n table 3.1 are defned as follows: X qanttes, reported n table.1, affectng the measrement reslt. As sad n 1.4 the ncertanty can be evalated n two separate ways: the frst way nvolvng the physcal qanttes sed for the scale defntons forces, lengths, tmes, veloctes etc., refers to the drect calbraton; the second way, nvolvng all the factors of nflence present n practce, refers to the ndrect calbraton. Notce that one cold sppose that ths second way contans all the ncertanty contrbtons, therefore can alone gve the ncertanty vale EURAMET cg-16, Verson.0 03/011 Page 7

12 reqred, bt ths s not always tre. For nstance t s possble to perform a very carefl ndrect calbraton that prodces an ncertanty lower than the ncertanty prodced by the tolerances accepted for drect calbraton []. For ths reason both ways shall be followed and the larger of the two ncertanty vales obtaned taken as the reslt. x estmate vales of the qanttes X x standard ncertantes of the estmates x. Some ways can be followed for determnng x. For the part connected wth the ncertanty of hardness scale defntons one shall take the tolerance felds of the defnton [3] as varablty felds, and evalate the ncertanty contrbtons of type B. Type B ncertantes shall be sed n any case when only a declaraton of conformty s avalable. For the part connected wth drect calbraton t s possble to determne x by the ncertanty declared n calbraton certfcates of the measrement nstrments sed for drect measrements. For the part connected wth ndrect calbraton, that s comparsons performed sng hardness blocks, the relevant ncertanty of type A shall be evalated. c s the senstvty coeffcent assocated wth the npt estmate x. The senstvty coeffcent c descrbes the extent to whch the hardness s nflenced by varatons of the npt estmate x. As sad before at the state of the art the mathematcal connecton between x and s nknown, therefore the senstvty coeffcents shall be evalated expermentally by the change n the hardness de to a change x n the npt estmate x as follows: c x X1 = x1,..., X n = x n The expermental evalaton of the senstvty coeffcents s sally tme consmng, therefore sally t s advantageos to se the expermental reslts gven n lteratre [4, 5] and adopted for the examples attached, bt one shall be carefl when the relevant factors depend on the characterstcs of the materal tested dwell tme and ndentaton velocty. In ths case some experments wth the specfc materal are necessary. s the contrbton to the standard ncertanty assocated wth the hardness resltng from the standard ncertanty x assocated wth the npt estmate x : = c x 3 d For ncorrelated npt qanttes the sqare of the standard ncertanty assocated wth the measred hardness s gven by: = n = 1 e Calclate for each npt qantty X the contrbton to the ncertanty assocated wth the hardness resltng from the npt estmate x accordng to Eqs. and 3 and sm ther sqares as descrbed n Eq. 4 to obtan the sqare of the standard ncertanty of the hardness. f Calclate the expanded ncertanty U by mltplyng the standard ncertanty assocated wth the hardness by a coverage factor k=: U = k 5 Shold the effectve degrees of freedom ν eff n exceptonal cases be less than 15, then calclate the coverage factor k accordng to EA/4-0, Annex E [1]. EURAMET cg-16, Verson.0 03/011 Page 8 4

13 g Report the reslt of the measrement as follows: n calbraton certfcates, the complete reslt of the measrement comprsng the estmate of the measrand and the assocated expanded ncertanty U shall be gven n the form ±U. To ths an explanatory note mst be added whch n the general case shold have the followng content: The reported expanded ncertanty of measrement has been obtaned by mltplyng the combned standard ncertanty by the coverage factor k= that, for a normal dstrbton, corresponds to a confdence level p of approxmately 95%. The combned standard ncertanty of measrement has been determned n accordance wth EA/4-0 [1]. 4 APPLICATION TO TE ROCKWELL C SCALE: EVALUATION AND PROPAGATION OF UNCERTAINTY The relevant standard docments [] reqre that both drect and ndrect calbraton methods be sed, at least wth new, revsed or renstalled hardness testng machnes. It s always good practce to se both calbraton methods together. 4.1 Calbraton ncertanty of hardness testng machnes drect calbraton method The drect calbraton method s based on the drect measrement of the hardness scale parameters prescrbed by ISO []. Even thogh t s not possble to establsh an analytcal fncton to descrbe the connecton between the defnng parameters and the hardness reslt [4], some experments [5] do allow, as descrbed n secton 3, to evalate ncertanty propagaton. Yet one shold be carefl n the applcaton becase some of the parameters are prmarly connected wth the measrng system prelmnary test force, total test force, ndentaton depth, ndenter geometry, frame stffness, whereas others refer to the measrand creep effect, stran-hardenng effect The measrand related parameters can be descrbed as an ndcaton based on reslts obtaned wth hardness reference blocks, bt shold be evalated drectly for the specfc measrand. The creep effect depends on both the measrng system and the materal characterstcs; the amont of creep s a fncton of the creep characterstc of the materal, also dependng on the tme reqred by the measrng system to regster the force. For a manal zerong machne, creep has generally stopped when zero s fnally reached. Even atomatc machnes are more or less prompt. A machne that takes 5 s to apply the prelmnary test force prodces a dfferent creep relaxaton than a machne takng only 1 s, and the strct observance of a 4 s force dwell-tme wll not help to obtan compatble reslts There s call for caton n nterpretng nmercal vales becase the reslts obtaned wth old manal machnes cannot represent those of a modern atomatc hardness testng machne, desgned to prodce ndentatons n the shortest possble tme The evalaton of ncertanty s descrbed n the relevant EA/4-0 docment [1]. The ncertanty calclaton mst be done n dfferent ways, dependng on the types of data avalable. The frst step s the evalaton of the approprate varances correspondng to the measrement parameters nvolved ndependent varables. EURAMET cg-16, Verson.0 03/011 Page 9

14 EURAMET cg-16, Verson.0 03/011 Page The measrement reslts gven n a calbraton certfcate, wth the ncertanty sally qoted for k= coverage factor, permt the calclaton of the standard ncertanty. It s sffcent to dvde the gven ncertanty by the stated coverage factor. Conformty declaraton can also be sed to evalate the standard ncertanty, takng the tolerance nterval ±a nto accont. A rectanglar dstrbton fncton shold be sed, wth eqvalent varance = a / The second step s the calclaton of the combned standard ncertanty. Theoretcally, f the hardness s the measrand dependent varable, t can be represented as a fncton of the measrement ndependent varables. The symbols sed are ndcated n table 4.1: ; ; ; ; ; ; ; ; ; 0 0 S N h v t t r F F f α = 6 More explctly, the eqaton s: x x S h N + = 7 where x are the ndependent varables n eq Usng the approprate senstvty coeffcents, namely the partal dervatves of the dependent varable aganst the ndependent varables x, one obtans the formla for evalatng the ncertanty propagaton n the approxmaton of ncorrelated ndependent varables: = = = n n x c In practce, the partal dervatves can be approxmated by the ncremental ratos: h h v v t t t t r r F F F F = α α The standard ncertanty can be evalated for dfferent condtons. As an example, Table 4. shows the evalaton of the standard ncertanty, and the expanded ncertanty wth coverage factor k=, for a conformty assessment of hardness testng machnes and ndenters to the relevant standard []. Ths was done sng the approprate tolerances to calclate type B standard ncertantes.

15 Table 4.1: symbols sed Measred hardness t total test force dwell tme d ncertanty of hardness scale defnton F0 Prelmnary test force v ndentaton velocty m ncertanty of prmary hardness standard machne F total test force h ndentaton depth s stablty ncertanty of calbraton machne r Indenter rads N constant nmber dependent by the scale α ndenter angle S constant nmber dependent by the scale f ν fttng ncertanty degrees of freedom t0 Prelmnary test force dwell tme b mean hardness measrement reslt of prmary hardness reference block sc Standard devaton of the measrements c b sngle hardness measrement reslt of prmary hardness reference block Sc Standard devaton of the measrements c bd Calbraton ncertanty of prmary hardness reference blocks consderng the scale defnton c Mean hardness vales of the scale of the calbraton machne bm Calbraton ncertanty of prmary hardness reference blocks consderng the ncertanty of the prmary hardness standard machne c Sngle hardness vales of the scale of the calbraton machne sb Standard devaton of the measrement b cdf Calbraton machne ncertanty consderng the scale defnton ncertanty and the fttng ncertanty sb Standard devaton of the measrements b cmf Calbraton machne ncertanty consderng the prmary standard machne ncertanty and the fttng ncertanty cd Calbraton ncertanty of the calbraton machne consderng the scale defnton cd Calbraton machne ncertanty consderng the scale defnton ncertanty and the calbraton reslts ncorrected cm Calbraton ncertanty of the calbraton machne consderng the ncertanty of the prmary hardness standard machne cm Calbraton machne ncertanty consderng the prmary standard machne ncertanty and the calbraton reslts ncorrected Correcton vale EURAMET cg-16, Verson.0 03/011 Page 11

16 Table 4.: Evalaton of the ncertanty propagaton for conformty assessment of the hardness testng machne and ndenter x a a x = 3 Senstvty coeffcents at dfferent hardness levels c = x Contrbtons to ²/RC² at dfferent hardness levels n n = c x = 1 = 1 0 to 5 40 to to 65 0 to 5 40 to to 65 F0 /N F /N α / r /mm h /µm v /µm/s t0 /s t /s TOTAL / RC = / RC Standard ncertanty /RC Expanded ncertanty U /RC = k /RC Table 4.3 shows the evalaton of standard and expanded ncertanty for calbraton certfcates for the hardness testng machne and ndenter. ere the example s for the hardness level 0 RC to 5 RC. Note that the dfferences between the parameter and nomnal vales are known, together wth ther ncertantes, and t s therefore possble to estmate both a correcton and ts ncertanty sng the same senstvty coeffcents as before Whlst n the case of type B ncertanty contrbtons the degrees of freedom ν of the varos parameters can be consdered large enogh to apply the Gassan dstrbton, n ths case ν depends on the adopted measrement procedre. Table 4.3 qotes typcal vales of ν. EURAMET cg-16, Verson.0 03/011 Page 1

17 Table 4.3: Evalaton of the ncertanty propagaton n calbraton certfcates for the hardness testng machne and for the ndenter for 0 RC to 5 RC hardness level Certfcate data X x U σ ν c = x Measred hardness 4 /ν RC RC RC 4 F0 /N F /N α / r /mm h /µm v /µm/s t0 /s t /s Total Standard ncertanty /RC 0.10 Degrees of freedom 15 Coverage factor k for confdence level p = 95%.13 Expanded ncertanty U /RC = k /RC 0. Where = c x and c x Ths method can only be sed correctly f nomnal vales are defned for the varos parameters. If, as s the case wth crrent standards, there are parameters whch are not defned as nomnal vales wth a gven tolerance bt as nform probablty ntervals, then the reference to a "nomnal vale" s not possble. In conseqence, the ncertanty calclated n ths way can only be accepted where there s a prelmnary agreement on the "nomnal vales" of the measrement parameters. 4. Calbraton ncertanty of the ndrect calbraton method The ndrect calbraton method s based on a metrologcal chan. A typcal seqence s cf. Fgre 1.1: a defnton of the hardness scale; b materalsaton of the hardness scale defnton by a prmary hardness standard machne; c calbraton of prmary hardness reference blocks for the dssemnaton of the hardness scale; d calbraton of a hardness calbraton machne for the ndstral prodcton of hardness reference blocks; e calbraton of hardness reference blocks; f calbraton of ndstral hardness testng machnes sng hardness reference blocks. g hardness measrement performed wth ndstral hardness testng machnes. EURAMET cg-16, Verson.0 03/011 Page 13

18 4..0. It s also possble to go drectly from step c to step f, or, after step e to add the calbraton of a frontlne hardness testng machne from the ndstral qalty system and, wthn the qalty system, to calbrate the hardness reference blocks necessary for the calbraton of other hardness testng machnes sed wthn the qalty system tself. Note that after step d the sbseqent steps are repettons of the prevos ones. In conseqence, the descrpton of the ncertanty evalaton can be restrcted to the frst for steps Uncertanty d of the Rockwell hardness scale defnton The evalaton of the ncertanty d of the hardness scale defnton and ts materalsaton s smlar to the evalaton of the ncertanty de to the drect calbraton method, takng the tolerances prescrbed by ISO [3] nto accont. Table 4.4 presents an example of ncertanty evalaton. Note that ncertanty contrbtons are of type B, therefore a coverage factor k= s sed. Table 4.4 : Evalaton of the ncertanty d de to the defnton of the Rockwell C Scale and ts materalsaton X a a x = 3 Senstvty coeffcents at dfferent hardness levels c = x Contrbtons to ²/RC² at dfferent hardness levels n n = c x = 1 = 1 0 to 5 40 to to 65 0 to 5 40 to to 65 F0 /N F /N α / r /mm h /µm v /µm/s t0 /s t /s TOTAL / RC = /RC d Standard ncertanty d /RC Expanded ncertanty U /RC = kd /RC The evalated vales are confrmed by reslts obtaned drng nternatonal comparsons, n partclar that nvolvng the largest nmber of partcpants, whch shows a spread of reslts of abot ±0.5 RC. 4.. Uncertanty of the materalsaton of the Rockwell hardness scale defnton To demonstrate an ncertanty evalaton for state of the art characterstcs of prmary hardness standard machnes, one may do a calclaton smlar to that n table 4.3, takng relevant ncertantes as shown n table 4.5 nto accont. The reslts are optmstc becase sgnfcant parameters, sch as the performance of the ndenter, are not acconted for, yet these mst be consdered as nherent n the ncertanty de to the defnton. It can be seen that the ncertanty of the machne s almost neglgble compared EURAMET cg-16, Verson.0 03/011 Page 14

19 to the effect of the tolerances gven by the defnton, wth the ncertanty contrbtons from nflencng qanttes mssng n the defnton tself. Table 4.5: Evalaton of the ncertanty m based on the state of the art of prmary hardness standard machnes for the 0 RC to 5 RC hardness level. Certfcate data X x U σ ν c = x Measred hardness 4 /ν RC RC RC 4 F0 /N F /N α / r /mm h /µm v /µm/s t0 /s t /s Total Standard ncertanty m /RC 0.03 Degrees of freedom 36 Coverage factor k for confdence level p = 95%.03 Expanded ncertanty U /RC = k /RC 0.06 Where = c x and c x 4... The vale of the ncertanty s therefore prmarly the reslt of tolerances of the measrng parameters prescrbed by relevant standards. Althogh table 4.4 does not take the contrbton de to the prmary hardness standard machne nto accont for the materalsaton of the defnton tself, t can stll be consdered a comprehensve evalaton Uncertanty of the calbraton of Rockwell prmary hardness reference blocks The prmary hardness reference block s calbrated by a prmary hardness standard machne makng fve hardness measrements b. The mean vale b s taken as the hardness vale of the block Repeatng the measrement reveals the effects of non-nformty of the reference block srface and the repeatablty of the prmary hardness standard machne, ncldng ts resolton. Other effects, sch as the hardness stablty of reference blocks, mst be estmated from experence wth the reference blocks and ther mantenance condtons Except for a possble drft that mst be evalated separately, the ncertanty bd or bm of b can be evalated from the ncertanty de to the scale defnton d, gven n Table 4.4, combned wth the standard devaton s b of b evalated sng the standard devaton s b of the measrements b The ncertantes bd or bm are gven by: EURAMET cg-16, Verson.0 03/011 Page 15

20 s 5 b b = 1 = b s b s b = = + s or bd d b = + s bm m b The calbraton certfcate shall at least state the vale of the standard ncertanty bd. Also reqred s the vale bm. Explct vales for the ncertanty contrbtons [5, 6] can be nclded for nformaton Uncertanty of the calbraton of Rockwell calbraton machnes The hardness reference block s calbrated by a hardness calbraton makng fve hardness measrements c. The mean vale c s compared wth the block hardness b to calbrate the machne for that scale and that hardness = c - b Repeatng the measrement reveals the effects of non-nformty of the reference block srface and the repeatablty of the hardness calbraton machne, ncldng ts resolton. Therefore, except for the stablty of the calbraton machne s that mst be evalated separately becase t depends on the workng condtons, the ncertanty cd or cm of c can be evalated by combnng the relevant ncertanty de to the hardness reference block bd or bm wth the standard devaton s c of c calclated sng the standard devaton s c of the measrements c To mnmse the ncertanty, the correcton shold be appled by the measred hardness. To derve the ncertanty cdf or cmf at any pont of the machne scale one shold nterpolate the reslts. The ncertanty de to fttng f depends on the strctre and the workng characterstcs of the calbraton machne, and shold therefore be determned to characterse the machne tself by a calbraton on fve hardness levels, comparng the least sqares parabola wth the parabola passng throgh the three ponts at the hardness level chosen for the sbseqent perodc checks For the ncertantes cdf or cmf we have: s 5 c c = 1 = c s c s c = = + s or cd bd c = + s cm bm c 15 + cdf = or cd f = + cmf cm f 16 f the correcton s not appled, the ncertanty cd and cm are calclated sng: cd = cd + or cm = cm The calbraton certfcate shall at least state the vale of the standard ncertanty cdf. Also reqred s the vale of cmf. Explct vales of the ncertanty contrbton [5, 6] can be nclded for nformaton. EURAMET cg-16, Verson.0 03/011 Page 16

21 4..5 Uncertanty of the calbraton of hardness reference blocks and testng machnes For the calbraton of hardness reference blocks and hardness testng machnes the same procedres are sed as those descrbed above for calbraton of prmary hardness reference blocks and hardness calbraton machnes. The formlae gven for those cases shall be sed Nmercal example The ncertanty evalaton can be set ot as n the followng example n Table 4.6. Table 4.6 Indrect calbraton chan - Uncertanty evalaton ardness level 0 to 5 40 to to 65 Defnton and standard machne ncertanty d see Table Prmary hardness reference block calbraton Nmber of ndentatons Non-nformty of prmary hardness reference block and machne reprodcblty. Relevant standard devaton s b Eq Standard devaton of the mean of ndentatons s b Eq Uncertanty of the hardness vale of reference blocks bd or bm Eq Calbraton of hardness calbraton machne Nmber of ndentatons Non-nformty of prmary hardness reference block and machne reprodcblty. Relevant standard devaton s c Eq Standard devaton of the mean of ndentatons s c Eq Fttng ncertanty f Uncertanty of the hardness scale of the calbraton machne cdf or cmf Eq.15 and Eq ardness reference block calbraton Nmber of ndentatons Non-nformty of hardness reference block and machne reprodcblty. Relevant standard devaton s b Eq Standard devaton of the mean of ndentatons s b Eq Uncertanty of the hardness vale of hardness reference blocks bd or bm Eq Effectve degrees of freedom ν Coverage factor Expanded ncertanty U EURAMET cg-16, Verson.0 03/011 Page 17

22 5 REFERENCES [1] EA/4-0: Expresson of the Uncertanty of Measrement n Calbraton, December 1999 [] ISO 6508-:1999: Metallc materals Rockwell hardness test Part : Verfcaton and calbraton of the testng machne [3] ISO :1999: Metallc materals Rockwell hardness test Part 3: Calbraton of reference blocks [4] Barbato, G.; Desogs, S.: The meanng of the geometry of Rockwell ndenters IMGC Techncal Report, No. R18, 1978, 6 [5] Petk, F.: The Unfcaton of ardness Measrement, BIML, Pars, 1991, p [6] OIML SP 19/SR 4: Compte-rend de la comparason nternatonale des échelles de dreté BIML, 1984 EURAMET cg-16, Verson.0 03/011 Page 18