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2 Chapter 7 Good Practce for Fatgue Crack Growth Curves Descrpton Sylwester Kłysz and Andrzej Lesk Addtonal nformaton s avalable at the end of the chapter 1. Introducton Fatgue lfe estmaton and crack propagaton descrpton are the most mportant components n the analyss of lfe span of structural components but t may requre tme and expense to nvestgate t expermentally. For fatgue crack propagaton studyng n cases when t s dffcult to obtan detaled results by drect expermentaton computer smulaton s especally useful. Hence, to be effcent, the crack propagaton and durablty of constructon or structural component software should estmate the remanng lfe both expermentally and by smulaton. The crtcal sze of the crack or crtcal component load can be calculated usng materal constants whch have been derved expermentally and from the constant ampltude crack propagaton curve, crack sze-lfe data and curve usng crack propagaton software. Many works n the feld of fracture mechancs prove sgnfcant development n the numercal analyss of test data from fatgue crack propagaton tests. A smple stochastc crack growth analyss method s the maxmum lkelhood and the second moment approxmaton method, where the crack growth rate s consdered as a random varable. A determnstc dfferental equaton s used for the crack growth rate, whle t s assumed that parameters n ths equaton are random varables. The analytcal methods are mplemented nto engneerng practce and are use to estmate of the statstcs of the crack growth behavor (Elber, 1970; Forman et al., 1967; Smth, 1986). Though many models have been developed, none of them enjoys unversal acceptance. Due to the number and complexty of mechansms nvolved n ths problem, there are probably as many equatons as there are researchers n the feld. Each model can only account for one or several phenomenologcal factors - the applcablty of each vares from case to case, there s no general agreement among the researchers to select any fatgue crack growth model n relaton to the concept of fatgue crack behavor (Kłysz, 2001; Pars & Erdogan, 1963; Wheeler, 1972; Wllenborg et al., 1971). Mathematcal models proposed e.g. by Pars, 2012 Kłysz and Lesk, lcensee InTech. Ths s an open access chapter dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

3 198 Appled Fracture Mechancs Forman, and further modfcatons thereof descrbe crack propagaton wth account taken of such factors as: materal propertes, geometry of a test specmen/structural component, the actng loads and the sequence of these loads (AFGROW, 2002; Kłysz et al., 2010a; NASGRO, 2006; Newman, (1992); Skorupa, 1996). Applcaton of the NASGRO equaton, derved by Forman and Newman from NASA, de Konng from NLR and Henrksen from ESA, of the general form (AFGROW, 2002; NASGRO, 2006): Kth n 1 da 1 f K C K dn 1 R K max 1 K c p q (1) has sgnfcantly extended possbltes of descrbng the crack growth rate tested accordng to the standard (ASTM E647). The coeffcents stand for: a crack length [mm], N number of load cycles, C, n, p, q emprcal coeffcents, R stress rato, K the stress-ntensty-factor (SIF) range that depends on the sze of the specmen, appled loads, crack length, K = Kmax - Kmn [ ], Kth the SIF threshold,.e. mnmum value of K, from whch the crack starts to propagate: 1 1 R 2 a 1 f Kth K a a 0 1 A RC th R C th (2) or 1 a 2 1 f Kth K0 / a a 0 1A0 1R 1C R th (2a) where: a 0 structural crack length that depends on the materal gran sze [mm], K0 threshold SIF at R0, K1 threshold SIF at R1, Cth curve control coeffcent for dfferent values of R; equals 0 for negatve R, equals 1 for R 0, for some materals t can be found n the NASGRO database, Kmax the SIF for maxmum loadng force n the cycle, Kc crtcal value of SIF, f Newman's functon that descrbes the crack closure:

4 Good Practce for Fatgue Crack Growth Curves Descrpton 199 f 2 3 max RA, 0 AR 1 AR 2 AR 3 for R0 A0 A1R for 2 R0 (3) where A0, A1, A2, A3 coeffcents are equal: A 0 2 S max ( ) cos, 2 0 A 1 max ( ) S, A 1 A A A, A 2A A (4) (5) (6) (7) Smax Newman's emprcal coeffcents. Determnaton of the above coeffcents for equaton that correctly approxmates test data s dffcult and causes some sngulartes descrbed below, when the Least Squares Method (LSM) s used. Fgure 1. Fatgue crack propagaton graphs: a) K=f(N); b) a=f(n) and c) da/dn=f(k)

5 200 Appled Fracture Mechancs The fatgue crack growth test results provde an llustraton of relatons such as: specmen stress ntensty vs. number of cycles (K=f(N)), crack growth vs. number of cycles (a=f(n)); crack growth rate vs. stress ntensty factor range (da/dn=f(k)). These expermental curves can be presented, for example, n the graphcal form shown n Fg. 1 (for a sngle specmen, two-stage test: stage I - decreasng K test, black curve; stage II constant ampltude test, blue curve). Specfcally, the da/dn=f(k) plots can be obtaned drectly from the materal test machne control software (e.g. by employng the complance method and by usng a clp gauge) or can be obtaned by dfferentatng the a=f(n) curve after correlatng t wth K=f(N). These plots, for sngle specmen tests, as well as for tests wth multple specmens under dfferent load condtons (e.g. varous stress rato R values), can be successfully descrbed analytcally when approprate mathematcal models and equatons are employed. 2. Test data Fatgue tests for structural components durablty analyss can be conducted wth the RCT (Round Compact Tenson) (Fg. 2a) or wth SEN (Sngle Edge Notch) (Fg. 2b) or other specmens accordng to the correspondng ASTM E647 standard. Fgure 2. RCT & SEN specmens for fatgue crack propagaton tests The general formula that descrbes the stress ntensty factor s as follows: K I P Y, (8) B W where: P appled force, B, W the specmen s thckness and wdth, Y the specmen s shape functon (ASTM E647, Fuchs & Stephens 1980, Murakam 1987): for the RCT specmen: a 2 W a a a a Y W W W W a 2 1 W (9)

6 Good Practce for Fatgue Crack Growth Curves Descrpton 201 for the SEN specmen: a a a a Y 1,12 0,231 10,55 21,72 30,39 W W W W (10) where a/w s a non-dmensonal crack length. The complance functon to compute the crack length n the RCT specmen has the form: a u u u u u W (11) for the SEN specmen (Bukowsk & Kłysz 2003): a u u u W (12) where: u complance descrbed by the followng formula: 1 u EBCOD 1 F 0.5, (13) E Young s modulus, COD Crack Openng Dsplacement. An example of the F-COD relatonshp has been plotted n Fg. 3. The plot has been ganed from the fatgue crack growth test conducted for the SEN specmen made from constructonal steel subjected to constant ampltude loadng wth overloads (Bukowsk & Kłysz, 2003). The records were taken n the course of statcally appled 2-cycle overloads of 40% order at subsequent stages of crack propagaton. Load base level and overload level were gradually reduced as the crack growth kept ncreasng and after non-lnearty (hysteress loop) had occurred n the F-COD plot. The objectve was to avod falure of the specmen n a subsequent overload cycle to be able then to contnue the crack-propagaton test. Any change n the angle of nclnaton of the rectlnear segment of each of the hysteress loops (.e. the F/COD proporton from formula (13)) s a measure of the specmen s complance u and proves the crack length n the specmen under examnaton keeps growng. Results presented below come from the examnaton of the 2024 alumnum alloy taken from the helcopter rotor blades (Kłysz & Lseck, 2009) or from the arcraft ORLIK s fuselage skns (Kłysz et al., 2010b) and are obtaned for three values of test stress rato R = 0.1; 0.5; 0.8, under laboratory condtons, wth loadng frequency 15 Hz. The crack length was measured wth the COD clp gauge usng the complance method. The crack growth rate was determned usng the polynomal method.

7 202 Appled Fracture Mechancs Fgure 3. Relatonshp of F-COD recorded n subsequent overload cycles of fatgue crack growth test 3. Data analyss Results of fatgue crack growth rate tests for 3 specmens (for R = 0.1; 0.5; 0.8) are presented n Fg. 4 (Lseck & Kłysz, 2007). (a) (b) Fgure 4. Fatgue crack growth rates n 3 specmens for dfferent R values a) test data, b) results of approxmaton

8 Good Practce for Fatgue Crack Growth Curves Descrpton 203 The NASMAT curve fttng algorthms use the least-squares error mnmzaton routnes n the log-log doman to obtan the correspondng constants usng the NASMAT module contaned wthn the NASGRO sute of software (NASGRO, 2006). The constants C and n,.e. the man ft parameters, are determned through the mnmzaton of the sum of squares of errors, where the error term correspondng to the -th data par (K, da/dn) s (Forman et al., 2005): p Kth 1 1 f da K e log logc nlog K log. dn 1 q R K max 1 K c (14) Values of da/dn are determned usng the method of dfferentatng the dependence a-n wth the secant or the polynomal method appled (AFGROW, 2002; ASTM 647; NASGRO, 2006). Generally the curve fttng of crack growth data s an terate process that conssts n usng establshed values of varous constants (other than C and n), specfyng the data sets that typfy the materal, applyng the least-squares algorthm to compute C and n, and plottng the data for varous R values wth the curve ft of each stress rato. The process s contnued by makng slght modfcatons n the entered values untl the best ft to the test data s obtaned. In general fttng the NASGRO equaton s really a mult-step process nvolvng: - fttng or defnng the threshold regon; - fttng or defnng the crtcal stress ntensty or toughness to be used at the nstablty asymptote; - makng ntal assumptons on key parameters such as p and q; - performng the least squares ft to obtan C and n; and fnally; - usng engneerng judgment to adjust the results for consstency and/or a desred level of conservatsm. For the LSM approxmaton of test data, analytcal descrpton thereof, and determnaton of coeffcents of approxmaton equatons, accordng to whch the crteron used n the analyss s the mnmum of the square sum: n 1 2 S y y (15) of devatons between values of the test data y and those of the approxmated functon. Ths method of approxmaton s characterzed wth the followng propertes that n some cases may be consdered as dsadvantages (Forman et al., 2005; Whte et al., 2005; Huang et al., 2005; Taher et al., 2003): - n respect of the order of magntude, value of the sum S ncreases as magntudes of approxmated values ncrease, e.g. f values of test data are of the order of magntude

9 204 Appled Fracture Mechancs 10, 1000, , wth the scatter of 10%, the summed dfferences are of the order of magntude 1, 100, , and hence, dynamc changes n the total value of the sum S depend on values of dfferences as a quadratc functon t s characterzed by a lnear functon of the dervatve, whch also means that for dfferences close to zero (e.g. 10-5, 10-8, etc.) ths dynamc change s much smaller than for dfferences of hgher magntudes, whch nfluences the flexblty of the performed approxmaton; - f the test data sgnfcantly dffer from each other n magntude (e.g. from 1 to or from 10-8 to 10-2 ), the approxmated values near the lower threshold contrbute much less to the total sum S than approxmated values near the upper threshold; ths means that, e.g. tens or hundreds of test data wth dfferences n magntude of 100% from value 1 are less sgnfcant n performng the approxmaton than one or a few data ponts whch dffer by 1% from value Accordng to the above stated example, the approxmaton s asymmetrc snce better approxmaton wll be acheved for hgher values of test data, neglectng dfferences around smaller values an example of such approxmaton s shown n Fg. 4b, where one can see a good ft of theoretcal descrpton of 3 curves for large values of da/dn (over 10-4 mm/cycle) whle there s an evdent msft for smallest values (below 10-5 mm/cycle). The presented approxmaton has been acheved by satsfyng the LSM crteron,.e. the mnmum value of the sum S. When the test data are wthn a wde range of values, e.g. 5 orders of magntude,.e. from 10-2 to 10-7 mm/cycle, then dfferences between the hghest values and the approxmatng functon wll have the largest effect on the square sum S of devatons whle dfferences for small values, sometmes of 2-3 orders of magntude, do not contrbute much to the total sum S. Hence, the msft of the approxmatng functon for low values of da/dn, practcally for values lower by only 1-2 orders of magntude than the maxmum values of da/dn. Wthn ths range the theoretcal descrpton s rather random and has rather no effect on the value of the sum S, whch ndcates that ths crteron s rather useless for ths type of analyss. It seems reasonable to use one of the followng crteron modfcatons, whch wll allow to remove the above stated problems: - changng the form of the crteron, or - usng logarthmc values of da/dn, n n 2 2 (16) S log y log y or S log y log y. 1 1 In the present study the frst varant has been examned (see secton 3.3) due to the fact that t s more general snce t does not lmt tself only to postve values of predcted y, whch s a requrement n the second varant. In the case of crack propagaton test data all the da/dn values are postve; therefore the second varant could also be used. Snce the crteron for fttng the theoretcal descrpton to the test data n the form of equaton (15) or (16), or any other, s closely connected wth the number of approxmated

10 Good Practce for Fatgue Crack Growth Curves Descrpton 205 ponts (n the case under dscusson, coordnates n the graph (da/dn,k)), the qualty of ft has to depend on: - the dstrbuton of the number of test ponts among partcular curves, - the dstrbuton of test ponts on partcular curves, not to menton - the scatter of test ponts and accuracy of fndng them. If the dstrbuton of ponts among partcular curves s not unform, the approxmaton wll show better ft to the curves wth a larger number of ponts than to those wth a smaller number of ponts the contrbuton thereof to the pooled error ncluded n the approxmaton crteron wll be greater; the mnmzaton thereof wll occur around the larger data cluster. Smlar stuaton occurs whle fttng the descrpton to a gven expermentally ganed curve where the data concentraton s larger, the approxmaton wll be better than where there s less data, or where the data are only ndvdual ponts. Therefore, essental to the analyss of test data and to descrpton thereof s the regular dstrbuton of the test data over the whole range to be subject to approxmaton. Snce t s sometmes beyond the reach of researchers whle recordng the test data drectly durng the testng work, some modfcaton or recalculaton of the test data set may prove ndspensable Data set modfcaton As clearly seen n Fg. 4, the number of ponts n the threshold and crtcal areas of the scope of the stress ntensty factor K s very small, whch results from the specfc nature of the performed test and data recordng. For crack growth rates lower than 10-6 mm/cycle the ncrement by 1 mm occurs after approx. 1 mllon cycles,.e. the process s a long-lastng one, and the recordng of the crack-length ncrement for nstance every 0.01 mm gves 100 ponts of test data only (whle n the case of takng records every mm, the number of ponts wll be 200). The testng work for even lower crack growth rates s stll more tme- and energy-consumng. Wth as lttle cracklength ncrements as these there s practcally no chance that n sngle load cycles any random jump wll occur n values of recorded data of the order of 0.01 or at least mm (.e. by approxmately 3 4 orders of magntude hgher than the crack growth rate under examnaton). Ths provdes relatvely regular recordng of crack lengths n the course of the testng work,.e. for subsequent ncrements 0.01, 0.02, 0.03, mm, etc. (even f measurements are taken for crack-length ncrements by only a fracton of a mllmetre,.e. n a shorter tme, whch means for the number of cycles lower than the above-mentoned 1 mllon). In the range of crtcal crack propagaton, at the crack growth rate hgher than 10-3 mm/cycle, the recordngs of the crack length ncrements every 0.01 mm (as above) take place more frequently than every 10 cycles. For load-applyng frequences of Hz ths means 1 s

11 206 Appled Fracture Mechancs long data-recordng ntervals n the course of the testng work. The fnal several mllmetres crack-length ncrement occurs as fast as over only several mnutes of the testng work, wth crack-length ncrements sgnfcantly ncreasng every cycle. Hence, at the testng rate gettng as hgh, the number of test ponts remans relatvely low and, because of these evergrowng ncrements, lower than the above-mentoned 100 or 200 ponts per every 1 mm of the crack length. In the ntermedate area of the graph (10-6 through 10-3 mm/cycle,.e. coverng 3 orders of magntude of the da/dn value) the above-mentoned exemplary crack-length ncrements every 0.01 mm take place on a regular bass, however, wth random fluctuatons typcal of the phenomenon under examnaton there are no dentcal data recordngs after 0.01, 0.02, 0.03, 0.04, mm of crack-length ncrement, snce nstantaneous readngs (varatons) from the measurng sensors may cause that the data recordng durng the test, wth the same recordng crteron assumed, can occur for ncrements of, e.g. 0.01, 0.028, 0.038, 0.057, mm, dsturbng at the same tme the regular bass of ncrements n the number of cycles between partcular measurements. Fg. 5a llustrates the non-unformty of such datarecordng practce; the arrows pont to where such dsturbances have occurred, and after whch the subsequent record s taken after the hgher number of cycles. Ths, n turn, affects the crack growth rate. Calculaton of the da/dn dervatve based on the n ths way recorded data must also be burdened wth a random scatter, Fg. 5b, larger than that resultng from the propertes of the materal under examnaton. To elmnate these ncdental dsturbances, the expermentally recorded tme functon may become smoothed by means of nterpolaton of results on the bass of any lnear regresson functon (wth ether a straght lne or a polynomal). Fg. 5c shows an example of such smoothenng: presented wth a full lne s result of the 7-pont regresson,.e. after havng nterpolated each pont (a;n), wth account taken of 6 adjacent ponts: 3 ponts n front of and 3 ponts behnd a gven pont (a;n). It s evdent that ths smoothed curve represents n a relable (or even better, n a more relable way) the expermentally recorded dependence between measured quanttes. On the other hand, the above-dscussed dsturbances have been removed from partcular measurements. Calculaton of the da/dn dervatve for any pont of the plot (a;n) can be carred out on the bass of lnear or polynomal regresson for e.g. 5, 7, or 9 adjacent ponts around a gven -th pont. Fg. 5d shows result of the 5-pont lnear regresson (2 ponts n front of the (a;n) pont, the (a;n) pont, 2 ponts behnd the (a;n) pont), of calculatons of the da/dn dervatve aganst the unsmoothed plot a-n. What n ths case s arrved at from the equaton for the lne of regresson y = mx + n (and more exactly, a = mn + n) s: da / dn y m. (17) In the case of lnear regresson wth polynomals of the 2nd (y = mx 2 +nx+l) or 3rd (y = mx 3 +nx 2 +lx+k) order, the crack growth rate s calculated from the formulae, respectvely: da / dn y 2 m x n, (18)

12 Good Practce for Fatgue Crack Growth Curves Descrpton da / dn y 3m x 2 n x l. (19) What becomes evdent s a consderable scatter of calculated values of the crack growth rate da/dn, and for ponts ndcated wth arrows t can be stated that: - any measurement dsturbance results n that the resultng (calculated) value of da/dn at one or two subsequent ponts s always lower than that for the pont n queston, - the measurement dsturbance s not expected to reflect the accelerated crack propagaton, even though n the form of a local maxmum, whch all the more confrms the correctness of treatng ths dsturbance as a random effect, - where the dsturbance occurs n the local-maxmum area, t magnfes ts value; however, the scale of ths ncrease may prove too large as compared to the actual crack growth rate. (a) (b) (c) Fgure 5. Calculated values of da/dn wth correspondng test data a-n: 5-pont lnear regresson, a) and b) output data; c) and d) smoothed data If calculatons are carred out for the smoothed curve a-n (Fg. 5c), the resultng da/dn curve presented n Fg. 5d takes the form of a sold lne. The scatter of values of the crack growth rate over the whole range of calculatons s much smaller. The local extremes have been mantaned, however, slghtly scaled down than n Fg. 5b. In the case the regresson used to calculate the da/dn dervatve s carred out for a greater number of ponts adjacent to a gven computatonal pont, the correspondng curves look (d)

13 208 Appled Fracture Mechancs lke n Fg. 5a (for 7-pont regressons: 3 ponts n front of the (a;n), the pont n queston (a;n) and 3 ponts behnd the (a;n)) and Fg. 5b (for 9-pont regressons: 4 ponts n front of the (a;n), the pont n queston (a;n) and 4 behnd the (a;n)). In all the cases the dervatve of da/dn has been found from equaton (17). (a) (b) Fgure 6. Calculated values of da/dn together wth correspondng expermental data a-n: lnear regresson, b) 9-pont lnear regresson a) 7-pont It s obvous that as the number of ponts taken nto account n the regresson analyss ncreases, the scatter of computatonal results gets reduced and the curve plotted for unsmoothed data (crcles n Fg. 6) ever more resembles the curve plotted for smoothed data (sold lne n Fg. 6). It s effected by the fact that the greater number of data accepted for regresson brngs the result closer to that of regresson for smoothed data. There s of course some dsadvantage: the greater number of data taken nto account n regresson analyses, the more reduced number of detals referrng to, e.g. local changes n value of da/dn are to be seen on the plotted curves. In the extreme, f all the ponts are subject to regresson at once, the smoothed curve a-n would be a straght lne and the da/dn curve would run horzontally. Another extreme conssts n that the whole curve a-n would be descrbed wth only one regresson equaton, whch n turn would provde the relable mappng of the whole a-n curve; the da/dn dervatve could be calculated by means of dfferentatng ths equaton. However feasble, t seems unpractcal, work-consumng, more of the art for art s sake category. Results presented n Fgs 5 and 6 could be consdered optmal: they provde good mappng of local changes n the approxmated curves and do not requre any complcated mathematcal apparatus. Characterstc of these plots (for both the unsmoothed and smoothed data) s that the calculated rates da/dn may be the same for dfferent numbers of cycles N (hence, for dfferent crack lengths a and dfferent values of K). Ths s the effect of more common, for ths range of crack growth rate da/dn, occurrences of changes n the monotoncty of curves a-n than n threshold or crtcal ranges of da/dn-k. Curves plotted n Fgs 5 and 6 correspond to approx. 1-mllmetre ncrement n the crack length (6.6 through 7.5 mm) and cover crack growth rates of mm/cycle. At the further stage of the crack growth as the crack length ncreases, the crack growth rate ncreases as well, and before the crack

14 Good Practce for Fatgue Crack Growth Curves Descrpton 209 reaches the crtcal growth range the calculated values of da/dn from the range 10-6 through 10-3 mm/cycle wll repeatedly appear n the calculatons. Hence, the number of measurng ponts recorded throughout the testng work for ths range of da/dn wll be hgher than for threshold or crtcal ranges of da/dn-k, what s to be seen also n Fg. 4. Moreover, n practce, the plottng of a complete crack propagaton curve da/dn-k,.e. startng from crtcal crack growth rates of 10-8 mm/cycle up to crtcal ones of 10-2 mm/cycle, s not performed n the course of one test only. Ths s closely related wth dfference n levels of K for the stage of the specmen s precrackng and the threshold range typcal of the rates of 10-8 mm/cycle. The precrackng usually fnshes at hgher values of K, snce t cannot proceed wth the threshold growth rate. The reason s that t would take much more tme than the test tself. Therefore, the test started after the specmen s precrackng stage from the threshold values of the crack growth (change n the loadng level from hgh to lower), would be connected wth the crack growth retardaton effect, whch - n turn - would dsturb test results n ths area,.e. t would not allow the researchers to gan the correct curve da/dn-k. Such tests are usually conducted as a two-stage effort see Fg. 1: I stage wth exponentally decreasng K (K=K0e -ga ), wth constant relatve gradent,.e. =, up to havng the left sde of the plot wthn the threshold range. The test starts from the level of loads hgher than those at the already completed stage of the specmen s precrackng, so as to elmnate the crack growth retardaton effect that appears as f the test s started at loads lower than those at the termnaton of the specmen s precrackng. The decreasng K, startng from some sutably hgh value, and the crack length both cause that the crack growth rate becomes reduced to reach then the threshold range of the plot. At ths stage, the a-n curve asymptotcally approaches the horzontal lne as the testng tme ncreases. The testng tme depends on the scentfcally and economcally justfed needs of the researcher, although n practce ths tme much more depends on senstvty of appled sensors, snce both the level of appled loads and the crack openng sze decrease for ths range to values comparable to electrc nose of the testng machne, whch usually results n the test beng automatcally nterrupted and the testng machne beng stopped for crack growth rates lower than 10-8 mm/cycle. The at ths stage obtaned curve a-n and the propagaton-curve secton da/dn-k may look lke e.g. those presented n Fg. 7 (to be also seen n Fg. 1b). II stage at constant ampltude load (CA test, constant ampltude test) up to the acquston of the rght sde wthn the crtcal range. The test s carred out at the level of loads hgher than the level at whch the stage I was completed; as the crack length ncreases, there s a systematc ncrease n the K, up to the moment the crtcal value s reached, at whch the specmen fals. The at ths stage obtaned curve a-n and the propagaton-curve secton da/dn-k may look lke e.g. those presented n Fg. 7 (to be also seen n Fg. 4). The total result of both the stages has been presented n Fg. 8 both the curves from Fgs 6 and 7 complement one another to full propagaton-curve plot a-n and da/dn-k: expermentally found ponts n the form of crcles, curves smoothed n the form of full lnes.

15 210 Appled Fracture Mechancs It s qute clear that the md secton (range) of the da/dn-k curve contans much more expermentally ganed ponts despte the same crteron for data recordng n the course of testng work for all three ranges, and also, ndependently of the fact that both the curves overlap over some specfc secton common to both of them. Furthermore, the plot presents the above-dscussed changes n the monotoncty of how they run, ndependently of whether the calculatons of the da/dn dervatve have been conducted for unsmoothed or smoothed data Fg. 8. (a) (b) (c) (d) Fgure 7. Curves a-n and da/dn-k for the frst (I) a), b), and the second (II) test stages c), d) 3.2. A Method of Regular Curves Mappng (MRCM) Dsturbances n the run, monotoncty of curves da/dn-k as well as dfferent measurng-data densty n partcular areas of the graph do not serve well any attempts to theoretcally descrbe these curves. As mentoned earler, the least squares methods better ft regresson curves to

16 Good Practce for Fatgue Crack Growth Curves Descrpton 211 areas where there s more approxmated ponts, n the case gven consderaton, n the mddle ranges of the da/dn-k curves. To elmnate ths effect, applcaton of the Authors Method of Regular Curves Mappng (MRCM) to approxmate the da/dn-k curves s advsable. (a) (b) (c) Fgure 8. Curves a-n and da/dn-k and how they run at the I and II test stages: results for data after the curve has been smoothed - a), b), and effect of havng appled the MRCM c) The MRCM technque of mappng test data conssts n fxng, at regular ntervals (along axes x or y), the k number of representatve ponts n the data set under analyss (upon the expermentally ganed curve). The followng actons are to be taken: a. determned are selected values of coordnates x (or y), for whch the above-mentoned ponts wll be fxed ( = 1,2,.,k), b. from the curve under analyss, pont x (or y ) s fxed, of coordnate value closest to the assumed value of x (or y), and 2m of adjacent ponts by assumpton, n half these are

17 212 Appled Fracture Mechancs ponts of values lower than x (or y) and n half - of hgher values, (m s equal to, e.g. 2, 3, 4 or 5), c. a set of n ths way ganed data 2m+1, (x -m, y-m) through (x +m, y+m) (or (x-m, y -m) through (x+m, y +m)) grouped around some selected value of x (or y) s subject to regresson wth any functon to determne the approxmated value of y * (or x * ) correspondng to the selected value of x (or y), d. the pont of coordnates (x, y * ) (or (x *, y)) s mapped on the curve under analyss as the -th representatve data tem found on the bass of the assumed crtera, e. steps b) through d) are repeated for subsequent k number of values determned n a), untl a set of k number of ponts that represent (map) the curve s obtaned. The effect of the n ths way performed mappng of values of da/dn, regularly dstrbuted wthn partcular ntervals (orders of magntude), n selected k = 37 ponts, for the curve shown n Fg. 8b, s presented n Fg. 8c. The ponts n queston: - well represent (map) the curve under analyss, - are equdense dstrbuted wthn the whole range of da/dn varablty, - do not show any more or less sgnfcant fluctuatons/scatter of values resultng from, e.g. random measurng-data dspersons. (a) (b) (c) Fgure 9. Expermentally ganed curves da/dn-k for 9 specmens tested at dfferent stress ratos R a) and the same curves havng been mapped wth ponts usng the MRCM, b) and wth extrapolated ponts that perform the mappng accordng to the MRCM, c) together wth approxmaton thereof wth the NASGRO equaton The set of ponts that map the curve seems to gve good bass, owng to the above descrbed features, for analyses of theoretcal descrpton of a gven, expermentally ganed curve. In the case of nne (9) curves that correspond to tests wth three (3) values of the stress rato R

18 Good Practce for Fatgue Crack Growth Curves Descrpton 213 Fg. 9a, the result of expermentally ganed data modfcaton wth the MRCM appled (ponts n the graph) s presented n Fg. 9b, together wth the data approxmaton by means of the NASGRO equaton, wth the LSM crteron used, accordng to formula (15). The MRCM technque also enables, f need be and wth scentfc correctness mantaned, the extrapolaton of the mappng ponts beyond the range of recorded test data,.e. nto the area of crack growth rates lower (the threshold range) or hgher (the crtcal range) than those recorded expermentally, wth ther tendency to change whch s pecular to those areas, on the bass of regresson at boundary (n the graph lower or upper) ponts of expermentally ganed curves the extrapolaton result has been shown n Fg. 9c. Applcaton of extrapolaton to prepare data for the analytcal modellng may prove advantageous n the case the partcular expermentally ganed curves show dfferent ranges of values, and hence, dfferent numbers of mappng ponts. After correctly performed extrapolaton one can arrve at the stuaton when they are equalzed, whch means the same power of each of the wth the regresson method approxmated curves Modfcatons of the LSM method crteron In order to elmnate the approxmaton msft as shown n Fg. 1 and to mprove the qualty of approxmaton, modfcaton of formula (15) takes the followng form: S * 2 2 n n y y y 1. 1 y 1 y (20) The fracton n brackets n formula (13), as a relatve error, s a measure of devaton ndependent of the order of magntude of compared values (approxmated and approxmatng ones), so that the contrbuton of all the test data s equally "strong" to the total error S, whch should have good effect on the approxmaton wthn the whole range, snce: - each value among test data y has equal contrbuton to the sum S*, ndependent of ts magntude 10-7, 10-2, 1 or (.e. t fts n any magntude range) always a devaton of e.g. 10-, 50-, 200-percent of approxmatng value wll gve a component of the sum S* equal to 0.01, 0.25, 4, respectvely; - the crteron assures that the acheved approxmaton s symmetrc,.e. the degree of approxmaton around lower and hgher values s the same; - dsadvantages of the crteron descrbed wth formula (15) are no longer vald. The crteron descrbed wth formula (20) has also some specfc property: f the approxmatng value equals zero (.e. for the approxmaton smaller by 100%) or t s twce as bg as the approxmated value (.e. for approxmaton larger by 100%), then ndependently of the approxmated value the component of the sum S* wll equal 1. In order to carry out the approxmaton of test data t s necessary to calculate coeffcents of the approxmatng equaton used to determne. Equaton (1), after applyng logarthms, takes the form:

19 214 Appled Fracture Mechancs da 1 f Kth K max log logc n log K p log 1 q log 1, dn 1 R K K c and can be presented n the followng general way: (21) y b 0 b 1 f 1 b 2 f 2 b 3 f 3. (22) Coeffcents b are drectly connected wth C, n, p and q (b0=log(c), b1=n, b2=p, b3=-q), whereas functons f depend on K and R and nclude all the remanng coeffcents of the NASGRO equaton. Coeffcents b of the approxmatng equaton are calculated from the mnmum condton of the equaton (20),.e.: S 1 b k n b0 b1 f1, b2 f2, b3 f3, y y b k 1,2,3,4. k 2 0 (23) Ths leads to the followng system of equatons: n S b0 b1 f1, b2 f2, b3 f3, y 1 2 0, b0 1 y y n S b0 b1 f1, b2 f2, b3 f3, y f 1 2 0, b1 1 y y n S b0 b1 f1, b2 f2, b3 f3, y f 2 2 0, b 2 1 y y n S b0 b1 f1, b2 f2, b3 f3, y f b 3 1 y y (24) It s a system of 4 lnear equatons wth 4 unknowns b, whch after transformaton takes a form: f f f n b b b b n n n n n 1 1 1, 2, 3, , 2 1y 1y 1 y 1 y 1 y n 2 f n n n n 1, f1, f1, f1, f2, f1, f3, y 1 y 1 y 1 y 1 y n f n n 2, f2, f1, f n 2 2, f n 2, f2, f3, b 2 2 b3 1 y y 1 y 1 y 1 y n 2 f n n n n 3, f3, f1, f3, f2, f3, f3, y 1 y 1 y 1 y 1 y n b b b b 0, n b b 0, n b b b b 0. (25)

20 Good Practce for Fatgue Crack Growth Curves Descrpton 215 and s easly solved by subtractng n the followng steps: - elmnatng b0 n n 2 1 f n n n n 1, f1, f 1, f2, f1, f2, n n y 1 y 1 y 1 y 1 y 1 y b n n 1 n n b2 n n 1 f1, 1 f 1, 1 f 1, y 1 y 1y 1 y 1y 1 y n f n 3, f1, f3, y 1 y b3 0, n n 1 f 1, y 1 y n n 2 1 f n n n n 2, f1, f1, f2, f2, f 2, n n y 1 y 1 y 1 y 1 y 1 y b n n 1 b n n 2 n 1 f 2, 1 f n 2, 1 f 2, y 1 y 1y 1 y 2 1y 1 y n f n 3, f2, f3, y 1 y b3 0, n n 1 f 2, 2 2 1y 1 y n n n n 1 f3, f1, f1, f3, n f n 2, f2, f3, n n y 1 y 1 y 1 y 1 y 1 y b n n 1 n n 1 f 3, 1 f b2 n n 3, 1 f 3, y 1 y 2 2 1y 1 y 1y 1 y n n 2 f3, f 3, y 1 y b3 0 n n 1 f3, 2 2 1y 1 y (26) what gves 3 equatons of the general form: Bk b1b1, k b2b2, k b3b3, k 0 k 1,2,3. (27) - elmnatng b1

21 216 Appled Fracture Mechancs B1 B B 2 1,2 B 2,2 B1,3 B 2,3 b2 b3 0, B1,1 B 2,1 B1,1 B 2,1 B1,1 B 2,1 B B 1 B 3 1,2 B 3,2 B1,3 B 3,3 b2 b3 0. B1,1 B 3,1 B1,1 B 3,1 B1,1 B 3,1 (28) what gves 2 equatons of the general form: Ck b2c2, k b3c3, k 0 k 2,3. (29) - elmnatng b2 C2 C C 3 3,2 C 3,3 b3 0, C2,2 C 2,3 C2,2 C 2,3 (30) hence: b 3 B1 B 2 B1 B 3 B1,1 B 2,1 B1,1 B 3,1 C2 C B 3 1,2 B 2,2 B1,2 B 3,2 C B 2,2 C 2,3 1,1 B 2,1 B1,1 B 3,1. C3,2 C3,3 B1,3 B2,3 B1,3 B3,3 C2,2 C 2,3 B1,1 B 2,1 B1,1 B 3,1 B1,2 B 2,2 B1,2 B 3,2 B1,1 B 2,1 B1,1 B 3,1 (31) Hence, coeffcent b2 can be calculated from one of the formulae (29); secondly, coeffcent b1 from one of equatons (27), and fnally, coeffcent b0 from one of equatons (25). The n ths way found coeffcents of the NASGRO equaton enable approxmaton of curves da/dn-k from Fg. 9 to the form shown n Fg. 10a. Consderable mprovement n the theoretcal (analytcal) descrpton for the whole range of plotted curves s evdent. Both crtera (15) and (20) have also some dsadvantage consstng n that f the approxmatng value s much smaller than the approxmated value y (.e. by 3, 5, 7 orders of magntude) or smply close to zero then the component of the sum S and S* s close to the squared value y (n case of (15)) or to 1 (n case of (20)), ndependently of how these two values dffer from each other. Obvously, t s mportant whether the approxmaton and behavor of the approxmatng curve near value y at the level of e.g and lower (.e. for strongly decreasng values wthn the threshold range of the graph) take place at the level of 10-8, or (what s not hard to acheve for curves showng strong vertcal courses on graphs plotted wth the

22 Good Practce for Fatgue Crack Growth Curves Descrpton 217 logarthmc scale appled); t s much better when the possble dfference between values and y s not too large. (a) (b) Fgure 10. Result of approxmaton of curves from Fg. 9 wth the NASGRO equaton wth the LSM crteron appled: a) by equaton (20), b) by equaton (32) Due to dynamc changes around value equal to zero (completely monotonc, as for the second-degree polynomal), functons (15) and (20) are practcally nsenstve to that the approxmated value equals e.g. 0.01, 10-5, 10-8 or Hence, t s most preferable f the LSM approxmatng crteron takes such cases nto account. Therefore, a modfcaton s proposed to transform the crteron nto the followng form: S ** n y y 11. (32) 1 y y Owng to ths for both large values (much dfferent from the approxmated value y) and small values (approachng zero) wth respect to value y, the components of the sum take sgnfcant values,.e. n both cases they gve a sgnfcant (although - as t can be seen - dverse/unsymmetrcal for each of the cases) contrbuton to the total approxmaton error as shown n Fg. 11. In order to make the S components of the sum (32) and the total sum S ** as an approxmaton crteron reaches the mnmum (not the maxmum, as n Fg. 11) and also, when the reversal of sgn takes place between the approxmated value y and the approxmatng value ), the followng form would be better:

23 218 Appled Fracture Mechancs Fgure 11. Component of the sum for the approxmaton crteron (32) Fgure 12. Component of the sum for the approxmaton crteron (32a) S ** n y y y y (32a) Both extremes of the S functon for both postve and negatve values of are the mnma shown n Fg. 12. Approxmaton crteron functons for (15), (20) and (32) (and ther components) as related to approxmatng values, for: - dfferent approxmated values y equal to 5; 2; 1; 0.25; 0.01; , - the same range of varablty of,.e. (-3y, 3y), n order to show the 0 effect, are shown n Fg. 13.

24 Good Practce for Fatgue Crack Growth Curves Descrpton 219 Fgure 13. Approxmaton crteron functons for dfferent approxmated values y for the (-3y, 3y) nterval arrow for equaton (32a) All advantages and dsadvantages of the above presented LSM approxmaton crtera can be seen on the graphs above, n partcular: - sgnfcant dependence of values of components of the sum S (formula (15)) on the approxmated value y;

25 220 Appled Fracture Mechancs - nvarablty of values of components of sums S* (formula (20)) and S ** (formulae (32) and (32a)) on all the graphs,.e. for any approxmated value y; - no response of the components of sums S and S* to the 0 effect and dynamc change n the components of the sum S** near value = 0. The only curve that changes n the graphs presented n Fg. 13 s the plot for components of the sum S graph,.e. for the standard form of the LSM. Result of approxmaton wth crteron (32a) appled s shown n Fg. 10b for data sets wth no extrapolaton ponts. The same approxmaton for only 1 specmen tested at dfferent R s shown n Fg. 14a, and for only 2 specmens tested at dfferent R - n Fg. 14b. Fgure 14. Approxmaton of da/dn=f(k) data n dfferent varants wth the NASGRO equaton, LSM modfed accordng to formula (32a) Exemplary results of approxmaton for dfferent test data (wth slghtly smaller scatter between ndvdual da/dn-f(k) curves) s presented n Fg. 14. Favorable effects of the approxmaton (n comparson wth results showed n Fg. 1) after mplementaton of the modfed LSM crteron can easly be seen. They tend to represent all the test data, wthn the whole range of data varablty, ndependently of ther absolute values, ndependently of the number of descrbed curves 3 (Fg. 14a), 6 (Fg. 14b), 9 (Fg. 10b). Ths effect has been acheved only by modfyng the LSM crteron, snce the dea underlyng the approxmaton method for all the presented graphs s dentcal the mnmum of the sum of squared devatons between the approxmated test data and the approxmatng values.

26 Good Practce for Fatgue Crack Growth Curves Descrpton Regresson of dependences n the NASGRO equaton Approxmaton of curves da/dn-k substantally depends on preset values of parameters Kc and Kth. Hence, t s very mportant whether they can be determned on the grounds of the test data only (f they cover the whole range of the curve,.e through 10-2 mm/cycle, whch s not always easy to reach), or whether they need any other method/way to be determned, e.g. formula (2), functonal dependences of the type Kth = f(r) and Kc = f(r), or the above-mentoned extrapolaton. The above-dscussed results of extrapolaton correspond to the case when both the parameters show constant values for all the approxmated curves. The plots for the test data show, however, that they depend on the stress rato R for each of nne expermentally ganed curves parameters Kth, and Kc, can be estmated and the data ganed can then be used to determne dependences Kth = f(r) and Kc = f(r), ncludng coeffcents for equaton (2). Formulae (2) and (2a) are specal cases of a general formula of the followng form: a 2 1R1A0 Kth K0 a a 0 1 f 1 CC R th. (33) Havng re-arranged ths formula, the followng s arrved at: 1 a 2 1R1A0 1R1A0 0 th a a 0 1 f 1 f log K log K Clog C Rlog th (34) and then: 1R1A0 1 a logk 1 th 2 log log K0 Clog a a 0 f 1R1A0 CthRlog 1 f (35) whch can be descrbed wth the lnear-regresson equaton as: where 1R1A0 1 f FR ( ) log, y m mfr ( ) mrfr ( ), (36) and the corrected value of the threshold range of the stress ntensty factor s: a y log 1 Kth 2 log. a a 0

27 222 Appled Fracture Mechancs Havng found coeffcents m0, m1, m2 of the regresson equaton (36) we can calculate coeffcents of equaton (33): C m, C m and K 10 m th 0 (37) at the same tme, value of the Kth functon s calculated from the regresson equaton by formula: 1 2 y a Kth 10. a a 0 (38) So, f we have data sets (R, Kth,, a) n the case under analyss there are 9 such sets we automatcally can fnd coeffcents by formula (37), thus reducng the number of coeffcents of the NASGRO equaton to approxmate the test data, whch we are lookng for. Snce there s no smlar dependence for the Kc, parameter, the relatonshp Kc = f(r) can be found n the same way (.e. usng the test data) from the ordnary lnear regresson Kc = mo+ m1r and also use t to descrbe 9 expermentally ganed curves. Functons Kth = f(r) and Kc = f(r) found n ths way wth the test data appled are shown n Fg. 15, whereas Fg. 16 llustrates effect of approxmatng curves da/dn-k n the case gven consderaton. (a) (b) Fgure 15. Functons a) Kth = f(r) by formula (33) and b) Kc = f(r) - regresson Evdent s good ft of analytcal descrpton n both the crtcal and threshold ranges, whereas worse - n the mddle secton. The ad hoc accepted lnear regresson for the expermentally found relatonshp Kc = f(r) not too precsely descrbes ths relatonshp (straght lne n Fg. 15, correlaton coeffcent reaches n ths case the 0.3 level). Optmsaton of values of the Kc coeffcent for R = 0.1; 0.5 and 0.8 (here denoted as Kc*) wth the LSM method to reach the mnmum devaton error (32) results n the da/dn-k curves

28 Good Practce for Fatgue Crack Growth Curves Descrpton 223 approxmatng courses as n Fg. 16b. The curve llustratng the Kc * = f(r) dependence s n ths case a broken lne shown n Fg. 15b, whch easy to see consderably strays away from the lnear dependence. Ths proves that, among other thngs, one cannot ad hoc mpose any form upon t. It can be assumed that for a larger number of expermentally ganed curves, ncludng the wder scope of values of R, the suggested method of determnng the relatonshp Kc = f(r) wll offer better results that better correspond to the actual dependence and wll reman useful for approxmatng the da/dn-k curves. The broken-lne curve, as that resultng from the optmsaton process, may be descrbed wth, e.g. a straght lne or a quadratc equaton (as n Fg. 17) and used as a component of the theoretcal (analytcal) descrpton of the test data wth the NASGRO equaton. In the case of a straght lne, the correlaton coeffcent ncreases up to approx for the polynomal. Obvously, wth three ponts Kc * the correlaton s complete, but f the scope of values of the asymmetry coeffcent was greater,.e. there would be more expermentally ganed curves of dfferent values of R (then the number of these ponts would ncrease), one should also expect hgh correlaton for the relatonshp Kc = f(r). Fgure 16. Approxmaton of curves da/dn-k wth the NASGRO equaton, modfed by formula (32) LSM, wth extrapolated mappng ponts accordng to the MRCM: a) wth regresson appled as n Fg. 15, b) wth optmsaton for values of coeffcents Kc*

29 224 Appled Fracture Mechancs Fgure 17. Lnear Kc = f(r) and polynomal Kc * = f(r) functons to optmse theoretcal (analytcal) descrpton of curves da/dn-k Fgure 18. Curves da/dn-k wth coeffcents Kth and Kc ndvdually ftted to each expermentally ganed curve

30 Good Practce for Fatgue Crack Growth Curves Descrpton 225 If we use values of Kth Kc coeffcents n forms determned not wth the above-mentoned regresson and optmsaton methods, but as ones ndvdually found for each of the expermentally ganed curves (Kth,nd Kc,nd), the theoretcal (analytcal) descrpton by means of the NASGRO equaton - wth the above-descrbed methodology of fndng other coeffcents appled - should gve even better result, see Fg. 18. Ths varant of the theoretcal (analytcal) descrpton s of only lttle practcal mportance, however, t shows that both the above-descrbed methodology of analyss and the way of fndng coeffcents of the NASGRO equaton result n correct descrpton of expermentally found curves of fatgue-crack propagaton and may be appled to ths and smlar categores of research ssues. 4. Concluson Applcaton of the Least Square Method n ts classcal form to determne coeffcents of the NASGRO equaton that descrbes fatgue crack propagaton curve s neffectve, snce data of the approxmated functon da/dn=f(k) take values from the range of a few orders of magntude, measurng ponts of the curves are rregular and n dfferent numbers dstrbuted n the graph (n threshold, stable-ncrease, and crtcal ranges), and subject to approxmaton are also several curves grouped n several sets (for dfferent values of R). The paper offers some technques to modfy the LSM crteron to sgnfcantly mprove approxmaton results. These nclude: - modfcaton of the approxmaton-method crteron, - smoothng of the expermentally ganed curves to elmnate slght random dsturbances resultng from, e.g. data recordng process, - dfferent varants of calculatng the dervatve da/dn, - regular mappng of the expermentally ganed curves n the form of selected ponts, - regresson for ponts that represent (map) the expermentally ganed curves to fnd coeffcents of the crack growth equaton, - regresson or optmsaton of the descrpton of partal dependences of the NASGRO equaton as based on expermental data. Values of parameters to be found as well as quanttatve and qualtatve results of performed approxmatons and theoretcal (analytcal) descrpton are affected by, among other thngs, the number of tests that produce expermental data, and confguratons thereof. They provde a wder or narrower range of varablty of parameters of sgnfcance that affect the courses of curves da/dn-k, and also enable determnaton of accuracy and repeatablty of obtaned results. Relablty of the theoretcal (analytcal) descrpton ncreases and the descrpton tself better characterses propertes of the materal under examnaton f there are tens of curves ganed expermentally from tests conducted for many (e.g. 5, 7, or 9) levels of the stress rato R, for a wder range thereof, e.g. 0.2 through 0.9.

31 226 Appled Fracture Mechancs The proposed modfcaton of the LSM crteron offers better ft of results of the test data approxmaton (unachevable wth the classcal LSM method). These effects are as follows: - the provson of equal weghts of each of the test data ponts n the total sum that determnes ths crteron (.e. the sum of dfferences between approxmated and approxmatng values) ndependently of the magntude of dfference between values of data subject to approxmaton and that of dfference between the approxmated and approxmatng values, - hgh effectveness whle approxmatng sngle, several, as well as a great number of sets/curves of test data, - t becomes even more precse as the test data from the same (research-testng) groups show smaller scatter, - may be used n other analyses of the same type related wth test data regresson, snce t offers an all-purpose approach not related to propagaton curves da/dn-k. Author detals Sylwester Kłysz Ar Force Insttute of Technology, Warsaw, Poland Unversty of Warma and Mazury n Olsztyn, Poland Andrzej Lesk Ar Force Insttute of Technology, Warsaw, Poland 5. References AFGROW Users Gude And Techncal Manual. (2002). AFRL-VA-WP-TR-2002-XXXX, Verson , James A. Harter, Ar Vehcles Drectorate, Ar Force Research Laboratory, WPAFB OH ASTM E647 Standard test method for measurement of fatgue crack growth rates Bukowsk, L. & Kłysz, S. (2003). Complance curve for sngle-edge notch specmen. Zagadnena Eksploatacj Maszyn, No. 2(134), pp Elber, W. (1970). Fatgue crack closure under cyclc tenson, Eng. Fracture Mechs., Vol.2, pp Forman, R.G.; Kearney, V.E. & Engle, R.M. (1967). Numercal analyss of crack propagaton n cyclc loaded structures, J. Bas. Engng, Vol.89, pp Forman, R.G.; Shvakumar, V.; Cardnal, J.W.; Wllams, L.C. & McKeghan, P.C. (2005). Fatgue crack growth database for damage tolerance analyss, DOT/FAA/AR-05/15, U.S. Dep. of Transportaton, FAA Offce of Avaton Research, Washngton, DC Fuchs, H.O. & Stephens, R.I. (1980). Metal fatgue n engneerng, A Wlley-Interscence Publcaton Huang, X.P.; Cu, W.C. & Leng, J.X. (2005). A model of fatgue crack growth under varous load spectra, Proc. of 7th Int. Conf. of MESO, Montreal, Canada, pp