ON THE THEORETICAL FRACTURE STATISTICS OF THE HERTZ INDENTATION TEST

Transcription

1 ON THE THEORETICA FRACTURE STATISTICS OF THE HERTZ INDENTATION TEST Gerrdo Díz R. () nd Pblo Kittl D. () () Deprtento de Cienci de los Mteriles, Fcultd de Ciencis Físics y Mteátics, Universidd de Chile, Csill 777, Sntigo, Chile. E-il: gediz@cec.uchile.cl () Colegio Ecológico. Pine, os Aroos # 07, Hospitl. Pine, R.M. CP: 95409, Sntigo, Chile. ABSTRACT The cuultive nd locl frcture probbilities of the Hertz indenttion test were deterined. When the teril is subjected to the Hertz indenttion test with sphere indenter the stress field used for clcultions depends on where the crcks originte the frcture, i.e. if the crcks begin in zone within the contct between the teril surfce nd sphere indenter or if crcks begin outside of such contct zone. Using the defined functions ethod for specific risk of Weibull s two or three-preter frcture functions, the Evns functions nd the cuultive nd locl frcture probbilities were deterined for ech zone. Additionly, using the integrl eqution ethod the respective specific risk of Weibull s frcture functions were deterined for cuultive nd locl frcture probbilities too. For the cuultive frcture probbility the respective integrl eqution ws solved by siple differentition nd its solution ws finite integrl differentil opertor pplied to the Evns function. For the locl frcture probbility two integrl equtions considering the frctures initited within or outside the contct zone were solved. The lst two integrl equtions were solved using finite differentil opertor pplied to ech function which cn be obtined fro the solution of n ordinry differentil eqution of second order with vrible coefficients. INTRODUCTION The clssicl proble of the contct between sphericl surfce nd flt plte ws solved by Hertz lost t the end of the nineteenth century, ssuing elstic behviour for the terils, the indenter nd the plte. Mny yers lter Frnk nd wn proposed theoreticl nlysis for the Hertzin frcture initition bsed on Griffith s criterion 3, using the energy blnce, nd the Hertz-Huber s stress field,4. More bout fundentl spects of Hertzin contct cn be found in 5,6. The study of the Hertzin frcture through indenttion of hrd sphere loded onto flt specien it is interesting considering both theoreticl nd prcticl points of view.

2 Fro theoreticl or scientificl stndpoint it is possible to develop odels to understnd the frcture behviour for brittle terils. On the other hnd, fro the prcticl or technologicl viewpoint Hertzin frcture is of interest becuse it is relted, for exple, to friction nd wer, bll illing nd bllistic ipct, ong others. One of the principl dvntges of the Hertz indenttion test is tht it llows us to deterine the frcture toughness of brittle terils by ens of this siple test 7 9. Using the ethodology proposed in reference n extended pproch with crck initition under Hertzin sliding contc cn be found in 0. The sttisticl frcture echnics, lso known s probbilistic strength of terils ws proposed by Weibull in 939 in order to tke into ccount the sctter observed experiently in the frcture strength of brittle terils. At first his forultion ws identified s wekest-link-theory nd ssocited with siil of chin where the frcture origintes in the wekest link. At present, it is not necessry to use such siil nd oreover, Weibull's theory hs been used in wide rnge of terils fro brittle terils to ductile terils, considering brittle-ductile trnsition too. After Weibull s work ny ppers hve been published using his forlis. The in objective is to deterine the cuultive frcture probbility for soe teril subjected to constnt or vrible stress field, eploying specific risk of Weibull s function with certin nuber of preters which cn be deterined experientlly for set of sples nufctured in the se nner nd subjected to equl stress field. Kittl nd Díz hve published pper in which they discussed differents terils nd deterined the Weibull preters of the specific risk of Weibull s function using sttisticl estition ethods nd deterining its respectives dispersions through the Fisher infortion ethod. A grphicl ethod to obtin the Weibull preters hs lso been developed in nd nother pproch considering diverse geoetricl sples subjected to diverse stress field ws published by Mtthews et.l. 3 nd by Evns nd Jones 4. In their work they introduced ethod to deterine the specific risk of Weibull s function, which Kittl lter ned Integrl Eqution Method, nd solved the integrl equtions directly without the introduction of interedi functions 5. A ore generl tretent bout the integrl eqution ethod pplied to Weibull s frcture sttistics ws developed by Kittl nd Díz 6. Specil pplictions of such ethod in torsion nd flexure were treted too 7-0. In the cse of Hertzin frcture Weibull's pproch hs been used widely 4,3,-6. The locl frcture probbility cn be deterined long with the deterintion of the cuultive frcture probbility using Weibull s distribution function. In this sense the in ide is to deterine the voluetric percentge of frctures initited t given point in soe teril subjected to certin stress field. Usully the defined ethod function is used to deterine the preters of the specific risk Weibull function. Most uthors hve used the two-preter specific risk Weibull function. There re only two differents pproches deling with this tter so fr. The first pproch ws proposed by Oh nd Finnie nd the second by Kittl nd Cilo 7. Both pproches re coincident when using Weibullin two-preter specific risk function but differ when three-preter Weibull function is used. Afterwrds Trustru 8 using the Oh nd Finnie pproch deterined Weibull odulus, one of the Weibull preters, fro bending, eploying both frcture position nd filure stress. The controversil pproches were discussed by Díz et.l. 9

3 showing by ens of nuericl evlution the differences in the results, when soe teril is subjected to flexure, using specific risk of Weibull s three-preter function, where Kittl nd Cilo s forultion present better fit thn Oh nd Finnie s forultion. For the Hertz indenttion test Oh nd Finnie pplied their forlis to deterine the locl frcture probbility obtining the distribution of cone crck loction for three preter Weibull distribution for filure stress. ter Trustru 6, using the se ethod developed less coplicted theoreticl nlysis to obtin soe predictions for the distributions of the criticl lod nd cone crck rdius for Weibullin twopreter specific risk function. The ppliction of Weibull sttistics is due to sctter of vlues observed for the cone crck rdius, for the contct rdius nd for the criticl lod. The objective of the present work is to deterine both the cuultive frcture probbility nd the locl frcture probbility for soe teril subjected to Hertz indenttion test using the specific risk of Weibull s two nd three preter functions nd pply the integrl eqution ethod to find ore generl specific risk of Weibull s functions. THE WEIBU DISTRIBUTION FUNCTION The cuultive probbility distribution function of frcture strength for brittle teril subjected to vrible unixil stress field considering volue brittleness is given by,: F = exp r dv V () 0 V where Vo is the unity volue, r is the vector position, V is the volue of teril subjected to stress, r is the vrible stress field, is the xiu strength, F is the cuultive frcture probbility nd is the specific risk of Weibull s function. The vrible stress field cn be expressed s follows: f r r = f r () where f is function obtined fro the elsticity theory which depends only on vector position nd is inor or equl to one. Considering eqution () nd rewritten eqution () fter Evns nd Jones 4 the following eqution is obtined: = ln = f r dv F V (3) 0 V 3

4 where () is clled Evn s function. This siplified nottion llows for n esier theticl tretents. Weibull proposed two nlyticl expressions for the specific risk of function tking into ccount nuerous experientl results over different terils subjected to different stress fields, nd the best fit ws obtined with the two or three-preter functions. When the specific risk function hs two preters its expression is given by: = 0 (4) where nd o re the Weibull preters, is known s Weibull s odulus nd o is norliztion preter. On the other hnd when the specific risk function hs three preters its respective expression dopted the following for: = 0 0 < (5) where nd o hve the se sense thn in eqution (4) nd the third preter is the stress under which the frcture does not occur or, equivlently, when the cuultive frcture probbility is equl to zero. Weibull s preters depend on the nufcturing process of the teril. Both previous equtions cn be used directly in eqution () or (3) to obtin the cuultive frcture probbility nd its respective Weibull preters. This ethod is clled the defined equtions ethod. There is second ethod to pply in the probbilistic strength of terils where the i is precisely to obtin generl expression for the specific risk of Weibull functions where it is not necessry to postulte the existence of certin nuber of preters. Tht ethod is known s integrl eqution ethod. Eqution (3) cn be considered s n integrl eqution where function is n unknown function which ust be deterined becuse the stress field is known through the elsticity theory nd () is known by ens of experientl dt. For ech specific stress field it is necessry to solve the integrl eqution (3). In soe cses the solution cn be deterined by pplying finite opertor over Evn s function i.e. by ens of finite nuber of theticl opertions nd the solution cn be expressed s n integro-differentil for. In other cses the solution cn not be solved s in the previous cse nd infinite opertor ust be pplied over Evn s function. When it is possible to pply Tylor s series expnsions the specific risk of Weibull s function cn be expressed s n infinite differentil opertor. Expnding () nd () in Tylor s series we obtin: 4

5 n (n) = 0 n = 0 n! n (n) = 0 n = 0 n! (6) Introducing equtions (6) in eqution (3) we obtin n eqution with two series nd both re equls when its respective coefficients re equl, so: 0 (n) (n) n 0 = f r dv V (7) 0 V Eqution (7) llows us to deterine (n) (0) which replced into expression of () in eqution (6) gives: = V 0 (n) n 0 (8) n n = 0 n! f r dv V Eqution (8) is generl solution of eqution (3) when it is eployed s n integrl eqution. Note tht the solution is n infinite differentil opertor pplied over Evn s function which is known through experientl dt. THE HERTZ INDENTATION STRESS FIED If hrd sphericl indenter loded onto plne surfce of brittle specien with surfce brittleness the circunferentil stress is copressive over the entire contct surfce, only the rdil stress ust be considered to deterine the frcture probbility. According to the Hertz theory,4 the stress field distribution on the surfce is non-unifor nd two zones ust be considered, within the contct re nd outside the contct re. Within the contct re the surfce of the teril loded is under copression. However, during the subergence under the indenter within the contct circle the xiu rdil tensile stress experienced by the plne surfce of the teril subjected to lod P is: br r = cr 0 < r < ; 0 (9) where R is the rdius of the sphere indenter, is the rdius of the contct circle, r is the rdil distnce fro the centre of the contct circle, which define the rdius of the cone crck when the crck ring is initited, is n ngle in polr coordintes. Constnts b nd c hve the following expression: 5

6 b = 3 c = 4 E E (0) In eqution (0) v, E nd v, E re Poisson s rtio nd Young s odulus of the sphere nd specien loded, respectively. Outside the contct re the surfce rdil stress field is tensile nd is given by: bp r = r r < ; 0 Mxiu tensile stress in the specien occurs on the surfce t r =, the border of the contct re, using eqution (9) this xiu is: () b r = = = () cr Fro eqution () it is possible to obtin the xiun tensite stress, nd is given by: bp r = = = (3) As both results in eqution () nd (3) ust be equl, then the expression of the contct circle rdius cn be deterined : 3 = crp (4) Eqution (4) shows when n elstic sphericl indenter of rdius R is pressed with lod P onto the surfce of sei-infinite elstic solid, circulr contct re increses in size with n incresing lod nd, in ddition, depends on Poisson rtio nd Young s odulus for the indenter nd the sei-infinite solid through constnt c given by eqution (0). Thus, the tensile stress field on the surfce of the specien cn be expressed in ters of the xiu tensile stress given by eqution (3). Then, tking into ccount eqution (9) nd () the tensile stress field is: r bp = 0 < r < ; 0 r = bp = r < ; 0 r (5) for within nd outside contct re of the specien, respectively. 6

7 THE CUMUATIVE PROBABIITY OF HERTZ INDENTATION FRACTURE In the cse of the Hertz indenttion test the crcks re initited on the surfce of the specien subjected to indenter, then the brittleness of the surfce ust be considered nd in the cuultive probbility of frcture the volue ust be chnged by surfce. After tht the Evns function given by eqution (3) is writen s follows: = ln = f r ds F S (6) 0 S where So is the surfce unity, S is the surfce subjected to stress field nd is the specific risk of Weibull surfce brittleness function. Defined function ethod for cuultive probbility When the specific Weibull risk function hs two preters s shown in eqution (4) nd tking into ccount the stress field given by eqution (5), then the Evns function dopts the following expression: r = rdrd rdrd S S 0 r = ; S0 0 (7) Eqution (7) llows for the deterintion of Weibull preters through the Weibull digr which is de by drwing ln () versus ln. Tking logrits to both sides of eqution (7) nd re-ordering: 3 ln = ln ln S0 0 (8) With series of experientl vlues to we cn find the best fit by ens, for exple, of the iniun squre ethod, obtining Weibull odulus s slope of the stright line, nd o cn be deterined through the free ter becuse the ters in squre brcket in eqution (8) re known, except o. Note tht () is known through experientl vlues. When the specific Weibull risk function hs three-preters s shown in eqution (5), in order to deterine the respective Evns s function it is neccessry to define geoetricl region where the integrtion ust be de. The restrictions on the stress field indicted in eqution (5) induce geoetricl restriction over the specien subjected to the stress field. Fro equtions (5) nd (5) there re two regions where the stress field is vlid. Within the contct re between the indenter sphere nd specien loded the stress field restriction is: 7

8 which iplies the following geoetricl restriction: r (9) r (0) Consequently, outside the contct re the respective stress restriction is: r () which iplies the following geoetricl restriction in this cse: r () Equtions (0) nd () show tht there re two regions where the crcks initited proote the Hertzin frcture, one within the contct re nd nother one outside the contct re of the specien subjected to the Hertz indenttion test. Not ll surfce contributes to Hertzin frcture. Then, with equtions (3), (5), (5), (0) nd () the Evns function is given by: r = rdr rdr S0 0 r Introducing the following chnges of vribles: (3) r = ; = r (4) Eqution (3) is finlly trnsfored into: = S 0 0 d (5) 8

9 Using eqution (5) the Weibull preters cn be obtined nd differents ethods hve been developed for the, for exple liner regression, xiu likelihood, iniun chi-squre. Kittl et.l. 30 hve discussed such ethods including nuericl siultions pproch. However there is n esier ethod clled noogrphic ethod which, prepring non-diensionl noogr llows us to deterine the Weibull preters. Such noogrphicl ethod hs been developed by Kittl nd eón 3. In generl, the eqution (5) cn be expressed s follows: S (6) S0 0 =,, is function given by: where, = d (7) Tking logrith to the ultiplictive fctor of function which hs the following expression:, nd denointed C S C = ln S 0 0 (8) Considering equtions (7) nd (8) nd tking logrith in both side of eqution (6) it is trnsfored into: ln = ln, C (9) Now, if we plot ln, ginst ln / for severl vlues of / nd we obtin non-diensionl digr resulting differents curves. Fro the experientl point of view the Evns functions () cn be known for set of experientl vlues. In tht cse the Weibull digr ln () ginst ln ust be de. The distributions of the experientl points is fitted to curve of the noogr by ens of superposition of the experientl digr, ln () ginst ln, nd this llows for preter to be deterined. Note tht this fit is equivlent to liner trsltion where the noogr is oved on Weibull digr. Eqution (9) corresponds to tht trsltion. Once tie preter is known the others preters cn be deterined by reding the coordintes of the origin of the noogr in respect to the Weibull digr. Then, the distnce between the xes ln () nd ln, llows for the deterintion of nd finlly the distnce C between the 9

10 xes ln nd ln/ llows for the deterintion of o in ccordnce with eqution (8) due to C being known directly fro the digrs, nd were previously deterined, SO is the unity of surfce nd S is the surfce subjected to stress field, both known too. This estition ethod ws used by Díz nd Morles 8 to deterine the Weibull preters in glss cylinders subjected to torsion. Integrl eqution ethod for cuultive probbility It ws sid tht eqution (3) cn be considered s n integrl eqution where the specific risk of Weibull's function is n unknown function to deterine. Then, ccording to the Evns function given by eqution (3) nd considering the stress field defined in eqution (5) for tensile stress within nd outside the contct surfce in Hertz indenttion test, the following eqution cn be writen: r = rdrd rdrd S0 S r 0 (30) where the first ter t the right side of eqution (30) tkes into ccount the crck initited whithin the contct re between the sphericl indenter nd the specien, nd the second ter considered the crck initited outside the contct re. Given the following chnges of vribles in eqution (9): r = ; = r After soe nipultion nd rerrnging eqution (30) is trnsfored into: S = d d (3) (3) Fro this eqution (3) it is esy to see tht the ethod to solve it is by siple differentition. Then, differentiting once nd rerrnging eqution (3) gives: S d = d 3 d (33) 0 0 Differentiting once gin nd rewriting we obtin the following differentil eqution: d S0 d d = = d 3 d d (34) Now, integrting eqution (34) it is possible to deterine the specific risk of Weibull s function, tht is to sy: 0

11 0 d = d (35) d Finlly, the solution to the integrl eqution given by eqution (30) is the following: S d d (36) 0 = d 3 d d 0 Note tht eqution (36) is finite integrl-differentil opertor pplied to Evns's function () which is known through experientl vlues. Eqution (36) is n exct solution to obtin the specific risk of Weibull s function for the cse of Hertzin indenttion test for specien with surfce brittleness subjected to Hertz contct. An pproxite solution cn be obtined by using the ethod of Tylor s series expnsion like eqution (8). In such sitution if we considered eqution (6) nd replce Tylor s serie expnsion to Evn s function, (), in eqution (36) the result is given by: n n n S (37) 3 n n! = 0 (n) 0 n = which requires the experientl vlues of Evns s function too nd its corresponding evlution of derivtes of () evluted in equl to zero. In eqution (37) the vlues of n ust be non equl to zero nd non equl to one due to its non convergence. The se solution given by eqution (37) cn be obtined fro integrl eqution (30) following the ethod explined in equtions (6), (7) nd (8). Obviously, if we replce the Evns function fro eqution (7) in eqution (36) we cn obtin the two preter Weibull s function s prticulr cse. If the Hertz indenttion test is pplied until the ppernce of the first frcture then stress nd point r in which it ppers cn be deterined, s well s the cuultive probbility of Hertzin frcture. The initition of the first crck cn be detected by sound, nd then through observtion with icroscope to deterine if the crck ws initited within or outside the contct surfce. However, if pplied lod P is fixed nd produces ore thn one crck, only locl frcture probbility cn be used, deterining the point r where the crck is initited nd seprting both cses when the crck is initited whithin the contct re or outside for brittle specien subjected to Hertz indenttion test. THE OCA PROBABIITY OF HERTZ INDENTATION FRACTURE The two pproches respect to wht forlis ust be used to deterine the locl probbility of frcture were discussed in the introduction of this pper. In ccordnce with tht we use the Kittl nd Cilo pproch 7 here. The se forlis ws used by Díz et.l. 3 to deterine both cuultive nd locl frcture probbilities of glss cylinders subjected to flexure.

12 The locl frcture probbility is given by: r dv n V,V V = = V r n V dv,v (38) where V is volue included in V where the crcks re initited, n(v ) is the nuber of frctures which re initited in volue V nd n(v) is the totl nuber of frctures. Eqution (38) yields the frction or percentge of frctures in the volue V of the specien t the stress. Due to the Hertz indenttion test the frctures occur t the surfce of the teril loded, then we ust use surfce S insted volue V. After tht, the locl probbility of Hertzin indenttion test is given by: r ds n S,S S = = S r n S ds,s (39) Here it is necessry to tret both the frctures initited within the contct surfce nd the frctures initited outside the contct surfce seprtely. Defined function ethod for locl probbility Eploying specific risk of Weibull s function the locl probbility of Hertzin frcture with two-preter function initited within the contct surfce, zone, between sphericl indenter nd specien loded nd considering the stress field in ccordnce with eqution (5), 0 r, is given by: n n r, r, r r = = r rdrd rdrd (40) nd evluting the integrls in eqution (40) we obtin: n r, r, r n r = = = n n (4) Note tht eqution (4) does not depend on both Weibull preter 0 nd stress.

13 In eqution (4) the left side is known through experientl vlues nd we cn ke Weibull digr. Tking logrith in both side of eqution (4) follows: n r r ln = ln (4) n Now, plotting ln(n(r)/n) ginst ln(r/) we cn estite Weibull preter becuse the slope of the Weibull digr for set of experientl vlues is equl to (+). When the frctures re initited outside the contct surfce, zone, nd considering the stress field in ccordnce with eqution (5), r, the locl probbility of Hertzin frcture is given by: n n r, r, r r 0 0 = = r 0 0 rdrd rdrd (43) evluting the integrls in eqution (43) it is trnsfored into: n r, r, n r = = = n r n (44) ike eqution (4) this eqution (44) is independent fro Weibull preter 0 nd stress. Rerrnging eqution (44) nd tking logrith in both side of eqution follows: n r ln = ln n r (45) Then, plotting ln n(r)/n ginst ln(/r) we cn estite the Weibull preter through deterining the slope of the Weibull digr for set of experientl vlues, which is equl to ( ). When we use specific risk of Weibull s three-preter function it is necessry to define geoetricl region where the frctures cn be initited. For the first zone, tht is to sy within the contct surfce for the Hertz indenttion test, the first zone continues to be defined by eqution (0). Then, considering eqution (5) nd (0) the locl probbility of Hertzin frctures is given by: 3

14 n n r, r, = = r 0 r rdrd 0 0 r rdrd 0 (46) Introducing in eqution (46) the vrible chnge = (r/)(/) given by eqution (4), then eqution (46) is trnsfored into: n n r, r, = = r d d (47) r r r r = =,, r where,,is non diensionl function. Note tht eqution (47) is independent fro Weibull preter 0. In order to deterine Weibull preters nd we cn plot (n (r,)/n) ginst (r/) where the left side of eqution (47) is known through experientl vlues. However it is necessry to use ore coplex ethod thn the Weibull twopreter function eployed fro eqution (4). It is posible to ke nuericl ethod siilr to tht developed fro eqution (6), in order to obtin non diensionl digr with different curves nd select tht better fittness with curve deterined for set of experientl dt which cn deterine the Weibull preters. For the second zone, zone, outside contct surfce of Hertz indenttion test nd using specific risk of Weibull three-preter function, s in the previous cse there is geoetricl region where the frcture occurs nd it is defined by eqution (). Then, considering eqution (5) nd eqution () the locl probbility of Hertzin frcture is given by: 4

15 n n r, r, = = 0 r 0 0 rdrd r r 0 rdrd (48) Introducing in eqution (48) the vrible chnge = (/r) (/) given by eqution (4), then eqution (48) is trnsfored into: r n r, r, = = n Note tht eqution (49) is independent fro Weibull preter o. Rewriting eqution (49) it is trnsfored s follows: r r d r = = n n r, r, r =,, d d d r where,,is non diensionl function. The se coents de bout eqution (47) re vlid to eqution (50). In order to deterine Weibull s preters nd we cn plot (n (r,)/n) ginst (r/) where the left side of eqution (50) is known through experientl vlues. However the greter coplexity of eqution (50) thn tht obtined for three-preter Weibull function in eqution (44) requires the use of nuericl ethods to deterine the Weibull preters. Consequently, it is posible to ke nuericl ethod siilr to tht developed fro eqution (6), in order to obtin non diensionl digr with different curves nd select tht better fittness with curve deterined for set of experientl dt. Using the defined function ethod with specific risk of two or three-preter Weibull function we need to estblish where the frctures re initited in order to deterine the locl frcture probbility for the Hertz indenttion test. The lod P pplied with sphericl indenter cn be vried for crrying out tests to obtin set of experientl vlues (49) (50) 5

16 for different lod conditions. At the se tie the size of the indenter cn be chnged in order to obtin both cuultive nd locl frcture probbilities. Integrl eqution ethod for locl probbility When in the locl probbility of frcture, given by eqution (39) the specific risk of Weibull s function is unknown, then this eqution ust be treted s n integrl eqution. Now we solve the integrl eqution for Hertz indenttion test considering both cses, when the frctures re initited within the contct surfce nd outside. Eqution (39) cn be writen s follows: n r, r ds = r ds n (5) S S where n(r,)/n is known through experientl vlues nd is n unknown function which ust be deterined. Note tht the frcture frction hs been expressed explicitly s function of stress given ore generl tretent. Then, such frcture frction depends on the position where the frcture initited nd the level of stress reched in tht position. When the frctures re initited within the contct surfce, previously referred to s zone, between sphericl indenter nd specien loded, nd considering the stress field in ccordnce with eqution (5) with 0 r, then the integrl eqution of the locl probbility of Hertzin frcture, tking into ccount eqution (5) is given by: r n r, r r rdrd = rdrd n (5) First, integrting eqution (5) respect to, then using the vrible chnge = (r/) given by eqution (3) nd differentiting prtilly respect to r nd finlly putting r =, fter soe nipultion the integrl eqution (5) yields: n r, d = r n (53) r = 0 In order to solve integrl eqution (53) we cn differentite prtilly respect to, hence this eqution becoes: r = 0 n r, n r, d = r n (54) r n Rendering the new following integrl eqution: r = 6

17 = d (55) 0 which obviously llows us to get () by differentition. Its solution is: = d d (56) Now, we ust obtin the expression of function () nd the initil proble of the solution of the integrl eqution of locl probbility of Hertzin indenttion test, for zone, within the contct surfce, cn be solved. Putting function () defined in eqution (55) in eqution (54) the following is obtined d d n r, d d r n = r = n r, d r n d r = (57) Eqution (57) cn be rerrnged s follows: n r, n r, = 0 r n r n r = r = (58) which is n ordinry differentil eqution of second order with vrible coefficients nd llows us to obtin function (). Then upon getting () fro eqution (58), it is introduced in eqution (56) to finlly obtin the solution to integrl eqution (5). Thus, this is the solution for zone where the frctures in the Hertzin indenttion test re initited within the contct re. Introducing eqution (4), which corresponds to the result of locl frcture probbility within the contct re when the specific risk of Weibull s functions hs two preters, into the differentil eqution (58) yields: = 0 (59) The differentil eqution (59) cn be solved esily by putting it s follows: = (60) integrting this eqution (60) nd using n ppropite constnt of integrtion we obtin: 7

18 = 0 nd integrting the solution once gin in this prticulr cse is given by: (6) = (6) 0 Then, replcing this eqution (6), which is the solution to differentil eqution (59), in eqution (56) it is obviously possible to obtin the specific risk of Weibull s two-preter function for locl frcture probbility when the specien is subjected to the Hertz indenttion test nd the frctures re initited within the contct re. When the frctures re initited in zone, tht it to sy outside the contct surfce between sphericl indenter nd specien loded, nd considering the stress field in ccordnce with eqution (5) with r, then the integrl eqution of the locl probbility of Hertzin indenttion test, tking into ccount eqution (5) is given by: r n r, rdrd = rdrd n r r 0 0 (63) Integrting this eqution (63) respect to nd with vrible chnge = (/r) given by eqution (3), integrl eqution (63) is trnsfored into: n r, d d = n (64) 0 r Differentiting prtilly integrl eqution (64) respect to r in order to solve it nd evluting r = this eqution yields: n r, d = r n (65) r = 0 If we differentited prtilly integrl eqution (65) respect to nd fter soe nipultion it becoes: n r, n r, (66) d = r n r n r = 0 r = et s consider the new following integrl eqution: 8

19 d (67) = which llows us to obtin () by siple differentition nd its solution is: 0 d = d (68) Anlogue coent de fter eqution (56) cn be de bout eqution (68). This eqution (68) is the solution to the integrl eqution (67) where we need to obtin the expression of function (). Thus, the integrl eqution of locl probbility of Hertzin indenttion test for zone, outside the contct surfce, cn be solved. Replcing the function () defined in eqution (67) in integrl eqution (66) fter soe nipultions gives: d d n r, d d r n = r = n r, d r n d r = (69) Finlly, eqution (69) cn be rerrnged nd it becoes: n r, n r, = 0 r n r n r = r = (70) which is, like eqution (58), n ordinry differentil eqution of second order with vrible coefficients nd llows us to obtin the function (). Then introducing () obtined fro eqution (70) in eqution (68) gives the solution of integrl eqution (64). After tht it is possible to know the specific risk of Weibull s function for zone where the frctures in the Hertz indenttion test re initited outside the contct re. Introducing eqution (44), which corresponds to the result of locl frcture probbility outside the contct re when the specific risk of Weibull s functions hs two preters, into differentil eqution (70) yields: = 0 Differentil eqution (7) cn be solved esily putting it s follows: = (7) (7) 9

20 integrting this eqution (7) nd using n ppropite constnt of integrtion we obtin: = 0 (73) nd integrting the solution once gin in this prticulr cse is given by: = (74) 0 Then, replcing this eqution (74), which is the solution of differentil eqution (7), in eqution (68) it is obviously possible to obtin the specific risk of Weibull s two-preter function for locl frcture probbility when the specien is subjected to the Hertz indenttion test nd the frctures re initited outside the contct re for non equl to one. Depending on where the crcks grow to initite the frctures, within or outside the contct re, the generl integrl equtions for locl probbilities of frcture given by equtions (5) nd (64), for the Hertz indenttion test, using finite differentil opertors pplied to respective functions were solved in both cses. These functions were obtined fro the solution of n ordinry differentil eqution of second order with vrible coefficients obtined fro equtions (58) nd (70). Being obtined, such functions were introduced in the finite differentil opertors given by equtions (56) nd (68), respectively. The finite differentil opertors re pplied to the percentges of frctures initited within or outside the contct re, respectively, n(r,)/n, which cn be obtined fro set of experientl dt. These functions follow the se role of the Evns function to solve the integrl eqution for cuultive frcture probbility. ACKNOWEDGEMENTS The uthors wish to thnks the Fcultd de Ciencis Físics y Mteátics de l Universidd de Chile for the support given to develop this work. REFERENCES.- HERTZ, H.: On the contct of elstic solids. Zeitschrift fur die Reine und Angewndte Mthetik, 9 (88) English trnsltion in Hertz s Miscellneous Ppers (trnslted by D.E. Jones nd G.A. Schott); Mcilln, ondon, UK, (986) 46-6, ch. 5 nd 6..- FRANK, F.C. nd AWN, B.R.: On the theory of Hertzin frcture. Proceedings of the Royl Society of ondon, A 9 (967) 9. 0

21 3.- GRIFFITH, A.A.: The phenoen of rupture nd flw in solids. Philosoficl Trnsction of Royl Society of ondon, A (90) HUBER, M.T.:Ann. Physik, 4 (904) AWN, B.R. nd WISHAW, T.R.: Indenttion frcture: Principles nd pplictions. Journl of Mterils Sciencie, 0 (975) AWN, B.R.: Indenttion of ceric with spheres: A century fter Hertz. Journl of Aericn Ceric Society, 8 (998) NADEAU, J.S.: Hertzin frcture of vitreous crbon. Journl of Aericn Ceric Society, 56 (973) WARREN, P.D.: Deterining the frcture toughness of brittle terils by Hertzin indenttion. Journl of Europen Ceric Society, 5 (995) ROBERTS, S.G., AWRENCE, C.W., BISRAT, Y., WARREN, P.D. nd HIS, D.A.: Deterintion of surfce residul stress in brittle terils by Hertzin indenttion. Theory nd experient. Journl of Aericn Ceric Society, 8 (999) MOUGINOT, R.: Crck fortion beneth sliding sphericl punches. Journl of Mterils Sciencie, (987) WEIBU, W.: A sttisticl strength of terils. Ingeniörs Vetenskps Akdeien Hndling, 5 (939) KITT, P. nd DIAZ, G.: Weibulls frcture sttistics, or probbilistic strength of terils: Stte of the rt. Res Mechnic, 4 (988) MATTHEWS, J.R., McCINTOCK, F.A. And SHACK, W.J.: Sttisticl deterintion of surfce flw density in brittle terils. Journl of the Aericn Ceric Society, 59 (976) EVANS, A.G. nd JONES, R..: Evlution for fundentl pproch for the sttisticl frcture. Journl of the Aericn Ceric Society, 6 (978) KITT, P.: Trnsfortion of flexurl stress Weibull s digr into trctionl one. Res Mechnic, (980) KITT, P. nd DIAZ, G.: On the integrl eqution of Weibull s frcture sttistics. Res Mechnic, 8 (986) KITT, P. nd DIAZ, G.: Integrl equtions in frcture sttistics of round be of brittle terils. Journl of Mterils Science etters, 3 (984) 9-3.

22 8.- DIAZ, G. nd MORAES, M.: Frcture sttistics of torsion in glss cylinders. Journl of Mterils Science, 3 (988) DIAZ, G. nd KITT, P.: Obtining the specific risk of torsionl frcture function of pristic br hving regulr polygon cross-section. Res Mechnic, 9 (990) MARTINEZ, V., KITT, P. nd DIAZ, G.: Frcture sttistics of torsion nd flexure in glss rectngulr brs. Journl of Mterils Sciencie, 7(99) OH, H.. nd FINNIE, I.: On the loction of frcture in brittle solids-i. Due to sttic loding. Interntionl Journl of Frcture Mechnics, 6 (970) FINNIE, I. nd VAIDYANATAN, S.: The initition nd propgtion of Hertzin ring crcks. In: Frcture Mechnics of Cerics, Vol.. Eds.: R.C. BRADT, D.P.H. HASSEMAN nd F.F. ANGE. Plenu Press, New York, 974, pp CONRAD, H., KESHAVAN, M.K. nd SARGENT, G.A.: Hertzin frcture of pyrex glss under qusi-sttic loding conditions. Journl of Mterils Science, 4 (979) KESHAVAN, M.K., SARGENT, G.A. nd CONRAD, H.: Sttisticl nlysis of the Hertzin frcture of pyrex glss using the Weibull distribution function. Journl of Mterils Science, 5 (980) RICHERSON, D.W., FINGER, D.G. nd WIMMER, J.M.: Anlyticl nd experientl evlution of bixil contct stress. In: Frcture Mechnics of Cerics, Vol. 5, Eds.: R.C. BRADT, A.G. EVANS, D.P.H. HASSEAN nd F.F. ANGE. Plenu Press, New York, 983, pp TRUSTRUM, K.: Reltion between position of cone crck nd criticl lod in the Hertz indenttion test. Journl of Mterils Science etters, 8 (989) KITT, P. nd CAMIO, G.M.: ocl probbility of filure in sttisticl theory of brittle frcture. Res Mechnic etters, (98) TRUSTRUM, K.: Estition of the Weibull odulus fro bending test using both frcture position nd the filure stress. Journl of Mterils Sciencie etters, 6 (987) DIAZ, G., MARTINEZ, V. nd KITT, P.: Contrdictory foruls of locl probbility of frcture. Applied Mechnics Review, 48 (995) S68-S KITT, P., ROSAES, M. nd DIAZ, G.: Nuericl ethods pplied to the probbilistic strength of terils. Cienci Abiert, 3 (00) -44 ( In Spnish.

23 3.- EON, M. nd KITT, P.: On the estition of Weibull s preters in brittle terils. Journl of Mterils Science, 0 (985) DIAZ, G., KITT, P., MARTINEZ, V. nd HENRIQUEZ, R.: Probbilistic strength of glss cylinders subjected to flexure: totl nd locl probbilities of frcture. Journl of Mterils Science, 37 (00)