Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Identfcaton of the Summarzed Webull model One tackles here the problem of the dentfcaton of the parameters of the Webull model on a sample of tests representatve of behavor wth fracture of a brttle materal (typcally, ferrtc steel of low temperature) The method of regresson lnear and the method of the maxmum of probablty are the two adopted methods One detals of t the prncple as well as the assocated methods of resoluton, beng based n both cases on an teratve process Lastly, one shows ther extenson f one of the two parameters of ths model (the stress of cleavage) depends on the temperature Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 2/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Contents Introducton 3 2 Recalls 32 The model of WEIBULL 322 Identfcaton of the parameters 3 3 Method of the lnear regresson 43 Prncple 432 Resoluton 5 4 Method of the maxmum of probablty 64 Prncple 642 Resoluton 6 5 Dependence of the parameters wth lnear temperature 75 Regresson 752 Maxmum of probablty 8 6 Concluson 8 7 Bblography 8 Descrpton of the versons of the document 8 Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 3/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Introducton 2Rappels When they call on the Webull model (cf POST_ELEM [U4822]), the study of modelzaton of the brttle fracture of steels n general requre a prelmnary dentfcaton of the parameters of ths model In order to avod a hard dentfcaton wth the hand of these parameters whch would requre to start agan repeatedly operaton POST_ELEM wth opton WEIBULL, an automatc procedure of retmng was establshed n Code_Aster In ths document, one brefly ponts out the equatons of the Webull model then one defnes the problem of dentfcaton posed One then descrbes the prncple of the two methods of resoluton adopted (lnear and maxmum regresson of probablty) by ncludng the case where one of the two parameters of the model depends on the temperature 2Le Webull model One consders a structure of behavor elastoplastc subjected to a thermomechancal request It s supposed that the m] probablty of cumulated fracture of ths structure follows the model of WEIBULL [bb] to two parameters followng: P f w = exp[ w éq 2- u statement n whch the modulus of WEIBULL m descrbes the tal of the statstcal dstrbuton of the szes of the s at the orgn of cleavage, u s the stress of cleavage and w s the stress of WEIBULL whch depends on the prncpal hstory of the stress feld n the plastczed zone of structure For example, n the case of a way of monotonc loadng, t s wrtten: w = m p the summaton relates to plastczed volumes of V p matter, I p Ip m V p V 0 éq 2-2 stress n each one of these volumes ( V 0 s a volume characterstc of the materal) 22Identfcaton of the parameters ndcatng the maxmum prncpal In a very general way, one consders an expermental base made up of tests of varous natures (type,2,, N), each type of test beng carred out n j tme so that the nombre total of tests rses wth: j=n N= n j j = Ths expermental base could for example be made up of tests on axsymmetrc test-tubes notched of radus of notch dfferent led to varous temperatures Takng nto account the random nature of the propertes wth fracture of the materal consdered, ths base consttutes only one sample The more mportant the number of these samples wll be, the more t wll be representatve of the behavor of the materal consdered Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 4/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Among the varous methods of dentfcaton suggested n the lterature (see for example [bb2]), we retan two of them: method of regresson lnear, often used, lke that of the maxmum of probablty recommended by the Structural European Integrty Socety (ESIS) [bb3] Note: A comparatve systematc study of the results gven by these two methods [bb2] accordng to the number of sample taken by chance on a theoretcal dstrbuton showed that the method of the maxmum of probablty led to a better estmate of the parameters of the Webull model The method of regresson lnear remanng nevertheless very much used, we ntegrated t nto our developments In the two adopted methods of retmng, one carres out the frst computaton of the stresses of WEIBULL wth a clearance of parameter gven (typcally m=20, s u =3000 MPa ) One classfes these N tests usng ther stress of WEIBULL reached at the nstant of the falure One thus has an ncreasng lst of stresses of WEIBULL w,, w,, N w, such as for each, the number of testtubes broken wth a lower stress of WEIBULL or equalzes wth w s n w Among the varous possble estmators of the probablty of cumulated fracture P f w [bb2], we choose that generally recommended: P f = N Note: (n general n w = ) correspondng wth In the cas partculer where the stress of WEIBULL depends on the temperature, the precedng rankng must be made temperature by temperature, each temperature correspondng to a dfferent statstcal model The estmator of the probablty of fracture precedent thus becomes: P f = N T temperature T, for whch there were tests N T, f the test-tube were broken wth the The two adopted methods of retmng are vald as long as [éq 2-] remans true If the dentfcaton s carred out on test results ansothermals whereas the stress of cleavage s supposed to depend on the temperature, ths condton s not checked any more (cf POST_ELEM [U4822]) In ths case, typcal case one wll not be able to thus apply the developments whch follow 3Méthode of the lnear regresson 3Prncpe the varaton theory-experment s measured by the statement: 2 LogLog P f W éq 3- LogLog P f ( Log ndcates the Naperan logarthm) One wants to mnmze ths varaton compared to (m, u ) Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 5/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 32Résoluton the method of retmng usually used leans on successve lnear regressons: wth the teraton k, the values ( m k, u k ) of the modulus and stress of cleavage are known It s thus possble, wth these values, to calculate the stresses of WEIBULL W k at varous tmes of fracture thanks to [éq 2-] One then classfes these new stresses of WEIBULL by ncreasng ampltude and one from of deduced the new estmates from the probablty of fracture P f k to the teraton k For these values of stresses of WEIBULL fxed, the mnmzaton of [éq 3-] s brought back to a smple lnear regresson on the group of dots ( Log W k, LogLog P f k ) snce f one defers LogLog P accordng to Log W, one obtans a lne of slope m whch cuts the x-axs n ( f Log u ) The new values ( m k, u k ) of these parameters are thus gven by (cancellaton of dervatves partal of [éq 3-] compared to each parameter): X N k Y j k Y k X k, j m k = éq 32- X N k X j k X 2 k, j u k =exp N X k m wth X k =Log W k Y k and Y k =LogLog, éq 32-2 P f k These teratons are repeated as long as the dfference between the clearances of parameter obtaned wth the teratons (K) and (k+) s sgnfcant (typcally, fve teratons) The measurement of ths varaton s gven by: Max[ m m k k m k, ] uk u k uk Note: If m s fxed, u k s always gven by [éq 32-2] On the other hand, f u s fxed, X k Y k m k s not gven any more by [eq 32-] but: m k = X k 2 log u X k Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 6/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 4Méthode of the maxmum of probablty 4Prncpe Let us note p f w the densty of probablty assocated wth the probablty of cumulated fracture P f w : p f w = m s m w u u exp[ w u The quantty p f w d W s equal to the probablty of breakng a test-tube subjected to a request correspondng to a stress of WEIBULL understood n the nterval [ W, W d W ] The probablty so that all the test-tubes of the base broke thus rases wth: p f W d w, éq 4- pm, u d w = p beng the functon of probablty The method of the maxmum of probablty then conssts n choosng the parameters of the Webull model so that the functon of probablty defned by [éq 4-] (n practce rather ts Naperan logarthm) that s to say maxmum m] 42Résoluton m k = N k = m k One uses an teratve process agan There stll, wth the teraton k, ( m k, u k ) as are W k known for them For these values of stresses of WEIBULL fxed, the maxmzaton of Log p condut to a new couple ( m k, u k ) gven by: =N m k = N = = N Log W k N f = N = W k =N = m k Log W k W k m k W k m k éq 42-2 =0 éq 42- A each steps, the resoluton of [éq 42-] can be realzed usng the method of Newton, the gradent of f m beng gven by: df m =- N dm m 2 =N W = m Log 2 W =N W = = N W = = N m W = m2 m Log W 2 Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 7/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Note: If m s fxed, u k s gven by [42-2] On the other hand, f u s fxed, m k s not any more soluton of [42-] but of: f m k = N = N Log W k W k m k =0 m k = u u Ths equaton can be agan solved usng the method of Newton, the gradent beng now gven by: = N df dm m =- N m 2 = W m Log 2 W u u 5Dépendance of the parameters wth the temperature If one wshes to fx ndependently the two parameters temperature by temperature, t s enough to break up the base of tests nto as much of under - bases by temperature and to apply to each one of these subbases the precedng methods If, on the other hand, one only wshes to vary the stress of cleavage u wth the temperature, one proceeds n the followng way lnear 5Régresson the estmate of the probabltes of fracture beng now carred out temperature by temperature (cf notces [ 22]), t s enough to fx the stress of cleavage on each group of dots assocated wth the varous temperatures (T) The equaton [éq 32-2] thus becomes: u k =exp N T T X k m T Y k ( N T ndcatng the number of tests for the subbase correspondng to the temperature (T)), the modulus of WEIBULL beng gven by: m k = T N T T, j T T N T X k Y j k X k X j k T, j T Y k X k X 2 k Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 8/8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 52Maxmum of probablty the stress of cleavage s gven for each temperature (T) consdered by: m k beng soluton of: f m k = N m k u k T = m k = N = N T T Log W k T W k T m k, N T T W k T m k Log W k W k m k =0 6Concluson command RECA_WEIBULL of the Code_Aster makes t possble to carry out the chock of the parameters of the Webull model [U48206] The user gves as starter ths command the concepts results assocated wth varous nonlnear computatons carred out The possble dependence of the stress of cleavage wth the temperature s mplctly specfed when dfferent temperatures are assocated wth each one of these concepts results (f all these temperatures are dentcal or f they are not specfed, t does not have there dependence wth the temperature of ths parameter) The user can carry out ths retmng by the method of the maxmum of probablty (METHODE: MAXI_VRAI ) or that of the lnear regresson (METHODE: REGR_LIN ) Quanttes determned by the command RECA_WEIBULL are deferred n an array n whch one fnds the value of the dentfed parameters, probabltes of fracture estmated startng from the expermental results as well as the probabltes of theoretcal fracture calculated wth the dentfed parameters 7Bblography [] F BEREMIN, A local crteron for cleavage fracture of has nuclear presses vessel steel, Metall Trans 4A, pp 2277-2287, 98 [2] A KHALILI, K KROMP, Statstcal propertes of webull estmators, Newspaper of Materal Scence, 26, pp 674-6752, 99 [3] ESIS, TC one Local Approach, Procedure to local measure and calculate approach crtera usng notched tensle specmens, P6, 998 Descrpton of the versons of the document Verson Aster Author (S) Organzaton (S) 05/0/00 R MASSON, W LEFEVRE (EDF/RNE/MTC) Descrpton of the modfcatons ntal Text Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Lcensed under the terms of the GNU FDL (http://wwwgnuorg/copyleft/fdlhtml)