Code_Aster. Identification of the model of Weibull

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1 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : PARROT Aurore Clé : R70209 Révson : Identfcaton of the model of Webull Summary One tackles here the problem of the dentfcaton of the parameters of the model of WEIBULL on a sample of tests representatve of behavour wth rupture of a fragle 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 constrant 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

2 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 2/8 Responsable : PARROT Aurore Clé : R70209 Révson : Contents Introducton 3 2 Recalls 3 2 The model of WEIBULL 3 22 Identfcaton of the parameters 3 3 Method of the lnear regresson 4 3 Prncple 4 32 Resoluton 5 4 Method of the maxmum of probablty 6 4 Prncple 6 42 Resoluton 6 5 Dependence of the parameters wth the temperature 7 5 Lnear regresson 7 52 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

3 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 3/8 Responsable : PARROT Aurore Clé : R70209 Révson : Introducton When they call on the model of WEIBULL (cf POST_ELEM [U4822]), the study of modelng 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 the operaton repeatedly POST_ELEM wth the opton WEIBULL, an automatc procedure of retmng was establshed n Code_Aster In ths document, one brefly ponts out the equatons of the model of WEIBULL 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 2 Recalls 2 The model of WEIBULL One consders a structure of behavor elastoplastc subjected to a thermomechancal request It s supposed that the m] probablty of cumulated rupture of ths structure follows the law of WEIBULL [bb] wth two parameters followng: P f w = exp[ w éq 2- u expresson n whch the module of WEIBULL m descrbed the tal of the statstcal dstrbuton of the szes of the defects at the orgn of cleavage, u s the constrant of cleavage and w s the constrant of WEIBULL whch depends on the hstory of the prncpal stress feld n the plastczed zone of the structure For example, n the case of a monotonous way of loadng, t s wrtten: w = m p The summaton relates to volumes of matter V p plastczed, I p Ip m V p V 0 éq 2-2 constrant n each one of these volumes ( V 0 s a volume characterstc of materal) 22 Identfcaton 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 full number of tests rses wth: j=n N= n j j = Ths expermental base could for example be made up of tests on notched axsymmetrc test-tubes of dfferent rays of notch led to varous temperatures Takng nto account the random nature of the propertes wth rupture of 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 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

4 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 4/8 Responsable : PARROT Aurore Clé : R70209 Révson : 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] Notce : 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 model of WEIBULL The method of regresson lnear remanng nevertheless very much used, we ntegrated t nto our developments In the two adopted methods of retmng, one typcally carres out the frst calculaton of the constrants of WEIBULL wth a game of parameter gven (, m=20, s u =3000 MPa ) One classfes these NR tests usng ther constrant of WEIBULL reached at the nstant of the falure One thus has an ncreasng lst of constrants of WEIBULL w,, w,, N w, such as for each, the number of test-tubes broken wth a constrant of lower WEIBULL or equalzes wth w s n w ) Among the varous possble estmators of the probablty of cumulated rupture P f wth w [bb2], we choose that generally recommended: P f = N Notce : (n general n w = correspondent In the typcal case where the constrant of WEIBULL depends on the temperature, the precedng rankng must be made temperature by temperature, each temperature correspondng to a dfferent statstcal law The estmator of the probablty of rupture precedent thus becomes: P f = N T temperature T, for whch there was N T tests, f the test-tube was broken at 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 ansothermes whereas the constrant 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 3 Method of the lnear regresson 3 Prncple The varaton theory-experment s measured by the expresson: 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

5 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 5/8 Responsable : PARROT Aurore Clé : R70209 Révson : 32 Resoluton The method of retmng usually used s based on successve lnear regressons: wth the teraton k, values ( m k, u k ) module and constrant of cleavage are known It s thus possble, wth these values, to calculate the constrants of WEIBULL W k at the varous moments of rupture thanks to [éq 2-] One then classfes these new constrants of WEIBULL by ncreasng ampltude and one from of deduced the new estmates from the probablty of rupture P f k wth the teraton k For these values of constrants of fxed WEIBULL, 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 f accordng to Log W, one obtans a lne of slope m whch cuts the x-axs n ( Log u ) New values ( m k, u k ) of these parameters are thus gven by (cancellaton of the dervatve 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 games 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 Notce : 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

6 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 6/8 Responsable : PARROT Aurore Clé : R70209 Révson : 4 Method of the maxmum of probablty 4 Prncple Let us note p f w densty of probablty assocated wth the probablty of cumulated rupture P f w : p f w = m s m w u u exp[ w u Quantty p f w d W s equal to the probablty of breakng a test-tube subjected to a request correspondng to a constrant 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 model of WEIBULL so that the functon of probablty defned by [éq 4-] (n practce rather ts Naperan logarthm) that s to say maxmum m] 42 Resoluton An teratve process agan s used There stll, wth the teraton W k m k = N k = m k k, ( m k, u k ) as well as are known For these values of constrants of fxed WEIBULL, the maxmzaton of Log p condut wth 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- Wth each step, 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

7 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 7/8 Responsable : PARROT Aurore Clé : R70209 Révson : Notce : 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 5 Dependence 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 constrant of cleavage u wth the temperature, one proceeds n the followng way 5 Lnear regresson The estmate of the probabltes of rupture beng now carred out temperature by temperature (cf notces [ 22]), t s enough to fx the constrant 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 module 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

8 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 8/8 Responsable : PARROT Aurore Clé : R70209 Révson : 52 Maxmum of probablty The constrant 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 6 Concluson The order RECA_WEIBULL Code_Aster allows to carry out the chock of the parameters of the model of WEIBULL [U48206] The user gves as starter ths order the concepts results assocated wth varous nonlnear calculatons carred out The possble dependence of the constrant 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 ( METHOD: MAXI_VRAI ) or that of the lnear regresson (METHOD: REGR_LIN ) Szes determned by the order RECA_WEIBULL are deferred n a table n whch one fnds the value of the dentfed parameters, probabltes of rupture estmated startng from the expermental results as well as the probabltes of theoretcal rupture calculated wth the dentfed parameters 7 Bblography [] F BEREMIN, wth room crteron for cleavage fracture of has nuclear presses vessel steel, Metall Trans 4A, pp , 98 [2] A KHALILI, K KROMP, Statstcal propertes of webull estmators, Newspaper of Materal Scence, 26, pp , 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) 0/05/00 R MASSON, W LEFEVRE (EDF/RNE/MTC) Descrpton of the modfcatons Intal 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 Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (

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