An identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites

Transcription

1 IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216 IOP Conf. Ser.: Mater. Sc. Eng Related content - he Blnd Identfcaton of Mult-Inputs and Mult-Outputs Shallow-Water Acoustc Channel R Y L J H Zhou and L Wang - Comparatve analyss of dfferent weght matrces n subspace system dentfcaton for structural health montorng H Shokrav and H Bakhary - Pezoelectrc mplant method Monner Y Jayet P Guy et al. Vew the artcle onlne for updates and enhancements. hs content was downloaded from IP address on 16/6/218 at :22

2 MEACS215 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng 124 (216) 1271 do:1.188/ x/124/1/1271 An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes V Zubov S Lure and Y Solyaev Computng Centre of Russan Academy of Scences 4 Vavlov st. Moscow Russa E-mal: lureccas.ru Abstract. hs paper consders the dentfcaton algorthm of parameters ncluded n a parabolc law that s often used to predct the tme dependence of the thckness of the nterfacal layers n the structure of composte materals based on a metal matrx. he ncubaton perod of the process and the speed of reacton and pressure are taken nto account. he proposed algorthm of dentfcaton s based on the ntroducton of a mnmzed objectve functon of a specal knd. he problem of dentfcaton of unknown parameters n the parabolc law s formulated n a varatonal form. he authors of the paper have determned the desred parameters under whch the objectve functon has a mnmum value. It s shown that on the bass of four known expermental values of the nterfacal layer thckness correspondng to dfferent values of temperature pressure and the tme of the nterfacal layer growth t s possble to dentfed four model parameters. hey are the actvaton energy a preexponental parameter the delay tme of the start of the nterfacal layer formaton and the parameter determnng the pressure effect on the rate of nterfacal layer growth. he stablty of the proposed dentfcaton algorthm s also studed. 1. Introducton he nfluence of nterfacal zones on physcal and mechancal propertes of composte materals based on a metal matrx can be sgnfcant [1 2]. A parabolc law s often used to evaluate the nterfacal layer thckness at the boundary of fbers and matrx n the metal composte materals [3-9]. hs law establshes the relatonshp that allows determnng the varaton of the nterfacal layer thckness dependng on the exposure tme of the composte sample at elevated temperature and elevated pressure. In accordance wth the parabolc law thckness of the nterfacal layer ht () s proportonal to square root of the process tme parameter t and the proportonalty coeffcent s rate constant k whch n ts turn depends on the temperature and the pressure of the envronment. o estmate the rate constant Arrhenus relatonshp s often used. hus the relatonshp used for estmaton of the nterfacal layer thckness at the boundary of fbers and the matrx s as follows: QP h( t) Kexp t t. (1) 2R where P s pressure (Pa); Q s the energy of actvaton of the reacton zone growth process (J / mol); s process temperature ( K); R = s a unversal gas constant (J / mol / K); t s exposure tme (sec); t s an "ncubaton" perod (sec); h s the nterfacal layer thckness (m); K s a pre- Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3. lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work journal ctaton and DOI. Publshed under lcence by IOP Publshng Ltd 1

3 MEACS215 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng 124 (216) 1271 do:1.188/ x/124/1/1271 exponental parameter (m/s -1/2 ); s a parameter determnng the effect of pressure on the rate of the nterfacal layer growth (m 3 /mol). he am of ths work s to develop an algorthm of dentfcaton of parameters whch are ncluded n the parabolc law and characterze the process of the nterfacal layer growth around the fbers n the matrx of a metal composte materal. It s assumed that there are data ponts each of whch s a set of four values: P t H. he task s to determne unknown parameters of the model K Q t usng known data ponts. 2. An dentfcaton algorthm of the parabolc model parameters of the nterfacal layer growth P t H Let us suppose that there are expermental data that are a set of ponts: 1 P t H 2 P t H. he problem of determnaton of mathematcal model parameters s formulated as follows: fnd such model parameters for whch an expermental set of ponts would be as lttle dfferent as possble from the theoretcal set of ponts. Let us pre-convert the equaton (1) used to estmate the nterfacal layer thckness at the boundary of the fbers and the matrx to the equvalent form: 1 P 1 ln h( t) Q lnt t (2) 2 Q Q/(2R) /(2R) are new unknown parameters of the model. We also ntroduce the functons Q t ) ( 1... ) : where ln K f ( 1 P f ( Q t) lnh ln h( t) Q ln H t t (3) where s the number of expermental ponts and the value of functon h ( t ) s determned by (1) wth the use of expermental values P t. As the objectve functon we used 1 ( Q t. (4) 2 1 lnh lnh( t ) f ( Q t 2 ) ) K 1 K 1 he model parameters n ths approach are determned by mnmzng the functon ( Q t). In case of the mnmum of the objectve functon ts partal dervatves are vanshed. hs leads to a system of nonlnear algebrac equatons n unknown parameters Q t. o determne the coeffcents of ths equaton system we wll ntroduce the followng -dmensonal vectors: V V 2 1/ / V 3 P 1 / 1... P / V.5/( t t )....5/( t t ) 4 1 H / t t... lnh t t V5 ln 1 1 /. ow the system of equatons can be wrtten as: A 1 A 2 Q A 3 D ( ) (5) 2

4 MEACS215 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng 124 (216) 1271 do:1.188/ x/124/1/1271 where A j m1 ( V ) m ( V ) j m ( V V ) D ( V V5 ) ( j ). (6) j A of matrx A depend on the unknown All the elements of vector D and elements A 41 A quanttes that makes a system of algebrac equatons (5) nonlnear. At the same tme elements ( j ) of matrx A do not depend on unknown varables. Consderng the above-mentoned feature of the system (5) we propose the followng algorthm to determne the dentfcaton parameters: Fnd the mnmum value of the expermental observaton tme.e. t t * mn. 1 Select value t [ t* ]. Consder a subsystem of the system of algebrac equatons (5) wth selected value t [ t* ] that conssts of the frst three equatons. hs subsystem s a system of lnear algebrac equatons for Q. he soluton of ths system allows determnng these parameters (for any gven value of t [ t* ]). Choosng the value of t n segment [ t ] * we strve to satsfy the last equaton of the A j system too (5). In addton for every t parameters Q should be determned agan. Snce the soluton of the last equaton of the system (5) can not belong to segment [ t *] nstead of solvng the last equaton t s recommended to mnmze objectve functon ( t ) [ ( ) ( ) t Q t ( t) t] of parameter t on segment [ t ] *. Here we can use any smple algorthm for mnmzng a functon of one varable. o solve the problem of dentfcaton of the model parameters t s necessary to select the data ponts n the way that vectors V 1 V V 2 3 V are lnearly ndependent and the number of data ponts 4 s not less than 4. It should also be noted that the ncrease of the number of data ponts helps to ensure more relable values of the dentfable parameters of the model. 3. Results Wth the proposed algorthm we performed the calculatons of the model parameters for the expermental data gven n [5]. In [5] t was supposed that and t are equal to and the data were presented for the two materals: for composte SC/ 2 Alb and for composte SC/super α 2. hese compostes renforced wth undrectonal fbers of slcon carbde of the SCS-6 brand were made wth the use of two dfferent ntermetallc matrces desgnated as 2Alb and super α2. he nvestgaton of possble chemcal reactons occurrng under the composte manufacture durng the formaton of the nterfacal layer at the boundary of the fber and matrx was conducted n [5]. able 1 shows the results of determnaton of the model parameters for composte SC / 2Alb (nne expermental ponts) and able 2 shows the results for composte SC / super α 2 (sx expermental ponts). Fg. 1 and 2 show the dependence of the nterphase layer thckness on the exposure tme of the sample. Dfferent fgures correspond to dfferent temperatures of the process. he ponts represent the expermental data the dark lne corresponds to the results of [5] and the brght lne represents the results obtaned usng the developed algorthm. It can be seen that the model parameters obtaned by the developed algorthm correspond to the expermental data rather than the parameters specfed n [5]. 3

5 MEACS215 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng 124 (216) 1271 do:1.188/ x/124/1/1271 able 1. Identfcaton of model parameters (1) for composte SC / 2Alb ( C ) t (h) h (µm ) he results of the dentfcaton: work [5] developed algorthm K (m/s ) Q (J/mol) able 2. Identfcaton of model parameters (1) for composte SC / super α 2 ( C ) t (h) h (µm ) he results of the dentfcaton: work [5] developed algorthm K (m/s ) Q (J/mol) a b Fgure 1. me dependence of the nterfacal layer thckness for the 2Alb composte a: 7 о С b: 8 о С. a b Fgure 2. me dependence of the nterfacal layer thckness for the super α2 composte a: 7 о С b: 8 о С. 4. Conclusons he algorthm whch allows dentfyng the knetc parameters of the parabolc law has been proposed. In order to obtan successful results of the estmatons one should use the expermental data consstng of not less than 4 ponts. More data ponts wll provde better results obtaned durng dentfcaton. 4

6 MEACS215 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng 124 (216) 1271 do:1.188/ x/124/1/1271 he expermental ponts should be selected n such a way that the ntroduced system of -dmensonal vectors would be lnearly ndependent. he approprate descrpton of the expermental data for the ntermettalc ttanum-based compostes has been shown. Acknowledgments he study has been performed wth the support of the grant from the Russan Scence Foundaton (project ). References [1] Clyne W and Wnters P J 1992 An Introducton to Metal Matrx Compostes (Cambrdge: Cambrdge Unversty Press) [2] alwa H 21 Handbook of Surfaces and Interfaces of Materals (Y: Academc Press) [3] Lu X et al 26 rans. onfer. Met. Soc. Ch [4] Yang Y Q et al 1998 Comp. Part A [5] Zhu Y et al 28 rans. onfer. Met. Soc. Ch [6] Guo Z X and Derby B 1994 Scrp. Metal. Mater [7] Martneau A et al 1984 J Mat. Sc [8] McLeod A D and Gabryel C M 1992 Metallurgcal ransactons A [9] Knyazeva A et al 214 AIP Conf. Proc