METHOD OF THE EQUIVALENT BOUNDARY CONDITIONS IN THE UNSTEADY PROBLEM FOR ELASTIC DIFFUSION LAYER

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1 Maerials Physics ad Mechaics 3 (5) 36-4 Received: March 7 5 METHOD OF THE EQUIVAENT BOUNDARY CONDITIONS IN THE UNSTEADY PROBEM FOR EASTIC DIFFUSION AYER A.V. Zemsov * D.V. Tarlaovsiy Moscow Aviaio Isiue (Naioal Research Uiversiy) Voloolamsoye shosse 4 А-8 GSP-3 Moscow 5993 Russia Isiue of Mechaics (omoosov Moscow Sae Uiversiy) Michurisy prospe Mosow 99 Russia * azemsov975@mail.ru Absrac. The proposed approach o solvig iiial-boudary problems is based o iegral relaios which coec righ sides of boudary codiios of differe ypes. Oe of hese soluios is assumed o be foud. I his case iegral relaios are cosidered as equaios as regard o righ sides which are equivale o oher codiios. Quadraure formulas are used o solve hese equaios.. Iroducio Solvig may of useady problems i coiuum mechaics icludig elasic diffusio problems is associaed wih serious mahemaical challeges. These are due o he eed of aplace rasform coversio used o solve problems of his ype. Depedig o cerai ypes of boudary codiios soluio for hese problems may be produced usig he Fourier rigoomeric series which sigificaly simplifies he origials fidig algorihm []. The disadvaage of his mehod is he resriced applicaio area which is due o he specifics of boudary codiios. I order o overcome his disadvaage i is proposed o build relaios amog boudary codiios of differe ypes. I his case i will be sufficie o solve a sigle (bechmar) problem while all oher problems will be reduced o i usig he relaios give below. The proposed mehod is demosraed hrough he example of a oe-dimesioal useady problem for elasic diffusio layer.. Algorihm geeral descripio e here be give a boudary-value problem which will be hereiafer referred o as origial problem: u u u x x a b () M u f M u f. () xa xb M ad M are liear differeial operaors ad is ime. Alog wih i a bechmar boudary-value problem is beig cosidered i which he required fucio saisfies equaio () ad boudary codiios N u g N u g (3) xa xb 5 Isiue of Problems of Mechaical Egieerig

2 Mehod of he equivale boudary codiios i he useady problem for elasic diffusio layer N ad N are liear differeial operaors. Problem soluio () (3) is preseed as follows: 37 u G x g d G x g d. (4) Here G ad G are he fucios saisfyig equaio () ad accordigly he followig boudary codiios ( - Dirac dela fucio): N G N G N G N G. xa xb xa xb Assumig ha fuciosg ad G are ow ad claimig ha g ad g saisfy he codiios () we arrive a a sysem of iegral equaios i regard o g ad g : M G x g d M G x g d f x a; x b. Subsiuio of soluio o his sysem i (4) leads o he soluio of he origial problem. u u x Remar. The proposed algorihm may be geeralized o vecor fucios u u x u x m x x x correspode marix differeial operaors M N ad righ sides of boudary codiios i he form of vecor fucios g. 3. Seig up he firs boudary-value problem of elasic diffusio e here be a homogeeous layer bouded by surfaces x3 ad x3 ( Ox xx 3 - he Caresia coordiae sysem). Oe-dimesioal physical ad mechaical processes i he medium are described usig elasic bouded diffusio model [ ]: u u D u ; (5) f u f f u f ; (6) u x x x x u (7) he prime ad he dos mea he spaial derivaive x ad ime derivaives accordigly. From ow o he followig dimesioless values i (5) (7) are used ( he same racig is used hey are mared wih a aseris which is omied i oher equaios): x3 u3 c * C D33g x u c D C c D33 * * 3333 f f f f c RT - ime; u3 - raslaio alog axis Ox 3; T - iiial emperaure; - coceraio icreme; ad - iiial ad curre coceraio of subsace; C ijl - elasic cosas esor compoes; - medium desiy; ij - coefficies deermied by

3 38 A.V. Zemsov D.V. Tarlaovsiy he ype of he crysal laice (he relaive volume chage due o diffusio); R - uiversal gas cosa; D - self-diffusio coefficies esor compoes. ij 4. Trasiio o equivale boudary codiios I order o solve problem (5) - (7) a auxiliary (bechmar) problem is beig cosidered which is defied by equaios (5) iiial codiios (7) ad boudary codiios (righ sides of he secod ad hird equaios coicide here ad i (6)): u D f u f u D f u f (8) x x x x Is soluio is foud i wor [] ad is give i he iegral form by (he aseris meas ime covoluio): l l l l l l l l (9) u G f G f G f G f l l Here G l Gl u - he Gree's fucio i problem (5) (7) (8) i.e. soluios of four problems ( l - heir umbers) which iclude equaios (5) iiial codiios (7) ad he followig boudary codiios: G DG G l l x l l x l G DG G l l x l l x l - Dirac dela fucio; i - Kroecer symbol i. Assumig ow ha he soluio for he bechmar problem saisfies he equaliies f f ad cosiderig ha f f f f we arrive a he Volerra equaios of covoluio ype i regard o he fucios f : G f G f G f G f f G f G f f G f G f. The Gree's fucio symmery specified i [] is ae io accou here. f ad The mai difficuly whe solvig sysem () is ha he fucios G ad G have sigulariy if. I order o research he aure of he sigulariies we proceed as G x is give by []: follows. I he aplace rasform domai he Gree s fucio G x s G s cos x s G P s s s D. () s 4 P s ()

4 Mehod of he equivale boudary codiios i he useady problem for elasic diffusio layer The followig rasformaio is made i formula () for fucio G : 39 G Q s s D P s 4 s D Q s Origial fuciog x s will have he he followig represeaio: x D G x 3 e G cos x 3 x q q cos x cos si 3 s 3 D 4. () G e A A A e A e ; s s are complex ad s 3 is he real roo of polyomial P s ; Re s ; Im s ; s s 3 s ; xq - Jacobi hea-fucio [3]; coefficies Aq q 4 are icluded i he followig formulas ( he prime meas he derivaive s ): A Re A Im A 3 A 4. Q s Q s Q s Q D 3 I wor [] approximae values of he polyomial roo were geeraed P s. Based G has order O o hese equaliies a coclusio may be made ha commo erm ad correspode series i () which coverge absoluely is bouded if while fucio x if 3. Fucio G G i he eighbourhood of has order iegrable sigulariy [3]. I his case i accordace wih [4] each equaio () is muliplied by d ad iegraed bewee ad. Thus we ge: K f d K f d K f d K f d K. G d G d d K (4) while erels K do o have sigulariies i zero ad. Whe solvig equaios (3) quadraure formulas are good o be used. Rage T of ime variable is spli io N iervals by dos i ih wih eve pich h T N ad mesh fucios y f K K are iroduced. i i i i Wih i each of he iegrals i (4) is approximaely subsiued wih he sum correspodig o he midpoi quadraure rule: i i l l l l l i i i i i j j j K f d hs hk y S K y l. (3)

5 4 A.V. Zemsov D.V. Tarlaovsiy he odes are icluded i he followig formulas: i i i hi i j i j hi j i N As a resul we ge a recurre sequece of simulaeous liear algebraic equaios ( i ): Ay b (5) i i K K y i b i A yi bi K K y i b i b h S S b h S S i. i i i i i i i y The soluio for equaio (5) aes he followig form: K b K b K b K b. i i i i i y i K K K K By usig his soluio i (9) we ge soluio for iiial problem (5) - (7). 5. Example As a example problem (5) - (7) is beig cosidered wih boudary codiios i he followig form: f H u u x x x x H - Heaviside sep fucio. The medium cosidered is alumiium havig he followig characerisics: N g m C T К D m. m m с Figure shows umerical calculaios for displacemes i disic pois of he layer from formulas (9) he umber of divisio pois is N ad he umber of erms i a Fourier series is M. Fig.. Time-displaceme depedece: x.5 - solid graph x. - doed lie x.8 - dashed lie. I should be oed ha if he subierval is half reduced while he umber of erms i a Fourier series is doubled he diagrams coicide.

6 Mehod of he equivale boudary codiios i he useady problem for elasic diffusio layer 4 Acowledgemes Fiacial suppor for his wor has bee provided by Russia Foudaio for Basic Research (projec 4-8-6) ad he Russia Federaio Preside gra NSh Refereces [] A.V. Zemsov D.V. Tarlaovsiy // Joural of Mahemaical Scieces 3() (4). [] D.V. Tarlaovsii V.A. Vesya A.V. Zemsov I: Ecyclopedia of hermal sresses (4) Vol. p. 64. [3] A.M. Zhuravsi Ellipic Fucio Referece Boo (USSR Academy of Scieces Мoscow 94) (i Russia). [4] A.D. Polyai A.V. Maghirov Iegral Equaio Referece Boo: Exac Soluios (Facorial Мoscow 998) (i Russia).