APPROXIMATE ANALYTIC SOLUTIONS OF A NONLINEAR ELASTIC WAVE EQUATIONS WITH THE ANHARMONIC CORRECTION

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 6 Number /5 pp 8 86 APPROXIMATE ANALYTIC SOLUTIONS OF A NONLINEAR ELASTIC WAVE EQUATIONS WITH THE ANHARMONIC CORRECTION V MARINCA R-D ENE B MARINCA Polehnca Unversy o Tmşoara Deparmen o Mechancs and Vbraon Tmşoara Romana Romana Academyener or Advanced and Fundamenal Techncal Research Deparmen o Elecromechancs and Vbraon Tmşoara Romana Polehnca Unversy o Tmşoara Deparmen o Mahemacs Tmşoara 6 Romana Polehnca Unversy o Tmşoara Deparmen o Elecroncs and Telecommuncaons Tmşoara Romana E-mal: remusene@upro In hs arcle we consder he propagaon equaon o he longudnal elasc wave n he presence o he volume orces akng no accoun he shear phenomena In order o nd an approxmae analyc soluons o he governng sysems we apply Opmal Varaonal Mehod (OVM) Ths approach nvolves he presence o some nally unknown convergence-conrol parameers whch are opmally deermned An excellen agreemen s ound beween approxmae and numercal soluons Key words: opmal varaonal mehod nonlnear paral derenal equaons hn elasc plae INTRODUCTION In spe o he grea deal o work o nonlnear phenomena he wave equaon wh he volume orces correcon has sll receved lle aenon n he connuum lm Bu he mahemacal modelng o physcal processes leads o nonlnear paral derenal equaons (PDE) whose analyc soluons are hard o nd A powerul general echnque or analyzng nonlnear PDEs s gven by he classcal Le symmery mehod [] [] Recenly n 9 Musaa and Massoud [] apply he Le symmery mehod o oban several exac soluons n an explc orm analyzng a nonlnear elasc wave equaon or longudnal deormaons wh hrd-order anharmonc correcons o he elasc energy Approxmae symmeres and prolongaon echnque were used o carry ou symmery analyss o some neresng cases o such nonlnear wave equaons by Alno e al n [4] The equaon o moon or longudnal deormaon wh he hrd-order anharmonc erms has also been analyzed n [5] by usng quadraures and asympoc seres where was concluded ha anharmonc correcons o he elasc energy are lkely o lead o soluons nvolvng medependen sngulares a ne mes A smlary analyss o a nonlnear wave equaon n elascy was suded n [6] where were obaned he exac soluon va he mehod o nvarans akng no accoun he hrd-order anharmonc correcons o he elasc energy In general such problem are no amenable o exac reamen and approxmae echnques mus be resored o Among hese he perurbaon mehods are n common use The perurbaon mehods are n prncple nended o solve problems nvolvng a small parameer [7] However n scence and engneerng here exs many nonlnear problems n whch parameers are no small The applcaon o he perurbaon mehods have been exended o oscllaons wh srong nonlneares Some exensons o he Lndsed- Poncare perurbaon mehod or he ncremenal harmonc balance mehod have been proposed by Cheung e al [8] or srongly nonlnear sysems An approach whch combnes he harmonc balance mehod and lnearzaon o nonlnear oscllaons equaon was presened n [9] There also exs a wde range o leraure dealng wh approxmae deermnaon o perodc soluons or nonlnear problems by usng a mxure o mehodologes: he opmal homoopy asympoc mehod [] an equvalen lnearzaon mehod [] he lnearzed Galerkn and arcal parameer echnques [] In 999 he varaonal eraon mehod was proposed by J H He [4] A varaonal prncple or he nonlnear oscllaons by consrucng he Hamlonan was suded n [5]

2 Approxmae analyc soluons o a nonlnear elasc wave equaons 8 In he presen work we propose a novel varaonal approach o nonlnear paral derenal equaon Our procedure nvolves an orgnal consrucon o Lagrangan whch s used o consruc approxmaons o solve nonlnear problems wh parcular emphass on elasc wave equaon wh he volume orces correcon The consrucon o an Lagrangan or dynamcal sysems wh more general Newonan orces are nowadays applcable only o sysems wh orce dervable rom a poenal uncon (namely conservave sysems) MATHEMATICAL MODEL OF NONLINEAR ELASTIC WAVE EQUATIONS WITH THE VOLUME FORCES CORRECTION The erms o order hgher han second n he sran ensor mus be consdered n he elasc energy or large values o he elasc deormaons These hgher-order erms generae nonlnear equaons o moon and hey are usually called anharmonc correcons o he wave equaon Bu he superposon o he soluons does no hold anymore or anharmonc correcons n general and he elasc waves exhb he combned-requency phenomenon and emporal resonances In he leraure he conrbuon o he nonlnear erms n hs equaon s reaed as a small perurbaon o he plane waves From he elascy heory [6 7 8] s known ha he nonlnear conrbuons o elascy appear rs hrough he ull expresson j k k γj ( ) j () j k o he sran ensor and secondly hrough hgher-order erms n he elasc energy where u ( u u u) s he dsplacemen vecor a poson o coordnaes x I j denoes he sress ensor and λ and μ represens he Lamé s maeral physcal consans hen he Hooke s law or elasc solds can be wren n he orm j j [ λδ δ μ( δ δ δ δ )] γ j () j kl k jl l jk where δj means he Kronecker s symbol The nonlnear equaons o moon o he elasc solds under he acon o he volume orces ( ) are wren as: kl j j j ρ ρ u ( x x x ) Ω R () where Ω [ a b] s he bounded doman and ρ s he densy o medum We assume ha he medum s soropc e λ μ and ρ are consans Frsly we noe ha he Ox axs concdes wh he propagaon drecon o he longudnal elasc wave e: u ( x ) u( x u ( x ) u u ( x ) u Takng no accoun Eqs () () and (4) Eq () can be wren as [5] [6]: (4) where u u u u u u u v v v x x x (5)

3 8 V Marnca R-D Ene B Marnca u u λ μ μ v v v ρ ρ ) cos( πx)sn( πv ) sn(4 πx) (4 πv) sn (4 πv ) (4 πv )sn(8 πx) cos (6) Equaon (5) s named he propagaon equaon o he longudnal elasc wave n he presence o he volume orces In Eq (6) v and v are longudnal and ransversal elasc velocy respecvely In he ollowng we sudy he nonlnear longudnal elasc wave o Eq (5) wh he ree boundary condons o sress e ( ) ( ) a b [] The nal condons are assumed o be o he orm: where v ( x) and ( x) u ) v( x) ) w ( x) x [] (8) w are known connuous uncons For lnear paral derenal equaons (5) and (5) we suppose ha he boundary and nal condons are respecvely: ( ) ( ) u) cos(4 πx) ) (9) ( ) ( ) u) cos(4 πx) ) () wh he soluons: u ) cos(4 πv )cos(4 πx) u ) cos(4 πv )cos(4 π x) () such ha Eq (6) becomes: ( x ) cos( πx)sn( πv) sn(4 π x) cos(4 πv)sn(4 π x) () (7) BASIC IDEAS OF OVM AND SOLUTIONS In order o develop an applcaon o he OVM we consder he general orm o a nonlnear paral derenal equaon: E u( x ux u )) () here u x ) and u ) denoe he paral dervaves wh respec o x and respecvely o he uncon u ) The varaonal prncple or Eq () can be esablshed here exss a unconal b a J L u ux u ))dd x (4) whch adms as exremals he soluons o Eq() where L s he Lagrangan o he sysem () x [ a b] [ ] Ths problem s based on he sudy o he condons under whch here exss a unconal

4 4 Approxmae analyc soluons o a nonlnear elasc wave equaons 8 L u( x ux u )) such ha Euler-Osrogradsk s equaon o unconal (4) concde wh he Eq () e: L L L Exuu ( x u ) (5) x In our procedure we assume ha he approxmae soluons o Eq () depends o a number o parameers C s : u u C ) (6) s such ha he acon unconal gven by Eq (4) becomes b s x a JC ( ) Lxu ( ( xc ) u( xc ) u( xc ))dx d s (7) The parameers C (whch wll be named convergence-conrol parameers) whch appear n Eq (7) can be deermned opmally applyng he Rz mehod [] [9]: (8) s We menon ha he approxmae soluon u gven by Eq (6) s chosen such ha he boundary and nal condons are ullled The expresson o he approxmae soluon (6) s no unque Remark The Rz mehod [] o obanng such "average" soluons can be derved rom calculus o varaons by seekng uncons ha mnmze a ceran negral Consder a uncon o he orm x( ) CΨ ( ) CΨ ( ) CnΨn ( ) J F( x x )d (9) where he Ψ k ( ) are prescrbed uncons all sasy he nal / boundary condons I x() s now nroduced or x( ) hen J J( C n ) and necessary condons or J o be mnmum are gven by Eqs (8) Ths gves n equaons o he orm or deermnng he n unknown consans J F F Ψ k d k C k Inegrang he las equaon we have: F F d F Ψ k kd C Ψ k d The rs erm s zero because Ψ n mus sasy he nal/boundary condons The expresson n brackes under he negral n he second erm s Euler s equaons The condon gven n Eqs (8) hen reduce o Ex ( ) Ψk d k n where E (x) s he derenal equaon o moon () Unlke he Rz mehod whn our procedure he prescrbed uncons should no sasy he nal / boundary condons On he oher hand he approxmae soluon x s subsued drecly no he unconal (5) no n he derenal equaon o moon

5 84 V Marnca R-D Ene B Marnca 5 The valdy o he proposed approach s llusrae on he Eq (5) wh he condons gven by Eqs (7) and (8) In our case he Lagrangan o Eq (5) can be wren as: L( xux () ux () x u()) x v ux ux u u (9) 6 I we consder s 5 n Eq (6) and v cos( π x) w n Eq (8) hen he approxmae soluon o he Eq (5) can be wren as: or u ) cosπ x Ccos( πx)( cos( π v)) Ccos( πx)( cos(6 π v )) Ccos( πx)( cos(8 π v )) C cos( πx)( cos(4 π v)) C cos( πx)( cos(8 πv)) 4 5 Also we can choose hs approxmae soluon n he orms: u ) cos π x ( C C cos πx)[ C ( cos π v ) C ( cos6 π v )] () 4 u ) cosπ x Ccos( πx)( cos( π v)) Ccos( πx)( cos(6 π v )) Ccos(4 πx)( cos(8 π v )) C cos(6 πx)( cos(4 πv)) 4 and so on The parameers C 4 5 whch appear n Eq () are obaned rom Eqs (8): () () 4 C5 () 4 NUMERICAL EXAMPLES In order o show he valdy and accuracy o he OVM we consder Eqs (5) and v v x [ ] [ ] (4) From Eq () n he condons (4) we oban he ollowng resuls: C (5) C The approxmae soluons o o Eq (5) n he condons (7) and (8) wh v cos( π x) w becomes: u ) cos( πx) ( cos(57 ))cos( π x) 6( cos(5655 )) cos( π x) 58( cos( π)) cos( πx) 9768( cos(6 π))cos( πx) 5966( cos(8 π))cos( πx) Fgures and presen a comparsons beween he presen soluon (6) and numercal resuls obaned usng he Wolram Mahemaca 6 soware or x [ ] and [ ] Fgures 4 and 5 presen a comparsons beween he presen soluon (6) or deren values o 7 x rom he doman [ ] and hree deren random values o : and respecvely and numercal 5 resuls I s easer o emphasze he accuracy o he obaned resuls n comparson wh he numercal resuls I can be seen ha he perodc soluons obaned by our procedure s n very good agreemen wh (6)

6 6 Approxmae analyc soluons o a nonlnear elasc wave equaons 85 numercal resuls s easer o emphasze he accuracy o he obaned resul n comparson wh he numercal resuls Fg The numercal soluon o Eq (5) Fg The approxmae soluon o Eq (6) or v v or v v Fg Comparson beween he numercal soluon o Eq (5) and approxmae soluon (6) or v v 5 / : numercal soluon; approxmae soluon Fg 4 Comparson beween he numercal soluon o Eq (5) and approxmae soluon (6) or v v / : numercal soluon; approxmae soluon Fg 5 Comparson beween he numercal soluon o Eq (5) and approxmae soluon (6) or v v 7 : numercal soluon; approxmae soluon / 5 CONCLUSIONS In hs paper we proposed and used Opmal Varaonal Mehod o deermne an analyc approxmae soluons o a nonlnear problems relaed o elasc wave equaons wh anharmonc correcon The proposed procedure s vald even he nonlnear equaon does no conan any small parameer Our

7 86 V Marnca R-D Ene B Marnca 7 consrucon s based o he consrucon o he approxmae soluon dependng o some convergence-conrol parameers C whch are opmally deermned Acually he capal srengh o he proposed procedure s s as convergence Our procedure s very eecve and accurae or nonlnear approxmaons convergng rapdly o exac soluons and provdes a convenen way o conrol he convergence o approxmae perodc soluon REFERENCES L V OVSIANNIKOV Group Analyss o Derenal Equaons Academc Press New York 98 P J OLVER Applcaons o Le Group o Derenal Equaons Sprnger-Verlag New York 986 M T MUSTAFA K MASOOD Symmery soluons o a nonlnear elasc wave equaons wh hrd-order anharmonc correcons Appled Mahemacs and Mechancs Engl Ed 8 pp E ALFINITO M S CAUSO G PROFILO G SOLIANI A class o nonlnear wave equaons conanng he connuum Toda case J Phys A pp B-F APOSTOL On a nonlnear wave equaon n elascy Physcs Laers A 8 pp A H BOKHARI A H KARA F D ZAMAN Exac soluons o some general nonlnear wave equaons n elascy Nonlnear Dyn 48 pp NAYFEH A Problems n perurbaons Wley New York Y K CHEUNG S H CHEN S L LAU A moded L-P mehod or ceran srongly non-lnear oscllaons In J Non-lnear Mechancs 6 pp W P WU P S LI A mehod or obanng approxmae analycal perods or a class o nonlnear oscllaors Meccanca 6 pp V MARINCA N HERIŞANU Nonlnear Dynamcal Sysems n Engneerng Some Approxmae Approaches Sprnger- Verlag Hedelberg H N ABRAMSON Nonlnear vbraon n: C M Harrs (Ed) Shock and Vbraon Handbook Mc Graw-Hll N Y 988 A BELENDEZ PASCUAL NEIPP e al An equvalen lnearzaon mehod or conservave nonlnear oscllaors In J Nonlnear Mechancs and Numer Smul 9 pp J I RAMOS Lnearzed Galerkn and arcal parameer echnques or he deermnaon o perodc soluons o nonlnear oscllaors Appl Mah and Compu 96 pp J H HE Varaonal eraon mehod a knd o nonlnear analycal echnque Some examples In J Nonlnear Mechancs 4 4 pp J H HE Varaonal approach or nonlnear oscllaorshaos Solons and Fracals 4 pp K R RAJAGOPAL A S WINEMAN New exac soluons n non-lnear elascy In J Engng Sc pp S MIROSLAV Mechancs and Thermodynamcs o Connuous Meda (Theorecal and Mahemacal Physcs) Sprnger Ed June 8 L LANDAU E LIFSCHITZ Theore de l Elasce Nauka Moscow V MARINCA N HERIŞANU Opmal Varaonal Mehod or Truly Nonlnear Oscllaors Appl Mahemacs ID 667 Receved March 4 4