Inverse Transient Quasi-Static Thermal Stresses. in a Thin Rectangular Plate

Transcription

1 Adv Theor Al Mech Vol 3 o 5-3 Iverse Trsie Qusi-Sic Therml Sresses i Thi Recgulr Ple Prvi M Slve Bhlero Sciece College Soer Ngur Idi rvimslve@hoocom Suchir A Meshrm Derme of Mhemics PGTD RTM Ngur Uiversi Ngur 44 Idi Asrc I his er he soluio of iverse rsie qusi-sic herml sresses i hi recgulr le occuig he sce - hs ee deermied The eressio of he emerure disriuio ukow emerure sresses fucio dislceme comoe d herml sresses comoe re oied i erms of Bessel s fucio d he resuls re illusred umericll d grhicll e words: Thi recgulr le iverse rsie qusi-sic herml sress Fourier rsform Llce rsform Iverse Fourier d Llce rsform

2 P M Slve d S A Meshrm Iroducio A cosise effor i he s hs ee mde umerous workers o sud vrious secs of herml sresses i differe odies Tigw Y Ishihr M Morishi H d wmur R [7] hve sudied heoreicl lsis of wo dimesiol hermoelsolsic edig deformio of le sujeced o rill disriued he sul Yoshiou Tigw d Ysuo omsur [9] hve discussed herml sress lsis of recgulr le d is herml sress iesi fcor for comressive sress field Ishihr M Tigw Y wmur R d Nod N [3] hve sudied heoreicl lsis of residul sress removig he he sul Furher Vihk VM Yuzvk MY d Ysikij AV [8] hve ivesiged he soluio of he le hermo-elsici rolem for recgulr domi Adms RJ d Ber CW [] hve deermied hermoelsic virio of lmied recgulr le sujeced o herml shock N L horgde d Deshmukh C hve sudied iverse qusisic herml deflecio rolem for hi clmed circulr le [] I his ricle em is eig mde o fid he soluio of he iverse rsie rolem of Qusi-sic herml sresses i recgulr le I order o oi he soluio of he goverig equio which is ril differeil equio he followig rocedures of lsis hve ee doed Normlizig of he goverig ril differeil equio sujec o rorie iiil oudr d ierior codiios Tkig he fiie Fourier d Llce rsform of he resulig equio wih resec o ime 3 Achievig he iverse Fourier d iverse Llce rsform mes of comle coour iegrio residue d covoluio heorem 4 The soluio d eressios of he emerure ukow emerure sress fucio dislceme d sress comoes re oied i erms of Bessel's fucio d he resuls re illusred umericll d grhicll The resuls oied m e useful i solvig egieerig rolems riculrl for idusril mchies sujeced o heig such s he mi shf of lhe uries d he rolls of he rollig mill Seme of iverse he coducio rolem Cosider hi recgle le occuig he sce D : - - Iiill he emerure of le is ke zero The differeil equio goverig rsie emerure disriuio T( i recgulr le is T T T - - > () sujec o he iiil oudr d ierior codiios s T ( ()

3 Iverse rsie qusi-sic herml sresses 3 T ± (3) T (4) T g( ukow (5) T ( f ( ) kow (6) where T re he emerure chge d herml diffusivi The Air s sress fucio U sisfies he equio s i [9] U E T α (7) where α d E re he lier coefficies of he herml esio d Youg s modulus of elsici of he merils of he recgulr le resecivel The dislceme comoes U d U i he d direcio rereseed i he iegrl form d he sress comoes i erms of U re give U U U T d E ν α (8) U U U T d E ν α (9) U () U () U d () where ν is he Poisso's rio of he meril of he le The equio () o () cosiues he mhemicl formulios of he rolem uder cosiderio 3 Soluio of he iverse he coducio rolem Deermiio of emerure T( d ukow emerure grdie g( O lig fiie Fourier d Llce rsform o he equio () o (6) d he usig heir iversios oe ois he eressio of he emerure disriuio d ukow emerure grdie resecivel s T ( cos ( ) ( ) si ( ) ( ) f ( ) ( ) ( G m N (3)

4 4 P M Slve d S A Meshrm g ( ( ) f ( ) ( )( ) si ( ) si ( ) N where G( m [ e( ( ) ] ( G m (4) is he Fourier rsform rmeer d erl ( ) defied s R ( ) ( ) (5) N R ( ) cos ( ) N where is he osiive roo of he rscedel equio si (6) 4 Deermiio of sress fucio d dislceme comoe Susiuig he vlue of emerure fucio from (3) i (7) we oi cos f U α E G( m (7) ( ) si( ( )) N Susiuig he vlue of T d U from (3) i (7) i (8) d (9) we oi cos f si U α v G m ( ) si( ( )) N U si ( ) ( ) si ( ) ( ) ( ) ( ) (8) f α v G( m (9) N 5 Deermiio of sress comoes Usig (7) i () () d () he sress fucios re oied s 3 cos ( ) ( ) si ( ) ( ) ( ) ( ) f αe G( m () N cos ( ) ( ) si ( ) ( ) ( ) ( ) f αe G( m () N

5 Iverse rsie qusi-sic herml sresses 5 si ( ) ( ) si ( ) ( ) ( ) ( ) f si αe G( m () N 6 Secil cse d Numericl resuls Se f ( ) (3) Alig he fiie Fourier rsform o he equio (3) oe ois f cos ( ) d (4) Now for fidig ou he soluio of iverse rsie qusi-sic herml sress seel (SN5C) hs ee ke s he meril of he le hvig he followig roeries : α 6 6 v 8 c δ -αe β -α(ν) γ αe 59-6 m s - E5 GP Susiuig (4) i (3) (4) (7) (8) (9) () () d () T g U ( ( ( δ cos ( ) ( ) si ( ) si ( ) ( ) ( ) si cos N N ( ) si ( ) ( ) ( ) cos ( ) ( ) cos N cos ( ) ( G m ( ) d ( G m ( ) d ( ) ( ) d ( G m (5) (6) (7)

6 6 P M Slve d S A Meshrm d G m N U cos si si cos β (8) d G m N U cos si si β (9) d G m N cos si cos 3 γ (3) d G m N cos si cos γ (3) d G m N cos si si si γ (3) 7 Grhs Fig : Temerure disriuio T versus Fig : Temerure disriuio T versus for fied 5 5 5for fied 5

7 Iverse rsie qusi-sic herml sresses 7 Fig 3 : Temerure disriuio T versus Fig 4 : Ukow Temerure disriuio g versus for fied vlue of for fied vlue of 5 Fig 5 : Sress fucio U/α versus Fig 6 : Sress fucio U/α versus for fied 5 for fied 5 Fig 7 : Sress fucio U/α versus Fig 8 : Dislceme Comoe U /β versus for fied for fied 5 Fig 9 : Dislceme Comoe U /β versus Fig : Dislceme Comoe U /β versus 5for fied for fied 3

8 8 P M Slve d S A Meshrm Fig : Dislceme Comoe U /β versus Fig : Dislceme Comoe U/β versus 5 3 5for fied for fied 5 Fig 3 : Dislceme Comoe U /β versus Fig 4 : Sresses comoe /γ versus for fied for fied 5 Fig 5 : Sresses comoe /γ versus Fig 6 : Sresses comoe /γ versus 5 for fied for fied 3 Fig 7 : Sresses comoe /γ versus Fig 8 : Sresses comoe /γ versus for fied for fied 5

9 Iverse rsie qusi-sic herml sresses 9 Fig 9 : Sresses comoe /γ versus Fig : Sresses comoe /γ versus for fied for fied 5 Fig : Sresses comoe /γ versus Fig : Sresses comoe /γ versus 5 for fied for fied 3 8 Discussio I his er equios (5) hrough (3) hve ee clculed o our oservios usig MhCAD grhs hve ee loed ccordigl s er he resuls oied d coclusios re eig drw Iiill he emerure of he recgulr le hs ee deermied usig he codiios give i he rolem d lig fiie Fourier d Llce rsform d is iverse Therefer he ukow emerure hs ee foud ou usig he emerure resuls Thus he vlue of sress fucio of he meril of he le is foud usig emerure T lier coefficies of he herml esio α Youg s modulus of elsici E d Poisso s rio of he meril v Fill he dislceme comoe hs ee rrived usig he sress fucio; d lsl he sress comoe i erms of U hs ee foud Now herml diffusivi d herml coducivi re wo imor herml roeries h eer he differeil equio of he coducio Therefore ccurc of he vlue chose for hese roeries ffecs he ccurc of he resuls i he-coducio rolems I isoroic recgulr le he chge of emerure does o led o chge i sher gles ece he sress d sri of he le These roeries re clerl refleced i he ove loed grhs I Fig hrough Fig he emerure disriuio T ukow emerure g sress fucio U dislceme comoe U d U d sress comoes

10 3 P M Slve d S A Meshrm of he recgulr le versus for differe vlues of d hve ee loed The emerure disriuio T ukow emerure g sress fucio U dislceme comoe U d U d sress comoes of he recgulr le versus for differe vlues of d hs lso ee cosidered I his rolem of iverse rsie qusi-sic herml sress i hi recgulr le he codiio h hs ee give is h is se is sujeced o he flu d iiill he emerure of he le hs ee ke zero Here we hve cosidered seel le (SN5C) s he mel d hece he grh shows riculr er resemlig he roeries of seel le (SN5C) Thus he followig coclusio c e drw: The emerure disriuio icreses s he ime icreses d is is oimum ime 5 for differe vlues of versus for fied 5; d ech lie coicides 4 Bu o icresig he ime ove 5 he grh refuses o show furher cosisec (refer fig ) Sme er i he grhicl rereseio c e oserved i ukow emerure (Fig4) sress fucio U (Fig5) dislceme comoe (Fig8 & ) d sress comoe (Fig4 7 & ) 3 The emerure disriuio is oimum 3 (refer Fig) fer which i srs decliig differe vlue of versus for fied ime 5 The sme er is oserved i sress fucio U (Fig6) d dislceme comoe U (Fig) All lies coicide 5 The sress comoe (Fig8) shows reverse er i he grh due o he egive sig i he sress comoe 4 The emerure disriuio ridl icreses s ime icreses for fied vlue of versus d coicides 5 (refer Fig3) The sme er is oserved i sress fucio U (Fig7) dislceme comoe U (Fig & 3) d sress comoe (Fig6 9 & ) 5 Dislceme comoe is mimum for differe vlues of versus for fied ime 5 (refer Fig9) Here ever lie coicides 5 Bu reverse er i he grh oied i sress comoe d (Fig5 & ) due o is egive sig

11 Iverse rsie qusi-sic herml sresses 3 Refereces [] Adms RJ d Ber CW Thermoelsic virios of lmied recgulr le sujeced o herml shock Jourl of Therml Sresses Vol (999) [] Ciell G A eesio of he fiie Hkel rsform d licio I J Eg Sci 3(965) [3] Ishihr M Tigw Y wmur R d Nod N Theoreicl lsis of residul sresses removl he sul [4] Ozisik MN Boudr vlue rolem of he coducio Ieriol Teook com Scro Peslvi (968) [5] Olcer NY O geerl he flow rolem J Mh Ad Phs Vol 46 No (967) 99-6 [6] Sedde IN The use of iegrl rsforms Mc Grw Hill New York (97) [7] Tigw Y Ishihr M Morishi H d wmur R Theoreicl lsis of wo-dimesiol hermoelsolsic edig deformio of le sujeced o rill disriued he sul TrsJSME Vol 6(996) No [8] Vihk VM Vuzvk MY d Ysikij AV The soluio of he le hermo-elsici rolem for recgulr domi Jourl of Therm sresses Vol (998) [9] Yoshiou Tigw d Ysou omsur Therml sress lsis of recgulr le d is herml sress iesi fcor for comressive sress field Jourl of Therml Sresses (997):57-54 [] Meshrm SA d Deshmukh C Theoreicl lsis of herml sresses i hi ulr disc due o rill disriued he sul Bul Pure & Alied Sc Vol 5 (6) [] horgde NL d Deshmukh C A iverse qusi-sic herml deflecio rolem for hi clmed circulr le Jourl of Therml Sresses 8(5) Received: Decemer 8