ASYMPTOTIC AND TOLERANCE 2D-MODELLING IN ELASTODYNAMICS OF CERTAIN THIN-WALLED STRUCTURES

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1 AYMPTOTIC AD TOLERACE D-MODELLIG I ELATODYAMIC OF CERTAI THI-WALLED TRUCTURE B. MICHALAK Cz. WOŹIAK Dpartmt of tructural Mchacs Lodz Uvrsty of Tchology Al. Poltrchk Łódź Polad Th objct of aalyss s a pla structurs rforcd by systm of prodcally dstrbutd th paralll rbs Fg.. Th am of cotrbuto s to drv a Dmacroscopc mathmatcal modl dscrbg lastodyamcs bhavour of ths structur. Th cosdratos ar basd o thos summarzd moographs Woźak Mchalak Jędrysak ds om applcatos of th tolrac avragg tchqu for th modllg of varous dyamc ad stablty problms for lastc mcrohtrogous structurs ar gv srs of papr by: Baro 003 Jędrysak ad Mchalak 0 Mchalak ad Wrowsk 0 agórko ad Woźak 00 Wągrowska ad Woźak 996 Wrzbck ad Woźak FORMULATIO OF THE MODELLIG PROBLEM Itroduc th orthogoal Cartsa coordat systm 3 O th physcal spac occupd by a plat udr cosdrato. Lt 0 L 0 b th mdpla L th symmtry pla of th structur. It s assumd that a thckss of th plat h s small compard to th mmum lgth dmso of th mdpla of th plat h m L L. At th sam tm th thckss h s supposd to b small compard to th wdth of th stffd rbs H h H Fg.. Fg.. A fragmt of a plat structur wth prodc systm of stffrs 487

2 ubsqutly t wll b assumd that umbr of th rbs s vry larg / ad th mamum dstac l btw rbs s vry small wh compard to L. Hc l L / wll b trtd as a mcrostructur lgth paramtr. At th sam tm th thckss h of th plat s supposd to b small compard to th mcrostructur lgth paramtr l h l. Fg.. A fragmt of a cross-scto of th stffd plat structur Th am of ths cotrbuto s to formulat D macroscopc modls of dyamcs bhavour of th plat udr cosdrato. Thr modls wll b rfrrd to as asymptotc ad tolrac rspctvly. By th -dmsoal macroscopc modl w shall udrstad mathmatcal modl govrd by avragd quatos of moto wth smooth coffcts ad ukows fuctos dpdt o coordats ad. Throughout th papr dcs k l ru ovr 3 dcs... ru ovr ad t stad for th tm coordat. ubsqutly w shall us dotatos / / suprscrpts.. PRELIMIARE 488. Th summato covto holds all aformtod sub-ad Th cosdratos ar basd o th wll-kow quatos for th pla strss stat th plat. It s assumd that th udformd mdpla of th plat occups rgo 0 L 0. Dotg by l dstac btw th rbs of th plat-structurs vry L whr wth ctr at l / l... / wll b rfrrd to th cll Fg.3. Lt 0 L ] wll b rgo th physcal spac [ occupd by plat-structur ad t -cross scto of by vry 0 L - pla. Lt subclls P P wll b parts of vry cll ; blogg to plat rbs-stffrs ad part blogg both to plat ad stffrs rspctvly. Th modl quatos for th dyamc bhavour of th plat-structur udr cosdrato wll b obtad for pla-strss stat th plat. P ubclls. Pla strss 0 hc

3 whr stra tsors. ubclls Fg. 3. Th basc cll of th stffd plat structur P. I ths rgo of th structur w cosdr 3D-strss stat whr h h h h h h ar Lam s costats. ubclls. Pla strss 0 hc 3 From codto of cotuty o trfacs 489 P

4 w drv h h 4 E h h E [ ]/[ ] 5 H H ad barg md that h H w shall assum appromato that h / H s glgbly small formula 4 ad th formula 4 tak th form. Hc w assum codtos; subcll ad 0 subcll P. Avragg formula 3 ovr h H / h H / w hav h he h [ HE h ] h 6 ad ovr h h P 7 w drv costtutv quatos for -dmsoal modl of th htrogous structur udr cosdrato. -dmsoal modl of th plat structurs wth th prodc dstrbuto of rbs. Lt dsplacmt of th mdpla of th plat wll b dotd by w t tral forcs by p t ad by th mass dsty avragd ovr th plat thckss rlatd to th mdpla. I th framwork of th lar thory for pla-strss stat w obta: - quatos of moto whr p w

5 h P 9 h H P 0 - costtutv quatos whch w shall wrt th form D whr D D D D D. It ca b s that th coffcts abov quatos ar dscotuous ad hghly oscllatg. Ths quatos ar to complcatd to b usd th grg aalyss ad wll b usd as startg pot th tolrac modllg procdur. 3. TOLERACE MODELLIG I ordr to drv avragd modl quatos w appld tolrac avragg approach. Ths tchqu basd o th cocpt of tolrac ad dscrblty rlatos ad o th dfto of slowly-varyg fuctos. Th gral modllg procdurs of ths tchqu ar gv books Woźak t al Th fudamtal cocpt of th modllg tchqu s th avragg a arbtrary tgrabl fucto f ovr th cll f f y dy y. 3 Th frst assumpto th tolrac modllg s mcro-macro dcomposto of th dsplacmt fld for ad t t 0 t. w t u t g V t 4 Th modllg assumpto stats that u V ar slowly-varyg fuctos wth rspct to th argumt 0 L. Fuctos u t V 49

6 V t V ar th basc ukows of th tolrac modl. Fucto g s kow dpdt o th mcrostructur lgth paramtr l fluctuato shap fucto. Lt g g stad for prodc appromato of g g rspctvly. Du to th fact that w t ar tolrac prodc fuctos t ca b obsrv that th prodc appromato of w h t ad w h t hav th form w w h y w h h y y t u t u t u t g y V t g y V t g y V t t t g y V t 5 for vry almost vry y ad vry t t 0 t. 49 Th modllg assumpto stats that f vry cll wll b df rsdual forcs r p w th th followg orthogoalty codtos holds r 6 T 0 g r T 0 7 whr oprator T stads for tolrac avragg ovr th cll. ubsttutg th rght-had sd of formula 3 to quatos 6 ad barg md orthogoalty codtos 7 w obta th followg systm of quatos of moto D u D u g V 0 D g D g u g D g V g D g V D g V p g V g D g V D g u g V p g g u g g V 0 Th abov rsults rprst th systm quatos for avragd dsplacmts u t 8 ad dsplacmts fluctuato ampltuds V t. Ths quatos togthr wth mcromacro dcomposto of dsplacmt flds 4 ad physcal codto that solutos hav to b slowly-varyg fuctos wth rspct to th argumt 0 L costtut th tolrac modl of structural plat udr cosdrato.

7 3. AYMPTOTIC MODEL For asymptotc modllg procdur w rta oly th cocpt of hghly oscllatg fucto. W shall ot dal wth th cocpt of th tolrac prodc fucto as wll as slowly-varyg fucto. Usg th asymptotc procdur w troduc paramtr /.... Lt l h H ad b ar a scald dmsos of th cll A scald cll s dfd by l / l / ad s a scald cll wth a ctr at. Th mass dsty ad tsor of lastc modul D ar assumd to b hghly oscllatg ad dscotuous fuctos 0 D HO for almost vry. If 0 D HO th for vry thr st fuctos y D y whch ar prodc appromato of fuctos D rspctvly. Th fudamtal assumpto of th asymptotc modllg s that w troduc dcomposto of dsplacmt as famly of flds y w y t u y t g V y t y t t0 t 9 whr g ar prodc appromato of hghly oscllatg fuctos g HO. From formula 9 w obta y y w y t u y t g V y t g V y t y w y t u y t g V y t 0 Barg md that by mas of proprty of th ma valu Jkov t al. 994 fucto g y / y s wakly boudd ad has udr 0 wak lmt. Udr lmt passag 0 for y w obta u y t u V y t V u y t u t O t O t O u V y y t u t V V y t V t O t O t O. By mas of w rwrt formula 9 ad 0 th form 493

8 w y w y t u t u t O y t g V t O. Usg formula for orthogoalty codtos 7 w obta quatos D D g u u D g D g V p g V 0 u 0 3 Elmatg V from quatos 3 V D g D g g u 4 w arrv th followg quato of moto for th avragd dsplacmts of th plat mdpla u t D D g D g u p g D g u 0. 5 Equatos 4 5 rprst th asymptotc modl of th structural plat udr cosdrato. REFERECE [] Baro E. O dyamc stablty of a uprodc mdum thckss plat bad J. Thor. Appl. Mch pp [] Jędrysak J. Mchalak B.O th modllg of stablty problms for th plats wth fuctoally gradd structur Th-Walld truct pp [3] Mchalak B. Wrowsk A. Dyamc modlg of th plat mad of crta fuctoally gradd matralsmccaca DOI 0.007/s z. [4] agórko W. Woźak Cz. oasymptotc modllg of th plats rforcd by a systm of stffrs Elctroc J. Polsh Agrc. Uv. Cvl Egrg [5] Wągrowska M. Woźak Cz. O th modllg of dyamc problms for vsco-lastc composts It. J. Egg c pp [6] Wrzbck E. Woźak Cz. O th dyamc bhavour of hoycomb basd compost solds Acta Mchaca pp [7] Woźak Cz. Mchalak B. Jędrysak J. ds. Thrmomchacs of Htrogous olds ad tructurs 008Wydawctwo Poltchk Łódzkj Łódź. 494