Three-Dimensional Theory of Nonlinear-Elastic. Bodies Stability under Finite Deformations

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1 Appld Mathmatcal Sccs ol. 9 5 o HKAR Ltd Thr-Dmsoal Thory of Nolar-Elastc Bods Stablty udr Ft Dformatos Yu.. Dmtrko Computatoal Mathmatcs ad Mathmatcal Physcs Dpartmt Bauma Moscow Stat Tchcal Uvrsty Baumaskaya Strt 5 55 Moscow Russa Copyrght 5 Yu.. Dmtrko. Ths artcl s dstrbutd udr th Cratv Commos Attrbuto Lcs whch prmts urstrctd us dstrbuto ad rproducto ay mdum provdd th orgal work s proprly ctd Abstract Th computato problm of lastc structurs stablty s o of ma problms of sold mchacs. Tradtoal mthods of stablty calculato ar basd o applyg th thory of two-dmsoal shll structurs gral th classcal Krchhoff-Lov thory. Th dvlopd mthods for solvg thr-dmsoal problms of th stablty thory allow us to xpad th frams of solvd stablty problms ad to cras th accuracy of obtad solutos. Th purpos of th papr s to drv gralzd thr-dmsoal quatos of th stablty thory of olarly lastc bods wth ft dformatos for a wd class of olar lastc modls. For ths th mthod of a vard cofgurato ad th uvrsal mthod of rprstato of olarly lastc cotua modls o th bas of rgtc coupls of strss ad stra tsors wr appld. t s show that for two of th tsor coupls th stablty thory rlatos gv a xplct aalytcal xprsso wthout calculato of gvalus of th strtch tsor. Kywords: thr-dmsoal thory of lastc stablty rgtc strss ad stra tsors ft lastc dformatos Equatos of th shll stablty thory wth small dformatos for dffrt cass ar usually drvd wth th hlp of a umbr of hypothss ad assumptos [ 6 7] bcaus th stablty quatos v for cotua wth small dformatos follow from gral olar quatos of th lastcty thory wth ft dformatos whch ar complcatd ough th gral statmt ad ot dftv. Du to dvlopmt of powrful computrs usg ft-lmt mthods thr appars a trst thr-dmsoal problms of th stablty

2 76 Yu.. Dmtrko thory. To drv gralzd thr-dmsoal quatos of th thory for olar-lastc solds wth ft dformatos w apply th advacd mthod of a vard cofgurato ad th uvrsal mthod of rprstato of sold modls o th bas of rgtc coupls of strss ad stra tsors [3 4]. Th vard cofgurato Lt us cosdr th gral cas of ft dformatos of lastc solds [3-5]. Togthr wth actual cofgurato K of a sold cotuum at tm t w troduc o mor actual cofgurato K whch s calld vard ad dffrs from th tru cofgurato K by a small dsplacmt. Th cofgurato K s usd for sarch of possbl ot uqu soluto th xstc of whch mas that thr appars a stablty of th body. Radus-vctor x of a pot M K s coctd wth x of th sam pot K by x x+ δ x whr δ x s th radusvctor varato dtrmd th followg way. Lt f ( b a smooth scalar fucto dfd wth trval m th a small ghborhood of pot ths fucto ca b cosdrd as th lar dpdc: df f ( f ( f( f( + f ( whr f ( ( lm s th drvatv d of th fucto zro. Wth us of ths rprstato lt us cosdr th radusvctor x as a fucto ot oly of Lagraga coordats X of th matral pot ad tm t but also of th addtoal paramtr (fcttous tm ad ths fucto s assumd to b lar: x x( X t x( X t + w ( X t x( X t x ( X t w ( d/ d x ( X t. Hc w fd th radus-vctor varato δ x as a lar fucto of : δ x w. Th body locato th actual cofgurato K s assumd to b kow (.. x s kow th th stablty thory problm cossts fdg th vard cofgurato K.. w (or δ x. Kmatcs of a vard cofgurato O dffrtatg th rlatos wth rspct to X w obta th local bass vctors cofgurato K : x x w w r + r + r ( E+ w. ( X X X X For r ˆ at pot w ca also us th lar rprstato rˆ r +r r d d r.o comparg ths rlatos wth ( w gt rcprocal bass vctors r ˆ hav th form ˆ. r r w. Th r r r w Th xprsso

3 Thr-dmsoal thory of olar-lastc bods 77 for th vctors th ghborhood of pot has th form d r r + r r r. ( d Comparg ths formula w obta th xprsso r r w. Ths rlatos allow us to rprst varatos of local bass vctors δr ad δr trms of gradt w : r r + δr rˆ r + δr δr r δr r. All formula for varatos of strss ad stra tsors hav th smlar form. Thrfor for fdg ths varatos t s suffct to fd oly drvatvs wth rspct to whch ar calld covctv drvatvs bcaus thy dtrm th chag of valus at pot M at passag from cofgurato K to K. Covctv drvatv of th stra gradt Th stra gradt F cofgur-ato K s dfd smlarly to th stra gradt K : F rˆ r whr r ar local vctors of th rcprocal bass th rfrc cofgurato. Wth accout of ( w th obta ts lar rprstato th ghborhood of pot : d F r r + w r r F+ w F F F w. (3 d Th vrs gradt F K s dtrmd by th formula: F r r d F F w F F F w. Th drvatv of smooth d scalar fuctos Φ ( F ad Φ ( F wth rspct to has th form Φ Φ ( F F ( ΦΦ Φ Φ + Φ Φ Φ Φ +ΦΦ. (4 F partcular choosg Φ g/ g whr g dt( gj g dt( gj ar dtrmats of mtrc matrcs: g r r j g rr j wth accout of th cotuty quato Lagraga dscrpto [4] ( j j g/ g dt F ad F F w obta ( g/ g ( g/ g F (dt F F w g/ g w F. (5 F Formula (4 ad (5 allow us to dtrm th covctv drvatv wthout us of th vard cofgurato K but mmdatly by formal ruls of dffrtato of tsors wth rspct to th fcttous tm chos as paramtr [4].

4 78 Yu.. Dmtrko Covctv drvatvs of gvctors ad gvalus of th strtch tsors Lt us dtrm covctv drvatvs λ p of gvalus λ ad gvctors of th rght strtch tsor p us th proprts of gvalus ad gvctors [3]: 3 λ U p p p p δ U F F ad drvatvs U ad λ U p p 3 U λ p p O. W U F F. (6 Dffrtato of th fourth formula (6 wth rspct to gvs (. U F ε w F Dffrtatg th scod formula (6 wth rspct to ad assumg thr w fd p p.. vctors p ad p ar orthogoal. Dffrtatg th thrd formula (6 wth rspct to w hav λ λ U p + U p p + p. (7 O multplyg ths quato by p w obta p U p+ pu p λ λ (/ λ (. λ p p + p p. Wth accout of (6 th scod summads at th rght ad lft sds of th quato vash thus of (4 w fd λ p U p. Hc wth accout p F ε w F p Usg th polar dcomposto F OU ad p Op [4] w gt Fp OUp λ p. Th w fd th fal formula for th covctv drvatvs of gvalus: λ λ p ε( w p ad λ λ p ε( w p. Multplyg th rlato (7 by p : p U p + p U p λ p p + λ p p (8 w obta (all λ ar assumd to b dffrt: p p pu p /( λ λ. Rsolvg th vctors p for bass p w hav 3 3 ( p U p p p p p p (9 Th fal formula for th drvatvs of gvctors has th form 3 p ε( w p p λλ p. (

5 Thr-dmsoal thory of olar-lastc bods 79 Usg th rsoluto of tsor for th gbass: p λ p w obta λ 3 λ p p + p p + p p p p p p p p bg th aalog of formula (8. Hc w hav p p 3 3 ( p p ( p ε p p p p w p p. Usg formula for λ p ad p w fd covctv drvatvs of strtch tsors U ad to th th powr ad th drvatv of th rotato tsor wth dffrtatg th formula of rsoluto of th tsors for th gbass: ( 3 3 ( ( λ λ λ( U ( U p p + p p + p p p ε p p p 3 ( ( ( 3 ( ( p ε( w p + ( p ε( w p p p O λ δ ( δ O Op p O ( λλ ( p ε p + p ε p λ λ ( ( δ ( ( δ λλ ( λ λ U λ δ + ( λ λ. Covctv drvatvs of rgtc ad quasrgtc stra tsors Usg formula ( ad gralzd rprstatos [4] for rgtc ad quasrgtc stra tsors ( C ad ( A w fd th xprssos for thr covctv drvatvs ( ( 4 C U ε( w ( ( ( 4 A ( 4 ε w + ε ( w (3 ( 3 ( ( 3 ( 4 4 U U p p p p p p p p. Th strss tsors th vard cofgurato Cosdr modls A of a olar gral asotropc cotuum whch ar (s dtrmd by costtutv rlatos [4]: T ϕ ( s ( s γc γ ( / C ( (s (s γ γ ( C ( r γ γ γc. Th wth accout of (5 w hav ϕ ψ/ γ ( s γ

6 7 Yu.. Dmtrko ( ( ( ( ( 4 ( s 4 ( s 4 T H C H U ε( w (s ψ γ +. ( ( C C ( r r 4 ( s (s (s H (s (s γc C ϕγ γ γ γ (4 To calculat th drvatvs T ad P of th Cauchy ad Pola-Krchhoff strss tsors w us th rlatos btw tsors ( T ad T P [4]: ( ( 4 ( ( ( 3 ( 4 4 T E T P E T E E p p p p ( ( 3 ( ( ( 4 4 ( ρ / ρ E E g/ g E / λ E F E p p p p. (5 ( 4 Compots of rgtc quvalc tsors E ( 4 ad E dpd oly o λ ad λ [4]. Dffrtatg th formula ( wth rspct to w gt ( ( ( ( 4 4 ( ( ( ( T E T+ ET P E T E T. (6 Formulato of th stablty problm for a olar-lastc body Lt us wrt th qulbrum quato rfrc cofgurato P+ ρ f whr f s spcfc mass forcs. Dffrtat th quato wth rspct to : P. aratos of th spcfc mass forcs vctor f ad also of xtral surfac forcs vctor S ad gv dsplacmts vctor u ar assumd to b zro. Th w obta th followg quato trms of th varatos vctor w : K : ( ( ( ( ( ( 6 T 4 ( s 4 6 T ( R T+ H U ε( w + R T ε ( w (7 ( 6 T whr R ar th traspos sxth-ordr tsors: ( 6 6 ( T T (5634 R ( R. Lar stra tsors ε( w ad ε ( w th rfrc cofgurato bass hav th form ε( w ( F w+ w F ε ( w ( F w+ w F. (8 For th cosdrd lastc body boudary codtos ar assumd to b gv as th forc vctor S at surfac part Σ σ ad th dsplacmts vctor u at part Σ u :

7 Thr-dmsoal thory of olar-lastc bods 7 P S u u. Dffrtatg quatos (5 ad usg xprsso ( for varato of th dsplacmts vctor u x x x+ w x u+ w ad u u w gt P w. Substtutg formula (6 to ths quatos w ca wrt th boudary codtos th form ( ( ( ( ( ( 6 T 4 ( s 4 6 T ( R T+ H U ε( w + R T ε ( w w. (9 Th quatos systm (7 (9 s th dsrd stablty problm statmt for a olar-lastc body. Ths problm s lar wth rspct to th ukow fuctos vctor w cluds drvatvs of th scod ordr ad s uform.. admts th trval soluto w. Solutos of th stablty problm ar just otrval solutos: w. Th trval soluto corrspods to a stabl qulbrum of th body ad otrval to th ostabl o. Th stablty problm statmt (7 (9 s cocrd to th class of problms o gvalus. Togthr wth th problm (7 (9 lt us cosdr th tal problm o body qulbrum K Lagraga dscrpto: P ( ( 4 P E T ( r ( ( (s T ϕγ γc( C C ( U E γ U F F F E+ u P µ S u µ u. ( Hr w troducd th scalar paramtr µ bg a multplr at vctors of xtral surfac forcs S ad dsplacmts u. Lt us assum that a soluto of th problm for th dsplacmt vctor u s foud for valus of µ from a crta trval ( µ µ th fuctos u ( µ ad ( T ( µ may b cosdrd. Substtut th strss tsor ( T ( µ to th stablty problm (7 (9 ad clud th paramtr µ to th umbr of ukows of th problm togthr wth w. Th th problm (7 (9 s formulatd as follows: o should fd such valus of µ that quato systm (7 (9 has th trval soluto w. So t s th problm o gvalus. Ths problm s solvd togthr wth th ma problm ( of th olar lastcty thory bcaus quatos (7 (9 cota tsor ( T ( µ whch s a soluto of th problm (. Stablty quatos for modls A ad A Formula (5 hav th smplst form for modl P g/ g A : T F T F T F. O dffrtatg ths quatos wth rspct to ad takg

8 7 Yu.. Dmtrko (3 ad (4 to accout th problm (7 (9 for A of a olar-lastc sold taks th form 4 ( s - H ε F T w F w T F ( ε w F+ F w ( + + ( 4 ( s - ( H ε F + T w+ ( F w T F w. ( P g/ gu TF. Dffrtat- For A formula (5 gv T F TF g ths quatos wth rspct to w gt o Σ u T F T F wf TF F TF w ( ( ( P g/ g U T F F ε F TF U TF w + w U TF Th systm (7 (9 for modl A of a olar-lastc sold taks th form 4 ( s - (( H ε F ε TF U TU w + ( F wu TF T ε ( w ( U w F + F w U w (3 4 ( s - (( H ε F ε TF U TU w + ( F wu TF Strss tsors T ad T problms ( (3 ar dtrmd by solvg th problm o a basc stat qulbrum (. For othr modls ( A A thr s a d to us th gral quatos (7 (9. Comparg th drvd quato systms ( (3 ad th gral systm (7 (9 for dffrt valus w gt that th larzd systms of stablty thory quatos prov dffrt for dffrt modls of th olar-lastc bhavor of a sold. Thus ultmat xtral loads ladg to th loss of solds stablty wll b dffrt as wll. Coclusos Th gralzd thr-dmsoal stablty thory of olar-lastc solds s suggstd for th cas of arbtrary ft dformatos whch s basd o th cocpt of a vard cofgurato of olar-lastc solds ad wth us of th gralzd modls of olar-lastc solds dvlopd by th author wth th hlp of fv rgtc coupls of strss-stra tsors. Th fal quatos of th thr-dmsoal stablty thory prov to b dffrt for dffrt modls of olar-lastc solds. Th xplct aalytcal quatos of th stablty thory ar foud for two typs of th modls cludg th rght Almas ad Cauchy- Gr stra tsors wh thr s o d to calculat gvalus of th strtch

9 Thr-dmsoal thory of olar-lastc bods 73 tsor. Th drvd quatos of th thr-dmsoal stablty thory hav th uvrsal charactr.. thy may b appld to calculatos of stablty of complcatd olar-lastc solds frams of thr-dmsoal aalyss of a strss-stra stat as wll as solds wth small lastc dformatos ad also for calculatg frams of two-dmsoal shll structurs. Ackowldgmts. Th rsarch s supportd by Russa Scc Foudato (grat Rfrcs [] B.E. Abal C. öllmck B. Woodward M. Kashtalya. Guz W.H. Müllr Thr-dmsoal lastc dformato of fuctoally gradd sotropc plats udr pot loadg Compost Structurs 8 ( do:.6/j.compstruct [] Z.P. Bazat L. Cdol Stablty of Structurs Oxford Uvrsty Prss Oxford 99. [3] Yu.. Dmtrko Novl vscolastc modls for lastomrs udr ft stras Europa Joural of Mchacs. A: Solds ( o [4] Yu.. Dmtrko Nolar Cotuum Mchacs ad Larg lastc Dformatos Sprgr Brl. [5] Yu.. Dmtrko Thrmomchacs of Compost Structurs udr Hgh Tmpraturs Sprgr Brl 6. [6] S. Sgh B.P. Patl Nolar lastc proprts of graphm sht udr ft dformato Compost Structurs 9 ( do:.6/j.compstruct.4.9. [7] S.P. Tmoshko J.M. Gr Thory of Elastc Stablty McGraw-Hll Nw York 96. Rcvd: Octobr 5 5; Publshd: Dcmbr 5