Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells

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1 Advances n Pure Mathematcs Publshed Onlne August 5 n ScRes Asymptotcally Confrmed Hypotheses Method for the Constructon of Mcropolar and Classcal Theores of Elastc Thn Shells Samvel Sargsyan Department of Physcal Mathematcal Scences Gyumr State Pedagogcal Insttute Gyumr Armena Emal: s_sargsyan@yahoocom Receved 5 January 5; accepted 6 August 5; publshed 9 August 5 Copyrght 5 by author and Scentfc Research Publshng Inc Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY) Abstract In the present paper asymptotc soluton of boundary-value problem of three-dmensonal mcropolar theory of elastcty wth free felds of dsplacements and rotatons s constructed n thn doman of the shell Ths boundary-value problem s sngularly perturbed wth small geometrc parameter Internal teraton process and boundary layers are constructed problem of ther ontng s studed and boundary condtons for each of them are obtaned On the bass of the results of the nternal boundary-value problem the asymptotc two-dmensonal model of mcropolar elastc thn shells s constructed Further the qualtatve aspects of the asymptotc soluton are accepted as hypotheses and on the bass of them general appled theory of mcropolar elastc thn shells s constructed It s shown that both the constructed general appled theory of mcropolar elastc thn shells and the classcal theory of elastc thn shells wth consderaton of transverse shear deformatons are asymptotcally confrmed theores Keywords Mcropolar Elastc Thn Shell Asymptotc Model Appled Theory Introducton Current methods of reducng three-dmensonal problem of theory of elastcty to two-dmensonal problem of theory of plates and shells are the followngs: ) hypotheses method; ) method of expanson by thckness; ) asymptotc method []-[9] From recent mportant papers on constructon of mcropolar elastc thn plates and shells must be noted papers [] [] where also revew of researches s done n the mentoned drecton The man problem of the general theory of mcropolar or classcal elastc thn plates and shells s n approx- How to cte ths paper: Sargsyan S (5) Asymptotcally Confrmed Hypotheses Method for the Constructon of Mcropolar and Classcal Theores of Elastc Thn Shells Advances n Pure Mathematcs

2 mate but adequate reducton of three-dmensonal boundary-value problem of the mcropolar or classcal theory of elastcty to two-dmensonal problem From our pont of vew for achevement of ths am []-[4] durng the constructon of appled theores of thn plates and shells man results of the asymptotc soluton of boundary-value or ntal boundary-value problem of three-dmensonal mcropolar or classcal theory of elastcty n correspondng thn domans can be used whch are formulated as hypotheses [5]-[8] Mcropolar and classcal theores of elastc thn plates and shells constructed on the bass of such approach are asymptotcally correct theores Ths problem s also essental n classcal theory of elastcty durng the constructon of mathematcal models of thn plates and shells wth the account of transverse shear deformatons: n paper [9] t s shown that one of the man theores of plates and shells of Tmoshenko s type where transverse shear deformatons are taken nto account s not asymptotcally consstent Problem Statement A shell of constant thckness h s consdered as a three-dmensonal elastc body Equatons of the statc problem of asymmetrc (mcropolar momental) theory of elastcty wth free felds of dsplacements and rotatons are the followngs [] []: Equlbrum equatons: mn mn nmk σ µ e σ () Physcal relatons: m m mk ( ) ( ) ( ) ( ) σ µ α γ µ α γ λγ δ µ γ ε χ γ ε χ βχ δ () mn mn nm kk nm mn mn nm kk nm Geometrcal relatons: k γmn mvn ekmnω χmn mωn () nm nm Here σ µ are the components of tensors of force and moment stresses; γmn κ mn are the components of n tensors of deformaton and bendng-torson; V are the components of dsplacement vector; ω n are the components of free rotaton; λµαβγε are physcal constants of the mcropolar materal of the shell; ndces mnk take values It should be noted that f α man equatons of the classcal theory of elastcty wll be obtaned from Equatons ()-() We ll consder three orthogonal system of coordnates α n ( H A( α R) H ) accepted n theory of shells [4] Boundary condtons of the frst boundary-value problem for front surfaces of the shell are accepted: σ p µ m on α ± h (4) ± ± n n n n Boundary condtons on the edge ΣΣ Σ of the shell are boundary condtons of the mxed boundaryvalue problem: σ µ Σ ω ω Σ (5) mnnm p n mnnm m n on Vn V n n n on ω n are the gven compo- where pn mn are the components of the gven loads and moments on Σ ; nents of dsplacement and free rotaton vectors on Σ V n Asymptotc Soluton (Constructon of Internal Problem) of Boundary-Value Problem of Three-Dmensonal Mcropolar Theory of Elastcty n Thn Doman of the Shell It s assumed that the thckness h of the shell s small compared wth typcal radus of curvature of the mddle surface of the shell ( h R) We ll proceed from the followng basc concept [4]: n the statc case general stress-stran state (SSS) of thn shell s composed of nternal SSS coverng all three-dmensonal shell and boundary layers localzng near the surface of the shell edge Σ On the bass of such approach and results of ntal approxmaton of nternal problem the constructon of general two-dmensonal (asymptotc) model of mcropolar thn shells wll be possble (n case of α also model of elastc shell by classcal theory of elastcty) 6

3 Queston of reducton of three-dmensonal statc problem of asymmetrc theory of elastcty for thn doman of the shell to two-dmensonal problem s consdered on the bass of asymptotc method wth boundary layer [4] ncludng the queston of satsfacton of boundary condtons on shell edge Σ At frst we ll consder the constructon of nternal nteractve process For achevement of ths am we ll pass to dmensonless coordnates n three-dmensonal Equatons ()-() of asymmetrc theory of elastcty: p l α ξ α ζ (6) Here quantty pl characterzes the varablty of SSS by coordnates; p l are ntegers l > p ; R s the characterstc radus of curvature of the shell mddle surface; λ s the bg constant dmensonless geometrc parameter determned wth the help of formula h l Followng dmensonless quanttes and dmensonless physcal parameters are also consdered: ± ± V σ µ pn ± mn ± R V σ µ pn mn R (7) R µ Rµ µ Rµ R E µ µ E α α β β γ γ ε ε (8) µ µ µ R µ R µ R µ On the bass of (7) (8) followng system of dmensonless equatons wll be obtaned nstead of system of Equatons ()-() Equlbrum equatons: Physcal-geometrcal relatons: Here σ σ σ σ λ L λ a λ L λ F pl l l pl R v v v λ K λ a λ a σ σ ( ) ( ) pl l l R l pl v l λ K λ Φ λ ( aσ aσ ) p V R V a λ V aσ va σ vσ A ξ AA α R E l V l V µ α µ α aaλ [ σ vaσ vaσ ] a λ ( ) a ω σ σ E 4µα 4µα V p R A µ α µ α a λ V ( ) aa ω aσ a A ξ AA α σ 4µα 4µα p V V µ α µ α a λ ( ) aa ω aσ aσ A ξ R 4µα 4µα p ω R A ω β γ β ( ) ( ) ( ) a λ ω aν aν ν A ξ AA α R γ β γ β γ ω β γ β λ aa ν aν aν ζ γ β γ β γ a ( ) ( ) ( ) l p ω R A γ ε γ ε λ ω aν aν A ξ AA α 4γε 4γε p ω ω γ ε γ ε l ω γ ε γ ε a λ aν aν aλ ν ν A ξ R 4γε 4γε 4γε 4γε (9) () 6

4 p p R A τ τ λ L A ξ AA α A ξ AA α ( σ σ ) ( σ σ ) σ σ σ σ L R R A AA A AA p p F σ σ ξ α ξ α p A v v K ( v v ) A ξ AA α A ξ AA p A v v ( ) v v v v λ ζ K Φ v v a R R A AA A AA R α p p l ξ α ξ α The case s consdered when dmensonless physcal parameters (8) have the followng values: α β γ ε ~ ~ ~ ~ µ R µ R µ R µ Followng replacements of unknown quanttes wll be done: τ λτ τ λτ τ λ τ τ λ τ τ λ τ l l l p c l pc pc ν λ ν ν λ ν ν λ ν ν λ ν ν λ ν lc lc lp lp pc V λ V V λ V ω λ ω ω λ ω lp lc lpc c at p l c p l at p l As a result followng system of equatons wll be obtaned: τ τ τ l c l λ L λ a a R ν ν ν l pc l l pc λ K λ λ τ τ a a R a τ ( ) a ( ) ( ) l p ν p c p c p c p c p c λ L F λ K λ Φ λ aτ λ aτ a 4µ λ µ µλ l pc λ τ e e λ τ a λ µ λ µ a λ µ a τ µ α µ α λ αω ( ) ( ) ( ) p t t a a V l p λ µ l c λ λ τ λ τ τ ζ aa µ λ µ µ λ µ a a ( ) ( ) V µ α µ α ζ a 4µα a 4µ l c l pc l pc l pc l pc ( ) λ ω λ τ λ τ λ g ( ) λ τ τ α a a a 4µα 4µα µ α τ λ ω τ ( ) ( ) p g a a µ α µ α µ α ( ) ( ) l pc κ κ λ γ ε γ ε a β γ β γ a β γ a l 4pc l pc β λ v λ v v ζ aa γ ( β γ ) γ ( β γ ) a a a 4γ β γ γβ β a v v v n n ω β γ ω γ ε γ ε a 4γε γ ε λ λ θ ζ γε γε γ ε γ ε l c l c v v v v a 4 a 4 a () () () (4) 6

5 where p τ A ξ AA α A ξ AA α L p ( τ τ ) ( τ τ ) τ τ τ τ p p F τ τ R R A ξ AA α A ξ AA α L v K v v A AA A τ p p ( ) ( v v ) ξ α AA v v v v K v v p p Φ R R A ξ AA α A ξ AA α V V V e V t V p p c λ A ξ AA α R A ξ AA α g n A v ξ α p V p cv ω p cω λ κ ω λ ξ R A ξ AA α R ω ω ω p c ω θ λ A ξ AA α A ξ R (5) Followng to the asymptotc method the queston s the followng: to reduce three-dmensonal Equatons (4) (wth free varables ξ ξ ζ ) to two-dmensonal ones (wth free varables ξ ξ ) Followng formulas wll be obtaned for dsplacements and rotatons force and moment stresses wth asymp- p l O λ on the bass of system (4): totc accuracy ( ) where V ω ω ω ω λ ζω V V λ ζv V V l pc l pc τ τ λ ζτ τ τ λ ζτ τ τ τ τ l pc l pc v v v v v v λ ζv v v λ ζv τ ( ) ( ) l pc l pc τ p c ζτ v v λ ζ v β γ β ω λ ν γ β γ γ β γ ( ) ( ) ( v v ) l pc µ α µ α p τ τ ( ) λ ω g ( τ τ ) 4µα 4µα 4µ λ µ µλ l pc λ p τ e e λ τ τ ( µ α ) t ( µ α ) t ( ) λ αω λ µ λ µ λ µ 4µ λ µ µλ λ c τ e e τ τ ( µ α ) t ( µ α) t ( ) λ ω λ µ λ µ λ µ ( ) 4γ β γ l pc γβ λ m m ν k k v ( γ ε ) n ( γ ε ) n β γ β γ β γ V V ω δ λ κ ω δ λ k p k k p k k c k k V k k A ξ AA α R A ξ AA α e k p k p k V k k ω A k t V n ω A ξ AA α A ξ AA α p c ω R k (6) 6

6 g V V ω ω k k k k k p c k c δkλ θ δkλ A ξ R A ξ R k p k p τ A τ L A ξ AA α A ξ AA α ( τ τ ) ( τ τ ) k k k k k p ( ( ) ) ( ) ( ) 4µα µ α ω τ g λ ω τ v K τ τ J µ α µ α τ (7) k p k p v A v K v v v v A ξ AA α A ξ AA α ( ) ( ) k k k k k 4γε γ ε 4γε γ ε v θ v v θ v γ ε γ ε γ ε γ ε p c V c τ λ L F v λ K Φ τ τ τ k k k p k p k τ τ k τ k τ k L F τ τ R R A ξ AA α A ξ AA α v v v v K Φ v v k k k k p k p k k k k R R A ξ AA α A ξ AA α The am s to construct asymptotcally strctly nteractve process for averaged along the shell thckness quanttes whch determne the stated problem (e dependng only on quanttes ξ ξ ) From ths pont of vew there s an opportunty to defne values from (6) of force stress τ and moment stress µ The approach s the followng: at the level of ntal approxmaton of the asymptotc method for quanttes τ and ν we have: p c ( ) ( ) ( ) τ τ ξ ξ ν ν ξ ξ λ ν ξ ξ (8) Keepng quanttes up to obtan: p l λ order n equlbrum equatons and ntegratng these equatons by ζ we ll l c l p ˆ τ τ λ ζτ λ ζ L (9) p c l c ˆ ν ν λ ζν λ ζ Φ ( τ τ ) () where τ and ν are constants of the ntegraton: τ τ λ ν λ Φ τ τ () c c L K R ˆ ν are equal to ze- It must be requred that averaged values along the shell thckness of quanttes ro: ˆ τ and ˆ ˆ τ dζ ν dζ () Substtutng (9) and () nto condtons () followng formulas wll be obtaned for l p l c τ λ L ν λ ( τ τ) 6 6 Φ τ and ν : 64

7 Thus for ˆ τ and ˆ ν we ll obtan: l c l p ζ ˆ τ λ ζ τ λ L 6 () Fnally for quanttes p c l c ζ Φ ( ) ν λ ζ ν λ τ τ 6 (4) τ and ν we ll have the sum of (8) () (4): ζ τ τ λ ζτ λ 6 l c l p L (5) p c l c ζ Φ ( ) ν ν λ ζν λ τ τ 6 It should be noted that averaged along the shell thckness quanttes for τ and ν at the level (8) and (5) (6) are equal Thus takng nto consderaton (5) (6) we ll have followng formulas for dsplacements rotatons force and moment stresses nstead of (6): ω ω ω ω λ ζω V V λ ζv V V l pc l pc τ τ λ ζτ τ τ λ ζτ τ τ l pc l pc v v v v v v λ ζ v τ τ ζτ l pc l c l p l pc L ν v λ ζv τ τ λ ζτ λ ζ 6 p c l c ζ ν ν λ ζν λ Φ ( τ τ) 6 The constructed asymptotcs (7) for nternal nteracton process of the stated problem gves an opportunty to reduce three-dmensonal problem to two-dmensonal one (what s already done for dsplacements rotatons force and moment stresses) As n the classcal theory nstead of components of tensors of force and moment stresses statcally equvalent to them ntegral characterstcs are ntroduced n mcropolar theory: forces T S N N moments M H L L L L and hypermoments Λ : h h h h T σ d z S σ d z N σ d z N σ d z h h h h h h h h M zσ d z H zσ d z L µ d z L µ d z h h h h h h h L µ d z L µ d z Λ zµ d z h h h Dsplacements and rotatons of ponts of the shell mddle surface are ntroduced as follows: u ω V w V Ω Ω ω ω ι ζ ζ ζ ζ ζ ζ Satsfyng boundary condtons (4) on shell surfaces z ± h takng nto consderaton (7) (7) followng system of equatons of two-dmensonal problem of mcropolar theory of shells wth free felds of dsplacements and rotatons wll be obtaned: (6) (7) (8) 65

8 Equlbrum equatons: T S N A α AA α A α AA α R ( T T ) ( S S ) ( p p ) M H A α AA α A α AA α ( M M ) ( H H ) N h( p p ) ( AN ) ( AN ) T T ( ) L L L ( L L ) ( L L ) ( ) ( N N ) ( m m ) A α AA α A α AA α R p p R R AA α α L L R R AA ( AL ) ( AL ) ( A ) ( A ) ( S S ) ( m m ) α α L Λ Λ H H h m m ( ) ( ) AA α α Elastcty relatons: N h µ α Γ h µ α Γ N h µ α Γ h µ α Γ Geometrc relatons: ( ) ( ) ( ) ( ) Eh ν T Γ vγ h( p p ) S h ( µ α ) Γ ( µ α) Γ v ν Eh h ν M K vk ( p p ) ν ( v ) h H ( µ α ) K ( µ α) K L h ( β γ) k β( k ι) L h ( γ ε) κ ( γ ε) κ L h ( β γ) ι β( κ κ ) 4γε γ ε m m h 4γε γ ε m m L h κ Λ l γ ε γ ε γ ε γ ε h u w u Γ u Γ u ( ) Ω A α AA α R A α AA α K ψ ψ ψ K ψ ( ) ι A α AA α A α AA α w u ι Γ Ω Γ Ω ( ) ( ) ϑ ψ ϑ l A α R A α Ω Ω A κ A α AA R κ Ω Ω Ω Ω Ω κ α A α AA α A α R System of equatons of thn shells of classcal theory wll be obtaned from system of Equatons (9)-() n case of α (e system of equatons of elastc thn shells of Tmoshenko s type []-[5] wth some dfference) 4 Constructon and Studyng of Boundary Layers We ll proceed from three-dmensonal Equatons ()-() of mcropolar theory of elastcty It s assumed that the surface of the shell edge Σ where stress state wll be consdered s gven wth the help of the equaton (9) () () 66

9 α α Replacng of free varables s done on the bass of formulas: α α ξ α ξ α ζ () l p l where quanttes R λ l p have the same meanng as n case of nternal problem Soluton of the obtaned system of boundary-value problem must satsfy homogeneous boundary condtons on surfaces α ± h of the shell: σ µ () n n We ll pass to dmensonless quanttes (7) (8) and ntroduce followng notatons: l l σ P µ Q V λ U ω λ ϖ (4) mn mn mn mn n n n n As a result three-dmensonal equatons of mcropolar theory n dmensonless form wll be obtaned from Equatons ()-() (wth consderaton of (7) (8)) p l At level O( λ ) of asymptotc accuracy boundary layer dvdes nto 4 ndependent systems of equatons: Force plane problem: P P P P U [ P νp νp ] A ξ A ξ A ξ Е U P ν ( P P ) [ P νp νp ] Е A U Force non plane problem: µ α µ α U µ α µ α P P P P ξ 4µα 4µα 4µα 4µα (5) P P U µ α µ α U µ α µ α P P P P A ξ A ξ 4µα 4µα 4µα 4µα ( µ α ) ( µ α ) ( µ α ) ( µ α) P P P P (6) Momental plane problem: Q Q ϖ γ ε γ ε ϖ γ ε γ ε Q Q Q Q A ξ A ξ 4γε 4γε 4γε 4γε ( γ ε ) ( γ ε ) ( γ ε ) ( γ ε ) Q Q Q Q (7) Momental non plane problem: Q Q Q Q ( β γ ) Q β ( Q Q ) A ξ A ξ A ϖ β γ β Q ( ) ( ) ( Q Q ) ϖ γ ε γ ε Q Q ξ γ β γ β γ 4γε 4γε ϖ β γ β ( ) ( ) ( ) ϖ γ ε γ ε Q Q Q Q Q γ β γ β γ A ξ 4γε 4γε where A A ξ The obtaned equatons of boundary layer n Cartesan coordnates ξ ζ ( ξ Aξ ) wth asymptotc accuracy ( λ ) descrbe SSS of plan and antplane force and momental ndependent problems of mcropolar p l theory of elastcty takng place n semband { ξ < ζ } Requrng that solutons (5)-(8) of boundary layers have fadng character when ξ we ll obtan that such solutons have followng mportant propertes: (8) 67

10 P dζ Q dζ U dζ ϖ dζ n ξ n ξ ξ ξ λ β U dζ ζp d ζ ϖ dζ ζq d ζ ( ) 4µ λ µ 4γ ( β γ ) ξ ξ ξ ξ µ α γ ε U dζ ζp dζ ϖ dζ ζq dζ ξ ξ ξ ξ 4µα 4γε U µ α U dζ dζ dζ dζ ξ P ξ ξ P ξ 4µα ζ µ α ϖ β γ d dζ ξ ζ Q ξ ζ 4γ β γ ( ) (9) From the above ntroduced relatons (specal for mcropolar theory of elastcty) followng mportant concluson can be done: when force and moment stresses are balanced n boundary layer dsplacements and free rotatons wll have the same property 5 Jontng of Asymptotc Expansons of Internal Interactons Process and Boundary Layer Consderng problem of ontng of nternal SSS and boundary layer followng symbolc formula must be ntroduced for the whole SSS of the shell: r ( SSS) ( SSS) ( SSS) а θ λ λ ( SSS) (4) whole n bl bl r θ are called ndcators of ntensty of plane and antplane boundary layers r θ must be chosen so that we can satsfy three-dmensonal boundary condtons on shell edge Σ Now the frst varant of three-dmensonal boundary condtons of mcropolar theory of elastcty wll be consdered when shell edge s loaded wth forces and moments ( Σ Σ Σ ) Satsfyng boundary condtons followng values wll be taken for quanttes r and θ : r θ l p c p l λ boundary condtons on ξ wll be as follows: At level ( ) l pc pc п( ) pc p a( ) p P p Q m τ λ ζτ λ λ ν λ λ l pc pc a( ) pc p п( ) p P p Q m τ λ ζτ λ λ ν λ λ p п( ) p l pc c a( ) c P p Q m τ λ λ ν λ ζν λ λ where p l p c l p c n µλ p n mn Rµλ m n Usng correspondng condtons from (9) and on the bass of (4) boundary condtons for system (9)-() of two-dmensonal equatons wll be obtaned: в (4) h h h α α α α α α α α α h h h T p d S p d N p d h h h α α α α α α α α α α α h h h М p d H p d L m d h h h α α α α α α α α Λ α α h h h L m d L m d m d (4) Let us study the second varant of three-dmensonal boundary condtons of mcropolar theory of elastcty when dsplacements and rotatons are gven on the shell edge ( Σ Σ Σ ) Satsfyng boundary condtons followng values wll be taken for quanttes r and θ n (4): r θ lp c p l λ boundary condtons on ξ wll be as follows: At level ( ) 68

11 V V U V l pc pc п( ) p c p a( ) p λ ζ λ λ ω λ ϖ λ ω l pc pc a( ) p c p п( ) p λ ζ λ λ ω λ ϖ λ ω p п( ) p l pc c a( ) c λ λ ω λ ζω λ ϖ λ ω V V U V V U V lp c lp c where Vn V n ωn λ ωn Wth the help of condtons from (9) boundary condtons for two-dmensonal model wll be obtaned: (4) u u d w u d d h h h α α n n h α α α α α α ω α h h Ω h h h ψ α α V α h V α h ι α α ω α h ω αh h h Mxed three-dmensonal boundary condtons are studed when hnged support takes place Followng values wll be taken for quanttes r and θ n (4): r l p c θ l p c p l λ boundary condtons on ξ wll be as follows: At level ( ) l pc l 4pc п( ) l pc p a( ) p P p Q m l pc pc a( ) pc c п( ) P p Q m p c pc п( ) p c l pc c a( ) c V U V Q m λ τ λ ζτ λ ν λ λ τ λ ζτ λ λ λ ν λ λ λ ν λ ζν λ λ l where V V l l p c p µλ p p µλ l p c p m Rµλ l m l p c m Rµλ m m Rµλ m In ths case usng condtons from (9) followng boundary condtons of hnged-support wll be obtaned for two-dmensonal model: (44) (45) w u d T p d S p d h h h α α h α α α α α α α h h h h h h α α α α α α α α α α α h h h М p d H p d L m d h h h α α α α α α Λ α α α h h h L m d L m d m dα (46) 6 Asymptotc Model of Mcropolar Elastc Thn Shells Thus two-dmensonal theory of mcropolar shells s constructed at level of ntal approxmaton of the asymptotc method System of equatons (9)-() and boundary condtons (4) (or (4) or (46)) ntroduce the asymptotc model of mcropolar elastc thn shells wth free felds of dsplacements and rotatons 7 Appled Theory of Mcropolar Elastc Thn Shells and Its Justfcaton Hypotheses method of constructon of classcal theory of elastc thn shells (e Krkhov-Love s or refned hypotheses) has an advantage above the asymptotc method from pont of vew of engneerng because some smplfcatons were put n the base of theory whch have physcal meanng and also vsblty and clarty Man problem of the constructon of appled theory of mcropolar elastc thn shells s the followng: to formulate such hypotheses that let us reduce three-dmensonal problem of mcropolar theory of elastcty to adequate twodmensonal boundary-value problem For achevement of ths am the use of qualtatve aspects of asymptotc soluton of three-dmensonal boundary-value problem ()-(5) of mcropolar theory of elastcty s approprate n thn doman of the shell In papers []-[4] the mentoned dea s developed: on the bass of qualtatve aspects of asymptotc soluton adequate hypotheses are formulated and as a result statc and dynamc appled theores of mcropolar elastc thn shells and plates are constructed The accepted hypotheses are the followngs: ) Durng the deformaton ntally straght and normal to the shell mddle surface fbers rotate freely n space 69

12 at an angle as a whole rgd body wthout changng ther length and wthout remanng perpendcular to the deformed mddle surface The formulated hypothess s mathematcally wrtten as follows: tangental dsplacements and normal rotaton are dstrbuted n a lnear law along the shell thckness: ( ) ( ) ( ) ( ) V u α α αψ α α ω Ω α α α ι α α (47) Normal dsplacement and tangental rotatons do not depend on coordnate α e V w α α ω Ω α α (48) ( ) ( ) It should be noted that from the pont of vew of dsplacements the accepted hypothess n essence s Tmoshenko s knematc hypothess n the classcal theory of elastc shells []-[5] Here hypothess (47) (48) n full we shall call Tmoshenko s generalzed knematc hypothess n the mcropolar theory of shells ) In the generalzed Hook s law () force stress σ can be neglected n relaton to the force stresses σ ; and analogcally moment stresses µ can be neglected n relaton to the moment stresses µ ) Durng the determnaton of deformatons bendng-torsons force and moment stresses frst for the force stresses σ and moment stress µ we ll take: ( ) ( ) σ σ α α µ µ α α (49) After determnaton of mentoned quanttes values of σ and µ wll be fnally defned by the addton to correspondng values (49) summed up obtaned by ntegraton of the frst two and the sxth equlbrum equatons from () for whch the condton wll be requred that quanttes averaged along the shells thckness are equal to zero α 4) Quanttes can be neglected n relaton to R Now we ll compare man equatons of appled statc theory of mcropolar elastc thn shells from paper [] whch are constructed on the bass of above formulated hypotheses wth analogcal Equatons (9)-() of the asymptotc model It s obvous that equlbrum Equatons (9) and geometrcal relatons () are the same Physcal relatons from paper [] dffer from physcal relatons () only wth underlned terms n relatons for T M L Λ It should be noted that underlned terms n relatons for T and M are the result of the fact that n case of asymptotc theory n relatons for γ quantty σ s not neglected n relaton to σ But as t s known such neglect s adopted n theores of thn shells Analogcal explanaton has also underlned terms n relatons for L and Λ Thus we can say that the general appled statc theory of mcropolar elastc thn shells constructed n paper [] s asymptotcally correct theory Concernng the dynamc theory of mcropolar elastc thn shells t should be noted that the correspondng asymptotc model s constructed n paper [6] and the appled model constructed on the bass of the above formulated hypotheses s ntroduced n paper [] If we compare these two models we ll see that moton equatons and geometrcal relatons (whch have form ()) are the same Concernng physcal relatons we can say that the dfference s underlned terms n relatons () for T M L Λ As n case α classcal model of elastc thn shells of Tmoshenko s type wll be obtaned from asymptotc model (Equatons (9)-()) and also from appled model of paper [] we can say that ths classcal appled refned model of thn shells s the asymptotcally correct model (such concluson can be also done n case of dynamc problem) It should be noted that n papers [7] [8] appled theores of mcropolar elastc thn plates and bars constructed n papers [4] [7] are ustfed on the bass of asymptotc method 8 Conclusons In the present paper the queston of reducton of three-dmensonal boundary-value problem of mcropolar and classcal theores of elastcty to general appled theores of thn shells s studed The asymptotcs of sngularly perturbed boundary-value problem of three-dmensonal mcropolar theory of elastcty s studed n thn doman of the shell The nternal teraton process and boundary-layers are constructed ontng of these two teraton processes s studed and boundary condtons are obtaned As a result two-dmensonal asymptotc model wth free felds of dsplacements and rotatons of mcropolar shells s constructed Transverse shear deformatons are 64

13 automatcally taken nto consderaton n the constructed model Partcularly classcal asymptotc theory of elastc thn shells wth consderaton of transverse shears can be obtaned from the above mentoned mcropolar model Hypotheses are accepted for the constructon of general appled theory of mcropolar elastc thn shells The hypotheses are adequate to the asymptotc behavor of the soluton of three-dmensonal problem Such approach ensures the asymptotc exactness of the constructed mcropolar and classcal theores of thn shells wth consderaton of transverse shears References [] Fredrchs KO and Dressler RFA (96) Boundary Layer Theory for Elastc Plates Communcatons on Pure and Appled Mathematcs - [] Green AE (96) On the Lnear Theory of Thn Elastc Shells Proceedngs of the Royal Socety Seres A [] Vorovch II (966) On Some Mathematcal Questons of Plate and Shell Theores Proceedngs of the II Unon Congress of Theoretcal and Appled Mechancs 6-6 (In Russan) [4] Goldenvezer AL (976) Theory of Elastc Thn Shells Moscow (In Russan) [5] Kaplunov JD Kossovch LYu and Nolde EV (998) Dynamcs of Thn Walled Elastc Bodes Academc Press [6] Agalovyan LA (997) The Asymptotc Theory of Ansotropc Plates and Shells Moscow (In Russan) [7] Rogacheva NN (994) The Theory of Pezoelectrc Plates and Shells Boca Raton London [8] Ustnov YuA and Shenev MA (978) On Some Drectons of Development of the Asymptotc Method of Plates and Shells Calculatons of Plates and Shells -7 (In Russan) [9] Sargsyan SH (99) General Two-Dmensonal Theory of Magnetoelastcty of Thn Shells Yerevan (In Russan) [] Altenbach H and Eremeyev VA (9) On the Lnear Theory of Mcropolar Plates Zetschrft fur Angewandte Mathematk und Mechank (ZAMM) [] Altenbach J Altenbach H and Eremeyev VA (9) On Generalzed Cosserat-Tape Theores of Plates and Shells: A Short Revew and Bblography Archve of Appled Mechancs [] Sargsyan SH () General Theory of Mcropolar Elastc Thn Shells Journal of Physcal Mesomechancs [] Sargsyan SH () The General Dynamc Theory of Mcropolar Elastc Thn Shells Reports of Physcs [4] Sargsyan SH () Mathematcal Model of Mcropolar Elastc Thn Plates and Ther Strength and Stffness Characterstcs Journal of Appled Mechancs and Techncal Physcs [5] Sargsyan SH (8) Boundary-Value Problems of Asymmetrc Theory of Elastcty for Thn Plates Journal of Appled Mathematcs and Mechancs [6] Sargsyan SH () The Theory of Mcropolar Thn Elastc Shells Journal of Appled Mathematcs and Mechancs [7] Sargsyan SH () The Constructon of Mathematcal Model of Mcropolar Elastc Thn Beams on the Bass of the Asymptotc Theory News of Hger Educatonal Inttutes The North Caucasus Regon Natural Scences 5-7 (In Russan) [8] Sargsyan SH () The Asymptotc Method of the Constructon of Mathematcal Models of Mcropolar Elastc Thn Plates Scentfc Proceedngs of GSPI 7-7 (In Russan) [9] Goldenvezer AL Kaplunov JD and Nolde EV (99) On Tmoshenko-Ressner Type Theores of Plates and Shells Internatonal Journal of Solds and Structures [] Palmov VA (964) Basc Equatons of the Theory of Asymmetrc Elastcty Appled Mathematcs and Mechancs 8 7- (In Russan) [] Nowack W (986) Theory of Asymmetrc Elastcty Pergamon Press Oxford New York Toronto Sydney Pars Frankfurt [] Pelech PL (977) Stress Concentraton around the Holes n Bendng Transversely Isotropc Plates Kev (In Russan) [] Grgolyuk EI and Kulkov GM (988) Multlayered Renforced Shells Calculaton of Pneumatc Tres Moscow (In Russan) [4] Grgorenko YM and Vaslenko AT (98) Theory of Shells of Varable Stffness Kev (In Russan) [5] Percev AK and Platonov EG (987) Dynamcs of Plates and Shells Lenngrad (In Russan) 64

14 [6] Sargsyan AA () Asymptotc Analyss of Dynamc Intal Boundary-Value Problem of Asymmetrc Theory of Elastcty wth Free Rotatons n Thn Doman of the Shell News of NAS Armena Mechancs (In Russan) [7] Sargsyan SH () Effectve Manfestatons of Characterstcs of Strength and Rgdy of Mcropolar Elastc Thn Bars Journal of Materals Scence and Engneerng