Here and further the following reduction of record is applied.

Transcription

1 MODL OF SRSSD-SRAIND SA OF MULILAYR MASSS WIH RGARD FOR NON-IDAL CONAC OF LAYRS V.G.Piskuov, A.V.Mrchuk We cosider lyered desig i rectgulr crtesi system o coordites. o xes o coordites x, y, there corresod umbers,,. A oit o level o idex desigtes oertio o dieretitio. Is mde summtio o lhbetic idexes. Stri d stress o lyer re coected by kow rities or orthotroic o medium. σ = ii ce (, i j =,,) ij ij () σ = ij Ge ( i j =,, ). ij ij For develomet o model o lyered mses used itertive method S.A.Ambrtcumi []. At the irst stge we set lier distributio movigs o thickess o lyered desig. u ( xy,, ) = i u ( xy, ) is ( ) is () ( xy,, ) = ( xy, ) ( ) w u where ( xy, ), ( xy, )-horiotl movigs o obverse surces lyered msses i i ( xy, ), ( xy, ) - verticl movigs o obverse surces o msses w u u w () d d i = / () = () i i () = / c d/ / c d () = (). Here d urther the ollowig reductio o record is lied. k r ( k) ( k) / c d + / c d ( k) ( k) r= ( ) = ( ) () r k = r ( k) / c d r= r ( kr, umber o lyer,, coordite o borders o lyer). r r At the secod stge we receive such exressios or movigs: ( xy,, ) = ( xy, ) ( ) + ( xy, ) ( ) u u w u ( x, y, ) = w ( x, y) ( ) u ( xy, ) q ( xy, ) q ( xy, ) w xy w xy ϕ i is is i, i where = + sum o exterl horiotl orces o obverse i i i surces o msses (, ), (, ) Fuctio o shit, i lyers isotroic, orces o 4 obverse surces o msses ( ) = ( ) = 4 ()

2 ( )/ d H c () H ()/ c / c d = d / 5 d c H () = () d () d ( ) = + ( ) d F ( c G ) ( c + G ) d () d i d () = d i ()/ i F = F F G ()/ d G i ()/ G i i d F G ϕ () = ()/ i d i ( )/ i i d ϕ () = 4 ()/ i d cii d () i F ii cii d () = () i d d i d d F F G F G i G ϕ () = d d d d i () +( () / ) We lso receive exressios or stress σ. σ w β ξ wii d d d d () () ( =5,, ). d () = ( xy,, ) = ( xy, ) ( ) + ( xy, ) ( ) ( =,..., 5) (4), d

3 γ () = d c () i i, φ υi i () = d γ (), γ c () = d φ ()/ i c i,, φ ( d ) d ( d ) i / () d ()/ β = c ξ = c υ c c ( d ) d d / c d d () () () ()., i i, he exressios () d (4) orm the bsis or costructio o model stressed-stried o coditio o lyered msses. We receive deormtios usig exressio or movigs (). We write dow stress so: = + + ( i j =, ) σ σ be be b ii ii ii ij ij i Ge = ( i j =,, ). ij ij ij σ We shll receive system o the dieretil equtios o the bsis o the Rysser s vritio ricile. δ R δ A = (6) { }[ ][ ][ ][ ][ ][ ][ ]{ } R = W d F d D d F d W dv V { }{ } A = U q ds { W} = { s s } S u u w [ ],, he ot ero members o mtrix (5) d : d (, ) =, d (, ) =, d(,) =, d(,) 4 = d / dx, d(,) 5 = d/ dy, d(,) 6 = d / dx, d(,) 7 = d / dy Not ero he members o mtrix F : F(,) = s(), F(,) 4 = ϕ(), F(,) = s(), F(,) 5 = ϕ (), [ ] F(, ) = ( ), F( 4, ) = β ( ), F( 46, ) = ξ( ), F( 47, ) = ξ ( ) the ot ero members mtrixes [] d : d(, ) = d / dx, d( 5, ) = d / dy, d( 7, ) = d / d, d(, ) = d / dy,

4 4 d(,) 6 = d / dx, d(,) 9 = d / d, d(,) = d / d, d(,) 8 = d / dx, d(,) 44 = he ot ero members o symmetric mtrix : D(,) = b, D(,) = b, D(,) 4 = b, D(,) = b, D(,) 4 = b, D(,) 4 =, [ D] D( 44, ) = b, D( 55, ) = G, D( 56, ) = G, D( 66, ) = G, D( 77, ) = G, D( 78, ) = G, D( 88, ) = G, D( 99, ) = G, D( 9, ) = G, D(, ) = G { U} ={ u, u, u, u, w, w } {}, q = { q, q, q, q, q, q} System o the dieretil equtios: [ d ][ F][ d][ D][ d][ F][ d ]{ Wd } = {} q he mtrix [ F ] such, s [ ] ( s =, = 4,,, ) mtrix [ d ] d [ d ] such, s [ ]. (7) F, but idexes s d re relced o s d d d [] q q q q q q q d but irst derivtive o coordites x d y egtive {} = { },,,,,,,. We shll demostrte oortuities o oered techique o ccout symmetric o thickess o three-lyered msses. he verge lyer ws shred o le o symmetry uder the l o msses. A msses i the l squre. O the bottom surce o msses verticl movigs re orbidde. O ed ces o msses is relied suorted by tie o Nvier. he chrcteristics o lyers ollowig: ( ) ( ) ( ) ( ) / = / = ν = ν = ν =,. he sum o thickess o outside lyers is equl to thickess o verge lyer. Height o msses is equl o hl o his legth. O the to surce the msses is loded by orce q( x, y) = q si( πx/ )si( π y/ ). he rictio i oe o strtiictio is wy. Sliery cotct i oe o strtiictio we simulte by thi lyer. he thickess o lyer i ive hudred time is more thi th thickess o msses. Its chrcteristics ollowig: G = / /( + ν ) = = G = G = G / ν = ν = ν = ν. ( I tble results o ccout re resulted t vrious ) / I lces o their mximum meigs (r - rectio o the bsis). ble U ( ) q r,69-7,4 -,4,754-7,59 -,85,755-7,546 -,85 xct,769-7,55 -,666 Good coormity o the oered roch exct three-dimesiol to the lyticl decisio is visible. As the secod exmle we shll cosider the bove-stted desig, but with reely sggig bottom surce. Results o ccout o oered techique d o exct threedimesiol to the decisio re resulted i tble.

5 5 ble U ( ) q,94-8,95,56-8,75,6-8,77 xct,44-8,744 As the third exmle we shll cosider symmetricl o thickess File with orthtroic by exterl lyers. heir chrcteristics ollowig: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = 7 * ΜΠ = = = = 6, 9 * ΜΠ G = G =, 8* ΜΠ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) G = G = G = G = 45, * ΜΠ ν = ν = ν = ν = ν = ν = 5, hickess o the bottom lyer - ( 4 )/ thickess o the to lyer ( 4 )/. A 4 verge lyer trsversely isotroic: = = ΜΠ ) = ΜΠ G = G = G =, 846* ΜΠ ν =, ν = ν =. Legth o msses twice is more th his thickess. O the to surce o msses orce q( x, y) = q si( πx/ )si( π y/ ) Works, the bottom surce is ree sgs. I tble results o ccout o such desig o re resulted to oered techique d uder the decisio []. he results re give i cetre o msses o the to surce. ble Oered he decisio [] U U q( ) q( ), 8,84,7 7,987 he cosidered exmles demostrte high ccurcy o oered techiques t ccout o lyered ltes d msses with rigid d sliery cotct o lyers. O the bsis o oered model esily to costruct iite elemet. We shll cosider costructio o rectgulr iite elemet. We shll reset movigs o obverse surces o msses s ollows: u = is { u} { V u} w = { u} { V w}, where { u}, { w} - kow uctios [] d { V } { } u u x y is m m V = w w ( x m, y m), w ( x y m, m), w ( x y m m, m), ( =,,, ),, { } { } = (, ), 4 required rmeters rorite to them i uits. Now the vector o required uctios c be writte dow so: =. (9) { W } [ ]{ V } he mtrix o rigidity ccets such kid: = s [] [ ][ ][ ][ ][ ][ ][ ][ ][ ] k d F d D d F d dds (8) ()

6 6 hus, mthemticl model stressed- stried o coditio o lyered msses is costructed. he model hs high ccurcy. It llows to simulte sliery cotct o lyers without rictio. hus ot the order o ermittig system o the equtios is icresed, d t its relitio the method o eite elemets does ot icrese qutity o required degrees reedom. he dieretil oertors icluded i system the equtios re similr kow i the clssicl theory o shells. It cilittes costructio o iite elemet. Presece i system o the dieretil equtios o derivtive o exterl orces llows to use her or the decisio o cotct roblems with sti o cotct commesurble with thickess o msses..ambrtcumi S.A. he theory isotroic o ltes. -Moskow: Sciece, (Russi).. Piskuov V.G., Sietov V.S., uimetov Sh.Sh. Solutio o the sttic roblem or lyered orthotroic ltes i sce sttemet//itertiol lied mechics V.6.- N.-P Piskuov V.G. Accout o lyered lt shells d ltes by method iite elemets.-kiev: Vish shkol, (Russi)