5th Iteratioa Coferece o Advaced Desig ad Maufacturig Egieerig (ICADME 15) Series form soutio for a graded composite strip with eastic variatio i spa directio Qig Yag 1,,a *, Weipig Liu 1,b, Muhuo Yu,c, Doga Xiu 1, Pig Cheg 1, Lie Jia 1, Peg Xu 1, Ra Wei 1, Jigjig Su 1 1. Shaghai Aircraft Maufacturig Co., Ltd, COMAC, Shaghai, 436;. Coege of Materias Sciece ad Egieerig, Doghua Uiversit, Shaghai, 16 agqig3@comac.cc; iuweipi@comac.cc; umuhuo@dhu.edu.c Ke words: Series soutio, Graded composite strip, Spa directio, Fourier series i compe form Abstract: A graded composite strip with eastic variatio i spa directio is ivestigated i the paper. So far, studies about materias variatio i spa directio are rare. Here, the strip is treated as a pae probem, ad the moduus varies epoetia with spa directio, ad Poisso ratio is costat. B usig Fourier decompositio i compe form, the series soutio for the strip is obtaied ad is verified b fiite eemet mode. Itroductio Ma researchers have performed ma studies o the mechaics behavior of ihomogeeous beams/strips ad reated structures [1-3]. Sakar [4] obtaied a easticit soutio of a fuctioa graded beam subjected to a trasverse siusoida oad, i which the eastic moduus ehibits epoetia variatio through the thickess. Zhog ad Yu [5] preseted a geera soutio of a fuctioa graded beam with arbitrar graded variatios of materia propert b the Air stress fuctio. Yag [6] ivestigated the mechaics behavior of a bi-aer fuctioa graded catiever beam with cocetrated oads, whose eastic moduus of each graded aer ca varies with the thickess as a arbitrar fuctio, respective. A Eastic soutio of a FG catiever beam with differet moduus i tesio ad compressio uder bedig oads is studied b Yag[7].Wag ad Li studied [8] the bedig probem of a bi-materia beam with a viscoeastic itermediate aer. There are ma researches about beams/strips with moduus varig aog thickess, but few works about moduus varig with spa directio. I this paper, a graded composite strip, treated as a ihomogeeous pae probem, ad with eastic gradatio i spa directio, is ivestigated. The strip is subjected to smmetrica trasverse oads ad its moduus is assumed to var epoetia i spa directio, ad its Poisso ratio is hod as a costat. B usig Fourier decompositio i compe form i spa directio, a cosed form soutio i series epasio is obtaied ad verified b fiite eemet resuts. Mode ad basic formuatio Cosider a graded composite strip subjected to smmetrica trasverse oads, i a Cartesia coordiate sstem, i Fig 1.Its moduus var epoetia aog the ogitudia directio ad Poisso ratio is hod costat. The thickess ad egth are h ad, respective. Fig.1 Schematic of graded composite strip. 15. The authors - Pubished b Atatis Press 1848
The mode is treated as pae easticit probem. Hece, i the absece of bod force the equatios are give as τ τ =, = (.1) where, ad τ represet the stress compoets. The reatios betwee strais ad dispacemets are u w u w ε =, ε =, γ = (.) where ε, ε, γ deote the strai compoets ad u, w deote the dispacemet compoets. The, the strai compatibiit equatio ca be derived from Eq. (.1) as ε ε γ = (.3) The costitutive reatios of strip are ε = s11 s1 ε = s1 s (.4) γ = s44τ where s are eastic compiace parameters. Comparig with homogeeous materias, s are fuctios of gradatio coordiate. As for two-dimesio ihomogeeous probem, s ca be further described b s = s (, ) (.5) I order to satisf the equatios of equiibrium, the stress compoets are defied i terms of Air stress fuctio ϕ (, ) as ϕ ϕ ϕ =, =, τ = (.6) B substitutig Eq.(.6), Eq.(.4) ito Eq.(.3) ad with cocer of Eq.(.5), we ca obtai the goverig equatio i Air stress fuctio 4 3 4 ( s ϕ 11 s ϕ 1 ) s ϕ 1 ( s ϕ 44 ) s ϕ = (.6) 4 Here, the eastic moduus is assumed to var epoetia aog the ogitudia directio. The, the fuctio of moduus, respected to ogitudia coordiate, ca be defied b E ( ) = Ee β (.7) where E is the basic moduus at =, ad β is the ihomogeeous factor i ogitudia directio. I pae stress probem, the reatio betwee s ad E ( ) ca be writte as 1 1 v v (1 v) (1 v) s11( ) = s( ) = = ; s E ( ) Ee β 1 ( ) = s 1 ( ) = = ; s E ( ) Ee β 44( ) = = (.8) E ( ) Ee β v If trasfer the probem to pae strai, we ca repace v b v* = i Eq.(.8). 1 v Noticig Eqs.(.7) ad (.8), the goverig equatio (.6) of the probem ca be give b 4 3 4 3 4 ϕ ϕ ϕ ϕ ϕ v ϕ β β β β ϕ = (.9) 4 3 4 Eq.(.9) usua is soved b method of separatio of variabes. We thus write Air stress fuctio ϕ (, ) i the separated-variabe form as 1849
ϕ (, ) = g ( ) f( ) (.1) i If take g ( ) = e ξ, we have () (, ) ( ) i = f e ξ ; (, ) ( ) i = ξ f e ξ (1) ; τ (, ) ( ) i = iξ f e ξ (.11) It is observed that Eq.(.11) coud satisf arbitrar tractio coditios described b Fourier series epasio i compe form o the boudaries =± h. Thus, it ma be the soutio of the probem. Now, substitutig Eq.(.1) ito Eq.(.9), we have (4) () ( i ) [ f ( ) ( i v ) f ( ) ( i ) f( )] e ξ ξ ξβ β ξ β ξ β = (.1) Eq.(.1) is a fourth-order iear differetia equatios with costat coefficiets of f( ), ad its characteristic equatio ca be give b 4 ω (ξ iξβ vβ ) ω ξ ( β iξ) = (.13) The, we obtai four characteristic roots about ω ω ω 1 ω 3 = = = vβ ξ ( iβ ξ ) β v β 4 vξ ( iβ ξ ) vβ ξ ( iβ ξ ) β v β 4 vξ ( iβ ξ ) vβ ξ ( iβ ξ ) β v β 4 vξ ( iβ ξ ) vβ ξ( iβ ξ) β v β 4 vξ( iβ ξ) ω4 = (.14) It is eas to be proved that whe β, v, ξ, Eq. (.13) has ot mutipe roots ( β = deote the case of homogeeous materias.). Therefore, the geera soutio of f( ) ca be writte as ω1 ω ω3 ω4 f ( ) = C e C e C e C e (.15) where j 1 3 4 C ( j = 1,,3,4) are compe costats to be determied. Therefore, the stress fuctio ϕ (, ) ca be determied b Eqs.(.13) ad (.19) iξ ϕ (, ) = e [ Ce Ce Ce Ce ] (.16) ω1 ω ω3 ω 1 3 4 4 The series soutios of a graded composite strip Cosider a graded composite strip subjected to trasverse oads i Fig.1 ad its boudar coditios is defied b ( h, ) = (, h) = q ( ), τ ( h, ) = τ (, h) = (3.1) I coordiate sstem give i Fig.1, the tractio oads ca be see as odd fuctios, therefore, the oads Eq.(3.1) ca be epaded b Fourier series i compe form as π i q ( ) = ce (3.a) = with i π c = q d =± ± ± I order to satisf the boudar coditios i the form of series, the stress fuctio ca be give i the form of superpositio of terms as ( )si( ), 1,, 3, (3.b) ω 1 3 1 ω ω 3 4 ω 4 iξ (3.3) = ϕ(, ) = ( C e C e C e C e ) e 185
where ξ = π. The, the stress compoets ca be derived as ω 1 3 4 (, ) ( 1 1 ω ω 3 3 ω 4 4 iξ ) = C e C e C e C e e = ω ω ω ω ω 1 3 4 (, ) ( 1 ω ω 3 ω 4 iξ ) = ξ C e C e C e C e e = ω 1 3 4 (, ) ( 1 1 ω ω 3 3 ω 4 4 iξ ) = i C e C e C e C e e = τ ξ ω ω ω ω (3.4) For coveiece, the stress fuctio ad stress compoets ca be epressed i the form of trigoometric fuctio. It is observed from Eq.(.14) that ω j (j= 1,,3,4) are compe, ad ω 1 ad ω are a pair of opposite umber, ad ω 1 ad ω are aother pair of opposite umber. So we have ω 1 = µ θi ω = θi (3.5) ω3 = η θi ω4 = η θi i e = cos( ) isi( ) B usig Euer formuatio, the stress fuctio i trigoometric form i e = cos( ) isi( ) ca be obtaied µ 1 = (3.6) η η iξ C3e (cos( θ) isi( θ)) C4e (cos( θ) isi( θ))] e ϕ(, ) = [ C e (cos( θ ) isi( θ )) C e (cos( θ ) isi( θ )) The cosequet stress compoets are iξ µ (, ) e [ C 1e ( i)(cos( ) isi( )) = = µ θ θ θ C e i i ( µ θ )(cos( θ ) si( θ )) C e i i η 3 ( η θ )(cos( θ ) si( θ )) C e i i η 4 ( η θ )(cos( θ ) si( θ ))] iξ µ (, ) i e [ C 1e (cos( ) isi( )) = = ξ θ θ C e (cos( θ ) isi( θ )) C e (cos( θ ) isi( θ )) η 3 C e (cos( θ ) isi( θ ))] η 4 iξ µ 1 = τ (, ) = iξ e [ C e ( µ θ i)(cos( θ ) isi( θ )) C e ( µ θi)(cos( θ ) isi( θ )) C e ( η θ i)(cos( θ ) isi( θ )) η 3 η C4e ( η θi)(cos( θ) isi( θ))] (3.7) Accordig to the tractio coditios (3.1) ad oticig Eq.(3.7), we have a set of equatios 1851
C e (cos( θ h) isi( θ h)) C e (cos( θ h) isi( θ h)) µ h h 1 C e (cos( θ h) isi( θ h)) C e (cos( θ h) isi( θ h)) = c ηh ηh 3 4 h µ h 1 (cos( θ ) si( θ )) (cos( θ ) si( θ )) C e h i h C e h i h C e (cos( θ h) isi( θ h)) C e (cos( θ h) isi( θ h)) = c ηh ηh 3 4 µ h h 1 µ θ θ θ µ θ θ θ C e ( i)(cos( h) isi( h)) C e ( i)(cos( h) isi( h)) C e ( η θ i)(cos( θ h) isi( θ h)) C e ( η θ i)(cos( θ h) isi( θ h)) = ηh ηh 3 4 h µ h 1 µ θ θ θ µ θ θ θ C e ( i)(cos( h) isi( h)) C e ( i)(cos( h) isi( h)) ηh ηh C3e ( η θi)(cos( θh) isi( θh)) C4e ( η θi)(cos( θh) isi( θh)) = (3.8) We ca sove 4 equatios for 4 ukows from Eqs.(3.8), the we obtai the eastic soutio of a graded composite strip with moduus varig aog ogitudia directio. Eampe ad discussio Sove the strip i Fig 1. Take =, h= 1, c= 1, v=.3, ad assume E = 1MPa. The boudar coditios of the probem are addressed b, < / 1 π q ( ) = si( ), / 1 / 1 (4.1), / 1< which depicts a discotiuous pressure oad. O the upper ad ower edges of the strip, the distributios of orma stress satisf Eq.(4.) which is smmetric with the cross sectio = 1 that is the positio with the maimum pressure, as show i Fig,3 Fig. The distributio of o Fig.3 The distributio of o the upper edges of strip the ower edges of strip Fig.4 ad Fig.5 show the distributios of stress i β = 1,,3 ad β = 1,, 3, respective. It is cear from Fig.1 that the biggest stress do t occur o the cross sectio = 1 with the biggest pressure, ad its positio moves toward the directio of bigger moduus, with β chagig. The smmetr of the orma stress distributio is destroed b ihomogeeit of materias, which is differet from the case of homogeeit. It is aso ca be see that Fig.4 ad Fig.5 are smmetric with each other. Whe β >, the curves i the right part of Fig.4 eperiece a process that deca from positive vaue to egative vaue first, the the vaish, ad vice versa i 185
the case of β <. Fig.4 The ifuece of β o Fig. 5 The ifuece of β o The cotour pots of stress distributio are cacuated b FEM ad series methods, respective, as show i Fig 13 Fig.6 Cotour Pot of b theor method Fig.7 Cotour Pot of b FEM From Figs.6 ad 7, it is cear that the stress fied is asmmetric with the ie of the maimum pressure oadig ad teds to move toward the directio with bigger moduus. The errors betwee FEM resuts ad theor resuts seem ver sma, which ca be verified b Fig.8 1853
Fig.8 The FEM ad series resuts of o the pae = Cocusios I this paper, a graded composite strip with eastic gradatio i spa directio is ivestigated b Fourier decompositio i compe form. First, a geera soutio of a fourth-order goverig equatio b Air fuctio is obtaied. Immediate, a series soutio for the probem of the strip subjected to trasverse oads is derived. From a eampe proposed here, it ca be foud that the ihomogeeit of materias, aog spa directio, has a sigificat ifuece o the mechaics behavior of a strip. Especia, whe the strip subjected to a pair of smmetrica orma tractios o the upper ad ower boudaries, the biggest stress withi the strip do t occur o the ocatio of the maimum tractio oadig, whie moves toward the directio with bigger moduus. Comparig the aatica soutios i series form with FEM resuts, a good agreemet has achieved. Ackowedgemets This work was supported b Natioa iteratioa scietific ad techoogica cooperatio projects of Chia (No. 13DFG54) ad the project of cooperatio of Chia ad EU (SAMC14-JS-13-1). Refereces [1]. S Suresh., A Mortese, Fudametas of fuctioa graded materias., IOM Commuicatios Limited, Lodo, 1998. []. D. Zekert, The Hadbook of Sadwich Costructio., Materias Advisor Services Ltd, Lodo, 1997. [3]. S. Abrate, Impact o Composite Structures., Cambridge Uiversit Press, Cambridge, 1998. [4]. B.V. Sakar, Compos. Sci. Tech. 61 (1) 689 696. [5]. Z. Zhog, T Yu, Compos. Sci. Tech. 67 ( 7) 481 8. [6]. Q. Yag, B.L. Zheg, K Zhag ad J.X. Zhu, Arch. App. Mech. 83(13) 455~466. [7]. Q. Yag, B.L. Zheg, K. Zhag ad J.X. Zhu, App. Math. Mode. 38(14) 143-16. [8]. J. Li, B.L. Zheg, Q Yag ad X.J. Hu, Compos. Struct.17 (14) 3-35. 1854