Effect of Material Gradient on Stresses of Thick FGM Spherical Pressure Vessels with Exponentially-Varying Properties

Transcription

1 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, Effect of Mateial Gadiet o Stesses of Thick FGM Spheical Pessue Vessels with Expoetially-Vayig Popeties M. Zamai Nejad a, *, M. Ghaibi a a Depatmet of Mechaical Egieeig, Yasouj Uivesity, Yasouj, Ia. ARTICLE INFO Aticle histoy: Received 5 Ap. 204 Accepted 23 May 204 Available olie 3 Aug. 204 Keywods: Thick sphee Pessue vessel Expoetial Fobeius seies method (FSM) Fiite elemet method (FEM) ABSTRACT Usig the Fobeius seies method (FSM), a aalytical solutio is developed to obtai mechaical stesses of thick spheical pessue vessels made of fuctioally gaded mateials (FGMs). The cylide pessue vessel is subjected to uifom iteal pessue. The modulus of elasticity is gaded alog the adial diectio accodig to powe fuctios of the adial diectio. It is assumed that Poisso s atio is costat acoss the cylide thickess. Pimaily, displacemets ad stesses have bee obtaied as closed-fom solutio. Next, the pofiles ae plotted fo diffeet values of ihomogeeity costat alog the adial diectio. Fially, the poblem was solved by usig the fiite elemet method (FEM). The obtaied esults of fiite elemet method wee compaed with those of the aalytical method. The aalytical solutios ad the solutios caied out though the FEM showed good ageemet. The values used i this study ae abitaily chose to demostate the effect of ihomogeeity o displacemets, ad stesses distibutios.. Itoductio I mateials sciece, fuctioally gaded mateials (FGMs) ae advaced composite mateials whose mechaical popeties vay cotiuously fom oe suface to aothe at maco level. Fom the pespective of solid mechaics, FGMs ae o-homogeeous elastic mediums. Usig this popety i the FGMs, a egiee ca desig composite mateials such that ay potio of the mateials eaches the same safety level []. Tutucu ad Oztuk [2], by usig the ifiitesimal theoy of elasticity, obtaied closed-fom solutios fo stesses ad displacemets i fuctioally gaded cylidical ad spheical vessels subjected to iteal pessue. Assumig that the mateial popeties except Poisso s atio ae vaiable as powe law of adius, Nayak et al. [3] ivestigated displacemets, stais, ad stesses of the FGM spheical vessel. Usig a accuate method, elastic aalysis of iteally pessuized thick-walled spheical pessue vessels of fuctioally gaded mateials was studied by You et al. [4]. Tutucu [5] deived the solutios of FGM thick cylidical shells with expoetially-vayig popeties. Assumig that the mateial popeties ae Coespodig autho: addess: m_zamai@yu.ac.i (Mohammad Zamai Nejad).

2 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, vaiable i ay abitay diectio, ad Poisso s atio is costat, by usig teso aalysis, Zamai Nejad et al. [6] obtaied a complete ad cosistet 3-D set of field equatios of FGM thick shells of evolutio with abitay cuvatue ad vaiable thickess. Che ad Li [7] obtaied the appoximate solutio fo thick cylides ad spheical pessue vessels. Tutucu ad Temel [8] gaied axisymmetic displacemets ad stesses i fuctioally gaded hollow cylides, disks ad sphees subjected to uifom iteal pessue usig plae elasticity theoy ad complemetay fuctios method. Assumig that the displacemet fuctio is ukow i the goveig equatio, Che ad Li [9] obtaied displacemet ad stesses i thick-walled cylides ad sphees made of fuctioally gaded mateials. Zamai Nejad et al. [0] peseted a aalytical solutio ad a umeical solutio fo stesses ad adial displacemet of paabolic FGM solid sphees with paabolic vayig popeties subjected uifom exteal pessue. I aothe study, usig a aalytical method, Zamai Nejad et al. [] obtaied a exact solutio fo stesses ad displacemets of pessuized thick sphees made of fuctioally gaded mateial with expoetiallyvayig popeties. I this pape, by usig the Fobeius seies method, a closed-fom aalytical solutio fo displacemets ad stesses of FGM thick spheical pessue vessels with expoetial vayig popeties has bee obtaied. Fo the umeical solutio, a commecial fiite elemet pogam ANSYS 2 has bee used. 2. Expeimetal Coside a fuctioally gaded hollow sphee with ie adius a ad oute adiusb. The sphee is subjected to iteal uifom pessue P. (Fig. ) The mateial is isotopic ad fuctioally gaded while the Poisso s atio emais costat thoughout the etie sphee. Modulus of elasticity E is assumed to vay expoetial fom as follows: a β E E b a ie [] Whee E i ad β ae modulus of elasticity i ie suface ad ihomogeeity costat, espectively. I the symmetical defomatio case, the equilibium equatio fo the adial ad cicumfeetial stess compoets ( σ ad espectively) disegadig the body foces σθθ ad ietia tems takes the followig fom: dσ σ σ + 2 θθ 0 [2] d Two adial ad cicumfeetial stai compoets ( ε ad ε θθ espectively) ca be expessed as du ε [3] d u ε θθ [4] Hee, the displacemet i the -diectio is deoted by u. The stess ad stai elatios fo ohomogeeous ad isotopic spheical shell ae σ A 2B ε E [5] θθ φφ B A B σ σ + ε θθ εφφ Whee A ad B ae elated to Poisso s atio ν as ( ν) A [6] + ν 2 ν B ( )( ) ν 2 [7] ( + ν)( ν) Usig Eqs. [] to [5], the Navie equatio i tems of the adial displacemet is 2 u ( ) h u 2( hν ) u 0 [8] Whee ν ν ν [9] β h b a Eq. [9] ca be solved by usig powe seies method with the solutio i the fom of + s u a [0] 0 Substitutig Eq. [0] ito Eq. [8] s a0 ( s )( s + 2) + ( + s )( + s + 2) a + s + h ( + s + 2ν ) a 0 [] Eq. [] has a idicial equatio ad a ecuece elatio, espectively. Sice a0 0

3 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, Fig.. Coss sectio of the FGM thick-walled spheical pessue vessel s + ( s )( s + 2) 0 [2] s2 2 h ( + s + 2ν ) a a [3] ( + s )( + s + 2) The idicial equatio has oots that diffe by a itege, thus oly oe of the solutios is i the fom of Eq. [0]. Expasio of the above ecuece elatio, the coefficiet a i tems of a 0 ad Gamma fuctios ae obtaied. 2 ( ) Γ ( + s + 2ν ) Γ ( s + 3) a0h a s( s + )( s + 2) Γ ( s + 2ν ) Γ ( + s) Γ ( + s + 3) [4] Fo the fist oot ( s +, a0 ), the fist solutio of Eq. [8] obtaied i the fom of ( ) + u + a s Whee 6( ) Γ ( + + 2ν ) a ( s ) h Γ ( 2 ν + ) Γ ( + ) Γ ( + 4) [5] [6] The secod solutio fo s 2 2 will be of the fom 2 2 l ( ) s u Qu + + C ( s2 ) [7] Whee 3 h ν ( )( ) ( ) ( ) ν 2ν Q lim s s s s 2 2 an s 3 d C ( s2 ) ( s s2 ) a ( s) ds Assumig that 2 ( ) Γ ( + s + 2ν ) Γ ( s + 3) ( + ) Γ ( + ν ) Γ ( + ) Γ ( + + ) L h s s s 2 s s 3 hece, [8] [9] L s2 2 ad ( ) Γ ( + 2ν ( ) ( ) Γ( ν ) h 2! 3! 2 2 d l L s s.5 2 ( ) ( ) ( ) Γ '( ) Γ '( + 2ν Γ ( ) Γ ( + ν ) ( ) ( ) ( ) ( ) 2 ds Γ ' 2ν 2 Γ ' 2 Γ ' + Γ 2ν 2 Γ 2 Γ + It is obvious that Γ '( z) Ψ ( z) Γ z ( ) [20] [2] [22] Whee Ψ(z) is called Psi (Digamma) fuctio. With substitutig Eq. [22] ito Eq. [2] d l ( L ) s s2.5 2 ( ) ( 2 2 ) ds + Ψ + Ψ + ν Ψ 2ν 2 Ψ 2 Ψ + ( ) ( ) ( ) [23] C s will be as Hece, the fial fom of ( 2 ) follows ( ) Γ ( + 2ν h C ( s2 ) 2! ( 3 )! Γ( 2ν.5 + 2Ψ ( ) + Ψ ( + 2ν ( 2 2) ( 2) ( ) Ψ ν Ψ Ψ + [24] Thus, the complete solutio fo u is expessed as + u cu + c2u2 c + a ( s ) + c2 Q + a s l + ( ) ( ) C ( s2 ) Substitutig Eq. [25] ito Eq. [5] AEc 2 Fa ( s ) σ + ν + ν + AEc2 Q ( + 2ν ) l ( ) [25] 3 + Q Fl ( ) + a ( s ) + GC ( s2 ) [26]

4 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, Whee F + + 2ν [27] G 2 + 2ν The costat coefficiets c ad c 2 ae foud usig the bouday coditios σ ( a) P ad σ ( b) 0 PD4 c AEi DD 4 D2D3 PD3 c2 AEi DD4 D2D3 [ ] [ ] Whee D + 2ν + Fa ( s ) a ν D 2 Q ( + 2ν ) l ( a) + 2 a [28] 3 Q Fl ( a) a ( s ) a GC ( s2 ) a 3 + ν + ( ) D 2 Fa s b ν D Q ( + 2ν ) l ( b) + 2 b Q Fl b + a s b + GC s b 3 ( ) ( ) ( 2 ) [29] We have the displacemet ad stesses equatios i the followig fom: P k u ( kq l ( ) ) [ ] AE 2 i D kd kq l ( ) a ( s ) k C ( s2 ) [30] a β b a Pe σ + 2ν k Q + 2ν l ] ( ) ] ( ) ( ) ( ) ( ) D kd2 ν + 2 F kq Fl a s k GC s [3] a β b a Pe σ φφ + 2ν k Q + 2ν l D kd 2 ν +ν ( ) ( ) ] F kq F l ( ) 3 ] a ( s ) k G C ( s2 ) +ν Whee D3 k D 4 F ν ( + 2) + G v ( ) + [32] [33] 3. Results ad Discussio Coside a thick spheical pessue vessel ude the iteal pessue of 40 MPa. The pessue vessel has the ie ad oute adii 30 cm ad 60 cm, espectively. I this study, it is supposed that the modulus of elasticity at the iteal adius has the value of 200 GPa ad the Poisso s atio, ν, has a costat value of 0.3. Fo a compaative study o umeical aalysis of this poblem, a geometic specime is modeled usig commecial fiite elemet code, ANSYS 2. Due to the geometical symmety i the sphee, oly a quate of the specime geomety i the fiite elemet model was cosideed. I ode to epeset the o-homogeeous specime, a 8-ode axisymmetic quadilateal elemet was used. The vaiatio i mateial popeties was implemeted by 20 layes, with each laye havig a costat value of mateial popeties, fo modelig of the FGM spheical pessue vessel. Fig. 2 shows the distibutio of elastic modulus i the adial diectio. The elastic modulus iceases as β iceases. Fig. 3 shows the distibutio of tesile adial displacemet alog the omalized adial diectio. It is obvious i this cuve that the adial displacemet deceases as β iceases at the same positio. I Fig. 4 distibutio of compessive adial stess alog omalized adial diectio is show. It is peceived that the adial stess

5 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, Fig. 2. Radial distibutio of modulus of elasticity Fig. 3. Radial distibutio of adial displacemet Fig. 4. Radial distibutio of adial stess iceases fo highe values of β. I Fig. 5 the tesile cicumfeetial stess alog omalized adial diectio fo diffeet values of β is plotted. Hee, it should be oted that i the same situatio, appoximately, fo a b a <, the value of the ( ) 0.5 cicumfeetial stess deceases as β iceases, wheeas fo ( a b a) > 0.5 this situatio was evesed. Futhe, alog the adial diectio, appoximately, fo β < the cicumfeetial stess deceases, while almost fo β >, the cicumfeetial stess iceases. It should also be oted that fo all the cosideed coditios, fo β the cicumfeetial stess emais almost uifom alog the adius of the sphee. The issue ca be a valuable facto fo cotollig the stess. Fo the pupose of studyig the stess distibutio alog the spheical pessue vessel adius, i Fig. 6 the vo Mises equivalet stess Fig. 5. Radial distibutio of cicumfeetial stess ( σ eq σ σφ ) is plotted i the adial diectio. The vo Mises equivalet stess deceases as the adius iceases fo all β values. I Figs. 7 to 0, the adial displacemets, adial, cicumfeetial ad vo Mises equivalet stesses values obtaied fom ANSYS commecial fiite elemets aalysis pogam ae epeseted. Fo veificatio, the Fobeius seies method esults ae compaed with those of the umeical solutio of Che ad Li method [9] (Figs ad 2 ( a 30 cm, b 60 cm, ν 0.3 )). 4. Coclusios The Fobeius seies method is a poweful techique fo obtaiig solutios of cetai diffeetial equatios which occu i applicatios. Based o basic equatios of elasticity ad usig FSM, closed-fom solutios have bee deived fo stesses ad the

6 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, Fig. 6. Radial distibutio of vo Mises equivalet stess Fig. 7. Radial displacemet obtaied fom ANSYS code i a FGM spheical pessue vessel ( β ) Fig. 8. Radial stess obtaied fom ANSYS code i a FGM spheical pessue vessel ( β ) Fig. 9. Cicumfeetial stess obtaied fom ANSYS code i a FGM spheical pessue vessel ( β ) Fig. 0. Vo Mises equivalet stess obtaied fom ANSYS code i a FGM spheical pessue vessel ( β ) displacemets of thick spheical pessue vessels made of fuctioally gaded mateials with expoetial vayig popeties. Followig Fig.. No dimesioal adial stess i spheical pessue vessel with a ie pessue P this, pofiles ae plotted fo diffeet values of ihomogeeity costat fo the adial displacemet, adial stess, ad cicumfeetial

7 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, Fig. 2. No dimesioal tagetial stess i spheical pessue vessel with a ie pessue P stess, as a fuctio of adial diectio. I this study, a umeical solutio usig a commecial fiite elemets code, ANSYS 2, is also peseted. Good ageemet was foud betwee the aalytical solutios ad the solutios caied out though fiite elemet code. The peseted esults show that the ihomogeeity costat has a sigificat ifluece o mechaical behavios of the thick spheical pessue vessels made of fuctioally gaded mateial with expoetial vayig popeties. Refeeces. M. Ghaad, M. Z. Nejad, Elastic solutio of pessuized clamped-clamped thick cylidical shells made of fuctioally gaded mateials, J. Theo. Appl. Mech., Vol. 5, 203, pp N. Tutucu, M. Oztuk, Exact solutio fo stesses i fuctioally gaded pessue vessels, Compos. Pat B-Eg., Vol. 32, 200, pp P. Nayak, S. C. Modal, A. Nadi, Stess, stai ad displacemet of a fuctioally gaded thick spheical vessel, It. J. Eg. Sci., Vol. 3, 200, pp L. H. You, J. J. Zhag, J. J. You, Elastic aalysis of iteally pessuized thick-walled spheical pessue vessels of fuctioally gaded mateials, It. J. Pes. Ves. Pip., Vol. 82, 2005, pp N. Tutucu, Stesses i thick-walled FGM cylides with expoetially vayig popeties, Eg. Stuct., Vol. 29, 2007, pp M. Z. Nejad, G. H. Rahimi, M. Ghaad, Set of field equatios fo thick shell of evolutio made of fuctioally gaded mateials i cuviliea coodiate system, Mechaika, Vol. 77, 2009, pp Y. Z. Che, X. Y. Li, Elastic aalysis fo thick cylides ad spheical pessue vessels made of fuctioally gaded mateials, Comp. Mate. Sci., Vol. 44, 2008, pp N. Tutucu, B. Temel, A ovel appoach to stess aalysis of pessuized FGM cylides, disks ad sphees, Compos. Stuct., Vol. 9, 2009, pp Y. Z. Che, X. Y. Li, A alteative umeical solutio of thick-walled cylides ad sphees made of fuctioally gaded mateials, Comp. Mate. Sci., Vol. 48, 200, pp M. Z. Nejad, M. Abedi, M. H. Lotfia, M. Ghaad, Exact ad umeical solutios fo stesses i pessuized FGM solid sphee with paabolic vayig popeties, Am. J. Sci. Res., Vol. 32, 20, pp M. Z. Nejad, M. Abedi, M. H. Lotfia, M. Ghaad, A exact solutio fo stesses ad displacemets of pessuized FGM thick-walled spheical shells with expoetial-vayig popeties, J. Mech. Sci. Techol., Vol. 26, 202, pp

8 M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204,