ELASTIC ANALYSIS OF CIRCULAR SANDWICH PLATES WITH FGM FACE-SHEETS

Transcription

1 THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS ELASTIC ANALYSIS OF CIRCULAR SANDWICH PLATES WITH FGM FACE-SHEETS R. Sbulati *, S. R. Atashipou Depatment of Civil, Chemical and Envionmental Engineeing, Univesity of Genoa, Italy Depatment of Civil, Envionmental and Natual Resouces Engineeing Lulea Univesity of Technology, Sweden * Coesponding autho (sbulati@dicat.unige.it) Keywods: functionally gaded mateials, sandwich plate, coating, elastic solutions Intoduction Functionally Gaded Mateials (FGMs) ae composite mateials in which the elastic popeties vay continuously with espect to pescibed diections. The advantages of these mateials ae to emove stess discontinuities that occu at the inteface of homogeneous multilaye composites. To investigate the elastic esponse of sandwich panels with FGM layes seveal analytical and numeical appoaches wee pefomed [-5]. In ecent yeas thee-dimensional elastic solutions as benchmak fo appoximate solutions have been studied and pemit us to highlight the advantages of gaded popeties [6, 7, 8]. The aim of this pape is to pesent ecent esults obtained by the authos to study the theedimensional elastic defomation of cicula sandwich plates in which functionally gaded mateial layes have been intoduced. In the famewok of the theedimensional elasticity theoy, we have found the analytical solutions of sandwich panels with thee diffeent laye systems: FGM coe and homogeneous mateial (HM) face-sheets [9], HM coe and FGM coating [] and HM coe and facesheets with intelayes in FGM between the coe and the face-sheets []. The specific mechanical poblems analysed concen the elastostatic esponse of cicula sandwich panels with axisymmetic tansvesal loading conditions. The HM layes ae assumed isotopic while in the FGM layes Young s modulus is consideed that vaying exponentially on the tansvese diection. Futhemoe, Poisson s atio is assumed constant and equal in all systems. The layes ae supposed pefectly bonded togethe. The geneal appoach to solve the poblem, on the basis of the elasticity theoy, uses suitable potential functions that educe the poblem to solve a system of patial diffeential equations (PDEs) fo all layes consideed (HM and FGM). Potential functions ae intoduced fo inhomogeneous and homogeneous mateial layes by using Plevako s epesentation fom [] to wite the stess and displacement fields. The compatibility conditions, which Plevako s functions must satisfy espectively fo homogeneous and inhomogeneous layes, give ise to a fouthode un-coupled PDE system. The solution of this system is witten in the fom of Bessel s expansions with espect to the adial coodinates [3]. In this way, we obtain a system of odinay diffeential equations fo any laye and the explicit solutions ae found in tems of fou unknown coefficients fo any laye. These coefficients ae detemined by using the specific bounday and inteface conditions. Finally, the explicit solutions ae obtained and numeical investigations ae pefomed to compae the elastic esponse with conventional homogeneous sandwich systems. In this pape we pesent the solution method and a compaative investigation of the esults obtained fo the HM coe and FGM coating case. Fomulation of the poblem We show in detail the solution pocedue fo the case of FGM coating and HM coe () but the methods to obtain the defomation of sandwich systems with FGM coe and HM face-sheets o HM coe/facesheets with FGM intelayes (FGMIL) is simila [9,].

2 Fig.. Thick cicula sandwich panel with FGM facesheets. By assuming an axisymmetic cylindical coodinate system as shown in Figue, we conside a thick homogeneous cicula plate of adius b and unifom thickness h, with two layes of thickness * h h h c on the top and the bottom of the plate. The sandwich plate is subected to an axisymmetic tansvesal loading. The elasticity modulus of the FGM laye on the top and bottom of the plate is supposed to vay exponentially though the thickness in the fom: k k E( ) Ee ( hc h), E ( ) E ( h h ), k k E3( ) Ee ( h hc ), * * hc E E E E k hhc hhc k ln( ) ln( ),, ( ) c c () whee E is the elasticity modulus of the homogeneous coe and E* is the elasticity modulus of the top and bottom sufaces; Poisson s atio is set to be constant and equal in the coatings and in the coe. Accoding to Figue in the following we indicate with the apex o ovescipts (), () and (3) espectively, the top coating laye, the homogeneous thick coe and the bottom coating. Since we compae the sandwich with FGM elements with a conventional sandwich panel, in Figue we also depict the behavio of Young s modulus fo the same panels. In this way, the bounday conditions fo the top and bottom sufaces ae Fig.. Young s modulus distibution in the thickness of the sandwich plate in and. p ( ),,,. () () () h h (3) (3) h h and, by assuming the layes pefectly bonded togethe, the inteface conditions ae () () () () u u w w hc () () () () hc () (3) () (3) u u w w c () (3) () (3),, hc,, hc,, h hc,. hc hc (3) On the mantle we equie only that the tansvesal displacement w (i) (b, ) fo i=,,3 is null [4,5]. 3 Elastic analytical solution 3. Basic Equations The linea elasticity equations fo the inhomogeneous mateials assume the following fom []: ( i) ( i) u u u w d + E i( ), Ei ( )

3 ELASTIC ANALYSIS OF CIRCULAR SANDWICH PLATES WITH FGM FACE-SHEETS w ( i) w whee Ei( ) i d E ( ), ( i) ( i) d (4) L (3.) L. (7) Ei( ) Ei( ) In the homogeneous mateial coe the equation (7) becomes u w u The index (i=,,3) indicates the tansvesal and adial displacement components espectively in inhomogeneous layes (i=,3) and, fo E ( ) =E constant, in homogeneous coe (i=). To solve the patial diffeential equation system we use Plevako s epesentation fom in which the displacement components ae witten in tems of a suitable potential function []. Then, we intoduce ( i ) the potential function L espectively as: u (, ) L ( ) L, Ei( ) ( ) w (, ) L Ei( ) i i ( ) L ( ) L Ei( ) ( ) ( ) The stess components assume the fom. (5) 3 ( i ) ( i ) ( i ) ( i ) (, ) L L L 4 (, ) L L L 3 L (, ) L (6) ( i ) 4 ( i ) ( i ) ( i ) (, ) L whee is adial Laplace opeato. The potential ( i ) functions L must satisfy the compatibility conditions that, in the coatings, assume the following fom 3. Solution Method 4 ( i) L. (8) To solve Equations (7) and (8) fo the cicula sandwich plate, we expand the unknown potential ( i ) functions L in the fom of Bessel seies whee () (, ) () ( ), L L J b (9) (), (=,, ) ae the oots of the eoode Bessel s function J. Also, the functions ( L i ) ( ) can be epesented in the following integal fom L L (, ) J ( ) d, () b bj( b) in which J is the fist-ode Bessel function. We emak that this choice satisfies the bounday condition that the tansvesal displacement is null on the lateal suface [8, 4, 5]. By substituting (9) in (7) and (8), we obtain the following thee odinay diffeential equations coesponding to the thee layes, in the foms: () () () L ( ) 4 kl ( ) ( k k ) L ( ) () 4 () 4 kk L ( ) k k k L ( ), L ( ) L ( ) L ( ), () () 4 () k k (3) (3) (3) L ( ) 4 kl ( ) ( k k ) L ( ) (3) 4 (3) 4 kk L ( ) k k k L ( ). The geneal solutions of equations () ae () 3

4 () ( k ) () () ( k ) () () L e [ A cos( ) B sin( )] e [ C cos( ) D sin( )] () () () () () L e ( A B ) e ( C D ) () (3) ( k ) (3) (3) ( k ) (3) (3) L e [ A cos( ) B sin( )] e [ C cos( ) D sin( )] ( ) ( ) ( ) whee i, i, i ( i A B C and ) D, (i=,,3) ae some coefficients which should be detemined by using bounday and inteface conditions (,3). Futhemoe, and ae two coefficients defined as 4 4 ( ) k k ( k ), ( ) k. (3) Now, we wite the distibuted applied load in the fom of Bessel s expansion: p () PJ( ) (4) whee, the constant coefficients P can be detemined using the following elation fo the abitay loading function p( ): b P p( ) J( ) d. (5) bj ( b) By using bounday condition () and (4) as well as the continuity conditions at the intefaces (3), we obtain a set of algebaic equations in tems of ( ) ( ) ( ) unknown coefficients i, i, i ( i A B C and ) D, (i=,,3) of equations (). To solve this set of equations, we use Maple s pogam and, the explicit fom of the coefficients is so obtained. Then, we substitute the expansion of Plevako s functions (9, ) in equation (5) and (6) to give the displacement and stess fields. 3. Explicit Solution We wite the displacement field in tems of the potential function fo the thee layes in this fom:, ( ) d ( ) Ei( ) u u J u L L and, ( ) 3 d d 3 Ei ( ) w w J w L L ( i) d ( i) d ( ) ( ) L L Ei Ei ( ) The stess components ae () i 4, L J( ) 3 d, L J( ), J( ) ( ) J ( ) d ( ) L ( ) d, J( ) ( ) L L ( ) J( ) ( ) d ( ) L ( ) L ( ) d ( ) L ( ) ( ) L ( )

5 ELASTIC ANALYSIS OF CIRCULAR SANDWICH PLATES WITH FGM FACE-SHEETS Fig.3. Compaison of the nomalied adial stess in the thickness of the plate fo and systems: thin plate and thick plate. The explicit foms of the functions L ae given by () and the coefficients ae witten in []. 4 Numeical Results In ode to obtain numeical esults in closed fom, in this section we conside a sandwich plate with tansvesal load as the fist tem of Bessel s expansion. The obective of these numeical investigations is to highlight the inhomogeneity effects of plate in the compaisons with homogeneous coating sandwich plate (). We assume Poisson's atio constant and equal to.3. Now we intoduce the following dimensionless paametes: Fig.4. Nomalied nomal stess in the thickness of the plate fo and in thin plate and thick plate in the cente of the plate. * E E u u,, E, (5) Hp p E i i i i whee H=h is the total thickness of the coated plate and p = is the intensity of the loading at the cente of the plate. Futhemoe, we compae the defomation esponse of the sandwich plate by using two diffeent atios: H/b=. (thin plate) and H/b=.6 (thick plate). Then in the following, we conside a atio h * / h. and a quite sevee Young s moduli atio E. In Figue 3 the though-thickness nomalied adial stesses ae shown fo thin plate and thick plate in = (equal to hoop stess). 5

6 u Fig.5. Compaison of the nomalied shea stess in the thickness of the plate fo the two diffeent coating systems: thin plate and thick plate. We obseve that, in a thin plate, an incease of the suface stess occus in the case of FGM with espect HM laye case. Contaiwise, fo a thick plate the give ise to a eduction of suface adial stess in compaison with. The continuity of the stess in with espect to HM system is moe evident in the thick plate. The incease of the adial stess fo is due to the global lowe stiffness of the sandwich plate in compaison with the. In Figue 4 we compae the nomal stess in the thickness of the plate fo =. Fo a thin plate, and FGM have simila linea behavio but fo a thick plate we obseve a diffeent slope in the coating and the coe due to the sevee moduli atio u Fig.6. Nomalied adial displacement in the thickness of the plate fo the two diffeent coating systems and, thin plate and thick plate. in conventional sandwich, that inceases with the thickness of the plate. In Figue 5 the shea stess is investigated fo =b/. These figues eveal the diffeent behavio of the shea stess in the thickness of the homogeneous face-sheet in a thin and thick plate. The pemits us to educe the shea stess in the thickness of the plate by obtaining a egula distibution given by the gaded popeties. In Figue 6 the adial dimensionless displacements ae shown in a thin plate and thick plate. We obseve the quasi-linea distibution in the thin plate in compaison with the non-linea behavio in the thickness of the coe in the thick plate. Then Figue 7 eveals a quasi-linea distibution fo tansvesal displacement in thin coated plates

7 ELASTIC ANALYSIS OF CIRCULAR SANDWICH PLATES WITH FGM FACE-SHEETS.5.5 E = E = E= E= w E = E= E= E= w Fig.7. Compaison of the nomalied tansvesal displacement in the thickness of the plate fo the two diffeent coating systems: thin plate and thick plate..5 E = E= E= E= which is simila to conventional plate theoies. Howeve, this vaiation becomes nonlinea in the coe when the thickness of the coated plate inceases. Futhemoe, in Figue 8 we investigate the influence of the elasticity modulus atio E on the elastic esponse of cicula plates; to this end, we assume h * h. and H b.6. The though- thickness vaiation of dimensionless adial, shea and nomal stess (c) fo E,,,. It can be obseved fom Figue 8a that the magnitude of adial stess on the sufaces of the stuctue inceases when the elasticity modulus atio E inceases. Howeve, the maximum value of adial stess in the homogeneous coe does not change tangibly (c) Fig.8. Nomalied adial stess, shea stess and nomal stess fo diffeent atio E fo thick plate. The maximum value of tansvese shea stess of Figue 8b deceases in the coe and inceases in the layes by inceasing the atio E. This maximum value occus in the homogeneous coe fo conventional values. 7

8 H-FG face sheet FG face sheet H face sheet H-FG face sheet FG face sheet Hfacesheet Fig.9 Nomalied adial stess, shea stess and nomal stess fo diffeent atio E fo thick plate. Figue 8c eveals that vaiation of E has almost no impotant effects on nomal stess component apat fom the fact that an incease of E educes the nonlineaity of though-thickness vaiation. Nevetheless, some studies have demonstated that the use of FGM layes to ealie coatings o coe o layes in a sandwich panel, may not be the best option [6]. As shown in this pape, the use of FGM coatings leads to the elimination of the discontinuity of some stess components acoss the intefaces but inceases the stess magnitude on the suface of the coatings fo thin plates. Fo this eason, in [] we find analytical solutions that show that the use of FGM intelayes between the HM coating and the HM coe solve the poblem of the elimination of the discontinuity of stess components without inceasing the stess magnitude at the sufaces of the coating as shown in Figue 9, whee the adial and hoop stess at the cente of the plate is pesented fo a thin and thick plate. A compaison between HM system (H face-sheets), FGM coating/hm coe (FG face-sheets) and HM face-sheets and coe with FGM intelayes (H-FG face-sheets) is investigated fo a cicula plate with popeties intoduced in this section. With continuous lines we obseve the eduction, in compaison with othe sandwich systems, of the stess on the sufaces in the pesence of FGM intelayes. 5 Conclusion An exact thee-dimensional solution fo sandwich panels with exponentially functionally gaded facesheets is obtained in the famewok of the elasticity theoy. Using Plevako's epesentation, the elasticity equations ae efomulated into some uncoupled fouth-ode patial diffeential equations with espect to some potential functions. The explicit solutions ae obtained by witing the potential functions as Bessel expansions. In this way the explicit solution is obtained The compaative study of vesus show that using can help eliminate the stess discontinuity poblem in the intefaces. Moeove, it was obseved that, except fo thin stuctues, eplacing the conventional system of with esults in a consideable decease in the maximum values of diffeent stess components in the stuctue. Acknowledgment This wok was suppoted by the Italian Ministy of Education, Univesity and Reseach (MIUR). Poect No.9XWLFKW: "Multi-scale modeling of mateials and stuctues". Refeences [] Yamanouchi, M., Koiumi, M., Hiai, T., Shiota, I. (eds.) Poceedings of fist Intenational Symposium on Functionally Gadient Mateials. Sendai, Japan 99. [] A.R Saidi, A. Rasouli and S. Sahaee Axisymmetic bending and buckling analysis of thick functionally gaded cicula plates using

9 ELASTIC ANALYSIS OF CIRCULAR SANDWICH PLATES WITH FGM FACE-SHEETS unconstained thid-ode shea defomation plate theoy. Composite Stuctues, Vol. 89, No., pp. -9, 9. [3] S. Chi and Y. Chung Mechanical behavio of functionally gaded mateial plates unde tansvese load: Pat Analysis. Intenational Jounal of Solids and Stuctues, Vol.43, pp , 6. [4] J. Zhao, Y. Li and X. Ai Analysis of tansient themal stess in sandwich plate with functionally gaded coatings. Thin Solid Films, Vol. 56, pp , 8. [5] Sh. Hosseini-Hashemi, H. Salehipuo, S.R. Atashipou and R. Sbulati On the exact in-plane and out-of-plane fee vibation analysis of thick functionally gaded ectangula plates: explicit 3-D elastic solutions. Composites Pat B: Engineeing, Vol. 46, No., pp.8-5, 3. [6] M. Kashtalyan and M. Menshykova Theedimensional elastic defomation of a functionally gaded coating/substate system. Intenational Jounal of Solids and Stuctues, Vol. 44, pp , 7. [7] M. Kashtalyan and M. Menshykova Effect of a functionally gaded intelaye on thee-dimensional elastic defomation of coated plates subected to tansvese loading. Composite Stuctues, Vol. 89, pp.67 76, 9. [8] R. Sbulati and L. Badella Thee-dimensional elastic solutions fo functionally gaded cicula plates. Euopean Jounal of Mechanics A/Solids, Vol. 3, pp. 9 35,. [9] R. Sbulati An axisymmetic elastic analysis fo cicula sandwich panels with functionally gaded coes. Composites Pat B, Vol. 43, pp.39 44,. [] R. Sbulati, S. R. Atashipou, Sh. Hosseini-Hashemi, Study on the effect of functionally gaded coating layes on elastic defomation of a thick cicula plate: a closed-fom elastic solution. Composite Stuctues, Vol. 99, No., pp.3-4, 3. [] R. Sbulati, S.R. Atashipou and S.A. Atashipou Impovement of though-thickness stess distibution of thick cicula coated plates using functionally gaded intelayes (submitted), 3. [] VP. Plevako On the theoy of elasticity of inhomogeneous media. Jounal of Applied Mathematics and Mechanics, Vol.35, pp.86-83, 97. [3] GN. Watson A teatise on the theoy of Bessel Functions, Cambidge Mathematical Libay. Cambidge Univesity Pess, 9. [4] R. Sbulati Thee-dimensional analytical solution fo an axisymmetic bihamonic poblem. Jounal of Elasticity, Vol. 95, pp , 9. [5] M. Levinson The simply suppoted ectangula plate: an exact, thee-dimensional, linea elasticity solution. Jounal of Elasticity,Vol.7, pp