Buckling analysis of single-layered FG nanoplates on elastic substrate with uneven porosities and various boundary conditions

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1 IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-issn: ,p-ISSN: X, Volume 15, Issue 5 Ver. IV (Sep. - Oct. 018), PP Bucklng analyss of sngle-layered FG nanoplates on elastc substrate wth uneven porostes and varous boundary condtons Nebojsa Radć 1, Dejan Jeremć Department of Appled Mechancs, Faculty of Mechancal Engneerng/ Unversty of East Sarajevo, Bosna and Herzegovna Correspondng Author: Nebojsa Radć Abstract: In ths nvestgaton the bucklng behavor of sngle-layered functonally graded nanoplates wth uneven porostes exposed to hygrothermal loads s modeled for the frst tme. Usng the Erngen s nonlocal elastcty theory wth one scale parameters, the small sze effect on the bucklng behavour of the functonally graded nanoplates s consdered. Based on the new frst order shear deformaton theory the equatons of equlbrum are obtaned from the prncple of mnmum potental energy. Also, elastc Pasternak foundaton s adopted to capture the foundaton nfluence on the crtcal bucklng load. The equatons of equlbrum are solved for varous boundary condtons usng Galerkn s method. The mpacts of nonlocal parameter, porosty dstrbuton and boundary condtons on the crtcal bucklng load are demonstrated. Keywords: Bucklng, FG nanoplates, Galerkn s method, Nonlocal elastcty theory, Porostes Date of Submsson: Date of acceptance: I. Introducton Functonally graded (FG) plates and nanoplates have an nhomogeneous structure and very good propertes under thermal loadngs due to the change of mechancal and thermal propertes n the thckness drecton. FG nanoplates are made from a mxture of ceramc and metal wth a change of mechancal propertes between two surfaces. FG nanoplates as well as other nanostructures have sze dependent mechancal propertes and behavours. In analyzng ther mechancal behavour a conventonal form of contnuum mechanc whch takes nto account the nfluence of sze effects s requred. The nonlocal theory of elastcty of Erngen s [1,] s a contnuous mechancal model that ntroduces nfluence of atomc length scale nto the consttutve relatons va materal (nonlocal) parameter. Daneshmehr et al. [3] studed the bucklng behavour of FG nanoplates wth dfferent boundary condtons usng the nonlocal elastcty and hgher order plate theores. Sobhy [4] nvestgated the bendng response, free vbraton and mechancal bucklng of sngle-layered FG nanoplate embedded n elastc medum usng the four-unknown shear deformaton theory ncorporated n nonlocal elastcty theory. Ansar et al. [5] appled nonlocal three-dmensonal theory to nvestgate to free vbraton of FG nanoplates restng on elastc foundaton. Khorshd and Fallah [6] presented the bucklng analyss of FG nanoplate based on nonlocal shear deformaton theory. Lu et al. [7] performed an analyss of nonlocal vbraton and baxal bucklng of doublevscoelastc FG nanoplate on the bass of nonlocal elastcty theory and the Kelvn model. Shahverd and Barat [8] nvestgated the vbraton beahavour of porous functonally graded nanoplates usng a general nonlocal stran-gradent theory. Barat and Zenkour [9] nvestgated the wave propagaton of nanoporous graded doublenanobeams based on the general b-helmholtz nonlocal stran gradent elastcty. She et al. [10] carred out the vbraton behavour of porous nanotubes based on the nonlocal stran gradent theory and refned beam theory whch ncludes effects of shear stress. Shafe and Kazem [11] used the modfed couple stress theory to analyze the bucklng behavour of functonally graded porous nano-/mcro-scaled beams. Radć [1] examned bucklng behavour of double-layered porous FG nanoplates n Pasternak elastc foundaton usng nonlocal stran gradent theory. To the authors best knowledge, the bucklng behavour of a sngle-layered FG nanoplates wth uneven porostes and varous boundary condtons has not been studed n the open lterature. II. Theoretcal formulaton.1 Porostes and thckness dependent materal propertes of FG nanoplate Consder a sngle-layered FG nanoplate restng on elastc Pasternak foundaton wth uneven porostes dstrbuton of unform thckness h, length L and wdth L assocated wth the z, x and y-axes of the coordnate x system. The present sngle-layered FG nanoplate s under n-plane mechancal and hygrothermal compressve load and embedded n Pasternak foundaton. DOI: / Page y

2 Fgure1: Confguraton of uneven porous sngle-layered FG nanoplate The FG nanoplate s assumed to be composed of mxture of Al and Al O 3 and exposed to hygrothermal envronment. For uneven dstrbuton of porostes the Young s modulus E s changed contnuously n the thckness drecton by the followng form. p z 1 z E( z) ( Ec Em ) Em ( Ec Em ) 1 h h (1) where p s the non-homogenety or power-law ndex (p s a non-negatve parameter) whch determne the materal dstrbuton across the nanoplate thckness.. New Frst Order Shear Deformaton Theory (NFSDT) The present dsplacement feld of new frst order shear deformaton theory can be expressed as [1] ux ( x, y, z) u( x, y) ( z z0 ) x uy ( x, y, z) v( x, y) ( z z0 ) y u ( x, y, z) w( x, y) z () where u, v and w are the dsplacement of md-plane of FG nanoplate along x, y and z-axs respectvely, and s rotaton parameter. The poston of the physcal neutral surface z 0 can be determned from the condton that the ntegral of the frst momentum of elastcty modulus Ez () n the drecton of z-axs s equal to zero. h h E( z)( z z ) dz 0 0 (3) Nonzero strans components of the observed nanoplate model can be deduced as u x x w v yz y y ( z z0), y y xz w xy u v x x y x xy (4) For the present bucklng case the prncple of mnmum of potental energy t gven as U V 0 (5) Where the stran energy s defned by U and work of n-plane loads and elastc foundatons s defned by V. The vrtual stran energy U for the new frst-order shear deformaton theory can be wrtten as DOI: / Page

3 U ( ) dv V xy xy xz xz yz yz (6) Substtutng Eqs. () and (4) n Eq. (6) yelds u u u u U N M N A M N xy M xy x x y y y x xy ( w ) ( w ) Qxz Qyz da x y (7) The nonlocal stress resultants N h / h / h / j s h / h / h / N, Q j and M z dz (,, xy; j xz, yz) dz Q K dz M are obtaned from the followng expresson (8) We can defne the varaton of the work done by appled loads n the ntegral form w w w w w w w w m T H m T H V N A N N N N N kww kg da x x y y x x y y (9) T H where, kw and k G are Wnkler and Pasternak parameter of elastc foundaton, N and N are the external thermal and hygro forced due to the changes of temperature and mosture h / T Ez () N () z Tdz h /1 (10) h / H Ez () N () z Hdz h /1 (11) The governng equatons of equlbrum for bucklng behavour are obtaned by substtutng Eqs. (7) and (9) n Eq. (5) when the coeffcents of w and are equal to zero Q Q xz yz w w m T H w m T H w w : kww kg N N N N N N 0 x y x y x y M M xy M Q Q xz yz : 0 x xy y x y (1).3 The nonlocal elastcty theory for FG nanoplates Makng certan assumptons presented by Erngen [1,] we wll assume the nonlocal dfferental consttutve equaton as 1 ( e0a) tj ( x) Cjkl kl, j, k, l x, y, z (13) where s the Laplacan operator whch s defned by ( / x / y ),and eas 0 the nonlocal parameter that takes nto account the small scale effects nto the dfferental consttutve equaton. DOI: / Page

4 Based on Eqs. (11) the stress-stran consttutve equatons of a rectangular FG nanoplates can be wrtten as t t Ez ( ) [1 ( e0 ) ] txy 0 0 (1 ) / 0 0 xy 1 t (1 ) / 0 xz xz t (1 ) / yz yz (14) where denote Posson s ratos..4 Equatons of equlbrum of sngle-layered FG nanoplates Equatons of equlbrum of nonlocal elastcty theory and new frst order shear deformaton theory for nvestgaton the bucklng behavour of sngle-layered FG nanoplates wth uneven porostes and hygrothermal n-plane loadngs can be wrtten n the terms of dsplacements as follows w w KF S x y x y w w m T H w m T H w 0 w G (1 ( e ) ) k w k N N N N N N 0 x y x y (15) w w D x x y y x y x y (16) where K sf h/ E( z z / 0) h Ez () D dz, F dz h/ 1 h/ (1 ) III. Soluton by Galerkn s method In ths secton, by usng Galerkn s method, the governng equatons of equlbrum for bucklng study are solved for smply supported (S) and clamped (C) boundary condtons. The analytcal soluton for dsplacement feld of Eqs. (15) and (16) can be ntroduced as: w( x, y) W X ( x) Y( y) mn ( x, y) mn X ( x) Y( y) (17) In ths study the sngle-layered FG nanoplate s assumed to have smply supported (S) and clamped (C) boundary condton or have combnatons of them, and the functons X( x ) and Y( y) that satsfy above boundary condtons can be wrtten as: SSSS: X ( x) sn( x), Y( y) sn( y) (18) CCCC: X ( x) sn ( x), Y( y) sn ( y) (19) SCSC: X ( x) sn ( x), Y( y) sn( y) where m L, n L x y Substtutng Eq. (17) nto Eqs. (15) and (16) and mplementng the Galerkn s method the equatons of equlbrum n terms of parameters W, can be obtaned from mn, mn DOI: / Page

5 11 1 Wmn 0 1 mn 0 (0) IV. Numercal results and dscussons In ths research the materal propertes of sngle-layered FG nanoplate (Al and Al O 3 ) are gven as: Ec 380 GPa, c 0.3, Em 70 GPa, m 0.3 In the present nvestgaton the followng dmensonless parameters are used: ˆ 4 NLx, kwlx, kglx N k 3 wn kgn Eh m D D where m m N N, N kn In Fg. the nondmensonal crtcal bucklng load s plotted as a functon of the uneven porosty volume fracton for SSSS boundary condtons. From Fg. t can be seen that the value ncrease of the uneven volume porosty fracton has a lnear decreasng effect on the nondmensonal crtcal bucklng load. Fgure : Varaton of nondmensonal crtcal bucklng load versus uneven porostes for SSSS boundary condtons From Fg. 3 t can be notced that n the case of SCSCS boundary condtons the behavour of a FG sngle-layered nanoplate wth the change of value of uneven porosty volume fracton and nonolocal parameter s the same as n the case of SSSS boundary condtons. It can easly be seen that n the case of SCSC boundary condtons bucklng, the value of nondmensonal crtcal bucklng load s hgher than the case of SSSS boundary condtons. Fgure 3: Varaton of nondmensonal crtcal bucklng load versus uneven porostes for SCSC boundary condtons DOI: / Page

6 Fg. 4 demonstrates the effects of the uneven porostes on value of nondmensonal crtcal bucklng load for CCCC boundary condtons. It can be concluded that the value of the nondmensonal crtcal bucklng load s reduced when the value of the uneven porostes volume fracton rses. Fgure 4: Varaton of nondmensonal crtcal bucklng load versus uneven porostes for CCCC boundary condtons V. Concluson We present n ths paper the bucklng behavour of the FG uneven porous sngle-layered nanoplates subjected to n- plane mechancal and hygrothermal and varous boundary condtons. The Galerkn s method has been used to solve the equatons of equlbrum for SSSS, SCSC and CCCC boundary condtons. Numercal results are presented to nvestgate the effects of uneven porosty volume fracton on nondmensonal crtcal bucklng load for three observed boundary condtons. It s notced that ncreasng the value of uneven porosty volume fracton wll decrease the value of nondmensonal crtcal bucklng load for the case of SSSS, SCSC and CCCC boundary condtons. References [1]. Erngen, AC. and Edelen DGB.,On nonlocal elastcty, Internatonal Journal of Engneerng Scence, 10, 197, []. Erngen AC., On dfferental equatons of nonlocal elastcty and solutons of screw dslocaton and surface waves, Journal of Appled Physcs, 54, 1983, [3]. Daneshmehr, A., et al., Stablty of sze dependent functonally graded nanoplates based on nonlocal elastcty and hgher order plate theores and dfferent boundary condtons, Internatonal Journal of Engneerng Scence, 8, 014, [4]. Sobhy, M., A comprehensve study on FGM nanoplates embedded n an elastc medum, Composte Structures, 134, 015, [5]. Ansar, R., et al., Nonlocal three-dmensonal theory of elastcty wth aplcaton to free vbraton of functonally graded nanoplates on elastc foundatons, Physca E, 76, 015, [6]. Khorshd, K. and Fallah A., Bucklng analyss of functonally graded rectangular nano-plate based on nonlocal exponental shear deformaton theory, Internatonal Journal pf Mechancal Scences, 113, 016, [7]. Lu, JC., et al., Nonlocal vbraton and baxal bucklng of double-vscoelastc-fgm-nanoplate system wth vscoelastc Pasternak medum n between, Physcs Letters A, 381, 017, [8]. Shahverd, H. and Barat MR., Vbraton analyss of porous functonally graded nanoplates, Internatonal Journal of Engneerng Scence, 10, 017, [9]. Barat, MR., and Zenkour A., A general b-helmholtz nonlocal stran gradent elastcty for wave propagaton n nanoporous graded double-nanobeam systems on elastc substrate, Composte Structures, 138, 017, [10]. She, GL., et al.,on vbratons of porous nanotubes, Internatonal Journal of Engneerng Scence, 15, 018, [11]. Shafe, N., and Kazem M., Nonlnear bucklng of functonally graded nano-/mcro-scaled porous beams, Composte Structures, 117, 017, [1]. Radć, N., On bucklng of porous double-layered FG nanoplates n the Pasternak elastc foundaton based on nonlocal stran gradent elastcty. Compostes Part B, 153, 018, Nebojsa Radć. Bucklng analyss of sngle-layered FG nanoplates on elastc substrate wth uneven porostes and varous boundary condtons. IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE), vol. 15, no. 5, 018, pp DOI: / Page