Curvature Effects on Thermal Buckling Load of DWCNT Under Axial Compression Force

Transcription

1 Jurnal f Slid Mechanics Vl. 3,. (0) pp. -8 Curvature Effects n Thermal Buckling Lad f DWCT Under Aial Cmpressin Frce A. Ghrbanpur Arani,,*, M. Mhammadimehr, M. Ghazi Department f Mechanical Engineering, Faculty f Engineering, University f Kashan, Kashan, Iran Institute f anscience & antechnlgy, University f Kashan, Kashan, Iran Received 0 December 00; 9 February 0 ABSTRACT In this article, curvature effects n elastic thermal buckling f duble-walled carbn nantubes under aially cmpressed frce are investigated using cylindrical shell mdel. Als, the small scale effect is taken int accunt in the frmulatin. The dependence f the interlayer van der Waals (vdw) pressure n the change f the curvatures f the inner and uter tubes at that pint is cnsidered. The effects f the surrunding elastic medium, curvature and the vdw frces between the inner and uter tubes increase the critical buckling lad under thermal and aial cmpressin lads, while small scale effect decreases it. 0 IAU, Arak Branch. All rights reserved. Keywrds: Curvature effect; Thermal buckling lad; Small scale effect; Aial cmpressin frce; DWCT. ITRODUCTIO C ARBO nantubes (CTs) discvered by Iijima in 99 []. Recently, many studies n the critical buckling lad f duble-walled carbn nantubes (DWCTs) under aial cmpressin r trsin lad have been presented (Ru [] and Ranjbartreh et al. [3], Han and Lu [4], Wang et al. [5], and Yang and Wang [6]). Als, the effects f the small scale and curvature had nt been cnsidered in their wrk. Mhammadimehr et al. [7] investigated the small scale effect n the trsinal buckling f a DWCT embedded n Winkler and Pasternak fundatins using the thery f nnlcal elasticity. Their results shw that the surrunding elastic medium increases the critical trsinal buckling, while the small scale effect decreases it. Using the nn-lcal elasticity thery, Mhammadimehr et al. [8] investigated Timshenk beam mdel t study the elastic buckling f DWCTs embedded in an elastic medium under aial cmpressin. The effects f the small scale, the surrunding elastic medium based n Winkler mdel and van der Waals (vdw) frce between the inner and uter nantubes are taken int accunt. Mrever, in rder t estimate the nn-lcal critical buckling lad f DWCTs under aial cmpressin, a simplified analysis is carried ut and the results are cmpared with thse btained using mlecular mechanics (MM).Ya and Han [9] studied the thermal effect n the aially cmpressed buckling f a multi-walled carbn nantube (MWCT). Mrever, they tk int accunt the effects f temperature change, the surrunding elastic medium and the vdw frces between tw adjacent layers. Their results shwed that at lw temperatures, the critical aial lad fr infinitesimal buckling f an MWCT increases by increasing the temperature. Ghrbanpur Arani et al. [0] investigated the trsinal and aial buckling analyses f the individual embedded MWCTs subjected t internal and eternal pressures. They cnsidered bth the small length-scale effect and the surrunding elastic medium. Their results shwed that the internal pressure increases the critical lad, whereas the eternal pressure tends t decrease it. Ghrbanpur Arani et al. [] studied the buckling analysis f a DWCT subjected t a unifrm internal pressure in a thermal field. Using the cntinuum cylindrical shell mdel, they cnsidered the * Crrespnding authr. Tel.: ; fa: address: aghrban@kashanu.ac.ir (A. Ghrbanpur Arani). 0 IAU, Arak Branch. All rights reserved.

2 A. Ghrbanpur Arani et al. effects f the temperature change, the surrunding elastic medium based n the Winkler mdel, small scale, and the vdw frces between the inner and the uter tubes. It is seen frm their results that the nnlcal critical buckling lad is lwer than the lcal critical buckling lad. It is als cncluded that, at lw temperatures, the critical buckling lad fr the infinitesimal buckling f a DWCT increases as the magnitude f temperature change increases; while, at high temperatures, the critical buckling lad decreases with increasing f the temperature. Qian et al. [] investigated the curvature effects f interlayer vdw frces n aially cmpressed buckling f a DWCT f diameter dwn t 0.7 nm. They cnsidered the dependence f the interlayer vdw pressure n the change f the curvatures f the inner and uter tubes at that pint. Hwever, s far, investigatin n the curvature effect n thermal buckling analysis f DWCTs embedded in an elastic medium has nt been fund in the literature. Mtivated by these ideas, this wrk aims t study curvature effect n thermal buckling lad f DWCT subjected t aial cmpressin lad. The effects f the small scale, the surrunding elastic medium, the vdw pressure between the inner and the uter tube, and the curvature n the elastic thermal buckling f DWCTs are emplyed in the duble-cntinuum cylindrical shells analysis. OLOCAL CYLIDRICAL SHELL MODEL The cnstitutive equatin f the nnlcal elasticity thery is written as fllws [3] (-ea ) = C : () C 0 is the elastic stiffness tensr f classical istrpic elasticity, and are the stress and the strain tensrs, respectively, e 0 dentes a material cnstant ( e 0 = 0.8), and a is the length f the C-C bnd ( a = 0.4 nm) [3], which ea 0 dentes the small scale effect in abve equatin. Cnsider a circular cylindrical shell with a middle radius r, the thickness h, Yung s mdulus E, and Pissn s rati. The crdinate system is cnsidered as the rigin at the middle surface f the shell; and are the aial and the circumferential crdinates f the cylindrical shell, respectively; and the z directin is the nrmal crdinate t the medium surface. The relatinships between the strain and the displacement fr cylindrical shell mdel are defined as the fllwing equatin [4]: 0 u 0, v w æ, u v ö = + = + - = + r r èr ø () where and are the initial nrmal strains prir t the buckling, the symbls, and dente the strain cmpnents at any pint thrugh the shell wall thickness; and u, v, and w refer t the displacement cmpnents f the middle surface f the shell in the aial, circumferential, and radial directins, respectively. In this paper, the nrmal stress z, its crrespnding strain z, and the shear strains z and z are assumed t be negligible. Subsequently, the cnstitutive equatins f the nnlcal elasticity thery are written as fr the case f the plane stress [] E ET -( ea ) = ( ) + - E E T -( ea ) = ( ) + - r E -( ea ) ( + ) = r - (3) where and are the nrmal stresses alng the, directins, respectively. is the shear stress in the - plane, and dente the thermal epansin cefficient in the aial and circumferential directins, respectively, and als, the temperature change T is als unifrmly distributed in the CTs. The resultant frces per unit length are written as the fllwing equatins 0 IAU, Arak Branch

3 Curvature Effects n Thermal Buckling Lad f DWCT Under Aial Cmpressin Frce 3 = h, = h, = h (4) where (,, ) are the assciated ttal membrane frces. Substituting Eq. (3) int Eq. (4) yields the fllwing resultant frces EhT -( e0a) = C( + )- Eh T -( e0a) = C( + ) - r -( e0a) ( + ) = C( -) r (5) where Eh C. The membrane frce can be written as [5]: M T -( e0a) = + ea r M T -( 0 ) = + (6a) M M where and are the membrane frce caused by mechanical lads. caused by thermal lads. Using Eqs. (5) and (6a), we have T and T dente the membrane frce M T M T = C( + ) EhT =- = C( + ) Eh T =- (6b) Fr the pst-buckling cnfiguratin, we have [7, 4, 5] = + 0 = + 0 = + 0 (7) where the superscript 0 dentes the pre-buckling state, and the superscript refers t infinitesimal increments f the crrespnding parameters during the buckling. Accrding t Ref. [7], the gverning equatin is written in the fllwing decupling frm [7]: 4 8 Eh w æ 4 4 w wö 4 D w+ + ( e ) (, ) 4 a - + p = r è r ø (8) where æ w w ö æ w w ö = p è r ø r è r ø (9) 0 IAU, Arak Branch

4 4 A. Ghrbanpur Arani et al. where = + r æ ö =. = r è ø æ ö 4 æ ö 4 æ ö æ ö =. = r è ø r è ø r è ø r è ø (0) and p (, ) is the additinal nrmal pressure after buckling. The surrunding elastic medium fr the uter nantube f a DWCT is cnsidered as fllws [7]: E E p = p0 - dw () E where p 0 is the pressure due t the surrunding elastic medium prir t the buckling and D e and w dentes the spring cnstant f Winkler mdel and radial displacement f the uter nantube, respectively. The vdw interactin pressure at any pint between the tw tubes fr DWCTs is assumed that it depends linearly n the change f the interlayer spacing and the change f the curvatures f the inner and the uter tubes at that pint. It is clear that the change f the curvature f the k th tube due t the deflectin w k is wk / rk + kwk, thus the interlayer vdw pressure n the inner tube with cnsidering the curvature change is written as fllws []: æ v w ö æ w ö p(, ) = c( w- w) + c w c w èr ø èr ø () v in which p (, ) are the pressure eerted n the inner tube by the vdw frce, c is a cnstant, w and w are the radial deflectins f the uter and the inner tubes, respectively, r and r dente the radii f the uter and the inner tubes, respectively. The interactin pressures between the inner and the uter tubes are equal in magnitude and are ppsite in sign; therefre, the fllwing equatin is written as []: v v p(, ) r =- p(, ) r (3) v where p (, ) is the pressure eerted n the uter tube by the vdw frce. Substituting Eq. () int Eq. (3) yields the interlayer vdw pressure n the uter tube v r é æ w ö æ w öù p(, ) =- c( w w) c w c w r r r ê è ø è ë øúû (4) The nrmal pressures n the inner and the uter tubes are written as p = p v, p v E = p + p (5) Substituting Eqs. (), () and (4) int Eq. (5) yield the nrmal pressure eerted n the inner and the uter tubes æw ö æw ö p = cw ( - w) + c + w - c + w è ø è ø r r (6a) 0 IAU, Arak Branch

5 Curvature Effects n Thermal Buckling Lad f DWCT Under Aial Cmpressin Frce 5 r é æw ö æw öù p =- êcw ( - w) + c + w - c + w ú+ p -Dw êë úû E 0 e r ê èr ø èr ú ø (6b) The gverning equatins f the equilibrium fr the inner and the uter tubes are written as Eh w æ w w ö D w + + ( e a) - + = p (, ) r è r ø Eh w æ w w ö D w + + ( e a) - + = p (, ) r è r ø (7a) (7b) where and are given by æ w w ö æ w w ö è ø è ø = p(, ) r r r æ w w ö æ w w ö è ø è ø = p(, ) r r r (8a) (8b) The ends f the inner and the uter tubes fr DWTs are cnsidered as simply supprted. Therefre, the buckling mdes are written as m m w = fsin sin n, w = fsin sin n (9) L L where f and f are tw real cnstants, m and n the aial half and circumferential wave numbers, respectively. Using Eqs. (6), (6) and (8), substituting Eq. (9) int Eqs. (7) yields the fllwing equatin: ì ïaw + Bw = 0 í ï ïî Aw + Bw = 0 (0) In rder t btain the critical buckling lad, it is necessary t set the determinant f the cefficient matri in Eq. (0) equal t zer: AB AB 0 - = () 3 RESULTS AD DISCUSSIO The mechanical, gemetrical and thermal data fr DWCT are cnsidered as fllws [, 5] r = nm, r =.4 nm, L= 0 r, = 0.34, E = 0 Pa, Eh= 360 J/m, D= J, c= /m, c =-3.96 kg/s, =. 0, = 0.8 0, e = 0.8, a= 0.4 nm 0 () 3. Thermal buckling f DWCTs withut cnsidering aial cmpressin frce Substituting mechanical, gemetrical and thermal data int Eq. () yields the critical temperature fr DWCTs withut cnsidering aial cmpressin frce. Fig. shws the critical buckling temperature versus the aial half 0 IAU, Arak Branch

6 6 A. Ghrbanpur Arani et al. wave number fr different radii. It is seen that the critical buckling temperature at the specified length decreases with increasing the uter radius fr DWCTs. Fig. illustrates the curvature effect n the critical buckling temperature. It is shwn frm the results that the critical buckling temperature with cnsidering the curvature effect is higher than that withut cnsidering it fr m 50, while this case reverses fr m 50. Fig. 3 depicts the small scale effect n the critical buckling temperature fr DWCTs. It is seen that the nnlcal critical buckling temperature is lwer than the lcal critical buckling temperature. 3.. Thermal buckling lad f DWCTs under aial cmpressin frce The buckling lad such as thermal lad and aial cmpressin frce is cnsidered as the fllwing equatin 0 0 =- Eh T - p, Eh T =- -p - - (3) Substituting Eq. (3) int Eqs. (7a) and (7b) yields the critical thermal buckling lad under aial cmpressin frce. Fig. 4 shws the effect f the interlayer vdw pressure n the thermal buckling lad under aial cmpressin frce. It is bserved frm this figure that the critical buckling lad under thermal and aial cmpressin lads increases with cnsidering the interlayer vdw pressure. Fig. 5 depicts the curvature effect n the thermal buckling lad under aial cmpressin frce. Fig. The effect f the uter radius n the critical temperature (m=70, n=6). Fig. The curvature effect n the critical temperature (n=6). Fig. 3 The small scale effect n the critical buckling temperature fr DWCTs (n=6). 0 IAU, Arak Branch

7 Curvature Effects n Thermal Buckling Lad f DWCT Under Aial Cmpressin Frce 7 It is seen that the critical buckling lad under thermal and aial cmpressin lads increases with cnsidering the curvature effect. Fig.6 illustrates the effect f the surrunding elastic medium based n the Winkler mdel n the thermal buckling lad under aial cmpressin frce. It is shwn that the critical buckling lad under thermal and aial cmpressin lads increases with cnsidering the surrunding elastic medium. Fig. 7 shws the small scale effect n the thermal buckling lad under aial cmpressin frce. It is seen frm the results that the critical buckling lad under thermal and aial cmpressin lads decreases with cnsidering the small scale effect. Fig. 4 The effect f the interlayer vdw pressure n the thermal buckling lad under aial cmpressin frce. Fig. 5 The curvature effect n the thermal buckling lad under aial cmpressin frce. Fig. 6 The effect f the surrunding elastic medium n the thermal buckling lad under aial cmpressin frce. Fig.7 The small scale effect n the thermal buckling lad under aial cmpressin frce. 0 IAU, Arak Branch

8 8 A. Ghrbanpur Arani et al. 4 COCLUSIOS In this paper, the effects f the small scale, the surrunding elastic medium, the interlayer vdw pressure between the inner and the uter tubes, and curvature n the elastic thermal buckling f DWCTs under aial cmpressin frce are investigated using cylindrical shell mdel. The fllwing cnclusins can be btained frm the results:. The critical buckling lad under thermal and aial cmpressin lads increases with cnsidering the interlayer vdw pressure.. The critical buckling lad under thermal and aial cmpressin lads increases with cnsidering the curvature effect. 3. The critical buckling lad under thermal and aial cmpressin lads increases with cnsidering the surrunding elastic medium. 4. The critical buckling lad under thermal and aial cmpressin lads decreases with cnsidering the small scale effect. ACKOWLEDGMETS The authrs wuld like t thank the Iranian antechnlgy Develpment Cmmittee fr their financial supprt. REFERECES [] Iijima S., 99, Helical micr tubes f graphitic carbn, ature 354: [] Ru C.Q., 00, Aially cmpressed buckling f a DWCT embedded in an elastic medium, Jurnal f the Mechanics and Physics f Slids 49: [3] Ranjbartreh A.R., Ghrbanpur A., Sltani B., 007, DWCT with surrunding elastic medium under aial pressure, Physica E 39: [4] Han Q., Lu G., 003, Trsinal buckling f a DWCT embedded in an elastic medium, Eurpean Jurnal f Mechanics A/Slids : [5] Wang X., Yang H.K., Dng K., 005, Trsinal buckling f multi-walled carbn nantubes, Materials Science and Engineering A 404: [6] Yang H.K., Wang X., 007, Trsinal buckling f multiwalled carbn nantubes embedded in an elastic medium, Cmpsite Structures 77: 8-9. [7] Mhammadimehr M., Saidi A.R., Ghrbanpur Arani A., Arefmanesh A., Han Q., 00, Trsinal buckling f a DWCT embedded n Winkler and Pasternak fundatins using nnlcal thery, Jurnal f Mechanical Science and Technlgy 4(6): [8] Mhammadimehr M., Saidi A.R., Ghrbanpur Arani A., Arefmanesh A., Han Q., 0, Buckling analysis f dublewalled carbn nantubes embedded in an elastic medium under aial cmpressin using nn-lcal Timshenk beam thery, Prceedings f IMechE, Part C: Jurnal f Mechanical Engineering Science, accepted. [9] Ya X., Han Q., 007, Investigatin f aially cmpressed buckling f a multi-walled carbn nantube under temperature field, Cmpsite Science and Technlgy 67: [0] Ghrbanpur Arani A., Rahmani R., Arefmanesh A., Glabi S., 008, Buckling analysis f multi-walled carbn nantubes under cmbined lading cnsidering the effect f small length scale, Jurnal f Mechanical Science and Technlgy : [] Ghrbanpur Arani A., Mhammadimehr M., Saidi A.R., Shgaei S., Arefmanesh A., 00, Thermal buckling analysis f duble-walled carbn nantube cnsidering small scale effect, Prceedings f IMechE, Part C: Jurnal f Mechanical Engineering Science 5: [] Qian H., Xu K.Y., Ru C.Q., 005, Curvature effects n aially cmpressed buckling f a small-diameter duble-walled carbn nantube, Internatinal Jurnal f Slids and Structures 4: [3] Eringen A.C., 983, On differential equatins f nnlcal elasticity and slutins f screw dislcatin and surface waves. Jurnal f Applied Physics 54: [4] Brush D.O., Almrth B.O., 975, Buckling f Bars, Plates and Shells, McGraw-Hill, ew Yrk, USA. [5] Ya X., Han Q., 007, The thermal effect n aially cmpressed buckling f a duble-walled carbn nantube, Eurpean Jurnal f Mechanics A/Slids 6: IAU, Arak Branch