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1 Citation: Thai, Huu-Tai, Vo, Thuc, Nguen, Trung-Kien and Lee, Jaehong (14) A nonlocal sinusoidal plate model for micro/nanoscale plates. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 8 (14). pp ISSN Published b: SAGE Publications URL: < This version was downloaded from Northumbria Research Link: Northumbria Universit has developed Northumbria Research Link (NRL) to enable users to access the Universit s research output. Copright and moral rights for items on NRL are retained b the individual author(s) and/or other copright owners. Single copies of full items can be reproduced, displaed or performed, and given to third parties in an format or medium for personal research or stud, educational, or not-for-profit purposes without prior permission or charge, provided the authors, title and full bibliographic details are given, as well as a hperlink and/or URL to the original metadata page. The content must not be changed in an wa. Full items must not be sold commerciall in an format or medium without formal permission of the copright holder. The full polic is available online: This document ma differ from the final, published version of the research and has been made available online in accordance with publisher policies. To read and/or cite from the published version of the research, please visit the publisher s website (a subscription ma be required.)

2 A nonlocal sinusoidal plate model for micro/nanoscale plates Huu-Tai Thai a,, Thuc P. Vo b, Trung-Kien Nguen c, Jaehong Lee d a School of Civil and Environmental Engineering, The Universit of New South Wales, Sdne, NSW 5, Australia b Facult of Engineering and Environment, Northumbria Universit, Newcastle upon Tne, NE1 8ST, UK. c Facult of Civil Engineering and Applied Mechanics, Universit of Technical Education Ho Chi Minh Cit, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh Cit, Vietnam. d Department of Architectural Engineering, Sejong Universit, 98 Kunja Dong, Kwangjin Ku, Abstract Seoul , Korea A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen s nonlocal elasticit theor and sinusoidal shear deformation plate theor. The small scale effect is considered in the former theor while the transverse shear deformation effect is included in the latter theor. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundar conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundar conditions are derived from Hamilton s principle. Analtical solutions for bending, buckling, and vibration of simpl supported plates are presented, and the obtained results are compared with the eisting solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated. Kewords: Sinusoidal shear deformation theor; Nonlocal elasticit theor; Bending; Buckling; Vibration Corresponding author. Tel.: address: t.thai@unsw.edu.au (H.T. Thai). 1

3 1. Introduction Nanostructures are being increasingl used in micro/nanoscale devices and sstems such as biosensor, atomic force microscope, micro-electro-mechanical sstems (MEMS), and nano-electro-mechanical sstems (NEMS) due to their superior mechanical and electronic properties 1. In such applications, small scale effects are eperimentall observed -4. It was found that when the thickness of these structures is close to the internal material length scale parameter, such effects are significant and have to be taken into account when studing their behavior. Conventional plate models based on classical continuum theories are not capable of describing such effects due to the lack of material length scale parameters. This motivated man researchers to develop plate models based on size-dependent continuum theories which account for the small scale effects. The nonlocal elasticit theor initiated b Eringen 5-7 is one of the promising size-dependent continuum theories. Unlike the classical continuum theories which assume that the stress at a point is a function of strain at that point, the nonlocal elasticit theor assumes that the stress at a point is a function of strains at all points in the continuum. In this wa, the small scale effects are included through the use of constitutive equations. Based on the nonlocal elasticit theor, a number of paper have been published in the last four ears, attempting to develop nonlocal plate models and appl them to analze the bending 8-1, buckling 13-1, and vibration -31 responses of nanoplates. All of these models were based on Kirchhoff plate theor 8, 1, 14-19, -7, Mindlin plate theor 9, 11,, 8-3, and Redd plate theor 1, 13, 31. It should be noted that the Kirchhoff plate theor (KPT) is onl applicable for thin plates. However, it underestimates deflection and overestimates buckling load as well as natural frequenc of moderatel thick plates where the transverse shear deformation effects are significant. The Mindlin plate theor

4 (MPT) gives accurate results for thin to moderatel thick plates, but it requires a shear correction factor to compensate for the difference between the actual stress state and the constant stress state due to a constant shear strain assumption through the thickness. The Redd plate theor (RPT) provides a better prediction of response of thick plate and does not require a shear correction factor, but its equations of motion are more complicated than those of MPT. The sinusoidal shear deformation theor of Touratier 3 is based on the assumption that the transverse shear stress vanishes on the top and bottom surfaces of the beam and is nonzero elsewhere. Thus there is no need to use shear correction factors as in the case of MPT. This theor was successfull applied to laminate plates 33 and functionall graded sandwich plates Therefore, it is useful to etend the application of this theor to the micro/nanoscale plates b accounting for the small scale effects. The aim of this paper is to etend the sinusoidal shear deformation theor of Touratier 3 to the micro/nanoscale plates. Equations of motion and boundar conditions are derived from Hamilton s principle based on the nonlocal constitutive relations of Eringen. Analtical solutions for deflection, buckling load, and natural frequenc are presented for simpl supported plates, and the obtained results are compared with the eisting solutions to verif the accurac of the present model.. Nonlocal plate model.1. Kinematics The displacement field of the sinusoidal shear deformation theor is chosen based on the assumption that the transverse shear stress vanishes on the top and bottom surfaces of the beam and is nonzero elsewhere. The displacement field is given as 3 3

5 w h z u1,, z, t u,, t z sin h w h z u,, z, t v,, t z sin h 3,,,,, u z t w t (1) where ( u, v, w) are the displacements at a point on the middle plane of the plate along the coordinates (,, z ); and are the rotation of the middle surface along the and directions, respectivel; and h is the plate thickness. The linear strain epressions associated with the displacement field in Eq. (1) are: u w h z z sin h u w h z z sin h (a) (b) u v w h z z sin h (c) z z z cos h z cos h (d) (e) It can be observed from Eqs. (d) and (e) that the transverse shear strains ( z, z ) are zero at the top ( z h/) and bottom ( z h/) surfaces of the plate, thus satisfing the traction free conditions for (, z )... Equations of motion z Hamilton s principle is used herein to derive the equations of motion. The principle can be stated in analtical form as 4

6 T U K dt (3) where U is the variation of strain energ; and K is the variation of kinetic energ. The variation of strain energ of the plate is calculated b h / U A h/ z z z z A N M P N M P (4) dadz u w v w u v w N M P Qz Qz da where N, M, P, and Q are the stress resultants defined as i h /,,, h / i (5a) N dz i i h /,,, h / i (5b) M z dz i h z Pi dz i h h / sin,,, h / i (5c) h / cos z Q,, i h / idz i z z h (5d) The variation of kinetic energ of the plate can be written as h / K u u u u u u dadz A h/ w w w w Iu u v v w w I A (6) w w w w J K da where dot-superscript convention indicates the differentiation with respect to the time variable t ; is the mass densit; and I, I, J, K are the mass inertias defined as 5

7 h / (7a) h / I dz h I h (7b) 3 h / z dz h / 1 J 3 h / zh z h sin dz h (7c) h / 3 K 3 h / h z h sin dz h (7d) h / Substituting the epressions for U and K from Eqs. (4) and (6) into Eq. (3) and integrating b parts, and collecting the coefficients of u, v, w,, and, the following equations of motion are obtained: u : N N I u (8a) v : N N I v (8b) M M M w: q N I w I w J P P w : Qz K J (8c) (8d) P P w : Qz K J (8e) The boundar conditions are of the forms u : Nn Nn (9a) v : Nn N n (9b) M ns w:vn Vn (9c) s 6

8 :Pn P n (9d) :P n P n (9e) w n : M nn (9f) where M M w w w V N N I J (1a) M M w w w V N N I J (1b) M M M n n M n n, M M n M n M n n (1c) ns nn.3. Constitutive relations The nonlocal theor assumes that the stress at a point depends not onl on the strain at that point but also on strains at all other points of the bod. According to Eringen 5-7, the nonlocal stress tensor at a point is epressed as 1 or (11) where is the Laplacian operator in two-dimensional Cartesian coordinate sstem; is the classical stress tensor at a point related to the strain b the Hooke s law; 1 is a linear differential operator; and ( ea ) is the nonlocal parameter which incorporates the small scale effect, a is the internal characteristic length and e is a constant appropriate to each material. The nonlocal parameter depends on the boundar conditions, chiralit, mode shapes, number of walls, and tpe of motion 37. So far, there is no rigorous stud made on estimating the value of the nonlocal parameter. It is suggested that the value of nonlocal parameter can be determined b eperiment or b conducting a comparison of dispersion curves from the nonlocal continuum mechanics 7

9 and molecular dnamics simulation For an isotropic micro/nanoscale plate, the nonlocal constitutive relation in Eq. (11) takes the following forms 9, 31, E (1 ) 1 (1 ) z z z (1 ) z z z (1) where E and are the elastic modulus and Poisson s ratio, respectivel. Using Eqs. (), (1) and (5), the stress resultants can be epressed in terms of displacements as where u N N 1 v N N A 1 1 N u v N w M 1 M 1 w M M D 1 F M M w w P 1 P 1 w P P F 1 H P P w Q z Q z s 1 A Qz Q z 1 (13a) (13b) (13c) (13d) 3 3 3,,, E, h, h, h A D F H h, A s Eh 4(1 ) (14).4. Equations of motion in terms of displacements The nonlocal equations of motion of the present theor can be epressed in terms of generalized displacements ( u, v, w,, ) b appling linear differential operator 8

10 on Eq. (8) u 1 u 1 v A I u u v 1 v 1 u A I v v 4 D w F q q N N 4 I w w I w w J w w K J w 1 1 s F H A w 1 1 s F H A w K J w (15a) (15b) (15c) (15d) (15e) Clearl, when the nonlocal effect is neglected (i.e. ), the present model recovers Touratier s sinusoidal shear deformation theor 3. Also, the equations of motion of the nonlocal KPT can be obtained from Eq. (15) b setting the rotations (, ) equal to zero. It is observed from Eq. (15) that the in-plane displacements ( uv), are uncoupled from the transverse displacements ( w,, ). Thus, the equations of motion for the transverse response of the plate are reduced to Eqs. (15c)-(15e). 3. Analtical solutions Consider a simpl supported rectangular plate with length L and width b under 9

11 transverse load q and in-plane load in two directions ( N N, N N, N ). 1 cr cr Based on the Navier approach, the following epansions of displacements are chosen to automaticall satisf the simpl supported boundar conditions of plate,, t X cos sin e m1 n1,, t Y sin cos e m1 n1 w,, t W sinsin e m1 n1 mn mn mn it it it (16) where i 1, m/ L, n / b,,, X Y W are coefficients, and is mn mn mn the natural frequenc. The transverse load q is also epanded in the double-fourier sine series as where q, Q sinsin (17) m1 n1 mn Lb 4 Qmn q, sinsin dd Lb (18) The coefficients Q mn are given below for some tpical loads: Q mn q for sinusoidal load of intensit q 16q for uniform loadof intensit q mn 4Q m n sin sin for point load Q at thecenter Lb (19) Substituting the epansions of (,, ) and q from Eqs. (16) and (17) into Eq. (15), w the analtical solutions can be obtained from the following equations s11 s1 s13 m11 m13 X mn s1 s s 3 m m 3 Ymn s13 s3 s33 k m13 m3 m 33 W mn Q mn () 1

12 where s 1 s 1 s11 A H, s A H 1 s1 H, s13 F, s3 F, cr, 1,,,, s D k N 33 1 m K m K m I I m J m J (1) The analtical solution of the nonlocal KPT can be obtained from Eq. () b setting coefficients ( X mn, Y mn ) equal to zero. Thus, the deflection w, buckling load N cr, and natural frequenc of the KPT are epressed as Qmn w, sinsin (a) m1 n1 D N cr D 1 (b) D I I (c) 4. Numerical results In this section, a simpl supported nanoplate made of single-laered graphene sheet (SLGS) is considered. The geometric and mechanical properties of the SLGS are 43 : E 1. TPa,.16,,5kg/m 3, h.34nm. The fundamental frequenc of simpl-supported armchair and zigzag square SLGSs with different side lengths L are presented in Table 1. The values of nonlocal parameter ea of the simpl-supported armchair and zigzag SLGSs are 1.16 nm and 1.19 nm, respectivel. The obtained results are compared with those predicted b molecular dnamics (MD) simulation 38 which is 11

13 one of the most widel used numerical methods related to the interaction between the atoms or molecules in a sstem. A good agreement between the results is observed for various sizes of the plate. To further validate the accurac of the present solutions, the obtained results are compared with those predicted b MPT in Table for simpl supported square plates with various values of side lengths L and nonlocal parameter ea= to. nm. The reason for choosing these values is that ea should be smaller than. nm for a singlewall carbon nanotube as pointed out b Wang and Wang 44. Since the maimum value of ea has not been eactl known for graphene sheet, it is assumed to be equal to that of the single-wall carbon nanotube. The shear correction factor used in MPT is taken as 5/6. The nondimensional deflection is obtained for the plate subjected to uniform loads, while the nondimensional critical buckling load is calculated for the plate subjected to biaial compression. The nondimensional deflection w, critical buckling load N, and fundamental frequenc are defined b 3 weh w, ql 4 N NcrL, 3 Eh L / E (3) h It can be seen that the present theor and MPT give almost identical results for all cases ranging from thin to thick plates confirming the accurac of present solutions. It should be noted that the present theor does not require shear correction factors as in the case of MPT. To illustrate the small scale effects on the responses of nanoplates, Figs. 1-3 plot the deflection, buckling load, and frequenc ratios with respect to the size of a simplsupported plate. The value of nonlocal parameter ea of a simpl-supported armchair nanoplate is 1.16 nm 38. The deflection, buckling load, and frequenc ratios are defined 1

14 as the ratios of those predicted b the nonlocal theor to the correspondences obtained b the local theor (i.e., ea=). It can be seen that the deflection ratio is greater than unit, whereas the buckling load and frequenc ratios are smaller than unit. It means that the local theor underestimates deflection (see Fig. 1) and overestimates buckling load (see Fig. ) and natural frequenc (see Fig. 3). This is due to the fact that the local theor ignores the small scale effect. In other words, the inclusion of the small scale effect leads to an increase in the deflection and a reduction of the buckling load and natural frequenc. The small scale effect is significant for thick plates (i.e. the size of the plate is small) especiall at the higher modes (see Figs. and 3). However, it will diminish for ver thin plates (i.e. the size of the plate is large). In addition to the small scale effect, the present nonlocal plate model also accounts for the shear deformation effect. The effect of shear deformation on the deflection, buckling load, and natural frequenc of a simpl-supported nanoplate is illustrated in Figs. 4-6, respectivel. The nonlocal parameter ea is taken as 1.16 nm 38. In these figures, the deflection, buckling load, and frequenc ratios are defined as the ratios of those obtained b the present nonlocal theor to the correspondences predicted b the nonlocal KPT where the shear deformation effect is omitted. It can be seen that the effect of shear deformation leads to an increase in the deflection and a reduction of the buckling load and natural frequenc, and this effect is significant for thick plates especiall at the higher modes (see Figs. 5 and 6). It means that the shear deformation effect makes the plate more fleible. 5. Conclusions A nonlocal plate model for bending, buckling, and free vibration of micro/nanoscale plates is developed based on the nonlocal differential constitutive relations of Eringen. 13

15 Equations of motion and boundar conditions are derived from Hamilton s principle. Analtical solutions for bending, buckling, and free vibration of a simpl supported plate are presented, and the obtained results are compared well with those generated b MD simulation and those predicted b the nonlocal MPT. As shown in this stud, the effects of small scale and shear deformation are similar. The inclusion of small scale and shear deformation effects makes the plate more fleible, and consequentl, leads to an increase in the deflection and a reduction of the buckling load and natural frequenc. These effects are significant for thick plates especiall at the higher modes, but the will diminish for ver thin plates. Acknowledgements The third author gratefull acknowledges financial support from Vietnam National Foundation for Science and Technolog Development (NAFOSTED) under Grant No , and from Universit of Technical Education Ho Chi Minh Cit. References 1. Li X, Bhushan B, Takashima K, Baek CW and Kim YK. Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscop. 3; 97: Chong ACM, Yang F, Lam DCC and Tong P. Torsion and bending of micron-scaled structures. Journal of Materials Research. 1; 16: Fleck NA, Muller GM, Ashb MF and Hutchinson JW. Strain gradient plasticit: theor and eperiment. Acta Metallurgica et Materialia. 1994; 4: Stolken JS and Evans AG. A microbend test method for measuring the plasticit length scale. Acta Materialia. 1998; 46:

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21 Fig. 1. Effect of small scale on deflection ratio of simpl supported plates under sinusoidal load Fig.. Effect of small scale on buckling load ratio of simpl supported plates under biaial compression

22 Fig. 3. Effect of small scale on frequenc ratio of simpl supported plates Fig. 4. Effect of shear deformation on deflection ratio of simpl supported plates under sinusoidal load 1

23 Fig. 5. Effect of shear deformation on buckling load ratio of simpl supported plates under biaial compression Fig. 6. Effect of shear deformation on frequenc ratio of simpl supported plates

24 Table 1. Fundamental frequenc (THz) of simpl support square SLGSs L (nm) Armchair SLGS, ea o 1.16 (nm) Zigzag SLGS, ea o 1.19 (nm) MD 38 Present Diff. (%) MD 38 Present Diff. (%) Table. Nondimensional deflection w, critical buckling load N, and fundamental frequenc of simpl supported square plates L/h ea(nm) o Deflection w Buckling load N Frequenc MPT Present MPT Present MPT Present