HOMONGENIZATION AND DIMENSIONAL REDUCTION OF THE NANO-SIZED - - PDF document

SLIDE 1

18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS

1 Introduction

Recently, as fabrication of the nano-sized structures has been realized with the development of manufacturing technology, nano-sized device and systems are now applied to various science and engineering fields [1,2]. Interestingly, functionality is emphasized in nano scale since unexpected properties appear in nano-sized device. Mechanical and thermal behavior show unique feature in nano size because nano-sized structures have very high ratio of the surface area to the volume. Such nano- sized structures exhibit different material behaviors compared with the macro-structures. Therefore, many researchers have tried to clarify these unusual size-dependent phenomena. The surface effect causes this phenomenon, and it comes from the difference of atomistic bonding states between the inner part and free surfaces of the nano-sized

• structure. This surface effect is negligible in the

macro-sized structures, however, it has a significant influence in the nano-sized structures. For the analysis of NEMS device or system the atomistic simulations are generally required. However, the atomistic simulation takes too much time to analyze the nano-sized structures so that alternative efficient analysis methods are needed. For this reason, the analysis method based on continuum theory has been developed by combining with surface elasticity has been developed [3,4,5] In this study, the mechanical behavior of the nano-sized honeycomb structures is investigated by combining the well-established homogenization

• method. For the analysis of the nano-sized

honeycomb structures considering the surface effect, the homogenization and dimensional reduction method is adopted, and the present method promotes the computational efficiency. Details

• f

homogenization can be found in Ref. [6, 8, 9]. 2 Homogenization and dimensional reduction For dimensional reduction of 3D problems with periodicity into 2D problems, the following assumption is used / / l L h L   (1) where L is the characteristic length, l is the periodic

• length. The non-dimensional parameter  defined in
• Eq. (1) is used to scale coordinates as follows

Fig.1. A periodic structure and a unit cell

1 2 3 1 2 3

( , , ) ( / , / , ) y y y x x x    (2) As shown in Fig. 1, the coordinate

3

y is chosen at

a mid-plane of a unit cell. The scaled coordinates in

• Eq. (2) are introduced by using the small parameter

 presented in Eq. (1). The displacement is

HOMONGENIZATION AND DIMENSIONAL REDUCTION OF THE NANO-SIZED HONEYCOMB STRUCTURES CONSIDERING SURFACE EFFECT

• Y. Lee1, J. Jeong2, M. Cho1*

1 Division of WCU multiscale Mechanical Design, School of Mechanical and Aerospace

Engineering, Seoul National University, Seoul 151-744, Republic of Korea.

2 Interdisciplinary Program in Automotive Engineering, Seoul National University, Seoul 151-744,

Republic of Korea.

* Corresponding author(mhcho@snu.ac.kr)

Keywords: Homogenization, Surface effect, Nano-sized honeycombs,

SLIDE 2

expanded with respect to  , and the strain and stress are asymptotically expanded from the displacement as shown in Eq. (3). For the 3D linear elastic problem, the governing equations (i.e. the equilibrium equation, the constitutive equation and the strain-displacement relationship) have the recursive form by employing the asymptotic expansion method, and the traction free condition at the upper and lower sides and the periodic condition at the lateral sides are applied.

 

1 2 2 1 1 1 1 1 1

( ) ( , ) ( , ) ( , ) ( ) ( ) ( ) ( ) ( ) ( , ) ( , ) ( , )

n n y x y

u x u x y u x y u x y e u e u e u e u x x y x y x y        

  

            . (3) By introducing two scales, the initial 3D elastic problem is separated into the macroscopic and microscopic problems. The homogenized stiffness are obtained by solving the microscopic problem. The solutions with respect to the leading order (- 1st

• rder) and the next order in the microscopic

problems are obtained directly from differential

• equations. The solutions to the 1st order are

decomposed into the rigid body displacement and warping displacement as follows

2 2

ˆ

i i

u u    . (4) Since the warping displacement cannot be solved directly, the virtual work principle is used, and the variational equation for the microscopic problem are written by

   

2 1 2 1 , ,

ˆ 0,

y y

u dy u dy        

 

 

 

1 3 ,

.

i ij y j i i k ij ijkl j j l

dy C y K dy y C E dy C dy y y y

   

                                              

   

(5) If the Kirchhoff-Love plate assumptions in Eq. (6) is applied, the dimensional reduction from 3D to 2D is accomplished.

1 1 1 3 2 3 3 3 3 3

ˆ ˆ ˆ ( ) ( ) ( ) 0, , ( ) ,

k i i i i

u x u x u x E x x x u x K E E K K x x

       

                  (6) In this study, the displacement solutions up to the 1st order are considered, and the equivalent membrane and bending stiffness of a unit cell are

• btained from the relationship between the stress

resultants (i.e. force and moment resultants) and the strains (i.e. strain and curvature) in the following equation.

   

   

1 T 1

N E K M                                            

h h h h

A B B D , (7) where

1

N and

1

M are in-plane force and moment resultants, respectively and E and K indicate macroscopic strain and curvature, respectively. In

• Eq. (7), the equivalent stiffness matrices are

expressed by

3 2 3 3

, , .

E h kl K h kl K h kl

A C C B y C C D y C y C

  

     

         (8) where the linear operator

is defined as

T 1 2 3 2 1 3 3 2 1

y y y y y y y y y                                        . (9) 3 The surface elasticity for capturing the size- dependent elastic properties In order to consider the surface effect, the surface elasticity proposed by Gurtin and Murdoch [3] is expressed by

SLIDE 3

, , , , ,

( )( ) ( )

s s s r r

u u u u

         

                . (10) where

 is the surface residual tension, and

s

 and

s

 are Lamé constants. These surface parameters

should be identified by the atomistic simulations. However, since the Gurtin and Murdoch’s model was derived from the isotropic materials, it cannot be applied to the single crystal structures which exhibit anisotropy with respect to the crystallographic orientations and directions. For single crystal structures, the surface elasticity can be extracted from the difference between two extensional stiffnesses in Eq. (10), and the detailed procedure is found out in [3].

* * 11 12 * * 12 22 * 66 1 12 2 21 12 21 12 21 1 2 21 12 21 12 12

(1 ) (1 ) . (1 ) (1 )

sur

C C C C C h C E h E h E h E h G h                                              (12) where the asterisk(*) denotes the values obtained by atomistic simulations.

• 4. Numerical example

The unit cell configuration of the nano-sized honeycomb structure having the regular hexagonal cells is shown in Fig. 2. Fig.2. Configuration of a unit cell As shown in Fig. 2, two distinct surfaces, where have the different crystallographic orientations,

• exist. Thus, each surface elasticity are estimated

through the atomistic simulations. In this study, the molecular dynamic simulations have been performed by open source code LAMMPS [7], and the obtained surface elasticity for the perpendicular wall and inclined wall in Fig.2 are given in Eqs. (13) and (14), respectively. 10.0930 24.2020 24.2020 10.0930 14.0045

sur blue

C            , (13) 2.6931 10.289 10.273 6.61585 0.8796

sur red

C            

. (14)

The free surfaces except the boundary areas are treated as the membrane finite elements, which does not have the thickness. As shown in Fig.3, the surface elasticity in Eq. (13) is applied to the blue- colored surfaces, and the red-colored surfaces correspond to the surface elasticity presented in Eq. (14). Fig.3. The mesh configuration of a unit cell including the surface elasticity The homogenized stiffness of the nano-sized honeycomb structure are examined with varying the wall thickness t , and the height of a unit cell is 300 nm. In Fig.4, the estimated homogenized stiffnesses of the nano scale honeycomb structure are normalized by those of the bulk, and it is shown that the size effect is represented strongly as the wall thickness

• decreases. The size-dependency with the wall

thickness is depicted overall, and the dramatic

SLIDE 4

increase of the rigidity against the in-plane shear or the twisting moment is especially remarkable. Fig.4 The normalized stiffness with varying wall thickness (height : 300 nm)

• 5. Conclusions

In this study, the analysis of the nano-sized honeycomb structure based on the continuum theory is presented. The simple atomistic simulations are carried out to identify the surface elasticity, and then it is embedded in the numerical homogenization method. In the numerical example, the homogenized extensional and bending stiffnesses with the size dependency are investigated. The present multiscale analysis method enables us to predict the mechanical behavior of the intractable size of structures in atomistic simulations. Acknowledgement This work was supported by the Korea Science and Engineering Foundation (KOSEF) grand funded by the Korea government (MES) (NO.2010- 00189202), and WCU (World Class University) program through the National Research Foundation

• f Korea funded by the Ministry of Education,

Science and Technology (R31-2010-000-10083-0). References

[1] M. Lillo and D. Losic “Pore opening detection for controlled dissolution of barrier oxide layer and fabrication of nanoporous alumina with through-hole morphology”, Journal of Membrane Science, Vol. 327, pp 11-17, 2009 [2] K. Honda, M. Yoshimura, T. Rao, D. Tryk, A. Fujishima, K. Yasui, Y. Sakamoto, K. Nishio, H. Masuda, “Electrochemical properties of Pt-modified nano-honeycomb diamond electrodes” , Journal of Electroanalytical Chemistry, Vol.514, pp 35-50, 2001 [3] M. Gurtin and A. Murdoch “A continuum theory of elastic material surfaces”, Archive for Rational Mechanics and Analysis, Vol. 59, pp. 389~390, 1975 [4] M. Cho, J. Choi and W. Kim “Continuum-based bridging model of nanoscale thin film considering surface effects”, Japanese Journal of Applied Physics

• Vol. 48, 020219

[5] J. Choi, M. Cho, W. Kim, “Multiscale Analysis of Nano-scale Thin Film Considering Surface Effects: Thermomechanical Properties”, Journal

• f

Mechanics of Materials and Structures, Vol. 5(1), pp 161-163, 2010. [6] N. Buannic, P. Cartraud and T. Quesnel “Homogenization of corrugated core sandwich panels”. Composite Structures, Vol. 29, pp 299- 312, 2003 [7] S. Plimpton, P. Crozier, A. Thompson, “2008, LAMMPS: large-scale atomic/molecular massively parallel simulator. Sandia National Laboratories.” [8] ́ , “Effective models of composite periodic plates – І. Asymptotic solution”, International Journal of Solids and Structures, Vol. 27, pp 1151- 1172, 1991 [9] ́ , “Effective models of composite periodic plates – Ш. Two-dimensional approaches”, International Journal of Solids and Structures, Vol. 27, pp 1151-1172, 1991