finite element method for linear micropolar elasticity and numerical study of some scale effects...

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International Journal of Mechanical Sciences 46 (2004) 1571–1587

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Finite element method for linear micropolar elasticity andnumerical study of some scale effects phenomena in MEMS

Lei Li�, Shuisheng Xie

Manufacturing Center, General Research Institute of Non-Ferrous Metal, Wai Da Road No. 2,

Xin Jie Kou, Beijing 100088, China

Received 6 April 2004; received in revised form 15 October 2004; accepted 19 October 2004

Abstract

The finite element formulation for micropolar elasticity theory is derived from the potential energyfunctional of the discrete system. According to this formulation, three kinds of elements are designed andthe effectiveness of these elements are investigated by a set of numerical tests. The scale effects in the purebending of ultra-thin beam and in the stress concentration problem of the plate with a hole are studied indetail. It is concluded that the beam’s bending stiffness will increase significantly when the beam’s thicknessis close to the material characteristic length parameter, and that the stress concentration factor will decreasewhen the hole radius is close to the material characteristic length parameter. The coupling parameter’sinfluences on the material mechanical behaviors are also studied.r 2004 Elsevier Ltd. All rights reserved.

Keywords: Micropolar elasticity; Finite element; Scale effects

1. Introduction

Materials are assumed to be homogeneous and continuous in classical continuum mechanics,generally, their intrinsic micro-structures are neglected. However, any macro-medium consists ofmass of grains at microscopic scale. Furthermore, materials always contain tremendous defects,

see front matter r 2004 Elsevier Ltd. All rights reserved.

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ress: [email protected] (L. Li).

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L. Li, S. Xie / International Journal of Mechanical Sciences 46 (2004) 1571–15871572

such as dislocations, grain boundaries and micro-flaws. These micro-structures and defects havegreat effects on materials’ mechanical behaviors at microscopic scale. Recent experiments [1–3]have shown that materials will display strong scale effect when the scale of deformation fieldassociated their characteristic length scale are in the order of microns. The trend ofminiaturization in engineering, such as micro-electro-mechanical systems (MEMS), electronicmaterials, ultra-thin films, etc., requires a comprehensive understanding of the effects of theintrinsic micro-structures on the material’s macroscopic responses.The classical continuum mechanics are unable to predict such scale-dependent phenomena since

their constitutive models contain no length scale parameter. The couple stress elasticity theorydeveloped by Toupin [4] and Mindlin [5] had recently been generalized to plasticity by Fleck andHutchinson [6] to explain the scale effects found in metallic materials. This theory is a special caseof micropolar elasticity proposed by Eringen [7–9]. The couple stress theory has the defects thatthe asymmetric couple stress tensors are undetermined, and the motions of material are alsounable to be fully determined. However, all components of stresses and motions of the materialare definite in micropolar elasticity. Therefore, micropolar theory is anticipated to be applied tothe fibrous materials, the granular and porous solid which would display scale effects atmicroscopic scale. In micropolar elasticity theory, each material point can rotate independently oftranslation and the material can transmit couple stress as well as the usual force. When the micro-rotations are constrained to be consistent with the vorticity of macro-displacements, micropolartheory degenerates to the couple stress theory.In practice, since the closed-form solutions are difficult to be obtained, numerical methods are

the main tools in engineering. Factually, most of the commercial finite element method (FEM)software widely adopted in the mechanical design are built on the classical continuum mechanicsand cannot be applied to the micropolar theory. Nakamura et al. [10] presented a finite elementformulation based on the principle of virtual work for micropolar elasticity. Providas et al. [11]designed a set of triangle elements for Cosserat theory, which can be easily changed formicropolar elasticity. Adachi [12] constructed a triangle element for the couple stress theory withpenalty function method. However, the validity of these elements is verified only by a plate stressconcentration problem. The element’s bending performance and sensibility to mesh distortion arenot examined. The boundary element method (BEM) is another powerful numerical tool for theanalysis of micropolar elasticity. Application of the BEM requires the so-called fundamentalsolution. A great deal of the literature had been devoted to study the existence theorems andfundamental solutions of various micropolar elasticity problems [13–17]. Kupradze [16] developedan efficient numerical scheme called generalized Fourier method for the boundary value problemsof differential equations. Based on Kupradze’s method, Schiavone and his co-workers [18–20],successfully extended this method to solve several problems in micropolar elasticity. Sladek andSladek [21–23] published a series of papers dealing with the boundary integral formulation ofmicropolar thermoelasticity problems. They divided the problems into six physically differentclasses and derived the fundamental solutions of governing differential equations for particularclasses of problems. Das and Chaudhuri [24] developed a two-dimensional boundary elementformulation based on the reciprocal theorem of micropolar elasticity. Schiavone [25] solved themixed boundary value problems in plane micropolar elasticity with the real boundary integralequation method. Liang and Huang [26,27] presented a boundary element method for two-dimensional elastic problems in micropolar elasticity, and successfully solved the pure bending

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L. Li, S. Xie / International Journal of Mechanical Sciences 46 (2004) 1571–1587 1573

problem of a plate and some thermoelasticity problems. Recently, Sladek and Sladek [28]presented a new meshless method for solving boundary value problems in micropolar elasticitybased on the local boundary integral equation method with the moving least squaresapproximation of physical quantities. El-Karamany and Ezzat [29] presented the directformulation of boundary integral equation method in Laplace transform domain for thegeneralized linear micropolar thermoelasticity, and derived the Green functions for thecorresponding differential equations.In the present paper, we developed an efficient element for micropolar elasticity and

studied the scale effects phenomena observed in MEMS. Three kinds of elements are designed andtheir performances, such as accuracy, shear locking and sensibility to mesh distortion, areexamined by detailed numerical tests. Eight-node element with a bubble function namedMQ8-l is recommended for application. This element is adopted to study scale effectsphenomena of the ultra-thin beams and stress concentration problem of a plate with a hole.The relationship between the couple stress theory, micropolar theory and classical elasticity arestudied in detail.

2. Review of micropolar elasticity

Consider a linear micropolar elastic material occupying a region O enclosed by a smoothboundary surface G in the Euclidean space. The material is assumed to be homogeneous andisotropic. Throughout this paper Cartesian coordinate system xi (i ¼ 1; 2; 3) is employed. Theusual summation convention for repeated indices and the comma convention representingderivatives with respect to coordinates are adopted.A complete derivation of the micropolar elasticity equations was given by Eringen [7]. For the

convenience of implementation of finite element process, governing equations are transformed tothe form slightly different from the original equations in [7]. The equilibrium equations for a staticproblem can be written as

ðsji þ tjiÞ;j þ pi ¼ 0; mjk;j þ ekijtij þ qk ¼ 0; (1)

where ekij is the permutation tensor (i; j; k ¼ 1; 2; 3), sji is the symmetric part of the asymmetricstress tensor tji and tji is the anti-symmetric part of tji; namely, sji ¼ ðtji þ tijÞ=2; tji ¼ ðtji � tijÞ=2:mjk is the couple stress tensor, pi is the body force and qk is the body couple.The kinematic relations are described as the following:

�ij ¼12ðui;j þ uj;iÞ; Zij ¼

12ðui;j � uj;iÞ � eijkok; wij ¼ oi;j; (2)

where eij is the symmetric part of the asymmetric strain tensor fij and Zij is the anti-symmetric partof fij : wij is the curvature tensor, ui is the displacement vector and oi is the micro-rotation vector.The constitutive laws are given by

sij ¼ ð2mþ kÞ�ij þ l�kkdij; tij ¼ kZij ; mij ¼ awkkdij þ bwij þ gwji; (3)

where dij is Kronecker delta, l; m are the Lame elastic constants from classical elasticity, k; a;b andg are additional material constants in micropolar theory.

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The strain energy density W ð�ij ; Zij; wijÞ of micropolar elastic material is

W ð�ij; Zij ; wijÞ ¼12 fl�ii�jj þ ð2mþ kÞ�ij�jig þ

12kZijZij þ

12 fawiiwjj þ bwijwji þ gwijwijg: (4)

Eringen [8] has shown that the necessary and sufficient conditions for the internal energy densityto be positive definite quadratic forms are 3lþ 2mþ k40; 2mþ k40; k40; 3aþ gþ b40 andbþ g40; g� b40:For plane problems of isotropic elastic micropolar solid, four material constants are required,

namely l; m; k and g: The following technical constants derived from them are more beneficial interms of physical insight. They are [30,31]: Young’s modulus E ¼ ð2mþ kÞð3lþ 2mþ kÞ=ð2lþ2mþ kÞ; shear modulus G ¼ ð2mþ kÞ=2; Poisson’s ratio n ¼ l=ð2mþ 2lþ kÞ; characteristiclength for bending l ¼ ½g=2ð2mþ kÞ1=2 and coupling number N ¼ ½k=ð2mþ 2kÞ1=2: Therange of Poisson’s ratio is from �1 to +0.5, the same as in the classical case. If k; g vanish,the solid becomes classically elastic. The case N ¼ 1 (its upper bound) is known as ‘couplestress theory’.Different types of boundary conditions are suggested depending on the nature of

problems. The displacement ui and the micro-rotation oi can be prescribed on the body surfaceGu; that is

ui ¼ ui; oi ¼ oi: (5)

Equivalently, we can prescribe surface tractions and couples on the body surface Gt

tjinj ¼ Ti; mjinj ¼ mi; (6)

where Ti and mi are surface stress and couple vectors, respectively, nj is the unit outward normalvector to Gt: It is also possible to prescribe displacements and micro-rotations on a portion of thesurface, and to prescribe tractions and couples on the remainder as mixed boundary conditions,such that, Gu [ Gt ¼ G; Gu \ Gt ¼ 0:

3. Finite element formulation and choice of shape functions for micropolar elasticity

The potential energy functional of elastic micropolar solid can be expressed as

Pðui;oiÞ ¼

ZO

W ð�ij; Zij ; wijÞdO�

ZO

ðpjuj þ qjojÞdO�

ZGt

ðTjuj þ mjojÞdG: (7)

The equilibrium equations (1) and the stress boundary conditions (6) can be derived fromvariation on Eq. (7).For plane problem, Eq. (7) is written in the matrix form

Pðu;xÞ ¼1

2

ZO

ðeTCe þ vTMv þ gTGgÞdO

�

ZO

ðuTp þ xTqÞdO�

ZGt

ðTT

u þ mTxÞdG; ð8Þ

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where

C ¼

lþ 2mþ k l 0

l lþ 2mþ k 0

0 0 mþ k2

264

375; M ¼

g 0

0 g

� ; G ¼

k2; (9)

e ¼ f�x �y �xygT; v ¼ fwzx wzyg

T; g ¼ fZg; (10)

e; v and g are engineering strain vectors.Three types of quadrilateral elements are designed for plane micropolar elasticity, viz., bilinear

4-node isoparametric element (MQ4), quadratic 8-node isoparametric element (MQ8) and 8-nodeelement with a bubble function (MQ8-l). The bubble function is introduced in order to improvethe element’s performances when the mesh is extremely distorted, and its value is zero on theboundaries of element. The element will not be sensitive to mesh distortion when bubble functionis introduced, as shown in numerical test 4.2.Generally, we consider the formulation of the element with a bubble function. The

displacement vector u and rotation vector x are interpolated as

u ¼ Nqq þ Nlk; (11)

x ¼ Nqr þ Nlh; (12)

where Nq and Nl are the node shape function and the internal bubble function, respectively, q=r isthe nodal displacement/rotation vector and k=h is the internal displacement/rotation parameter.Substituting (11) and (12) into (8) gives

P ¼1

2½dT qTT

Kdd Kdr

KTdr Krr

" #d

q

� � ½dT qTT

F

0

" #; (13)

where d ¼ ½qT rTT; q ¼ ½kT hTT; F ¼ ½PTþ PT Q

Tþ QT

T; detailed expressions of Kdd ;Kdr;Krr

are omitted for concision.The functional stationary condition dP ¼ 0 gives

Kdd Kdr

KTdr Krr

" #d

q

� ¼

F

0

" #: (14)

From (14), we obtain

q ¼ �K�1rr KT

drd: (15)

Internal parameter q vector is eliminated at the element level by substituting (15) into (14), wehave

P ¼ 12

dTKd � dTF; (16)

where the element stiffness matrix is

K ¼ Kdd � KdrK�1rr KT

dr: (17)

For an element without bubble function, its element stiffness matrix is Kdd :

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A set of quadrilateral elements are designed based on the formulation derived above. To obtainthe elements’ shape functions, variables are defined as follows:

x0 ¼ xix; Z0 ¼ ZiZ; (18)

in which xi and Zi are coordinate values of node i under natural coordinate system.For 4-node isoparametric element, as shown in Fig. 1a, the shape functions are

Nqi ¼14ð1þ x0Þð1þ Z0Þ: (19)

For 8-node isoparametric element, as illustrated in Fig. 1b, the shape functions are at cornernodes

Nqi ¼14ð1þ x0Þð1þ Z0Þðx0 þ Z0 � 1Þ; i ¼ 1; 3; 5; 7 (20)

at mid-side nodes

Nqi ¼x2i2ð1� x2Þð1þ Z0Þ þ

Z2i2ð1� Z2Þð1þ x0Þ; i ¼ 2; 4; 6; 8: (21)

For 8-node element with a bubble function, its shape functions at nodes are the same as 8-nodeisoparametric element, and the additional bubble function is selected as

Nqi ¼ ð1� x2Þð1� Z2Þ: (22)

4. Numerical tests and study of some scale effects phenomena with micropolar elasticity

4.1. Micropolar elastic beam’s pure bending

Ultra-thin beams are widely utilized in MEMS. Their thickness ranges typically fromsubmicrons to ten microns. Within this range, materials will exhibit strong scale effects.Microbend experiment of elastic epoxy polymetric beam performed by Lam et al. [3] revealed thatthe beam bending stiffness increases significantly when the beam thickness decreases from 115 to

(a)

3

x

y

ξ

η

u

v ω

1

2

4

(b)

x

y ξ

ηu

v ω

1 2

3

4

5

6

7

8

Fig. 1. (a) Four-node element (MQ4) and (b) 8-node element (MQ8) and 8-node element with a bubble function (MQ8-

l).

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25mm: The experimental results are promising to be explained by the micropolar theory asnumerical examples shown below.The analytical solution of micropolar elastic plate under pure bending has been given by

Gauthier [32], and this problem is simplified to a cantilever in plane strain status, as shown in Fig.2. It is adopted to examine the validity of the elements.Two numerical tests are designed. The first one is the thin beam test that is the generalization of

the famous test proposed by Macneal and Harder [33] for classical elasticity. This test willexamine whether the element has spurious mechanisms or not, such as shear locking, as shown inFigs. 3 and 4. Three types of meshes are used, rectangle mesh, parallelogram mesh and trapezoidone. Twelve-element meshes are used for MQ4 element (Fig. 3a–c), and 6-element meshes are usedfor MQ8 and MQ8-l (Fig. 4a–c). The second test is used to investigate the element’s sensibility tomesh distortion. Only two elements are employed in calculation, and the degree of element’sdistortion is represented by e ¼ S=ðL=2Þ; as shown in Fig. 5. In computation, Poisson ration ¼ 0:3; material characteristic length parameter l ¼ H=2 and coupling parameter N ¼ 0:5 aretaken.Numerical results of the thin beam test are listed in Table 1. From this table, we find that both

MQ8 and MQ8-l give excellent results for all three kinds of meshes (rectangle, parallelogram andtrapezoid mesh), while the results of MQ4 element are poor for all the three kinds of meshes.Especially, when irregular meshes are taken, MQ4 element exhibits severe shear locking, as shownin Table 1.It is observed from Table 1 that the beam’s deflection in classical elasticity is five times larger

than that in micropolar elasticity. This means that the beam’s bending stiffness of micropolarelasticity is more rigid than that of classical elasticity. The phenomena are similar to the situationobserved in the microbend experiment of elastic epoxy polymetric beam [3]. Factually, thisphenomenon will occur when the beam’s thickness H is close to the material’s characteristic lengthl. If the beam’s thickness is larger than l, such as 10 times, the deflection will tend to be the same asthat of classical elasticity.Fig. 6a and b show the elements’ sensibility to the mesh distortion. We can conclude that both

MQ4 and MQ8 are sensitive to the mesh distortion, while MQ8-l is nearly not affected by thedistortion factor e, when e varies from 0 to 0.98. The reason that MQ8-l is insensitive to the meshdistortion is that its interpolation polynomials are complete up to P2(x, y) in physical coordinatesystem while MQ8 is only complete up to P1(x, y).

P

P

H

L

Fig. 2. Cantilever beam under pure bending.

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0.2

6.0

A

(b)

0.2

45o

6.0

A

(c)

0.2

45o45o

6.0

A

(a)

Fig. 3. Twelve-element mesh for QM4 element, (a) rectangle meshes; (b) parallelogram meshes; and (c) trapezoid

meshes.

(a)

0.2

6.0

A

(b)

0.2

45o

6.0

A

(c) 6.0

0.2

45o45o

A

Fig. 4. Six-element mesh for QM8 and QM8-l element, (a) rectangle meshes; (b) parallelogram meshes; and (c)

trapezoid meshes.


From the numerical tests above, it is evident that MQ8-l’s performance is best among thesethree kinds of elements. Next, we will analyze the scale effects of stress concentration problemwith this element.

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S

L=10.0

H=

2.0

S

L= 10.0

H=

2.0

(a)

(b)

Fig. 5. (a) Two-element mesh for QM4 and (b) 2-element mesh for QM8 and QM8-l:

Table 1

Displacement vA/microrotation oA at point A in thin beam test (E ¼ 1e+8N/m2; n ¼ 0:3; l ¼ H/2, N ¼ 0.5,

P ¼ 5e+4N)

Rectangle mesh Parallelogram mesh Trapezoid mesh

MQ4 0.2947/0.0970 0.1342/0.0430 0.1102/0.0348

MQ8 0.4723/0.1561 0.4727/0.1561 0.4646/0.1532

MQ8-l 0.4730/0.1561 0.4730/0.1562 0.4729/0.1560

Micropolar elasticity solutions [34] 0.4726/0.1575

Classical elasticity solutions 2.4573/—


4.2. Scale effects of an infinite elastic plate with a hole

We consider an infinite plate containing a circular cylindrical hole of radius a. The plate issubjected to remote uniform tension p0; as shown in Fig. 7. The closed-form solution is obtainedin article [34].The symmetry of geometry and boundary conditions implies that a quarter of plate is adequate

for analyzing. The size of the analyzed domain is taken to be 50a (see Fig. 8). Totally 120 MQ8-lelements and 407 nodes are used in calculations, as shown in Fig. 8. The numerical process iscarried out with n ¼ 0:3; l ¼ a, N ¼ 0.5.Figs. 9 and 10 show that the distributions of the normalized normal stress ty and couple stress

my along radical line y ¼ 90�; respectively. Fig. 11 shows the variation of the normalized shear

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(a) (b) 0.0 0.2 0.4 0.6 0.8 1.0

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

v A

MQ4 MQ8 MQ8-λ analytical solution [34]

e

0.0 0.2 0.4 0.6 0.8 1.0

0.0350

0.0325

0.0300

0.0275

0.0250

0.0225

0.0200

0.0175

0.0150

0.0125

ωA

MQ8 MQ8-λ analytical solution [34]

e

Fig. 6. Study of element distortion sensitivity, (a) vA vs. e; (b) oA vs. e (E ¼ 1:5eþ 5N=m2; n ¼ 0:3;l ¼ H=2;N ¼ 0:5;P ¼ 1:0eþ 3N; e ¼ S=ðL=2Þ).

θ

2ax

y

0p

0p

r

Fig. 7. Infinite plate with a hole subjected to remotely uniform tension.


stress try and tyr along the radical line y ¼ 45�: The analytical solutions of micropolar elasticityand that of classical elasticity are also plotted in the figures for comparison. From these figures,we can find that all numerical results exactly agree with the analytical solutions.It is noted from the above figures that the stresses near the hole surface are greatly different

from that of classical elasticity. The stress concentration factor ty=p0��r¼a;y¼90� is only 2.65, while it

is 3.0 under classical elasticity (see Fig. 9). Two more evident differences can be observed fromthese figures. The first is that no couple stresses appear under classical elasticity, while they arequite large and cannot be neglected near the hole under micropolar elasticity (see Fig. 10). Thesecond is that the surface of the hole is shear stress free under classical elasticity, while it issubjected to the shear stress tyr under micropolar elasticity (see Fig. 11). These two phenomenamay contribute to explain the micro-void growth and coalescence. Another difference is that shearstress tryatyr near the hole, while they are equal to each other under classical elasticity. Actually,the above differences between the micropolar elasticity and classical elasticity only exist in a small

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Fig. 8. Finite element mesh for micropolar elastic plate.

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.01.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.0

t θ/p 0

ν=0.3l=aN=0.5

r/l

MQ8-λmicropolar elasticity solutions [34]classical elasticity solutions

Fig. 9. Distributions of the normalized normal stress ty along the radical line y ¼ 90�:


area around the hole surface, and they tend to disappear when the distance is away from the holesurface. Normal stress ty in micorpolar elasticity will be equal to that in classical elasticity whenr=l43:0: Shear stress try will be equal to tyr and couple stresses will disappear when r=l48:0:Hence, we can conclude that the scale effects phenomena only exist in very small area around thehole.The influences of coupling parameter N on the stresses around the hole surface

ðr ¼ a; y 2 ½0�; 90�) are plotted in Figs. 12–14. The results of classical solutions and couple stress

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2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0-0.75

-0.70

-0.65

-0.60

-0.55

-0.50

-0.45

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

mθ/(

a2p 0

)ν=0.3l=aN=0.5

r/l

MQ8-λmicropolar elasticity solutions [34]

Fig. 10. Distributions of the normalized couple stress my along the radical line y ¼ 45�:

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

ν=0.3l=aN=0.5

r/l

trθ

/p0 MQ8-

tθr

/p0 MQ8-

micropolar elasticity solutions [34] classical elasticity solutions

λλ

Fig. 11. Distributions of the normalized shear stresses try and tyr along the radical line y ¼ 45�:


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0 10 20 30 40 50 60 70 80 90-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t θ/p

0

ν=0.3l=a

θ /(˚)

MQ8-λ λ (N=0.1)

MQ8- (N=0.5) MQ8-λ (N=0.9) classcial elasticity solutions couple stress theory solutions [35]

Fig. 12. Influence of N on distributions of the normal stress ty around the hole surface (r ¼ a; y 2 0; 90�½ ).


theory solutions [35] are also presented for comparison. Scatters represent the numerical solutionsby MQ8-l: Solid line curve is classical elasticity solution and the dash line curve is the couplestress elasticity solution. It is observed that values of stresses range from the classical elasticity tocouple stress theory while N varies from 0.1 to 0.9. When N ! 0, the distributions of stresses tendto be the classical elasticity solutions, while N ! 1, they tend to be the same as couple stresstheory solutions. Hence, both classical elasticity and couple stress theory can be taken as specialcases of micropolar theory.Next, we study the influence of material characteristic length l on material’s mechanical

behaviors while l varies from 0:1a to a, as shown in Fig. 15. When l ¼ 0:1a; the scale effects isfairly weak and the normal stress ty around the hole is almost equal to that of the classicalelasticity. With l increasing, ty decreases, indicating that the scale effects will become stronger if l

increases with hole radius a fixed. In other point of view, if the hole radius a is close to the materialcharacteristic length l, the scale effects will exhibit strongly. While if hole radius a is great largerthan this length, such as a410l; the scale effects can be neglected. Material characteristic length isvery small since it is interpreted as the measurement of material intrinsic micro-structure. For Ni,Stoken and Evans [2] estimate it is 6mm; therefore, the scale effects cannot be observed formaterial with large hole whose radius is larger than material characteristic length.

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0 10 20 30 40 50 60 70 80 90-1.3

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

t θr/p

0

ν=0.3l=a

θ/(°)

MQ8-λλ

λ (N=0.1) MQ8- (N=0.5) MQ8- (N=0.9) classical elasticity solutions couple stress theory solutions [35]

Fig. 13. Influence of N on distributions of the couple stress my around the hole surface (r ¼ a; y 2 0; 90�½ ).

0 10 20 30 40 50 60 70 80 90-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

mθ/(

a2 p 0)

ν=0.3l=a

θ/(°)

MQ8-λ λ

(N=0.1)MQ8- (N=0.5)MQ8-λ (N=0.9)

classical elasticity solutionscouple stress theory solutions [35]

Fig. 14. Influence of N on distributions of the shear stress tyr around the hole surface (r ¼ a; y 2 ½0; 90�).


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0 10 20 30 40 50 60 70 80 90-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0t θ/

p 0

= 0.3Ν=0.5

θ/(°)

classical elasticity solutionsMQ8- (l=0.1a)MQ8- (l=0.6a)MQ8- (l=1.0a)

ν

λλλ

Fig. 15. Influence of l on distributions of the normal stress ty around the hole surface (r ¼ a; y 2 ½0; 90�).


5. Conclusion

The finite element formulation for micropolar elasticity is derived from the potential energyfunctional. Subsequently, three kinds of elements are designed based on this formulation. Theperformances of these elements are examined by a set of tests. MQ8-l element is recommended forapplication. Two typical scale effects phenomena in MEMS are analyzed numerically. Somevaluable mechanical behaviors different from the classical elasticity are studied in detail. Theelement developed in this paper is reliable and will facilitate the study of mechanical behaviors ofmicropolar media.

Acknowledgements

The authors are grateful for the reviewers’ helpful suggestions. This work was supported by theNational Natural Science Foundation of China (No. 50175006, 10172078).

References

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1994;42:475–548.

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ARTICLE IN PRESS


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finite element method for linear micropolar elasticity and numerical study of some scale effects...

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