Acta Mech. Sin. (2011) 28(1):91–103 DOI 10.1007/s10409-011-0540-y
RESEARCH PAPER
C 1 natural element method for strain gradient linear elasticity and its application to microstructures Zhi-Feng Nie · Shen-Jie Zhou · Ru-Jun Han · Lin-Jing Xiao · Kai Wang
Received: 23 December 2010 / Revised: 7 June 2011 / Accepted: 17 October 2010 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012
Abstract C 1 natural element method (C 1 NEM) is applied to strain gradient linear elasticity, and size effects on microstructures are analyzed. The shape functions in C 1 NEM are built upon the natural neighbor interpolation (NNI), with interpolation realized to nodal function and nodal gradient values, so that the essential boundary conditions (EBCs) can be imposed directly in a Galerkin scheme for partial differential equations (PDEs). In the present paper, C 1 NEM for strain gradient linear elasticity is constructed, and several typical examples which have analytical solutions are presented to illustrate the effectiveness of the constructed method. In its application to microstructures, the size effects of bending stiffness and stress concentration factor (SCF) are studied for microspeciem and microgripper, respectively. It The project was supported by the SDUST Spring Bud (2009AZZ021) and Taian Science and Technology Development (20112001). Z.-F. Nie (¬) · L.-J. Xiao College of Mechanical and Electronic Engineering, Shandong University of Science and Technology, 266510 Qingdao, China e-mail:
[email protected] S.-J. Zhou School of Mechanical Engineering, Shandong University, 250061 Jinan, China R.-J. Han Department of Scientific Research, Shandong University of Science and Technology, 266510 Qingdao, China K. Wang Jinan Vocational and Technical College in Engineering, 250200 Jinan, China
is observed that the size effects become rather strong when the width of spring for microgripper, the radius of circular perforation and the long axis of elliptical perforation for microspeciem come close to the material characteristic length scales. For the U-shaped notch, the size effects decline obviously with increasing notch radius, and decline mildly with increasing length of notch. Keywords Strain gradient linear elasticity · C 1 natural element method · Sibson interpolation · Microstructures · Size effects 1 Introduction Microstructures have been used widely in micro-electromechanical-systems (MEMS). The size of microstructures ranges from submicron to micron. Within this range, materials manifest strong size effects. For example, torsion test of thin copper wires showed that the normalized torsion hardening increases with decrease in the wire diameter [1]. Indentation test revealed that the hardness increases as the size of the indentor is decreased [2]. Micro-bending test also showed that the normalized bending hardness increases as the beam thickness decreases [3]. Size dependencies were also observed in some elastically deformed polymers, for example, micro-bending test exhibited that the bending rigidity increases with decrease in the beam thickness [4]. Classical continuum constitutive models possess no material characteristic length scales, and are thus incapable of describing such a behavior. In order to predict size effects, some sizedependent continuum theories have been developed, such as strain gradient theory. Starting from the Cosserat theory [5], some strain gradient continuum theories have been developed. Toupin [6]
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proposed a strain gradient elasticity theory by incorporating the strain gradient term into the constitutive equations. Later, Mindlin [7] proposed a more general theory that encompassed the theories of Cosserats and of Toupin as special cases. Fleck and Hutchinson [8, 9] extended the Toupin and Mindlin theories to the deformation and flow theories of plasticity. Variants of strain gradient elasticity and plasticity theories have also appeared [4, 10, 11]. The generalized continuum theories mentioned above fall within the class of higher-grade (or higher-order) theories, in which the higher-order stresses are included. In order to simplify the constitutive models, some gradient theories without higherorder stresses were proposed [12–20]. A comparison of various higher-grade (or higher-order) theories can be found in Ref. [9], and that of various gradient theories without higherorder stresses can be found in Ref. [21]. For higher-grade (or higher-order) continuum theories, analytical solutions are restricted to a few very simple problems; solutions of practical engineering problems have to resort to numerical methods. Within the framework of finite element method, it is difficult to construct the C 1 -continuous elements, and the available elements are limitable [22–24]. In order to degrade the requirement of continuity, some efforts have been made to replace the C 1 -continuous elements, such as mixed finite element formulations [25–27], discontinuous Galerkin formulations [28]. However, these methods either have some drawbacks, such as either a high number of degrees of freedom (DOF), or must be used in approximation fields, or the computational cost is high. Meshless methods are alternative numerical methods, which are based on the particle-based interpolation. Meshless methods based on the moving least-square (MLS) approximation like the element-free Galerkin method (EFG) [29], the meshless local Petrov–Galerkin method (MLPG) [30] and the reproducing kernel particle method (RKPM) [31] and similar ones can construct higher-order continuous shape functions, however, EBCs can not be applied directly because of the noninterpolating properties of the shape functions. A few meshless methods have been used in the analyses of gradient-dependent elasticity or plasticity problems. For example, Askes et al. [32] used EFG to carry out the gradient elasticity analysis of plane strain problems. Tang et al. [33] developed a true rotation-free numerical approach based on MLPG for materials within the strain gradient elasticity. Manzari et al. [34] adopted RKPM to study the gradient plasticity modeling of geomaterials. The natural element method (NEM), which is based on Sibson’s [35] C 0 interpolant, was proposed by Braun and Sambridge [36]. It was used for data approximation in geophysical applications [36], and successfully explored for solid mechanics problems [37], and also applied to the simulation of crack propagation [38]. C 1 NEM [39] combines the easy achievement of higher continuity with the direct application of EBCs, which is a smooth extension of the NEM. C 1 NEM and NEM are the Galerkin-based meshless methods that are
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built upon the notion of NNI. Fischer et al. [40] applied C 1 NEM to non-linear gradient elasticity, and compared the performances of C 1 finite elements and the C 1 NEM. In the present paper, C 1 NEM for strain gradient linear elasticity is built when C 1 NEM is applied to the strain gradient linear elasticity problems, and some typical examples which have the analytical solutions are used to check the convergence and computational accuracy of the built method. In its application to microstructures, size effects for microgripper and microspeciem are studied. The paper is outlined as follows. In Sects. 2 and 3, strain gradient linear elasticity and C 1 NEM are reviewed. C 1 NEM for strain gradient linear elasticity is constructed in Sect. 4, in which, meshless method implementation is described and two numerical examples are used to illustrate the effectiveness of the constructed method. Size effects for microgripper and microspeciem are analyzed in Sect. 5. Some concluding remarks are mentioned in Sect. 6. 2 Review of strain gradient linear elasticity Mindlin [7] developed strain gradient theory for linear elastic materials whereby the strain energy density w is taken to be a function of both strain tensor εi j and the second gradient of displacement tensor ηi jk . Therefore, w can be written as w = w(εi j , ηi jk ) 1 λεii ε j j + μεi j εi j + a1 ηi j j ηikk + a2 ηiik ηk j j 2 +a3 ηiik η j jk + a4 ηi jk ηi jk + a5 ηi jk ηk ji ,
=
(1)
where λ and μ are the usual Lame’s constants, ai (i = 1, 2, · · · , 5) are the five additional material constants. εi j and ηi jk are given by 1 (ui, j + u j,i ), 2 = uk,i j ,
εi j =
(2)
ηi jk
(3)
where ui are the displacement fields. The strain energy variation in the domain V of a deformed isotropic elastic body can be written as δwdV = (σi j δεi j + τi jk δηi jk )dV, (4) δW = V
V
where the stress tensor σi j conjugated to εi j and the double stress tensor τi jk conjugated to ηi jk are given by σi j =
∂w = λεkk δi j + 2μεi j , ∂εi j
τi jk =
∂w ∂ηi jk
= a1 (ηipp δ jk + η jpp δik ) 1 1 +a2 ηkpp δi j + η ppi δ jk + η pp j δik 2 2
(5)
C 1 natural element method for strain gradient linear elasticity and its application to microstructures
+2a3 η ppk δi j + 2a4 ηi jk + a5 (ηk ji + ηki j ).
(6)
The governing equations and boundary conditions of strain gradient elasticity can be expressed as fk + (σ jk − τi jk,i ), j = 0,
in V,
uk = u¯k ,
Duk = v¯k ,
on S u ,
and tk = t¯k ,
Rk = R¯k ,
on S t ,
Ci (x) =
n φk,α (x) , ik φk (x) k=1
(8)
−D j ni τi jk − ni D j τi jk ,
(9)
Rk = ni n j τi jk ,
(10)
where D j = (δ jp − n j n p )∂ p is a surface-gradient operator, ni is the i-th component of the unit surface normal vector. u¯k , v¯k , t¯k and R¯k represent prescribed displacements, normal derivatives of the displacements, tractions and double stress tractions on the boundary, respectively.
(α, β = x, y).
|i|=3
where bi is the B´ezier ordinate, multi-index i is n-tuples of nonnegative integers, i.e. i = (i1 + i2 + · · · + in ); B3i (x) is the cubic Bernstein–B´ezier basis function, which can be expressed as [39] ⎛ ⎞ ⎜⎜ 3 ⎟⎟ 3 Bi (x) = ⎜⎜⎜⎝ ⎟⎟⎟⎠ φi11 (x)φi22 (x) · · · φinn (x). (12) i The first- and second-order derivatives of ten as [39] (B3i (x)),α = B3i (x)Ci (x),
(14b)
uh (x) =
3n
ψ j (x)u j ,
(15)
j=1
where ψ3I−2 (x), ψ3I−1 (x) and ψ3I (x) are the shape functions for node I that are associated with the nodal DOF uI , uI,x , and uI,y , respectively. The first- and second-order derivatives of Ψ (x) can be obtained as follows [39] (Ψ ,α (x))T = (B ,α (x))T T ,
(16)
T
(Ψ ,αβ (x)) = (B ,αβ (x)) T ,
As for the NEM, the Sibson interpolant is C k (k ≥ 1) continuity everywhere, except at the nodes where it is C 0 continuity [35,37,41,42], therefore, it is globally C 0 continuity. Farin [41] embedded the C 0 Sibson interpolant in the Bernstein–B´ezier patch, and C 1 continuity at the nodes is achieved. Sukumar and Moran [39] proposed a computational methodology that renders the C 1 interpolant amenable to numerical implementation for the solution of PDEs. Let point x in R2 have n natural neighbors, with φI (x) the natural neighbor shape function of node I. By introducing the Sibson interpolant φI (x) in the Bernstein–B´ezier patch, a C 1 interpolant is constructed in the form of Ref. [41] B3i (x)bi , (11) uh (x) =
B3i (x)
can be writ-
(17)
where T is the transformation matrix, which maps the nodal function and gradient values to B´ezier ordinates. 4 C 1 NEM for strain gradient linear elasticity 4.1 C 1 NEM formulations for strain gradient elastic analysis As C 1 NEM interpolant has the interpolating properties to nodal function and nodal gradient values, EBCs can be imposed directly. Therefore, the weak form of the governing Eq. (7) and associated boundary conditions Eq. (8) can be written as ( fk + σ jk, j − τi jk,i j )δuk dV + (t¯k − tk )δuk dS V
+
S
S
(R¯k − Rk )Dδuk dS = 0.
(18)
Using divergence theorem and integrating by parts, we can write the first integral term of Eq. (18) as fk δuk dV− (σ jk δuk, j + τi jk δuk,i j )dV V
V
+
S
δuk σ jk n j dS −
(13a)
(14a)
In order to render the C 1 natural neighbor interpolant, originally developed for data interpolation [41], amenable to numerical computations, Sukumar and Moran [39] proposed a computational methodology to get the standard C 1 shape functions, and interpolation formulation can be expressed in another form [39]
T
3 Review of C 1 NEM
(13b)
n φk (x)φk,αβ (x) − φk,α (x)φk, β (x) , ik φ2k (x) k=1
(Ci (x)), β =
tk = ni (σik − τi jk, j ) + ni n j τi jk (D p n p )
(α = x, y),
(B3i (x)),αβ = (B3i (x)), βCi (x) + B3i (x)(Ci (x)), β ,
(7)
where V is the problem domain bounded by the boundary S = S u + S t . fk is the body force per unit volume of the body V, tk and Rk are the traction and the double stress traction per unit area of surface S , respectively. D = n p ∂ p is the surface normal-gradient operator. tk and Rk are in equilibrium with σi j and τi jk according to
93
+
S
δuk, j τi jk ni dS .
S
δuk τi jk,i n j dS (19)
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The gradient of displacement variation on the boundary surface can be decomposed into a surface-gradient and a normal-gradient, i.e. δuk, j = D j δuk + n j Dδuk .
(20)
Taking into account Eqs. (9), (10), (19) and (20) and assuming that the boundary integral would be carried out over a smooth surface, we can write the variational weak form of governing Eq. (7) as (σ jk δuk, j + τi jk δηi jk )dV V
=
V
fk δuk dV +
S
(t¯k δuk + R¯k Duk )dS .
(21)
4.2 C 1 NEM discretization and numerical implementation For the plane strain problems, the non-zero displacement fields are u = [u x uy ]T .
(22)
Non-zero strain tensor and the second gradient of displacement tensor can be written in vector form as ε = [ε xx εyy 2ε xy ]T = L1 u,
(23) T
η = [η xxx ηyyx 2η xyx η xxy ηyyy 2η xyy ] = L2 u,
(24)
where the differential operator matrixes L1 and L2 are defined as ⎡ ∂ ⎤⎥⎥T ⎢⎢⎢ ∂ 0 ⎥ ⎢⎢⎢ ∂x ∂y ⎥⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥ , L1 = ⎢⎢ (25) ⎢⎢⎢ ∂ ⎥⎥⎥⎥ ∂ ⎣ 0 ⎦ ∂y ∂x ⎡ 2 ⎤T ∂2 ∂2 ⎢⎢⎢ ∂ ⎥⎥⎥ 0 0 0 2 ⎢⎢⎢ ⎥⎥⎥ ∂x∂y ⎢⎢⎢ ∂x2 ∂y2 ⎥⎥⎥ L2 = ⎢⎢⎢ ⎥⎥⎥ . (26) 2 2 2 ⎢⎢⎢ ⎥⎥⎥ ∂ ∂ ∂ ⎣ 0 ⎦ 0 0 2 ∂x∂y ∂x2 ∂y2 The constitutive relations for the stress tensor, double stress tensor, as expressed in Eqs. (5) and (6), can be written in vector form as σ = [σ xx σyy σ xy ]T = D σ ε, T
τ = [τ xxx τyyx τ xyx τ xxy τyyy τ xyy ] = D τ η,
⎡ ⎢⎢⎢ D1 ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢ D τ = ⎢⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ symmetry
D2
0
0
0
D4
0
0
0
D6
D5
D3
D4
D2 D1
⎤ D3 ⎥⎥ ⎥⎥⎥ ⎥ D5 ⎥⎥⎥⎥ ⎥⎥⎥ 0 ⎥⎥⎥⎥⎥ ⎥⎥⎥ , 0 ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 ⎥⎥⎥⎥ ⎥⎥⎦ D6
(30)
where D1 = 2(a1 + a2 + · · · + a5 ),
(31a)
D2 = a2 + 2a3 , a2 D 3 = a1 + , 2
(31b)
D4 = 2(a3 + a4 ), a2 + a5 , D5 = 2 a1 a5 + a4 + . D6 = 2 2
(31d)
(31c)
(31e) (31f)
Next, a surface traction vector and a surface double stress traction vector are defined as t¯ = [t x ty ]T ,
(32)
R¯ = [R x Ry ]T .
(33)
The discretization of displacement vector can be expressed as u = [u x uy ]T =
n
ΨI uˆ I ,
(34)
I=1
where
⎡ ⎢⎢ ψ3I−2 Ψ I = ⎢⎢⎢⎣ 0
ψ3I−1
ψ3I
0
0
0
0
ψ3I−2
ψ3I−1
u¯ I = [uI x uI x,x uI x,y uIy uIy,x uIy,y ]T .
⎤ 0 ⎥⎥⎥ ⎥⎥⎦ , ψ3I
(35a) (35b)
(27)
After substituting Eqs. (23), (24), (27), (28) and (34) into Eq. (21), neglecting body force f , and using the arbitrariness of nodal variations, a discrete system of equations can be obtained as
(28)
K uˆ = f ext ,
where the constitutive matrixes D σ and D τ for an isotropic linear material are defined as ⎤ ⎡ 2 (1 − ν) 2ν ⎢⎢⎢ 0 ⎥⎥⎥⎥ ⎢⎢⎢ 1 − 2ν ⎥⎥⎥ 1 − 2ν ⎢⎢⎢ ⎥⎥⎥ ⎢ ⎥⎥⎥ , ⎢ (1 2 − ν) D σ = G ⎢⎢⎢ (29) ⎥⎥⎥ 0 ⎢⎢⎢ ⎥⎥⎥ 1 − 2ν ⎢⎣⎢ ⎦ symmetry 1 in which G is shear modulus and can be expressed by E . Young’s modulus E and Poisson’s ratio ν, i.e. G = 2(1 + ν)
where K IJ = f ext I
V
(36) T T B σI D σ B σJ + B τI D τ B τJ dV,
∂Ψ T ∂Ψ TI I + ny Ψ TI t¯ + n x = R¯ dS . ∂x ∂y S
(37a) (37b)
In Eq. (37a), B σI , B τI are the matrixes of shape function derivatives which are given by B σI = L1 Ψ I
C 1 natural element method for strain gradient linear elasticity and its application to microstructures ⎡ ⎢⎢⎢ ψ3I−2,x ⎢⎢⎢ = ⎢⎢⎢⎢⎢ 0 ⎢⎢⎣ ψ3I−2,y
ψ3I−1,x
ψ3I,x
0
0
0
0
ψ3I−2,y
ψ3I−1,y
ψ3I,y
ψ3I−1,y
ψ3I,y
ψ3I−2,x
ψ3I−1,x
ψ3I,x
0
⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ , ⎥⎥⎥⎥ ⎦
95
(38a)
B τI = L2 Ψ I ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ = ⎢⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎣
ψ3I−2,xx
ψ3I−1,xx
ψ3I,xx
0
0
0
ψ3I−2,yy
ψ3I−1,yy
ψ3I,yy
0
0
0
2ψ3I−2,xy
2ψ3I−1,xy
2ψ3I,xy
0
0
0
0
0
0
ψ3I−2,xx
ψ3I−1,xx
ψ3I,xx
0
0
0
ψ3I−2,yy
ψ3I−1,yy
ψ3I,yy
0
0
0
2ψ3I−2,xy
2ψ3I−1,xy
2ψ3I,xy
⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ . ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦
(38b) 4.3 Numerical examples to verify the constructed method In this section, two numerical examples, are presented to verify the effectiveness of C 1 NEM for strain gradient linear elasticity. i.e. (1) a bimaterial system subjected to remote uniform shear stress and (2) an infinite plate with a central circular hole subjected to remote biaxial tension. 4.3.1 Boundary layer analysis Consider a bimaterial system composed of two perfectly bonded half planes of elastic strain gradient solids, subjected to a remote shear stress of P, as shown in Fig. 1. The shear modulus μ of material No. 1 is taken to be twice that of material No. 2, i.e. μ1 = 2μ2 . For each material i (i = 1, 2), material constants are assumed as a1 = a2 = a5 = 0, and 1 a3 = a4 = μi l2i , here li is usually called the characteristic 2 length scales for materials No. i, and l1 = l2 = l has been used in the analysis. An analytical solution is presented by Shu et al. [26] and the boundary conditions are dealt with as did in Refs. [26, 33]. A numerical model with several nodal distribution schemes is described in Fig. 2, where the domain with dimensions of 500l × 100l is taken to model the bimaterial system, and both uniform and non-uniform distributions of nodes are considered.
Fig. 2 Nodal discretization schemes for boundary layer analysis. a Uniform grid; b Non-uniform grid (105 nodes); c Uniform grid; d Non-uniform grid (217 nodes); e Uniform grid; f Non-uniform grid (533 nodes)
Define an average shear strain as ε xy =
P(μ1 + μ2 ) . μ1 μ2
(39)
The results of 533 nodes discretization scheme are presented in Fig. 3. It is obvious that the numerical solutions agree well with the analytical ones. Some numerical errors occur in the coarse nodal cases (105 and 217 nodes), however, the numerical results converge quickly to the analytical solutions with an increasing refinement of nodal distances, as shown in Fig. 4. Perhaps Voronoi diagram and Delaunay triangulation can accommodate non-uniform densities of nodes, there is little discrepancy in the numerical solutions between the uniform nodes and non-uniform nodes for all cases. 4.3.2 Infinite plate with a central circular hole
Fig. 1 Geometry of a bimaterial system subjected to remote uniform shear stress P
Consider an infinite plate with a central circular hole of radius a subjected to a biaxial tensile load of P at infinity, as shown in Fig. 5. Owing to symmetry, only the upper right quadrant of the plate with dimensions of 20a × 20a is modeled, and, the nodal distributions used in the calculation are shown in Fig. 6, where the total number of nodes is 650. Plane strain condition is assumed with material constants 1 in Eq. (6) are chosen as a1 = λl2 , a2 = a3 = 0, and 2 1 a4 = a5 = μl2 . The exact solutions for stresses and dis2
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placement can be found in Ref. [27], and the definition of the boundary conditions can be found in Refs. [27, 33].
Fig. 6 Nodal discretization for the plate with a central circular hole and its local enlarged diagram
The numerical simulations are carried out for ν = 0.3 and a = 3l along AE boundary. The numerical solutions of the normalized displacement and normalized stresses agree well with the exact solutions, as shown in Figs. 7 and 8. Fig. 3 Shear strain in the bimaterial system
Fig. 4 Accuracy study in the boundary layer analysis
Fig. 7 Variation of displacement for the plate with a central circular hole along AE boundary
Fig. 5 Plate with a central circular hole subjected to biaxial tension
Fig. 8 Variation of stresses for the plate with a central circular hole along AE boundary
C 1 natural element method for strain gradient linear elasticity and its application to microstructures
97
5 Size effects on the microstructures
From the above numerical examples, it is clear that C 1 NEM is an effective method to solve strain gradient elasticity problems. Therefore, C 1 NEM for strain gradient elasticity is applied to practical engineering problems in which the exact solutions are difficult to be obtained, and size effects on microgripper and microspeciem are analyzed. 5.1 Size effects of the bending stiffness for microgripper Microgripper capable of handling tiny objects is of interest for use in laser printing and projection display systems. Compared to the parallel-plate-capacitor, the drive capacitance of the electrostatic comb driver [43] is linear with displacement of the structure, resulting in a force which is independent of movement. An electrostatic comb driver has been modified for the gripper [44], and a cantilever with a large number of comb teeth has been designed to increase the force and to achieve the maximum displacement. Figure 9 shows the layout of the polysilicon microgripper consisting of two movable gripper tines driven by three electrostatic comb drivers. The gripping force is the main parameter, which is defined as the difference between the electrostatic driving force and the elastic recovery force. Once the applied voltage is given, the electrostatic driving force is assumed to be a fixed value, while the elastic recovery force is proportional to the bending stiffness of the gripping arm, therefore, in order to increase the gripping force, it is necessary to reduce the bending stiffness of the gripping arm. Li Yong et al. [45] adopted S-shaped spring flexible structure to link the drive arm and the anchor, and the bending stiffness of the gripping arm decreases drastically as the number of the springs increases. The cross-sectional view of the S-shaped spring is shown in Fig. 10. In fact, besides the number of the springs, other parameters such as the width and height of the spring, and the distance between two springs would also affect the bending stiffness. Furthermore, when the sizes of the microgripper are of the order of micron, the size effects of some physical parameters should be considered.
Fig. 10 S-shaped microspring (from Ref. [45])
The microgripper fabricated by Li Yong et al. [45] can be simplified as a cantilever beam with S-shaped spring, as shown in Fig. 11. The eletrostatic force (per unit surface area) acting on the cantilever beam is [45] p=
εU 2 , 2(b + d)d
(40)
where ε is the permittivity of free space, U is the applied voltage, b is the width of the comb teeth, and d is the gap between the drive-arm comb teeth and the driver-comb teeth. According to the definition of the bending stiffness [46], it is defined in this paper as the ratio of the applied resultant force to the deflection corresponding to the point whose x coordinate is L − Ld /2, where L and Ld are the length of the whole cantilever beam and the electrostatic comb driver, respectively, as shown in Fig. 11. Because of good mechanical properties and adaptation to extreme environments, more and more metal materials are introduced to fabricate the microstructures. Zavracky et al. [47] used nickel to fabricate the micromechanical switches, nickel is also used in this paper, and its material parameters are as follows: E = 207 GPa, ν = 0.312, and l = 5.2 μm [3]. Nodal discretization of the cross-sectional view for the microgripper arm is demonstrated in Fig. 12, where the nodes in the left surface are imposed on all constraints, and the nodes in the top surface are imposed on the eletrostatic force p. During the numerical simulation, the structure parameters are prescribed as follows.
Fig. 11 A cross-sectional view of the microgripper arm modeled as a cantilever beam with S-shaped spring
Fig. 9 Schematic of the microgripper (from Ref. [44])
Fig. 12 Nodal discretization of the cross-section of the gripping arm
98
The height of spring H = 60 μm, the distance between two springs S = 5 μm, the number of springs n = 4, the thickness of drive arm h2 = 10 μm when the impact of the width of spring h1 on bending stiffness Keff is studied. When the impact of H on Keff is studied, other parameters are chosen as h1 = 10 μm, S = 5 μm, n = 4, and h2 = 10 μm. Similarly, when the impact of S on Keff is studied, other parameters are chosen as H = 60 μm, h1 = 10 μm, n = 4, and h2 = 10 μm. When the impact of n on Keff is analyzed, other parameters are chosen as H = 60 μm, h1 = 10 μm, h2 = 10 μm, and S = 5 μm. For all cases, the depth of the gripping arm is taken to be 100 μm, and the ratio of depth to width is so large that plane strain state is assumed. With the width of the comb teeth b = 6 μm and the gap d = 6 μm, using Eq. (40), the eletrostatic force can be obtained as p = 276.75Pa. The relations between the structure parameters and the bending stiffness (or the normalized bending stiffness) are illustrated in Figs. 13–20. It should be noted that the solutions of the classical theory can be obtained when l = 0.0 is set in strain gradient theory.
Fig. 13 The bending stiffness versus the width of spring
Fig. 14 The normalized bending stiffness versus h1 /l
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The bending stiffness of the gripping arm decreases drastically when the S-shaped spring flexible structure is adopted to link the drive arm and the anchor, moreover, the values in strain gradient theory are larger than those in the classical theory when the strain gradient effects are considered, as shown in Fig. 13. The normalized bending stiffness versus h1 /l is plotted in Fig. 14. It shows that the normalized bending stiffness becomes very large when the width of spring is less than or close to the material characteristic length scales, which implies that size effects are rather strong in this case. However, the normalized bending stiffness decreases with increase in the width of spring, and size effects become weak gradually. The bending stiffness of the gripping arm decreases with increase in the height of spring, moreover, the values in strain gradient theory are larger than those in the classical theory when strain gradient effects are considered, as shown in Fig. 15. The normalized bending stiffness versus H/l is plotted in Fig. 16. It is obvious that the normalized bending stiffness decreases slowly with the increase of H/l.
Fig. 15 The bending stiffness versus the height of spring
Fig. 16 The normalized bending stiffness versus H/l
C 1 natural element method for strain gradient linear elasticity and its application to microstructures
Figure 17 demonstrates that the bending stiffness decreases with increase in the distance between two springs. As the distance of the two springs increases, there is a sharp decrease in the bending stiffness initially, afterward, this decrease slows down gradually. Similarly, the bending stiffness in strain gradient theory is larger than that in the classical theory when strain gradient effects are considered. The normalized bending stiffness versus S /l is plotted in Fig. 18. It is clear that the normalized bending stiffness decreases steadily with the increase of S /l.
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is plotted in Fig. 20. It is clear that the normalized bending stiffness is not sensitive to the number of springs.
Fig. 19 The bending stiffness versus the number of spring
Fig. 17 The bending stiffness versus the distance of two sections
Fig. 20 The normalized bending stiffness versus n
5.2 Size effects of the SCF for microspeciem
Fig. 18 The normalized bending stiffness versus S /l
Figure 19 demonstrates that the bending stiffness decreases with increase in the number of the springs. n = 0 implies that no spring is adopted, in this case, the bending stiffness is very large. Even if only one spring is adopted, the bending stiffness would decrease abruptly. As the number of the springs increases, there is a slow decrease in the bending stiffness, while the difficulty of technics is raised quickly. Therefore, it is not necessary to employ too many springs in design. The normalized bending stiffness versus n
Microspeciem is usually used to measure physical parameters such as Young’s modulus, Poisson’s ratio, fracture toughness, fracture strength, and so on in microsample tension tests. Central perforations have an important influence on the distribution of the structure strength; investigation of the influence of the shapes and sizes of perforations on the structure strength is of interest to the design and experimental research of microspeciem. Chasiotis et al. [48] performed tensile tests to investigate the mechanical strength and fracture toughness of polysilicon films, and to study size effects associated with the elliptical and circular perforations. Pugno et al. [49] proposed a novel experimental-theoretical method to investigate the strength and fractures of MEMS structures having complex defect geometries, such as micro-
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fabricated sharp cracks, blunt notches and re-entrant corners. The geometry of perforated microspeciem is shown in Fig. 21, the unit is μm. Different perforated shapes (circular perforations; elliptical perforations whose ellipticities γ are 2 and 3, respectively; U-shaped notches) and perforated sizes are illustrated in Fig. 22. The symmetry of the problem reduces the discretization area to one quarter of the complete geometry, irregular discretization scheme is adopted with denser nodes used in the area of interest, as shown in Fig. 23 (for the microspeciem with elliptical perforation). The boundary conditions are prescribed as (1) uy = v x = 0 on the bottom surface, (2) u x = vy = 0 on the left surface, (3) ty = R x = Ry = 0, t x 0 on the right surface,
Fig. 23 Nodal discretization and its local enlarged diagram
(4) other surfaces are free surfaces. Microspeciem is made of polycrystalline copper, and material parameters are defined as: E = 119 GPa, ν = 0.326, and l = 3.7 μm [8]. It should be noted that the solutions of the classical theory can be obtained when l = 0.0 is set in the strain gradient theory. 5.2.1 Size effects associated with the circular perforations Fig. 21 The geometry of perforated microspeciem
Radii of the circular perforations a are adopted as 4 μm, 8 μm, 12 μm, 36 μm, respectively. Numerical calculation is carried out along the left surface, and the result is plotted in Fig. 24. It is obvious that SCF in strain gradient theory decreases significantly in comparison with that in classical theory when the radius is close to the material characteristic length scales, which implies that size effects are very strong in this case. With the increase of the radius of the circular perforation, the normalized value of SCF becomes large, and size effects become weak gradually.
Fig. 22 Perforation at the center of the gauge section, with a Circular hole; b Elliptical hole; c U-shaped notch
Fig. 24 Normalized SCF of microspeciem with circular perforation versus a
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5.2.2 Size effects associated with the elliptical perforations
5.2.3 Size effects associated with the U-shaped notch
For the microspeciem with the elliptical perforations, numerical results of ellipticities γ = 2 and γ = 3 are plotted in Figs. 25 and 26. Compared to the microspeciem with circular perforations, size effects on the elliptical perforations are stronger; for example, the normalized values of SCF are 0.26 and 0.29 (for the case of γ = 2 and 3, respectively) when the long axis t = 4 μm, which are less than 0.476 for the case of microspeciem with circular perforation when a = 4 μm. Similarly, the normalized values of SCF are 0.68 and 0.53 (for the case of γ = 2 and 3, respectively) when the long axis t = 36 μm, which are less than 0.9 for the case of microspeciem with circular perforation when a = 36 μm. It is clear that the size effects become strong when the long axis of elliptical perforation comes close to the material characteristic length scales.
Numerical results of microspeciem with U-shaped notch are illustrated in Figs. 27 and 28. It is obvious that both the root notch radius a and the length of notch h have influences on SCF. With the increase of h, SCF gets larger and larger accordingly. With the increase of a, SCF in the classical theory becomes small gradually; however, SCF in strain gradient theory increases significantly at first, and then increases slowly, the trend predicted by strain gradient theory agrees well with the experimental results [49], as shown in Fig. 27. Figure 28 demonstrates that with the increase of notch radius, size effects decline obviously; while with the increase of the length of notch, size effects decline slightly.
Fig. 27 SCF of microspeciem with U-shaped notch versus a and h Fig. 25 Normalized SCF of microspeciem with elliptical perforation versus t (for γ = 2)
Fig. 28 Normalized SCF of microspeciem with U-shaped notch versus a and h
Fig. 26 Normalized SCF of microspeciem with elliptical perforation versus t (for γ = 3)
In addition, for the case of circular and elliptical perforations, when a and t are close to the material characteris-
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tic length scales, SCF in strain gradient theory is about 1.3, which approximates the stress distribution in far field. In this case, such factors as the shapes of perforations, ellipticity and location of perforations, have little influence on SCF. 6 Conclusions Based on C 1 NEM, a meshless method has been developed for strain gradient linear elasticity. Because of the interpolating properties of shape functions in C 1 NEM, EBCs can be easily dealt with. Several typical examples which have analytical solutions are used to illustrate the effectiveness of the developed method. With the increase of nodes, numerical results converge quickly to the analytical ones, and they are in good agreement with the analytical ones, which reveals that C 1 NEM is an effective method for the solutions of strain gradient elasticity problems. In its application to microstructures, some valuable results can be obtained: (1) Adopting the S-shaped spring flexible structure to link the drive arm and the anchor is a feasible way to decrease the bending stiffness and to increase the gripping force. (2) The bending stiffness increases with increase in the width of spring, while decreases with increase in the height, the distance between two springs and the number of springs. (3) Strain gradient effects make the material “harden”, the bending stiffness in strain gradient theory is larger than that in classical theory, and the size effects become rather strong when the width of spring comes close to the material characteristic length scales. (4) For microspeciem with circular and elliptical perforations, when the radius and the long axis are close to the material characteristic length scales, the microspeciem exhibits obvious size effects. For the U-shaped notch, the size effects decline obviously with increasing notch radius, and decline mildly with increasing length of notch. References 1 Fleck, N.A., Muller, G.M., Ashby, M.F., et al.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994) 2 Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853–863 (1995) 3 St¨olken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109–5115 (1998) 4 Lam, D.C.C., Yang, F., Chong, A.C.M., et al.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003) 5 Cosserat, E., Cosserat, F.: Theorie des Crops Deformables. Hermann et Fils, Paris (1909) 6 Toupin, R.A.: Elastic materials with couple stresses. Arch. Rational Mech. Anal. 11, 385–414 (1962)
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