Abstract The classical continuum theory not only underestimates the stiffness of microscale structures suchas microbeams but is also unable to capture the size dependency, a phenomenon observed in these structures.Hence, the non-classical continuum theories such as the strain gradient elasticity have been developed. Inthis paper, a Timoshenko beam finite element is developed based on the strain gradient theory and employedto evaluate the mechanical behavior of microbeams used in microelectromechanical systems. The new beamelement is a comprehensive beam element that recovers the formulations of strain gradient Euler–Bernoullibeam element, modified couple stress (another non-classical theory) Timoshenko and Euler–Bernoulli beamelements, and also classical Timoshenko and Euler–Bernoulli beam elements; note that the shear-lockingphenomenon will not happen for the new Timoshenko beam element. The stiffness and mass matrices ofthe new element are derived in closed forms by following an energy-based approach and using Hamilton’sprinciple. It is noted that unlike the classical beam elements, the stiffness matrix of the new element has asize-dependent nature that can capture the size-dependent behavior of microbeams. The shape functions ofthe newly developed beam element are determined by solving the equilibrium equations of strain gradientTimoshenko beams, which brings about a size-dependent characteristic for them. The new beam element isemployed to evaluate the static deflection of a microcantilever, and the results are compared to the experimentaldata as well as the results obtained by using the classical beam element and the couple stress plane element.The new beam element is also implemented to calculate the static deflection, vibration frequency, and pull-involtage of electrostatically actuated microbeams. The current results are compared to the experimental data aswell as the classical FEM outcomes. It is observed that the results of the new element are in excellent agreementwith the experimental data while the gap between the experimental and classical FEM results is significant.
1 Introduction
Microscale mechanical components, such as microbeams, are the main building blocks of microelectromechan-ical systems (MEMS) [1,2] and atomic force microscopes (AFMs) [3,4]. Hence, investigating the mechanicalbehavior of such components has always been an important issue among researchers. Due to the complications
M. H. Kahrobaiyan · M. Asghari · M. T. Ahmadian (B)School of Mechanical Engineering, Sharif University of Technology, Tehran, IranE-mail: [email protected].: +982166165503
M. T. AhmadianCenter of Excellence in Design, Robotics and Automation (CEDRA),Sharif University of Technology, Tehran, Iran
M. H. Kahrobaiyan et al.
existing in microscale systems such as the presence of the complex forces like electrostatic, Casimir, Van DerWaals and capillary forces, complex geometry, or some other issues like existence of the squeeze film damping,the exact analytical solutions may not be achieved for the behavior of the mechanical components; so, someapproaches other than the analytical one are required. One of the most popular approaches is the finite elementmethod (FEM). The FEM is utilized by many researchers in order to investigate the mechanical behavior ofmicroscale systems. For example,
• Analysis of piezoelectric cantilever-type beam actuators [5]• Study of the mechanical behavior of conducting polymer electromechanical actuators (CPEA) [6]• Investigation of the static behavior and pull-in voltage of electrostatically actuated cantilever microswitches
[7]• Analysis of the mechanical response of microswitches with piezoelectric actuation [8]• Investigating the dynamic pull-in of an electrostatically actuated micro-/nanoplate considering geometrical
nonlinearities and fluid pressure by employing a nine-node plate element [9]• Modeling the MEMS subjected to electrostatic forces by developing a non-conforming element [10]
It is noted that all of the above-mentioned works are based on the elements developed on the basis of theclassical continuum theory.
The experimental observations have indicated that the classical continuum mechanics not only underesti-mates the stiffness of microscale structures but is also incapable of justifying the size dependency observedin these structures [11–13]; note that the size dependency is a peculiar phenomenon in which the normalizedmechanical quantities of microscale structures that the classical continuum theory predict to be independentof the structure size significantly changes by the size. Hence, during past years, some non-classical contin-uum theories such as the strain gradient theory [12,14] and the couple stress theory [15,16] have been found,developed, modified, and employed to study the mechanical behavior of the microscale structures.
Mindlin [14] proposed a higher-order gradient theory for elastic materials by considering the first and thesecond gradients of the strain tensor effective on the strain energy density. Fleck and Hutchinson [17–19] usedMindlin’s formulations and expressed that the strain energy density of an elastic material is a function of notonly the first but also the second gradient of the displacement field (a function of strain and the first derivativeof strain). The aforementioned theory is named the strain gradient theory. By utilizing the equilibrium equationof moments of couples in addition to the classical equilibrium equations of forces and moments of forces,Lam et al. [12] introduced a modified strain gradient theory, which became a popular non-classical theoryin microsystems area. Henceforth, wherever the “strain gradient” is used in the paper, it refers to the straingradient theory modified by Lam et al. [12].
The strain gradient theory is employed to formulate bar, beam, and plate models [20–29]. Three materialparameters called the length scale parameters (l0, l1, and l2) appear in this theory in addition to the two classicalparameters, i.e., elastic modulus and Poisson ratio, which enables the theory to capture the size dependency.In order to determine the length scale parameters for a specific material, some typical experiments such asmicrobend test, microtorsion test, and micro-/nano-indentation test can be carried out [11–13,30]. It is noted thatletting l0 = l1 = 0, the formulations of the strain gradient theory reduce to another non-classical theory calledthe modified couple stress theory proposed by Yang et al. [31]. This theory has been employed to develop beamand plate models [32–38]. It is also utilized to investigate the characteristics of some microsystems [39–43].
According to the necessity of utilizing the finite element method in MEMS due to the force and geomet-rical complications, and also the necessity of using the non-classical continuum theories due to the inabilityof the classical continuum theory to evaluate the accurate stiffness and justify the size dependency of MEMScomponents, developing new structural finite elements based on the non-classical continuum theories seemsto be crucial.
This paper presents a new non-classical comprehensive Timoshenko beam element capable of capturingthe size dependency of microscale systems. The stiffness and mass matrices of the new beam element arederived in a closed form using Hamilton’s principle, and the shape functions are determined by solving theequilibrium equations of a strain gradient Timoshenko beam model. It is noted that the new beam elementis comprehensive such that the stiffness and mass matrices of strain gradient Euler–Bernoulli beam element,modified couple stress Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko andEuler–Bernoulli beam elements can be recovered from the present formulations. The new beam element isemployed to model the mechanical behavior of microbeams. It is utilized to evaluate the load-end deflectioncurve of short microcantilever subjected to a concentrated force at its free end. The results of the new elementare compared to the experimental data as well as the classical FEM results, and it is observed that the results ofthe new beam element are in excellent agreement with the experimental results, whereas the gap between the
Strain gradient Timoshenko beam element
experimental and classical FEM results is significant. In another example, the new beam element is employedto calculate the static pull-in voltage of an electrostatically actuated microswitch. The results are compared tothe experimental and classical FEM results, and it is indicated that while the outcomes obtained by employingthe new beam element successfully coincide with the experimental data, the attempts of the classical beamelements to be in good agreement with the experimental results have been in vain.
2 Preliminaries
The strain energy U of a linear elastic strain gradient continuum occupying volume � is expressed as [12]
U = 1
2
∫
�
(σi jεi j + piγi + τ
(1)i jk η
(1)i jk + ms
i jχsi j
)d�, (1)
where εi j , η(1)i jk, χ
si j , and γi are the kinematic parameters: εi j , η
(1)i jk , andχ s
i j , respectively, denote the componentsof strain, deviatoric part of stretch gradient, and symmetric part of rotation gradient (curvature) tensors, and γi
represents the components of dilatation gradient vector. In addition, σi j , pi , τ(1)i jk and ms
i j stand for the workconjugates of the aforementioned kinematic parameters noted that σi j and ms
i j are known as the components
of stress and couple stress tensors, and pi and τ (1)i jk are known as the components of higher-order stress tensors.The components of the strain tensor can be related to the components of displacement vector filed ui as
εi j = 1
2
(ui, j + u j,i
), (2)
and the other kinematical parameters are related to the strain tensor components as [12]
γi = εmm,i , (3)
η(1)i jk = 1
3
(ε jk,i + εki, j + εi j,k
) − 1
15δi j
(εmm,k + 2εmk,m
)
− 1
15
[δ jk
(εmm,i + 2εmi,m
) + δki(εmm, j + 2εmj,m
)], (4)
χ si j = 1
2
(Eimnεnj,m + E jmnεni,m
), (5)
where Ei jk refers to the permutation symbol. The work conjugates are related to the kinematic parameters asfollows [12]:
σi j = νE
(1 − 2ν) (1 + ν)εkkδi j + 2μεi j , (6)
pi = 2μl20γi , (7)
τ(1)i jk = 2μl2
1η(1)i jk, (8)
msi j = 2μl2
2χsi j , (9)
in which δi j represents the Kronecker delta, and E, μ and ν, respectively, denote the elastic (Young) modulusand shear modulus, and Poisson’s ratio noted that E = 2(1 + ν)μ. In addition, l0, l1 and l2 represent thematerial length scale parameters that are the additional material properties enabling the theory to capture thesize dependency.
3 Deriving the stiffness and mass matrices
In this section, the mass and stiffness matrices of the new strain gradient Timoshenko beam element are goingto be derived following an energy-based approach and Hamilton’s principle. To that end, the components ofthe displacement field vector u of a Timoshenko beam model depicted in Fig. 1 are introduced as
M. H. Kahrobaiyan et al.
Fig. 1 A Timoshenko beam model: loading, kinematic parameters, and coordinate system
u1 = −zψ (x, t) , u2 = 0, u3 = w (x, t) , (10)
where u1, u2, and u3 represent the displacement of an arbitrary point along the x, y and z axes, respectively. Inaddition, z, ψ and w, respectively, denote the lateral distance of an arbitrary point from the neutral axis, rota-tion angle of the beam cross sections, and lateral deflection of the beam. Substituting Eq. (10) into Eqs. (2)–(5),the nonzero kinematic parameters are obtained as [26]:
ε11 = −z∂ψ
∂x, ε13 = ε31 = 1
2
(∂w
∂x− ψ
), (11)
γ1 = −z∂2ψ
∂x2 , γ3 = −∂ψ∂x, (12)
χ s12 = χ s
21 = −1
4
(∂ψ
∂x+ ∂2w
∂x2
), (13)
η(1)111 = −2
5z∂2ψ
∂x2 , η(1)113 = η
(1)311 = η
(1)131 = 4
15
(∂2w
∂x2 − 2∂ψ
∂x
),
η(1)122 = η
(1)133 = η
(1)212 = η
(1)221 = η
(1)313 = η
(1)331 = 1
5z∂2ψ
∂x2 , (14)
η(1)223 = η
(1)232 = η
(1)322 = 1
15
(2∂ψ
∂x− ∂2w
∂x2
), η(1)333 = 1
5
(2∂ψ
∂x− ∂2w
∂x2
).
Substitution of Eqs. (11)–(14) into Eqs. (6)–(9) gives the nonzero work conjugates (stresses, couple stresses,and higher-order stresses) as [26]:
σ11 = −Ez∂ψ
∂x, σ13 = σ31 = kμ
(∂w
∂x− ψ
), (15)
p1 = −2μl20 z∂2ψ
∂x2 , p3 = −2μl20∂ψ
∂x, (16)
ms12 = ms
21= −μl2
2
2
(∂ψ
∂x+ ∂2w
∂x2
), (17)
τ(1)111 = −4
5μl2
1 z∂2ψ
∂x2 , τ(1)113 = τ
(1)311 = τ
(1)131 = − 8
15μl2
1
(2∂ψ
∂x− ∂2w
∂x2
),
τ(1)122 = τ
(1)133 = τ
(1)212 = τ
(1)221 = τ
(1)313 = τ
(1)331 = 2
5μl2
1 z∂2ψ
∂x2 , (18)
Strain gradient Timoshenko beam element
τ(1)223 = τ
(1)232 = τ
(1)322 = 2
15μl2
1
(2∂ψ
∂x− ∂2w
∂x2
), τ(1)333 = 2
5μl2
1
(2∂ψ
∂x− ∂2w
∂x2
).
where k denotes the shear correction factor in Timoshenko beam model. By insertion of the kinematic parame-ters and their work conjugates mentioned in Eqs. (11)–(19) into Eq. (1), one can express the potential energyof the strain gradient Timoshenko beam as
U = 1
2
L∫
0
∫
A
(σi jεi j + piγi + τ
(1)i jk η
(1)i jk + ms
i jχsi j
)d Adx
= 1
2
L∫
0
{k1
(∂2ψ
∂x2
)2
+ k2
(∂ψ
∂x
)2
+ k3
(∂2w
∂x2 + ∂ψ
∂x
)2
+ k4
(2∂ψ
∂x− ∂2w
∂x2
)2
+ kμA
(∂w
∂x− ψ
)2}, (19)
in which the length and cross-sectional area of the beam are, respectively, denoted by L and A. In addition, theshear deformation factor, accounting for the variation in the shear stress along the cross section of Timoshenkobeam model, is denoted by k, which depends on the geometry of the cross section (e.g., k = 5/6 for rectangularcross section) [44]. Moreover,
k1 =(
2μl20 + 4
5μl2
1
)I, k2 = E I + 2μAl2
0 , k3 = 1
4μAl2
2 , k4 = 8
15μAl2
1 , (20)
where I refers to the area moment of inertia of the beam cross section. The kinetic energy T of the Timoshenkobeam can be given as
T = 1
2
∫
A
L∫
0
ρ
[(∂u1
∂t
)2
+(∂u2
∂t
)2
+(∂u3
∂t
)2]
dxd A
= 1
2ρ
L∫
0
[I
(∂ψ
∂t
)2
+ A
(∂w
∂t
)2]
dx, (21)
in which ρ stands for the beam density. The work W of external distributed force and moment exerted to theTimoshenko beam (see Fig. 1) is expressed as
W =L∫
0
(F (x, t) w − M (x, t) ψ) dx =L∫
0
{wψ
}T {F (x, t)
−M (x, t)
}dx, (22)
It is noted that the negative sign, i.e., “−”, appears in Eq. (22) since the direction of the applied moment isin opposition to the ψ direction. In order to derive the governing equations of motion of the strain gradientTimoshenko beam model, the Hamilton’s principle can be employed as
t2∫
t1
δ (T − U + W ) dt = 0. (23)
Substitution of Eqs. (19), (21), and (22) into Eq. (23) reveals the governing equations as
− k1∂4ψ
∂x4 + k2∂2ψ
∂x2 + k3
(∂3w
∂x3 + ∂2ψ
∂x2
)+ 2k4
(2∂2ψ
∂x2 − ∂3w
∂x3
)
+ kμA
(∂w
∂x− ψ
)− M (x, t) = ρ I
∂2ψ
∂t2 , (24)
M. H. Kahrobaiyan et al.
Fig. 2 A two-node Timoshenko beam element: geometry, coordinate system, and nodal degrees of freedom
− k3
(∂4w
∂x4 + ∂3ψ
∂x3
)+ k4
(2∂3ψ
∂x3 − ∂4w
∂x4
)+ kμA
(∂2w
∂x2 − ∂ψ
∂x
)
+ F (x, t) = ρA∂2w
∂t2 . (25)
In order to develop the new element, consider a two-node strain gradient Timoshenko beam element depictedin Fig. 2. The length of the element is represented by L , and two degrees of freedom are assigned to each node:1) lateral deflection and 2) rotation angle of the cross section. So, the nodal displacement vector δ of the newelement can be expressed as
δ =
⎧⎪⎨⎪⎩w1ψ1w2ψ2
⎫⎪⎬⎪⎭ , (26)
where the subscripts 1 and 2 refer to the node number in the beam element. For the new beam element, thedisplacement and rotation fields of the element, w and ψ , can be related to the nodal displacement vector byutilizing the shape function matrices:
{wψ
}2×1
=[
Nw1×4
Nψ1×4
]2×4
δ4×1, (27)
where
Nw = [Nw
1 Nw2 Nw
3 Nw4
], Nψ =
[Nψ
1 Nψ2 Nψ
3 Nψ4
], (28)
in which Nw and Nψ represent the shape function matrices, matrices with one row and four columns whosecomponents are the appropriate shape functions of the new element that will be derived later. By substitutingthe displacement and rotation angle fields mentioned in Eq. (27) into Eqs. (19), (21), and (22), the potentialenergy, kinetic energy, and work of external loads can be rewritten in more appropriate forms as
U = 1
2δ
T Kδ, (29)
T = 1
2δ
T Mδ, (30)
W = δT f, (31)
where the dot symbol refers to derivation with respect to time and
Strain gradient Timoshenko beam element
K =L∫
0
{k1
(d2Nψ
dx2
)T (d2Nψ
dx2
)+ k2
(dNψ
dx
)T (dNψ
dx
)
+ k3
(d2Nw
dx2 + dNψ
dx
)T (d2Nw
dx2 + dNψ
dx
)+ k4
(2
dNψ
dx− d2Nw
dx2
)T (2
dNψ
dx− d2Nw
dx2
)
+ kμA
(dNw
dx− Nψ
)T (dNw
dx− Nψ
)}dx, (32)
M =L∫
0
ρ I(Nψ
)TNψdx +
L∫
0
ρA(Nw
)T Nwdx, (33)
f =L∫
0
{Nw
Nψ
}T {F (x, t)
−M (x, t)
}dx =
L∫
0
{(Nw
)TF (x, t)− (
Nψ)T
M (x, t)}
dx . (34)
By substituting Eqs. (29)–(31) into the mathematical formulation of Hamilton’s principle, i.e., Eq. (23), thegoverning equation of the new element can be achieved as
Mδ + Kδ = f . (35)
Regarding Eq. (35), it is inferred that M, K and f , respectively, denote the mass matrix, stiffness matrix, andnodal force vector of the new beam element. Considering Eqs. (32)–(34), it is clear that by having the propershape functions, the stiffness and mass matrices as well as the nodal force vector of the new beam elementcan be fully determined. Hence, hereafter, the procedure of deriving the shape functions of the new beamelement is explained. The shape functions of the new beam element are derived by solving the equilibriumequations of strain gradient Timoshenko beam model and then applying the proper boundary conditions. Theequilibrium equations can be obtained from Eqs. (24) and (25) by dropping the external force and momentand also considering ∂/∂t = 0 in these equations as
−k1d4ψ
dx4 + k2d2ψ
dx2 + k3
(d3w
dx3 + d2ψ
dx2
)+ 2k4
(2
d2ψ
dx2 − d3w
dx3
)+ kμA
(dw
dx− ψ
)= 0, (36)
d
dx
{−k3
d2
dx2
(dw
dx+ ψ
)+ k4
d2
dx2
(2ψ − dw
dx
)+ kμA
(dw
dx− ψ
)}= 0. (37)
In order to solve the equations, two new variables, γ and θ , are introduced as
γ = 2ε13 = 2ε31 = dw
dx− ψ, θ = −2θ2 = dw
dx+ ψ. (38)
Equations (36) and (37) can be rewritten with respect to the new variables as follows:
{(k2
4− k3
)θ ′′ +
[−3
2k4γ
′′ + kμAγ
]}′= 0 ⇒
(k2
4− k3
)θ ′′ +
[−3
2k4γ
′′ + kμAγ
]= c1, (39)
[−k1
2θ(4) +
(k2
2+ k3 + k4
)θ ′′
]+
[k1
2γ (4) −
(k2
2+ 3k4
)γ ′′ + kμAγ
]= 0, (40)
where the prime symbol refers to derivative with respect to x . In addition, c1 denotes the constant of integration.The non-dimensional analysis of Eqs. (39) and (40) will be helpful in order to derive the shape functions. Tothat end, a dimensionless parameter can be introduced as x = x/L . Substitution of x into the second term ofEq. (39), i.e.,
[− (3/2) k4γ′′ + kμAγ
], and also recalling k4 from Eq. (20) results in
− 3
2k4γ
′′ + kμAγ = kμA
(− 4
5k
(l1L
)2 d2γ
dx2 + γ
). (41)
M. H. Kahrobaiyan et al.
10-5
10-4
10-3
10-2
10-1
100
101
1020
1
2
3
4
5
6
7
h/l
Err
or%
L/h=2L/h=3L/h=5L/h=7
Fig. 3 Effects of h/ l and L/h on the error of ignoring (l1/L)2 in the evaluation of the static end deflection of a microcantilever
Since for most of the common materials, the length scale parameter l1 is in the order of a few microns,the term of (l1/L)2 is very small even for microbeams used in MEMS. Hence, the coefficient of d2γ /dx2
is negligible compared to 1, i.e., the coefficient of γ . In order to investigate the aforementioned assump-tion on neglecting (l1/L)2, consider a microcantilever with length L having rectangular cross section withthickness h and width b = 2h subjected to a concentrated force at its free end. The governing equationsare solved both with and without the aforementioned assumption in order to obtain the maximum deflec-tion of the microcantilever. An error parameter can be defined as Error% = ∣∣wmax
2 − wmax1
∣∣ /wmax1 where
wmax1 and wmax
2 refer to the maximum deflection of the microcantilever obtained, respectively, without andwith the aforementioned simplification. Since (l1/L)2 can be rewritten as (l1/h)2 (h/L)2, it would be help-ful to investigate the effect of the ratio of the thickness to the length scale parameter h/ l and the ratio ofthe beam length to the beam thickness L/h. Assuming l0 = l1 = l2 = l, which is a common assump-tion in strain gradient theory, the error parameter, i.e., the error of ignoring (l1/L)2, is depicted in Fig. 3,and the effects of h/ l and L/h have been assessed. The figure shows that for high values of h/ l, the errorapproaches zero since the deflections evaluated both with and without the aforementioned simplification willcoincide on the results predicted by the classical beam theory. In addition, it is observed that for low val-ues of h/ l, the error will be saturated, the saturated mechanical behavior of microbeams in low values ofh/ l is a noticed issue [45], which will be explain later in this article. Moreover, the figure indicates that asL/h increases, the error increases too in a way that for a very short beam, L/h = 2, the error is about6 %. In conclusion, since the error is negligible in the case study, the neglecting of (l1/L)2 in shape func-tion derivation seems to be reasonable. It is noted that later in this section, the shape functions derived bythe aforementioned simplification (neglecting (l1/L)2) are compared to those numerically derived withoutsimplification.
Regarding the previous explanations, Eq. (39) reduces to
γ = c1
kμA+ 1
kμA
(k3 − k4
2
)θ ′′. (42)
Substitution of Eq. (42) into Eq. (40) gives
1
kμA
(k3 − k4
2
)k1
2θ(6) −
[k1
2+ 1
kμA
(k2
2+ 3k4
) (k3 − k4
2
)]θ(4) +
(k2
2+ 2k3 + k4
2
)θ ′′ = −c1.
(43)
Strain gradient Timoshenko beam element
Now, by recalling x , the non-dimensional form of Eq. (43) can be obtained as follows:
μ
k E
[1
4
(l0L
)2
− 4
15
(l1L
)2] [(
l0L
)2
+ 2
5
(l1L
)2]
d6θ
dx6
−{μ
E
[(l0L
)2
+ 2
5
(l1L
)2]
+ 1
k
[1
2+ μ
E
(l0r
)2
+ 8
5
μ
E
(l1r
)2] [
1
4
(l2L
)2
− 4
15
(l1L
)2]}
d4θ
dx4
+1
2
[1 + μ
E
(2
(l0r
)2
+ 8
15
(l1r
)2
+(
l2r
)2)]
d2θ
dx2 = −c1L2, (44)
where r denotes the gyration radius of the beam cross section, i.e., I = Ar2. According to the justification madefor simplifying Eq. (41), it is inferred that the coefficients of d6θ/dx6 and d4θ/dx4 are negligible comparedto the coefficient of d2θ/dx2. Hence, Eq. (44) reduces to
θ ′′ = − 2c1
E I (1 + α), (45)
where
α = μA
E I
(2l2
0 + 8
15l21 + l2
2
)= μ
E
[2
(l0r
)2
+ 8
15
(l1r
)2
+(
l2r
)2]. (46)
The general solution of Eq. (46) is obtained as
θ = − C
E I (1 + α)x2 + c2x + c3, (47)
in which c2 and c3 represent the constants of integration. Substituting Eq. (47) into Eq. (42) and recalling k3and k4 from Eq. (20), one can express
γ = c1
kμA
1 + β
1 + α, (48)
where
β = μA
E I
(2l2
0 + 16
15l21 + 1
2l22
)= μ
E
[2
(l0r
)2
+ 16
15
(l1r
)2
+ 1
2
(l2r
)2]. (49)
Substitution of Eqs. (47) and (48) into Eq. (38) gives the rotation angle ψ and lateral deflection was
ψ = 1
2(θ − γ ) = 1
2
(− c1
E I (1 + α)x2 + c2x + c3 − c1
kμA
1 + β
1 + α
), (50)
w = 1
2
∫(γ + θ) dx = 1
2
{c1
kμA
1 + α/2
1 + αx − c1x3
3E I (1 + α)+ c2
2x2 + c3x
}+ c4 . (51)
In order to obtain the shape functions of the new two-node Timoshenko beam element, the following boundaryconditions should be applied:
w (0) = w1, ψ (0) = ψ1, w (L) = w2, ψ (L) = ψ2. (52)
By applying the aforementioned boundary conditions, the integration constants, c1, …, c4 are determinedas
c1 = E I (1 + α)
L3 (1 + ϕ){−12w1 − 6Lψ1 + 12w2 − 6Lψ2} ,
c2 = 2(ψ2 − ψ1)
L+ c1L
E I (1 + α), c3 = 2ψ1 + c1
kμA
1 + β
1 + α, c4 = w1,
M. H. Kahrobaiyan et al.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/L
Fir
st d
efle
ctio
n s
hap
e fu
nct
ion
Polynomial (L/h=3)Higher-order (L/h=3)
Polynomial (L/h=7)
Higher-order (L/h=7)
Polynomial (L/h=13)Higher-order (L/h=13)
Fig. 4 First shape function of the lateral deflection field, Nw1 : numerically obtained without ignoring (l1/L)2 (higher-order shape
functions) and analytically obtained with ignoring (l1/L)2(polynomial shape functions)
where
ϕ = 12E I
kμAL2 (1 + β) = 12E
kμ
( r
L
)2(1 + β) . (53)
Substituting Eq. (53) into Eqs. (50) and (51) and comparing the results with those mentioned as matrix formsin Eqs. (27) and (28), the shape functions of the new beam element are derived as
Nψ1 = − 6
L (1 + ϕ)
( x
L
) (1 − x
L
), Nψ
2 =(
1 − x
L
) [1 − 3
(1 + ϕ)
( x
L
)],
Nψ3 = 6
L (1 + ϕ)
( x
L
) (1 − x
L
)Nψ
4 =( x
L
) [1 − 3
(1 + ϕ)
(1 − x
L
)],
Nw1 = 1 + 1
1 + ϕ
[2
( x
L
)3 − 3( x
L
)2 − ϕ( x
L
)],
Nw2 = L
2 (1 + ϕ)
[2
( x
L
)3 − (4 + ϕ)( x
L
)2 + (2 + ϕ)( x
L
)], (54)
Nw3 = 1
1 + ϕ
[3
( x
L
)2 − 2( x
L
)3 + ϕ( x
L
)],
Nw4 = L
2 (1 + ϕ)
[2
( x
L
)3 + (ϕ − 2)( x
L
)2 − ϕ( x
L
)].
It is observed that the shape functions of the new beam element have the size-dependent nature, i.e., theyare the functions of the ratio of the gyration radius to the length scale parameters. Here, in order to jus-tify the simplification made to derive the shape functions mentioned in Eqs. (54) and (55), i.e., neglect-ing (l1/L)2, the first two polynomial shape functions of the lateral deflection, i.e., Nw
1 and Nw2 , obtained
based on neglecting of (l1/L)2 are depicted in Figs. 4 and 5 and compared to the shape functions numer-ically derived without the aforementioned simplification for different values of L/h. The figures indicatethat as L/h increases, the difference between the higher-order and polynomial shape functions decreases.It is observed that the effect of ignoring the coefficients of (l1/L)2 in solving the equilibrium equations isnegligible. Hence, the assumption of neglecting the coefficients of (l1/L)2, which is made in the processof deriving the shape functions mentioned in Eqs. (54) and (55), is justifiable. It is noted that the negli-gible difference is observed between all the 8 shape functions expressed in Eqs. (54) and (55) and thosehigher-order shape functions derived without simplification, but only Nw
Fig. 5 Second shape function of the lateral deflection field, Nw2 : numerically obtained without ignoring (l1/L)2 (higher-order
shape functions) and analytically obtained with ignoring (l1/L)2 (polynomial shape functions)
Now, by substitution of the shape functions mentioned in Eqs. (54) and (55) into Eq. (28), the shape func-tion matrices, Nw and Nψ , can be determined, and subsequently, by inserting Nw and Nψ into Eqs. (32) and(33), the stiffness and mass matrices of the new beam element can be determined as
K = E I (1 + α)
L3 (1 + ϕ)
⎡⎢⎢⎣
12 6L −12 6L(4 + ϕ) L2 −6L (2 − ϕ) L2
12 −6LSymm. (4 + ϕ) L2
⎤⎥⎥⎦
+ 36μI
L3 (1 + ϕ)2
[2
(l0L
)2
+ 4
5
(l1L
)2] ⎡
⎢⎢⎣4 2L −4 2L
L2 −2L L2
4 −2LSymm. L2
⎤⎥⎥⎦ , (55)
MT .I. = ρAL
210 (1 + ϕ)2
⎡⎢⎢⎢⎣
(70ϕ2 + 147ϕ + 78
) L4
(35ϕ2 + 77ϕ + 44
) (35ϕ2 + 63ϕ + 27
) − L4
(35ϕ2 + 63ϕ + 26
)L2
4
(7ϕ2 + 14ϕ + 8
) L4
(35ϕ2 + 63ϕ + 26
) − L2
4
(7ϕ2 + 14ϕ + 6
)(70ϕ2 + 147ϕ + 78
) − L4
(35ϕ2 + 77ϕ + 44
)Symm. L2
4
(7ϕ2 + 14ϕ + 8
)
⎤⎥⎥⎥⎦ ,
(56)
MR.I. = ρ I
30L (1 + ϕ)2
⎡⎢⎢⎣
36 L (3 − 15ϕ) −36 L (3 − 15ϕ)L2
(10ϕ2 + 5ϕ + 4
) −L (3 − 15ϕ) L2(5ϕ2 − 5ϕ − 1
)36 −L (3 − 15ϕ)
Symm. L2(10ϕ2 + 5ϕ + 4
)
⎤⎥⎥⎦ , (57)
where “symm.” refers to the symmetric nature of stiffness and mass matrices. Moreover, MT .I. and MR.I.,respectively, represent the tensors of the transitional and rotary inertia that together make the total mass matrixof the Timoshenko beam element M as
M = MT .I. + MR.I.. (58)
The new strain gradient Timoshenko beam element is a comprehensive beam element that recovers the formula-tions of strain gradient Euler–Bernoulli beam element, modified couple stress Timoshenko and Euler–Bernoullibeam elements, and classical Timoshenko and Euler–Bernoulli beam elements. By lettingα = 0 in Eq. (55), theformulations of the classical Timoshenko beam element can be achieved [47] noted that the condition of α = 0happens either when one utilizes the classical continuum theory so considers l0 = l1 = l2 = 0 in the formula-tions or when the dimensions of the structure are large, e.g., in macroscales, and consequently the ratio of thelength scale parameters to the gyration radius of the beam cross section l0/r, l1/r and l2/r becomes negligible.
M. H. Kahrobaiyan et al.
Fig. 6 A microcantilever subjected to a concentrated force at its free end: loading, geometry, and coordinate system
Moreover, the formulations of a modified couple stress Timoshenko beam element can be obtained by lettingl0 = l1 = 0 and l2 = l in the formulation of the new beam element. It will be indicated that the formulations ofthe new beam element reduce to the formulations of a two-node strain gradient Euler–Bernoulli beam elementwhen the ratio of the beam length to the gyration radius of the beam cross section is large, i.e., ϕ = 0 in Eqs.(55)–(57). It is noted that letting l0 = l1 = 0 and l2 = l, in addition to ϕ = 0, the stiffness matrix of a modifiedcouple stress Euler–Bernoulli beam element can be achieved [48]. In addition, assuming l0 = l1 = l2 = 0 inaddition to ϕ = 0, the stiffness matrix of a classical Euler–Bernoulli beam element can be obtained [49].
Since when the ratio of the beam length to the gyration radius of the beam cross section is large, i.e.,ϕ → 0, the stiffness matrix of the new Timoshenko beam element reduces to the stiffness matrix of a straingradient Euler–Bernoulli beam element, one can be sure that the shear-locking phenomenon will not happenfor the new beam element. To sum up, the new strain gradient Timoshenko beam element is a comprehensivebeam element that recovers the formulations of strain gradient Euler–Bernoulli beam element, modified couplestress Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko and Euler–Bernoullibeam elements.
4 Examples
In this section, it will be indicated how the new beam element can be employed to investigate the mechanicalbehavior of microscale systems. In order to assess the validation of the new beam element, the results obtainedby utilizing the new beam element are compared to the experimental data as well as the classical FEM outcomes.It is observed the new beam element results are in excellent agreement with the experimental observations,whereas the attempts of the classical FEM to capture the experimental data have been in vain.
4.1 A microcantilever static deflection
Consider a microcantilever with length Lb having a uniform rectangular cross section with height h and widthb subjected to concentrated force P at its free end (see Fig. 6). The microcantilever is modeled by the newbeam element, and its static deflection is evaluated. Assuming the microcantilever is made of epoxy withE = 1.44 Gpa and ν = 0.38 [12] and also considering l0 = l1 = l2 = l, the normalized end deflection ofthe microcantilever, 3E Iw (L) /P L3, has been depicted in Fig. 7 versus the ratio of the beam thickness to thelength scale, h/ l, for various values of the ratio of the beam length to the beam thickness, L/h. As it can beobserved in Fig. 7, there exist three separate regions: 1) classical saturated region for high values of h/ l, inwhich the results of the new beam element approach those predicted by the classical beam theories, noting thatfor slim beams (high values of L/h), the normalized deflection is close to what the classical Euler–Bernoullibeam theory predicts (value of 1), but for short beams (low values of L/h), the difference (between theevaluated results and those the Euler–Bernoulli beam model predict) increases as the ratio of L/h decreases,noting that the evaluated results approach the results predicted by the classical Timoshenko beam theory; 2)strain gradient transitional region, in which the effect of h/ l is dominant; and 3) strain gradient-saturatedregion for low values of h/ l, where the normalized deflection increases as L/h decreases. The existence ofthese three distinct regions first discovered by Darrall et al. [45] in an article on FEM modeling of couple stressmedia with plane elements. They expressed that the beams show “nearly pure shear deformation” behaviorat very low values of h/ l. They indicated that there is a saturated equivalent stiffness for microbeams at low
Strain gradient Timoshenko beam element
10-4
10-3
10-2
10-1
100
101
102
10-3
10-2
10-1
100
h/l
w (
3EI/p
L3)
L/h=5L/h=10L/h=20L/h=50
0 10 20 30 40 50 60 70 80 90 1000.9
0.95
1
1.05
1.1
1.15
Fig. 7 Normalized end deflection of the microcantilever subjected to a concentrated force at its free end
values of h/ l: the stiffness is proportional to μA/L [45]. So, the saturated deflection of the microcantileverobserved in Fig. 7 at low values of h/ l can be justified, and it can be inferred that the normalized end deflectionof the microcantilever will be proportional to (E/μ) (h/L)2 at low values of h/ l.
For a microcantilever with E = 2 and ν = 0 subjected to a unit lateral deflection at its free end [45], theequivalent stiffness of the beam (end reaction force divided by end deflection) is evaluated using the new straingradient beam element (assuming l0 = l1 = 0 and l2 = l), and the results are compared to those evaluatedby employing the couple stress plane elements developed by Darrall et al. [45]. The normalized equivalentstiffness of the beam is depicted in Fig. 8 versus h/ l for L/h = 20 and 40. The comparison indicates that theresults of the new beam element are in good agreement with the results of the plane couple stress element. Itis noted that the three distinct regions can be observed in this figure. The lateral deflection of the cantilever isevaluated by employing the new beam element, and the results are compared with those obtained on the basisof the couple stress plane elements developed by Darrall et al. [45] in Fig. 9. A good agreement between theresults obtained by new beam element and couple stress plane element is observed in this figure.
As another example, the static deflection of a microcantilever made of epoxy is evaluated by implementingthe new beam element, and the results are compared to the experimental data extracted from the work doneby Lam et al. [12]. They conducted a bending test on a microcantilever made of epoxy with the followingmechanical properties: the elastic modulus: E = 1.44 Gpa and Poisson ratio: ν = 0.38. It is noted that thelength scale parameter of epoxy is evaluated to be l0 = l1 = l2 = l = 11.02µm [12]. Moreover, in theaforementioned bending test, the geometrical properties of the microcantilever are reported as follows: thethickness: h = 38µm, the width: b = 0.235 mm, and the ratio of the length to the thickness: Lb/h = 10
M. H. Kahrobaiyan et al.
10-4 10-3 10-2 10-1 100 101 10210
0
101
102
103
104
Ratio of the beam thickness to the length scale parameter (h/l)
No
n-d
imen
sio
nal
Sti
ffn
ess
L/h=20 (The current work)L/h=40 (The current work)L/h=20 (Darrall et al., 2013)L/h=40 (Darrall et al., 2013)
Fig. 8 Normalized stiffness of the cantilever: a comparison between the results obtained by using the present beam element andthose obtained by using couple stress plane element
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
x/L
Lat
eral
Def
lect
ion
(w
)
h/l=0.0001 (Current work)h/l=2 (Current work)h/l=10000 (Current work)h/l=10000 (Darrall et al., 2013)h/l=2 (Darrall et al., 2013)h/l=0.0001 (Darrall et al., 2013)
Fig. 9 Lateral deflection of the cantilever: a comparison between the results obtained by using the present beam element andthose obtained by using couple stress plane element
[12]. In order to bend the microcantilever, Lam et al. [12] used a nanoloading system (Hysitron Triboindenter)to produce a concentrated force at the free end of the cantilever. After that, they graphically reported theexperimentally measured end forces versus the end deflections. In order to validate the new non-classicalbeam element, the aforementioned microcantilever is modeled by using 10 new beam elements. The stiffnessmatrices of the elements are assembled, and the boundary conditions, i.e.,w (0) = ψ (0) = 0, are applied. Theend force applied to the microcantilever is depicted in Fig. 10 versus the end deflection of the microcantilever,wLb . In this figure, the results obtained by the FEM based on the newly established beam elements have beencompared to the experimental results and the results obtained by applying the classical beam elements. It isobserved that the outcomes based on the new beam elements are in good agreement with the experimentalresults while there is a significant difference between the experimental and classical FEM results. It is alsoobserved that the results of the strain gradient Timoshenko beam element are closer to the experimental data thanthe results of the strain gradient Euler–Bernoulli beam element. Since for the aforementioned microcantilever,the ratio of the beam length to the beam height is 10 and the above-mentioned observation is justifiable.
To sum up, it seems that in order to model the microscale structures, employing the non-classical elementsis essential. The good agreement between the current and the experimental results implies that the newly
Strain gradient Timoshenko beam element
0 200 400 600 800 1000 1200 1400 1600 1800 20000
50
100
150
200
250
300
End deflection (nm)
En
d F
orc
e (µ
N)
Strain gradient Timoshenko Strain gradient Euler-BernoulliClassical Timoshenko Classical Euler-BernoulliExperimental Data (Lam et al., 2003)
Fig. 10 End-force versus the end deflection of the microcantilever: a comparison between the FEM results and experimental data.The FEM results are obtained by employing strain gradient and classical Timoshenko and Euler–Bernoulli beam elements
developed beam elements are valid, reliable, and can be successfully employed to deal with the mechanicalproblems in micron and submicron scales. It is seen that the non-classical beam elements predicts the beamsstiffer than those the classical beam theories.
4.2 An electrostatically actuated microcantilever: static deflection, frequency, and pull-in voltage
In this section, the new beam element is implemented to investigate the static and vibration behavior ofelectrostatically actuated microbeams.
Consider an electrostatically actuated microcantilever with length Lb having a uniform rectangular crosssection with height h and width b and an initial distance d from a fixed substrate subjected to electrostatic forcewith the voltage V as shown in Fig. 11. Assuming the aforementioned microcantilever is made of poly-silicon,which is an isotropic material with elastic modulus of E = 150 Gpa and Poisson’s ratio of ν = 0.23 [46],and also considering the geometrical properties of the microcantilever to be tabulated in Table 1, the staticdeflection of the microcantilever under electrostatic force is calculated using the new beam element. For themicrobeam depicted in Fig. 11, the distributed electrostatic force-per-unit length is expressed as
F(x) = ε0bV 2
(d − w)2
(1 + 0.65
d − w
b
), (59)
Fig. 11 An electrostatically actuated microcantilever: configuration, geometry, and coordinate system
M. H. Kahrobaiyan et al.
Table 1 Geometrical properties of the microbeam tested by Jensen et al. [46]
represents the vacuum permittivity and V stands for the applied voltage betweenthe beam and the substrate. Substituting F from Eq. (59) into Eq. (34) and considering M to be zero, the nodalforce vector can be calculated.
Assembling the stiffness matrices and force vectors of all elements and also satisfying the boundaryconditions, i.e., w (0) = 0 and ψ (0) = 0, the equation of the static deflection of the electrostatically actuatedmicrocantilever is determined as follows:
δ = K−1f, (60)
where the symbol “ ∼ ” refers to the assembled version of the respected quantity. Equation (60) is a nonlinearequation in which the force vector f is a nonlinear function of the displacement vector δ. This equation canbe solved using an iterative method as follows. At first, the electrostatic load on the un-deformed microbeam(i.e., δ
0 = 0, which yields w0(x) = 0) is calculated as
F0(x) = ε0bV 2
d2
(1 + 0.65
d
b
). (61)
Given F0(x), the force vector f0 and consequently the first estimation of the nodal displacement vector (i.e., δ1)
and subsequently the displacement field (i.e.,w1(x)) can be calculated using Eq. (60). Now, the next estimationof the electrostatic load and force vector can be obtained by substitutingw1(x) into Eq. (59), and subsequentlythe next estimation of the nodal displacements vector and displacement field can be found. This procedure willbe stopped when the convergence is observed or pull-in phenomenon is happened. The convergence criterionis considered as
errori < errordesired, (62)
in which
errori =∣∣∣δi − δ
i−1∣∣∣∣∣∣δi
∣∣∣, errordesired = 10−8, (63)
and pull-in happens if wmax ≥ 1.The static deflection of the microcantilever obtained by employing the new beam element is depicted in
Fig. 12 for various values of applied voltages and compared with the results achieved by utilizing the classicalbeam elements as well as the experimental results extracted from work of Jensen et al. [46]. It is noted that thelength scale parameter of poly-silicon is assumed to be l = 0.14 µm. The figure shows that there is a goodagreement between the results evaluated by implementing the new beam element and the experimental datawhile the difference between the experimental findings and the classical FEM modeling outcomes is notable.It is noted that the difference increases as the applied voltage increases.
As another example, the newly developed beam element is employed to evaluate the pull-in voltage of elec-trostatically actuated microbeams. As the applied voltage increases, the attractive electrostatic force betweenthe fixed substrate and the microcantilever increases and so does the deflection of the microcantilever; as a con-sequence, at a certain voltage, the microcantilever tends to collapse to the fixed substrate. The aforementionedbehavior is well known in the literature as the pull-in phenomenon, and the corresponding voltage is recognizedas the pull-in voltage denoted by VP . It is noted that determining the accurate values of pull-in voltage is anextremely important issue in design of microswitches. Hence, the static pull-in voltage of a microcantilevermade of silicon is determined here by employing the newly developed beam elements.
Strain gradient Timoshenko beam element
0 100 200 300 400 500 600 700-2.5
-2
-1.5
-1
-0.5
0
x (µm)
w (
µm)
V=6.96 volts
V=7.81 volts V=7.38 volts
Fig. 12 Static deflection of an electrostatically actuated microcantilever for different applied voltages: a comparison with theexperimental and classical FEM results. The solid line represents the results of the new (strain gradient) beam element; the dashedline represents the results of the classical beam element; and green hollow square marks represents the experimental results (colorfigure online)
Table 2 Specifications of the silicon microcantilevers tested by Osterberg [50,51]
Specification Group 1 Group2
Crystal direction along beam length 110 010Elastic modulus along beam length 169.2 GPa 130.4 GPaPoisson’s ratio in side plane of the beam 0.239 0.177The range for length (L) 75–250µm 75–225µmHeight (h) 2.94 µm 2.94 µmWidth (b) 50 µm 50 µmDistance from the base (d) 1.05 µm 1.05 µm
In order to indicate the advantages of the new beam element, the results obtained using the new beamelement are compared with those reported in the experimental research performed by Osterberg [50]. Thespecifications of the microbeam tested by Osterberg [50] are presented in Table 2. It is noted that the microbeamswere fabricated in two different directions of silicon crystal: 110 direction, i.e., the length of the beam is alongthe 110 direction and the side plane of the beam normal to 110 direction of silicon crystal, and 010 direction,i.e., the length of the beam is along the 010 direction and the side plane of the beam normal to 010 direction ofsilicon crystal [50,51]. Silicon can be assumed as an orthotropic material [50–52], and the FEM formulationderived in this paper is for isotropic materials, but since for bending of slim microbeams, the uniaxial straincondition can be supposed and the deflection of the microbeam is considered to be in x−z plane only, thecurrent formulation can be employed as long as one uses the effective elastic modulus in x direction andeffective Poisson’s ratio in x−z plane. These effective mechanical properties are experimentally determinedfor silicon by Osterberg [50], and the same values are used in FEM modeling of the current work.
Figures 13 and 14 compare the results of the pull-in voltage evaluated by the new beam element with thoseevaluated based on the classical FEM and also the experimental observations [50] for silicon microbeams,respectively, in 110 and 010 directions. It is noted that in order to generate the graphs of Figs. 13 and 14, thelength scale parameter of the silicon is considered to be l110 = 0.31 µm and l010 = 0.38 µm in crystal planesnormal to 110 and 010 directions, respectively (since the silicon is anisotropic material, its length scale willbe different for different directions). The figures indicate that the classical FEM underestimates the pull-involtage of the microcantilever. On the other hand, the pull-in voltages evaluated by the new beam element arein good agreement with the experimental observations [50].
In another example, the first vibration frequency of a clamped–clamped microbeam subjected to electro-static force is obtained using the new FEM formulation. In order to evaluate the frequency, the eigenvalues ofM−1K should be calculated. The clamped–clamped microbeam used for numerical modeling is assumed to bemade of poly-silicon with elastic modulus of E = 150 Gpa, Poisson’s ratio of ν = 0.23 [46,53], and the length
M. H. Kahrobaiyan et al.
50 100 150 200 2500
10
20
30
40
50
60
70
80
( )bL mµ
New beam element ( 0.31l µ= )
Experimental data, Osterberg, 1995
Classical beam element
()
PV
v
m
Fig. 13 Comparing the present and the experimental results of static pull-in voltage for silicon microbeams fabricated in 110direction
50 100 150 200 2500
10
20
30
40
50
60
70
80
( )bL mµ
New beam element ( 0.38l mµ= )
Experimental data, Osterberg, 1995
Classical beam element
()
PV
v
Fig. 14 Comparing the present and the experimental results of static pull-in voltage for silicon microbeams fabricated in 010direction
Table 3 Geometrical properties of the microbeams tested by Tilmans and Legtenberg [53]
scale parameter of l = 0.14 µm (as assumed in the first example). The geometrical properties of the microbeamcan also be found in Table 3. The frequencies obtained by employing the new beam element are comparedto the experimental data found by Tilmans and Legtenberg [53] in Fig. 15 noted that the midplane stretching,i.e., (E A/2L)
∫ L0 (∂w/∂x)2 dx which appears in beams with immovable supports such as clamped–clamped
[54] is also taken into account in FEM modeling. The figure shows that there is a good agrement betweenthe current and experimental results. In this figure, it is observed that as the voltage increases, the frequencydecreases until the pull-in happens.
In this section, the new strain gradient beam element is implemented to evaluate the static deflection, staticpull-in voltage, and vibration frequency of electrostatically actuated microbeams. The results are comparedto the experimental results and those evaluated by the classical FEM modeling. It is observed that unlike theclassical FEM results, the results of the new element are in good agreement with the experimental results. Itcomes to the conclusion that the new strain gradient beam elements can reduce the gap between the experimental
Strain gradient Timoshenko beam element
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Applied Voltage (V)
Fre
qu
ency
(kH
z)
L=210 µm (Present work)
L=210 µm (Experimental)
L=310 µm(Present work)
L=310 µm (Experimental)
L=410 µm(Present work)
L=410 µm (Experimental)
L=510 µm (Present work)
L=510 µm (Experimental)
Fig. 15 First vibration frequency of the electrostatically actuated clamped–clamped microbeam: a comparison with experimentalfindings extracted from Tilmans and Legtenberg [53]
observations and simulation results, and therefore, the necessity of using the non-classical continuum-basedFEM such as the present strain gradient beam element will be comprehensible.
5 Summary and conclusion
Since the classical continuum theory is neither able to capture the size dependency observed in microscalestructures nor able to evaluate the accurate stiffness of these structures, developing structural finite elementsbased on non-classical continuum theories such as the strain gradient theory seems to be crucial. In this paper,a comprehensive beam element is developed on the basis of the strain gradient theory. The formulations of thenew beam element recover the formulations of strain gradient, modified couple stress and classical Timoshenkoand Euler–Bernoulli beam elements. An energy-based approach is utilized to obtain the stiffness and massmatrices of the new beam element, and the shape functions are derived by solving the governing equations ofstrain gradient Timoshenko beams and applying appropriate boundary conditions. The characteristics of thenew beam element can be outlined as:
• The new beam element can capture the size dependency of microscale systems unlike the classical beamelements;
• The new beam element predicts the accurate stiffness for microscale structures where the classical beamelements underestimate the stiffness;
• The formulations of the new strain gradient Timoshenko beam element reduce to the formulations ofstrain gradient Euler–Bernoulli, modified couple stress Timoshenko and Euler–Bernoulli, and also classicalTimoshenko and Euler–Bernoulli beam elements in special circumstances;
• The stiffness and mass matrices of the new beam element are presented in closed forms;• The shape functions of the new beam element are derived by solving the static equilibrium equations and
consequently have size-dependent nature;• The shear-locking phenomenon will not happen for the new beam element.
In order to validate the new beam element, the static deflection of a microcantilever is evaluated by utilizingthe new beam element, and the results are compared to the experimental data as well as the results obtained bythe classical FEM and the couple stress-based plane element. The static deflection, pull-in voltage, and vibrationfrequencies of electrostatically actuated microbeams are evaluated by employing the new beam elements, andthe results are compared to the experimental data as well as the classical FEM results. It is observed that theresults based on the new beam elements are in excellent agreement with the experimental results, whereas thegap between the classical FEM and experimental outcomes is notable. It is noted that when the dimensions ofthe beams increase, the results of the new beam element approach the results of classical FEM.
M. H. Kahrobaiyan et al.
References
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