Received: 15 November 2018 Revised: 25 March 2019 Accepted: 26 March 2019

DOI: 10.1002/nme.6073

R E S E A R C H A R T I C L E

A simple unsymmetric 4-node 12-DOF membrane elementfor the modified couple stress theory

Yan Shang1 Zheng-Hua Qian1 Song Cen2 Chen-Feng Li3

1State Key Laboratory of Mechanics andControl of Mechanical Structures, Collegeof Aerospace Engineering, NanjingUniversity of Aeronautics andAstronautics, Nanjing, China2Department of Engineering Mechanics,School of Aerospace Engineering,Tsinghua University, Beijing, China3Zienkiewicz Centre for ComputationalEngineering and Energy Safety ResearchInstitute, College of Engineering, SwanseaUniversity, Swansea, UK

CorrespondenceYan Shang, State Key Laboratory ofMechanics and Control of MechanicalStructures, College of AerospaceEngineering, Nanjing University ofAeronautics and Astronautics, Nanjing210016, China.Email: shangyan@nuaa.edu.cn

Zheng-Hua Qian, State Key Laboratory ofMechanics and Control of MechanicalStructures, College of AerospaceEngineering, Nanjing University ofAeronautics and Astronautics, Nanjing210016, China.Email: qianzh@nuaa.edu.cn

Funding informationNational Natural Science Foundation ofChina, Grant/Award Number: 11702133and 11872229; Natural ScienceFoundation of Jiangsu Province,Grant/Award Number: BK20170772

Summary

In this work, the recently proposed unsymmetric 4-node 12-DOF(degree-of-freedom) membrane element (Shang and Ouyang, Int J NumerMethods Eng 113(10): 1589-1606, 2018), which has demonstrated excellentperformance for the classical elastic problems, is further extended for themodified couple stress theory, to account for the size effect of materials. Thisis achieved via two formulation developments. Firstly, by using the penaltyfunction method, the kinematic relations between the element's nodal drillingDOFs and the true physical rotations are enforced. Consequently, the continuityrequirement for the modified couple stress theory is satisfied in weak sense, andthe symmetric curvature test function can be easily derived from the gradientsof the drilling DOFs. Secondly, the couple stress field that satisfies a priori therelated equilibrium equations is adopted as the energy conjugate trial functionto formulate the element for the modified couple stress theory. As demonstratedby a series of benchmark tests, the new element can efficiently capture thesize-dependent responses of materials and is robust to mesh distortions. More-over, as the new element uses only three conventional DOFs per node, it can bereadily incorporated into the standard finite element program framework andcommonly available finite element programs.

KEYWORDS

finite element method, modified couple stress theory, penalty function, size effect, unsymmetric

1 INTRODUCTION

There is mounting evidence that strong size effects may be observed when the deformation takes place at the microscale orbelow. For instance, Fleck et al1 conducted a torsion test on copper wires and discovered that the shear strength increasesas the wire diameter decreases; Lam et al2 reported that the bending rigidity of a micro epoxy beam varies with respect tothe structure size. Similar phenomena have also been observed in other experiments.3-7 The classical continuum theory

Int J Numer Methods Eng. 2019;119:807–825. wileyonlinelibrary.com/journal/nme © 2019 John Wiley & Sons, Ltd. 807

808 SHANG ET AL.

cannot capture such size effects, because it assumes that the material is homogeneous and only the force can be trans-mitted on infinitesimal surfaces, which may no longer be valid at small length scales. Therefore, new continuum theoriesare needed to describe these size-dependent deformation behaviors.

Since the pioneering work of Cosserat brothers,8 substantial efforts have been made to investigate the size effects relatedto material deformation, resulting in a diverse range of continuum theories. Mindlin9 developed a microstructure theorythat contains 18 material constants for isotropic materials, and it was later simplified into the strain gradient theory10

that contains two Lamé coefficients and five additional material constants. Fleck and Hutchinson11 further reformulatedthis theory and extended it into the plastic strain gradient theory. Aifantis12 also developed a nonlocal theory for plasticmaterials. Lam et al2 modified the strain gradient theory so that it only requires three additional material constants. Inaddition, by eliminating the difference between the microrotation and macrorotation in the Cosserat theory,8 Koiter,13

Mindlin and Tiersten,14 and Toupin15 proposed the couple stress theory that contains only two additional constants, whichhas been proved to be a special case of the strain gradient theory.10

To support practical use in engineering applications, many attempts16-20 have been made to further reduce the numberof additional material constants required to describe the size-dependent behaviors. Among them, Yang et al20 proposedthe modified couple stress theory, in which only the symmetric terms of the curvatures are considered for contribution tothe deformation energy. The modified couple stress theory requires just one additional material parameter to describe thesize-dependent deformation and has become increasingly popular in recent years due to its simplicity and verifiability.

It is hard to solve analytically the size-effect related problems, because the size-dependent continuum theories are muchmore complicated than the classical elasticity theory. The finite element method (FEM), which is generally regarded asthe most efficient and popular numerical tool21 for modeling solids, provides a promising solution.22-26 However, due to thepresence of the second-order derivatives of displacements in these size-dependent continuum theories, a straightforwarddisplacement-based finite element implementation would require continuity in both the displacements and their firstderivatives, ie, C1 continuity. This brings significant challenges to the element construction and will make the elementformulations much more complicated. For instance, Zervos et al27 and Papanicolopulos et al28 developed strict C1 elementsfor the strain gradient theory but, due to some inherent drawbacks,29 these complex C1 elements can only be used incertain restricted cases.

Alternatively, Ma and Chen30,31 developed 3-node triangular and 4-node quadrilateral hybrid stress element modelsthat have six DOFs (degrees of freedom) per node, including four displacement derivative DOFs in addition to twodisplacement DOFs. In these hybrid elements, the C1 continuity requirement is satisfied in weak sense. Zhao et al32,33

and Wang et al34,35 also proposed similar elements based on the refined nonconforming element method and thequasi-conforming element method, respectively. However, such treatments with both displacements and their gradientstaken as nodal DOFs may significantly increase the computation cost at the element level. Moreover, it is inconvenientto incorporate such elements into the standard finite element (FE) program framework or commercial FE software. Aneffective approach to keep the element formulation relatively simple is to introduce the independent rotation or drillingDOFs.36,37 In this approach, the penalty function method38,39 or the Lagrangian multiplier method29,40 is usually employedto constrain the kinematic relations between the independent rotation DOFs and the physical rotations derived fromdisplacements, so that the C1 continuity requirement is met in weak sense.

Recently, Shang and Ouyang41 proposed a simple but very robust 4-node 12-DOF quadrilateral membrane elementUS-Q4θ for the classical elastic problems based on the unsymmetric FEM.42-44 The unsymmetric FEM, which has beensuccessfully applied to various applications in past years,45-49 is characterized of employing two different interpolations torespectively determine the test function and trial function. Demonstrated by numerical tests, the element US-Q4θ exhibitsquite good numerical accuracy and resistance to mesh distortions. In particular, it can still work well even when theelement shape severely distorts into concave quadrilateral or triangle. In this work, the unsymmetric element US-Q4θ,originally designed for classical elastic problems, is further developed to model size-dependent material responses basedon the modified couple stress theory.20 Since the element US-Q4θ has already employed the drilling DOFs to enhance ele-ment behaviors, the key step here is to introduce a proper penalty function to ensure the nodal drilling DOFs effectivelyapproximate the true physical rotations. Then, the test function for the symmetric curvature in the virtual work principlecan be readily determined as the derivatives of these drilling DOFs. Meanwhile, the trial function for the couple stress,which conjugates with above the curvature test function, can be directly formulated following the analytical trial func-tion method50 and the quasi-conforming technique.51,52 Developed for the modified couple stress theory, the new element,renamed as US-Q4θ-CS, is expected to work with size-dependent problems with good numerical precision and high resis-tance to mesh distortions. Moreover, as the new element still has only two translational DOFs and one drilling DOF pernode, it can be easily incorporated into the standard FEM framework and commercial FEM software.

SHANG ET AL. 809

The remainder of this paper is organized as follows. The modified couple stress theory proposed by Yang et al20 is brieflyreviewed in Section 2. Next, the element formulation is explained in Section 3, and several numerical benchmark testsare presented in Section 4 to validate the new element's capacity. Finally, some conclusions and discussions are drawn inSection 5.

2 OVERVIEW OF MODIFIED COUPLE STRESS ELASTICITY THEORY

2.1 General governing equationsIn the modified couple stress theory,20 the strain components 𝜀ij and the rotation components 𝜔i are defined as the spatialderivatives of the displacement ui

𝜀ij =ui,𝑗 + u𝑗,i

2, 𝜔i =

12

eijkuk,𝑗 , (1)

where eijk is the Levi-Civita symbol. Different from the classical couple stress theories13-15 in which both the symmetricand antisymmetric parts of rotation gradients contribute to the deformation energy, only the symmetric part is consideredin the modified couple stress theory of Yang et al.20 Then, the symmetric curvature components 𝜒 ij are defined as

𝜒ij =𝜔i,𝑗 + 𝜔𝑗,i

2. (2)

For a linear elastic isotropic material, the stress 𝜎ij and couple stress mij, which are work conjugates of the above strainand curvature, respectively, can be obtained using the constitutive relations

𝜎ij = 𝜆𝜀kk𝛿ij + 2G𝜀ij, mij = 2Gl2𝜒ij, (3)

in which 𝜆 and G are the two Lamé constants in Cauchy elasticity, and l the additional material length scale parameter,and 𝛿ij the Kronecker delta. The higher-order equilibrium equations are expressed as

𝜎ik,i −12

ejlkmij,il + 𝑓k = 0, (4)

where fk denotes the body force per unit volume. The body couple force is not included in Equation (4), because it can bedecomposed into an equivalent system of body force and surface force.16

2.2 Two-dimensional problemAs discussed by Choi and Lee,53 the curvature terms produced by the thickness-direction displacement will contributeto the deformation energy in the modified couple stress theory; therefore, the plane stress state cannot be accuratelysimulated through a two-dimensional simplification. Thus, only the plane strain state is considered in this work.

In the following sections, x and y are used to represent the two in-plane directions, while z refers to the thicknessdirection. Correspondingly, the three displacement components are denoted as u, v, and w, respectively. Under the plainstrain assumption, the nonzero strain and curvature terms can be expressed in the following vector forms:

𝛆 =⎧⎪⎨⎪⎩𝜀x

𝜀𝑦

𝛾xy

⎫⎪⎬⎪⎭ =

⎧⎪⎪⎨⎪⎪⎩

𝜕u𝜕x

𝜕v𝜕𝑦

𝜕u𝜕𝑦

+ 𝜕v𝜕x

⎫⎪⎪⎬⎪⎪⎭, (5)

𝛘 ={

2𝜒xz

2𝜒yz

}=⎧⎪⎨⎪⎩

𝜕𝜔z

𝜕x

𝜕𝜔z

𝜕𝑦

⎫⎪⎬⎪⎭ , (6)

in which

𝜔z =12

(𝜕v𝜕x

− 𝜕u𝜕𝑦

). (7)

810 SHANG ET AL.

The coefficient “2” used in Equation (6) is to consider the symmetry of the curvatures. Besides, the stress and couple stresscan be obtained by using the constitutive equations

𝛔 =⎧⎪⎨⎪⎩𝜎x

𝜎𝑦

𝜎xy

⎫⎪⎬⎪⎭ = Dn𝛆, (8)

m ={

mxz

myz

}= Dc𝛘, (9)

with

Dn = E(1 + 𝜈)(1 − 2𝜈)

⎡⎢⎢⎣1 − 𝜈 𝜈 0𝜈 1 − 𝜈 00 0 (1 − 2𝜈)∕2

⎤⎥⎥⎦ , (10)

Dc =

[Gl2 00 Gl2

], (11)

in which E denotes Young's modulus, 𝜈 is Poisson's ratio, and G = E/2(1+𝜈).

3 FINITE ELEMENT FORMULATION

3.1 Virtual work principle for unsymmetric FEMThe unsymmetric FEM is developed following the virtual work principle and is characterized of employing differentinterpolations for the element test and trial functions. With respect to the new membrane element model for the modifiedcouple stress theory, the virtual work principle incorporating the penalty term can be expressed as

∫∫Ω𝛿𝛆T��tdΩ + ∫∫Ω

𝛿𝛘TmtdΩ − ∫∫Ω𝛿uTf tdΩ − ∫Γ

𝛿uTRtdΓ

− ∫Γ𝛿𝜔zMztdΓ + k∫∫Ω

𝛿ΛΛtdΩ = 0,(12)

in which Ω is the domain bounded by Γ; t is thickness; f denotes the body force vector; R and Mz respectively representthe prescribed external force and couple force at the boundary; u is the test function for the displacement, and 𝛆 and 𝛘 arethe test functions for the strain and symmetric curvature; 𝜔z is the test function for the physical rotation, which will beindependently interpolated by the element nodal drilling DOFs; �� and m are the trial functions for the stress and couplestress, respectively.

Moreover, the last term in Equation (12) is the penalty function term that is used to constrain the kinematic relationsbetween the independently assumed rotation 𝜔z and the one derived from displacements u. By adding the penalty func-tion term, the rotation 𝜔z can effectively reproduce the true physical rotation. This will be discussed in more details inSection 3.2.3.

3.2 The new unsymmetric element US-Q4 θ-CSAs stated above, the proposed new element is directly developed from the existing high-performance 4-node elementUS-Q4θ,41 which is proposed for classical elastic problems. As shown in Figure 1, the new element still has three DOFsper node

qe = [u1 v1 𝜃1 u2 v2 𝜃2 u3 v3 𝜃3 u4 v4 𝜃4]T. (13)

SHANG ET AL. 811

u2

v2

u1

v1

4

1 2

3 u3

v3

u4

v4

4

3

1

2

FIGURE 1 Unsymmetric 4-node 12-degree-of-freedom quadrilateral membrane element

3.2.1 The element's test functionsIn this work, the test function for the displacement is still determined by using the concise interpolations proposed byShang and Ouyang,41 which can meet the requirements of interelement compatibilities for any distorted geometry

u ={

uv

}= Nqe, (14)

withN =

[N1 N2 N3 N4

], (15)

Ni =

[Ni 0 − 1

2Ni(𝑦 − 𝑦i)

0 Ni12

Ni(x − xi)

], i = 1 ∼ 4, (16)

where (xi, yi) are the Cartesian coordinates of node i; Ni (i = 1∼4) are the shape functions of the standard 4-nodeisoparametric element {

N1 = 14(1 − 𝜉)(1 − 𝜂), N2 = 1

4(1 + 𝜉)(1 − 𝜂)

N3 = 14(1 + 𝜉)(1 + 𝜂), N4 = 1

4(1 − 𝜉)(1 + 𝜂),

(17)

in which (𝜉, 𝜂) are the isoparametric coordinates. Then, by substituting Equation (14) into the displacement-strainequations, the test function for the strain can be obtained

𝛆 =⎧⎪⎨⎪⎩𝜀x

𝜀𝑦

𝛾xy

⎫⎪⎬⎪⎭ = Bnqe, (18)

withB

n=[

Bn1 B

n2 B

n3 B

n4

], (19)

Bni =

⎡⎢⎢⎢⎣Ni,x 0 − 1

2Ni,x(𝑦 − 𝑦i)

0 Ni,𝑦12

Ni,𝑦(x − xi)

Ni,𝑦 Ni,x12

Ni,x(x − xi) − 12

Ni,𝑦(𝑦 − 𝑦i)

⎤⎥⎥⎥⎦ , i = 1 ∼ 4. (20)

As previously discussed, 𝜔z is independently interpolated by the element nodal drilling DOFs

𝜔z = N𝜔

qe, (21)

withN

𝜔=[

0 0 N1 0 0 N2 0 0 N3 0 0 N4]. (22)

Because the penalty function method is employed to minimize the difference between𝜔z and the physical rotation derivedfrom the displacement u, the physical rotation can be approximately replaced by𝜔z for the finite element implementation.

812 SHANG ET AL.

Thus, the test function for the symmetric curvature can be easily obtained by

𝛘 ={ 2𝜒xz

2𝜒yz

}=⎧⎪⎨⎪⎩

𝜕𝜔z

𝜕x𝜕𝜔z

𝜕𝑦

⎫⎪⎬⎪⎭ = Bcqe, (23)

in whichB

c=[

Bc1 B

c2 B

c3 B

c4

], (24)

Bci =

[0 0 Ni,x

0 0 Ni,𝑦

], i = 1 ∼ 4. (25)

3.2.2 The element's trial functionsSimilar to the original element US-Q4θ,41 the stress trial function of the new element, as work conjugate of the strain testfunction, is also formulated based on the Airy stress solutions. Since the detailed discussions can be found in the work ofShang and Ouyang,41 the derivation procedure is only briefly summarized here. First, the stress field �� in Equation (12)is initially assumed as

�� =⎧⎪⎨⎪⎩��x

��𝑦

��xy

⎫⎪⎬⎪⎭ = Hn𝛂n, (26)

in which

Hn =⎡⎢⎢⎣

0 0 2 0 0 2x 6𝑦2 0 0 6x 2𝑦 0 00 −1 0 0 −2x −2𝑦 0

⎤⎥⎥⎦ , (27)

𝛂n =[𝛼1 𝛼2 𝛼3 𝛼4 𝛼5 𝛼6 𝛼7

]T. (28)

Second, to determine the relations between the above unknown coefficients in Equation (28) and the element nodal DOFsin Equation (13), the following quasi-conforming conditions is employed:

∫∫ΩHnT (𝛆 − D−1

n ��)

tdΩ = 𝟎, (29)

in which 𝛆 and Dn are defined by Equation (18) and Equation (10), respectively. Then, the stress �� can be rewritten interms of qe

�� = Snqe, (30)

in whichSn = HnM−1

n Vn, (31)

with

Mn = ∫∫ΩHnTD−1

n HntdΩ, (32)

Vn = ∫∫ΩHnTB

ntdΩ. (33)

The trial function for the couple stress of the new element, which is work conjugate of the curvature test function shownin Equation (12), can be obtained by following a similar procedure. The couple stress field m is initially assumed as thefollowing form:

m ={

mxz

myz

}= Hc𝛂c, (34)

SHANG ET AL. 813

in which

Hc =[

1 0 x 0 𝑦 00 1 0 x 0 𝑦

], (35)

𝛂c =[𝛼8 𝛼9 𝛼10 𝛼11 𝛼12 𝛼13

]T. (36)

Then, the quasi-conforming technique is employed once again to determine the relationships between the unknowncoefficients in Equation (36) and the element nodal DOFs, as follows:

∫∫ΩHcT (𝛘 − D−1

c m)

tdΩ = 𝟎. (37)

By substituting Equation (23) and Equation (34) into above Equation (37), we can obtain

𝛂c = M−1c Vcqe, (38)

where

Mc = ∫∫ΩHcTD−1

c HctdΩ. (39)

Vc = ∫∫ΩHcTB

ctdΩ. (40)

Finally, substitution of Equation (38) back into Equation (34) yields

m = Scqe, (41)

with

Sc = HcM−1c Vc. (42)

It should be noted that the stress field �� defined in Equation (30) and the couple stress field m defined in Equation (41)can satisfy a priori the related equilibrium equation, making them a good choice for the stress interpolation.

3.2.3 The penalty functionSince the rotation test function is independently interpolated by the element nodal drilling DOFs, as shown inEquation (21), there will be in general differences between 𝜔z and the physical rotation that is derived from thedisplacement u. Specifically, Λ in Equation (12) has the following expression:

Λ = 12

(𝜕v𝜕x

− 𝜕u𝜕𝑦

)− 𝜔z, (43)

By substituting Equation (14) and Equation (21) into Equation (43), we can rewrite it as

Λ = NΛ

qe, (44)

with

NΛ=[

NΛ1 N

Λ2 N

Λ3 N

Λ4

], (45)

NΛi = 1

2

[−Ni,𝑦 Ni,x

12

Ni,x(x − xi) + 12

Ni,𝑦(𝑦 − 𝑦i) − Ni

], i = 1 ∼ 4. (46)

The penalty function method is used in this work to enforce the constraint Λ = 0. The penalty parameter should be setlarge enough to enforce the constraint at an acceptable level, and, in general, the constraint will be satisfied more strictlyas the penalty parameter k increases. On the other hand, the penalty parameter should not be overly large because it maymake the stiffness matrix ill conditioned. According to dimensional analysis, the penalty parameter k should be propor-tional with G. Parametric studies have been performed, by varying the ratio k/G from 10 to 1028, to test the limits of the

814 SHANG ET AL.

penalty parameter with respect to the mesh size and mesh distortion. The numerical results indicate that the numer-ical solutions are independent from the penalty parameter when 104 ≤ k/G ≤ 1024. Note that the quadruple-precisionfloating-point format is used for the FE program implementation.

3.2.4 The element stiffness matrixBy substituting the related equations into the virtual work principle in Equation (12), the element stiffness matrix andthe equivalent nodal load vector can be obtained as

Ke = ∫∫ΩB

nTSntdΩ + ∫∫Ω

BcT

SctdΩ + k∫∫ΩN

ΛTN

ΛtdΩ, (47)

Pe = ∫∫ΩN

Tf tdΩ + ∫Γ

NT

RtdΓ + ∫ΓN

𝜔TMztdΓ. (48)

After solving the element nodal DOFs qe, the stress and couple stress at an arbitrary point within the element can becalculated using Equation (30) and Equation (41).

Note that, as the independently assumed rotation𝜔z and the physical rotation derived from displacements have differentorders of interpolation, severe locking behavior may be observed when the full-integration scheme is used for the lastpenalty stiffness in Equation (47). To overcome this problem, the selective reduced integration procedure suggested inthe work of Garg and Han38 is employed here: the last penalty term is calculated by using the one-point Gauss quadraturestrategy while other integrations are operated by using the full quadrature scheme.

4 NUMERICAL VALIDATION

In this section, several benchmark problems are solved to examine the performance of the proposed new elementUS-Q4θ-CS for modeling the size effect. The penalty parameter is set as k/G = 105 in all benchmark tests, except whereotherwise stated.

4.1 The test for rigid body rotationFigure 2 shows a square block with an edge length L =

√2 mm modeled by using four elements. To produce a rigid body

rotation, two cases with different boundary conditions are considered. The first one is to make u1 = v1 = 0 and 𝜃1 = 0.1at node 1, and the second one is to make u1 = v1 = 0 at node 1 while v2 = 0.1 mm at node 2. In the context of smalldeformation problems, the rotational angles should be 0.1 at all nodes in these two cases. The numerical results listed inTable 1 verify that the proposed new element can correctly reproduce the rigid body rotation motion.

E

l

x

y

1 (0, 0) 2 (1 mm, 0)

3 (0, 1 mm)

4 (-1 mm, 0)

5 (0, -1 mm)

FIGURE 2 The test for the rigid body rotation

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TABLE 1 Rotation results of the test forthe rigid body rotation

Node 1 2 3 4 5

𝜃 Case A 0.1 0.1 0.1 0.1 0.1Case B 0.1 0.1 0.1 0.1 0.1

4.2 The cantilever thin beamAs shown in Figure 3, the benchmark proposed by Choi and Lee53 is investigated, in which the cantilever thin beam issubjected to a tip shear load. The reference bending rigidity for the modified couple stress theory is given by53

DCS = 1bh3

( EI1 − 𝜈2 + GAl2

), (49)

where A is the cross-sectional area and I is the area moment of inertia. Besides, the bending rigidity obtained by the finiteelement analysis can be evaluated by

DFEM = PL3

3vmaxbh3 , (50)

in which vmax is the maximum vertical tip displacement.First, the convergence property of the new element is tested. The beam height h is set as 20μm. Three cases with dif-

ferent material length scale parameters l = 17.6μm, 13.2μm, and 8.8μm are considered. The computations are repeatedby successively refining the basic mesh 1 × 10 given in Figure 4A into 2 × 20, 4 × 40, and 8 × 80. The relative errors ofthe bending rigidity are shown in Figure 5, in which the results from another 4-node 12-DOF quadrilateral element pro-posed by Garg and Han38 are also plotted for comparison. It can be observed that Garg and Han's element experiencesthe locking problem in the coarse mesh 1 × 10. The main reason for this is that the element's displacements are interpo-lated by the standard isoparametric interpolation. In contrast, numerical results show that the new element US-Q4θ-CSconverges much more rapidly and is free of shear locking.

y

x

P

L

h

FIGURE 3 The microcantilever thin beam

(A)

(B)

(C)

FIGURE 4 Three typical basic meshes 1 × 10 for the microcantilever thin beam. A, Regular mesh; B, Parallelogram mesh; C, Trapezoidalmesh

816 SHANG ET AL.

FIGURE 5 The relative errors of the bending rigidity using the regular mesh shown in Figure 4A with h = 20μm [Colour figure can beviewed at wileyonlinelibrary.com]

FIGURE 6 The relative errors of the bending rigidity using different typical meshes (h = 20μm, l = 17.6μm, and Δ = 10μm) [Colour figurecan be viewed at wileyonlinelibrary.com]

Second, the influence on the convergence from the mesh distortion is checked by using the distorted meshes as illus-trated in Figures 4B and 4C. The beam height h is 20μm, the material length scale parameters l is 17.6μm, and the distortionparameter Δ is set as 10μm. As shown by the results in Figure 6, the new element can still converge well in distortedmeshes.

Moreover, to further assess the new element's robustness to mesh distortions, the coarse trapezoidal mesh 1 × 10 isanalyzed once again, by varying the distortion parameter Δ from −20μm to 20μm. Figure 7 shows the variation of thenormalized bending rigidity with respect to the distortion parameter. It can be seen that the maximum deviation is lessthan 4%, confirming again that the proposed new element has low susceptibility to mesh distortion.

Next, the effect on the bending rigidity from the structure size is studied by using the refined regular mesh 4 × 40.Four different values of height h = 20μm, 38μm, 75μm, and 115μm are examined. As shown in Figure 8 and Table 2,the numerical results are in good agreements with the theoretical reference values, proving that this new element caneffectively capture the size effect.

Finally, parametric studies are performed to check the limits of the penalty parameter with respect to the mesh size andmesh distortion. The material and dimension parameters are set as h = 20μm and l = 17.6μm. The ratio of the penalty

SHANG ET AL. 817

FIGURE 7 The normalized bending rigidity versus the distortion parameter using the trapezoidal mesh 1 × 10 with h = 20μm andl = 17.6μm [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 8 The bending rigidity versus the beam thickness using the regular mesh 4 × 40

TABLE 2 The relative errors of the bending rigidity using theregular mesh 4 × 40

h = 20𝛍m h = 38𝛍m h = 75𝛍m h = 115𝛍m

l = 17.6μm 0.127% 0.686% 1.065% 1.154%l = 13.2μm 0.386% 0.881% 1.132% 1.175%l = 8.8μm 0.725% 1.064% 1.173% 1.186%

parameter k to the shear modulus G is varied from 10 to 1028. Table 3 lists the results obtained by using regular mesheswith different mesh sizes, while Table 4 summarizes the results obtained by using the three typical meshes 1 × 10 shownin Figure 4. It can be concluded that the solutions are independent from the penalty parameter when 104 ≤ k/G ≤ 1024.

4.3 The simple shear problemAs shown in Figure 9, a slender rectangular panel is clamped at its bottoms. On the top surface, the rotation and y-directiondisplacement are restrained while the x-direction displacement is constrained to have the prescribed constant value

818 SHANG ET AL.

TABLE 3 The normalized bending rigidity with different penalty parameters anddifferent mesh sizes of the regular mesh (h = 20μm and l = 17.6μm)

k/G 10 102 103 104 105 106 107 … …1 × 10 1.00627 1.00673 1.00678 1.00678 1.00678 1.00678 1.00678 1.006782 × 20 1.00240 1.00278 1.00282 1.00283 1.00283 1.00283 1.00283 1.002834 × 40 0.99974 1.00090 1.00122 1.00127 1.00127 1.00127 1.00127 1.001278 × 80 0.99863 0.99989 1.00016 1.00019 1.00019 1.00019 1.00019 1.00019k/G … … 1022 1023 1024 1025 1026 1027 1028

1 × 10 1.00678 1.00678 1.00678 1.00678 1.00678 1.00678 1.00664 1.006512 × 20 1.00283 1.00283 1.00283 1.00283 1.00283 1.00282 1.00226 1.002824 × 40 1.00127 1.00127 1.00127 1.00127 1.00129 1.00124 0.99910 1.000108 × 80 1.00019 1.00019 1.00019 1.00019 1.00027 1.00020 0.99136 1.00349

TABLE 4 The normalized bending rigidity with different penalty parameters and differentbasic meshes 1 × 10 (h = 20μm and l = 17.6μm)

k/G 10 102 103 104 105 106 107 … …Mesh (a) 1.00627 1.00673 1.00678 1.00678 1.00678 1.00678 1.00678 1.00678Mesh (b) 1.02515 1.02562 1.02567 1.02568 1.02568 1.02568 1.02568 1.02568Mesh (c) 1.01206 1.01251 1.01256 1.01256 1.01256 1.01256 1.01256 1.01256

k/G … … 1022 1023 1024 1025 1026 1027 1028

Mesh (a) 1.00678 1.00678 1.00678 1.00678 1.00678 1.00678 1.00664 1.00651Mesh (b) 1.02568 1.02568 1.02568 1.02568 1.02568 1.02567 1.02561 1.02395Mesh (c) 1.01256 1.01256 1.01256 1.01256 1.01256 1.01256 1.01266 1.01208

y

x

L

h

U

FIGURE 9 The simple shear problem

U = 1μm. The reference analytical solution for the modified couple stress theory have been proposed by Park and Gao54:

u(𝑦) = C1 + C2𝑦 + C3e2l𝑦 + C4e−

2l𝑦, (51)

v = 0, (52)

in which ⎧⎪⎨⎪⎩C1 = 𝛼

2

(e−

2lh + e

2lh − 2

), C3 = 𝛼

2

(1 − e−

2lh)

C2 = 𝛼

l

(e−

2lh − e

2lh), C4 = 𝛼

2

(1 − e

2lh),

(53)

𝛼 = U[−2 +

(1 + h

l

)e−

2lh +

(1 − h

l

)e

2lh] . (54)

l is the material length scale parameter. Accordingly, the analytical shear strain can be expressed as follows:

𝛾xy = C2 +2l

C3e2l𝑦 − 2

lC4e−

2l𝑦. (55)

This rectangle is meshed into 10 × 100 elements and three different material length scale parameters, ie, l = 176μm,17.6μm, and 8.8μm, are considered. Figures 10 and 11 respectively present the distributions of the displacement u andthe shear strain 𝛾xy along the y-axis. It is obvious that the numerical results make good agreements with the referencesolutions, confirming once again that the proposed new element can correctly capture the size effect.

SHANG ET AL. 819

FIGURE 10 Distribution of displacement u along x = 0 of the simple shear problem using the mesh 10 × 100

FIGURE 11 Distribution of shear strain 𝛾xy along x = 0 of the simple shear problem using the mesh 10 × 100

ppx

y

2L

r

E=1.44 GPa, =0.3, r=0.1 mm, L=3.6 mm, width b=10r

FIGURE 12 Square plate with a hole under uniform tension

820 SHANG ET AL.

FIGURE 13 Meshes used for the square plate with a hole. A, 128 elements; B, 288 elements; C, 512 elements

FIGURE 14 The stress concentration factor versus l/r obtained by using different meshes

FIGURE 15 The stress 𝜎x near the hole, obtained by using 512 elements. A, l/r = 1; B, l/r = 0.01

SHANG ET AL. 821

FIGURE 16 The couple stress mx near the hole, obtained by using 512 elements. A, l/r = 1; B, l/r = 0.01

4.4 The square plate with holeIn this example, a large square plate containing a circular hole under uniform uniaxial tension, as shown in Figure 12, isconsidered to check the size effect. Owing to symmetry, only a quarter of the structure is modeled and the correspondingsymmetric boundary conditions are applied. Figure 13 depicts the biased meshes employed for the FE analysis.

In Figure 14, the stress concentration factors for different ratios of the material length scale parameter to the circleradius are plotted. It is observed that the stress concentration factor increases with decreasing the length scale parameter.Moreover, it is interesting to look at the contours of the stress 𝜎x and the couple stress mx in the vicinity of the holewhich are obtained by using 512 elements, as shown in Figure 15 and Figure 16. It can be observed that as the materiallength scale parameter increases, the couple stress concentration becomes more pronounced while the concentrationphenomenon of the stress was relieved.

4.5 The bracketThe bracket model proposed by Choi and Lee53 is modified in this work to assess the proposed element's performance forpractical cases. As shown in Figure 17, the bracket is fixed at its left hole while the right hole is constrained with a verticaldisplacement U. To study the size effect, two cases with different material length scale parameters l = 176μm and 17.6μmare considered. The contours of the shear stress 𝜎xy and the couple stress mx calculated using 1822 elements are providedin Figures 18 and 19. One can clearly see that, as the material length scale parameter increases, the shear stress reducesin magnitude while the couple stress increases, implying that the couple stress plays an increasingly significant role inenergetics.

6 mm

4 mmR=1 mm

R=1 mm

r=0.4 mm

r=0.4 mm

E=1.44 GPa, =0.38, b=10 mm, U=0.1 mm

U

FIGURE 17 The geometry of the bracket

822 SHANG ET AL.

FIGURE 18 The shear stress 𝜎xy of the bracket. A, l = 176μm; B, l = 17.6μm

FIGURE 19 The couple stress mx of the bracket. A, l = 176μm; B, l = 17.6μm

5 CONCLUSIONS

In this paper, an unsymmetric 4-node 12-DOF quadrilateral membrane element is proposed for the modified couple stresstheory. This new element is directly developed from the existing high-performance element model US-Q4θ with drillingDOFs.41 Specifically, the nodal drilling DOFs in the original element formulation are constrained by the penalty functionmethod to appropriately approximate the physical rotation. Therefore, the test function for the symmetric curvature can besimply derived from these nodal drilling DOFs. Meanwhile, the trial function for the couple stress, which conjugates withabove the curvature test function, is formulated following the analytical trial function method. Moreover, to overcomethe locking problem, the penalty stiffness term is evaluated by using the one-point Gauss integration strategy.

The proposed new element, named as US-Q4θ-CS, has the following characteristics.

(i) This element is developed within the framework of the unsymmetric FEM. The construction procedure is straight-forward and the resulting element formulation is concise. Numerical benchmarks verify that this element caneffectively simulate the size-dependent responses and exhibits quite good robustness to mesh distortions.

(ii) According to dimensional analysis, the penalty parameter k should be proportional with G. Parametric studies havebeen performed to check the limits of the penalty parameter with respect to the mesh size and mesh distortion. Thenumerical results indicate that the solutions are independent from the penalty parameter when 104 ≤ k/G ≤ 1024

for all cases tested in this paper. Note that the quadruple-precision floating-point format is used for the FE programimplementation. For practice applications, the ratio k/G can be simply set as a value larger than 104, which can beregarded as an approximate reference lower limit for the penalty parameter.

(iii) Different from Garg and Han's 4-node 12-DOF element38 in which the rotation DOFs are used only to approximatethe physical rotation, the proposed new element employs the nodal rotation DOFs not only to determine the testfunction for the physical rotation but also to enhance the test function for the displacement. Besides, the new ele-ment's trial functions for the stress and couple stress are designed using the analytical functions which can a priori

SHANG ET AL. 823

satisfy the equilibrium equations. Numerical results show that the new element has much better performance thanGarg's model.

(iv) Compared with those elements,30-35 which employ both displacements and displacement derivatives as nodal DOFs,the new element has only three conventional DOFs per node. Thus, it is computationally cheaper at the element leveland can be readily incorporated into the standard finite element program for practical applications. For instance, thiselement can be easily implemented in the commonly used commercial software Abaqus55 through the user-definedelement (UEL) subroutine.

(v) As the new unsymmetric element is directly developed from the virtual work principle, it can be easily extendedto the geometric nonlinear analysis. The key points are to use the updated Lagrangian framework and adopt anappropriate algorithm for updating the stress trial functions at each iterative step.46

(vi) The new element can also be extended to the material nonlinear analysis by using the corresponding incrementalformulation. Note that the main difference between the new unsymmetric element and the classical finite elementformulation is that the present method formulates the stress trial function based on the stress functions that cana priori satisfy the related governing equations. In fact, it can be easily proven that the stress functions shown inEquation (27) can be obtained by using only the equilibrium equations. That means these stress functions can alwayssatisfy the equilibrium equations, including the one expressed in the incremental form. Thus, when the present ele-ment formulation is extended to the material nonlinear analysis, these stress functions can be employed to properlyderive the trial function for the stress increment. Meanwhile, the test function for the strain increment and otherrelated variables are still obtained by following the same procedure with the classical FEM. Moreover, the constitu-tive relations should also be updated at each iterative step. This topic will be discussed in more details in our futureworks.

ACKNOWLEDGEMENTS

The authors would like to thank for the financial supports from the National Natural Science Foundation of China (grants11702133 and 11872229) and from the Natural Science Foundation of Jiangsu Province (grant BK20170772).

ORCID

Yan Shang https://orcid.org/0000-0001-9869-5039Song Cen https://orcid.org/0000-0002-8674-4005Chen-Feng Li https://orcid.org/0000-0003-0441-211X

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How to cite this article: Shang Y, Qian Z-H, Cen S, Li C-F. A simple unsymmetric 4-node 12-DOFmembrane element for the modified couple stress theory. Int J Numer Methods Eng. 2019;119:807–825.https://doi.org/10.1002/nme.6073