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State Key Laboratory of Mechanics and Control of Mechanical Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract: The modified couple stress theory（MCST）is applied to analyze axisymmetric bending and buckling behaviors of circular microplates with sinusoidal shear deformation theory. The differential governing equations and boundary conditions are derived through the principle of minimum total potential energy，and expressed in nominal form with the introduced nominal variables. With the application of generalized differential quadrature method（GDQM），both the differential governing equations and boundary conditions are expressed in discrete form，and a set of linear equations are obtained. The bending deflection can be obtained through solving the linear equations，while buckling loads can be determined through solving general eigenvalue problems. The influence of material length scale parameter and plate geometrical dimensions on the bending deflection and buckling loads of circular microplates is investigated numerically for different boundary conditions.

Key words：circular microplates；size-effect；modified couple stress theory（MCST）；general differential quadrature method（GDQM）

0 Introduction

Recent years，the micro- and nano-scale circu?lar and annular plates are widely applied in micro-/nano-electro-mechanical systems such as accua?tors［1-2］，sensors［3］，resonators［4-5］and so on. Both experimental tests and molecule simulation have shown that the mechanical responses of micro- and nano-scale structures are size-dependent. Classical continuum mechanics fails to capture the size-depen?dent behaviors of micro- and nano-scale structures due to the absence of intrinsic length parameters.Several high-order continuum mechanical models，e.g. modified couple stress theory（MCST）［6-7］and strain gradient theory（SGT）［8-9］，have been devel?oped to address the size-dependent behaviors.

SGT：Gousias and Lazopoulos［10］derived the in-close form solution for static bending of clamped and simply-supported circular Kirchhoff plates. Ji et al.［11］compared the bending and vibration responses of circular Kirchhoff plate with different strain gradi?ent theories. Li et al.［12］applied general differential quadrature method（GDQM）to study the nonlinear bending of circular Kirchhoff plate. Ansari et al.［13-14］applied GDQM to study thermal stability of annular Mindlin microplates and the nonlinear bending，buckling and free vibration of Mindlin plate. Mo?hammadimehr et al.［15］applied GDQM to study the dynamic stability of annular Mindlin sandwich plates. Zhang et al.［16］applied GDQM to study the bending，buckling and vibration of third-order shear deformable circular microplates.

MCST：Combining the orthogonal collocation point method and Newton-Raphson iteration meth?od，Wang et al.［17］studied the nonlinear bending be?havior of clamped and simply-supported circular Kirchhoff plates. Ke et al.［18］employed GDQM to study the bending，buckling and vibration behaviors of annular Mindlin plate. Arshid et al.［19］studied the influence of Pasternak foundation on the bending and buckling response of annular/circular sandwich Mindlin sandwich microplate. Reddy and Ber?ry［20］and Reddy et al.［21］derived the differential gov?erning equations and developed finite element model for nonlinear axisymmetric bending of functionally graded circular Kirchhoff and Mindlin plates，respec?tively. Zhou and Gao［22］applied Fourier-Bessel se?ries to study the linear bending of clamped circular Mindlin plate. Eshraghi et al.［23］applied GDQM to study the bending and free vibrations of thermally loaded FG annular and circular micro-plates based on Kirchhoff plate，Mindlin plate and third-order shear deformation theories. Sadoughifar et al.［24］em?ployed GDQM to study the influence of Kerr elastic foundation on the nonlinear bending of thick annular and circular microplate based on two-variable shear deformation theory.

In this paper，MCST is applied to study the ax?isymmetric bending and buckling circular micro?plates based on sinusoidal shear deformation theory.The differential governing equations and boundary conditions are derived through the principle of mini?mum potential energy. Several nominal variable are introduced to simplify the mathematical expression，and the governing differential equations and bound?ary conditions are discretized with GDQM. The ef?fect of material length scale parameter and plate di?mensions as well as boundary conditions on bending deflections and buckling loads is investigated numer?ically.

1 Mathematical Modeling

The annular plate with thicknessh，inner radi?usaand outer radiusbis defined in a cylindrical co?ordinate system（r，z）where ther-axis is on the midplane and thez-axis is parallel to the thickness direc?tion，as shown in Fig.1. Notice that the annular plate turns to be solid circular plate fora=0.

Fig.1 Schematic diagram of annular plate

The displacement field of circular plate based on sinusoidal shear deformation theory is assumed as［25］

whereβ=π/hand“ ′”denotes the differentiation with respect tor.

According to the modified couple stress theo?ry［7］，the nonzero strainεijand symmetric curvature componentsχijcan be expressed as

The virtual strain energy of the plate can be cal?culated as

where

The virtual work done by the external axisym?metric loads is given by

whereqandPare the distributed transverse load and inplane radial compressive force，respectively.

Based on the principle of minimum potential en?ergy， the differential governing equations and boundary conditions can be expressed as

Based on the modified couple stress theory，the relation between general stress and strain compo?nents can be expressed as

whereE，Gandνare Young’s modulus，shear mod?ulus，Poisson’s ratio，respectively，andlis the ma?terial length scale parameter which describes the mi?crostructural effect.

Combination of Eq.（4）with Eqs.（2）and（8）gives

Taking into account Eq.（9），the differential governing equations and boundary conditions can be expressed as

2 Numerical Solution Based GDQM

In order to simplify the mathematical expres?sion，the following nominal variables are introduced

Therefore，the differential governing equations and boundary conditions can be expressed in nomi?nal form as

whereηα=α+η.

Based on the basic procedure of the GDQM，the nominal radial coordinate is discretized byNnodes

wherei=1，2，…，N.

According to Bellman et al.［26］，and Wu and Liu［27］，the functionΦandWcan be approximated as

whereLkandψkare Lagrange and Hermite interpola?tion basis functions which are defined explicitly in［26-27］，and

Performing derivative respect toηon Eq.（17），one obtains

whereX(i)andY(i)are the weighting coefficients of theith-order derivative and Einstein summation con?vention is adopted in this paper.mandnvary from 1 toNandN+2，respectively.

Fig.2 illustrates the influence ofR/hon nomi?nal bending deflection（NBD）and nominal buckling load（NBL）of circular solid plates without micro?structural effect under clamped and hinged boundary conditions，where CM indicates current model and

The differential governing equations and bound?ary conditions can be expressed in discrete form，and there are 2N+2 linear equations. One can ex?press the discrete linear equation in matrix form as

whereK，n，andMare stiffness，geometrical stiff?ness and mass matrices based on GDQM，respec?tively.d=[δWδΦ]T，qis the load vector.

For a circular plate（a=α=0），according to the L’Hospital’s rule，the boundary conditions（Eq.（15））atη=0 can be approximated as

Therefore，the boundary condition atη=0 for solid circular plates can be expressed in discrete form as

3 Numerical Results and Discussion

In this section，the influence of the material length scale parameter and geometrical dimensions on the bending and buckling responses of circular plates is investigated numerically. The material pa?rameters are adopted as following［28］：E=1.44 GPa，ν=0.38，l=17.6 μm.

Meanwhile，results based on Kirchhoff and Mindlin plate theories［29］are plotted for comparison.Fig.2 shows that，with the increase ofR，bending deflection and buckling loads based on CM approach to those based on Kirchhoff plate theory. In addi?tion，compared with bending deflection based on Mindlin plate theory，CM would provide higher and lower prediction for bending deflection of clamped and hinged plates，respectively.

Fig.2 Validation of current model NBD and NBL of solid circular plates without microstructural effect

Fig.3 illustrates the influence ofh/lon normal?ized bending deflection and buckling load of circular solid microplates under clamped and hinged bound?ary conditions，in which，normalized bending deflec?tion and buckling load are defined as the ratio be?tween with and without microstructural effect.Meanwhile，solid and dash lines represent results for solid circular hinged and clamped microplates. It can be seen that bending deflections decrease and buckling loads increase with the decrease ofh/l. In addition， the microstructural effect on clamped plates is larger than on hinged plates.

Fig.3 Microstructural effect on normalized bending deflec?tion and buckling load of solid circular plates

Fig.4 illustrates the influence of buckling order on normalized buckling load of circular solid micro?plates under clamped and hinged boundary condi?tions，where solid，dash-dot and dash lines repre?sent data forh/l=2，h/l=5 andh/l=10；1—6 de?note buckling order. It can be seen that the micro?structural effect increases with the decrease ofh/land the increase of buckling order.

Fig.4 High-order normalized buckling loads of clamped and hinged solid circular plates

Fig.5 illustrates the influence ofR/handa/hon nominal buckling loads of annular microplates un?der clamped-clamped and hinged-hinged boundary conditions，in which，red，green and blue lines rep?resent data forh/l=2，h/l=5 andh/l=10. It can be seen that NBLs increase consistently with the in?crease ofR/hand decrease ofa/h.

Fig.5 Nominal buckling loads of clamped-clamped and hinged-hinged annular plates

4 Conclusions

Static bending and elastic buckling of circular microplates are investigated on the basis of modified stress couple theory and sinusoidal shear deforma?tion theory. The differential governing equations and boundary conditions are derived through the principle of minimum total potential energy. Several nominal variables are introduced to simplify the mathematical expression. L’Hospital’s rule is ap?plied to deal with the boundary conditions of plate center for circular microplates. The general differen?tial quadrature method is applied to discretize the dif?ferential governing equations and the boundary con?ditions，and a set of linear equations are obtained.Validation is performed through comparing the re?sults of the proposed model without microstructural effect with those of the methods based on classic Kirchhoff and Mindlin plate theories.

Based on numerical results of this study，one can obtain the following conclusions：

（1）For circular solid microplates，bending de?flections increase and buckling loads decrease with the increase ofh/l，and nominal bending deflections and buckling loads approach to constants with the in?crease ofR/h.

（2）For annular microplates，nominal buckling loads increase with the increase ofR/hand decrease ofa/h.

AcknowledgementsThis work was supported in part by

the National Natural Science Foundation of China（No.12172169） and the Priority Academic Program Develop?ment of Jiangsu Higher Education Institutions.

AuthorProf. QING Hai received the B.S. degree in engi?neering mechanics from Xi’an Jiaotong University in 2002 and Ph.D.degree in solid mechanics from Tsinghua Universi?ty in 2007，respectively. From 2007 to 2011，he was a post?doctoral research fellow in Ris? National Laboratory，Tech?nical University of Denmark，Denmark. From 2011 to 2014，he was senior engineer of Siemens Wind Power in Denmark. Since 2014，He has been with Nanjing University of Aeronautics and Astronautics（NUAA），where he is a currently full professor. His research has focused on compos?ite materials and structures，computational solid mechanics and microscale continuum mechanics.

Author contributionsProf. QING Hai designed the study，complied the models，interpreted the results and wrote the manuscript. Ms. WEI Lu conducted the analysis and contributed to the discussion and background of the study. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing inrests.